= OF

N A S A
CONTRACTOR
REPORT
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4
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A COUPLED COMPUTER CODE FOR
THETRANSIENT THERMAL RESPONSE
A N D ABLATION OF NON-CHARRING
HEAT SHIELDS A N D NOSETIPS
by Curl B. Moyer, Lurry W. Anderson,
und Thomus J. Duhm
Prepared by
AEROTHERMCORPORATION
Mountain View, Calif.
for Langley Research Center
N A T I O N AAL E R O N A U T I C AS N SD P A C AE D M I N I S T R A T I O N
W A S H I N G T O N , D. C.
OCTOBER 1970
5 A S A CR-1630
I.,'
A COUPLED COMPUTER CODE FOR THE
TRANSIENT THERMAL RESPONSE AND ABLATION O F
NON-CHARRING HEAT SHIELDS AND NOSE TIPS
By C a r l B. Moyer, Larry W. Anderson,
and Thomas J . Dahm
n c-.
c
- I
Issued by Originator. as ReporbNo-. 69 -60
Prepared under Contract No. NAS 1-8501 by
AEROTHERM CORPORATION
Mountain View, Calif.
for Langley Research Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
~~
For sale by the Clearinghouse for Federal Scientific and Technical Information
Springfield, Virginia 22151
CFSTI price $3.00
-
ABSTRACT
The report presents the analysis background and numerical details
computer code for the transient thermal response and ablation
of noncharring heat shields and nose-tips. The controlling sub-code is the
in-depth transient heat conduction routine;
it treats two-dimensional axisymmetric bodies using the explicit finite-difference method. It is coupled
to a completely general thermochemistry sub-code for the surface state and
ablation computation. This routine allows for full equilibrium considering
a large number of species, and can also account for any number of nonequilibrium effects. The code also features automatic coupled calculations
of the pressure around the body, using a variation of modified Newtonian
relations coupled to Prandtl-Meyer expansions. The subsonic region pressure
description includes empirical corrections necessary on very blunt bodies.
In addition, the code automatically computes the boundary layer edge static
enthalpy, the recovery enthalpy, and the convective transfer coefficient,
employing appropriate computed property values any
for edge gas.
of a
iii
TABLE OF CONTENTS
Title
Section
Page
iii
vii
ix
ABSTRACT
LIST OF FIGURES
LIST OF SYMBOLS
GENERAL
1
INTRODUCTION
1.1 PROBLEM STATEMENT
1.1.1Background
1.1.2
Present Computer Code
1.1.3
Purpose of This Report
2REPORT
COMPUTER
PROGRAM
AND
OVERVIEW
OF
4
3
6
6
4
IN-DEPTH TRANSIENT THERMAL CALCULATION
3.1
GENERAL REMARKS
3.2
NODAL LAYOUT AND GEOMETRY
3.2.1
General Pattern
3.2.2
Geometry
3.2.2.1
Location of Nodal Centers
3.2.2.2
Path Lengths for Conduction
3.2.2.3
Side Areas
3.2.2.4
Volume
3.2.2.5
Geometric Effects of Surface Recession
3.2.2.6
Surface Shape
3.3
INTERNAL CONDUCTION PARAMETERS
3.4
IN-DEPTH CONDUCTION SOLUTION
3.5
TEMPERATURES O F SURFACE NODES AND SURFACE POINTS
IV-V COUPLING TO SURFACE THERMOCHEMISTRY SOLUTION
GENERAL ASPECTS
COMPUTATIONAL APPROACHES
4.2.1
Pre-calculated Table
4.2.2
Direct-Coupled Table
4.2.2.1
General Description
4.2.2.2
Some Dangers in the Direct-Coupled
Table Approach
4.2.2.3
The Other Independent Variables in the
Direct Coupled Table
4.2.3
Summary
4.3
THE CONVECTIVE SURFACE ENERGY BALANCE EQUATION
4.4
CONVECTIVE SURFACE ENERGY BALANCE ITERATION
PROCEDURE!
4.5
AVOIDING THE SURFACE STATE SOLUTION
4.1
4.2
5
ACE
CALCULATION
STATE
SURFACE
- P-SE IV
5.1 INTRODUCTION
OF 5 . 2
USES
PHYSICAL
PROBLEMS
TREATED
BY THE
ACE
PACKAGE
5.3
5.3.1
Closed System in Equilibrium
5.3.2
Open Systems in Equilibrium- Simplified
Case
Open Systems in Equilibrium- Unequal Diffusion
5.3.3
Coefficients
Open Systems in Equilibrium- With Condensed
5.3.4
Removal
Material
Surface
Phase
Non-Equilibrium Effects in Open System Calculation
5.3.5
V
6
8
8
9
10
11
11
12
15
16
17
21
21
23
23
26
26
28
31
32
32
34
35
36
36
36
37
37
42
45
49
51
Title
Section
Page
5.3.5.1
5.4
6
Removal of Selected Species from
the System
5.3.5.2
Isolated Species
5.3.5.3
Isolated Subgroups of Elements
5.3.5.4
Low Temperature Surface Equilibrium
Suppression
5.3.5.5
Full Treatment of Kinetics
CODING DETAILS
Basic Solution Technique
Restriction on Corrections
Ease Species
SOME
5.4.1
5.4.2
5.4.3
52
52
53
53
53
55
55
58
58
COUPLING THE TRANSFER COEFFICIENT CALCULATION TO THE SURFACE
THERMOCHEMISTRY SOLUTION (111- IV COUPLING) AND TO THE INDEPTH TRANSIENT SOLUTION (111
- V COUPLING)
60
COUPLING THE FLOW-FIELD (OR BOUNDARY LAYER EDGE STATE)
SOLUTION (PHASE 11)TO THE SURFACE STATE SOLUTION (I1
- IV
COUPLING) AND TO THE TRANSFER COEFFICIENT SOLUTION
(I1 - I11 COUPLING)
61
8
THE BOUNDARY LAYER EDGE STATE SOLUTION
8.1
THERMOCHEMICAL COMPUTATIONS
8.2
STATIC PRESSURE DISTRIBUTION
8.2.1
Subsonic Region
8.2.2
Downstream of the Sonic Point
8.2.3
Examples of the Accuracyof the Method
8.3
EVALUATION OF OTHER PROPERTIES
63
63
64
64
68
70
75
9
THE TRANSFER COEFFICIENT EVALUATION
76
OVERVIEW OF COMPUTER PROGRAM EVENTS
82
APPENDIX
87
7
10
A
92
REFERENCES
vi
LIST OF FIGURES
Title
Figure No.
Page
4
1
Sketch
of
Problem
2
Sketch
of
Typical
3
Sketch
of
One-Dimensional
4
Sketch of Nodal
5
Illustration
of
Path
Lengths
6
Nomenclature
of
Nodal
Sides
7
Sketch
of
Surface
Nodal
8
Sketch
of
Surface
Points
9
Sketch of Surface Geometrical Relationships (exaggerated)
14
10
Sketch
15
11
Sketch of Implicit and Explicit Temperature Links in
Finite Difference Solution, for Two Typical Situations
18
12
Representation
21
13
Representation of Course of Independent
Surface History, One Body Point
of
Considered
Nodal
Center
Thermal
of
6
Layout
Thermal
Location
for
Network
Ablating
8
Material
8
9
10
Box
Undergoing
12
Recession
13
Resistance
Surface
RC
Nomenclature
Energy
Terms
During
Ablation
Variables
in
24
Sketch Indicating Solution Space and Solution "Sheets"
Stored at any Instant During Solution
28
15
Ablation Rate B' Versus Temperature for Carbon in Air
29
16
Ablating
42
17
Diagram
18
Pressure
Distribution
on 60°
a Sphere
19
Pressure
Distribution
on
20
Pressure
Distributionon an
21
Pressure
Distribution
22
Heat
23
Overall
24
Surface Energy Balance
(Subroutine SURFB)
14
Surface
of
Mass
Nosetip
Transfer
Balance
Showing
a
on
MACABRE
Code
Flow
Cone
with
Sonic
Corner 71
72
Sphere
73
Ellipse
Ellipse
74
Prediction
81
an
Coefficient
65
Nomenclature
83
Chart
Operations
vii
Flow
Chart
84
, ,
. .,, ., ...... ..- ....
.. - - - . .
. .....
-.
LIST OF SYMBOLS
A
area of side of nodal box
AT
a ratioof collision integrals based
on a
Lennard-Jones intermolecular potential (see
Ref. 3 )
ft2
"_
nodalboxsideandcenterlinelengths,Figure 7, Equation (10)
in
constants effectively defined by Equation (22)
"_
a2 b2
constants in Equation ( 2 4 )
Btu/f t2sec OR
and
Btu/ft2sec
Bm
pre-exponential factor
(see Equation ( 7 4 ) )
f
bl
f
B'
for
kinetic
reaction
m
irrQ/peueCM)
thermochemical
defined as (BA ablation parameter
-"
defined as mc/peueCM,totalablationparameter
"_
definedas
mr II/peueCM
"-
"_
same as B'
see
a,b,c
nodal capacity, Equation (14)
Stanton number for heat transfer (corrected
f o r "blowing", if necessary)
C
HO
'ki
cM
and OR
Stanton number for heat transfer not corrected for blowing
number of k atoms in molecule i
Stanton number for mass transfer, corrected
, if necessary
for "blowing"
ix
Btu/OF
"-
SYMBOLS (continued)
specific
cP
heat
capacity
at constant
pressure
Btu/lb OR
Zi averaged heat capacity (see Equation
(A-15)
Btu/lb OR
C
specific
Btu/lb OF
C
see a,b,c
D
Fick's
D
constant definedby Equation (62)
ft 2/sec
binary
ft 2/sec
SP
heat
in
law
diffusion
Em
activation
F
radiation
Fi,F.
7
-
diffusion
coefficient
energy
view
ft'/set
coefficient
for
kinetic
reaction
m
factor
empirical factors appearing in Equation
(62)
F
ratio of summations defined
by Equation (67)
f
denotes general functional relationship
H
total enthalpy (sensible+ chemical
Hr
recovery
h
static
h
heat transfer
difference
hC
hw
I
Btu/lb mole
+
u2/2)
enthalpy
Btu/lb
coefficient
enthalpy of
temperature
ablating
enthalpy
gases
total number of
in the system
Btu/lb
Btu/lb
enthalpy
of
"_
on temperature
based
wall
adjacent
candidate
chemical species indices
X
material
at wall
to
gas
the
wall
phase
species
Btu/f t
Btu./lb
Btu/lb
"-
OR
2sec
SYMBOLS (continued)
diffusional
mass
flux
of
species
i
lb
total diffusional mass flux
of element k regardless of molecular configuration
K
total
number
of
chemical
Ki
mass fraction of species
i
elements
in
the
i/ft2sec
lb/ft2sec
system
"-
I
Kk
total mass fraction of
molecular configuration
K
Pi
equilibrium constant (see Equation
(39))
various
k
thermal conductivity
Btu/ft sec OR
kFm
forward reaction rate constant
tion m (see Equation(72))
L
total number of candidate
cies in the system
L
path lengths in a nodal box, Figure
5
Le
Lewis
L*
length scale used in heat transfer analysis,
defined in Equation (126)
ft
M
Mach
"-
m
system
mi
k
regardless
for
kinetic
condensed
phase
of;b k/lb
reacvarious
spe-
ft
number
number
molecular
molecular
weight
corner
and
lb/lb mole
weight
cXiVb
of
species
i
m
node
m
mass flow rate per unit area
thermochemical effects only
mC
lb/lb mole
"-
center
row number
from
the
surfacelb/ft2sec
total mass flow rate of "char" or main ablating
lb/ft2sec
material per unit surface area, all effects
(thermochemical plus condensed phase mechanical
removal)
xi
I
element
SYMBOLS (continued)
flow rate of condensed phase II
removed from surface
m
rL
lb/ft2sec
mechanically
a
for
N
representing
species
the
molecular
n
node
and
center
P
total
pi
partial pressure of species
i
lb/f t
Pr
Prandtl
”-
corner
heat
formula
column
-”
number
”-
lb/f t
pressure
number
Btu/ft2sec
flux
rate of energy input to solid surface
by diffusional processes in the boundary layer
rate of energy
at the surface
qcond
qrad
in
out
ccnduction
into
rate
of
energy
radiated
away
from
the
sec
0
F/Btu
Btu/lb mole
constant
universal
gas
forward
Btu/ft2sec
surface Btu/ft2sec
(17), (18)
Rm
net
Rmax
maximum of either the
as defined in Figure
17
RN
material Btu/ft2sec
rate of energy input to the surface
by radiation from the boundary layer or from outside
the boundary layer, same as qrad
thermal
R resistance,
see
Equations
R
solid
Btu/ft2sec
rate
of
reaction
m
(see
OR
Equation
(72) 1
nose
nose
radius
% or
R*
ft
ft
radius
R*
contact thermal resistance, Equation(17);
sonic point radius defined in Figure
17
ft2sec OR/Btu,
ft
r
radius
ft
xii
. .
..
. ..
SYMBOLS (continued)
recovery factor
surface location, seeAS
ft
Schmidt number, see Equation(A-17)
entropy
surface
Btu/lb OR
running
length
ft
temperature
OR
failure or mechanical
a surface species
removal
temperature
for
OR
wall (surface) temperature, general term for
Ts ,n
0
TW
U
velocity
ft/sec
U*
velocity used in heat transfer
(124)- (126)
defined in Equations
V
nodal
V
gas
X
analysis,
ft/sec
ft3
volume
velocity
R
normal
amount of
tion ( 4 3 )
condensed
streamwise
location
to
surface
phase
II defined in Equa-
or
coordinate
ft/sec
moles I
I
/
moles gas
ft
mole fraction of species
j
m o l j/mol
general
various
variable
independent
defined
in
variables
Section
4.3
Btu/lb
Y
ratio defined by Equation(13)
"_
Y
distance
ft
normal
to
surface
xiii
SYMBOLS (continued)
axial
coordinate
unequal diffusion
Equation (61)
ft
quantity
for
species
i,
diffusion driving force see Equation (60),
and (61)
unequal diffusion quantity for element
gardless of molecular confiquration
k
re-
unequal diffusion driving potential quantity
for elementk regardless of molecular configuration (see Equation ( 5 9 )
GREEK
a
heated surface absorbtivity (taken equal Eto
~ )
“k j
mass fraction of element k in species j
lb k/lb j
6
angle between
nsrmal
radians
Y
a mass fraction- Z fraction
nent, see Equation (60)
Y
isentropic
A
denotes
AS
change
A0
time step in finite difference solution
E
emissivity
E
error
E
an empirical diffusion factor
exponent (see Equation (63))
rl
input multiplicative safety factor
step calculation, Equation(19)
body
centerline
and
surface
weighting
-”
expo-
exponent
change
in
or
surface
or
point
location
A0
during
sec
emittance
departure
from
xiv
zero
various
correlation
in
time
SYMBOLS (continued)
time
sec
angle between
to a shock
velocity
vector
and
a
line
normal
radians
constant in Equation (128)
dimensionless
(90) and (96)
dynamic
length
ratio
defined
by
Equations
viscosity
lb/ft sec
stoichiometric coefficient for species
j of
kinetic reaction m (see Equation
(72))
"_
quantities defined by Equations (A-4), (A-5) ,
and (A-14), respectively
and lb/lb
mol; ib mol/lb
density
1b1f.1-3
.
lb/ft2sec
.
m = mC
mass
flow at boundary
heat
transfer
convective
mass
transfer
convectivefilm coefficient
Stefan-Boltzmann
layer
edge
film
coefficient
constant
lb/f t2sec
lb/ft2sec
lb/ft2sec
Btu/ft sec
energy thicknesses definedby Equation (105)
and (106)
ft
cp
parameter defined by Equation(127)
"_
m
'
rate coefficicht
temperature exponent in the
for kineticreaction m (seeEquation (74))
SUBSCRIPTS
indices for path lengths
L (Figure 5 ) , nodal s
side areas A (Figure 6) and thermal resistances
R (Figure 10)
"_
OR&
SYMBOLS (continued)
denote side lengths, Figure
7
see A , B , C , D
denotes
back
wall,
comunicating
with
reservoir
see A , B , C , D
denotes
nodal
column
denotes
nodal
box
denotes tofal
rial; see m C
center
line
corner
amount
of
"char"
or
ablating
mate-
D
see A , B , C , D
e
denotes boundary layer outer edge, or boundary
layer edge gas
FD
denotes
fail
refers to material failing (being removed) from
heated durface in condensed phase,
e.g., mechanical
removal, melting
g
denotes
H
see CH
i
station
i,j
denotes any identifiable species: atom, ion, molecule
(i reserved for gas phase)
int
internal contribution
Equation ( A - g()A, - 1 2 )
k
denotes
9,
index of condensed phase
removed from surface
flat
gas
disc
phase
index
to
thermal
conductivity,
element
xvi
species
mechanically
SYMBOLS (continued)
M
see CM
M
match
m
general index
m
node
mix
refers to total mixture property, Equation
(A-9)
mono
refers to pure species, Equations (A-lO), (A-11)
mono-mix
refers to simplified mixture rule, Equation (A-10)
N
denotes
N
see RN
n
node
r
see
r
denotes recovery, radiation
rad
see
ref
refers to an
Equation ( 6 3 )
res
"reservoir"
S
denotes
t
total; transition point
W
denotes wall, i.e., heated surface
0
see
point
corner
and
node
center
row number
index
center
corner
and
center
column
r?.umberindex
m
rE
q
rad
in' qrad out
empirical
reference
communicating
heated
with
value
back
of
wall
surface
C
HO
112
see x1,x2; also denote "earlier" and "later"
xvii
molecular
weight,
SYMBOLS (concluded)
at times 8 and O s r respectively
denotes
free
stream
condition
SUPERSCRIPTS
(prime)
see B ' ,
BAr
B;C
' (prime)
denotes at new time 8 ' = 8
' (prime)
at the
P
reaction
products
R
reaction
reactants
enthalpy
datum
TW
reference
+
A8
enthalpy
temperature- wall
temperature
normalized by stagnation point pressure; also see
6,F,
denotes total
configuration
*
sonic
appearance
of element
independent
of
5
molecular
point
*
*
**
x$ denotes the current estimated
during an iterative solution
x** denotes the new estimated
aA iterative solution
see
6
xviii
value
of the
value
of
variable ix
the
variable
xi
during
SECTION 1
GENERAL
1.1
PROBLEM
STATEMENT
1.1.1
Background
INTRODUCTION
The computationof the thermal and dynamic historya vehicle
of
entering
a planetary atmosphere has a number of important features, which may be
crudely summarized as follows:
I. Trajectory analysis
11. Inviscid flow field calculation
111. Boundary layer calculation
IV. Interactions between vehicle surface and boundary layer
V.
In-depth response
Complete calculation of vehicle response over a period of time would involve
a coupled calculation of all five phases, each computation phase providing,
in effect, the time dependent boundary conditions for the adjacent phases.
A recent survey (Ref.1) has carefully examined the current computational state-of-the-art for each phase and for the coupling between phases.
Major deficiencies were identified in Phase 111, boundary layer calculations,
where general solution procedures for reacting multicomponent boundary layers
did not exist, in Phase IV, where sufficiently general chemical routines did
not exist,and in Phase V, where multidimensional in-depth solution programs
did not exist. In addition, no 111-IV-V coupled programs existed. Programs
with IV-V coupling existed, but were limited to one space dimension for the
in-depth response.
Efforts to upgrade the computational state-of-the-art will probably
not attempt to couple together large "complete" computer routines for all of
the five phases; the resulting computer code would be unwieldly, uneconomical,
and unreliable. Since, in general, the coupling aspects of the predictive
problem are more interesting and important than the degree of computational
detail in the individual phases, a more judicious computational approach seeks
to couple detailed calculations of only one or perhaps two problem
to phases
less detailed approximate or correlation calculations of several of the other
phases. The resulting computer code would include the important coupling
effects without becoming excessively large, and would still offer good detail
in at least one phase
of the total problem.
1.1.2
Presentcomputer
Code
l a t t e r approach.
The work r e p o r t e d h e r e e x e m p l i f i e s t h i s
The code
d e v e l o p e du n d e rt h i sp r o g r a mi n c l u d e sl a r g es u b r o u t i n e sf o rt h ed e t a i l e d
computationof
the t r a n s i e n t i n - d e p t h t e m p e r a t u r e r e s p o n s e o f a n a b l a t i n g
h e a ts h i e l d( P h a s e
V in the
schemeabove)and
a l s oo ft h et h e r m o c h e m i c a l
the e n v i r o n m e n t g a s a n d t h e h e a t e d s u r f a c e o f t h e h e a t
i n t e r a c t i o n sb e t w e e n
s h i e l dm a t e r i a l( P h a s e
IV).
The p r o g r a mf e a t u r e s
more s i m p l i f i e da p p r o a c h e s
111), f o r which a con-
fortheboundarylayertransportcalculations(Phase
ectivetransportcoefficientapproach
f l o wf i e l dc a l c u l a t i o n s( P h a s e
was a d o p t e d , a n d f o r t h e i n v i s c i d
11), f o r w h i c h e m p i r i c a l c o r r e l a t i o n
m o d i f i c a t i o n so fN e w t o n i a np r e s s u r ed i s t r i b u t i o n s
around the a b l a t i n g body.Thesecombine
tion to define the
w i t ht h ei s e n t r o p i ce x p a n s i o n
assump-
b o u n d a r yl a y e re d g es t a t e .
The completedcode
a “I1 - 111 - I V - V c o u p l e d c o d e “ ,
may thus be termed
i n which t h e r o u t i n e s f o r p h a s e s
I V and V a r e r e l a t i v e l y c o m p l e t e , g e n e r a l
and l a r g e , w h i l e t h o s e f o r p h a s e s
fied.
and v a r i o u s
are used t o f i n d t h e p r e s s u r e
,
I1 and I11 a r e r e l a t i v e l y s m a l l a n d s i m p l i -
The n a t u r eo ft h ei n d i v i d u a lp h a s er o u t i n e s
may besummarizedas
follows :
Phase
Phase V - In-Depth
Response
Code C h a r a c t e r i s t i c s
2-D A x i s y m m e t rF
i ci n iD
t ei f f e r e n c e
Transient Heat Conduction
Code,
E x p l i c i t ,M u l t i - m a t e r i a lC a p a b i l i t y ,
G e n e r a l Geometry b u t some r e s t r i c t i o n s on g r i d l a y o u t
make u s e on
verysharpbodiesdifficult,
charringmaterials,
non-
some t y p e so f
a n i s o t r o p yc o n s i d e r e d
Phase I V
-
Surface Response
General thermochemical state routine
allowsforablationof
m a t e r i a li n
any one
any environment, con-
sidersfullequilibriumplus
any
number of heterogeneous and
homo-
g e n e o u s( b u ts u r f a c ea r e as c a l e d )
kineticallycontrolledreactions;
i d e n t i f i e st h e n n o c h e m i c a l l yc o n t r o l lingsurfacematerial,detectsfailing
o r m e l t i n g compounds
2
I
Phase
-
Phase I11
BoundaryLayer
Code Characteristics
Transfer Coefficient routine based
onenergyintegraltechnique,
laminaronly
at present t i m e ,
boundary layer radiation not evaluated
Phase I1
-
Flow F i e l d
Modified Newtonian pressure for
sharp bodies, blending into empirical
pressurecorrelationsforblunt
b o d i e s combined with Prandtl-Meyer
expansionsforsupersonicregions;
isentropic expansion used for boundary
l a y e re d g ea n dr e c o v e r ys t a t ec a l c u l a t i o n s ,f l o wf i e l dr a d i a t i o n
notevaluated
Phase I - T r a j e c t o r y
Not i n c l u d e di np r o g r a m ,
time
historiesofstagnationpressure
andenthalpytobeinput
1 . 1 . 3P u r p o s eo fT h i sR e p o r t
T h i sr e p o r tp r e s e n t s
a d e s c r i p t i o no ft h ec o u p l e dc o d e .
I t describes
of t h e codeand theflowofinformationfromonecomputation
p h a s et ot h eo t h e r .
The r e p o r t a l s o d e s c r i b e s t h e p h y s i c a l
modelsused
in
e a c hc o m p u t a t i o n a lp h a s e ,a n do u t l i n e si n
some d e t a i l t h e n u m e r i c a l a n a l y s i s
u s e di ne a c hp h a s e .
The r e p o r td o e sn o ti n c l u d e
a u s e r sg u i d e ;t h i sh a sb e e n
p u b l i s h e ds e p a r a t e l y( R e f e r e n c e
2).
t h eg e n e r a ln a t u x e
3
SECTION 2
OVERVIEW OC
FOMPUTER
PROGRAM AND REPORT
Figure 1 illustrates the general physical problem considered by the
code.
Computation
Physics
Main stream
radiation
of
Calculation
flux to wall
D e t e r m i n a t i o n of
b o u n d a r yl a y e re d g e
state
Boundary l a y e r e d g e
Account f o r b o u n d a r y l a y e r t r a n s p o r t o f mass andenergy t o and
from s u r f a c e
Determinechemical
state at surf a c e and perform energy balance
s lution
c o u p l i n gt oi n - d e p t h
Substrate
F i g u r e 1.
-In-depththermalresponse
calculation
SketchofProblemConsidered
I nt h ec o u p l e dc o d e ,t h er o u t i n ef o rt h ei n - d e p t ht h e r m a lr e s p o n s e
c a l c u l a t i o n s( P h a s e
r e f e r r e d t o as
MACABRE,
BoundaryLayers
after"MaterialAblationwithChemicallyActive
i n Re-Entry".
The couplingbetweenPhases
t h r o u g ht h es u r f a c ee n e r g yb a l a n c ec a l c u l a t i o n ,w h i c h
s e p a r a t es u b r o u t i n e ,
It is
f u n c t i o n sa st h em a i n ,c o n t r o l l i n gr o u t i n e .
V)
SURFB.
V and I V e f f e c t e d
i s performed by a
T h i se n e r g yc a l c u l a t i o nn a t u r a l l yr e q u i r e s
(among
o t h e rt h i n g s )i n f o r m a t i o na b o u tt h ee n t h a l p yo ft h eb o u n d a r yl a y e rg a s e s
w a l l ; t h i s i n f o r m a t i o n i s c a l l e d f o r by SURFB when needed
a packageof complex t h e r m o c h e m i s t r y s u b r o u t i n e s c o n t r o l l e d
by s u b r o u t i n e EQUIL.
I na d d i t i o n ,
SURFB r e q u i r e sv a l u e sf o r
a convective
adjacent to the
and obtained from
t r a n s f e rc o e f f i c i e n t ,o b t a i n e da sn e e d e df r o ms u b r o u t i n e
pressure,obtained
as neededfromsubroutine
RUNLP,
e n t h a l p y ,o b t a i n e d
as neededfromsubroutine
PARBL.
and RUCH c o n s t i t u t e I V
mentsofthecode.
4
-
I11
-
RUCH,
forthelocal
and f o r t h e l o c a l r e c o v e r y
C a l l st o
PARi3L ,
RUNLP ,
I1 c o u p l i n g , f i l l i n g o u t t h e c o u p l i n g r e q u i r e -
S e c t i o n s 3 through 9 b e l o w d e s c r i b e t h e v a r i o u s c o m p u t a t i o n a l p h a s e s
and t h ec o u p l i n gl i n k sb e t w e e n
S e c t i o n 10 t h e n p r e s e n t s
some o v e r a l l f l o w c h a r t s
them.
T h i sn e c e s s a r i l yi n v o l v e s
much d e t a i l .
a recapitulationofthisintroductorysectionand
of thecomputercodeoperationswhichshouldillus-
t r a t e more c l e a r l y a l l o f t h e i m p o r t a n t f e a t u r e s o f t h e p r o g r a m .
5
I
I
SECTION 3
IN-DEPTH
3.1
TRANSIENT
THERMAL
CALCULATION
GENERAL REMARKS
The in-depth transient temperature calculation serves as the controlling logic unit for the entire coupled computer code. (It also consumes most
of the computing time.) This calculation is of the familiar finite difference, lumped-capacity type. The following sections describe the nodal grid
employed for this calculation and the basic equations employed.
3.2 NODAL LAYOUT AND GEOMETRY
3.2.1
General Pattern
2, is imagined
The geometric shape considered, illustrated in Figure
to be divided up into a grid pattern in the customary manner for finite difwithin each grid"box" will be termed
fernnce computation schemes. The
the
a "node"; this is in contrast to terminology which calls eachofcorner
grid network a node. The thermal capacity of each node
is imagined to be
lumped at a single point within the box; this point will be termed the nodal
center. It will be convenient to assume for the moment that the nodal center may be located anywhere within the nodal box. Strategies for optimizing
the location of the nodal centers will be discussed later.
area
For convenience, the nodes are imagined to be quadrilaterals,
so that
the entire nodal network is an assemblage of quadrilaterals. For bookkeeping
Insulated
Heated
Surface
Surface
Insulated
Surface
Back Wall
Convective
Surface
h e a t shield
Figure 2.
