AFATL-TR-85-12 Viscous Unsteady Flow Analysis Applied to ...
117
AFATL-TR-85-12 Fully Viscous Two-Dimensiona Unsteady Flow Analysis Applied to Detonation Transition in ch Porous Explosives . tHerman Krier Laurence S Samuelson DEPARTMENT OF MECHANICAL AND INDISTRIAL ENGINEERING UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN JUE 1985 DTIC FNiALREPORT FOR PERIOD MAY1982- MARCH 1985 LECT it Approved for public release; distribution unlimited - .- -* Air Force Armament Laboratory AN FORCE SYSTEMS COMMAHD * UNITED STATES AI FORCE * EGLIN AiR FORCE BASE, RORIDA , 85 06 21, 01 --
Transcript of AFATL-TR-85-12 Viscous Unsteady Flow Analysis Applied to ...
AFATL-TR-85-12Fully Viscous Two-Dimensiona
Unsteady Flow Analysis Appliedto Detonation Transition in
ch Porous Explosives
. tHerman KrierLaurence S Samuelson
DEPARTMENT OF MECHANICAL AND INDISTRIAL ENGINEERINGUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
JUE 1985
DTICFNiALREPORT FOR PERIOD MAY1982- MARCH 1985 LECT it
Approved for public release; distribution unlimited
- .-
-* Air Force Armament LaboratoryAN FORCE SYSTEMS COMMAHD * UNITED STATES AI FORCE * EGLIN AiR FORCE BASE, RORIDA
, 85 06 21, 01--
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I I TITLE f(,aide Sec-t.y CIlj(IIsi iu lion) 62602F I2543 19 64A Fully Viscous Tvo-Dimensiona1 Unsteady ______I
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Herman Krier and Martin R. Dahm13. TYPE OF REPORT 1311 TIME COVERED l4- DATE OF REPORT (Yr. Mo. Day) 15. PAGE COUNT
Final IFROM _5/82 TO/§5 1 1984 Feb 1151B. SUPPLEMENTARY NOTATION
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1) COSATI CODES JI& S".B.Cr TERMS (Continue on ,,. e .suw ndim by b6uei RumbtleFIELD IGROUP - sutN G R. ITwo-Dimenaional Flow Porous Explosives
____________21_____ Fully Viscous Unsteady Flow Warhead DamageDetonation Transition Warhead Deflagratiogs
III ABSTPACT IWonf~nuir an aWl' f ee l nd idej,tify by block nowmbero
This report describes significant new progress towards solving the two-dimensional fullyvi~cous unsteady flow In a reactive solid/gas mixture. The results demonstrate that itmay be possible to predict whether a warhead with case failure (and filled with fragmentedhigh explosive) will produce a low order detonation which is weaker than the one requiredto damage a structure. The loss of mass, momentumn, and energy through a warhead caseopening is a multi-dimensional problem even though the reaction front may progressaxially through the damaged explosive. 14nss is ejected in a itasically radial directionand nh~cessitates a multi-dimensional prcblem formulation in order to model the deflagratia,to detonation transition event accurately enough to make meaningful predictions. Theappropriate equations of state and continuity equations are formulated and are solved
* by a finite differencing scheme. Run-up length to detonation in the fragmented explosivebed from a case opening through which mass is ejectell should be married with a model ofprobabilistic case failure onm impact to predict warhead lethality.
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Itncldi Anre Cc-delMIark A. Amend (904) 882-8302 AFATL/DLYV
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Item 11, Continued: Flow Analysis Applied To Detonation Transition In Porous Explosives
Va
44.
.-
'a
PkEFACE
This report docunents work performed during the period May 1982 to MarchL1985 by the University of Illinois, Urbana-Champzign, Illinois, under Research
Contract F08635-82-K-0321 with the Air Force Armament Laboratory, Eglin AirForce Base, Florida. Mr Mark Amend moniLtofed the program for ~the ArmamentL Laboratory.
The Public Affairs Office has reviewed this report, and it is releasableto the National Technical Information Service (NTIS), where it will be avail-able to the general public, including foreign nationals.
This technical report ha3 been reviewed and is approved for publication.
FOR THE CO'N4ANDER
NMILTON D KINGCAID, Colfiel, USAF-hief, Analysis & Stra ic Defense Division
U
(The reverse of t his page i~s blaak)
TABLE OF CGNTENTS
Section 1i' Page
I NTRODUCT ION ...... .... .....
II FUNDAMENTAL CONCEPTS
FOR THE TWO DIMENSIONAL MODEL ........................ 5
! nt roduct ion .................................... 5
S!'Assumptions. , . .. . . . . . . . . . . . . . .5
Domain of the Model .......................
Conservation Equations.......................... 8
Constitutive Equations (Closure) ............... 13
Boundary Conditions ............................ 14
Initial Conditions ............ ................. 17
III DERIVATION OF TIHE TWO DIMENSIONAL
AXISYMMETRIC CONSERVATION EQUATIONS ................. 18
IV NUMERICAL INTEGRATION TECHNIQUE ...................... 39
- I nt roduct ion ....
Domain Discretization .................. .. 39
The Finite Difference Equations ................ 43
N. Boundary Conditions ............................ 43
Stability ............... ...................... 53
,-- i
TABLE OF CONTENTS (CONCLUDED)
Section Title Page
v PRELIMINARY COMPUTATIONS ......................... 57
The Compu~ter Program......................... 57
The Baseline Case,..................I...5
Test Computations......v........... 62
Possible Follow Up ................... 65
REFERENCES............ *..***................. ..... * 68
* APPENDIX
A AUXILIARY DATA BASE AND CONSTITUTIVE RELATIONS ......... 70
r, FINITE DIFFERENCE FORM OF IIECONSERVATION EQUATIO NS......................... . .74
C COMPUTER CODE LISTING.........*~ .......... ..... ....... ooo7
iv
4I>
LIST OF FIGURES
Figure Title Page
I Damage to GP Warhead from Impact withHardened Target (Notice the assumed metal casefailure and internal high explosive fracture.] ......... 2
2 Schematic Representation of the Two DimensionalDomain with a Ring-Shaped Opening ........... 9
3 Boundary Conditions Imposed on the Two DimesionalDomain (0 < r < R and 0 < z < L)...................15
4 Typical Two Dimensional Finite Difference CellUtilizing the Staggered Grid ........................ 40
5 The Staggered Grid Applied to the Two DimensionalDomain [Heavy border indicates the locationof the boundaries while dashed border representsthe opening in the container wall.] ................... 42
6 Symmetry Boundary Conditions Applied atthe Center Line, r = 0 ................................ 45
7 Implementation of Synmetry Boundary Conditionsat the Near End Wall, z = 0 ........................... 45
8 Application of the Radiative Boundary Conditionsto the Far End Wall, z = L .................... 47
9 No-Slip Boundary Condition Employed Along withthe Circumferential Wall, r = R, for 0 < a <zand zc + AZc < z < L ...........
i0 Boundary Conditions Applied at the Opening inthe Circumferential Wall, r = R forz < Z < Zc + Zc..*.....** .$***.....t............52
11 Two Dimensional Baseline Case InitialTemperature Profile ................................... 61
12 Two Dimensional Baseline Case InitialPressure Profile ................ . . ................. 61
13 Comparison of the Flame Front Locations Predicted byInviscid One Dimensional Simulations .................. 63
v
--,
. .. ... ., .- - , . . . . . . .. -- . . - . . . . . . . . .. . . . . . .. .. ...
LIST OF FIGURES (concluded)
14 Flame Front Locus Showing the Transition from aTwo-Dimensional Initiation to One Dimensional Fl(;w....64
15 Viscous Gas Radial Velocity Profile at t = 11.33 us[Notice the severe oscillations which occur ahead oftne steep gradient.] ................................. 66
vi
... .......
SECTION I
I NTRODUCT ION
General purpose M) warhead. applied against enemy installations can
produce devastating effects. The best results are obtained through maximum
relIiability of conventional explosive penetration and detonation. However,
advances in target hardening technology have created conditions where such
warheads may fail to meet this goal. Prior to fuze initiation, metal contain-
ment wall failure and fragmentation of the high explosive can result from the
impact forces between the warhead and a hardened target as shown in Figure
1. The detonation that is expected to occur under totally confined conditions
may be reduced to an unsteady, rapid deflagration with much less damage to the
target. The principal motivation for the work presented here is the develop-
ment of a model which ca,. be used to predict whether detonation will occur
within the ruptured warhead filled with fragmented high explosives.
Understanding the fluid mechanics of flame spreading through a porous
reactive solid medium has been the subject of considerable research at the
University of Illinois at Urbana-Champaign. Under the direction of Dr. Herman
Krier, much work has been done in the analysis of these complex flows. A
strong background for the research presented in this report was provided by
the earlier efforts of: Van Tassell and Krier (Reference 1); Krier and
Gokhale (Reference 2); Krier, Rajan, and Van Tassell (Reference 3); Dimitstein
(Reference 4); Krier, Dimitstein, and Gokhale (Reference 5); Krier, Gokhale,
and Hughes (Reference 6); Krier and Kezerle (Reference 7); Butler, Krier, and
Lembeck (Reference 8); and Krier, Dahm, and Butler (Reference 9).
A 4 UAA,-L - . -
4L. 0 4,.-LCL
,,-C
X0* F
ea"
cu
s-
1
4-' Wn4
0
4)
rSaoS.a, u.
:Em
The culmination of these previous studies has been the devclopment of one
dimensional, numerical models of the transient events of deflagration to
detnnation transition. The computational representation solves the inviscid
conservation equations using finite difference techniques. The codes devel-
oped in References 7-9 can accurately simulate shock formation within the
*' fragmented propellant or explosives, and are used to depict DDT-deflagration
to detonation transition. The delay of DDT resulting from the loss of mass
through the container walls was modeled by a quasi one-dimensional anaiysis by
including sink terms in the mass and momentum conservation equations (Refer-
ence 9). Significant detonation delays for a variety of input conditions have
been calculated from this quasi one-dimensional code (Reference 8, 9).