Sketch of Typical Nodal Layout
6
I
convenience, the network
i s imagined t o b e " c o m p l e t e " , even though the shape
ofthematerial
may d i c t a t e t h a t some of t h e mesh boxes are empty.Both
nodes(boxes
orcenters)and
mesh c o r n e r s are numbered i n a row andcolumn
w i l l b e l a b e l e d as an m-n system,where
m . d e n o t e s a row and
m-n mesh scheme may b e o r i e n t e d a r b i t r a r i l y w i t h
* s o t h a t i n g e n e r a l t h e rowr e s p e c tt ot h ep h y s i c a l
r - Z coordinatescheme,
column d e s c r i p t i o n m i g h t seem t o b e m e r e l y
a d e s c r i p t i v e a r t i f i c e ; however,
in exploiting the simplicities associated with
low c u r v a t u r e b o d i e s a n d h e a t
s h i e l d s it hasbeenassumed
in constructing the program that the heated surface
is a tt h et o p
of t h e c o l u m n s ,a si n d i c a t e di nF i g u r e
2.
Thus as s u r f a c e
recessionoccurs,theboxes
a t t h et o po fe a c h
column s h r i n k ; t h e o t h e r b o x e s
r e m a i nf i x e d .T h i sl i m i t a t i o nt h a tt h eh e a t e ds u r f a c eb el o c a t e da tt h et o p
of each column i s merely a c o n v e n i e n c e p r o c e d u r e e x p l o i t e d f o r t h e s p e c i a l
case of h e a t s h i e l d s t y p i c a l
of f a i r l y b l u n t c o n e s . F o r h i g h c u r v a t u r e b o d i e s
s u c ha ss h a r pc o n e st h i sr e s t r i c t i o n
becomes i n c o n v e n i e n t ,a l t h o u g hR e f e r e n c e
systemwhich
d e n o t e s a column.The
n
g i v e sa ne x a m p l eo ft h es u c c e s s f u lu s eo ft h ec o d e
2
on .a verysharpbody.
The s i d e w a l l s o f t h e n o d a l n e t w o r k a r e p r e s u m e d i n s u l a t e d
and t h e back
w a l l communicatesthrough a s i m p l e h e a t t r a n s f e r c o e f f i c i e n t
law ( p l u s r a d i a t i o n ) t o a " r e s e r v o i r " a t Tres.
Thus t h eb o u n d a r yc o n d i t i o na tt h i sf a c e
doesnotinvolvethermochemistry.
t o thelayout
of computingthermal
c o n d u c t a n c e sb e t w e e nn o d a lc e n t e r st h a tt h e
mesh scheme i s n e a r l y o r t h o g o n a l ,
s o t h a t c o n d u c t a n c e may b e t a k e n a s c o n d u c t i v i t y
times s i d e - a r e a d i v i d e d by
**
l e n g t h .S e c o n d l y ,
i t i s presumed t h a to n l yo n em a t e r i a la b l a t e s
, and t h a t
o n l yo n em a t e r i a l
i s exposed t ot h eh e a t e ds u r f a c e .T h i r d l y ,
i t i s presumed
thattheprincipaldirections
of t h e r m a l a n i s o t r o p y a r e a l i g n e d w i t h t h e n o d a l
mesh.
T h i ss i m p l i f i e sc o m p u t a t i o n sa n dr e d u c e si n p u tr e q u i r e m e n t s F
. o rt h o s e
a p p l i c a t i o n s i n which t h e h e a t e d s u r f a c e i n t e r s e c t s t h e p r i n c i p a l d i r e c t i o n s
o fa n i s o t r o p y
at "difficult"anglesthisrestrictioncould
become a majorinconvenienceand
more g e n e r a l schemeswouldhave
tobedevisedforthoseproblems.
T h r e eo t h e r" c o n v e n i e n c el i m i t a t i o n s "h a v eb e e na p p l i e d
of t h e n o d a l g r i d . F i r s t ,
*
it i s assumed f o rt h ep u r p o s e s
T h a t i s , t h e n o d a l mesh scheme may b e a b o v e o r b e l o w t h e Z - a x i s l i n e ,
o r i e n t e d i n anygeneraldirection,and
may b e " b e n t " o r s h a p e d i n a n y
convenienttotheuser(subjecttothelimitationsdescribedintheparagraphsfollowingbelow).
**
A s u i t a b l e more g e n e r a l c o n d u c t a n c e c a l c u l a t i n g s c h e m e c o u l d
may b e
manner
remove t h i s
r e s t r i c t i o n w i t h o u t any e s s e n t i a l change t o t h e program.
7
3.2.2
Geometry
3 . 2 . 2 . 1L o c a t i o n
of NodalCenters
The studysummarized
i n Reference 3 showed t h a t t o o b t a i n a node
d r o p p i n g schemewhichproducesmooth
surface temperature histories,
it w a s
necessarytoavoidassociatingthermalcapacitywiththesurfacetemperature.
Intermsoftheconventionalthermal
d i m e n s i o n a l schemewouldlook
means t h a t a oneAll t h ec a p a c i t y
RC n e t w o r k , t h i s
l i k eF i g u r e
3.
~
node 1
w
-"
"
"
node 2
-
node 3
~~
node 5
node 4
"'112
-+=>==+
I
I
-
Surf
Figure 3 .
ofthesurface
SketchofOne-DimensionalThermal
RC Network
node i s l o c a t e d i n t h e c e n t e r o f t h e f i r s t n o d a l z o n e ,
and
zone i s i n t e r p o s e db e t w e e nt h es u r f a c e
h a l fo ft h et h e r m a lr e s i s t a n c eo ft h i s
t e m p e r a t u r ea n dt h et e m p e r a t u r eo ft h ef i r s tc a p a c i t y
lump.
Note t h a t f o r
m
m+l temperature points, in contrast to most
Thus f o r an m x n n o d a ln e t w o r ks c h e m e ,t h e r e
w i l l be m x n nodal
nodes o r b o x e s t h e s y s t e m h a s
schemes.
*
plus n
w i t h thermac
l apacity)
(not associated with thermal capacity)
t e m p e r a t u r e s( a s s o c i a t e d
s u r f a c et e m p e r a t u r e s
.
as
rN and Z N r t h ec o o r d i n a t e s
shown i n . F i g u r e 4 may b e r e p r e -
Denoting
the
n o d acl e n t ecr o o r d i n a t e s
f o r node m, n w i t h c o r n e r c o o r d i n a t e s a s
sentedin
terms of t h e c o r n e r c o o r d i n a t e s
by E q u a t i o n s (1) and ( 2 ) :
Corner
m+l n
box
m, n
center
Corner
m. n
Figure 4 .
1
Corner
m + l , n+l
Corner
m, n + l
S k e t c ho fN o d a lC e n t e rL o c a t i o nF o rA b l a t i n g. M a t e r i a l
*The c a p a c i t y l o c a t i o n scheme d e s c r i b e d h e r e
recommended i n Reference 3
8
.
i s animprovementover
that
r
-
N, m, n
5,
m, n
r
c,m,n
+
'c,m+l,n
+
'c,rn+l,n+l
+
rc,m,n+l
4
-
'c,m,n
+
'cIm+l,n
+
'c,m+l,n+l
+
'c,m,n+l
4
*
(Nodes i n t h e a b l a t i n g m a t e r i a l
w i l l automaticallybecentered;the
MACABRE
user has the option to put back-up material nodal centers
anywhere i n t h e
nodal box. )
3.2.2.2
P a t hL e n g t h sf o rC o n d u c t i o n
w i l l have a g e n e r a l
I n t h i s and t h e f o l l o w i n g s e c t i o n s , t h e n o d a l c e n t e r
s .oc ro m p u t i nt g
hermal
conl o c a t i o n rN,m,n' 2 N,m,n f oi lrl u s t r a t i vpeu r p o s e F
d u c t a n c e s , it w i l l b e n e c e s s a r y t o h a v e t h e r m a l p a t h l e n g t h s b e t w e e n n o d e s .
it i s c o n v e n i e n t
f i r s tt oc o n s i d e rp a t hs e g m e n t si n s i d ee a c h
box. I n g e n e r a l t h e r e
are f o u r
p a t hs e g m e n t so fi n t e r e s ta s
shown i n F i g u r e 5 . The programcomputes
the
S i n c em a t e r i a lp r o p e r t i e sa r ea s s o c i a t e dw i t he a c hn o d a lb o x l
lengths L
mln,B
m+l, n
GeneralNodal
CenterPoint
F i g u r e 5.
I l l u s t r a t i o no fP a t hL e n g t h s
and L
i nt h e
m d i r e c t i o n as t hde i s t a n c ebs e t w e e tnhne . d acle n t e r
mlnlD
m + l and m faces of t h e box.(This
i s b e c a u s et h e
and t h ec e n t e r so ft h e
n o d a l c e n t e r i s u s u a l l y on t h e l i n e j o i n i n g t h e c e n t e r s o f t h e s e
two f a c e s . )
Thus t h ep a t h s
B and D h a v et h el e n g t h s
*
Except for the first
column, i n w h i c h n o d a l p o i n t s
are c e n t e r e d o n t h e f i r s t
m l l and m + l , l ) u n d e r t h e a s s u m p t i o n t h a t
sideofthebox(betweencorners
this l i n e o f s i d e s w i l l end a t t h e body s t a g n a t i o n p o i n t .
9
rc,m+l,n
'm,
'c,m+l,n
+
rc,m,n+l
c,m,n
2
N,m.n
+ r
[[
r c,m,n
=
+
=N,m,n
=
I
+
rc,m+l,n
l2
-
2
N,m, n
+ zc,m+l,n
zc,m,n
2
[[rc,m+,,n+,
+
Lm,n,c
-
2
N,m,n
rc,m,n+l
2
+I
+ z
'c,rn+l,n+l
c,m,n+l
2
- r
1'1%
i'
1'1%
N , m, n
N,m. n
Side
areas
Areas of t h e s i d e s
of each box are a l s o r e q u i r e d f o r t h e r m a l c o n d u c t a n c e
c a l c u l a t i o n s .F o rt h es i d e sl e t t e r e d
Figure 6 .
10
-
d i r e c t i o np a t hl e n g t h s
'm,n,A
3.2.2.3
l2
-
'c,m+l,n+l
2
=
Lrn,n,D
n
N,m,n
2
+I
S i m i l a r l y , for t h e
-
rc,m+l,n+l
+
n, B
as shown i nF i g u r e
NomenclatureofNodalSides
6 , t h ea r e a s
are
given by e l e m e n t a r y g e o m e t r y ( F i r s t
Theorem ofPappus)
+ r
1"
+
Only t h e s e two a r e a sn e e d
('c,m,n+l
-
zc,m,n
t o be computed f o r e a c h b o x , s i n c e t h e a r e a s
on t h e
A and B o f a d j a c e n t
othersidesofthenodalboxareidenticalwiththeareas
nodes.
3.2.2.4
,.I+
Volume
The volume of t h e e l e m e n t a l
box i s by t h e Second Theorem of Pappus,
+ zc , m + l , n + l
+
3.2.2.5
r2c,m+l,n
-
~,m,n+l]
+
Zc,m,n+l
Lr
c,m,n+l
.
G e o m e t r i cE f f e c t so fS u r f a c eR e c e s s i o n
F o rn o d e sa d j a c e n tt ot h eh e a t e ds u r f a c e ,s i d e
B
may
move
due t o
s u r f a c er e c e s s i o n( a b l a t i o n ) .T h i sr e c e s s i o nr e d u c e st h et h e r m a lr e s i s t a n c e
b e t w e e nt h es u r f a c ep o i n ta n dt h ea d j a c e n tn o d a lp o i n t ,i n c r e a s e st r a n s v e r s e
t h e r m a lr e s i s t a n c e s( a sd i s c u s s e db e l o w )
associatedwiththenodalcenterofthenodal
, and
r e d u c e st h et h e r m a lc a p a c i t y
box a d j a c e n t t o t h e s u r f a c e .
11
s o as t o m a i n t a i n t h e
The program assumes t h a t t h e s u r f a c e r e c e s s i o n o c c u r s
ratios
S1,SZ = s u r f a c e
points
1-earlier center
s u r f a c e node
of
c e n t e r of
node Surfacem,n+l
Figure 7.
a s shown i nF i g u r e
S k e t c ho fS u r f a c eN o d a l
7.
a
c22
b2
al
bl
Box UndergoingRecession
c1
The b l i n ej o i n st h ec e n t e r so ft h e
and s e r v e s t o d e f i n e t h e l o c a t i o n o f t h e s u r f a c e p o i n t
m+l,n and m + l ,
n + la r el o c a t e da c c o r d i n g l y ,
a sb e f r j x e ,e x c e p tt h a tf o rt h e s en o d e s
The n o d a l p o i n t o r c e n t e r
S
*
'
A
m planes
The moving c o r n e r s
andvolumescomputed
.
and t h e a r e a s
m + l and
m,n,A
Am,n-l,C'
i s a l s o presumed t o move so t h a t i s i s
the nodalbox
always i n t h e a r i t h m e t i c c e n t e r o f
*
.
is
Thus t h en o d a lc e n t e r
" s q u e e z e d "t o w a r dt h eb a c kw a l lo ft h en o d a lb o xa sr e c e s s i o np r o c e e d s .
The
p a t hl e n g t h sf o rc o n d u c t i o na r ea d j u s t e da c c o r d i n g l y .
Surface
Shape
3.2.2.6
by F i g u r e 7 , it is m o s t c o n v e n i e n t t o
As n o t e d a b o v e a n d i l l u s t r a t e d
considerthesurfacepoints
t h en o d a l
on t h e l l e a t e d s u r f a c e a s b e i n g a l w a y s l o c a t e d o n
column c e n t e r l i n e
*
,
t h a t i s , o nt h el i n ej o i n i n gt h ec e n t e rp o i n t s
m + l and m p l a n e s( " p a r a l l e l "t ot h eh e a t e ds u r f a c e )
, where m is
t h e row i n d e x o f a nodalbox a t t h e s u r f a c e
and m + l i s t h e l o c a l c o r n e r i n d e x
o ft h e
of t h eh e a t e ds u r f a c e
( s e e F i g . 7)
.
is
.The l o c a t i o no ft h es u r f a c ep o i n t s
a l lt h ei n f o r m a t i o nn e e d e da b o u tt h es u r f a c ef o rt h o s ea b l a t i o np r o b l e m sw i t h
a s p e c i f i e d ,i n p u ts u r f a c er e c e s s i o nr a t ea s
a b o u n d a r yc o n d i t i o n( t h ev a r i o u s
ablationproblemboundaryconditionoptionsaredescribedinSection
s i n c ef o rt h o s ep r o b l e m s
i t i s mostconvenient
s i o nh i s t o r ya l o n gt h en o d a lc e n t e rl i n e .
t os p e c i f y ,
The n o d a lg r i d
4 below)
as i n p u t , recesas i n p u tt h u ss e r v e s
*
E x c e p t , as n o t e d a b o v e , f o r t h e f i r s t
column o f n o d a l b o x e s , i n
which t h e
A o fF i g u r e
6 ) o ft h en o d a l
n o d a lp o i n t sa r ec e n t e r e d
on t h e f i r s t s i d e ( S i d e
S i s located a t thecorner m + l , l (Figure 7 ) .
boxand
thesurfacepoint
12
. . ...
.
,
todefinethevariousnodal
column c e n t e r l i n e s , a n d t h e r e c e s s i o n h i s t o r y
f o r e a c h column t h e n d e f i n e s t h e h i s t o r y o f t h e s u r f a c e p o i n t s i n
a perfectly
s t r a i g h t f o r w a r d manner.However,
a n o t h e ri m p o r t a n th e a t e ds u r f a c eb o u n d a r y
conditionoptioninvolvesnotinputrecessionsbutvariousenergyand
chemm a s s loss fromenergybala n c ec o n s i d e r a t i o n s( O p t i o n
l, d i s c u s s e d i n S e c t i o n
4.3 below)
T h i sa b l a t i o n
c a l c u l a t i o nd o e sn o tp r o d u c er e c e s s i o n
rates d i r e c t l y , o f c o u r s e ; i n s t e a d
it p r o d u c e sr a t e so f
mass loss from t h e s u r f a c e .
I t w i l l p r o v ec o n v e n i e n t ,
nevertheless, to adhere to the concept of the surface point
which moves a l o n g
t h e column c e n t e r l i n e as r e c e s s i o np r o g r e s s e s ,
T o d e f i n et h es u r f a c ep o i n t
motionwithcomputed
mass loss r a t e s d e t e r m i n e d f r o m t h i s g e n e r a l e n e r g y balance-determinedoptionrequiresknowledge
of t h e a n g l e b e t w e e n t h e l o c a l
normal t o t h e s u r f a c e
and t h e column c e n t e r l i n e , s i n c e c o m p u t e d
mass loss
may b e t r a n s l a t e d d i r e c t l y i n t o r e c e s s i o n a l o n g
a normal t o t h e s u r f a c e .
Rec e s s i o na l o n gt h en o r m a l
may b e p r o j e c t e d i n t o t h e n o d a l
column c e n t e r l i n e
once t h ea n g l eb e t w e e nt h e s et w ol i n e s
i s known.
The d i r e c t i o n of t h e s u r f a c e normal may c o n v e n i e n t l y b e d e t e r m i n e d f r o m t h e s l o p e o f t h e s u r f a c e ,
t h a t is t h e s u r f a c e s h a p e , a n d o f c o u r s e t h e d i r e c t i o n
of t h e column c e n t e r
l i n e i s known.
istry information sufficient to calculate surface
.
I t i s o b v i o u s l yn o ts a f et ou s et h es l o p eo ft h eh e a t e d( t o p )s u r f a c e
it is
o ft h es u r f a c en o d a lb o xt oo b t a i nt h es u r f a c es l o p e ,s i n c ei ng e n e r a l
neitherpossiblenoralwaysdesirabletolayout
a nodalgrid
"conform" t o t h e " r e a l " s u r f a c e f o r t h e e n t i r e
which w i l l
p r o b l e mh i s t o r y .T h e r e f o r e
t h e MACABRE p r o g r a m i n c l u d e s v a r i o u s s p e c i a l c u r v e - f i t s u b p r o g r a m s w h i c h
compute a s u r f a c e s h a p e a t e a c h
t i m e stepduringthesolutionafterexamining
t h el a y o u to ft h es u r f a c ep o i n t s .R e f e r r i n gt oF i g u r e
programprogramcomputes
8 , onecurve
f i t sub-
the
Z
F i g u r e 8.
s l o p eo ft h es u r f a c e ,d r / d Z ,
b e t w e e snu r f a cp
e o i n tns - 1
S k e t c ho fS u r f a c eP o i n t s
a t s u r f a c ep o i n t
and
n,
and
n
n
and
n+l.
a st h ea v e r a g eo ft h es l o p e s
Thus
13
Note t h a t n-1,
n , and n + l a r e s u r f a c e p o i n t s , n o t n o d a l c o r n e r p o i n t s .
(Problemswithonlyonenodal
column h a v eo n l yo n es u r f a c ep o i n t .T h i s
r e q u i r e st h ep r o g r a mt oa b a n d o nt h ea v e r a g es l o p es c h e m e
and t o u s e t h e n o d a l
box heated surf ace slope as the surface slope.)
Othercurve
fit r o u t i n e s a r e a v a i l a b l e
i ns u c c e s s i v ei n t e r v a l s .
choiceof
whichinvolvefittedquadratics
The User’s Manual g i v e sr e c o m m e n d a t i o n sf o rt h e
f i t routinesfordifferent
body s h a p e s .
Specialformulasareusedfortheslopesatthefirst
s u r f a c ep o i n t s ,h o w e v e r ,i no r d e rt oh a r m o n i z et h ea r r a y
v a l u e sw i t h
some b a s i c a s s u m p t i o n s
r o u t i n e s .S i n c et h ef i r s t
made i n t h e c o n v e c t i v e t r a n s f e r c o e f f i c i e n t
column s u r f a c ep o i n t
i s presumed t o b e
the s t a g -
is s e t t o a v e r yl a r g ep o s i t i v ev a l u e .
n a t i o np o i n t ,t h es l o p eh e r e
assumed t h a t t h e s u r f a c e
andsecond
of s u r f a c e s l o p e
I t is
is s p h e r i c a l b e t w e e n t h e f i r s t a n d s e c o n d p o i n t s ,
i s givenby
so that the second point surface slope
$1,
1 - y
2
2y
=
where
Y
=
-
22
r
21
2
With s u r f a c e s l o p e d e t e r m i n e d , t h e s u r f a c e
movement AS computed d u r i n g
the time s t e p may b e p r o j e c t e d o n t o t h e n o d a l b o x c e n t e r l i n e ,
Figure 9 .
14
SketchofSurfaceGeometrical
R e l a t i o n s h i p s( e x a g g e r a t e d )
and t h e n t h e
new n o d a l volumecomputed,
as i n d i c a t e d i n t h e e x a g g e r a t e d s k e t c h o f F i g u r e
9.
A c t u a l s u r f a c e movement duxing a time s t e p i s l i m i t e d t o a small f r a c t i o n o f
the nodal thickness to ensure
agood
approximationtotheconservationof
column i s an e x c e p t i o n , s i n c e t h e s u r f a c e p o i n t
Noteagainthatthefirst
mass.
is
l o c a t e d on t h e l e f t c o r n e r , n o t i n t h e c e n t e r o f t h e t o p s u r f a c e .
3.3
INTERNAL CONDUCTION PARAMETERS
The t h e r m a l r e s i s t a n c e s b e t w e e n n o d e s a n d t h e t h e r m a l c a p a c i t y o f t h e
t i m e i n t e r v a l from t h e m a t e r i a l p r o p e r t i e s o f t h e
time.
The m a t e r i a l p r o p e r t i e s
( d e n s i t y ( p ) , s p e c i f i c h e a t ( c ) , c o n d u c t i v i t y ( k ) , and e m i s s i v i t y ( E ) a r e i n p u ta st a b l el o o k
up f u n c t i o n so ft e m p e r a t u r e .L i n e a ri n t e r p o l a t i o n
is emn o d e sa r ec a l c u l a t e de a c h
nodecorresponding
t o i t s t e m p e r a t u r ea tt h a t
ployedformaterialpropertydeterminationattemperaturesintermediateto
t h o s et a b u l a t e d .C o n s t a n tt h e r m a lc o n t a c tr e s i s t a n c e s
may b es p e c i f i e db e tweenany
o ra l ln o d e s .
The n o d a lc a p a c i t i e s
and r e s i s t a n c e s are c a l c u l a t e d
as follows :
R
m,n,A,
e
R
m,n,B,e
-
1
-
1
Lm,n,C
Lm,n+l,A
km,n+l,B
km,n,B
+
+
-
'm+l,n,D
kA+l, n,
F i g u r e 1 0 shows t hl eo c a t i o norsfe s i s t a n c e s
R
m,n,A,e
+
+
R*
m, n, B
R*
e
m,n,A
and R
m,n,B,O'
The
I
<
Rm, n, B
I
m, n-1
I
m,n
m
0
Figure 10.
,
I
Rm,n,A
n
+
l
m-l,n
SketchofThermalResistanceNomenclature
15
two s i d e s of t h e n o d a l
resistanceson the other
t h es u r f a c e
, however ,
box are c a l c u l a t e d when the
are c a l c u l a t e d .F o rn o d e sa d j a c e n tt o
q u a n t i t i e sf o rt h ea d j a c e n tn o d e s
#
Am ,n+l ,A g e n e r a l l y( r e f e rt oF i g .
6 f o ra r e a
nomenclature);forthesenodes
Rm,n,A
Lm, n ,
c
Lm,n+l, A
+
Am,n,C kmen, e
+
%,n+l,~
km,n+l,
e
Rm*,n,B
Am,n,
I t s h o u l db en o t e dt h a ta n i s o t r o p i ct h e r m a lc o n d u c t i v i t yv a l u e s
IN-DEPTH
k
(17)
and k'
(15) and ( 1 6 ) , r e s p e c t i v e l y .
a r eu s e di nE q u a t i o n s
3.4
c
CONDUCTION SOLUTION
The i n - d e p t hc o n d u c t i o ns o l u t i o n
is theexplicitfinitedifferencetype
o f t e n employed f o rt r a n s i e n th e a tc o n d u c t i o na n a l y s i s .
The t e m p e r a t u r eo f
node m,n a t t i m e 8' ( T m , n l e l ) i s o b t a i n e d b y a p p l i c a t i o n o f t h e f i n i t e d i f f e r e n c ee n e r g yb a l a n c e
and r a t e e q u a t i o n s t o t h e n o d a l
Solvingfor
1
Tmlnle,,
1
Rm,n-l,Al 8
Inthe
+
R
o n eo b t a i n s
1
+
1
.
m , n , ~R, em , n , ~R, em - l r n r ~ , e
program t h i s e q u a t i o n
a l l n o d e se x c e p tf o r
and f o r backwallnodes.
volume.
1
)]e+
T
m, n, 8
is usedtoobtain
mrnre
"new" t e m p e r a t u r e s f o r
two nodes i n each column a d j a c e n t t o t h e h e a t e d s u r f a c e
The n o d e s n e a r t o t h e h e a t e d s u r f a c e a r e l i n k e d t o
thesurfacetemperatureimplicitly
andhence
a specialprocedure
is u s e d f o r
t h et e m p e r a t u r eo ft h e s en o d e s ,a sd e s c r i b e di nt h en e x ts e c t i o n .
Back w a l l nodes include
i n s i d et h eb r a c k e t so fE q u a t i o n
zero.
16
(18)
a quantity
(18) , andhave
T
m-1 ,n, 8
f o r m a l l ye q u a lt o
The e x p l i c i t r e l a t i o n
t h e t i m e s t e p s i z e A0 =
(18) imposes the f a m i l i a r s t a b i l i t y r e s t r i c t i o n
The MACABRE programautomaticallyemploys
8'8
-.
on
a
conservative stability equation for interior nodes:
AB
=
q
Normally f o rs t a b i l i t y ,t h ei n p u tp a r a m e t e r
n o d e sa r en o tc o n s i d e r e df o r
below:back
TI
i s less t h a nu n i t y S
. urface
t i m e intervalcalculation,as
wallnodesincludethe
terms Am , n , D ( h / 2
f
w i l l b ee x p l a i n e d
~ U E ~ ~ T : , ~
i n, t~h )e
denominator.
The a u t o m a t i c s t a b i l i t y c r i t e r i o n c a l c u l a t i o n
may b e s u p p r e s s e d f o r a n y
node i f t h e u s e r
is surethattheallowed
time s t e p f o r t h a t
node w i l l n e v e r
b et h e
minimum one f o rt h es y s t e m .T h i ss a v e s
some computer t i m e .
Alternatively,thestabilitycriterioncalculation
~f it i s n o tu s e d ,t h e n
A 8 m u s tb es p e c i f i e d .
entirely.
3.5
TEMPERATURES
SURFACE
OF
NODES
can b es u p p r e s s e d
AND SURFACE
POINTS
Temperaturesofsurfacepointsaredeterminedeither
by assignment
(Option 2 ) o r by s u r f a c e e n e r g y b a l a n c e s d e s c r i b e d i n t h e n e x t s e c t i o n ( O p t i o n s
1 and 3 ) .
t h en t h
The s u r f a c ee n e r g yb a l a n c ed e t e r m i n e st h e
column TA
new s u r f a c et e m p e r a t u r eo f
w i t ha ni m p l i c i ti t e r a t i o nt e c h n i q u e .S t a b i l i t yc o n s i d e r -
ations dictate thst the first
andsecond node t e m p e r a t u r e s a l s o b e t r e a t e d
i m p l i c i t l y , and t h a t a n yt r a n s v e r s eh e a tc o n d u c t i o nl i n k( a c r o s sc o l u m n s )
f o rs u r f a c et e m p e r a t u r e sm u s tb ei m p l i c i t .T h i sl a t t e rr e q u i r e m e n t
n o tl i n k e dt r a n s v e r s e l y .F i g u r e
c o n d u c t i o np a t h sf o r
is a
the s u r f a c et e m p e r a t u r ep o i n t s
11 shows t h e i m p l i c i t and e x p l i c i t h e a t
complex one t o m e e t ascolumnsrecede:hence
are
two t y p i c a l s i t u a t i o n s .
17
Boundary Layer
Edge
SurfaceTemperatures
F i r s t Nodes
Second Nodes
Implicit
Links
n-1
n
n+1
_"""""-
BoundaryLayerEdge
Explicit
Links
Surface
n-1
n
n+1
F i g u r e 11.
Ingeneral
node
S k e t c ho fI m p l i c i t
and E x p l i c i tT e m p e r a t u r eL i n k s
Two T y p i c a l S i t u a t i o n s
inFiniteDifferenceSolution,for
terms, t h e e q u a t i o n f o r t h e
is
new t e m p e r a t u r e o f
a surface
( n o t a s u r f a c ep o i n t )
-
Tm,n, 8 1
Tm,n+l,e
Tm,n,e
Rm,n,A, e
+
Tm,n-l,e
Tm,nre
R
m, n-1, A, 8
(20)
This equation links the
new t e m p e r a t u r eo ft h en e x tn o d e
18
new f i r s t node temperature
down T m - l , n , e , .
Tm , n , 6 '
ana
to the
A n o t h e re q u a t i o na l s ol i n k s
T
and Tm-l,n,e,;
t h i s i s t he n e r g y
b a l a n co
en
t h see c o n d
node
down.
m,n,8'
Thishasthe
same form as E q u a t i o n (18) w i t h T
replaced
m,n, e and Tm-l ,n ,e
r i n t h e conductance terms :
by. T
m,n, 8' ana T m - l , n , e
+
-+
Tm-2,n,~- T m - l , n , ~
Tm,n,e'
Tm-l,n,e'
Rm-2 ,n , B ,0
Rm-l,n,B,B
1
'm-1
,nt e
E q u a t i o n s ( 2 0 ) and ( 2 1 ) between them h a v et h r e e
new unknown temperat i m e s t e p : Ts , n ,e ' , T m , n , e, , and Tm-l
e , (where m h e r e
d e n o t e st h e row i n d e x o f t h e s u r f a c e m d a l b o x i n
Column n ) . The two e q u a t i o n s
may becombined t o e l i m i n a t e T m - l , n , e , ,
leaving a simplelinearlinkbetween
thesurfacepointtemperature
Ts
and t h e s u r f a c e nodetemperature T m,n, 0 ' '
,nr8'
Formally w e have a r e l a t i o n b e t w e e n t h e
two unknowns as
turesateach
(as d e s c r i b e d i n t h e n e x t
The s u r f a c ee n e r g yb a l a n c eh a st h eg e n e r a lf o r m
section)
convectionandchemicalenergy
+
terms (T
s,n,e')
radiation to w a l l - radiationout
(T
s,n,e')
-- . l c t i o n a lr e l a t i o n s h i p .
w h e r e . t h ep a r e n t h e s e sd e n o t e
on a u n i ta r e ab a s i s .R e l a t i o n
(22)
The e q u a t i o n i s w r i t t e n
may b es u b s t i t u t e di n t oE q u a t i o n
(23)
togivethenon-linearsurfaceenergybalanceEquationfor
convectionandchemicalenergy
+
radiation t o wall
-
T
s,n,e':
terms ( T
s,n,e')
radiationout
(T
s,n,e' )
19
When T
T
h ab
s een
found
from
t h ie
s q u a t i o nt,h e
srnr8'
may be
found
from
Equation
(22).
m,n,e'
t h e method f o r f i n d i n g
T
T
is d e s c r i b e di nS e c t i o n
4 below.
( I nO p t i o n 2 ,
s , n , 0'
i s known immediately by user
assignment
and
Equations
( 2 0 ) and ( 2 1 )
s,n,e'
then determine
20
new s u r f a c e node
temperature
F osru r f a ceen e r g b
y a l a n coe p t i o n s ,
Tm , n , e r
and T m - l , n , e l
a t once.)
SECTION 4
IV-V
COUPLING TO SURFACE
THERMOCHEMISTRY
SOLUTION
GENERAL ASPECTS
4.1
The d i r e c t c o m p u t a t i o n a l l i n k b e t w e e n t h e s u b s u r f a c e s o l u t i o n a n d t h e
is e f f e c t e dt h r o u g ht h es u r f a c ee n e r g yb a l a n c ec a l c u l a -
s u r f a c es t a t es o l u t i o n
*
t i o n .T h i sl i n k a g e
solutioninSection
was d i s c u s s e df r o mt h ep o i n to fv i e wo ft h ei n - d e p t h
3 a b o v ea n dt h ee n e r g ye q u a t i o nw r i t t e n
down i n s c h e m a t i c
terms as E q u a t i o n s (23) and (24)
detail.
The e n e r g yb a l a n c e
Figure 1 2 .
w h e r et h ei n d i c a t e dc o n t r o l
f l u x e sl e a v i n gt h ec o n t r o l
.
W e now must d i s c u s st h i se q u a t i o ni n
i s i l l u s t r a t e di nF i g u r e
more
12,
R e p r e s e n t a t i o no fS u r f a c eE n e r g y
During Ablation
Terms
volume i s f i x e dt ot h er e c e d i n gs u r f a c e .E n e r g y
volume i n c l u d e c o n d u c t i o n i n t o t h e m a t e r i a l , r a d i a -
t i o n away from t h e s u r f a c e , e n e r g y i n
any f l o w o fc o n d e n s e dp h a s em a t e r i a ls u c h
a sm e c h a n i c a lr e m o v a l ,a n dg r o s sb l o w i n ga tt h es u r f a c e .E n e r g yi n p u t st ot h e
c o n t r o l volume i n c l u d e r a d i a t i o n i n f r o m t h e b o u n d a r y l a y e r a n d e n t h a l p y f l u x
due t o t h e c h a r f l o w r a t e .
The f i n a l i n p u t i n t h e s k e t c h
i s denoted -9,.
I t i n c l u d e sa l ld i f f u s i v ee n e r g yf l u x e sf r o mt h eg a s - p h a s eb o u n d a r yl a y e r .
were coupled t o an e x a c t boundary-layer
is,
of c o u r s e , a complex f u n c t i o no ft h eb o u n d a r y - l a y e rs t r u c t u r e .I f ,o nt h e
o t h e rh a n d ,t h ei n - d e p t hr e s p o n s e
i s b e i n gc o u p l e dt o
a s i m p l i f i e db o u n d a r y l a y e r scheme suchas
a c o n v e c t i v e f i l m c o e f f i c i e n tm o d e l ,
as w i l l bedonehere,
t h e n t h e t e r m -qaassumes
t h ef o r mo f
a correlationequation.
Ifthein-depthresponsecomputation
solution, the term
-9,
wouldbeobtainedfromthatsolutionprocedureand
The computation of t h e s u r f a c e e n e r g y b a l a n c e r e q u i r e s f r o m t h e i n depthsolution
a relationbetween
the s u r f a c e t e m p e r a t u r e
e n e r g yc o n d u c t i o ni n t ot h em a t e r i a l ,q c o n d .
and t h e rate o f
As n o t e d i n S e c t i o n
3 above, t h i s
relationderivesnacurallyfromthefinitedifferenceenergybalanceforthe
*For Option 1 c a l c u l a t i o n s .
The s i m p l e rs u r f a c eo p t i o n s
4 . 5 and 4.6 below.
discussedinSections
w i l l be
21
the node j u s t u n d e r t h e s u r f a c e a n d
i s a simple linear equation
q cond = a2Tw
With t h i s i n f o r m a t i o n , t h e s u r f a c e e n e r g y b a l a n c e c o n s i d e s a t i o n s a l l o w d e t e r -
rate
minationofthethermochemicalerosion
I t w i l l be useful to keep in
mind t h a t
,
mc
andsurfacetemperature
TW '
from t h i s p o i n t o f v i e w , t h e p u r p o s e
a t any i n s t a n t i s t o p r o v i d e i n f o r m a t i o n a b o u t
may b e w r i t t e n as
q c o n d ( T w ) . The s u r f a c ee n e r g yb a l a n c ee q u a t i o n
o ft h ei n - d e p t hs o l u t i o n
where
The r e l a t i o n q cond = f (Tw) i s d e l i v e r e d by t h e i n - d e p t h s o l u t i o n .
O t h e rd e p e n d e n c i e so fi n t e r e s ta r e
out
F o rt h eo t h e rt e r m s ,
Tw, -qa
,
in
If the
h,,
--
'rad (Tw)
w r i t e i n general
we
may
qrad,
'rad
out
2' mrr
a
hi
=
f u n c t i o n so fb o u n d a r y l a y e r - e d g ee n t h a l p y ,
p r e s s u r e ,l o c a l
boundary-layer solution,
l a w sf o rc o n s e r v a t i o n
ofchemicalelements,
chemicalequilibria
and/or kinetic relations,
upstream e v e n t s , mc
boundarylayertransportaspectsoftheproblemare
modeled
by a f i l mc o e f f i c i e n ts c h e m e ,t h e nE q u a t i o n( 2 5 )c a nb en o r m a l i z e d
on t h e
mass t r a n s f e r c o e f f i c i e n t i n t h e c u s t o m a r y
22
manner:
+
b2'
and r e l a t i o n s h i p ( 2 9 ) becomes
-9,
%'p eue cM *
in
"
p u c
eeM
hw
,
hk = f u n c t i o n s of bounda r y l a y e r e d g e enp r e s s ut rhea,l p y ,
l a w s forconser3.ation
of chemical e l e m e n t s ,
chemical e q u i l i b r i u m
and/or k i n e t i c relations,
ee
' cM
a
B;
The g e n e r a t i o n o f t h e s e r e l a t i o n s h i p s
i s t h eg o a lo fP h a s e
s u r f a c et h e r m o c h e m i s t r ys o l u t i o n ,t ob ed i s c u s s e db e l o w .F o rt h ep r e s e n t ,
IV, the
it may
an i n i t i a l
g u e s so ft h ed i m e n s i o n l e s s
mass removal r a t e BA i s o b t a i n e di n some manner.
With t h i s BL, t h eq u a n t i t i e s -4, /peueCM, h w , x rirQhQ/peueCM, and Tw a r e
b eo b s e r v e dt h a tt h ee n e r g yb a l a n c ec o u p l i n gp r o c e e d sa sf o l l o w s :
o b t a i n e df r o mt h es u r f a c et h e r m o c h e m i s t r ys o l u t i o n .
The q u a n t i t i e sh c
and
so o b t a i n e d . The s u r f a c ee n e r g y
qrad out
b a l a n c e i s t h e nc o m p u t e d ,t h e
qcond as a f u n c t i o no f
Tw havingbeenprovidedby
t h ei n - d e p t hs o l u t i o n .I ng e n e r a l ,h o w e v e r ,t h e
sum o ft h e
terms w i l l n o te q u a l
z e r o ,b u t
some a r r o r . An i t e r a t i o n p r o c e d u r e is t h e nu s e dt o
select success i v e l yb e t t e r estimates of B L w h i c hd r i v et h ee r r o rt oz e r o .E x p e r i e n c e
shows
t h a t Newton's,.rocedure,
i n which t h e d e r i v a t i v e o f t h e e r r o r w i t h r e s p e c t t o
E:
i s u s e d t o compute t h en e x tg u e s sf o r
g i v e s good r e s u l t s .
a r et h e nf o r m u l a t e du s i n gt h e
Tw
€
3
;
4.2
COMPUTATIONAL APPROACHES
4.2.1
Pre-celculated T a b l e
It is evidentthateachiterationinthesearchfor
a s u r f a c ee n e r g y
b a l a n c e ,i fp e r f o r m e d
as d e s c r i b e da b o v e ,w o u l dr e q u i r e
a new s u r f a c e c h e m i s t r y
solution,generallyinthenearneighborhoodof
many s u c h . p r e v i o u s s o l u t i o n s .
If this
s t a t e s o l u t i o n is t o b e o b t a i n e d f r o m
a l a r g es u b p r o g r a m( r a t h e rf r o m ,
, t h i sc o n s t a n tr e p e t i t i o no ft h e r m o c h e m i s t r yc a l c u -
s a y ,s i m p l ec o r r e l a t i o n s )
l a t i o n s would cqnsume an
t h a t a tabularapproachin
hand f o rp r e a s s i g n e d
BL
-
l a c c e p t a b l e amount ofcomputer
time.
which t h e s u r f a c e s t a t e s o l u t i o n s a r e
T h i ss u g g e s t s
donebefore-
v a l u e sw o u l do f f e rs i g n i f i c a n tc o m p u t a t i o n a l
economies. As anexampleof
t h i sa p p r o a c h ,c o n s i d e ro n e
body p o i n t .
f o rs i m p l i c i t y ,t h a tc h e m i c a lk i n e t i c sa r en o ti m p o r t a n t :t h e nt h ei n d e p e n d e n t
v a r i a b l e si nE q u a t i o n( 3 1 )r e d u c et op r e s s u r e
enthalpy H
variables.
r'
and B h .
Assume,
P , b o u n d a r yl a y e rr e c o v e r y
F i g u r e1 3r e p r e s e n t st h es p a c eo ft h e s et h r e ei n d e p e n d e n t
The c o u r s eo f
a t y p i c a lt r a n s i e n ts o l u t i o nm i g h tb ea sr c p r e s e n t e d
23
. ....
.. ..
i nt h es k e t c h .
As time p r o c e e d s ,s o l u t i o n sp r o g r e s s
t
F i g u r e 13.
RepresentationofCourseofIndependent
VariablesinSurfaceHistory,
OneBody
Point
t h r o u g ht h es p a c eo ft h et a b l ei n d e p e n d e n tv a r i a b l e s
BA,
H r , and P .
dotsinthepicturerepresentsolutionssatisfyingthesurfaceenergybalance
a s time p r o g r e s s e s .S u r r o u n d i n gt h e s ep o i n t sa r e
The
a number o f o t h e r p o i n t s
e x a m i n e dd u r i n gt h ei t e r a t i o n st os a t i s f yt h es u r f a c ee n e r g yb a l a n c e .F o r
s o t o s p e a k ,w i t h i n
any i t e r a t i o n , t h e s o l u t i o n p r o c e d u r e f i n d s i t s e l f ,
cubeformedby
t h eb r a c k e t i n gt a b u l a rv a l u e so f
d e p e n d e nqt u a n t i t i e s
BA,
Hr,
and P .
a
The
h /peueCM have
been
prer51 R
calculatedforthesetabularpoints;relevantvaluesofthesequantities
Tw, -9,
/peueCM, hw and
f o rt h ec u r r e n ti t e r a t i o nv a l u e s
of B;,
H r , and P
canthenbeformed
by
cube and t h e s u r f a c e e n e r g y b a l a n c e e q u a t i o n c a l c u l a t e d .I ft h ee n e r g yb a l a n c e
is notsatisfiedto
some p r e s e l e c t e dd e g r e e
o fa c c u r a c y ,
a new v a l u eo f
B h c a nb es e l e c t e da n dt h ep r o c e s sr e p e a t e d .
i n t e x p o l a t i o ni n s i d et h e
The i n t e r p o l a t i o n f e a t u r e
of t h e t a b u l a r a p p r o a c h d r a s t i c a l l y r e d u c e s
same time a l l o w i n g
The p r e c a l c u l a t e dt a b l ea p p r o a c hh a st h ef o l l o w i n g
t h e number o f s u r f a c e s t a t e c a l c u l a t i o n s w h i l e a t t h e
s u f f i c i e n ta c c u r a c y .
*
advantages :
1.
I np a r a m e t r i cs t u d i e s ,t a b l e so n c eg e n e r a t e da r eu s e a b l e
f o r many d i f f e r e n t p r o b l e m s , y i e l d i n g e v e n g r e a t e r
*
24
economy.
Forexample,
a t y p i c a l 2 0 0 0 t i m e s t e pp r o b l e mw i t ht h eu s u a l
5 iterations
which
p e r time s t e p would r e q u i r e 1 0 , 0 0 0 s u r f a c e s t a t e c a l c u l a t i o n s ,
7 0 9 4 s p e e dc l a s s .
A single
r e q u i r e 1 t o 3 s e c o n d se a c hf o rm a c h i n e si nt h e
300 s t a t e
e n t h a l p y table, on t h eo t h e rh a n d ,m i g h ti n v o l v eo n l ya b o u t
30 improvement.
s o l u t i o n s (30 B; v a l u e s times 1 0 p r e s s u r e s ) , a f a c t o ro f
M u l t i p l ee n t h a l p yt a b l e sr e d u c et h i sa d v a n t a g e ,b u tu s u a l l yo n l y
a few
e n t h a l p i e sa r er e q u i r e d .
2.
Formostproblems,
it i s d i f f i c u l t t o s p e c i f y
a p r i o r i an
of i n d e p e n d e n t t a b u l a r v a l u e s EA, H,-,
- and P.
An e x a m i n a t i o n o f t h e p r e c a l c u l a t e d s u r f a c e
tables b e f o r e
e x e c u t i o no ft h ea c t u a la b l a t i o n - p r o b l e m - p l u s - i n - d e p t h - s o l u t i o n
c a nr e v e a l if t h e r e a r e a n y " h o l e s " i n t h e t a b l e s i n
areas
whereenergy
terms a r ev a r y i n gr a p i d l y .D e s i r a b l e
table p o i n t s
c a nb ea d d e db e f o r et h ei n - d e p t hr e s p o n s er u n .
"adequate" array
3.
The s u r f a c e t a b l e s a r e f r e q u e n t l y o f i n d e p e n d e n t i n t e r e s t
inthemselvesforjudgingtheablationeffectsundervarious
conditions.
4.
Finally,withoutthetabularapproach,theoccasional
vergentsurfacechemistrysolutionwouldstoptheentirein-
noncon-
d e p t hs o l u t i o np r o c e s s .
With t h ep r e c a l c u l a t e dt a b l ea p p r o a c h ,
s u c hs o l u t i o n sa r ea u t o m a t i c a l l y
weeded o u t o f t h e t a b l e s
w i t h o u t damage t o t h e s u b s e q u e n t i n - d e p t h s o l u t i o n s .
On t h e o t h e r s i d e o f t h e l e d g e r ,
some b i g d i s a d v a n t a g e s i n t h e
p r e c a l c u l a t e dt a b l ea p p r o a c ha r ee v i d e n th o w e v e r :
1.
F i g u r e 1 3 , which i s a r e a l i s t i c s c h e m a t i c view
Of
a typical
calculationhistory,indicatesthat
m o s to ft h el a b o r i o u s l y
c a l c u l a t e d a n da s s e m b l e ds u r f a c es t a t ep o i n t si nt h et a b l e
a r en e v e ru s e di nt h ec o u r s e
2.
of a g i v e n s o l u t i o n .
state solution
The " m e c h a n i c a l "l i n k a g eb e t w e e nt h es u r f a c e
and t h e i n - d e p t h s o l u t i o n ,
i.e., thetransferof
punchedcard
surfacestateoutputtoinput
of t h ei n - d e p t hp r o g r a m ,l e a d s
t o computingdelaysandoccasionally
(wrongdecks,missingpartsofdecks
togrossinputblunders
,
etc.)
.
3.
Too m-xh s t o r a g e i s r e q u i r e d by t h e l a r g e p r e c a l c u l a t e d
table.
4.
The s y s t e m c a n n o t r e a d i l y b e g e n e r a l i z e d t o a c c o m o d a t e
more
p a r a m e t e r s .B o t hs t o r a g er e q u i r e m e n t sa n dc o m p u t a t i o n
time
grow o u t o f r e a s o n a b l e b o u n d s .
T h i s l a s t d i f f i c u l t y wouldprove
to be
a great stumbling block in
t h ea b l a t i o nc o d ed e s c r i b e dh e r e ,w h i c hc o n s i d e r sc h e m i c a lk i n e t i c si n
additiontotheotherindependentvariables
BA, H,,
and P d e s c r i b e d a b o v e .
" K i n e t i c s "c o n t r i b u t e a f o u r t hp a r a m e t e ri nt h ef o l l o w i n g
manner:
The
a number o f s i m u l t a n e o u s e q u a t i o n s i n c l u d i n g
and k i n e t i c a l l y c o n t r o l l e d r e a c t i o n s
Each k i n e t i c a l l yc o n t r o l l e dr e a c t i o ne q u a t i o nh a so n e
chemicalstateroutineconsiders
elementbalances,equilibriumrelations,
( R e f e r e n c e s 4 and 5 ) .
forward r a t e c o e f f i c i e n t , which i n t u r n c o m p r i s e s o n e p r e - e x p o n e n t i a l f a c t o r
B m ( f o r t h e masuch r e a c t i o n ) and a t e m p e r a t u r ed e p e n d e n tq u a n t i t y .
All
25
I
e q u a t i o n sa r en o r m a l i z e d
on t h e t r a n s f e r c o e f f i c i e n t
desiredindependentvariable
B'
peueCg t h i s y i e l d s
intheelementbalances
n o r m a l i z e dk i n e t i cr a t ec o e f f i c i e n t s
Bm/peueCM.
the
and a l s o p r o d u c e s
Thus a n yf o r w a r dr a t ec o e f -
ficientsusedinthestateroutineareinputinthe
form Bm/peueCM, and t h e
whole s e t o f Bm/peueCM v a l u e s c h a r a c t e r i z e s t h e r o l e o f k i n e t i c s . 1 n
a given
s e t of B,'s
i s onlyoneparameterzaling
the k i n e t i c e f f e c t s . E x ~ a l t l y
what t h i sp a r a m e t e r i s c a l l e d i s n o ti m p o r t a n t ;
l e t i t betermed B1/peueCM where B1 r e p r e s e n t s t h e f i r s t o f t h e s e t o f k i n -
problemthe
is f i x e d ; t h u s r e a l l y t h e r e
e t i c a l l yc o n t r o l l e dr e a c t i o n sc o n s i d e r e d .
Thus t h e f o u r p r o b l e m v a r i a b l e s , w h i c h w o u l d b e t h e i n d e p e n d e n t
table, are:
v a r i a b l e si nt h ep r e p a r e d
1.
A b l a t i o nr a t e
2.
K i n e t i c sp a r a m e t e r
3.
Recovery
enthalpy
4.
Pressure
BA
B1/peUeCM
F o u ri n d e p e n d e n tv a r i a b l e sa r ep l a i n l yv e r yt r o u b l e s o m e .
t h es a v i n g si nc o m p u t a t i o n
time a r e s e r i o u s l y
First,
compromised, s i n c e t h e
number
o fp r e p r e p a r e ds o l u t i o n sh a sg o n e
up by a l a r g e nvrmber.
Second,machine
are now i n c o n v e n i e n t ;s e v e r a ld e p e n d e n tv a r i a b l e sm u s t
b es t o r e da te a c hp o i n ti na na r r a yw h i c hm i g h ti n c l u d e
30 x 1 0 x 1 0 x 1 0
=
3 0 0 0 0 p o i n t s .T h e s ed i f f i c u l t i e si nt h ep r e - c a l c u l a t e dt a b l ea p p r o a c h
lead to the invention of
a revisedtabularapproachdescribedinthenext
sub-section.
s t o r a g er e q u i r e m e n t s
4.2.2
4.2.2.1
Direct-Coupled
Table
G e n e r aD
l escsiption
I t is obviousthatthedifficultiesinvolvedintheprecalculated
tableapproachto
a " f o u r - d i m e n s i o n a lt a b l ep r o b l e m "c o u l db e
removed w i t h
a " c l o s e - c o u p l i n g ' 'p r o c e d u r ew h i c ho n l yc a l l e df o rc a l c u l a t i o no ft a b u l a r
*
v a l u e sa st h e ya r en e e d e dd u r i n gt h ea c t u a li n - d e p t hs o l u t i o n
.
A t any
i n s t a n t ,t h es u r f a c e - e n e r g y - b a l a n c e - p l u s - i n - d e p t hr e s p o n s es o l u t i o np r o c e d u r e
f i r s t examines t h ec o r n e r so ft h e" s q u a r e "( o r" c u b e " ,d e p e n d i n g
number o fi n d e p e n d e n tv a r i a b l e s )
it f i n d s i t s e l f i n , t o
on t h e
see i f t h e n e c e s s a r y
d e p e n d e n tq u a n t i t i e sh a v eb e e nc o m p u t e da n ds t o r e d .I fn o t ,t h ei n - d e p t h
routinereturnsto
v a l u e s be s u p p l i e d .
a m a s t e rc o n t r o lr o u t i n e
and a s k s t h a t
any m i s s i n g
The m a s t e rc o n t r o lr o u t i n ed e t e r m i n e sw h i c hc o r n e r s
*Another way o u t , n o t c o n s i d e r e d h e r e ,
is t o r e d u c e t h e number o f p a r a m e t e r s
fromfourto
twoby
i g n o r i n g k i n e t i c s andbyassumingequaldiffusion
coF o rt h i s" c l a s s i c a l "s i m p l ec a s e ,a s
is w e l l known,
e f f i c i e n t s and CM = CH.
theedgestatedoesnotenterthedeterminationofthesurfacestate,
and
B ' and P a r e t h e
onecanusethe
sametwo
parametersurfacetableinwhich
independentquantities,forallsurfacelocations
and a l l tfmes.
26
I.
"
.
.
I.
."..... ......""""._
needfilling,
calls t h e s u r f a c e s t a t e program t o f i l l them,andthenreturns
tothein-depthroutine,whichcontinueswiththeproblemsolution.
Suchanapproachminimizes
the number o f s t a t e p o i n t s c a l c u l a t e d ,
since the four parameters in the
list above shrink to only three because
a n de d g es t a t e )
can now
time = t h e s o l u t i o n p r o g r e s s e s , a l t h o u g h
when
t h es e c o n d ,t h i r d ,a n df o u r t hv a r i a b l e s
be directly associated with
(B1/peueCM
t h i s i s done a x i a l s t a t i o n s m u s t b e d i s t i n g u i s h e d f r o m e a c h o t h e r , s i n c e
each station w i l l have different
t i m e h i s t o r i e s . T h e r e i s a n e t d e c r e a s e of
one i n t h e number o f i n d e p e n d e n t v a r i a b l e s . I f t h e
t a l e s were t o b e p r e c a l c u l a t e d ,t h et h r e ev a r i a b l e sw o u l d ,o fc o u r s e ,r e q u i r es e p a r a t ea r r a y s
s p e c i f i e di na d v a n c e ,s i n c et h et r u eh i s t o r i e s
c o n s e q u e n c e so ft h es o l u t i o nd u et o
of p u C andedge s t a t e a r e
eeM
body s h a p e e f f e c t s .
With d i r e c t
c o u p l i n g , E / p u C and t h ee d g e s t a t e a r e e f f e c t i v e l y known f u n c t i o n so f
1 eeM
t i m e foreachstationsincethebodyshape
is to be calculated
a t each
instant.
Thus t h ei n d e p e n d e n tv a r i a b l e
1.
A b l a t i orna t e
2.
A x i aslt a t i o n
l i s t becomes:
E'
T i3m
. e
Thus t h ed i r e c tc o u p l e da p p r o a c hb r i n g s
s y s t e ma n dt h e
u s back t o a t h r e e - p a r a m e t e r
same generaleconomicsasbefore.Anothercomputational
economy i s i n t r o d u c e d s i n c e i n g e n e r a l t h i s a r r a y
filledinthecourseof
w i l l notbecompletely
a t r a n s i e n tp r o b l e m .
A d d i t i o n a le c o n o m i e so fc o m p u t e rs t o r a g ea r er e a l i z e ds i n c eo n l y
two t a b l e v a l u e s o f
t i m e areinvolvedinthecalculationat
I n s t e a d of h a v i n g t o s t o r e
F i g u r e 13, t h e programneeds
o fo n eb o xa si n d i c a t e di nF i g u r e
a wholeboxofsolutionsofthesort
to store at
14.
any one i n s t a n t .
shown i n
any one t i m e o n l y a d j a c e n t " s h e e t s "
A s s o o na st h e
time p a s s e so u to ft h e
i n t e r v a lb e t w e e nt h e
two s h e e t s , t h e e a r l i e r
( O n ) o ft h e s e
t w o sheets is
d i s c a r d eadn d
a new s h e e t a t
( t e m p o r a r i l y empty o f any dependent
s o l u t i o nv a l u e s )a d d e d .
Thus t h e d i r e c t c o u p l e d a p p r o a c h p u t s t h e
"more-than-three-parametert a b l e " p r o b l e mw i t h i nr e a c he c o n o m i c a l l ys p e a k i n g ,a n dr e n d e r s
it more
27
4
%
8 dimensionincludes
effects of t r a n s i e n t ,
shapedependentedge
s t a t e and p,ueCM
2
4
%
+,
v)
rl
d
Figure 1 4 .
S k e t c hI n d i c a t i n gS o l u t i o nS p a c ea n d
S o l u t i o n" S h e e t s "S t o r e da t
Any I n s t a n t
During Solution
e c o n o m i c a lt h a nt h ep r e c a l c u l a t e dt h r e e - p a r a m e t e rt a b l ea p p r o a c h .F u r t h e r more, it r e d u c e sc o m p u t e rs t o r a g ec o n s i d e r a b l ya n dr e d u c e sm e c h a n i c a lh a n d l i n g
problemsofpunched
tables.
However, i n t h i s d i r e c t - c o u p l e d
schemetwo
i m p o r t a n tp r o c e d u r a l
w i l l b ed e s c r i b e di nt h ef o l l o w i n g
m a t t e r sh a v et ob ec o n s i d e r e d .T h e s e
sub-section.
Some Dangers i nt h eD i r e c t - C o u p l e dT a b l eA p p r o a c h
4.2.2.2
is
The g e n e r a l a t t r a c t i v e n e s s o f t h e d i r e c t - c o u p l e d t a b l e a p p r o a c h
somewhatcompromisedbytwo
1.
potentialproblems:
The u s e r w i l l probablywant
t a b u l a ri n d e p e n d e n t
Bh
t os p e c i f yi na d v a n c et h ea r r a yo f
values.Butwhat
if he makes a p o o r
c h o i c e , s o t h a tt h ei n - d e p t hs o l u t i o nh a st r o u b l ef i n d i n g
a
s u r f a c e - e n e r g yb a l a n c ew i t ht h et a b u l a rv a l u e sp r o v i d e d ?