However, the loss of mass, momentum, and energy through an opening in the
casing is a multi-dimensional phenomenon. Even though the reaction front
progresses axially through the damaged warhead, the mass is basically ejected
in the radial direction. This turning of the flow is accomplished by changes
in momentum and energy which cannot be described accurately by a one-dimen-
sional representation. A multi-dimensional model is therefore needed to
4 provide a much more accurate description of the effects of mass loss prior to
the prediction of any steady detonation. The research described in the fol-
lowing chapters is intended to provide the foundation for such an analysis.
Three-dimensional flame spreading analyses are now being investigated in
great detail by Oahm (Reference 10) and have been reported by Markatos and
Kirkcaldy (Reference 11). The Dahm version simulates flow through a frag-
mented solid explosive inside a container by solving all the inviscid con-
%servation equations. These relations are a coupled set of non-linear hyper-
holic partial differential equations which are solved using finite difference
3
techniques. However, and as expected, the numerical scheme requires enormous
amounts of computer memory due to the fire grid resolution needed and other
intricacies of the problem. Consequently, computation costs are very high.
rhe Markatos and Kirkcaldy model attempts to simulate flame spreading in a bed
. of solid propellant grains used as a gun charge. That study utilized many of
the concepts presented in Reference 3, and lacks the complexities associated
with radial venting.
This study presents the analysis for the two dimensional f'ow inside a
ruptured bed of granulated high explosive. Although requiring the assumpticn
of a circumferential crack, the model does account 'for radial components of
the flow. Therefore, it will be possible to describe the turning transitions
taking place as mass vents through the rip in the casing surrounding the
explosive bed. True non-planar burn initiation and early flame spreading
transients can also be modeled, but most importantly, the twoi dimensional
model features will also include viscous gas effects. This creates a more
realistic flow simulation by satisfying the no-slip velocity boundary con-
dition at the gas-wall interface, a process not considered by either Dahm
(Reference 10) or Markatos and Kirkcaldy (Reference 11). This greatly com-
plicates the differential equations representing momentum and energy conser-
vation, because new second derivative terms must be numerically differenced.
The complete two-dimensional model is presented in the following sec-
tion. Included are the necessary assumptions, boundary conditions, and con-
servation equations for the problem. The detailed derivation of these govern-
V... ing relations is analyzed in Section Il. Section IV deals with the manner in
which these equations are solved, by describing the numerical integration
scheme. The computer code, preliminary results, and topics for possible
future work are presented in Section V.
4
SECTION II
FUNDAMENTAL CONCEPTS FOR THE TWO-DIMENSIONAL MODEL
INTRODUCTION
The two-dimensional model described below is a mathematical representa-
tion of the two-phase (solid-gas) fluid mechanics within an impact-damaged
warhead. First, initial and boundary conditions are imposed on the problem.
Then, partial differential equations and algebraic relations describing the
flow are applied to all points in the domain. Finally, a numerical differ-
encing method is utilized to solve the system. This section will present the
formulation of the two dimensional model, while a derivation of the governing
conservation equations is examined in the next chapter.
ASSUMPTIONS
Before attempting the representation of two dimensional flame spreading,
it is necessary to make several assumptions about the flow conditions. These
are:
1. Warhead damage occurs prior to fuze initiation.
The model presented here simulates the fluid mechanics in a ruptured bomb
only and does not include any treatment of the time-dependent case failure.
The impact with the target, case fracture, and fragmentation of the high
explosive ta.e place prior to time t=O.
5
2. A circumferential crack is produced by the force of the concission.
A ring-shaped opening must be considered in order to yield a two-dimen-
sional representation of the damaged weapon. While this effectively slices
the casing into two portions, it must be assumed that structural integrity is
still maintained, and the size of the crack remains constant.
3. The explosive particles are treated as uniform spheres.
The shock of the warhead's impact causes granulation of the solid explo-
sive. The resulting fragments have a high surface-to-volume ratio, and can be
considered to be millimeter or sub-millimeter spheres.
4. The separated two-phase analysis is used.
This formulation considers each component as an individual fluid, with
the particles treated as a pseudo-fluid (Reference 7). Each phase has its own
set of conservation equations, with coupling terms to account for interactions
between the solid and the gas.
5. The particles are uniformly distributed at time t=O.
It is assumed that the spherical particles are initially packed in a
modified face centered cubic arrangement, with porosity (the ratio of gas
volume to total volume) typically ranging from 0.2 to 0.3. However, porosity
will change with time, due to combustion and rapid pressure increases which
compress the granules.
6. The particles burn in compliance with an assumed ignition criteria and
combustion rate law.
6
If a particle exceeds a specified input energy (or temperature), it
ignites and starts producing gas. The burning rate of the explosive is
described by ' = bPn with b and n being material constants. When the poros-
ity reaches approximately 0.99, the paticles are assumed to be burned out and
gas production ceases. The fuse initiation at time t=O is simulated by a zone
of particles with Lemperatures already high enough to satisfy the ignition
condition.
7. The gas phase obeys a non-ideal equation of state.
The two dimensional model utilizes a co-volume approach to account for
molecular interactions of the gas.
8. Interphase heat transfer is accomplished through convection only.
Energy transport due to conduction and radiation effects is neglected in
this analysis. Heat transfer takes place solely through convection. (See
Reference 7.)
9. The flow through the damaged warhead is generally laminar.
The major source of turbulence is the two-phase nature of the flow. Gas
moves around the spherical granules creating local disturbances and insta-
bilities in its wake. However, the fluid properties are averaged over the
entire control volume, and the turbulent fluctuations cancel out.
DOMAIN OF THE MODEL
The domain of the two dimensional model is prescribed by the geometry of
the damaged warhead. The warhead is presented as a right circular cylinder
7
having solid, closed ends. The circumferential wall is also imoermeable,
except at the opening where mass can be ejected. This crack in the casing
must he ring-shaped in order to preserve the two-dimensional nature of the
prohlem.
The domain is axisymetric as illustrated in Figure 2. The radial coor-
dinate ranges from r=O at the center line to r=R at the circumferential con-
tainer wall. The axial coordinate extends from the near end, at z-O, to the
far end, at z=L. The crack begins at z=zc and its length is Azc
CONSERVATION EQUATIONS
The fundamental relations needed for the analysis of fluid motion in the
ruptured bomb are presented here and are derived in Section III. These equa-
tions which govern the conservation of mass, momentum, and energy must be
satisfied at all points in the domain. Separated flow concepts require that a
coiplete set of these relations be written for each phase with coupling terms
- added to represent particle-gas interaction. If viscous gas effects are also
included, this forms a system of eight second order nonlinear partial dif-
ferential equations.
The relations listed below were derived in cylindrical coordinates due to
the axisymmetric geometry of the problem. The necessary interphase coupling
terms (see Reference 7) are represented by:
rc - the gas production rate from the burning particles,
D - the interphase drag,
and (1 - the convective heat transfer between the two constituents.
8
2'DINENSION'ALMOE
N.,
Figure 2. Schematic Representation of the Two DimensionalDomain with a Ring-Shaped Opening
9
Subscripts g anid p indicate the gas and particle phases respectively, while r
and z denote radial and axial dir-ections. Bulk densities are defined as:
*Pg 1
ID2 p
where the porosity, *,as stated earlier, represents the ratio of the gas
volume to the total volume. The two-dimensional conservation equations are
V (see Section 111):
Conservation of Mass (Gas Phase)
1pr a. (pw + r. (3)
Conservation of Mass (Particle Phase)
aP2 I a (~ru) rc~w~ (4)
Conservation of Momentum (Gas Phase)
Radial Direction
Tt (plug) r 7F 9 az
[+ (rug:] , (rug) (5)
10
.. N14
+-j 2 + r u - Dz "cp r ar
Axial Direction
" ( lw g (pl u g (p.w.2
Sp1. a2w ru 1 aaw]
rar az4+iT 'a r 'g 3z
+rW - D - i(Pg) (6)c p z 3z k 9
Conservation of Momentum (Particle Phase)
Radial Direction
_. _ (,2 u~p). - j (2uop)
- r up + D - . [Pp(1 -)] (7)
Axial Direction
T (02wp) r ar (P2 rUW) - (p2Wp)
" r wp + Dz -z[Pp(1 - *)I (8)
.- z1-
'. Conservation of Energy (Gas Phase)
E [PlrUg(Eg + - [PlWg(Eg +
at (p. .( 9).0)] a z~ ' p 9.)
au2~ av 2 au 9u aw
+) + ( )] _a + +
9 r 3) ar r aZ)
+ !! + -!.!!22 a aa2+ r 3) Ug [ -j (rug)] + Ug 7
az
2
u2 g w a
i Conservation ,of EnergLy (Particle Phase)
~ ('t pE ) = r u +~ p + ]- -k f rWp9) + w3p a r r r 9 zrpr a
+~I + + Dr p +D
P
P
a + - a- ()
2 2 2
In Equations (9) and (10), the total energies are dfined as:
2 12
Eg = C T (1)
U2 WEp = C T p + - + (12)
CONSTITUTIVE EQUATIONS (Closure)The conservation of mass, momentum, and energe in the two dimensional
model is described by eight nonlinear partial differential equations con-
taining eleven unknown quantities, namely: pg. pp, Ug, Up, Wg, Wp, Pg# Pp, Tg,
Tp, and #. Consequently, three additional relations are required to provide
closure for the system:
1. An equation of state for the gas phase
2. An equation of state for the particle phase
3. A procedure for determining porosity as a function of local stress
conditions.
In additon, several constitutive relations are necessary in order to
accurate'y represent the source/sink terms. These include:
1. An equation determining the gas production rate from the burning
explosive
2. An equation for the gas viscosity coefficient
3. A relation describing the interphase drag
4. A gas-particle convective heat transfer equation.
13
.... . . . -
The specific relations which were used in this work were taken from Reference
8 and are described in Appendix A.
BOUNDARY CONDITIONS
The conservation relations presented in Section II are nonlinear partial
differential equations. In general, there are a very large number of entirely
different possible solutions for such a system , Finding a unique solution
for these equations requires specific knowledge of the physical characteris-
tics of the problem. This information is provided by the boendary conditions.
For the two dimensional model shown in Figure 3, five conditions must be
specified. These are:
1. At the Center Line (r=O, O< z <L)
Since the damaged warhead is depicted as a right circular cylinder with a
ring shaped opening, the domain is symmetric about the center line. Conse-
quently, there can be no flux across this border, although the flow can move
parallel to it. All radial velocity components are set equal to zero, as are
all gradients taken across this boundary.
* For example, the functions
u = x2 - y2 u = ex cosy u - Ln (x2 + y2)are all possible solutions of the second order Laplace equation,
a 2u/ax 2 + 32u/ay2 = 0
14
a ~ . P 0__
0 C: 0 :1 N c "Moo",
0 N F avi N.. r-= &
N v .j ; L.
C; i U..c
o N ? UN C
.J0 1 < M3
00V
V N N1 O ---)-
No. +
IfI0 N
E CE 0
o0 0,
0 : 0 0toi
0-
i!!0-- -
0
E) CLr
0 0) I0 if
4- UI
15~
..... ....
2. At the Near End (z=O, O< r _R)
It is asiumed that symmetry conditions also prevail along this bound-
ary. While this would actually require the inclusion of an additional crack
located outside the domain, these conditions will also be used to represent a
. solid wall where slip is allowed. Flow can move along the boundary but cannot
pass through it, since axial velocity components and gradients are equal to
zero.
3. At the Far End (z=L, 0< r <R)
Here a radiative boundary condition is applied, allowing all disturbances
to pass through the wall as if it did not exist. The solid end is assumed to
be farther downstream outside the domain.
4. At the Circumferential Wall (r=R, 0< z <zc Zc+AZ< z <L)
This is a solid wall, requiring the radial velocity component to be set
equal to zero. However, the no-slip condition is also applied at this bound-
ary. Note that now the axial velocities are forced to zero as well.
5. At the Opening (r=R, zc < z < zc + AZc)
It is very difficult to define a boundary condition applicable at the
crack due to the unknown status of the flow beyond the opening. The radial
V velocity of the gas at the hole is determined by assuming the flow is
choked. This maximum value is located at the center of the crack, and a
linear profile is assumed, in order to lower the velocity to zero at the edges
16
of the opening. The radial component of the particle velocity can then be
found by applying the interphase drag constitutive relation as described by
Krier, Dahm, a):, Samuelson (Reference 12). The drag force on the solids
produces a change in momentum which is used to predict a particle velocity at
the tear.
The specific details of how these boundary conditions are included in the
two dimensional model are explained in Chapter Four which deals with the
numerical methods employed to solve the problem.
INITIAL CONDITIONS
LI.itial conditions represent boundaries with respect to time and are
applied at the start of the simulation, Initially, all velocities are set
equal to zero. Above-ambient temperatures and pressures are determined from
polynomial profiles, with several particles assumed to exceed the ignition
temperature criteria and thus burning. Initial gas density values are
computed from the equation of state, while the solid phase density is an input
parameter. The porosity and particle radius are also initially prescribed and
assumed constant throughout the bed at time t=O.
9.
17
-EW
SECTION III
DERIVATION OF THE TWO DIMENSIONAL AXISYMMETRIC CONSERVATION EQUATIONS
The relations describing the conservation of mass, momentum, and energy
in the two-dimensional model were presented in the previous section as Equa-
tions (3) - (10). The derivation of these governing equations is discussed
here, and follows much of the logic used by Krier and Kezerle (Reference 7)
for the one-dimensional flow process. But additional complexities due to the
treatment of a viscous gas and the cylindrical geometry of the domain will now
be incorporated into the analysis. The resulting system consists of eight
second order, coupled, nonlinear part4 al differential equations.
1. CONSERVATION OF MASS
a. Gas Phase Continuity
Consider the control
4 volume shown, with gas
flowing through the control
P2 L surfaces
18
Si nce,
- Rate of Mass Rate of Rate of Rate of Mass
Accumulation Mass in - Mass out + Generation
Then,3(PgVg (p"gA, (pgUgAlg wr
at (PgUgAg)r" )r + Ar+ (pgwgA2g)z
- (pgWgA2g)Z + +rcV (13)
where: Pg = gas density
Alg = cross sectional area of gas-radial direction
A2g = cross sectional area of gas-axial direction
Vg = volume of gas within control volume
V = total volume of control volume
Ug = radial velocity of gas
Wg = axial velocity of gas
rc = gas generation due to combustion of solid particles
From the geometry of the control volume:
V =r Ar Aq Az
A1 = raeAz (14)
A rArAo
J'- VNow with the definition of porosity *
19
A Ig = r~o~z (15)
A 2g # rArae
Substitute back for Al., A2g, Vg, and V to get
at -p (Pgu+r,&eaz) r (p 9 u9 rae,&z) r+Ar
+ (p 9w 9 r~rAe) Z- (p 9w 9 r&r&) Z+A + r arAeAz (6
Letting *pg z p and dividing by r~r,&o~z gives
a 1 (p1 u gr)r- (Plugr)r+Ar (p1 W)- (PIW
(17)
Now take the limit as ar, Az + 0 to get the final form of the gas continuity
equation
rpr (p~u r) - -1(plw) + r (18)
20
b. Solid Phase Continuity
From a control volume analysis similar to that used for the gas phase, one can
write:
a(p Vp)
at pp Ip r- (PPuPAIp) r+Ar
+ (ppwpA2p)z - (ppWpA2p)Z+AZ - rcV (19)
Here pp = solid particle density
Alp = cross sectional area of particles-radial direction
A2p = cross sectional area of particles-axial direction
Vp = volume of solid particles within control volume
Up = radial velocity of particles
Wp = axial velocity of particles
Notice that the rcV term is negative here, since the mass of particles
decreases as gas is generated. Since
V V -V 1 V(.P 9 1(
Therefore
I.V = (1 - *) rar&ez
A 1P (I-,) rAeAz (21)
A2p (1-.) rArAo
21
Now substitute for Alp, A2p, Vp and V to get
- t 0 [rrleAz u ) rAAz~r [ppUp(1 - *) rAebZlr+Ar
+ [pW(l - *) rAraO1z - [PpWp(1 - ) rAr&O]z+,z
- r rAreAz (22)
letting (1 - *)Pp - P2 and dividing by rAuAeaz gives,
3' P2 I (P2upr) - (P2upr) r+Ar (pWp)z (p z )+z+ -z r,:at r [ Ar Az " rc
(23)
Again taking the limit as &r, &z + 0, the final form of the solid phase con-
tinuity equation is:
82_- 1 a (p upr) - -(pep) - rc (24)
-a r ar 02ur p -a(0w)
2. CONSERVATION OF MOMENTUM
a. Gas Phase Momemtum
In order to include the gas viscous effects, it will first be necessary
to consider the distortion of a moving fluid volume in three dimensions. The
reader is referenced to any of a number of graduate level textbooks on fluid
mechanics for detail.
22
'k
Extension Strain:
Radial Direction
dr + -- ) dta dr -dr
I t 4jfJtdrdt3r 'at, a (arr "t, dr JU-
£rr ar ar (25)
Azimuthal Direction
rde + -A (r-1-) dt de- rdeC7dtae at)
rde
Y, 1 a (r -er dt
coo To- at)r aeat eat
+ 1 avr r ae (26)
Axial Direction
dz + -i (ML) dt dz dz4' dt= 3Z at a (z - 3
, dz = 4 abt
-' I II! I _ ' _z= .
. 3 .M 2 .w (27)4Z III 22 at 3z
Shear Strain: r - e Strain
N £redt = [TAN.1 r dr
23
+ TAN-1 39 a) dtdterdo
[r -A (1) + ] (28)
o - z Strain
dt = 1 - "A (N-)_ dtdoE dt =- [TANrdo
M(*)t do ez 2 d
r -TAN 1 dtdzdz
: " 1 aw av
9.,-+,Z (9
z - r Strain
a ar dtdzCrz dt = [TA NC dz
dtdrSTAN- I dr
-
Cr [.L + LW(30)
Since it is assumed that the gas phase behaves as a Newtonian viscous fluid,
the stress tensor can be written as:
24
Tij = -2 ucij - A kk6ij (31)
,where x - j by Stoke's hypothesis and 6 is the Kronecker delta.
Consequently the components of the stress tensor are:
T. ., = - 2 -L a ( r u ) + + Lwr -ar 3 r ar r a -
V I
T19.= - 2 u + - _ I a(ru)+-Ay+ Ir-% r r 3 r ar r a Z)
aw 22- .? 1 rul + 1 av + 1)
zz P~ az 3 'r ar r ae 3
Tr r~ + - -ar~(-. u
=T 3e r a
T1 zT [.W+2- (32)zr rz " Oz-
Now for the two-dimensional model, the azimuthal gradients and velocity coef-
ficient are zero. Hence the stress components reduce to:
(2 aw 2 1 a(ru)+ _W
ZZ 3Z 3 r ar
25
Ire Tor 0
T.W 3U+!(
.zr rz ar (33
(1) Gas Phase Momentum-Radial Direction
To determine the changes in momentum in the radial direction, consider
the forces acting on the control volume:
S-
,a.
-; 26
i--S=
Rate of Rate of Rate of Sum of forces acting on
Momentum = Momentum - Momentum + the control volume including:
Accumulation In Out Shear Forces
Gas-Particle Interaction Forces
- Pressure Gradient Forces
Forces associated with mass
addition or loss
-= (Pgu g) gUg A1g)r+hr +(gugWgA2g)z" pguggA2g)z+&z
SrrAl) (TrrAi)r+ar+ 2(teA 3)sin(-A)
+ (trzA2)z (TrzA2)z+Az - DrV
+ (PgAlg)r - (PgAig)r+hr + 2 (PgA 3g) sin(-) + ru pV (34)
Introduced here are: Pg = gas phase pressure
Dr = radial component of interphase drag
A3g = cross sectional area of gas-azimuthal direction
It is assumed that the pressure gradient force acts through the area of the
gas only. It is also envisioned that combustion takes place at the particle
velocity, imparting momentum to the product gases. This is accounted fcr by
the rcupV term.