Can
the p r o b l e m s o l u t i o n b y a u t o -
a scheme b e d e v i s e d t o r e s c u e
m a t i c a l l yi n s e r t i n q
new t a b u l a r
E;
v a l u e si nu n f o r e s e e nc r i t i c a l
ordifficultareasofthetable?
2.
What c a nb ed o n ea b o u tt h eo c c a s i o n a ln o n c o n v e r g e n ts u r f a c e
s t a t es o l u t i o no b t a i n e d
from t h e s u r f a c e s t a t e
shows t h a t i t i s n o t p o s s i b l e t o g u a r a n t e e
fromcomplex
a r r a yu s u a l l y
s u r f a c ec h e m i s t r yr o u t i n e s :i n
t h es u r f a c es t a t e
28
any s u r f a c e t a b l e
some of t h e p o i n t s p r o v e t o b e u n o b t a i n a b l e w i t h o u t
some hum.an i n t e r v e n t i o n i n t h e
schememust
program?Experience
a c o n v e r g e ds o l u t i o n
program.
form o f b e t t e r f i r s t q u e s s e s f o r
I nt h ec o u p l e dp r o g r a m ,t h ei n - d e p t h
beabletoadjusttothisfact
a n dc o n t i n u ec a l c u l a t i n g
I
of a c o n v e r g e d e n t r y i n t h e s u r f a c e
even i n t h e o c c a s i o n a l a b s e n c e
s t a t e table.
an automaticanswer
The c o d e d e s c r i b e d h e r e d o e s n o t c o n t a i n
to the
f i r s t problem. A somewhat s i m i l a rc o u p l e dc o d e( R e f e r e n c e
6) u s e df o r
charringmaterialsofvery
complex compositionhas shown t h a t s u c h a f e a t u r e
i s less necessarythanmightbesupposed
a priori.In
w o r t h w h i l es c h e m ew o u l db ev e r yd i f f i c u l tt od e v i s e .
any e v e n t , a g e n e r a l l y
The codedoescontain,
however, a v e r y v a l u a b l e f e a t u r e w h i c h s u c c e s s f u l l y
meets most of t h e d i f f i c u l t i e s which a r i s eu n d e rp o i n t
number 1. above.Carbon
a b l a t i o ni na i r
p r o v i d e s a worthwhileexample.Figure
o fc a r b o na b l a t i o nr a t e si na i r .
1 5 shows t h e dependence on t e m p e r a t u r e
Over a widerangeoftemperature
B& i s f o r
a l lp r a c t i c a lp u r p o s e si n d e p e n d e n to ft e m p e r a t u r e ;i nt h i sr a n g et h eo n l y
u s e f u li n d e p e n d e n tv a r i a b l ei nt h es u r f a c et h e r m o c h e m i c a ls o l u t i o np r o c e s s
i s t e m p e r a t u r e ,s i n c e
i t wouldbenearlyimpossible
t o s e l e c t a sequenceof
Bd v a l u e s d i s t r i b u t e d a l o n g t h e p l a t e a u .
Therefore , the description presented above, in
which BG was usedas
o n eo ft h ei n d e p e n d e n tv a r i a b l e s ,s h o u l db ee x t e n d e dt oi n c l u d et e m p e r a t u r e
a s an i n d e p e n d e n tv a r i a b l ei np l a c eo f
BG, a t l e a s t i n
some c a s e s .F i g u r e
a l s o shows,however,
t h a tt e m p e r a t u r e
15
i s n o t a good i n d e p e n d e n t v a r i a b l e i n
some i n s t a n c e s .
I
I
I
+
K i n e t i cPsl1a t e a u
Regime
Regime
F i g u r e 15.
A t h i g ht e m p e r a t u r e s
T
I
I
I
Sublima.tion
Regime
A b l a t i o nR a t e
B& VersusTemperature
f o r Carbon i n A i r
B & becomes stronglydependenton
T , s o t h a t BA i s t h e
preferredindependentvariable.
T h i sg e n e r a lp r o b l e m
i s r a t h e rt y p i c a l ,
i t t u r n so u t .C o n s e q u e n t l y
thesingleindependentvariablerepresented
by B b i n F i g u r e 14 h a s , i n t h e
c o d e , a more complexdualnature.
The v a r i a b l eb e g i n sw i t h
a s t r i n go f
a s c e n d i n gs u r f a c et e m p e r a t u r e( n o t
B&) values(chosen
by t h eu s e r )a n d
concludeswith a string of ascending
B b values(alsochosen
by t h e u s e r )
.
29
MACABRE code calls f o r s u r f a c e s t a t e s o l u t i o n s
During the s o l u t i o n p r o c e s s , t h e
a t t h e minimum ( f i r s t ) B;
Bi s t r i n g a n d n o t e s t h e t e m p e r a t u r e
valueineach
d i s c o v e r e df o rt h i sp o i n t .T e m p e r a t u r ep o i n t si nt h et e m p e r a t u r es t r i n ga b o v e
t h i st e m p e r a t u r ea r ei g n o r e d .I ns o l v i n gf o rt h es u r f a c ee n e r g yb a l a n c ea t
a givenpointat
a g i v e n t i m e , t h e codecompares
f a c et e m p e r a t u r e
T
o ft h e
W
t ot h e
thepreviously
formed s u r f i r s t B l v a l u e sf o re a c h
two t e m p e r a t u r e s a t t h e
two t i m e s h e e t s c u r r e n t l y s t o r e d . I f
TW i s less t h a n e i t h e r o f t h e s e
w i l l refer only to
two v a l u e s , t h e s u r f a c e e n e r g y b a l a n c e i t e r a t i o n p r o c e s s
t h eu s e r - a s s i g n e dt e m p e r a t u r ep a r to ft h et a b l e( w i t h
BGls and o t h e rd e p e n d e n t
informationateachtabularpointdiscoveredasnecessary
by c a l l s t o t h e
w i l l refer t ot h e
c h e m i s t r yr o u t i n e s ) .O t h e r w i s e ,t h ei t e r a t i o np r o c e s s
user-assigned B ' p a r to ft h et a b l e( w i t h
T w ' s and o t h e rd e p e n d e n ti n f o r m a t i o n
d i s c o v e r e d i n a s i m i l a r manner a s n e e d e d ) .
Thisdualnature
mechanism s o l v e s many o f t h e d i f f i c u l t i e s
o t h e r w i s ea r i s eu n d e rp o i n t
1. above.
caution,however.Forexample,
The u s e rm u s t
w h i c hn i q h t
s t i l l e x e r c i s e some
i nt h ec a s ei l l u s t r a t e d
the
by F i g u r e1 5 ,
code u s e r m i g h t b e g i n h l s s e q u e n c e o f i n d e p e n d e n t v a r i a b l e s w i t h t e m p e r a t u r e
e n t r i e ss t a r t i n ga tt h el o w e s tt e m p e r a t u r eo fi n t e r e s t( s a y ,
e x t e n d i n g up t o ( a s
a minimum o b j e c t i v e ) t h e p o i n t w h e r e
53O0R) and
Bd b e g i n s t o
a b o v et h ep l a t e a uv a l u ef o rt h el o w e s tp r e s s u r et ob ee n c o u n t e r e d .
rise
I t would
"rise
b ec o n s e r v a t i v ea n dp r e f e r a b l et oa d dt e m p e r a t u r e se v e nb e y o n dt h e
pomt"atthehighestpressureincasethesepointsshouldbeneededduring
thesolution.
Afterthesetemperaturevalues,
rangeof
t h ep l a t e a uv a l u e .
plateauinthe
1.
Bdls may beadded
The u s e r m u s t b e c a r e f u l t o p i c k
Bd 1s.
to span the expected
a minimum Bd s l i g h t l y above
A Bd v a l u e n e a r t h e s t a r t o f t h e p l a t e a u o r o f f t h e
low t e m p e r a t u r e r e g i o n
The l o w e s ta s s i g n e d
is e x t r e m e l y u n d e s i r a b l e f o r
two r e a s o n s :
BG d e t e r m i n e st h eb r e a kb e t w e e na s s i g n e d
Bg s o l u t i o n s ; a low Bd w i l l
t e m p e r a t u r es o l u t i o n sa n da s s i g n e d
ineffect"disqualify"alloftheassignedtemper--.%urepoints
a c r o s st h ep l a t e a u
and l e a v e a l a r g e " h o l e " i n t h e a r r a y o f
s u r f a c et h e r m o c h e m i c a ls o l u t i o n s .
2.
F o re q u i l i b r i u mc a l c u l a t i o n s ,a s s i g n e d
below t h e p l a t e a u h a v e
Bd s o l u t i o n s a t
Bd v a l u e s
a highprobabilityofnon-convergence
difficultiesinthechemistryroutines.
The s e c o n d p r a c t i c a l d i f f i c u l t y c i t e d a b o v e f o r t h e d i r e c t c o u p l e d
t a b l ea p p r o a c hc o n c e r n sp o s s i b l en o n - c o n v e r g e n ts o l u t i o n si nt h ec h e m i s t r y
r o u t i n e s .S t r i c t l ys p e a k i n g ,e v e no n es u c hs o l u t i o ns h o u l ds e r v et oh a l t
theentiresolutionprocess,butfortunatelythesituationusuallydoesnot
demand s u c hd r a s t i ca c t i o n .
of iterations to find the
30
The c h e m i s t r yr o u t i n e sa l l o w
a c e r t a i n number
state solution;failuretoconvergewithinthepre-
s e t a l l o w e d number of i t e r a t i o n s i s r a r e a n d u s u a l l y
means t h a t t h e s o l u t i o n
e i t h e r is oscillating in the very near neighborhood of the true solution
or has arrived near to the solution
and i s p r o c e e d i n g v e r y s l o w l y t o w a r d
it.
Therefore,usuallythe
MACABRe code may s a f e l y a c c e p t a c e r t a i n number o f '
non-converged answers. Beyond t h i s l i m i t , t h ec o m p u t a t i o n i s s t o p p e d as a
are u s u a l l y d u e t o l a r g e
safetymeasure,sincerepeatedfailurestoconverge
input blunders.
4.2.2.3
The O t h e r I n d e p e n d e n t V a r i a b l e s i n t h e
S u b - s e c t i o n 4.2.2.2
o ft h e
TJBA
abovehasdescribedin
Direct Coupled T a b l e
some d e t a i l t h e u s e r i n p u t
s t r i n go fi n d e p e n d e n tv a r i a b l ev a l u e s .T h e r ea r e ,o fc o u r s e ,
The f i r s t o f t h e s e
two
are q u i t e s i m p l e .
additional independent variables, but fortunately these
i s simply body s t a t i o no rs u r f a c ep o i n t .T h e s e
i t s own s t r i n g o f TJBA v a l u e s . Thus
wouldappear i n F i g u r e 1 4 a s a
i t s own h o r i z o n t a l
number o f s t r i n g s o f s o l u t i o n p o i n t s , e a c h w a n d e r i n g i n
p l a n e ( t h e t i m e -BA p l a n e ) .
a r ek e p ts e p a r a t e ,e a c hp o i n th a v i n g
thehistoriesofallthesurfacepoints
t i m e dimension, is represented
t i m e v a l u e s . The same a r r a y w i l l b e u s e d f o r a l l
stations.
I t is n o t d i f f i c u l t f o r t h e u s e r t o
s e l e c t a s e t ofvaluesappropriate to the transient nature of
a givenproblem.
The l a s t dimension t o b e c o n s i d e r e d , t h e
by a u s e r s e l e c t e d s t r i n g o f
any g i v e n time, t h e
Thus,inevaluatingthesurfaceenergybalanceat
surface energy balance routine refers
to s t o r e d t a b u l a r v a l u e s o f
T w and
enthalpyquantitiesatthe
two t a b u l a r v a l u e s o f
t i m e s u r r o u n d i n gt h ec u r r e n t
time.
I ft h er e q u i s i t ei n f o r m a t i o nh a sn o ty e tb e e nc a l c u l a t e d
and s t o r e d ,
theFILLroutinecallsthethermochemicalstateprogramtosupplythisinformation.
4.2.2.1
I f a " p r e v i o u s time" p o i n t( a t
andFigure
naturallyevaluated
8
i nt h en o m e n c l a t u r eo fS e c t i o n
is
of t h e l o c a l i n d e p e n d e n t v a r i a b l e s i n t h e
1 4 ) i s beingfilledin,thethermochemicalstate
a t storedvalues
*
c h e m i s t r ys o l u t i o n ,p r e s s u r ea n dr e c o v e r ye n t h a l p y ,a tt h a tp r e v i o u s
I f a " f u t u r e time" t a b l e p o i n t is b e i n g f i l l e d i n , t h e F I L L r o u t i n e
to obtain the state solution at future local values of pressure
e n t h a l p y .I ft h eu s e r
i s n o ts u p p l y i n gt h e s ev a l u e sa sf u n c t i o n so f
time.
is obliged
andrecovery
time
( a n dt h e r e b ys u p p r e s s i n g
some of t h ea u t o m a t i cc o u p l i n gf e a t u r e so ft h ec o d e ;
see S e c t i o n 7 b e l o w ) ,t h e nt h e s ef u t u r ev a l u e sm u s tb ee s t i m a t e d .F u t u r e
l o c a l p r e s s u r e is e s t i m a t e d as t h e f u t u r e s t a g n a t i o n p r e s s u r e
times t h e p a s t
ratio of local pressure to stagnation pressure
(i.e. , basedonthecorrect
.
f u t u r es t a g n a t i o np r e s s u r e ,b u tt h ep a s t
body s h a p e )
S i m i l a r l y ,f u t u r e
l o c a lr e c o v e r ye n t h a l p y
is e s t i m a t e da st h ef u t u r es t a g n a t i o ne n t h a l p y
times
the ratio of the past local recovery enthalpy to the past stagnation enthalpy.
*
The t r a n s f e r c o e f f i c i e n t p ueCM i s a l s o a n i n d e p e n d e n t v a r i a b l e , a s d i s c u s s e d
above; i f k i n e t i c s a r e b e i g g c o n s i d e r e d
it i s h a n d l e d d i f f e r e n t l y , a s
will
be explained below.
31
refers o n l y t o " f u t u r e t i m e " communicationwiththe
state
r o u t i n e , f o r which an a p p r o x i m a t e f u t u r e v a l u e
of H r i s q u i t e a d e q u a t e s i n c e
H r h a s o n l y a s l i g h t e f f e c t on t h e s u r f a c e t h e r m o c h e m i c a l
state evaluation:
inthesurfaceenergy,balanceequationevaluation
i t s e l f , the"properly"
computed c u r r e n t r e c o v e r y e n t h a l p y
i s used.)
(Thisapproximation
4.2.3
Summary
a surface state
Threeprocedureshavebeenconsideredforcoupling
s o l u t i o nt ot h et w o - d i m e n s i o n a li n - d e p t hs o l u t i o n :d i r e c tc a l c u l a t i o na t
tables, and d i r e c t c o u p l e d t a b l e s f i l l e d i n
e v e r yi t e r a t i o n ,p r e c a l c u l a t e d
The t h i r d w a s shown t o b e t h e o n l y r e a s o n a b l e
a st h es o l u t i o np r o g r e s s e s .
p r o c e d u r ef o r
a t w o - d i m e n s i o n a li n - d e p t hs o l u t i o n .
A s u i t a b l ec o u p l i n g
a logicroutinetodeterminewhich
table p o i n t s n e e d
fillingduringthesurfaceenergysolutionprocess,plus
some means o f
procedurerequires
table i n d e p e n d e n tv a r i a b l e s .I nt h ec o d e ,t h e
s p e c i f y i n gt h ev a l u e so ft h e
userspecifies
f o re a c h
a priorithesequenceof
body s t a t i o n , p l u s
v a l u e sb ec o n s i d e r e da p p r o p r i a t e
B'
a series o f time p o i n t s a t
w h i c hs o l u t i o n s
are
to be performed.
4.3
THE CONVECTIVE SURFACE ENERGY BALANCE EQUATION
w e havebeenabletodiscussthe
Up t o t h i s p o i n t ,
thein-depthsolution
and t h e s u r f a c e t h e r m o c h e m i c a l
IV-V l i n k b e t w e e n
s t a t e s o l u t i o n , which
i s e f f e c t e dt h r o u g ht h ec o n v e c t i v es u r f a c ee n e r g yb a l a n c eo p e r a t i o n , w i t h o u t
discussingthesurfaceenergyequation
,
g i v e n by Equations(23)
(24)
t oc o n s i d e rt h i se q u a t i o ni n
,
itself except in
(25) , ( 3 0 ) , and ( 3 1 ) .
more d e t a i l .
its generalnature,
I t i s now a p p r o p r i a t e
As n o t e dp r e v i o u s l y ,t h es u r f a c e
l i n k a g e s are b u i l t upon a t r a n s f e r c o e f f i c i e n t a p p r o a c h : t r a n s f e r c o e f f i c i e n t
equationsareusedfor
mass d i f f u s i o n i n t h e e l e m e n t b a l a n c e e q u a t i o n s
and
i s employed
a transfer coefficient type of surface energy balance equation
i n t h e IV-V l i n k a g e .
The p r o p e r c h o i c e o f e n e r g y e q u a t i o n c o n s t i t u t e s
a subject f a r t o o
l a r g e and d i f f i c u l t t o d i s c u s s c o m p l e t e l y i n t h i s r e p o r t .
The MACABRE code
u s e so n eo ft h r e ed i f f e r e n tf o r m so ft h ec o n v e c t i v ee n e r g ye q u a t i o n ,
d e p e n d i n go nu s e rc h o i c e .I na s c e n d i n go r d e ro fc o m p l e x i t y ,a n dd e s c e n d i n g
o r d e ro f
" w e l l f o u n d e d n e s s ", t h e s ee q u a t i o n sa r e
(1) F o re q u a l
mass d i f f u s i o nc o e f f i c i e n t sf o rb o u n d a r yl a y e r
moleculesand
Hr
-
f o r C M = C H ( L e = 1) :
( l + B ' ) h w+ B ' h
Fo_
E
-__
PeuecH
32
c c
Tw4
=
+
OLWqradin
'eUecH
qcond
'eUecH
T h i se q u a t i o n
i s a " s t a n d a r d " (see, f o r example,Reference
7) one
and r e q u i r e s n o d i s c u s s i o n .
(2)
CM # CH ( L e # 1):
F o re q u a ld i f f u s i o nb u t
+
awqrad i n
-
4
FU€TW
~
-
'cue%
'eUecH
SO
#
C
forchemicallyactiveboundarylayershasbeen
remainsopen(References
(3)
7, 8 , 9)
CH e f f e c t s , a n dd o u b t l e s s
M
The accuracy
i s frozen.
if t h eb o u n d a r yl a y e r
(33)
'eUecH
T h i se q u a t i o ns u p p o s e d l ya c c o u n t sf o r
does
=
qcond
Of
Equation ( 3 3 )
much d i s c u s s e d a n d t h e q u e s t i o n
.
F our n e q u adl i f f u s i o n :
4
FU ETw
+ - - -OLwqrad
-"-
'eUecH
"euecH
-~
qcond
'eu,%
Thisequation,introducedinReference
=
(34)
10 , a p p e a r s t o b e u s e f u l f o r
m o d e l i n gu n e q u a ld . f f u s i o n
effects, atleastinfrozenboundarylayers,and
currently is b e i n g s t u d i e d i n d e t a i l t o e s t a b l i s h
i t s validity(Reference
9).
All three o ft h e s ee q u a t i o n s
( 3 2 ) , ( 3 3 ) , and ( 3 4 ) s e r v e t o p r o v i d e
i nE q u a t i o n ( 3 0 ) . The
conexpressionsforthediffusionenergyflux
-q
a
, whichincludeh.
s t i t u e n t s o f -9,
ZZ; h y , and 2 Z* h?
e
iw
Wedge
gas
are g e n e r a t e d by t h e c h e m i s t r y r o u t i n e s a l o n g w i t h
hw and Tw.
The s u b r o u t i n e
FILL,which
c a l l st h ec h e m i s t r yr o u t i n e s
, thencombines
the q u a n t i t i e s i n
the a p p r o p r i a t ef a s h i o na sd i c t a t e d
by E q u a t i o n ( 3 4 ) o r o n e o f t h e s i m p l e r
forms ( 3 3 ) and ( - 2 ) . Forexample,suppose
the FILL r o u t i n ed i s c o v e r s
a
r e q u i r e m e n t f o r a s t a t e s o l u t i o n a t a t a b u l a r B ' and f o r a g i v e n p r e s s u r e
C
a n dr e c o v e r ye n t h a l p y .
FILL c a l l s t h e s t a t e r o u t i n e s , c o n t r o l l e d by E Q U I L ,
whichperform.the
s t a t e s o l u t i o n ,e v a l u a t ek e yp r o p e r t yv a l u e s ,
and r e t u r n
t o FILL v a l u e s Tw , hw , P Z* hiTw , hw,
Zz
h?
FILL t h e n ,r e f e r r i n g
e
le
W
.
33
tothetemperature
structsthe
Tw, looks up t h e a b l a t i n g m a t e r i a l e n t h a l p y h c
and con-
term
of i n t e r e s t . The t e m p e r a t u r e
(On t h ea s s i g n e d
EQUIL r e t u r n s BA and t h i s i s s t o r e d . )
and s t o r e s i t as o n eo ft h ed e p e n d e n tq u a n t i t i e s
Tw i s s t o r e da st h es e c o n dd e p e n d e n tq u a n t i t yo fi n t e r e s t .
Tw l o w e r p a r t o f t h e t a b l e ,
4.4
CONVECTIVE SURFACE ENERGY BALANCE I T E R A T I O N PROCEDURE
A t everysurfacepoint
a t which a c o n v e c t i v e e n e r g y b a l a n c e
i s t o be
o b t a i n e d ,t h ee n e r g yb a l a n c es u b r o u t i n em u s t
decidewhether
Tw o r B '
i s theproperindependentvariable
to iterate
s e l e c t a trial v a l u e o f t h i s i n d e p e n d e n t v a r i a b l e
refertostoreddependentvaluesofenergyquantity
Tw ( o r B;
andtemperature
Y
if in first part of table), inter-
p o l a t i n ga sn e c e s s a r y
on BA and 8 , and f i l l i n g i n needed
FILII
t a b l ep o i n t sa sn e c e s s a r yb yc a l l i n q
look up aW(TW)and
C H , and H r ( e )
EW
(Tw)
,
grad
(8), l o c a l F , p l u s CM,
in
(which i n many cases a r e g e n e r a t e d
s u b r o u t i n e sd e s c r i b e di nS e c t i o n s
8 and 9 below)
f o r m u l a t et h el e f t - h a n ds i d eo fE q u a t i o n
f r o mz e r o( e r r o r
(34), n o t e d e p a r t u r e
E)
if e r r o r i s t o o l a r g e , c o r r e c t
, go t o ( c )
variable
by s p e c i a l
-
G e n e r a l l yt h i sp r o c e s sc o n v e r g e sv e r y
t r i a l v a l u eo fi n d e p e n d e n t
f a s t , u s u a l l yw i t h i nt h r e e
o ff o u ri t e r a t i o n s .S t e p( f ) ,t h ec o r r e c t i o ns t e p ,
a t t h e same t i m e t h e e r r o r
i s b a s e d upon t h e
i s beingcomputed,
Newton-Raphson
procedure:
the derivative
of t h e e r r o r w i t h r e s p e c t t o t h e i n d e p e n d e n t v a r i a b l e
a l s of o r m e d ,u s i n gd e r i v a t i v e s
34
is
of t h e t a b u l a r q u a n t i t i e s
Y and Tw ( o r B b )
i s t h e n computed d i r e c t l y :
The c o r r e c t i o n i n t h e i n d e p e n d e n t v a r i a b l e
or
E
.
4.5
AVOIDING THE SURFACE STATE SOLUTION
The c o n v e c t i v e b o u n d a r y c o n d i t i o n d e s c r i b e d
so f a r i n t h i s s e c t i o n
d o e sn o ti n c l u d e
a l l p r o b l e m so fi n t e r e s t .C o n s e q u e n t l yt h es u r f a c eb o u n d a r y
c o n d i t i o nt r e a t m e n th a sb e e nf o r m u l a t e d
as a t r i p l e o p t i o n .
The f i r s t o f
t h e s e( c a l l e dO p t i o n
i nS e c t i o n
4.1-4.4
1) c o n s t i t u t e s the c o n v e c t i v eb o u n d a r yc o n d i t i o nd e s c r i b e d
above.
A s e c o n do p t i o n( O p t i o n
2) m e r e l yr e f e r st oi n p u t
time-dependentquantitiesforuserassignedvaluesofsurfacetemperature
a n dr e c e s s i o n
rate. A t h i r do p t i o n( O p t i o n
3 ) r e t a i n st h ee n e r g yb a l a n c e
aspectsofOption
1, b u t e l i m i n a t e s t h e c o n v e c t i v e h e a T i n g t e r m s a n d d o e s
n o ta l l o ws u r f a c er e c e s s i o n .T h i so p t i o n
i s u s e f u lf o r
"cool-down'' o r
"soak - o u t " p o r t i o n s o f p r o b l e m s , a n d c a n a l s o
f l u xp r o b l e m si ng e n e r a l( t h r o u g ht h eq r a d
be u s e d f o r a s s i g n e d h e a t
in v a r i a b l e ) i f s u r f a c e r e c e s s i o n
i s n e g l i g i b l e . The h e a t i n g o p t i o n may bechanged
a tt h eu s e r ' sc h o i c ea t
any t i m e f o r a g i v e n column o r s u r f a c e s t a t i o n ,
and t h e h e a t i n g o p t i o n a s s i g n ments may v a r y i n anymanneraround
t h e body a t a given t i m e .
35
SECTION 5
SURFACE
STATE
5.1
CALCULATION
-
PHASE I V
INTRODUCTION
Section 4 abovehasdescribedtheconvectivesurfaceenergybalance
the u s e made i n t h i s b o u n d a r y c o n d i t i o n
calculationofcomputersubroutinesforevaluatingvariousheatedsurface
b o u n d a r yc o n d i t i o na n dd e s c r i b e d
thermochemical s t a t e v a r i a b l e s , c h i e f l y e n t h a l p y a n d m o l e c u l a r c o m p o s i t i o n .
it would be possible to avoid the use
of c h e m i s t r y r o u t i n e s f o r t h i s c a l c u l a t i o n a n d i n s t e a d r e l y
on e n t h a l p y
correlationsandthelike,butforthepresentcoupledcode
it w a s s t r o n g l y
d e s i r e d t o have a g e n e r a l a b l a t i o n p r o g r a m w h i c h c o u l d t r e a t a n y n o n - c h a r r i n g
a b l a t i n gm a t e r i a le x p o s e dt o
any a r b i t r a r ye n v i r o n m e n t .F o r t u n a t e l y ,
suitablethermochemistryroutinesalreadyexistedwhichcouldbecoupledto
For a restricted range of problems
t h ei n - d e p t hr e s p o n s ec o d e :t h ep r o g r a mr e p o r t e dh e r ed i dn o td e v e l o pt h e s e
c h e m i s t r yc o d e s .S i n c et h e s ec o d e sa r ef u l l yd e s c r i b e de l s e w h e r e( R e f e r e n c e s 4 and 5 ) t h i s s e c t i o n
physicalproblemstreated
w i l l seekonlytogive
a g e n e r a lo v e s v i e wo ft h e
by t h e c h e m i s t r y r o u t i n e s .
as a package are r e f e r r e d t o a s
ACE,
a f t e r AerothermChemicalEquilibriumcode.This
i s an i n a p p r o p r i a t e
acronym s i n c e (1) t h er o u t i n e sc o n s i d e rn o n - e q u i l i b r i u mc h e m i s t r y
as w e l l
a s e q u i l i b r i u m , and ( 2 ) i n t h e c o u p l e dc o d en or o u t i n eh a st h e
name ACE;
t h ec o n t r o l l i n gr o u t i n e
i s c a l l e d EQUIL.
F o rh i s t o r i c a lr e a s o n s ,h o w e v e r ,
w e shallcontinuetocallthethermochemistrypackage
ACE.
The t h e r m o c h e m i c a lr o u t i n e s
USES
OF
5.2
ACE I N MACABRE
The ACE package i s c a l l e d upon f o r a number o f c o m p u t a t i o n a l t a s k s
i nt h ec o u p l e dc o d e .
i s i ng e n e r a t i n gs u r f a c e
I t s mainuse,ofcourse,
thermochemical s t a t e s o l u t i o n s .