Like the earlier analysis of the geometry done in Equations (14) and (15)
27
A3 =ArAz (35)
A3g * ArAz (36)
Now since Ao is small, then sin (__) . * Substituting this and the rela-
tions in Equations (14), (15), (35), and (36) into the momentum Equation (34)
yields:
~jap rrAeizu ~ 2 2at _ - (pg9ug9 rAgaz )r - (P9 u 9 rhAzAZ
+ (pgugwg rAre)z - (pu gg,rArAe)z+Az
+ (rr rAeAz)r - (T rrrAeAZ)r r
rr r rz rA" +Z 2('ee"r'z)-' + (TrzrArAO)rI
- (' rzrarao)r+Ar - DrrArAGz
+ (Pg+raz)r - (Pg#rAeaZ)r+Ar + 2 (Pg#ArAz) Ae
+ rc u prArAeaz (7)
Rewriting Equation (37) in terms of bulk density P, = pp and dividing by
volume rarAeAz gives:
(plU -) 2 ( )- (pIU )r+Ar (pjw U >z" (ptw U )z+z
" = I -PI'- S.[AI
28
1 _ Trrr r - (rrr)r+Ar + + Trz Z rzZ+AZAr r &Z
1(P~r 40 (P +r) r+&r ,g D + r r r + r + r up (38)
-r-r9
Now take the limit as ar, Az + 0 and simplify the pressure terms:
(Pl r) (l ) TP+!Trz
at ,rPugr) - (p1U9 Wg) - [ r (rrrr) - r
Dr - * (Pg*) p (39)
Substituting the results from (33) for the stress terms and rearranging yields
the final form of the radial direction gas morentum equation:
- (p,u'r) - pl +~ Iigij**+at r Tr 1g -z (P4Ugzg) + g r ar g)I + z
+ fl - (rUg) + !!1 + rcu- D - 1 (Pg0) (40)+ r ir ar " z r ar
(2) Gas Phase Momentum-Axial Direction
Again consider d :ontrol volume but now examine the forces acting axially
on it. (See figure on following page.)
"at (pguwgAg r (PgugwgAlg)r+Ar +
29
9w A2)Z- (PgwWLA 2() ~ + (z A'O
:44
rz 1r+Ar+ (tzzA 2)z- (TzzA2)z+az
D Z + (PgA2 P gZ4
+ r w V (41)
where: D = axial component of interphase drag
As done previously for the radial momentum component, substitute geomet-
ric relations Equations (14) and (15), divide by the vol ime rsrseAz, and let
Pl = *Pg to get:
3(P 1 (puwr)- (puw r)r+ar (P2 2)Z"at 1 (14 r lagrhr I z ( 1
(rrzr) (t r)r+r (T ( )
+ A r rz tr+[ Z zz Z+Az ]
(Pg)z (P Z +z] + rW (42)-z +[ az +c p P4Z
Again let &r, Az * 0
__(PlW9) 1 3 1 w 3 rA a 2 1 z ]
a at = -n r '- Pg'gr " i (P g) - [ : 9 r (rz r +--
D z *(Pg)+ rcWp (43)
30--S..
Now substitute in the stress terms from Equation (33) and rearrange to
get the final form of the axial direction gas momentum equation:
3(P.wg) I1(Pluwr)- -1a 2 1 a r 2 Wat "-g r " (oWg) + g- -E r) + a z
az
+ u z 1 + r w D --1(P) (44)
b. Solid Phase Momentum
The derivations of the solid phase momentum equatior; are very similar to
those done earlier for the gas phase, but the complexities of viscous shear
are not included. Instead, interphase drag forces become source terms while
combustion effects are now nomentum sinks. Finally, assume that solid phase
pressure acts through the particle area only.
(1) Solid Phase Momentum-Radial Direction
From a control volume approach similar to that used for the gas phase:
21221p = (p~u2A1 )r" (p-u2A p)r+Ar+ Aat p p lpr-p p P rr = ppp p2p
- (PpupWpAzp)z+z+ D rV + (P pAp)r
-(Pp Alp)r+ar + 2 (PpA3 )sin(A-) - r upV (45)
with: Pp = particle phase pressure
As in the geometry analysis done in Equations (20) and (21)
31
A (-)Araz (46)3p
Now substitute for A A , A~ )1 an_.Lt i A to ban
a[P(-) rArAeG&zuI[puj*)ezr+f
D' [Ppu 2 *rAA~
+ Or~r~I~z+ [P(1-)ree zraoz [PP(l*)r orAe~
at r-rAGz)
at~~ r +A] [Ap+Z
+ [P[u w (- )rvjz [p (-*
+D 2[ Ar e~ (48)
Nowrtkte litit(7 is erms oZ bu0landeimpiy the presure term tovige the
I. a ( P~u 1 92 p u 2 r ) - (P2uw) 2 r ~ ~ +D [~1.I (9
p pP )~a] +[(Pu~p Z (32
atr4.A PLA
(2) Solid Phase Momentum-Axial Direction
Again using the control volume approach, the axial momentum equation is
written as:
p(p V w ) 2at p p PIP (ppUpWpAip)r+ar+ (PpwP)z- (Pp p)z+AZ@t =(pUpWA p) ppr-ZA
+ DzV + (PpA2p)z- P pA2p)z+,Z- rcw pV (50)
Substitute geometric relations Equations (14) and (21), divide by the volume
rArAOAz, and let P2 (l-)Pp to get:
a (p2wp) - p r) r- (P2 w r) r+rA _[_ _2 0 )z" (P2p2 )z+AZ]"at --r I'Lr 1+Az
[Pp(1-0) ]Z [Pp(l-0) ]z+&z}+ Dz + I z-. 1z rcw p (51)
Take the limit as ar, Az 4 0 to arrive at the final form of the axial direc-
tion solid phase momentum equation:
a(~~2w ~w r ~) 2)±(w r w + D--2 [P (1-*']*at r r Pupwpr 'al 'P2wp c p z_ az p
(52)
33
3. CONSERVATION OF ENERGY
a. Gas Phase Energy
til; 4,0
At From the first law of
thermodynamics for a viscous
gas passing through the control
surfaces, we know that:
Rate of Energy Rate of Energy Rate of Energy
Accumulation In Out
Heat addition Work done by the
+ from + surroundings including:
combustion or pressure, viscous shear,
convection and interphase drag
Thus,
at (Pt (gUgA 1gEg)r" (P u A gE r+Ar
+ (PgwgA2gEg) z- (a gg2gg)z+Az- OV
34
;!2 w2
(EceA 4r !, + ( A Uc g 22 (P9 1gUgr
(PgAigUg)r+Ar+ (PgA2gWg)z- (P 9A2gWg)z++z
* + (TA u ) (TA u ) + (T zzA 2W) z- (zzA2Wg)z+Az
+ (T rzA lwg)r- (r rzA2U) z (rzA 2 Ug )z+Az
- DrupV - z wzpV (53)
2 2
where: Eg = total gas energy CvgTg +-q +
=rate of convective heat transfer from gas to solid
E Egc he m = gas chemical energy released during combustion
Now substitute in the geometry relations of Equations (14) and (15) and divide
by the volume. If p, Pg#* energy Equation (53) becomes:
. a(pjEg) 1 (P1U gE gr)r - (plu E r)r. + (p WgEg) z- 6W 9E zat r ar 3z
.-ch + P + ( g )r" (P +ur) rrC+ - ( E + c9 2 2 [ r ]t-
+ " 9 Z - .+ 'I (T rr Ugr)r" "( r _ ~~
k Iw (T~ W1 (Trwur) -(T u r)
1 q zz Z+l + r...+..- +:+ I A+ z [_.." g r 1 ~ r r (rz gr r+Ar]
j5
+ [ ( T 'rzz ) , , .r rz U - )Z+Az) - D U - 9,w (54)&Z rp z (54
Combine energy and pressure terms and take the limit as Ar, az + 0 to get:
at r rar [plrug(Eg+ -2 j [PWg Eg+ Iat T 99 9 [lgEg g
[!! (rrU r ) +--(t w ) + 1 - A( r ) + A (TrzUg)
: 2 2
OUp wp C 9 2 2(55)
Substitute Equation (33) for the stress terms and take the derivatives.
Rearrange to get the final form of the gas energy equation:
a(pE 9) 1 -i[r E
at T ar - I -ruOlg(Eg~ ) 9 E+ A~
au9 2 awg 2 2 a+ 2+ 1g2 _ ar)2+ (__%+ I :a - 3:.mUr %z
9ia az r) ar) Tgl r rg4
2 2
uw- au 9.2 au
3ar -9 zr g a r rar ar
3 az rar (u rp zp22
+ a~ (Ech + D (U6
2 2+r36
• -" • " m m i m v m 1" . .. • I m i , " ' . . . .
3
Al
b. Solid Phase Energy
The solid phase energy equation is derived using a conrol volume analy-
sis similar to the one used for the gas phase. Once again, complications from
particle viscosity are neglected as they were in solid phase momentum equa-
tions.
a(p V E.)*at - u (UAlpEp )r- (ppUpA1oEp)r Ar* p ppA2pEp)z
(.pw A2 E)z + Qv + r - -che u
+ (PpAlu ) (PApU)r+ r+(PA W)p pr- p lp p ra p 2p Pz
(PpA2pwp)z+Az ' U V+DWpV (57)
2 2
where: E = total particle energy CvT + +
p h particle chemical energy released during combustion
Substitute in the ceometry Equations (20) and (21) divide by the volume,
and let P2 Pp(1-t) to get:
a(P2Ep(PU r) r)(P 2 WpEp)z+P2 PE r- (P2UpEp r A (P2 p p z- Pw +A
at =- Ar rr + Az -
rc(E hem_ _ 1 [P (1 )u r
p 2 2 + r
37
'-?..- S-~-
+ {[P p (1-#)w PIZ -[Pp(I'$)WP]z+-Z-I + Dru + D w (58)
A Z r z p
Combine pressure and energy terms and take the limit as Ar, Az + 0 to yield
the final fom of the solid phase energy equation
a(pE) P
at T [perup(Ep+ _Ep2
- P p, p [ 2wp(E p
: ' - "Eche - uP )(g+ D u+ + DrUp+ zWp+ o (E p 2 (59
.38
J-
)%, 38
------------------------------------a----.-- ~
SECTION IV
NUMERICAL INTEGRATION TECHNIQUE
INTRODUCTION
In the previous sections, the mathematical foundations of the two-dimen-
sional model were derived and discussed in great detail. The system of par-
tial differential equations, constitutive relations, initial conditions, and
boundary conditions form the basis of a well-posed problem, indicating that a
solution is possible. However, analytical methods for solving this set of
complex, coupled, non-linear partial differential equations do not currently
exist. Therefore, a computational numeric method must be utilized in order to
determine the solution of the system. The implementation of such a tinite
differencing scheme is discussed here.
DOMAIN DISCRETIZATION
Before a numerical integration technique can be used to solve a system of
partial differential equations, the domain of the problem must be re-
defined. The continous region is replaced by a set of controi cells, which
represent locations where values of dependent variables are stored. These
discrete noaes make it possible to approximate derivatives which would ordi-
narily be unknown.
The two-dimensional model utilizes a staggered grid to represent the
domain which was described in Section II. in this arrangement, velocity
components are calculated at points that lie on the edges of the control cells
while scalar quantities are defined at the centers. As shown in Figure 4,
39
Jz
0 Radial Velocity (U) Node
a Axial Velocity (W) Node
+ Pre.sure, Temperature, Density,Porosity Node
Figure 4. Typical Two Dimensional Finite DifferenceCell Utilizing the Staggered Grid
40
--- Y-1.- ' l j T W -4
radial velocities are determined at the sides that are normal to the r direc-
tion, and axial velocities are calculated at faces perpendicular to the z
direction.
Figure 5 illustrates how the staggered grid is implemented in the two-
dimensional model. The mass continuity equations (Equations (3) and (4))
which are used to determine p, and P2' and the energy equations (Ecuations (9)
and (19)) which are used to find E and Ep, are solved at the scalar nodes.
The radial momentum equations (Equations (5) and (7)) which solve for ug and
u p are applied at the displaced radial velocity nodes while the axial momentum
equations (Equations (6) and (8)) that determine wg and wp are employed at the
* staggered axial velocity nodes. The grid is -.rranged so that the boundaries
of the domain pass through sites containing normal velocity components, al-
though the entire mesh extends beyond these borders. These exterior nodes
will be shown to be very useful in dealing with the boundary conditions in
finite difference form.
There are three important advantages to be gained by using a staggered
grid instead of a conventional one. First, it provides second-order accuracy
for the centered space finite differencing used here. Because flow informa-
tion is known at more locations, due to the displaced velocity nodes, deriva-
tive approximations can be made over smaller distance, as explained by Dahm,
Samuelson, and Krier (Reference 12). The second advantage is that the pres-
sure difference between two adjacent nodes becomes the natural driving force
for the velocity component located between these grid points. According to
Patankar (Reference 13) this feature eliminates the problems which can arise
in the momentum equation due to a wavy pressure field. Finally, as will be
shown, the staggered grid makes the necessary boundary conditions much easier
to implement in finite difference form.
41--.44
4.44
w-Iz
0000t~ w U (D
to 49 0) = =
Z 0.
* ZL
4...
+ ++
cu2(
+ wSI
THE FINITE DIFFERENCE EQUATIONS
Once the domain has been discretized, the continuous conservation equa-
tions (3) - (10) are cast into finite difference form using the "leapfrog"
scheme (Reference 14). This centered time, centered space, explicit method
evaluates unsteady terms at time levels t + At and t - At, while the fluxes,
sources, and sinks are determined at the current time t. Spatial derivatives
are taken in a similar manner, by evaluating properties it grid sites adjacent
to the location in question -nd dividing by the distance between the nodes.
The finite difference forms of the conservation of mass, momentum, and ennergy
are listed in Appendix B.
The leapfrog scheme was chosen for the two dimensional model based upon
the successful results reported by other authors. Kurihara (Reference 14)
showed that this method could be used to integrate the time dependent waveequation. Williams (Reference 15) applied this scheme to the three dimen-
sional conservation equations for an incompressible, single-phase, viscous
fluid. In addition, Dahm (Reference 10) is currently obtaining successful
results by employing this technique to solve the two-phase, reactive flow
equations in three dimensions, a problem similar to the one being investigated
here.
BOUNDARY CONDITIONS
When partial differential equations are applied at a boundary of the
4.' domain, special conditions must be defined in order to yield a unique solu-
tion. This is also true when dealing with the finite difference form of the
equations. Hence the boundary conditiois must now be stated in a finite
difference configuration.
41
- r6 .
2%
Recall from Section II that there are five boundary conditions which must
be specified for the two dimensional model. Each one is considered sepa-
rately:
1. At the center line (Figure 6)
The symmetry conditions applied along this boundary specify that all
radial velocity components and derivatives in the radial direction must vanish
at the center line. Consequently, the staggered grid is applied so that the
u-velocity nodes are positioned on the boundary with their values initialized
to zero. Since only the radial momentum equations are evaluated at these
points, halting the integration prior to reaching the center line leaves the
u-velocity components unchanged. Their values remain at zero and thus satisfy
the boundary condition. Note that this procedure eliminates the problems of
evaluating the lir terms as r 0 0. The singularities which normally occur at
the center line do not have to be dealt with, since no computations are per-
formed on the nodes at this location. The values at these grid points are
preset to zero and then left alone.
It is also necessary to be able to evaluate the continuity, energy, and
axial momentum equations along the boundary, although the nodes containing the
axial velocities and the scalar quantities do not lie direccly on the border
at r = 0. Consequently, the symmetry conditions are applied by using reflec-
tion points about the centerline. Recall from Paragraph 2 and Figure 5, that
the mesh of grid points extends beyond the limits of the domain. The values
stored in these exterior nodes are set equal to their adjacent counterpart
directly across the boundary. Therefore, all derivatives taken radially at r
- 0 are equal to zero and satisfy the boundary condition.
44
~~ U 1 0 ~CENTER
REFLECTED
REFLEC.TGDAXIALVELOCITY NODES
Figure 6. Symmetry Boundary Conditions Applied atthe Center Line, r =0
REFLECTED ., -RADIALVELOCITYNODES+
REFLECTED a.- .SCALAR-NODES
0 Ij
V SYMMETKYWALL
Figure 7. Implementation of Symmuetry BoundaryConditions at the Near End Wall, z =0
45
It should be pointed out that the two dimensional model requires that the
values stored in the reflected nodes consistently match those located within
the domain. This is due to the existence of the second derivative, viscous
force terms in the gas phase momentum equations. Dahm (Reference 10) explains
how the radial flux terms are automatically set to zero when a staggered grid
is applied. However, because he is solving inviscid conservation equations
containing only fluxes, sources, and sinks, he can store arbitrary values in
his external nodes, since eventually they get multiplied by radial velocities
known to be equal to zero. But viscous gas effects are due to forces rather
than fluxes, and the second derivatives are not multiplied by these velocity
components. Consequently, the symmetry of the reflected nodes must be main-
tained in order to provide an adequate boundary condition for these higher
order terms.
2. At the near end wall (Figure 7)
Symmetry conditions are also applied along this end wall. Here the axial
velocity components are set equal to zero, as are all gradients taken across
the boundary. The displaced w-velocity nodes lie directly on the wall and are
initialized to zero. These values remain unchanged since the axial momentum
equations are not applied at these grid points. Reflected external nodes are
again implemented to provide symmetry conditions for the second derivative
terms.
3. At the far end (Figure 8)
In crder to allow flow disturbances to propagate through this boundary, a
radiative end condition was utilized. This involved using a Taylor series
46
0~ ~ NODESPKEDICTIED
i -40 BY TAYLOR0u. .j 'a SERIES AND
ONE - SIED
0--
RADIATIVEBOUNDbARY
Figure 8. Application of the Radiative BoundaryConditions to the Far End Wall, z L
-FALSE' EXTEPIOR NODES
I J '\ ~ I . NO-SLIP1.9 A. 11'SOLID WALL
. oI. . ' o,.,WinO
: . lij- + , , +,
1"j. "" i -+,
Figure 9. No-Slip Boundary Condition Employed Alongwith the Circumferential Wall, r = R, for0 < a < z and z + Az < z < L
,' 47
-k
expansion with one-sided finite differences to determine the quantities to be
stored in the external nodes. These predicted results are then used in the
conservation equations to calculate values inside the domain.
For example, assume that "A" represents a generalized dependent variable
which is stored in either a scalar or a velocity node. As illustrated in
-^ Figure 8, the nodes at (j), (j-1), (j-2), and (j-3) are located inside the
domain while (j+1) is an external grid point. Using a Taylor series expan-
sion, the value of A at the outside node is:
A -A. + AZ+ z + 2 az2 (60)
One-sided differencing is used to evaluate the derivatives in this approxi-
mation:
aA 3A.- 4A. +A
-I z 2 -z (61)
2 2 A 2A- 5Aj. + 4A .- A .z2 -AZ) Z _73 (62)
which when substituted into Equation (60) yields the predicted value to be
stored in the exterior node:
7A,- 9A. + 3A A,.A j+I=j-2 J 3 (63)
48
4
.* 4. At the circumferential wall (Figure 9)
Recall that this is a solid wall where the no-slip condition is imposed
upon the flow. Once again the normal, radial velocity nodes are positioned on
the boundary and the values are intialized to zero. Integration of the radial
momentum equation is halted prior to reaching these points, in order to leave
the u-velocity components unchanged during the numeriral simulation.