A s w i l l b ed i s c u s s e db e l o w ,t h e s ea r eo p e n
s y s t e mc a l c u l a t i o n si n v o l v i n gi n t e r a c t i o n sb e t w e e ne d g eg a sa n dt h em a i n
a b l a t i n g material.
terms l i k e hw
The s u r f a c ee n e r g ye q u a t i o n s
and CKi
gas edge
h'liy
e
or
CZ;
hp
(33) and (34) i n c l u d e
, h o w e v e r , w h i c ihn d i c a t easn o t h e r
. e
n e e d e dt h e r m o c h e m i c a lc a l c u l a t i o n :t h em o l e c u l a r
make-up o ft h eb o u n d a r y
l a y e re d g eg a sm u s tb ei d e n t i f i e da n dc e r t a i np r o p e r t i e se v a l u a t e d( s u c h
as Z ? )
1
.
Each c a l l t o ' ACE f o r a s u r f a c e s t a t e s o l u t i o n i n f a c t i n v o l v e s t h e
follo8ing computations :
36
a closed system equilibrium calculation
on edge gas
a l o n e t o d e t e r m i n e Ki
and Z? ; t h i sc a l c u l a t i o n
'e
done a t a s s i g n e d p r e s g u r e a n d r e c o v e r y e n t h a l p y * .
is
anopensystemablationcalculation
a t assignedpressure,
peueCM ( f o r k i n e t i c e f f e c t s ) , and e i t h e r B& o r Tw
(depending on l o c a t i o n i n i n d e p e n d e n t v a r i a b l e a r r a y ) :
this calculation
may i n c l u d e k i n e t i c e f f e c t s
a f r o z eenv a l u a t i oonf
t e m p e r a t u r e TW.
hw
and
edge
gas
, Z 2; h p
Wedge
gas
asdescrlbedinSections
The q u a n t i t i e s h
e
from ACE andemployed
at
h?
CZ*
'e
, hw,
and
h?
C 2;
are o b t a i n e d
W
4.3 and 4 . 4 above.
(The ACE r o u t i n e may also be used by t h e p r e s s u r e d i s t r i b u t i o n
and t r a n s f e r c o e f f i c i e n t r o u t i n e s t o g e n e r a t e c e r t a i n t h e r m o d y n a m i c p r o p e r t i e s and t r a n s p o r tp r o p e r t i e s of t h eb o u n d a r yl a y e rg a s .T h e s ep r o p e r t i e s
are s t o r e d for f u t u r e u s e i n a M o l l i e r C h a r t
(as w i l l b e d e s c r i b e d i n more
detailinSections
8 and 9 below) i n which P andhe
a r et h ei n d e p e n d e n t
variables;therequired
ACE c a l c u l a t i o n s a r e t h e r e f o r e c l o s e d s y s t e m e q u i l i b r i u m c a l c u l a t i o n s of t h e e d g e g a s a t a s s i g n e d p r e s s u r e a n d e n t h a l p y . )
The f o l l o w i n g s e c t i o n s d e s c r i b e t h e
manner i n which c a l c u l a t i o n s
a r e done i n t h e ACE package.
5.3
P H Y S I C A L PROBLEMS TREATED BY THE ACE PACKAGE
The ACE p a c k a g e o f s u b r o u t i n e s
t.reats a widerange of thermoc h e m i c a lp r o b l e m s .S i n c et h e s ep r o b l e m sa r ec o m p l e x ,t h ec a p a b i l i t i e so f
t h e ACE p a c k a g e c a n p e r h a p s b e s t b e i l l u s t r a t e d
by a s e t o f d e s c r i p t i o n s
o fi n c r e a s i n gg e n e r a l i t y .
The f o l l o w i n gs e c t i o n sd e s c r i b es u c h
a graded
seriesofproblems.
5.3.1
Closed
System
Eiqnu i l i b r i u m
Closedsystems are d e f i n e d as t h o s e f o r which t h e r e l a t i v e
amounts o fc h e m i c a le l e m e n t sa r ep r e s p e c i f i e d .
The e d g eg a ss t a t ec a l c u l a t i o n sa r eo ft h i st y p e .
Consider K chemicalelements,
Good argumentscanbe
made f o r u s i n g
p r o v e si n c o n v e n i e n ti nt h e
code.The
any c a s e .
Nk,
introducedinto
a previously
the s t a t i c e n t h a l p y i n s t e a d , b u t t h i s
d i f f e r e n c ei nr e s u l t s
i s s m a l li n
37
e v a c u a t e dc o n t a i n e r .I ng e n e r a l ,t h e s ee l e m e n t s
number o fc h e m i c a ls p e c i e s
*
I f enough t i m e h a s e l a p s e d
, Ni
(gasphase)and
w i l l i n t e r a c t t o form a
N R ( c o n d e n s e dp h a s e s ) .
s o t h a t thermodynamicandchemicalequilibrium
is e s t a b l i s h e d , t h e t h e r m o d y n a m i c s t a t e o f t h e s y s t e m , i n c l u d i n g t h e r e l a t i v e
i s completelydetermined i f two i n d e a m o u n t so fc h e m i c a ls p e c i e sp r e s e n t ,
pendentthermodynamicvariablesare
known
**
.
T h i sc o n d i t i o n
mathematicallybyexaminingthegoverningequationsforsuch
andshowing
that the
number of i n d e p e n d e n t e q u a t i o n s
may b es t a t e d
a system,
is equal to the
number
of unknown q u a n t i t i e s .
of t h eg a s e o u sc h e m i c a ls p e c i e s
as f o l l o w s :
R e l a t i o n se x p r e s s i n gt h ef o r m a t i o n
from t h e g a s e o u s c h e m i c a l e l e m e n t s
may b e w r i t t e n
S i m i l a r l y ,f o r m a t i o no fc o n d e n s e dp h a s es p e c i e sf r o mt h eg a s e o u se l e m e n t s
is written:
K
'kll
(38)
Nk
K= 1
Intheabove,
o fs p e c i e s
Cki
number of atoms of e l e m e n t k i n a molecule
representsthe
R (condensed).
i ( g a s )o rs p e c i e s
If t h e g a s p h a s e s p e c i e s a r e
assumed t o i n d i v i d u a l l y b e h a v e a s
t h e r m a l l yp e r f e c tg a s e s ,t h e nt h ee q u i l i b r i u mr e l a t i o nc o r r e s p o n d i n gt o
r e a c t i o n( 3 7 )
is
k=l
I n Pi
-
Cki
I n Pk = I n K
k= 1
where Pk d e n o t e sp a r t i a lp r e s s u r ea n d
f o rt h ef o r m a t i o nr e a c t i o n( 3 7 )
**
38
K
Pi
Pi
(T)
( T ) i s t h ee q u i l i b r i u mc o n s t a n t
of s p e c i e s Ni.
Foreachcandidatecondensed
" C h e m i c a ls p e c i e s "a su s e dh e r ei n c l u d e sm o l e c u l a r ,a t o m i c ,i o n i c ,
electronspecies.
Duhem's Theorem.
(39)
and
p h a s es p e c i e s
K
k=l
where
=
i n d i c a t e st h ee x i s t e n c eo ft h ec o n d e n s e dp h a s es p e c i e s
N E i ne q u i l i b r i u mw i t hg a sp h a s es p e c i e s ,a n d
<
NE w i l l not
i n d i c a t e st h a tt h ec o n d e n s e dp h a s es p e c i e s
bepresentinequilibrium.
F o re a c hc h e m i c a le l e m e n ti n t r o d u c e di n t ot h es y s t e m ,t h ec o n s e r v a t i o n
of atoms d i c t a t e s that t h e amount ofanyelement
k i nt h eg a s
andcond e n s e dp h a s e s( r e g a r d l e s so fm o l e c u l a rc o n f i g u r a t i o n )m u s t
sum t o t h e t o t a l
amount o fe l e m e n t
k i nt h es y s t e m .M a t h e m a t i c a l l y ,t h i s
may b ew r i t t e n ,
f o re a c he l e m e n t
k , as
Mass f r a c t i o n
of e l e m e n t k
inpu
to
ht e
=
sys t e m
where
nf,
x
I
i=l
L
s i p i
+
Ec ~ ~ x Q
11=1
74 i s a c o m p o s i t es y s t e mm o l e c u l a rw e i g h t *d e f i n e d
andwhere
Xi
i s amole
xi
by
f r a c t i o n of condensedphasespecies
=
m o l e c u l e s of c o n d e n s e d s p e c i e s
totalgasphasemolecules
i
This is themolecularweightappropriate
i f c o n d e n s e dp h a s e sa r ep r e s e n t .
R
d e f i n e da s
9.
to theidealgasequation
(43)
of s t a t e
39
,
In addition, for the gas phase species
thereexiststherequirementthat
system pressure
sum t o t h e t o t a l
partial pressures must
the
= P
,
Mixture thermodynamic properties
relatedtothespeciesconcentrations
such as specific enthalpy
,
are
b ye q u a t i o n so ft h ef o r m
Consider now t h e number o fi n d e p e n d e n te q u a t i o n sf o rt h es y s t e m .
is e q u a lt ot h e
number o fg a sp h a s ee q u i l i b r i u mr e l a t i o n s( 3 9 )
g a sp h a s es p e c i e s
aretrivial
I minus t h e number o fe l e m e n t s
when i = k )
f o re a c ho ft h e
.
I na d d i t i o n ,t h e r ee x i s t s
K
The
number of
( b e c a u s eE q u a t i o n s( 3 9 )
(40)
a r e l a t i o ns u c ha s
c a n d i d a t ec o n d e n s e dp h a s es p e c i e si nt h es y s t e m .N o t e
L
t h a tt h es y s t e mt e m p e r a t u r e
i s c o n t a i n e di m p l i c i t l yi nE q u a t i o n s( 3 9 )a n d
t h r o u g ht h et e m p e r a t u r ed e p e n d e n c eo ft h ee q u i l i b r i u mc o n s t a n t s .T h e r e
K
c o n s e r v a t i o no fe l e m e n t sE q u a t i o n s
i n t r o d u c e di n t ot h es y s t e m .
t o t h es y s t e mp r e s s u r e
(44)
(41)
,
one f o re a c ha t o m i ce l e m e n t
The r e q u i r e m e n t t h a t t h e p a r t i a l p r e s s u r e s
c o n t r i b u t e so n ea d d i t i o n a le q u a t i o n .
(45)
sum
For any
a d d i t i o n a l thermodynamic p r o p e r t i e s o f t h e m i x t u r e ( e n t h a l p y , e n t r o p y ,
t h e r ee x i s t se q u a t i o n ss u c ha s
(40)
are
.
etc.),
of t h e
I s p e c i e si nt h eg a sp h a s ea r e
Considernextthevariablesappropriatetothisformulation
problem.
The r e l a t i v ec o n c e n t r a t i o n so ft h e
g i v e n by t h e P i ' s
and t h e amountsof
s p e c i e sa r eg i v e n
by X L (most o r a l l o f
the
l a t i o n ,t h ec o m p o s i t es y s t e mm o l e c u l a rw e i g h t
a r e oneeach
L
candidatecondensedphase
which may b e z e r o ) . I n t h i s
,%
of t h em i x t u r et h e r m o d y n a m i cv a r i a b l e s
number o f v a r i a b l e s a n d a v a i l a b l e i n d e p e n d e n t e q u a t i o n s
40
formu-
i s also a v a r i a b l e .T h e r e
T , P, h , s , e t c .
The
may besummarizedas
VARIABLES
NO. OF SUCH
VARIABLES
EQUATION
NUMBER
TO. O F SUCH
EQUATIONS
I
(39)
I - K
L
L
K
(44)
1
1
1
of the type
n
1I
total
variables
T h u s ,t h e r ea r e
(45)
total
equations
I+L+n+3
two less e q u a t i o n st h a n
dynamic s t a t e o f t h e s y s t e m
I+L+n+l
there a r e v a r i a b l e s ,
i n d e p e n d e n tv a r i a b l e sa r es p e c i f i e d( e . g . ,
e l e m e n t a lc o m p o s i t i o n ,t h e nc l o s u r e
n
and s o i f two
P and T ) i n a d d i t i o n t o t h e
i s obtainedandthechemicalandthermo-
may, i n p r i n c i p a l , b e d e t e r m i n e d .
The ACE p r o g r a mp e r f o r m st h i sd e t e r m i n a t i o n .T h a t
the e q u i l i b r i u m t h e r m o d y n a m i c a n d c h e m i c a l s t a t e o f
i s , t od e t e r m i n e
any . c l o s e d s y s t e m , o n e
needsonly
tofurnishthe
ACE p r o g r a m w i t h t h e e l e m e n t a l c o m p o s i t i o n , t h e
*
candidategaseousandcondensedphasespecies,
andtwo
thermodynamicproperties.
One o f t h e s e p r o p e r t i e s
m u s tb ep r e s s u r e ,a n dt h eo t h e r
e i t h e rt e m p e r a t u r e ,e n t h a l p y ,o re n t r o p y .
program w i l l c a l c u l a t e a n do u t p u tt h e
may b e
Given t h i si n f o r m a t i o n ,t h e
ACE
mole f r a c t i o n so fe a c hc a n d i d a t e
s p e c i e s , the t e m p c L a t u r e ,e n t h a l p y ,e n t r o p y ,d e n s i t y ,e f f e c t i v em o l e c u l a r
w e i g h t ,e q u i l i b r i v m
and f r o z e n s p e c i f i c h e a t s , t h e i s e n t r o p i c e x p o n e n t , a n d
a few other quantities of potential interest.
A=
a l r e a d yn o t e d ,
a n de n t h a l p ys o l u t i o n s .
opensystemcalculation
MACABRE o n l y c a l l s
The e d g e s t a t e
upon ACE f o r a s s i g n e d p r e s s u r e
i s s a v e dm o m e n t a r i l yw h i l et h e
i s performed.
*The ACE p r o g r a m m u s t b e s u p p l i e d c e r t a i n b a s i c t h e r m o d y n a m i c d a t a ( d e s c r i b e d
i n R e f e r e n c e s 2 and 5) f o r e a c h c a n d i d a t e s p e c i e s t o b e c o n s i d e r e d i n
a given
i s c o n t a i n e d i n a t h r e ec a r d s e t , one s e t f o r e a c hs p e c i e s .
s y s t e m .T h i sd a t a
A c e r t a i n amount ofjudgement
is r e q u i r e d o n t h e p a r t o f t h e u s e r r e l a t i v e
t o w h i c hc a n d i d a t es p e c i e ss h o u l db ei n c l u d e di n
a g i v e ns y s t e m .F r e q u e n t l y
t h i s judgement i s a v o i d e d b y s i m p l y i n p u t t i n g d a t a f o r a l l s p e c i e s c o n t a i n i n g
c o m b i n a t i o n so ft h ei n p u te l e m e n t s .
41
5.3.2
Open Systems i n E q u i l i b r i u m
-
S i m p l i f i e dC a s e
The b a s i c t h e o r y u n d e r l y i n g t h e t r e a t m e n t o f o p e n s y s t e m s
may b e s t b e
i l l u s t r a t e d by e x a m i n i n g t h e e q u a t i o n s e x p r e s s i n g t h e c o n s e r v a t i o n
e l e m e n t s and energy a t t h e a b l a t i n g s u r f a c e . I f t h e b o u n d a r y l a y e r
no m a t e r i a l i s removedfrom
t e r i z e d by e q u a l d i f f u s i o n c o e f f i c i e n t s , a n d i f
the surface in
( i.e.
a condensed phase
r e m o v a l ) ,t h e nt h e s ee q u a t i o n st a k e
,
no mechanical erosion or liquid layer
on a p a r t i c u l a r l y s i m p l e
form f o r e q u i w i l l b ec o n s i d e r e di nt h i ss e c t i o n
l i b r i u ms y s t e m s .T h i ss i m p l i f i e ds i t u a t i o n
and t h e s e c o n s i d e r a t i o n s
of chemical
is charae-
w i l l beextendedto
more g e n e r a l i z e d c a s e s i n
Sections 5.3.3 t o 5.3.5.
of chemicalelements
( k ) e n t e r i n ga n d
l e a v i n g a c o n t r o ls u r f a c ea f f i x e dt ot h ea b l a t i n gs u r f a c e .
The s o l i d m a t e r i a l
may b ev i s u a l i z e da s
moving i n t o t h i s s u r f a c e
a t a r a t e 6 . I f it i s assumed
C o n s i d e rf i r s tt h ef l u x e s
t h a t no m a t e r i a l i s b e i n g removed i n a c o n d e n s e d p h a s e , t h e n t h e s u r f a c e
and t h e f l u x e s o f t h e
kth
chemicalelement
Figure 1 6 .
may b e i l l u s t r a t e d
A b l a t i n gS u r f a c e
Terms s u p e r s c r i p t e d by a t i l d e ( - )
Mass Balance
representthetotal
o fe l e m e n tk i, n d e p e n d e n to fm o l e c u l a rc o n f i g u r a t i o n .
as
mass f r a c t i o n o r f l u x
Thus
I
'kW
=
'k?iw
i=l
42
I
I
‘kW =
ukijiw
i=l
where k p e r t a i n st oe l e m e n tk ,
mass f r a c t i o no e
f lement
k
(47)
i s the
i p e r t a i n st os p e c i e s
i , and
o’ki
i ns p e c i e s
i. F l u x e o
s ef l e m e n t
k away from
thesurfaceconsistofboundarylayerdiffusionandgrossmotionofthefluid
a d j a c e n tt ot h es u r f a c ed u et ot h ei n j e c t i o nf l u x
AC.
From the a b o v es k e t c h ,
it i s s e e n t h a t c o n s e r v a t i o n o f c h e m i c a l
e l e m e n t sr e q u i r e st h a t
jkw + ( P V ) K
=
kw
Summing E q u a t i o n ( 4 8 ) o v e ra l le l e m e n t s
equation(forthecase
kc
k
y i e l d st h et o t a l
mass c o n t i n u i t y
when t h e r e i s nocondensedphasematerialremoval)
(Pv),
An
r;l K
=
(49)
mC
i m p o r t a n tf u n d a m e n t a lo ft h ep r e s e n tm a t h e m a t i c a lm o d e l i n go ft h ea b l a t i o n
p r o c e s s i s t h ee x p r e s s i o no ft h ed i f f u s i v eh e a t
a transfercoefficientformulation.
In
t h i fs o r m u l a t i o nt,h e
diffusional
and mass f l u x e s i n
term
-j
terms o f
is written
kW
U t i l i z i n gt h i st r a n s f e rc o e f f i c i e n tf o r m u l a t i o n
f l u xi nt h ee l e m e n t a lb a l a n c eE q u a t i o n
( 5 0 ) f o rt h ed i f f u s i o n
(48) yields
43
.1
..1
.".
... ,..,
-_".
I n terms of d i m e n s i o n l e s s a b l a t i o n r a t e
(51) for t h e t o t a l
B& a n ds o l v i n g
mass f r a c t i o n of e l e m e n t k a t t h e w a l l y i e l d s ( f o r e q u a l d i f f u s i o n c o e f f i c i e n t s
and no condensedphase
material removal)
K,
Given t h e r e l a t i v e
chemicalandthermodynamic
kc
=
a m o u n t so fc h e m i c a le l e m e n t ss p e c i f i e db y( 5 2 )
I the
t o theablatingsurface
state ofthegasesadjacent
may be c a l c u l a t e d f r o m e q u i l i b r i u m r e l a t i o n s .
Sincethegases
are i n e q u i l i b r i u m w i t h t h e a b l a t i n g s u r f a c e
(53)
k=l
i f 9. r e p r e s e n t st h es u r f a c es p e c i e s ,a n d
(54)
I n t h ep r e s e n tf o r m u l a t i o n ,
(54) may be t h o u g h to fa sb e i n gu s e dt oi d e n t i f ys u r f a c es p e c i e s
R for
f o r a l l o t h e rc a n d i d a t ec o n d e n s e dp h a s es p e c i e s .
which(53)applies.
The e q u i l i b r i u mr e l a t i o n s
f o r g a sp h a s es p e c i e s
the form
K
and t h e p a r t i a l p r e s s u r e s
mustobey
the relation
I
& = P
i-1
I
The e l e m e n t a l mass f r a c t i o n sa d j a c e n tt ot h es u r f a c e l
related to the species partial pressures
, Pi,
K
I
k"
of ( 5 2 )a r e
by r e l a t i o n s s u c h a s
i s of
Thus, i f P i s known and BL i s s p e c i f i e d ( t h i s may b e v a r i e d p a r a m e t r i c a l l y
, and
as w i l l b e d i s c u s s e d s u b s e q u e n t l y )
i f Tw and Pi
number o f unknowns a n a e q u a t i o n s a v a i l a b l e
I
I.
NO. OF
SUCH
UNKNOWNS UNKNOWNS
.
.
~~
%
.nc
Tw
-
EQUATION
:QUATION
NO.
.
I
T h u s ,c l o s u r e
NO. O
SF
UCH
EQUATIONS AVAILABLE
.
I
(55)
I-K
K
K
1
(53)
1
1
(52)
K
(56)
1
1
(57)
K
1
Unknowns
are unknowns, t h e n t h e
may b e summarized as
-1'0"
-11
I+K+2
T
0
Eqquuaa tt ii o
on
n ss 1
t
'
Z
K ++ 22
II ++ K
-
Y
i s o b t a i n e da n d ,i np r i n c i p a l ,t h et e m p e r a t u r e
and
t o t h e s u r f a c e may bedetermined
From t h ep r e s s u r e ,t e m p e r a t u r e ,
andchemical
c h e m i c a lc o m p o s i t i o no ft h eg a s e sa d j a c e n t
P and EA are s p e c i f i e d .
c o m p o s i t i o n ,t h ec a l c u l a t i o no fo t h e rt h e r m o d y a n m i cp r o p e r t i e s( e n t h a l p y ,
etc.) i s straightfornard.
if
The ACE p a c k a g e o f p r o g r a m s h a s t h e c a p a b i l i t y
t o d e t e r m i n et h e
s t a t e o ft h eg a s e sa d j a c e n tt oa na b l a t i n gs u r f a c e i n a f a s h i o ns i m i l a rt ot h a td i s c u s s e dh e r e .I nt h ec o u p l e d
MACABRE
c o d e ,t h e
ACE group i s t y p i c a l l yc a l l e dw i t hp r e s s u r e
P
known and B '
C
a s s i g n e da sa ni n d e p e n d e n tv a r i a b l e
by t h eu s e r .
The t e m p e r a t u r e T and
chemicalandthermodynamic
W
o t h e r s t a t e d a t a are
t h e nd e t e r m i n e d .A l t e r n a t i v e l y ,
( i nt h el o w e rt e m p e r a t u r ep a r t so ft h et a b l e s ,
4.2,2.2
above)and
B'
Tw may b ea s s i g n e d
as d i s c u s s e d i n S e c t i o n
i s determined.
C
5.3.3
O p e n S y s t e m s i nE q u i l i b r i u m
-
U n e q u a lD i f f u s i o nC o e f f i c i e n t s
w a s l i m i t e d t o opensystems
material
i n t h e c o n d e n s e dp h a s e .T h e s es i m p l i f i c a t i o n s
were made i n o r d e r t o r e n d e r
t h eb a s i ct h e o r y
e a s i e r t o e x p l a i n .W h i l et h i ss i m p l em o d e l
i s reasonably
The d i s c u s s i o n o f t h e p r e v i o u s s e c t i o n
withequalspeciesdiffusioncoefficientsandnoremovalofsurface
accuratefor
many a b l a t i o n s i t u a t i o n s , t h e s e a s s u m p t i o n s a r e i n a p p r o p r i a t e
f o ro t h e r s .F o rt h i sr e a s o n ,c a l c u l a t i o n sp e r f o r m e d
by t h e ACE package
( a n d h e n c e t h e MACABRE program) are n o t r e s t r i c t e d t o any of t h e s e s i m p l i f i cations.
The e x t e n , s i o n t o u n e q u a ld i f f u s i o nc o e f f i c i e n t s
w i l l b ed i s c u s s e d
first, in this section.
45
The u n e q u a l d i f f u s i o n c o e f f i c i e n t
model h a s a l r e a d y b e e n b r i e f l y i n t r o -
4.3 above , i n whichEquations
(33) and (34) may becompared.
mass f r a c t i o n d u r i n g f o r c e s
Ki
*
e
w i t hm o d i f i e dd r i v i n gf o r c e s
Zi
A s i m i l a rm o d i f i c a t i o nm u s tb e
made i n
t h e ACE t r e a t m e n t .
The d i f f u s i G e flu ofelement
k a t the w a l l i s now
modeled by
duced i n S e c t i o n
I n s i m p l i s t i c terms, t h e model r e p l a c e s t h e
.
*
'kw
e e M :z;
= p u c
-2;)
e
In (58),
i s , i n - e f f e c t , a w e i g h t e da v e r a g eo ft h em o l e
Z;
of element k .
The Z;
a r ed e f i n e d
andmass
fractions
by
~
i=l
(59)
ZTK1-Y
z;
=
I
(where y
Ref. 10 1
P
2/3, see
Ki/Fi
i=l
where t h e Fi a r e d i f f u s i o n f a c t o r s d e f i n e d
by t h e f o l l o w i n g r e l a t i o n f o r
thebinarydiffusioncoefficients
where
5
i s a c o n s t a n tf o r
weakly on t e m p e r a t u r e .
a g i v e nt e m p e r a t u r ea n dp r e s s u r ea n dt h e
The -0
tj
mustobey
( 6 2 ) i no r d e rf o rt h e
l a y e rs p e c i e sd i f f u s i o ne q u a t l o n st or e d u c et o
b ei n f e r r e d
by s i m i l a r i t ya r g u m e n t s .R e f e r e n c e
binarydiffusioncoefficientsfor
c o r r e l a t e d by ( 6 2 ) .
F
i
is
( 5 8 ) can
11 d e m o n s t r a t e st h a tt h e
a v a r i e t yo fc h e m i c a ls y s t e m sa r ea c c u r a t e l y
T h i sr e f e r e n c ea l s o
r e l a t i o ne q u a t i o nf o rt h e
46
a formfromwhich
F depend
i
boundary
shows t h a t a r e a s o n a b l y good cor-
when 5 i s t a k e n as t h e s e l f - d i f f u s i o n c o e f f i c i e n t o f
02. A d d i t i o n a l d i s cussionrelativetothisunequaldiffusioncoefficientformulation
i s cont a i n e d i n Appendix A.
C o n s i d e r a t i o no fu n e q u a ld i f f u s i o nc o e f f i c i e n t sa f f e c t st h es u r f a c e
elementalbalancerelationshipswhichareneededto.determinetheequilibrium
thermochemical s t a t e a tt h es u r f a c e .S u b s t i t u t i n g( 5 8 )i n t o( 4 8 )y i e l d s
and t h e "unknowns" h e r e a r e Kk
I
and Z;
,
each
of
which
may beexpressed
partiaY pressur&
i nt e r m so ft h es p e c i e s
4
and
I
where 7n
9
defi*ned as
is t h e mean m o l e c u l a rw e i g h to ft h eg a sp h a s e
and
F is a mean Fy
i
i=l
47
Substitutingtheseexpressionsinto
(64) and utilizing the definition of
B'C'
yields anexpressionforthespeciespartialpressuresatthesurfacein
terms of q u a n t i t i e s a t t h e
Note t h a t( 6 8 )r e d u c e st o
boundary l a y e r e d g e a n d i n t h e m a t e r i a l
( 5 2 ) when t h ed i f f u s i o nc o e f f i c i e n t sa r ee q u a l .
When p e r f o r m i n g u n e q u a l d i f f u s i o n c o e f f i c i e n t o p e n s y s t e m c a l c u l a t i o n s ,
t h e ACE program u t i l i z e s ( 6 8 ) r a t h e r t h a n
,
e q u a t i o n s .O t h e rt h a nt h i s
d i s c u s s e di nS e c t i o n
5.3.2.
(52)
a st h ee l e m e n t a l
t h es o l u t i o np h i l o s o p h y
mass b a l a n c e
is e s s e n t i a l l ya s
The d i f f u s i o n f a c t o r s u t i l i z e d i n t h e
ACE pro-
gram may b e c a l c u l a t e d i n t h r e e w a y s , a t t h e u s e r ' s o p t i o n
a .d i f f u s i o nf a c t o r s
e a c hs p e c i e s
b.
Fi may b ei n p u ti n d i v i d u a l l yf o r
i
d i f f u s i o nf a c t o r s
may b ec a l c u l a t e da c c o r d i n gt o( 6 3 )
withtheuserspecifyingthereferencemolecular
weight,
c.
r e f , and t h ee x p o n e n t
E
i ft h eu s e rd o e sn o t h i n gs p e c i a l ,t h ep r o g r a m
ref = 23.4
and
c a l c u l a t e Fi a c c o r d i n gt o( 6 3 )w i t h
E
will
= 0.431.
The a c t u a l p r o g r a m i n p u t f o r t h e s e a l t e r n a t i v e s
i s discussedinReference
2.
It should also be pointed out that the diffusion factors have an effect on
theothertransportpropertiescalculated
briefly discussed in
by t h e ACE program,andthese
Appendix A.