However, the no-slip requirement can not be imposed through the staggered
grid, but must be incorporated into the finite difference scheme. While the
boundary conditions force the axial velocities to be equal to zero at the
wall, the grid points which store these quantities are located elsewhere, as
illustrated in Figure 9. Hence the axial velocity nodes can not simply be
initialized to zero and then left alone. Instead, their values must be deter-
mined from the discretized z-momentum equation, with special considerations
made for the solid wall and the no-slip conditions. This is done by using
"lopsided" finite differences for approximating the radial derivatives repre-
senting viscous gas effects.
For example, assume that the axial velocity gradient - must be deter-ar
mined at node (i) as shown in Figure 9. At any other boundary, the derivative
could be determined by utilizing the centered difference of:
w -waw i i+1 i-i 64ar i" Ar (
This type of approximation is possible because the velocities stored outside
the domain at (i+1) are known from symmetry or predicted by a Taylor series.
However, the values in the exterior nodes along the solid wall are unknown, so
49
.5'
center differencing can not be applied. Instead, an imaginary axial velocity
node is created on the boundary at (I +1/2), and a lopsided approximation is
used such that:
4wi - 3wi- wi(6-r l" 3mr(65)
ar 3&r
A derivation of this formula can be found in a variety of textbooks dealing
with numerical integration methods. Now the no-slip condition can be imple-
mented by setting the value of Wi +i/2equal to zero to get:o2
aw -3w i-iI° w-Ii(66)
ar * ~
Following a similar line of reasoning, second derivative values are approxi-
mated at the solid wall as:
-12wi+ 4w-16
ar2 3(Ar)
Viscous gas terms, consisting of gradients evaluated in the axial direc-
tion, require no special treatment at the solid wall. Centered space differ-
encing is used to approximate these derivatives, since all the necessary grid
points are located inside the domain.
The radial flux terms in the conservation equations are satisfied at the
boundary by utilizing "false" exterior nodes, as described by Dahm (Reference
10). Recall that grid points located beyond the solid wall were not required
for the evaluation of the gas viscosity terms, since lopsided differencing was
employed. In addition, when a flux is determined across the boundary, the
so
quantities stored in these exterior nodes are multiplied by radial velocity
components, which are known to be equal to zero. Consequently, arbitrary
values can be stored in these false exterior nodes in order to facilitate the
integration scheme.
5. At the opening (Figure 10)
The flow conditions imposed at the rip in the circumferential wall are
based upon an assumed linear profile for the radial component of the gas
velocity. These values are stored in the boundary nodes at (i+I) shown in
Figure 10:
,,a z - z c FO A<z< z _c
gi+ = Az FOR Zc <z.< zc (8
z - (zc+ &zC ) AzC
u a i Z FOR Zc+ -r < Z < Zc+ AZc
Here aI is the gas speed of sound, which is a function of local temperature
and pressure below the hole:
Ca 1+2bP )(+b (69)
Cvg9g i Ig g
This formula is derived from the non-ideal equation of state in the paper by
Dahm, Samuelson, and Krier (Reference 12).
In order to predict the values of the particle velocity components normal
to the opening, the interphasE drag constitutive relation was utilized. It
was assumed that this force acts to pull the particles through the hole, such
that:
51
NODES PREOICTED 8B' TAYLORSERIES Aib, ONG-SIbED DIFFERLNCES
,IHOLE- - : g I / It.t,.j, BOUNDARY
- RADIAL VELOCITY VALUESDETERMINED FROM LINEAR IOFILE
tJ
" ~ O--- 0. 4,,
iij
t-2,j- O-..j O;.z,j.I
.3-;,J4.1 + S'
Figure 10. Boundary Conditions Applied at the Openingin the Circumferential Wall, r = R for
<z < ZZ + AzC- - C C
52
(P2U)tt=At = t D r (2wr 2 + (P2 up)tt (70)
The reader is again referred to Reference 12 for a derivation of this rela-
tion.
The remaining scalar quantities and the axial velocity components are
determined by using a Taylor series predictor, similar to the one derived for
the end wall radiative boundary conditions. If "A" is a generalized dependent
variable, then2 '.
-7A.- 9Ai_1+ 3Ai A
A 7A i-i i-2" Ai-3 (71)i+1 2
These predicted values are stored in the exterior nodes at (i+1) as shown in
Figure 10. The conservation equations are then applied to the grid points at
(i) using the standard centered differencing techniques.
STABILITY
_' The stability of the finite difference scheme is always a problem when
using numerical methods to integrate partial differencing equations. Solu-
tions which oscillate or grow without bound can develop due to the necessary
mathematical approximations required to discretize the system. But steps can
be taken to minimize the effects of these inherent instabilities. The metho-
dology used for the two dimensional model is discussed below.
-Recall from Section IN that the explicit, centered time, centered space,
leapfrog technique wds used to cast the conservation equations into finite
difference form. In any scheme such as this, a bound exists on the maximun,
53
size of the time step. If this limit is exceeded, oscillations will soon
develop and the predicted solution will become unstable. The conditions which
determine this maximum time step can be found from the Von Neumann criteria
which is described by Richtmeyer and Morton (Reference 16). However, the
complexity of the governing system of equations makes a complete analysis of
this kind extremely unwieldy and difficult. Consequently, a trial and error
method was utilized to find a suitable time step, although a Von Newmann
stability condition should probably be derived in the future.
Another source of instabilities is called time splitting. Although this
is a direct consequence of the centered time differencing used in the leapfrog
scheme, this method was selected because Kurihara (Reference 14) showed that
% it provided less damping of the kinetic energy. If "A" is a generalized
: dependent variable and f(rz) represents the fluxes, sources, sinks, and
viscous gas terms, centered time differencing can be represented schematically
as:
At+At_ At-At t2t = f(r,z) (72)
Rearranging the terms gives:
A t+t = 2&t f(r,z)t + AtAt (73)
Notice that oscillations can develop if the value of At+At is less than the
value of At . These instabilities grow until eventually, two separate solu-
tions form at the even and odd time steps. However, this problem can be
controlled through the use of an intermediate smoothing step. The time filter
developed by Robert (Reference 17) was used for the two-dimensional model:
54
A*t+At 2At f(r,z) *t + At at (74)
At A*t + c(A*t +ht - 2A*t + At- At) (75)
where the starred terms represent unsmootid quantities, and c is the filter
parameter. First, Equation (74) is used to predict values at time t + At, by
employing centered differencing and the unfiltered quantities at time t.
Then, the smoothing step of (75) is implemented to produce a filtered value
at t. Finally, the iteration moves to the next time level and the process
repeats. This method has been shown to provide excellent damping of the time
splitting oscillations of the leapfrog scheme (Reference 18).
-Instability due to the finite differencing of non-linear partial differ-
ential equations appears through the mechanism of aliasing. This misinterpre-
* tation of the high frequency waves is explained in the paper by Krier, Oahm,
and Samuelson (Reference 12). Aliasing is controlled by evaluating the con-
servation equations as derived, in their flux form (as opposed to the convec-
tive form with expanded derivatives) (Reference 19).
SThe development of shocks in the computational domain is another source
of instability in the numerical integration. The extremely steep gradients
produced in the vicinity of the wave can cause explosive growth of the solu-
tion. The viscous gas effects do help to reduce the shock discontinuities,
however, the second derivatives cannot provide sufficient damping of these
strong pertubations. Therefore, artificial diffusion terms a,. introduced
into the conservation equations in a manner similar to the one described by
Sod (Reference 20). In order to ensure stability, It is necessary to evaluate
-55
these quantities based upon values calculated at the previous time step (Ref-
erence 21). The magnitude of the artificial diffusion terms is determined by
numerical experimentation.
6'4..
-- t
'.
4' "
.4 4:
I...
SECTION V
PRELIMINARY COMPUTATIONS
THE COMPUTER PROGRAM
The fluid dynamics concepts and numerical integration techniques which
are described in the previous chapters were incorporated into a FORTRAN-V
computer program called DDT2D. This algorithm features:
1. An external data file
This enables a user to vary the irnut parameters without having to re-
compile the entire program, resulting in a lower overall computation cost.
2. A variable time step
For each integration, the distance between two adjacent grid points is
divided by the local sound speed in the solid phase. That result is then
multiplied by an input factor to yield the size of the next time increment.
This feature provides added stability and produces better results when steep
gradients develop. The DDT2D program automatically reduces the time step so
that the rapid changes occurring near a disturbance can be accurately repre-
sented without generating oscillations in the solution.
3. A choice of one or two dimensional burn initiation
The user can selict from two polynomial profiles for the initial temper-
ature distribution in the solid phase. One option leads to a one dimensional
57
initiation, by producing an area of burning particles along the entire near
end wall. The alternative generates a localized ignition zone at the r = 0, z
= 0 corner of the domain, resulting in a two dimensional problem.
4. The ability to switch off the viscous gas effects
- . The viscous force and dissipation terms in the gas momentum and energy
equations can be set to zero by simply changing the value of an input para-
meter. This feature is particularly useful when examining the effects of gas
viscosity on the flow near the opening in the outer wall. An identical but
inviscid case could be used to highlight these aspects by providing con-
trasting results.
These features make DDT2D a versatile program which should be easy to use. A
listing of the current version of this code is given in Appendix C.
.. The DDT2D program requires a great deal of computer memory due to the
complex nature of the fluid dynamics problem. At each node, it is necessary
to store the values of eight primary variables at three time levels, as well
as the values of 18 secondary variables at a single time level. For the CDC
CYBER-175 corouter currently in use at the University of Illinois at Urbana-
Champaign, the maximum core memory available is 131,000 words. After allrwing
4' enough space for the program itself, the remaining storage limits the size of
the computational domain to under 3000 nodes. While this should be sufficient
for most cases, additional memory can be obtained by using the more expensive,
off-line storage techniques.
58
I
THE BASELINE CASE
All the tests descrihed in this chapter were conducted for a single set
of fluid characteristics and initial conditions. These input parameters for
the two-dimensional baseline case are listed in Table 1.
Recall from Section II that the initial temperature and pressure condi-
tions were prescribed by polynomial profiles imposed over the domain. The
ones utilized for the two-dimensional baseline case are shown in Figures 11
and 12. Notice that at several points located near r = 0 , z = 0 the temper-
atures satisfy the ignition criteria and the particles are assumed to be
burning. If a one-dimensional initiation is desired, a similar set of pro-
files are used, with variation occurring only in the axial direction.
The baseline case does not include any mass loss through the container
wlls. This was done in order to simplify the problem during the computer
program evaluation stage. When the computer code can accurately simulate
flame spreading in a totally confined bed, the opening in the circumferential
wall can be re-introduced into the problem.
45
.1"
59
1%
TABLE 1. BASELINE CASE INPUT PARAMETIRS
Parameter Symbol Nominal Value
Length of bed L 3.0 cm
Radius of bed R 0.5 cm
Particle radius r 100 Jrmrp0
Porosity 4 0.30
Particle density po 1.675 g/cm 3
Ambient gas temperature T 300 K
Ambient particle temperature TPO 300 K
Ambient gas pressure Pg 100 kPa
Ambient particle pressure PPo 100 kPa
Particle ignition energy Eig n 4.621 x 109 erg/g
Gas viscosity 1.8 x 10 "4 poise
Prandtl Number Pr 0.7
Gas constant R' 2967.9 erg/Kggmol
Gas phase specific heat Cv 1.51 x 107 erg/g'K
Particle phase specific heat Cv 1.51 x 101 erg/g*K
Particle chemical energy El m 5.74 x 1010 erg/g'K
Burning rate index n 1.0
Burning rate coefficient b 36.81Cm/s)I[dynelcM2]n
Gas equation of state coefficient bI 4.0 cm3/g
Axial space increment Z I mm
Radial space increment Ar 1 mm
Time filter coefficient £ 0.25
60
I I I I-----------------I I Y-.--, I I-- - I I -I I I * -- -- r,. i-
424
'1Sts W
J og 0
42%0
4Wig
43"X
Figure 11. Two Dimensional Baseline Case InitialTemperatuire Profi le
* 4P,
pZia
4.P4
4.,.4
Figure 12. Two Dimensional Baseline Case InitialPressure Profile
61
TEST COMPUTATIONS,.- s,.
Before any results can be obtained from the two dimensional model, the
computer program should be able to successfully reproduce the solutions of
other simpler cases. The failure to do so may indicate the need for some
numerical fine tuning or variations of input parameters. (On the other hand,
the inability to duplicate previous results may be symptomatic of a more
serious problem with the code.) For these reasons, comparisons with other
models are essential for the development of a viable two dimensional repre-
sentation of the viscous gas flow in the damaged warhead.
The first case which was examined was the one dimensional inviscid, two-
phase, reactive flow problem described by Butler. Lembeck, and Krier (Refer-
ence 8). In order to model these flow characteristics, viscous gas effects
were switched off and initiation was fixed to take place in one dimension.
"- The resulting propagation of the flame front is plotted in Figure 13. Notice
the excellent correlation between the three different versions of the same
problem. The Dahm (Reference 10) three dimetisional code (run with a one
dimensional initiation) predicts almost identical flame front locations while
the purely one dimensional model produces slightly lower values. These
2 results indicate that the DDT2D computer code can accurately model one dimen-
sional, inviscid flows.
However, this is not the case for the more complicated two dimensiona)
.AN problems. This time the DDT2D program produced the locus of ignition fronts
shown in Figure 14. Note that initially the flame front location varies in
both the axial and radial directions. As time progresses, the variations in
the r-direction become less and less, until the flow is nearly one dimen-
* sional, after t - 1.97 us. While this result is qualitatively correct, the
62
430
cl 0
La t 0
43 U *'
g ~ 4.i
-4
Zqfa CD0 0
00 J cd. '3
* I.'
Vo 41o0
(w)
"Y I
0
0
La
%.. - 4-
J
"-C
o
64 C
event takes place much too fast. Based iipon results obtained by Dahm (Refer-
ence 10), the transition to one dimensional flow should not occur until nearly
50 us. Obviously, there is too much energy in the system causing the parti-
cles to ignite prematurely.
Therefore, it was decided to attempt to run a case with a two dimensional
initiation and gas viscosity effects. It was assumed that the inclusion of
the viscous dissipation terms in the gas energy equation would slow the
advance of the flame front. Unfortunately, no meaningful results could be
obtained since almost imediately, spatial oscillations developed. These wild
swings ir, the radial velocity components (shown in Figure 15) eventually
forced the progra!' to stop after 11.33 us. Despite numerous attempts at
damping these numerical oscillations with adjustments to the input parameters,
no significant progress was made.
POSSIBLE FOLLOW UP
Although the current algorithm used in the DDTD code may not be entirely
free from all errors, the foundations are correct and work on this two-dimen-
sional, viscous gas model should be continued. Reasonable results have been
obtained for an inviscid one dimensional analysis. For this reason, it is
felt that this prog-am can eventually be used to solve the two-dimensional
viscous flow problem without major alterations. Suggested areas for possible
changes in DDT2D are,
I. Provide better initial conditions
The initial temperature ard pressure profiles which are currently used in
the two dimensional model actually do not satisfy the conservation equations
65
. - i I ! i
kc 4-S
Ct d
ItC
-J
2. 0
7*4-i 4-i
' Vt:'g
h-c
'A°
"0
JU"*r r
r G
o 'M-ds=
-~Mae
at time t 0 0. Consequently, the flow must quickly adjust in order to meet
the requirements of these governing relations. This momentary instability may
be the trigger which touches off the oscillations which develop downstream.
Therefore, better results may be obtained from initial conditions based upon
established flovis from inviscid models, such as a two-dimensional version of
the Dahiri code, given in Reference 10.
2. Determine an optimal time step
It may be advantageous to use a Von Neumann stability analysis (Reference
16) to determine the formula for the optimal size time step. This could then
be incorporated into the computer program, enabling it to determine the ideal
time increment for each integration, and insuring the stability of the
differencing scheme.
3. Change the finite differencing scheme
The inviscid conservation relations form a set of hyperbolic partial
differential equations which can be solved numerically by using the leapfrog
scheme. However, when the second derivative terms representing viscous forces
are included, the nature of the governing equations may become parabolic or
elliptic. If this is the case, then a different finite differencing scheme
must be implemented.
67
.*A
REFERENCES
1. Krier, H. and Van Tassel, W. F., "Combustion and Flame SpreadingPhenomena in Gas Permeable Explosive Material," International Journal ofHeat and Mass Transfer, Vol. 8, pp. 1377-1386, 1975.
2. Krier, H. and Gokhale, S., "Vigorous Ignition of Granulated PropellantReds by Blast Impact," International Journal of Heat and Mass Transfer,Vol. 19, pp. 915-923, 1976.
3. Krier, H., Rajan, S., and Var Tassel, W. F., "Flame Spreading andCombustion in Packed Beds of Propellant Grains," AIAA Journal, Vol. 14,pp. 301-309, 1976.
4. Dimitstein, M., "A Separated-Flow Model for Predicting the PressureDynamics in Highly Loaded Beds of Granulated Propellant," MS Thesis,Dept. of Aero/Astro Engineering, University of Illinois at Urbana-Champaign, 1976.
5. Krier, H., Dimltstein, M., and Gokhale, S., "Reactive Two Phase Flow.y- Models Applied to the Prediction of Detonation Transition in GranulatedSolid Propellant," Technical Report AE 76-3, UILU-Eng 76-0503, UIUC,Dept. of Aero/Astro Engineering, July 1971.
6. Krier, H., Gokhale, S., and Hughes, E. D., "Modeling of Convective ModeCombustion Through Granulated Solid Propellant to Predict PossibleDetonation Transition," AIAA paper 77-857, Presented at the 19th,AIAA/SAE, Propulsion Conference, Orlando, FL, July 1977.
7. Krier, H., and Kezerle, J. A., Seventeenth Combustion Symposium, TheCombustion Institute, pp. 23-24, 1979.
8. Rutler, P. B., Lembeck, M. F., and Krier, H., "Modeling of ShockDevelopment and Transition to Detonation Initiated by Burning in PorousPropellant Beds," Combustion and Flame, Vol. 46, pp. 75-93, 1982.:£ "Modelin
9. Krier, H., Dahm, M. R., and Butler, P. B., ing the Burn-to-ViolentReaction to Simulate Impact Damaged GP Warheads," Proceedings of theFirst Symposium on the Interactions of Non-Nuclear Munitions withStructures_, U.S. Air Force Academy, Colorado, pp. 37, May i9-.
10. Vahm, M. R., "Numerical Integration of the Unsteady Two-PhaseConservation Equations to Predict Multi-Dimensional Flame Spreading inPartially Confined Porous Propellant Beds," MS Thesis, Dept. ofMechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 1984.
11. Markatos, N. C., and Kircaldy, D., "Analysis and Computation of Three-Dimensional, Transient Flow and Combustion Through GranulatedPropellants," International Journal of Heat and Mass Transfer, Vol. 26,No. 7, pp. 1037-1051, 1983.
68
12. Krier, H., Dahm, M. R., and Samuelson, L. S., "Multi-DimensionalReactive Flows to Simulate Detonation Delays in Damaged Warheads,"Technical Report to Air Force Armament Laboratory (to be publishedSeptember 1984).
13. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Washington:Hemisphere Publishing Co., pp. 118-120, 1980.
14. Kurihara, Y., "On the Use of Implicit and Iterative Methods for the TimeIntegration of the Wave Equation," Monthly Weather Report, Vol. 93, No.1, pp. 33, June 1965.
15. Williams, G., "Numerical Integration of the Three-Dimensional Navier-Stokes Equations for Compressible Flow," Journal of Fluid Mechanics,Vol. 37, Part 4, pp. 727-750, 1969.