F o ru n e q u a ld i f f u s i o nc o e f f i c i e n t s ,t h et r a n s f e rc o e f f i c i e n tf o r m u l a -
is E q u a t i o n( 3 4 ) .C o n s i s t e n tw i t h( 3 4 ) ,
t i o nf o rt h es u r f a c ee n e r g ye q u a t i o n
f o ru n e q u a ld i f f u s i o nc o e f f i c i e n tp r o b l e m s ,t h es u r f a c et h e r m o c h e m i s t r y
q u a n t i t i e s p r e p a r e d by t h e ACE p r o g r a m i n c l u d e t h e q u a n t i t y
2
:
2
i=l
hiTW
W
i na d d i t i o nt ot h eq u a n t i t i e sp r e v i o u s l yd i s c u s s e d .
48
N o t ea g a i n ,t h a tf o r
are
equaldiffusioncoefficients
Similarly, the quantities
and
frozen
hw
edge gas
are computed a f t e r r e f e r e n c e t o t h e p r e v i o u s a u t o m a t i c a l l y
computedand
storededgesolution.
5.3.4
all the
Open Systems In Equilibrium
M a t e r i a l Removal
-.
With Condensed Phase Surface
Considerationstothispointhavebeenbased
on t h e a s s u m p t i o n t h a t
mass l e a v i n g a n a b l a t i n g s u r f a c e
i s i nt h eg a sp h a s e
and t h i s flux
hasbeendenoted
by (pv),.However,
f o r many a b l a t i o n s i t u a t i o n s o f i n t e r e s t ,
some o f t h e m a t e r i a l l e a v i n g t h e s u r f a c e
i s i n a condensedphase
-
e-g.,
l i q u i d l a y e r removal o r" m e c h a n i c a la b l a t i o n " .
The condensedphaseremoval
a f f e c t sb o t ht h es u r f a c e
m a s s and e n e r g yb a l a n c e s( R e f e r e n c e
1 0 ) . For t h e
s u r f a c ee l e m e n t a lb a l a n c e ,t h e r e
i s an a d d i t i o n a l f l u x
term
L
'L'k,
L=1
l e a v i n gt h es u r f a c e .T h i sr e s u l t si nt h es u r f a c ee l e m e n t a lb a l a n c e( i n
terms o ft h es p e c i e sp a r t i a lp r e s s u r e s
-
analogous t o( 6 8 )h a v i n gt h ef o r m
49
element k .
foreachatomic
Summing o v e r a l l e l e m e n t sy i e l d st h eo b v i o u sn e t
mass c o n t i n u i t y r e l a t i o n
S i m i l a r l y ,t h e r e
is an a d d i t i o n a le n e r g yf l u x
L
term
II=1
l e a v i n gt h es u r f a c e .
The s u r f a c ee n e r g yb a l a n c ee q u a t i o n
becomes ( a n a l o g o u s
t o (34)
r
peuecH ( H r- hw) f r o z e n
(2;
-
to this
EA h c
W
The condensedphasesurfacematerialremoval
p o r a t e di nt h e
+
) hTW
i
2;
edgegas
model c u r r e n t l y i n c o r -
ACE program i s termed a “ f a i lt e m p e r a t u r em o d e l “ .A c c o r d i n g
model, a “ f a i l t e m p e r a t u r e ” , T f a i l - & ,
L.
candidatecondensedphasespecies
f r o mt h es u r f a c e( i n
may b e p r e - a s s i g n e d t o
T h i ss p e c i e s
a condensedphase)
a t a rate
w i l l t h e nb e
ma if
any
removed
t h es u r f a c et e m p e r a -
t u r e is g r e a t e r t h a n , o r e q u a l t o , t h e f a i l t e m p e r a t u r e f o r t h i s s p e c i e s .
A d d i t i o n a l comments r e l a t i v e t o t h e c o m p u t a t i o n a l a s p e c t s
briefdiscussion
of t h e f a i l t e m p e r a t u r e c o n c e p t .
The f a i l t e m p e r a t u r e
i.e.,
w i l l follow a
is the temperature at
which t h e m a t e r i a l
“ f a i l s “,
w i l l not remain affixed to
which a l i q u i d l a y e r i s u n s t a b l e , t h e
thetemperatureabovewhichthematerial
t h es u r f a c e .F o rm e l t i n gs o l i d sf o r
f a i lt e m p e r a t u r e
is s i m p l yt h e
m e l t t e m p e r a t u r e .F o rt h e n o c h e m i c a l l y
erodingsolids,thefailtemperaturemightbe
s e t a t some t e m p e r a t u r e
c h a r a c t e r i s t i c of t h e m a t e r i a l y i e l d t e m p e r a t u r e a t t h e a p p r o p r i a t e e x t e r n a l
l o a d i n g .A l s o ,t h e
ACE programhas
f a i lt e m p e r a t u r ef o ra l l
temperaturemightcorrespond
When f a i l i n g o c c u r s ,
a maximum
maximum f a i l
to the fail temperature of the substrate.
an a d d i t i o n a l e q u i l i b r i u m r e l a t i o n
able f o re a c hc o n d e n s e dp h a s es p e c i e s
50
a p r o v i s i o nf o ri n p u t t i n g
c o n d e n s e ds p e c i e s .P h y s i c a l l y ,t h i s
II f o r which mII > 0 .
is avail-
A l s ot h es u r f a c e
temperature must be consistent with the assignment fail temperatures:
~w
~w
'
'
*fail
'fail
for the surface species
f o r any s p e c i e s f o r w h i c h
ma.
> 0
Theseadditionalequilibriumrelations,surfacetemperaturerestrictions,
andassignedfailtemperaturesaresufficienttosolvefortheadditional
mass b a l a n c e r e l a t i o n s .
unknowns rhk i n t r o d u c e di n t oe l e m e n t a l
Ha.
,
thermochemicaldata.
s o l i d , and i f Tfail
<
I f Tfail
Tmelt,
The e n t h a l p y ,
removed is d e t e r m i n e df r o mt h es p e c i e s
o ft h ec o n d e n s e dp h a s eb e i n g
Ha. is t a k e na st h ee n t h a l p yo ft h e
Tmelt,
Ha. i s t a k e na st h ee n t h a l p yo ft h el i q u i d .
The
conventionforassociatingfailtemperatureswitheachorallcandidate
condensedphasespecies
w i l l bespecifiedinReference
Incommunicatingback
to
MACABRE,
terms ( p v )wHw and
2
*
.
t h e ACE packageamalgamates
the
L
by s u i t a b l ym o d i f y i n gt h er e t u r n e dv a l u eo f
quantity Y asdescribedinSection
Hw.
Thus FILL c o n s t r u c t s t h e
4.3 without specific reference to failing
events.
5.3.5
Non-Equilibrium E f f e c t s i n Open S y s t e mC a l c u l a t i o n
To calculatetheequilibriumstate
of a c h e m i c a ls y s t e m ,i n f o r m a t i o n
r e l a t i v et oa l lc h e m i c a lr e a c t i o n s
i s n o tn e e d e d .T h i sf a c tp e r m i t s
a
significantsimplificationinthe
ofthe
p r o b l e mf o r m u l a t i o n ,a n dt h o s ef e a t u r e s
ACE p a c k a g e d i s c u s s e d t h u s f a r t a k e a d v a n t a g e o f t h e s e s i m p l i f i c a -
t i o n s .F o r
some s y s t e m so fi n t e r e s t ,h o w e v e r ,t h e
etics may n o tb en e g l e c t e d .
A g e n e r a ls o l u t i o no f
reactionkineticseffectsareimportant
l e a s t ' t w or e a s o n s :a )t h e r e
e f f e c t s o fr e a c t i o nk i n complexproblems
f o r which
is p o t e n t i a l l y d i f f i c u l t f o r a t
are s i g n i f i c a n tc o m p u t a t i o n a l
andbookkeeping
problemsassociatedwiththeanalyticaltreatmentofmixedequilibriumand
n o n e q u i l i b r i u ms y s t e m s ,
and b ) f o r
theratecontrollingreactionsarenot
f o rt h e s er e a c t i o n sa r eu n a v a i l a b l e .
many s y s t e m s o f e n g i n e e r i n g i n t e r e s t ,
w e l l known and/or r a t e c o n s t a n t s
The ACE p a c k a g eo fr o u t i n e se f f e c t i v e l y
* F o rc a r b o na b l a t i o ni n
a i r , t h ef a i lt e m p e r a t u r ec o n c e p t
i s o fa l m o s tn o
interest.Graphitesloadedwithmetals,
however , t e n d t o f o r m t h i n
oxidation resistant oxide coatings
which g e n e r a l l y a b l a t e by m e l t i n g .
The ACE package w i l l p r o p e r l y i d e n t i f y any suchoxidesformed,andthe
f a i l t e m p e r a t u r e mechanism a l l o w s them t o b e a b l a t e d r e a l i s t i c a l l y . F a i l .
t e m p e r a t u r e s are u s e f u l f o r m e t a l h e a t s h i e l d s , o f c o u r s e .
51
surmounts the f i r s t of t h e s e p r o b l e m s as d i s c u s s e d i n R e f e r e n c e
4 and as
w i l l b e b r i e f l y summarizedsubsequently.
However, d i f f i c u l t i e s f a l l i n g i n t o
t h es e c o n dc a t e g o r y
above f r e q u e n t l yp r e e m p t
any s u c h " e x a c t " a n a l y t i c a l
t r e a t m e n t .I nt h e s ec a s e s ,v a r i o u sa p p r o x i m a t et r e a t m e n t sc a no f t e ny i e l d
u s e f u li n f o r m a t i o n .T h e s ea p p r o x i m a t et r e a t m e n t se s s e n t i a l l yc o n s i s to f
notallowingcertainspecies,orgroupsofspecies,toreactwith
w i s ee q u i l i b r a t e ds y s t e m .
A few examplesof
thissort
an o t h e r ofapproximate
t r e a t m e n t w i l l b ed i s c u s s e di nt h ef o l l o w i n gp a r a g r a p h s .T h e s ed i s c u s s i o n s
w i l l be followed
bya
b r i e f summary o f t h e g e n e r a l k i n e t i c s o p t i o n o f t h e
ACE program.
5.3.5.1
Removal o fS e l e c t e dS p e c i e sf r o mt h eS y s t e m
the c r u d e s t , way t o a p p r o x i m a t ec e r t a i n
i s t o simply removefrom t h e e q u i l i b r i u m s y s t e m t h o s e
The s i m p l e s t ,a n dp e r h a p s
kinetic effects
s p e c i e s whose f o r m a t i o n i s , i n r e a l i t y , s u p p r e s s e d
by r e a c t i o n k i n e t i c
are removed m e r e l y byremoving
e f f e c t s .C o m p u t a t i o n a l l y ,t h e s es p e c i e s
t h et h e r m o c h e m i c a ld a t ac a r d sf o rt h a tm o l e c u l ef r o mt h es p e c i e st h e r m o chemicaldatadeck.(Reference
2 d e s c r i b e st h i sd e c ki nd e t a i l . )
5.3.5.2
Isolated
Species
C e r t a i nc h e m i c a ls p e c i e s
c o n c e n t r a t i o n .T h i st r e a t m e n t
may b e f r o z e n i n d i v i d u a l l y a t
any given
i s s o m e t i m e su s e f u lf o rs p e c i e sw h i c ha r e
r e l a t i v e l yn o n r e a c t i v eb e c a u s eo fc h e m i c a lk i n e t i ce f f e c t s .F o re x a m p l e ,
c o n s i d e rt h ef l o wo f
a C-0-H gasover
c a r b o ns u r f a c e .I s o l a t i o no f
would provide amore
e q u i l i b r i u mw e r e
a r e l a t i v e l y low t e m p e r a t u r e a b l a t i n g
H 2 0 a t i t s b o u n d a r yl a y e re d g ec o n c e n t r a t i o n
realisticsurfacethermochemistrytablethaniffull
assumed ( s i n c e t h e w a t e r - g a s r e a c t i o n
r e l a t i v e l ys l o wa tl o w e rt e m p e r a t u r e s ) .I nt h e
o fs p e c i e s
a t thesurface
is
ACE p r o g r a m ,t h i si s o l a t i o n
i s achieved by t r e a t i n g t h a t p a r t i c u l a r s p e c i e s a s
anatomic
a f i c t i t i o u s a t o m i c number s o t h a t i t cannot
e l e m e n t , anelementwith
r e a c tc h e m i c a l l yw i t h
any o t h e r members o ft h ec h e m i c a ls y s t e m s .
Com-
p u t a t i o n a l l y ,t h i sr e q u i r e ss p e c i f i c a t i o no ft h ei n p u to ft h ea p p r o p r i a t e
q u a n t i t y of t h i s" e l e m e n t "( a c c o m p a n i e d
amounts o f o t h e r a c t u a l e l e m e n t s )
52
by a p p r o p r i a t e r e d u c t i o n s
i n the
and t h e a l t e r a t i o n o f t h e s p e c i e s
set for that species
t h ef i c t i t i o u s" e l e m e n t . "
t h e r m o c h e m i s t r yd a t ac a r d
one"atom"of
,
so t h a . t i t c o n t a l n s o n l y
5.3.5.3IsolatedSubgroupsofElements
a subgroup of e l e m e n t s
w i t h eachother,butthesubgroup
is notinchemical
e q u i l i b r i u mw i t ht h es y s t e ma s
a whole.The
i s o l a t i o n o f a subgroupof
e l e m e n t sc a nb ea c h i e v e d
by c o n s i d e r i n g t h e s e e l e m e n t s t o b e d i f f e r e n t
"eleI n some p r a c t i c a l a p p l i c a t i o n s , t h e r e e x i s t s
which a r e i n e q u i l i b r i u m
m e n t s "w i t hp r o p e r t i e s
the same a s , b u t w i t h a t o m i c
numbers d i f f e r e n t f r o m ,
the c o r r e s p o n d i n ga c t u a le l e m e n t s( e . g . ,
add 1 0 0 t o t h e a t o m i c numbers o f
t h ee l e m e n t si n
the i s o l a t e ds u b g r o u p ) .F o rt h i st y p eo fc a l c u l a t i o n ,t h e
subgroup t o b e i s o l a t e d
i s consideredtobeonecomponent,andthe
com-
p o s i t i o no f
this component is s p e c i f i e d a s t h e a c t u a l r e l a t i v e e l e m e n t a l
compositionofthesubgroup
to be isolated, but in
terms o f t h e a l t e r e d
e l e m e n t s .I no r d e rt h a tt h e s ea l t e r e de l e m e n t s
o t h e r ,t h es p e c i e st h e r m o c h e m i c a ld a t ac a r d
which may b ef o r m e df r o mt h e s ea l t e r e de l e m e n t s ,
atomic number c o n v e n t i o n .
5.3.5.4
s e t m u s tc o n t a i n a s e t o f s p e c i e s
i . e . , specieswiththe
same
Low T e m p e r a t u r eS u r f a c eE q u i l i b r i u mS u p p r e s s i o n
A s p r e v i o u s l yd i s c u s s e d ,e q u i l i b r i u m
c i o na t
may e q u i l i b r a t ew i t he a c h
low s u r f a c et e m p e r a t u r e s( e . g . ,b e l o w
problems.
t u r er a n g e s
i s a ni n c r e a s i n g l yp o o r
2000'R)
assump-
i n some a b l a t i o n
However, i n most s u r f a c et h e r m o c h e m i s t r yt a b l e s ,l o w e rt e m p e r a -
are u s u a l l y o n l y o f m i n o r i n t e r e s t ( e . g . , d u r i n g
a v e r ys h o r t
i n i t i a lp e r i o do f
a t r a n s i e n ta b l a t i o np r o b l e m ) .N o n t h e l e s s ,t h es u r f a c e
tables must f r e q u e n t l y e x t e n d t o t h e s e l o w e r t e m p e r a t u r e s i n o r d e r t o b e
c o m p a t i b l ew i t ht h eh e a tc o n d u c t i o n
and a b l a t i o ns o l u t i o n .F o rt h e s e
c a s e s , t h e ACE p a c k a g ec o n t a i n sa no p t i o n
by w h i c hs u r f a c e( h e t e r o g e n e o u s )
e q u i l i b r i u m w i l l besuppressedandsurfacerecessionnulled
(fit = 0 ) below
some i n p u t l i m i t t e m p e r a t u r e f o r s o l u t i o n s i n t h e a s s i g n e d t e m p e r a t u r e
p a r to ft h es u r f a c et a b l e s .
The g a s e sa d j a c e n tt ot h es u r f a c ea r e
assumed
to beinequilibrium,butheterogeneousequilibrium
is not required for
t h i ss o l u t i o n .F o r
many c a s e s ,s u r f a c et h e r m o c h e m i s t r yt a b l e sg e n e r a t e d
inthisfashionareclosertorealitythanfullequilibriumsolutions
down
t o low t e m p e r a t u r e s .
The i n p u tc o n v e n t i o nf o ri m p l e m e n t i n gt h i so p t i o n
i s d i s c u s s e di nR e f e r e n c e 2 .
5 . 3 . 5 . 5F u l lT r e a t m e n to fK i n e t i c s
The p r e v i o u s d i s c u s s i o n h a s d e s c r i b e d v a r i o u s a p p r o x i m a t e t r e a t m e n t s
when a g e n e r a l s o l u t i o n , i n c l u d i n g
an " e x a c t " t r e a t m e n t o f r e a c t i o n
r a t e e f f e c t s , is e i t h e r i m p o s s i b l e o r
i m p r a c t i c a l . The ACE p r o g r a md o e s ,h o w e v e r ,p o s s e s st h ec a p a b i l i t yo f
a c c u r a t e l y t r e a t i n g a number o f k i n e t i c a l l y c o n t r o l l e d r e a c t i o n s i n o t h e r w i s e
e q u i l i b r a t e ds y s t e m s( e i t h e rc l o s e do ro p e n ) .
The r e m a i n d e ro ft h i ss e c t i o n
ofkineticseffectswhichareoftenuseful
53
w i l l be devoted
t o a brief description
of t h i s "exact" t r e a t m e n t of k i n e t i c s .
is
The i n c l u s i o n o f k i n e t i c a l l y c o n t r o l l e d c h e m i c a l r e a c t i o n s
( 3 9 ) from t h e s e t of equa-
a c c o m p l i s h e db yr e m o v i n ge q u i l i b r i u mr e l a t i o n s
tionsforcertainspeciesparticipatinginreactionsthat
are t o b e
are r e p l a c e d by k i n e t i c r a t e
t h e A r r h e n i u sf o r m )f o re a c hk i n e t i c a l l yc o n t r o l l e dr e a c t i o n .
k i n e t i c a l l yc o n t r o l l e d .T h e s ee q u a t i o n s
e q u a t i o n s( o f
This i s accomplishedby
thereactions
first i d e n t i f y i n g t h e p r i m a r y r e a c t i v e s p e c i e s i n
andsecond,byallow-
which are t o b e k i n e t i c a l l y c o n t r o l l e d ,
rate
ing these species to be created or destroyed only via the kinetic
e q u a t i o n s .T h i sa p p r o a c hr e q u i r e s
a r e l a b e l i n go fs p e c i e st o
be c o n s i d e r e d
i nt h ek i n e t i c a l l yc o n t r o l l e dr e a c t i o n s .T h e s es p e c i e sa r ec a l l e dp s e u d o elementssincetheybehavelikeelementsexceptthatthey
ordestroyed
at rahesspecified
by t h e r e a c t i o n
may be c r e a t e d
rate equations.
C o m p u t a t i o n a l l y ,t h ei n c l u s i o no fk i n e t i c sr e s u l t si nt h ea d d i t i o n
term t o t h e e l e m e n t a l b a l a n c e e q u a t i o n s
unknowns t c t h es y s t e m
e q u a l i n number t o t h e number o f s p e c i e s whose c o n c e n t r a t i o n s a r e k i n e t i i . e . , thepseudo-elements.
The r e l a t i v ec r e a t i o n and
c a l l yc o n t r o l l e d ,
d e s t r u c t i o n r a t e s of a l l pseudo-elements i n a g i v e n r e a c t i o n are r e l a t e d
by stoichiometry,however,
s o t h e number o f a d d i t i o n a l unknowns remaining
is e q u a lt ot h e
number o f k i n e t i c a l l y c o n t r o l l e d r e a c t i o n s .
The r e a c t i o n
r a t e s , fromwhich t h e p s e u d o - e l e m e n t c r e a t i o n o r d e s t r u c t i o n
rates d e r i v e ,
are g i v e n by
of a r a t e - o f - c r e a t i o n o r d e s t r u c t i o n
f o rt h e s ep s e u d o - e l e m e n t s .T h i sa d d sa d d i t i o n a l
f o re a c hk i n e t i c a l l yc o n t r o l l e dr e a c t i o n
m
o ft h ef o r m
where t h e sums a n d p r o d u c t s a r e o v e r t h e s p e c i e s a n d p s e u d o - e l e m e n t s
p are t h e s t o i c h i o m e t r i c c o e f f i c i e n t s ,
c o n s t a n tf o rr e a c t i o n
54
,K
i s t h ee q u i l i b r i u m
Pm
( 7 3 ) I and kFm i s t h ef o r w a r d r a t e c o n s t a n t . I n t h e
r e a c t a n t s a n dp r o d u c t sr e s p e c t i v e l y I. n
N
and s u p e r s c r i p t s R and P d e n o t e
(72)
1'
present formulation,
kFm i s r e p r e s e n t e d b y
an A r r h e n i u s t y p e f u n c t i o n
Cpm i s t h et e m p e r a t u r ee x p o n e n t ,a n d
B,
i s a c o n s t a n tr e p r e s e n t i n g a m u l t i t u d e of phenomena.
Em, C
p
, and Bm
must be input to the program for each kinetically controlled reaction.
T h e s ec o n s t a n t ss h o u l db eb a s e do ne x p e r i m e n t a ld a t a .
The u n c e r t a i n t y i n ,
orunavailabilityof,thesedatafor
many c h e m i c a l s y s t e m s o f i n t e r e s t i n
ablationproblemsfrequentlyrepresent
a significant constraint on the
a p p l i c a t i o n o f the k i n e t i c s o p t i o n of ACE program t o t h e s e p r o b l e m s .
More
detaileddiscussionofthecomputationaltreatmentofkinetics
is presented
i nR e f e r e n c e s 4 and 11. R e f e r e n c e 2 d e s c r i b e st h ei n p u td e t a i l sf o rt h e
kineticallycontrolledreactiondata.
where Em is t h e a c t i v a t i o n e n e r g y ,
5.4
SOME CODING DETAILS
are employed i n t h e ACE program t o s o l v e
The c o m p u t a t i o n a l p r o c e d u r e s
t h ee q u a t i o n s s e t f o r t hi nS e c t i o n5 . 3 ,b r i e f l yd i s c u s s e dh e r e .C o n s i d e r a b l y
g r e a t e r d e t a i l i s presentedinReference
4 and i n p a r t i c u l a r , T a b l e
I of
that reference.
5 . 4 . 1B a s i cS o l u t i o nT e c h n i q u e
The b a s i c s o l u t i o n t e c h n i q u e
may b e i l l u s t r a t e d
e x a m p l e ,a no p e ns y s t e mw i t hu n e q u a ld i f f u s i o nc o e f f i c i e n t s ,
p h a s em a t e r i a lr e m o v a l ,a n d
5.3.3)
.
by c o n s i d e r i n g , f o r
no condensed
no k i n e t i c s ( i . e . , a s d i s c u s s e d i n S e c t i o n
F o rt h i ss y s t e m ,t h eb a s i ce q u a t i o n sd e f i n i n gt h e. p r o b l e ma r et h e
e l e m e n t a lc o n s e r v a t i o ne q u a t i o n s
( 6 7 ) , t h et o t a lp r e s s u r ee q u a t i o n( 5 6 ) ,
thereactionequilibriumequations(55),
r e l a t i o n( 5 3 )
.
and o n e h e t e r o g e n e o u s v a p o r p r e s s u r e
shows t h a t t h e r e are as many
so c l o s u r e is obtained.
The t a b l eo fS e c t i o n5 . 3 . 3
knowns a s unknowns i n t h e s e e q u a t i o n s
Summarizing these equations
,
kPi T
P = 0
55
K
k= 1
K
The number o f unknowns could immediately be reduced
by I - K t h r o u g h
thedirectsubstitutionof(77),
as s o l v e d f o r P i , i n t o t h e o t h e r r e l a t i o n s .
I t provesadvantageous,however,
t o c o n t i n u e t o t r e a t t h e f u l l s e t of
equations,andtosubsequentlyutilizethissubstitutionduringtheitera-
t i v e convergenceprocedure.
a l g e b r a i ce q u a t i o n s
The s o l u t i o no ft h e s es i m u l t a n e o u sn o n l i n e a r
i s basedon
Newton-Raphson i t e r a t i o n . S i n c e t h i s p r o a more l i n e a r form
e t c . ) it i s w e l l t o examine t h e s e t
c e d u r e i s a c c e l e r a t e d by c a s t i n g t h e e q u a t i o n s i n t o
( v i at r a n s f o r m a t i o n s ,s u b s t i t u t i o n s ,
of e q u a t i o n sa b o v e .
With t h eb o u n d a r yl a y e re d g e ,c h a ra n dp y r o l y s i sg a s
compositiongivenas
w e l l a st h e
B', ( 7 5 )a n d( 7 6 )a r el i n e a rr e l a t i o n s
F i s r e a s o n a b l yc o n s t a n t I. n
con9
(77)and(78)
are l i n e a r r e l a t i o n s b e t w e e n t h e i n
P i , En Pk and
thelattervariablebeingapproximatelylinearin
1/T.
b e t w e e nt h e
trast,
Rn Kpi,
Pi
a n dp r o v i d i n gt h a t
The ACE p r o g r a m t a k e s a d v a n t a g e o f t h i s s i t u a t i o n
s p e c i e s which a r e s i g n i f i c a n t i n t h e
( 7 6 ) i n terms o f Pi
by t r e a t i n g t h o s e
mass a n d p r e s s u r eb a l a n c e s( 7 5 )a n d
less s i g n i f i c a n t s p e c i e s i n
andthe
terms of t h e i r
i n Pi.
The Newton-Raphson p r o c e d u r e , a s a p p l i e d
by t h e ACE program,canbe
summarized by c o n s i d e r i n g a s e t o f e q u a t i o n s o f t h e g e n e r a l f o r m
f
. (x1,x2,. . ., x i , . . . )
3
A t any p o i n t i n t h e s o l u t i o n p r o c e d u r e t h e r e e x i s t s
x?, for all the variables
r e l a t i o n s and w i l l l e a dt o
a s e t of e s t i m a t e s ,
which w i l l i n g e n e r a l n o t s a t i s f y
a non-zerovalue
Newton-Raphson methodproceeds
evaluatingthechangeineach
= 0
a l l ofthe
of the f j , namely,
t o" d r i v e "t h e s ee r r o r st o w a r dz e r o
E
j.
The
by
unknown v a r i a b l e , Axi, whichwouldreduce
a l lt h ee r r o r st oz e r oi ft h ef u n c t i o n s ,
f , were l i n e a r . The l i n e a r
j
approximation i s based on t h e c u r r e n t v a l u e s o f t h e
unknown v a r i a b l e s
and t h e c o r r e s p o n d i n g a r r a y o f v a l u e s o f t h e p a r t i a l d e r i v a t i v e s a f . / a x i
56
I
Thus
which is l o c a l l y c o r r e c t a n d
i n the l i n e a r approximation.
is i n t e g r a t e d t o
The s o l u t i o n of
(80) is
of p a r t i a l d e r i v a t i v e s a p p e a r i n g i n
(81) i s s i m p l yt h em a t r i x
( 8 0 ) . In t h e ACE program t h e f o r m u l a t i o n o f t h e
p a r t i a ld e r i v a t i v e s
u s e s the v a r i a b l e s , Iln Pi
Iln T and I n
and ( 8 1 )
y i e l d s ,f o re x a m p l e ,
w h e r et h ea r r a y
i n v e r s e of t h e a r r a y i n
m
which i f t a k e n a s l i n e a r
sincethe
error.
a l l t h e way t o s o l u t i o n y i e l d s
desired change i n t h e f u n c t i o n s
+n e q u a l ? ye x a c tr e l a t i o no b t a i n e d
is s i m p l y t h e n e g a t i v e o f t h e
from (82) i s
tThechoiceof
En P . p e r m i t s a m a t r i x r e d u c t i o n by t h e u s e o f t h e s i m p l e
algebraic substituhon, previously mentioned after
(78) , p r i o r t o m a t r i x
inversion.
57
which i f t a k e n
as l i n e a r a l l t h e way t o s o l u t i o n y i e l d s
The ACE programuses
(85) f o r a l l s p e c i e s s i g n i f i c a n t i n
mass balancesand
( 8 3 )f o rt h eo t h e r s .
5 . 4 .R
2 estriction
on C o r r e c t i o n s
The s e t of c o r r e c t i o n (Axi)
*
canbethoughtof
as a v e c t o r i n t h e
s p a c eo ft h ei n d e p e n d e n tv a r i a b l e sw h i c h
i s added t o t h e c u r r e n t v e c t o r
E x p e r i e n c eh a s shown t h a t
approximation x* t o y i e l d a new e s t i m a t e x:*
1
it i s f r e q u e n t l y u n w i s e t o p r o c e e d a l o n g t h i s c o r r e c t i o n v e c t o r t h e f u l l
.
amount i n d i c a t e d by ( 8 3 )o r
(85)
R a t h e r , it i s b e t t e r t o p r o c e e d
same d i r e c t i o n .
way, a l t h o u g hp r e s e r v i n gt h e
todepartfromthisvector,
.
A t o t h e r times, i t i s e x p e d i e n t
and seek a n o t h e r b a s e d
some v a r i a b l e a n d e l i m i n a t i n g
a limited
on f r e e z i n g t h e v a l u e o f
a correspondingequation.
The s c a l i n g o f t h e c o r r e c t i o n v e c t o r
i s s u c ha st o
l i m i t changes i n
the partial pressures of major species to increases of one order of
magni-
t u d ea n dd e c r e a s e so ft h r e eo r d e r so fm a g n i t u d e ,a n dc h a n g e so ft e m p e r a t u r e
t oa p p r o x i m a t e l y
20 p e r c e n t .
M o l e c u l a rw e i g h t ,t e m p e r a t u r e
correctionsarefrozen
and condensedphaseconcentration
and a new c o r r e c t i o n v e c t o r g e n e r a t e d i f t h e i n i t i a l
set ofcorrectionsindicateexcessivetemperatureormolecularweight
e x c u r s i o n s , a c o n t r a d i c t o r yt e m p e r a t u r ec h a n g e ,o rn e g a t i v ec o r r e c t i o n so n
newly i n t r o d u c e d c o n d e n s e d s p e c i e s .
The f o r m u l a t i o n of t h e s e and o t h e r s c a l i n g a n d f r e e z i n g
i s an e s s e n t i a l f e a t u r e o f t h e
criteria
Because o f t h e s e f e a t u r e s ,
w e l l f o r m u l a t e d ,p h y s i c a l l yu n i q u e
ACE program.
convergence i s v i r t u a l l y a s s u r e d f o r
problems.
5.4.3
Base S p e c i e s
The d i s c u s s i o n o f S e c t i o n
5.3 d e s c r i b e d t h e e q u i l i b r i u m r e a c t i o n
e q u a t i o n sa se q u a t i o n sg i v i n gt h ef o r m a t i o n
Thus t h e r e a c t a n t s a r e e l e m e n t s
of a s p e c i e sf r o ma t o m i ce l e m e n t s .
and t h e p r o d u c t s a r e u s u a l l y m o l e c u l e s .
T h i s scheme h a s t h e a d v a n t a g e o f f o r m a l s i m p l i c i t y , s i n c e t h e s t o i c h i o m e t r i c
coefficientsneededintheequilibriumequationsaregivendirectly
58
by t h e
productspecieschemicalformula.This
scheme c a nh a v ec o m p u t a t i o n a ld i s -
a d v a n t a g e s ,h o w e v e r ,s i n c et h ea t o m i ce l e m e n t s
are f r e q u e n t l y n o t p r e s e n t
t o any g r e a t e x t e n t i n t h e e q u i l i b r i u m s y s t e m . T h i s i n i t s e l f
results i n
no d i s a d v a n t a g e . I f
one mass b a l a n c e ( e . g .
d e f e a tc o n v e r g e n c e .
, however , a molecule (e. g. , CO) dominates more t h a n
, C and 0) , l o s s of s i g n i f i c a n t f i g u r e s c a n slow o r
I t i s more d e s i r a b l e t o w r i t e t h e e q u i l i b r i u m r d a c t i o n s
( 3 8 ) as w e l l as t h e m a s s b a l a n c e s i n
(37) and
terms o fr e a c t a n ts p e c i e sw h i c h
are
a c t u a l l yp r e s e n ti na p p r e c i a b l ea m o u n t s .T h e s es p e c i e sa r et e r m e d" b a s e
s p e c i e s "( f r o mR e f e r e n c e
1 2 ) s i n c ea l lo t h e rs p e c i e sa r et a k e nt ob e
formed
from them.
The ACE program s e l e c t s t h e b a s e s p e c i e s f r o m t h e c a n d i d a t e s p e c i e s
t h e r m o c h e m i c a ld a t aw h i c ha r ei n p u ta ss p e c i f i e di nR e f e r e n c e
2.
programautomaticallyselectsasbasespeciesthefirstset
s a t i s f y i n gt h er e q u i r e m e n tt h a t
thisbasespeciesset
may beformedfrom
and ( 2 ) t h a t no b a l a n c e dr e a c t i o nc a nb e
i n v o l v i n go n l yb a s es p e c i e s .
eachelement.
(1) a l l o t h e r s p e c i e s
One b a s es p e c i e s
The
of s p e c i e s
written
may b ec o n s i d e r e dt or e p r e s e n t
T h u s , t h eb a s es p e c i e sa r ee s t a b l i s h e d
t h e u s e r i n p u t st h ec a n d i d a t es p e c i e st h e r m o c h e m i c a ld a t a .
by t h e o r d e r i n
which
The c a l c u l a t i o n
o ft h es t o i c h i o m e t r i cc o e f f i c i e n t s
and e q u i l i b r i u m c o n s t a n t s a p p r o p r i a t e t o
any s e t o f b a s e s p e c i e s
is h a n d l e d a u t o m a t i c a l l y by theprogram.
While t h e s e l e c t i o n
computationaleconomies
whichoccur
of b a s e s p e c i e s
may b e r e a l i z e d
is n o tu s u a l l yc r i t i c a l ,c e r t a i n
i f t h e basespeciesarethosespecies
i n r e l a t i v e l yh i g hc o n c e n t r a t i o n si nt h es y s t e m .
Also t h e t r e a t -
ment of a c h e m i c a lk i n e t i c s( S e c t i o n
5 . 3 . 5 ) may havean
i n f l u e n c e on t h e
s e l e c t i o n of t h eb a s es p e c i e s .T h e r e f o r et h eU s e r s G
' u i d e( R e f e r e n c e
2)
recommends c e r t a i n i n p u t p r o c e d u r e s f o r f o r c i n g t h e s e l e c t i o n o f d e s i r a b l e
basespecies.
59
SEmION 6
COUPLING THE TRANSFER
COEFFICIENT
CALCULATION TO THE
SURFACE
THERMOCHEMISTRY
SOLUTION
(111 - I V COUPLING) AND
TO THE IN-DEPTH
TRANSIENT
SOLUTION
(111
V COUPLING)
-
5
S e c t i o n 5 above makes it p l a i n t h a t t h e t r a n s f e r c o e f f i c i e n t a p p r o a c h
t o modelingdiffusionalfluxes
is builtintothesurfacethermochemistry
s o l u t i o np r o c e s s ,s i n c et h ec h e m i s t r yr o u t i n e su s e
i n j e c t i o n and t h e B; c o n c e p t d e p e n d s n a t u r a l l y o n
idea.
An a d d i t i o n a l ,
B ' as t h em e a s u r eo f
mass
C
the t r a n s f e r c o e f f i c i e n t
more e x p l i c i t , l i n k t o t h e t r a n s f e r c o e f f i c i e n t
model i s
peueCM. The p r e - e x p o n e n t i a lf a c t o r s Bm
t h es c a l i n go fk i n e t i ce f f e c t so n
are i n p u t d i r e c t l y t o t h e c o u p l e d c o d e , a n d t h e s e
are normalizedon peueCM
ACE package, as n o t e di nS e c t i o n
5 a b o v e .T h i sr e q u i r e st h a t
a
i n s i d et h e
peueCM v a l u e b e c o m m u n i c a t e d d i r e c t l y t o t h e
f o ru s ei nn o r m a l i z i n ga n d
B 's.
ACE packagewithevery
call
O t h e rc o u p l i n gl i n k sa p p e a ri nt h es u r f a c e
m
t e m p e r a t u r ed e p e n d e n c ea n dt h e" b l o w i n gr e d u c t i o n "o ft h et r a n s f e rc o e f f i c i e n t :
t h e s e are d i s c u s s e d i n S e c t i o n
The I11
-
V couplingbetweenthetransfercoefficientand
d e p t hs o l u t i o no c c u r s r o f
t h es u r f a c ee n e r g ye v e n t s .
sentedinSection
60
9 below.
the i n course, i n t h e t r a n s f e r c o e f f i c i e n t e q u a t i o n f o r
The r e l e v a n te q u a t i o n sh a v ea l r e a d yb e e np r e -
4 . 3 andrequirenofurther
comment.
SECTION 7
SOLUTION
COUPLING THE FLOW-FIELD (OR BOUNDARY LAYER EDGE STATE)
STATE
SOLUTION
(I1 - I V COUPLING)
(PHASE 11) TO THE SURFACE
COEFFICIENT
SOLUTION
(I1 - I11 COUPLING)
AND TO THE TRANSFER
B e f o r ed i s c u s s i n gt h eP h a s e
I1 and I11 c a l c u l a t i o n s t h e m s e l v e s ,
w i l l be of interest to
examine t h e c o u p l i n g l i n k s b e t w e e n t h e P h a s e
I1
solution,whichreferstothespecificationoftheboundarylayerouter
edge s t a t e a r o u n dt h e body ( p r e s s u r e , s t a t i c e n t h a l p y , r e c o v e r y e n t h a l p y ,
and t r a n s p o r t p r o p e r t i e s ) , and t h e P h a s e I11 andPhase
I V solutions.
it
The b o u n d a r yl a y e re d g es t a t ei n f l u e n c e st h es u r f a c et h e r m o c h e m i s t r y
s t a t es o l u t i o n ,a sh a sa l r e a d yb e e nd i s c u s s e di nS e c t i o n
thedirect
communication of l o c a l p r e s s u r e
package.
The edge s t a t e i n t u r n
as w i l l bediscussedinSection
5
above,through
and r e c o v e r y e n t h a l p y t o t h e
i s i n f l u e n c e db yt h ec u r r e n t
ACE
body s h a p e ,
8 below.
The b o u n d a r y l a y e r e d g e s t a t e i n f l u e n c e s t h e t r a n s f e r c o e f f i c i e n t
calculationthroughthedistributionsofpressure
body,asmightbeexpected.This
a n de n t h a l p ya r o u n dt h e
i s d i s c u s s e di nS e c t i o n
9 below.
it i s a l s o u s e f u l
A l l ofthesecouplinglinksareofinterest,but
a tt i m e st ob ea b l et os u p p r e s s
some o r a l l o f
them.Such
c o u p l i n gs u p p r e s sion is particularlydesirableinparametricstudiesattemptingtostudy
,
one e f f e c t a t a t i m e , r a t h e r t h a n c o u p l e d e f f e c t s
it is d e s i r e d t o
m a t c he x p e r i m e n t a ld a t ai n
s u r f a c et e m p e r a t u r e s ) .T h e r e f o r e ,t h e
a l l o wv a r y i n gd e g r e e so fd e c o u p l i n g .
foreachsurfacepoint
a.
and i n s t u d i e s i n
which
some r e s p e c t( s u c ha sm e a s u r e d
MACABRE c o d eh a sb e e nw r i t t e nt o
A l l o ft h ef o l l o w i n go p t i o n se x i s t
on a body a t anytime:
S u r f a c et e m p e r a t u r ea n dr e c e s s i o n
r a t e d i s c o v e r e d by
e n e r g yb a l a n c e ,o r
b .S u r f a c et e m p e r a t u r ea n dr e c e s s i o n
f u n c t i o n s of time by u s e r( O p t i o n
r a t e a s s i g n e d as
2)
Under ( a ) w e have
a . 1 .F u l ls u r f a c ee n e r g yb a l a n c ew i t hc o n v e c t i o na n dt h e r m o c h e m i s t r(yO p t i o n
a.2.
1)
No c o n v e c t i o n ,n or e c e s s i o n ,a s s i g n e dh e a tf l u x( O p t i o n
2)
61
Under (a.1.) w e have
L o c a lp r e s s u r ea n d / o rl o c a lr e c o v e r ye n t h a l p ya s s i g n e db yu s e r
a.l.1
as a f u n c t i o n of t i m e
a . 1 . 2L o c a lp r e s s u r ea n d / o rl o c a lr e c o v e r ye n t h a l p y
computedby
a p p r o p r i a t es u b r o u t i n e sa u t o m a t i c a l l y
F o rb o t h( a . l . l ) a n d
1
a.1. (?) .1
(a.1.2) w e have
Localheattransfercoefficientassigned
functionof
a.1.
1
(7).2
by u s e r as a
time
Localheattransfercoefficient
computedautomatically
b ya p p r o p r i a t es u b r o u t i n e s
Forboth
(a.1.
(3)-1) a n d( a . 1 .
($) . 2 )
correctiontothetransfercoefficient.
62
w e haveanydegreeof"blowingreduction"
SECTION 8
THE BOUNDARY LAYER EDGE STATE SOLUTION
state i s
of t h e body s h a p e ,
shockshape , t h es h o c kl a y e rf l o w ,a n dt h eb o u n d a r yl a y e rf l o w .F o r
many
e n g i n e e r i n gp u r p o s e s ,h o w e v e r ,
a detailedanalysisofthissort
i s n o tp o s s i b l e , and c e r t a i n s i m p l i f y i n g a s s u m p t i o n s
are n e c e s s a r y which b r i n g t h e
a n a l y s i s t o a more t r a c t a b l e l e v e l of s o p h i s t i c a t i o n .S i n c et h ea p p l i c a t i o n
fortheanalyticaltechniquebeingdiscussedhere
i s primarilyre-entry
v e h i c l e n o s e t i p s and h e a t s h i e l d s , t h e f i r s t s i m p l i f y i n g a s s u m p t i o n
is t h a t
theentireregionofinterest
i s immersed i n a boundarylayeredgeflowwhich
h a sp a s s e dt h r o u g ht h ed e t a c h e d
bow shock a t o r n e a r t h e s t a g n a t i o n p o i n t .
Thus, it i s n o t n e c e s s a r y t o c a l c u l a t e
a shockshapeandshocklayerflow
s i n c et h es h o c kl a y e rg a so fi n t e r e s t
i s assumed t o p a s s t h r o u g h
a normal
shock. The n e x ts i m p l i f y i n ga s s u m p t i o n
i s t h a tt h eb o u n d a r yl a y e re d g eg a s
s t a t e i s determined by a n i s e n t r o p i c e x p a n s i o n f r o m t h e
body s t a g n a t i o n
condition.
With t h ee d g eg a se n t r o p yf i x e da tt h es t a g n a t i o nv a l u e ,t h e
completeedgegasthermodynamic
s t a t e is determined by t h e s p e c i f i c a t i o n of
one more p r o p e r t y ,t y p i c a l l yt h el o c a ls t a t i cp r e s s u r e .T h u s ,t h ed e t e r m i n a t i o no ft h ee d g eg a s
s t a t e r e d u c e s t o a problemof
firstcalculatingthe
s t a g n a t i o ne n t r o p y ,w h i c h
i s r e l a t i v e l ys t r a i g h t f o r w a r d ,t h e nd e t e r m i n i n g
t h ep r e s s u r ed i s t r i b u t i o na r o u n dt h ea r b i t r a r y
body s h a p e , and f i n a l l y
evaluatingtheotheredge
s t a t e p r o p e r t i e s of i n t e r e s t by c a r r y i n g o u t t h e
i s e n t r o p i ce x p a n s i o nt ot h e
known l o c a lp r e s s u r e .
The t e c h n i q u e su s e dt o
carryoutthesesteps
are d e s c r i b e d i n t h e s u b s e c t i o n s b e l o w .
A c o m p l e t ea n da c c u r a t ea n a l y s i so ft h eb o u n d a r yl a y e re d g e
a v e r y complexproblem
8.1
requiringsimultaneousconsideration
THERMOCHEMICAL COMPUTATIONS
Thermochemicalcomputations
are r e q u i r e d i n
a number o f i n s t a n c e s i n
state calculations. As
mentioned i n S e c t i o n 5 , t h e s e c a l c u l a t i o n s a r e c a r r i e d o u t
by t h e ACE r o u t i n e .
i t hasbeenfoundmostconvenient
to store
Forboundarylayergasproperties,
t h ec o u p l e dc o d e ,i n c l u d i n gb o u n d a r yl a y e re d g e
ACE s o l u t i o n s f o r t h e r e q u i s i t e
t a b u l a r formandsupply
thermodynamicand
transportpropertiesin
thistableofpropertiestothe
programasinput.
The o p t i o n a l s o e x i s t s
of m e r e l y s p e c i f y i n g t h e d e s i r e d v a l u e s o f t h e i n d e pendent variables in this'Wollier Chad'
and c a l l i n g f o r ACE c a l c u l a t i o n s t o
d e t e r m i n et h er e q u i r e dp r o p e r t i e s .
The p r o p e r t i e s a t t h eg a s
s t a t e o fi n t e r e s t
d u r i n gt h ec o u r s eo f
a solutionarefound
by i n t e r p o l a t i o n b e t w e e n t a b u l a r
63
Chart approachhas a number o fa d v a n t a g e s
similar t ot h ep r e c a l c u l a t e ds u r f a c et a b l e s :r e u s e a b i l i t y ,e x t e r n a le v a l u a tion of the
t a b l e a c c u r a c ya n ds u f f i c i e n c y ,a n de l i m i n a t i o no fn o n - c o n v e r g e d
solutions.
However, c a r e m u s tb et a k e nt og u a r a n t e et h a ti n t e r p o l a t i o nw i t h i n
the chart w i l l y i e l d s u f f i c i e n t l y a c c u r a t e state properties.
values.
The p r e c a l c u l a t e d M o l l i e r
The M o l l i e r C h a r t u s e s p r e s s u r e a n d e n t h a l p y a s i n d e p e n d e n t v a r i a b l e s .
Dependentvariablesfound
a t e a c h( P - h )c o m b i n a t i o na r et e m p e r a t u r e ,d e n s i t y ,
P r a n d t l number, v i s c o s i t y ,e n t r o p y ,
and i s e n t r o p i ce x p o n e n t .L i n e a ri n t e r p o -
l a t i o n is usedeverywherewithinthechart,howeverpressuresanddensities
i s carried
limit t h e number o fp r e s s u r e - e n t h a l p y
areconvertedtologarithmicformbeforethislinearinterpolation
out.Sincethecomputercodedimensions
i t i s recommended t h a t t h e u s e r
combinationswhichcanbeinput,
check t h e
a c c u r a c y of i n t e r p o l a t e d s o l u t i o n s b e f o r e g o i n g a h e a d w i t h p r o b l e m s o l v i n g .
Thiscanbeaccomplished
by p l o t t i n g t h e p r o p e r t i e s o f i n t e r e s t , d e t e r m i n i n g
theregionofmostlikelyerrorwithinthelinearinterpolationapproximation,
andcomparingan
interpolatedpointwithan"exact"solution
from t h e ACE
program.
i s t y p i c a l l y s t a r t e d by u s i n g t h e
The edge s t a t e s o l u t i o n p r o c e d u r e
specifiedvalues
of s t a g n a t i o n p r e s s u r e
and s t a g n a t i o n e n t h a l p y t o d e t e r m i n e
t h ee n t r o p yl e v e lf r o mt h eM o l l i e rC h a r t .
bodymust
8.2
The s t a t i c p r e s s u r e a r o u n d t h e
thenbefound,asdiscussedinthenextsection.
STATIC PFiESSUFiE DISTRIBUTION
The t e c h n i q u e u s e d t o c a l c u l a t e t h e s t a t i c p r e s s u r e d i s t r i b u t i o n
i n c l u d e s a v a r i a t i o n of Newtoniantheory
inthesubsonicflownearthestag-
nationpoint,matchedto
a P r a n d t l Meyer e x p a n s i o n d o w n s t r e a m o f t h e s o n i c
point.
i nt h e s er e g i o n sa r ee x p l a i n e db e l o w .
The methodsused
8.2.1
Subsonic
Region
The p r e s s u r e d i s t r i b u t i o n i n t h e s u b s o n i c r e g i o n
is found by a v a r i a -
1 3 , i n which t h e Newtonian
t i o no ft h et e c h n i q u es u g g e s t e di nR e f e r e n c e
is modified t o i n c l u d e blunt bodies w i t h s o n i c
The b a s i c Newtonian p r e s s u r ee x p r e s s i o n i s
pressure distribution relation
corners.
P
=
-
Pm
+ (1 - Fa,
cos2 B
wherebarredquantitiesarenormalizedbythestagnationpressure,and
t h ea n g l eb e t w e e nt h e
atthepointofinterest.
clature.
64
body a x i s ofsymmetryand
The sketchbelow
B is
a normal t o t h e body s u r f a c e
w i l l helptoclarifythis
nomen-
external
flow
R* = r a d i u s t o s o n i c p o i n t
along a s u r f a c e n o r m a l
F i g u r e 1 7 . Diagramof
N o s e t i p Showing Nomenclature
are smoothlycontoured
i s subsoniceverywhere
ahead of aconvex
s h a r p c o r n e r and t h e f l o w e x p a n d s t o t h e s u p e r s o n i c
cond i t i o n a t t h ec o r n e r .F o rt h e s es h a p e s ,t h eN e w t o n i a np r e s s u r ed i s t r i b u t i o n
is g r o s s l yi n a c c u r a t es i n c e
it d e p e n d so n l yo nl o c a ls l o p e .
I t is obvious
t h a t h ep r e s s u r es h o u l dv a r ys m o o t h l yf r o mt h es t a g n a t i o nv a l u e
= 1.0
atthestagnationpointtothesonicpointpressure
P = P* a t t h e s h a r p
c o r n e r . However, f o r a sphere-coneshapeas
shown i nF i g u r e1 7a b o v e ,t h e
Newtonian d e s c r i p t i o no fp r e s s u r e
would c a l l f o r a c o n s t a n td e t e r m i n e d
by
t h es u r f a c es l o p ea l o n gt h ee n t i r ec o n es u r f a c e .
I t was found i nR e f e r e n c e 13
t h a t a modification of the Newtonian distribution involving an empirical
A class of nosetips of particular interest
bluntbodieshavinglocalslopessuchthattheflow
65
expressionforpressureon
same d i a m e t e r ( s o n i c p o i n t
a f l a t d i s c o ft h e
d i a m e t e r )w o u l dg r e a t l yi m p r o v ep r e s s u r ep r e d i c t i o n s .
The s u g g e s t e dp r e s s u r e
is
d i s t r i b u t i o ne x p r e s s i o n ,m o d i f i e df o rn o n z e r oa m b i e n tp r e s s u r e ,
1
where PFD i s t h e f l a t d i s c p r e s s u r e d i s t r i b u t i o n e x p r e s s i o n ,
r u n n i n ql e n g t h ,
n o s er a d i u so r
%
R*,
i s t h en o s er a d i u s ,
a sd e f i n e di nF i g u r e
and Rma,
17.
expressionsforthesonicpointpressure
s i s t h es u r f a c e
i s the maximum o f e i t h e r t h e
I t i s n e c e s s a r yt os p e c i f y
F* and t h e f l a t d i s c p r e s s u r e
FFD
I t i s i nt h i sl a s te x p r e s s i o nt h a tt h e
i no r d e rt ou s et h i se x p r e s s i o n .
p r e s e n ta n a l y s i sd i f f e r sf r o mt h a to fR e f e r e n c e
13.
The s o n i c p o i n t p r e s s u r e h a s b e e n c h o s e n t o b e d e s c r i b e d
by t h e i d e a l
g a se x p r e s s i o n
The i s e n t r o p i ce x p o n e n t
y i s evaluatedatthestagnationpoint
t o r e m a i nc o n s t a n to v e rt h er e g i o no fi n t e r e s t .
and i s assumed
The assumption of a c o n s t a n t
y i n t r o d u c e s a n e g l i g i b l ys m a l le r r o ri n t ot h ea n a l y s i s .A n o t h e rq u a n t i t y
of i n t e r e s t i s t h el o c a t i o no ft h es o n i cp o i n t ,
s*.
F o rt h es h a r pc o r n e r e d
b o d i e sd i s c u s s e da b o v e ,
s* i s m e r e l yt h ed i s t a n c et ot h ec o r n e r .
However, a s
these s h a r p c o r n e r e d b o d i e s a b l a t e o r f o r o t h e r s m o o t h l y - c o n t o u r e d b o d i e s ,
m u s tb el o c a t e di no r d e rt ou s eE q u a t i o n( 8 7 ) .
The Newtonian
p r e s s u r e d i s t r i b u t i o n i s employedhere t o g i v e t h e s u r f a c e a n g l e a t t h e s o n i c
t h es o n i cp o i n t
p o i n t of
B * = cos -l
F
1-Pm
Thus t h e s u r f a c e r u n n i n g l e n g t h
66
s*
canbefound
by u s i n g t h e
(89)
known s u r f a c e
slopesandrunninglengthsalongthesurface
and i n t e r p o l a t i n g b e t w e e n
The f l a t d i s c p r e s s u r e d i s t r i b u t i o n u s e d i n R e f e r e n c e
x =
them.
13 i s
s
-4
(90)
I t w i l l be shown t h a t t h e h e a t t r a n s f e r c o e f f i c i e n t a t t h e s t a g n a t i o n p o i n t
depends upon t h e s t a g n a t i o n p o i n t v e l o c i t y g r a d i e n t
(see E q u a t i o n( 1 2 0 ) ) .
Thisgradientinturndepends
upon t h e s e c o n d d e r i v a t i v e o f p r e s s u r e w i t h
d i s t a n c ea c c o r d i n gt ot h ef o l l o w i n gr e l a t i o n
O'S
( 8 7 ) twice y i e l d s
Differentiatinq Equation
d2P
ds
which f o r
RN
- + -2
2
RNRmax .
appioachinginfinityyields
n a t i o np o i n ti fE q u a t i o n
d2FFD
2
ds
a zerovelocitygradientatthestag-
( 9 0 ) i s employed a s i t s t a n d s .
T o c o r r e c t t h i s un14 were e v a l u a t e d t o o b t a i n
a flat
disc stagnation point velocity gradient expression
realisticresult,thedataofReference
(93)
By combiningEquations
(911,
( 9 2 ) , and(93)
,
t h ef o l l o w i n gr e s u l t
is
obtained
67
This expression provides a more correct stagnation point pressure distribution
( g o ) , has been modified to blend
The flat disc pressure distribution, Equation
smoothly with the new second derivative requirements,
and the
definition
A has
of
been
x
5
=
changed
G
slightly:
p
Equations ( 9 5 ) and ( 9 6 ) are the final forms of the
f l a t d A > cpressure distribution relations which are used
in the present analysis.
8.2.2
Downstream of the Sonic, Point
( 8 7 ) above, the local static
Using the expression given in Equation
pressure is exactly equal
to the Newtonian pressure for all body shapes
at
the sonic point. The Newtonian description of pressure
is then used downstream of the sonic point until the Prandtl-Meyer match point is reached,
at which point a Prandtl-Meyer expansion expression
is used for the remainder
of the running length. The match point
is determined by the requirement of
continuity of pressure and pressure gradient along the surface:
( 9 7)
Prandtl-Meyer
It is more
pressure :
convenient
work in terms
to
of
the
angle
B and
dimensionless
Prandtl-Meyer
The
68
Newtonian
pressure
expression be
candifferentiated to give
For a perfectgas,thePrandtl-Meyerpressuregradient
- -
-
is
yFM
2
(100)
Prandtl-Meyer
while the relation between pressure
Equatingthe
and Mach number i s
two g r a d i e n t e x p r e s s i o n s y i e l d s t h e f o l l o w i n g e q u a t i o n f o r t h e
match point:
T h i s i s an i m p l i c i t e q u a t i o n f o r
s o l u t i o n i s foundby
Pressuresonthe
carryingout
B M , s i n c e F and M are f u n c t i o n so f
a Newton-Raphson
B.
The
iteration procedure.
bodydownstream
of the
match p o i n t a r e f o u n d
by
a Prandtl-Meyerexpansionfromthematchpointconditionsto
t h el o c a ls u r f a c es l o p e .
The g o v e r n i n ge q u a t i o n si nt h i sr e g i o na r e
E q u a t i o n (101 ) above r e l a t i n g p r e s s u r e t o
Mach number,and
which r e l a t e s Mach number t o l o c a l s u r f a c e a n g l e .
Once a g a i n ,t h ee q u a t i o n s
are i m p l i c i t s i n c e M i s a f u n c t i o n o f B , t h e r e f o r e a Newton-Raphson i t e r a t i o n
procedure w a s d e v i s e d t o s o l v e t h e s e e q u a t i o n s .
69
Examples oftheAccuracy
8.2.3
of t h e Method
The a c c u r a c y o f t h e p r e s s u r e p r e d i c t i o n t e c h n i q u e d e s c r i b e d a b o v e
i s demonstratedwithfoursampleproblems.
h a l fa n g l es p h e r ec o n ew i t h
a s h a r pc o r n e r
The f i r s t o f t h e s e
as shown i n F i g u r e
i s a 60"
1 8 . With t h i s
is fixedatthesharpcorner,thereand Rmax a r e known from t h e geometry.
The p r e s s u r e p r e d i c t i o n
s h a p e ,t h el o c a t i o no ft h es o n i cp o i n t
fore B*
,
s*
,
for this shape
is compared i n F i g u r e 18 t o t.be d a t a o f R e f e r e n c e
15
,
and
a l s o t c t h e N e w t o n i a np r e s s u r ed i s t r i b u t i o nf o rt h e
same s h a p e . Agreement
is s e e n t o b e e x c e l l e n t o v e r m o s t o f t h e s u r f a c e , w h i c h
is tobeexpected
sincetheempiricalportionsofthetheory
were i n v e n t e d t o s o l v e t h i s v e r y
problem.
The s h o r t c o m i n g so fN e w t o n i a nt h e o r ya r ea l s or e a d i l ya p p a r e n ti n
this figure
.
Figure 1 9 d e m o n s t r a t e st h eu t i l i t yo ft h i sp r e s s u r ep r e d i c t i o nt e c h -
of p r e s -
n i q u e on a s p h e r i c a ls h a p e .F o rs p h e r e s ,t h eN e w t o n i a nd e s c r i p t i o n
s u r ei nt h es u b s o n i cr e g i o n
exception.
i s g e n e r a l l yv e r yg o o d ,
and t h .c a s e
i s no
The d i f f e r e n c e sb e t w e e nN e w t o n i a na n dt h ec u r r e n t l yp r o p o s e d
t h e o r y on a s p h e r e i n t h e s u b s o n i c r e g i o n a r e v e r y s l i g h t , h o w e v e r ,
methodsgive
andboth
good a g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a .
F i g u r e s 2 0 and 2 1 demonstratetheagreementbetweentheoryand
e x p e r i m e n tf o r
a b l u n t and a s h a r pe l l i p s e ,r e s p e c t i v e l y .A g a i n ,t h e r e
is
very l i t t l e differencebetweenNewtonianandthecurrenttheories,with
eithergivingsatisfactoryagreementwiththedata.
I n summary, t h e p r e s e n t t h e o r y o f f e r s
good agreementwithNewtonian
theoryforsphericalandellipticalshapes,butalsoprovidesaccurate
d e s c r i p t i o n so fp r e s s u r eo v e rf l a t - d i s c - t y p es h a p e sw h e r eN e w t o n i a nt h e o r y
f a i l se n t i r e l y .T h i s
makes t h et h e o r yp a r t i c u l a r l ys u i t a b l ef o rt h el a r g e
cone a n g l e b l u n t s h a p e s b e i n g c o n s i d e r e d f o r p l a n e t a r y e n t r y .
70
1 .c
0.9
0.e
0.7
0
n
\
n
0.6
D A T AF R O MR E F ERENCE 15
C . 5I N C H E S
0.5
-
60'
2.283 INCHES
0.4
t
4
0
0.1
0.2
0.3 0 . 5
0.4
0.6
0.7
0.8
0.9
1 .o
s/s*
Figure 18.
Pressure Distribution on a 60° Sphere Cone w i t h Sonic Corner
1 .o
I
-
0.9
&'
_"_
NEWTONIAN
THEORY
M
\
0.8
I
THEORY
0
\
I
-
-
DATA FROM R E F . 1 6
-
6.0
0.7
I
0.6
e
0
0.5
n
0.4
0.3
0.. 2
0.1
0
0.2
0.4
0.6
0.8
1.o
1.2
1.4
S/RN
Figure 19.
72
Pressure Distribution on a Sphere
1.6
1 .o
0.8
0.6
0
n
\
n
0.4
0.2
0
10
20
30
40
50
60
8-DEGREES
Figure 20.
Pressure Distribution on an Ellipse
70
80
90
1
.o
-.
b
0.8
~
-
t
M
=
4.00
b/a
=
0.5
Po
=
1 . O ATM
H,
=
5000 B T U / L B
4
-
0.6
- THEUKY
0
a
\
a
""
-
0.4
0
NEWTONIAN
THEORY
DATA F R O M R E F . 1 6
0.2
"
0
10
20
L
30
640
0
-0
"
50
0-DEGREES
Figure 21.
Pressure Distsibution on an Ellipse
70
I
1
1
80
90
8.3
EVALUATION O F OTHER PROPERTIES
Once t h e p r e s s u r e
i s assumedequal
i s known a t anygiven
body p o i n t a n d t h e e n t r o p y
tothestagnationvalue,othergaspropertiesofinterest
may b e f o u n d b y r e f e r r i n g t o t h e M o l l i e r C h a r t t a b l e s
S e c t i o n8 . 1 .
as d e s c r i b e d i n
Some c a r e i s n e c e s s a r y i n c a r r y i n g o u t t h i s o p e r a t i o n ,
The M o l l i e r C h a r t l o o k u p s u b r o u t i n e s h a v e b e e n w r i t t e n
however.
t o usepressureand
e n t h a l p ya st h ei n d e p e n d e n tv a r i a b l e si nt h el o o k u pp r o c e d u r es i n c et h e s e
a r e t h e known q u a n t i t i e s a t t h e s t a g n a t i o n p o i n t . I n o r d e r t o m i n i m i z e t h e
error resulting
w h e r ep r e s s u r e
from l i n e a r i n t e r p o l a t i o n , p r o p e r t i e s a t o t h e r
and e n t r o p y a r e
known a r e f o u n d i t e r a t i v e l y
body p o i n t s
by e s t i m a t i n g
e n t h a l p y ,l o o k i n gi nt h e
tables a t t h e known P a n de s t i m a t e d
h , determining
e n t r o p y s , comparingwiththe
known s , r e v i s i n gt h e
h e s t i m a t e , and so on.
T h i sp r o c e d u r ep r e v e n t s
were t oi n t e r p o l a t e
on
t of i n dh .
a compounding o f e r r o r s
P
and
h
t of i n d
s,
whichcouldariseif
then
interpolate
P
one
and
s
75
SECTION 9
THE TRANSFER COEFFICIENT
&-I
energyintegraltechnique
EVALUATION
is u s e d t o e v a l u a t e t h e n o n a b l a t i n g
body shapewhich
heat transfer coefficient along the surface of an arbitrary
starts a t a s t a g n a t i o np o i n t .
The e n e r g yi n t e g r a lt e c h n i q u e
is c o m p l e t e l y
d i s c u s s e d i n R e f e r e n c e 1 7 , and w i l l b e o n l y b r i e f l y
summarize2here.
f l a t p l a t e r e l a t i o n s h i p bei s assumed t o b e v a l i d f o r a l l
In the energy integral technique, the
tween energy thickness and heat transfer rate
i s t h e nu t i l i z e dt os o l v et h ee n e r g y
t y p e so ff l o w s .T h i sr e l a t i o n s h i p
i n t e g r a le q u a t i o nf o ra na r b i t r a r yf l o w .
ful to establish
method, it i s use-
To o u t l i n e t h i s
a few b a s i c d e f i n i t i o n s .
heat transfer coefficient:
a
CHI
(104 )
p ue (Hr-h W 1
energy thickness :
referenceenthalpy:
h' =
0.23he
+
0.19Hr
+
0.58hw(laminar)
h' =
0.36he
+
0.19Hr
+
0.45hw
(turbulent)
(107)
where
Hr
=
h
e
+
?
:u
V
2
2
2
=
(pr 1 )
2
=
(Pr') ' I 3( t u r b u l e n t )
(laminar)
(108)
1
(109)
and a l l p r i m e d q u a n t i t i e s a r e e v a l u a t e d a t
thereferenceenthalpycondition.
76
It w i l l b e r e c a l l e d t h a t f o r f l a t p l a t e f l o w s w i t h
layer,theheattransfercoefficientexpression
where
a = 0.332
m = 0.5
or
no i n i t i a l boundary
is
I
laminar
a = 0.0296
turbulent
m = 0.2
Combining t h i s w i t h t h e d e f i n i t i o n
which r e l a t e s h e a t f l u x t o
of C H I r e s u l t s i n
x for a flat plate.
i s o t h e r m a lw a l l s ,t h ee n e r g yi n t e g r a le q u a t i o n
Combining Equations111and
F o rt h ef l a tp l a t ew i t hn e a r l y
is
1 1 2 a n da s s u m i n gs m a l lv a r i a t i o ni n
hw compared
w i t h Hr-hwl t h e f l a t p l a t e e n e r g y i n t e g r a l e q u a t i o n c a n b e s o l v e d t o y i e l d
r e l a t i o n s h i pb e t w e e ne n e r g yt h i c k n e s sa n d
By cornbinihgEquations
heatflux
a
x:
1 1 3 and 111, w e h a v e a n e x p l i c i t r e l a t i o n s h i p b e t w e e n
a n de n e r g yt h i c k n e s s :
Thisexpression
i s now assumed t o b e v a l i d f o r p l a n a r b o d i e s o r b o d i e s
ofrevo-
lution,witharbitrarypressuregradient.
77
Themore
generalboundrylayerenergyequationinintegralform
Combining E q u a t i o n s 1 1 5 and114
,
t h er e s u l t i n gf o r m
ci.
is
.e i n t e g r a t e d d i r e c t l y
to give
(x
1
+
[rPeue (Hr-hw)]
A J
x
i
where X i s a dummy v a r i a b l er e p r e s e n t i n gr u n n i n gl e n g t h .T h i sr e l a t i o nb e t w e e n
9 and x canbecombinedwithEquation
114 above t o g i v e t h e g e n e r a l r e l a t i o n
x.
b e t w e e nh e a tf l u x( o rh e a tt r a n s f e rc o e f f i c i e n t )a n d
s t a r t i n g from a s t a g n a t i o n p o i n t , t h e f i n a l e x p r e s s i o n
Forlaminar
flows
is
0.332p'p'rue(Hr-hw)
PeUeCH
=
(Pr')2/3
'117)
{ l"
S i m i l a r l y ,f o ra l lt u r b u l e n tf l o w ,
0.0296 p ' ~ ~ ( " r ) l / ~ ( H ~
-h
)lI4
PeUeCH
=
(Pr')2'3
I
W
~ ' ~ ~ ( u ' ) ~ / ~ r ~
5/4dX
/ ~ ( H ~ - h ~ )
F i n a l l y ,f o rt r a n s i t i o nf r o ml a m i n a rt ot u r b u l e n tf l o w
I
p'u'ue(Hr
-
+
0.0161
JX
X
t
78
(118)
at x
=
Xt '
For a stagnation point,
x
*
w e use the laminar expression and take the
l i m i t as
o with
r
=
X
=
const.
HZ-hw =
const.
p'p'
The h e a t t r a n s f e r c o e f f i c i e n t r e d u c e s t o
1/2
Consistentwiththepressuredistribution
stagnationpointvelocitygradientcanbe
d i s c u s s e d i n S e c t i o n8 . 2 . 1 ,t h e
e x p r e s s e d as
whichbecomes
InthepresentversiolL
expressionE
, quations
of t h e c o d e , o n l y t h e l a m i n a r h e a t t r a n s f e r
( 1 1 7 ) and ( 1 2 0 ) , are i n c l u d e d F
. or
programming p u r p o s e s ,
i t i s most c o n v e n i e n t t o w r i t e t h e h e a t t r a n s f e r e x p r e s s i o n i n t h e
form
where
u* * 4 "due
R
(point ds
stagnation
79
or
u*
=
U
e
away from
stagnation
point
S
J
(Hr-hw)r2ueds
p
The running integral term for
E* away from the stagnation point is evaluated
by treating each linear term in the integrand it
as varied
if
linearly bep', y ' , Hr-hw, r,
tween body stations where its value is known. That is,
and ue are all approximated as linear functions
of s between each pair of
of the
body stations, thus making direct integration possible. The value
integral at each body station is stored and merely added to the increment
in R* at each new body station. All primed quantities are evaluated by referring to the Mollier Chart lookup routine with the known pressure and
reference enthalpy.
AI-I
example
of
the
accuracy
of
the
energy
integral
method
is
provided
in Figure 22, where results of the present theory applied
to a spherecone configuration are compared with results from Aerotherm's multicomponent,
18). The two programs
nonsimilar, boundary layer analysis program (Reference
were supplied with identical pressure and wall temperature distributions Over
the length of interest. It is apparent that the energy integral approach gives
satisfactory results for the
a sphere-cone type of body.
nonablating
heat
transfer
coefficientatfor
least
For use in the MACABRE code, the nonablating heat transfer coefficients computed with the analysis outlined above are corrected to account for
the effects of ablation ("blowing").If we denote this nonablating coefficient
A = 0) as peueCHOl then both data and simple analyses (see, for
(that is, for
example, Reference 7) indicate that the dependence of peueCH on is given
fairly accurately by
where
cP=
This correction
MACABRE code.
80
is
built
into
the
2xi
'euecH0
surface
energy
balance
operations
of
the
I
3.6
I
I
SPHERE-CONE
I
BODY
3.2
2.8
2.4
2.0
I
u
,
"1 . 6
01
Q
1.2
0.8
0.4
0
0.4
0.8
1.2
2 . 8 1 . 62 . 4
2.0
S/RN
Figure 22.
Heat Transfer Coefficient Prediction
81
SECTION 10
OVERVIEW
OF
COMPUTER
PROGRAM
EVENTS
The preceding sections have described the individual computing phases
of the coupled computer code and the linkages between the phases. This secclarify further the
tion presents two simple flow charts for the to
code
operation of the code. Figure2 3 gives an overall chart of code events, and
of the surface energy balance sub-package.
Figure 24 shows some details
82
!
I
I N P U TI ,N I T I A L I Z A T I O N
NODAL THERMAL R E S I S T A N C E S AND THERMAL
COMPUTE NET HEAT FLUXESTO
EACH
SUBSURFACE NDDE (BUT
RESERVING
SURFACE
NODES AND NEXT NODE DOWN F O RS P E C I A L
I M P L I C I T TREATMENT)
~.
CALL
SURFACE
ENERGY
BALANCE PACKAGE
TODETERMINE
NEW SURFACETEMPERATURES
AND R E C E S S I O NR A T E S( S E EF I G U R E2 4 )
COMPUTE NEW TEMPERATURESFORALLINDEPTHNODES,NOTINGSPECIALPROCEDURES
FORSURFACENODES
AND NEXT NODE DOWN
t
~~
OUTPUT,
IF
(1) T H I S I S F I R S PT A S S
(2T
) HIS
I S A MANDATORY SURFACE
STATE
SOLUTIONTIME
( 3 )T H I S
(4)
I S REGULAR
A
OUTPUT
TIME
T H I S I S T HF
EI N A T
LI M E
I F (1) OR ( 2 ) , V O I D OUT ONE OF THE TWO
T I M ES H E E T SO F
THERMOCHEMICAL
DATA,
INCREASEINDEXOFPRESENTSETOF
2 MANDATORYSURFACESTATESOLUTIONTIMES
BY ONE
Figure 2 3 .
O v e r a l l MACABRE C o d e Flow C h a r t
a3
CALLED
.
I
FROM
MACABRE
1
rn
+
COMPUTE ALLOWABLE TIME INCREMENT AS
INFLUENCED BY SPACING OF TIME TABLE
ENTRIES, SPACING OF MANDATORY SURFACE STATE SOLUTION TIMES, RECESSION
RATES, OUTPUT TIMES
*
COMPUT CURRENT SURFACE SHAPE,
S L O P E S 1
MAIN LOOP OVER SURFACE POINTS (COLUMNS)
SET COLUMN INDEX =I 1
1
-
m
b
DETERMINE TIME TABLE ASSIGNED
TO THIS
COLUMN, LOOKUP ALL TIME DEPENDENT
QUANTITIES IN TABLE
.
i
CALL PARBL
(I), RETURNS ANY OR ALL
OF
CURRENT LOCAL PRESSURE, CURRENT LOCAL
RECOVERY ENTHALPY, CURRENT LOCAL CONVECTIVE TRANSFER COEFFICIENT, AS
REQUIRED
~~
DETERMINE CURRENT LOCAL HEATING OPTION:
GO TO RELEVANT "PREPARATIONS" SECTIONS
ACCORDING TO OPTION
-
-
OPTION 2
NOT
OPTION 3
~
CONVECTIVE
OPTION 1
SURFACE ENERGY
BALANCE
~
~
OPTION
~~~~~
v
CONTINUED ON NEXT
Figure 2 4 .
84
PAGE
Surface Energy Balance Operations Flow Chart
(Subroutine SURFB)
I”
-
1
IF HAVE JUST ARRIVED AT NEXT MANDATORY
SOLUTION TIME, LOOK
IN RELEVANT TIME
TABLE FOR THIS STATION
TO FIND FUTURE
LOCAL PRESSURE, FUTURE LOCAL RECOVERY
ENTHALPY, AND FUTURE LOCAL TRANSFER
COEFFICIENT; ALL OF WHICH WILLREBE
QUIRED IN FUTURE-TIME CALLS OF THE
SURFACE STATE PACKAGE. IF SOME OF
THESE QUANTITIESARE NOT ENTERED AT
THE FUTURE TIME, USE SPECIAL RULES
TO
ESTIMATE THEM FROM CURRENT DATA
ABLE
DATA
INCLUDING
Tw
THIS STATION TO TEMPERATURES LOWIN
EST ASSIGNED BA ENTRIES IN TABLES.
IF CURRENTTW isGREATER, GO TO
BG
SECTION OF TABLES AND ITERATE E;.ON
IF CURRENTT, IS LESS, GO TO Tw SEC-
ASSIGNED BA
ITERATION
SURFACE
ENERGY
BALANCE
ITERATION
SHOWN
NOT
t
CONTINUED
Figure 2 4 .
ON
NEXT
PAGE
(continued)
85
I
1
IF CURRENT B& IS ZERO HERE, ESTIMATE
B& FROM BOTTOMBA’S IN TABLE
I
A
CHECK TABLE ENTRIES SURROUNDING THE CURRENT ESTIMATE OF
B& AND THE CURRENT TIME.
IF THERE ARE ANY VOIDS AMONG THE DEPENDENT VARIABLE VALUES FOR THESE POINTS,
CALL ACE PACKAGE TO FILL IN. FOR SOLUTIONS AT PAST TIME, USE SAVED PAST VALUES
OF P and H, IN ACE CALL; FOR SOLUTIONS
AT FUTURE TIME, USE PREVIOUSLY UP
SET
VALUES (OR ESTIMATES) OF P AND
H, IN THE
ACE CALL.
+
CONSTRUCT SURFACE ENERGY BALANCE EQUATION, NOTE “ERROR“ (DEPARTURE FROM
ZERO).
CALCULATE A CORRECTION TO
B& BY NEWTONRAPHSON METHOD
~
BALANCED TO ACCEPTABLE
TOLERANCE?
I
t
CORRECT B& LIMITING CORRECTIONTO TABLE
INTERVAL WIDTHS, REPEAT
TO CONVERGENCE
-
I
L
ADJUST
SIZE
e
OF SURFACE
NODAL
BOX
~
ON
I
I
LAST
POINT?
*
INCREMENT COLUMN INDEX: nI I
+
1
I
-
I
RETURN TO MACABRE
Figure 2 4 .
86
SURFACE
I
(concluded)
APPENDIX A
TRANSPORT
PROPERTIES
FROM
THE ACE PACKAGE
When t h e ACE package of s u b r o u t i n e s i s used t o g e n e r a t e t r a n s p o r t
property data for subsequent use in the transfer coefficient subroutine
RUCH,
the t r a n s p o r t p r o p e r t i e s a r e c a l c u l a t e d
by ACE f r o m e x p r e s s i o n s
which d e r i v e f r o m s i m p l e k i n e t i c t h e o r y a n d t h e p a r t i c u l a r m u l t i c o m p o n e n t
d i f f u s i o nr e p r e s e n t a t i o np r e v i o u s l yd i s c u s s e di nS e c t i o n
5.3.3.
The
d e v e l o p m e n to ft h e s ee x p r e s s i o n s
is discussedindetailinReference
A b r i e f summary o f t h i s d e v e l o p m e n t , a n d t h e r e s u l t i n g e x p r e s s i o n s , a r e
19.
presentedinthissection.
DiffusionCoefficients
-
InSection
tion for binary diffusion coefficients
multicomponentdiffusion
5.3.3, a b i f u r c a t i o n approxima-
was mentionedwhichcharacterized
phenomena w i t h r e a s o n a b l e a c c u r a c y w i t h o u t u n d u l y
c o m p l i c a t i n gt h es y s t e mo fe q u a t i o n st ob es o l v e d .T h i ss i m p l i f i c a t i o n
a c h i e v e dt h r o u g h
is
a correlationforbinarydiffusioncoefficientsofthe
form
where 5 i s a r e f e r e n c e d i f f u s i o n c o e f f i c i e n t
and t h e Fi a r e d i f f u s i o n f a c -
t o r s .T h e s eq u a n t i t i e sa r ed i s c u s s e di nd e t a i li nR e f e r e n c e
i n c o r p o r a t i o no f
( A - 1 ) i n t h e Stefan-Maxwell r e l a t i o n f o r
f l u x e si n d i c a t e st h a tt h ed i f f u s i o nf l u xo fs p e c i e s
terms o fo n l yp r o p e r t i e s
of s p e c i e s
i
i
19.
The
mass d i f f u s i o n
may b ew r i t t e ni n
and g l o b a ls y s t e mp r o p e r t i e s .
j e c t t o a few s i m p l i f y i n ga s s u m p t i o n s( R e f e r e n c e
Sub-
19), t h i s e x p r e s s i o n
f o r j i may b e w r i t t e n
where
87
i s examined i n R e f e r e n c e 1 9 f o r a v a r i e t y o f
I t i s shown t h a t t h e B i j c a l c u l a t e d
by E q u a t i o n (A-1) re-
The a c c u r a c y o f t h i s f o r m u l a t i o n
chemicalsystems.
present a very substantial
improvementoverequaldiffusioncoefficients
when compared t o exact v a l u e sc a l c u l a t .
calculationofthemixtureviscosity
on t h e d i f f u s i o n f a c t o r s g i v e n
'1 d i r e c t l y from k i n e t i c t h e o r y .
a n dt h e r m a lc o n d u c t i v i t y
by (A-1) I a n d t h e s e
The
is based
w i l l bedl-scussed
in the
following paragraphs.
MixtureViscosity
-
The expression employed by the
calculatethemixtureviscosityderives
t h e o r y( R e f e r e n c e
cussedinReference
20)
I
ACE program t o
fromrigorousfirstorderkinetic
subject t o a few s i m p l i f y i n ga s s u m p t i o n s ,a sd i s -
19.
1
I
'mix
=
i=
1
where
ui is
t h ev i s c o s i t yo ft h ep u r es p e c i e s
i n terms o f t h e s e l f d i f f u s i o n c o e f f i c i e n t s
where AZi
The
vi
may be e x p r e s s e d
Bii
is a r a t i o o f c o l l i s i o n i n t e g r a l s b a s e d
on a L e n n a r d - J o n e s i n t e r ( A - 1 ) and (A-7) i n t o (A-6) r e s u l t si nt h e
m o l e c u l a rp o t e n t i a l .S u b s t i t u t i n g
88
i.
p-
following expression for the viscosity of the multicomponent mixture.
(A-8)
and t h i s i s t h e e x p r e s s i o n u t i l i z e d t o c a l c u l a t e t h e F i x t u r e v i s c o s i t y o u t put by the
ACE program.
-
MixtureThermalConductivity
a t o m i cg a sm i x t u r e
may b e r e p r e s e n t e d b y ( R e f e r e n c e
kmix = k
where kmono-mix
The t h e r m a l c o n d u c t i v i t y i n
mono-mix
+
is thethermalconductivityin
a poly-
20)
(A-9)
kint
a m i x t u r e computed n e g l e c t i n g
allinternaldegreesoffreedom
andkint
is thecontributiontothethermal
c o n d u c t i v i t y of t h e m i x t u r e due t o t h e i n t e r n a l d e g r e e s o f f r e e d o m
of t h e
molecules.
A s i m p l i f i e de x p r e s s i o nf o rt h e
canbederivedin
mono-mixturethermalconductivity
a manner s i m i l a r t o t h e p r o c e d u r e p r e v i o u s l y d i s c u s s e d f o r
t h em i x t u r ev i s c o s i t y .T h i ss i m p l i f i e de x p r e s s i o n
is (fromReference
r
19)
1
Xiki
mono
kmono-mix
i=l
xi
+
1.475
(A-10)
where k i
is the
t h e r m acl o n d u c t i v i t y
otfh e
pure
species
a liln t e r n adl e g r e e o
s ffr e e d o m
otfh e
molecule.
i n terms o f t h e
pi
i neglecting
The ki
may b e x p r e s s e d
a sp e r
(A-11)
The c o n t r i b u t i o n t o t h e t h e r m a l c o n d u c t i v i t y f r o m t h e i n t e r n a l d e g r e e s o f
89
freedom may b e e x p r e s s e d
as ( f r o m R e f e r e n c e 1 9 )
(A-12)
By combining (A-1)
kmix
e
with (A-9)
through ( A - 1 2 ) ,
t h em i x t u r et h e r m a lc o n d u c t i v i t y
as
may b e w r i t t e n
[
pl
1 5 xi R
4 Fi
"
x.F
l i (6Ali
i=l 1.475
+- 5
p1
-
(A-13)
where p 1 and
vq
a r eg i v e n
L
by ( A - 4 )
and ( A - 5 )
r e s p e c t i v e l y ,a n d
(A-14)
2
T
a.
c
P
Thus,
=
zicpi
(A-15)
i=
1
t:he e x p r e s s i o n u t i l i z e d t o c a l c u l a t e t h e m i x t u r e t h e r m a l
( A- 13) i s
conductivity output
by t h e ACE program.
A l s o c a l c u l a t e d and o u t p u t by t h e ACE program are t h e P r a n d t l a n d
Schmidtnumberswhich
are d e f i n e d h e r e a s
Pr
=
L Cp-frozen
k
(A-16)
(A-17)
The t r a n s p o r t p r o p e r t i e s c a l c u l a t e d
by t h e ACE program are a l l b a s e d
on t h e b i f u r c a t i o n a p p r o x i m a t i o n f o r t h e b i n a r y d i f f u s i o n c o e f f i c i e n t s e x p r e s s e di n
(A-1).
casediffusion
andthermodynamic
90
T h i s is so even f o rc l o s e ds y s t e mc a l c u l a t i o n s( i n
which
phenomena n e e d n o t b e c o n s i d e r e d t o c a l c u l a t e t h e c h e m i c a l
s t a t e o ft h es y s t e m )
and f o r o p e ns y s t e mc a l c u l a t i o n sf o r
which e q u a ld i f f u s i o nc o e f f i c i e n t sa r e
assumed ( S e c t i o n5 . 3 . 2 ) .
From t h e
it may b e o b s e r v e d t h a t t h e p r o p e r t i e s c a l c u l a t e d a r e
h i g h l yd e p e n d e n to nt h ed i f f u s i o nf a c t o r s ,
Fi.Three
a l t e r n a t e methods f o r
p r e s c r i b i n g t h e Fi were d i s c u s s e d i n S e c t i o n
5.3.3.
The u s e o f t h e d i f f u s i o n
equationspresented,
f a c t o rc o r r e l a t i o n
( 6 3 ) w i t hr e s i d e n tv a l u e so f
nref
and
E
(whichwere
d e r i v e dp r i m a r i l yf r o mc o n s i d e r a t i o no fs p e c i e sd i f f u s i o nc o e f f i c i e n t s )s h o u l d
r e s u l t i nr e a s o n a b l ya c c u r a t ev a l u e so fo t h e rt r a n s p o r tp r o p e r t i e s .
Altern a t e l y ,t h ec o r r e l a t i o n
(63)
may be u s e d w i t hv a l u e so f
nref
and
E
derived
by c o r r e l a t i n g a v a i l a b l e t r a n s p o r t p r o p e r t y d a t a f o r t h e p a r t i c u l a r s y s t e m
of i n t e r e s t . I f t r a n s p o r t p r o p e r t i e s o f
maximum a c c u r a c ya r et o
be c a l culated,thenthediffusionfactorsshouldbeinputindividuallyforeach
s p e c i e s .T h e s e
data may b eo b t a i n e df r o mt a b u l a t i o n ss u c ha sR e f e r e n c e
21.
91
REFERENCES
1.
Bartlett, E. P., Rindal, R . A . and Kendall, R. M.:
A Critical Evaulation
of Recent Developments and Future Requirements for the Prediction
of
Ablation of Manned Reentry Vehicles at Superorbital Velocities. Vidya
Division, Itek Corporation, Palo Alto, California, Vidya Report
No. 1 7 4 ,
February 4 , 1 9 6 5 .
2.
Aerotherm Corporation, Mountain View, California, Input Guide, Computer
Program for Material Ablation with Chemically Active Boundary Layers in
Reentry (MACABRE), September, 1 9 6 9 .
3.
Moyer, C. B.: Axi-Symmetric Transient Heating and Material Ablation
Program (ASTHMA), Description and User's Manual, Aerotherm Corporation,
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