16. Richtmeyer, R. D., and Morton, K. W., Difference Methods for InitialValue Problems, New York: Interscience Publishers, 1967.
17. Robert, A. J., "The Intergration of a Low Order Spectral Form of the
Primitive Meteorological Equations," Journal of the MeteorologicalSociety of Japan, Ser. 2, Voi. 44, No75, pp. 237-245, Toyko, ctober1966.
18. Asselin, R., "Frequency Filter for Time Integrations," Monthly WeatherReview, Vol. 100, No. 6, pp. 487-490, June 1972.
19. Arakawa, A., "Computation Design for Long Term Numerical Integration ofthe Equations of Fluid Motion, Two-Dimensional Incompressible Flow.Part 1," Journal of Computational Physics, Vol. 1, pp. 119, 1966.
20. Sod, G. A., "A Survey of Several Finite Difterence Methods for Systemsof Nonlinear Hyperbolic Conservation Laws," Journal of ComputationalPhysics, Vol. 27, pp. 1-37, 1978.
21. Wilhelmson, R., ATMOS 405-A, Numerical Methods in Fluid Dynamics:
Lecture Notes, University of Illinois. Dept. of Atmospheric Sciences,Fall 1983.
22. Butler, P. B., "Analysis of Deflagration to Detonation Transition inHigh-Energy Solid Propellants," Ph.D. Thesis, Dept. of Mechanical andIndustrial Engineering, University of Illinois at Urbana-Champaign,1984.
23. Coyne, D. W., Butler, P. B., and Krier, H., "Shock Development from* Compression Waves Due to Confined Burning in Porous Solid Propellants
and Explosives," AIAA Paper 83-0480, Presented at the 21st AerospaceSciences Meeting, Reno, NV, January 1983.
24. Jones, D. P., and Krier, H., "Gas Flow Resistance Measurements ThroughPacked Beds at High Reynolds Numbers," Journal of Fluids Engineering,Vol. 105, pp. 168-173, June 1983.
69
x.
APPENDIX A
AUXILIARY DATA BASE AND CONSTITUTIVE RELATIONS
The eight conservation equations presented in Section II contain 11
unknown quantities. Determining the values of these parameters requires the
addition of three state relations, thereby providing closure to the system.
Moreover, constitutive equations are needed to describe the gas viscosity and
the interactions between phases. This appendix details these necessary sup-
plemental relations which were used for this report.
1. Gas Phase Equation of State
A non-ideal equation of state was utilized:
P.-RT P (1 + b (A-i)
,=.g Rgpg blPg)
The value of the coefficient bI is based upon a fit of CJ detonation dataA produced by the TIGER code (Reference 22) and is assumed to remain constant.
2. Solid Phase Equation of State
The density of the explosive particles was specified by applying the Tait
equation, (see Coyne, Butler, and Krier, Reference 23):
3P 1/" Pp 0 p K + i] (A-2)
00
"d
70
...
3. Porosity-Stress Relation
7here are actually no suitable correlations for the dynamic porosity-
stress relation for a packed bed of compressible solid particles over th2
q, range of conditions imposed by this problem. Consequently, an approximate
numerical predictor method was used. At each time step, an equilibrium con-
.i dition is applied, setting the particle pressure equal to that of the gas.
Porosity is then determined by dividing the particle density from the previous
step into the current value of the solid phase bulk density (predicted by
Equation (4)). Since the time 3teps are small, and the changes in particle
density are very gradual, the errors resulting from this approximation are not
signi ficant.
4. Gas Production Rate
The burning particles produce gas according to:
13
r c =-- ( -) P (A-3)c r p
where the particle surface to volume ratio for the spheres is and ther r
- .~ psurface burning rate was described as:
bp (A-4)g
with b and n being known imposed constants.
71.
A'-
5. Gas Viscosity Coefficient
The viscosity of the gas phase was assumed to be a standard function of
temperature only, and was given as:
TV _g (A-5)90
A relation which considers the coefficient as a function of both temperature
and pressure would have been more desirable here because the extreme pressures
will have some influence on the viscosity.
6. Interphase Drag
The equations used to predict the drag forces between the gas and par-
ticles were:
1) ~~u ) f(A6p
* ,fD R (.7 f (A-7)A, 4r w)p
where the friction factor fp P9 s defined as:
-fpg 2 [150 + 1.75 1T- (A-8)
a Here Re is the flow Reynolds number based upon the particle diameter:
2r Vp,Re = Ug (A-9)
72
wherel V represents the relative velocity of the gas as it moves over the
solids, such that:
V (u 9- up) + (Wg9- p2 ] 1/2 (A-10)
This drag relationship was derived from semi-empirical results by Jones and
Krier (Reference ?4) for steady fiow over a packed beJ of soherical heads. It
assumes a constane porosity and is valid for Reynolds rumber ranging from 733
to 126670. Although the problem presented in this study is unsteady with
V "variable porosity, the Jones and Krier correlation produces reasonable values
for the interphase drag in the two dimensional model.
7. Convective Heat Transfer
The rate of heat transfer between the gas and the particles is described
by:
r = (l-)(Tg, TP hpg (A-I1)
where the heat transfer coefficient h is the classical Denton formula also
used by Krier and Kezerle (Reference 7):
h 9 0.58 r~ Re0 Pr0 3 (A- 12)pg
73
APPENDIX B
FINITE DIFFERENCE FORM OF THE CONSERVATION EQUATIONS
Numerical computation methods must be employed to solve the unsteady,
two-phase conservation equations derived in Section I1. After the domain has
been divided into discrete nodes, the governing partial differential equations
are rewritten in finite difference form using the explicit, centered time,
centered space, leapfrog scheme. The resulting algebraic relations are
applied at all node points, producing a system which can be solved by a digi-
tal computer.
The following shorthand notation is used in the finite difference equa-
tions which are presented here. If "A" is considered to be a generalized
dependent variable while "x" is an independent variable, then:
x+ Ax+ Ax-A Ax
A + A -2A6 XxxA - x+Ax x-Ax x (B-2)
(Ax)
A x + A xX+ - (B--3)- x 2 2
2
Thus the conse-rvation equations in finite ditference form are:
1. Conservation of Mass
a. Gas Phase Continuity (applied at scalar rodes)
- 74
6t - 6r r rug) - zlZ Wg) +(-4)
b. Particle Phase Continuity (applied at scalar nodes)
t. 1 (pr -- (B-5)tP2 r rP2 ru) - 6zP 2z wp) c
2. Conservation of Momentum
a. Gas Phase Momentum
(1) Radial Direction (applied at radial velocity nodes)
6t(lr g) = -- z uzgr Ugr -u
+ 6rI- 6r(rug)] + 6zzUg + L 6r(rug) + 6 W
+ Tc Up - Dr -6r(Pg) (B-6)
(2) Axial Direction (applied at axial velocity nodes)
tz Wg) r I uf w r wP Wgr)- zP ;r gZ)
- + 6 r (r 'rWg ) + 6 zzWg , [6z Sr(ru ) + 6 zWg l
+ Tc W n - D z 6z(Pg* ) (R-7)
75'o/
b. Particle Phase
(1) Radial Direction (applied at radial velocity nodes)
6t(- 2 rup) = , r(P2 -r upr) U- r)
rr Up + Dr [Pp(l-) ] (B-8)
(2) Axial Direction (applied at axial velocity nodes)
St(0 2 w1) 6r(02 r 'Pw pr) -z(p2pZ ;pZ)
SV# p + Dz - 6z[Pp(1-.)] (B-9)
3. Conservation of Energy
4. a. Gas Phase (applied at scalar nodes)
- t I ru r Tr r z
tI g) r 6r r g - zI wg( g 9 j)
+ .g [(6u) 2 + !j_- 2 2 zg r g r) + 6zwg
2 ~ 2) + 6 W + Lj
-" 2 _L 2 +~ zr" + r zg + 6gr
+- r 1 -r) + ur r: .. g 6r[ rr UgH ] 9 +- zz Ugr
g r r rrr8r Z-r+~ ~~ i r+ (r :Jgr. + 6z gr + -!9- 6 (r gZ
76
: II
a ;W-z + - 6r(r U 9Z wW
rp r2 2
' / E~ ~C h e m + w-- + )( - 0_ + r~ +
P2r p) -cl Pa (appliedPrat sEcz1 noes) I ,z6 ZP P-)____ - ' --
P Pr' r2 _ z2 P
r2 w
(B-li)
In Equations (B-10) and (B-1I) the total energies are defined as:
TC CT =VT +( + (B-12)
r= C -r 02 +3
E- z C v + 2 - (B-14)
whEre a subscript is added to denote gas or particle phase. In addition, the
..vPriges of the interphase drag terms must be described as:
77
r r_ r, f
Z 4r2 u9 up P
p
z- = " gr upr fp (z5
4r? (;qwg Pz f p9gB6
~P
while the remainder of the source/sink terms present no significant problems
resulting from discretization of the system.
• 78
IL,
°78
APPENDIX C
COMPUTER CODE LISTING
PROGRAM DDT2DA( INPUT, OUTPUT, TAPE6 ,TAPE7 ,TAPE8)CCCC WRITTEN BY LAURENCE S. SAMUELSON AND HERMAN IRIERC UNIVERSITY OF ILLINOIS AT URBANA-CHAKPAIGNC LATEST VERSION--AUGUST 1984cCCC THIS IS DDT2DA, A PROGRAM WHICH MODELS A DAMAGED GENERAL PURPOSEC WARK"EAD IN TWO DIMENSIONS AND INCLUDES THE EFFECTS OF GASC VISCOSITY. THE PROGRAM USES A CENTERED TIME, CENTERED SPACE,
* C EXPLICIT DIFFERENCING SCHEME ON A STAGGERED GRID. TO REDUCE THEC TIME SPLITTING OSCILLATIONS INHERENT IN CENTERED TIME METHODS,C A SMOOTHING STEP IS ALSO INCLUDED.CC THE MAIN PROGRAM SETS THE SIZE OF THE TIME STEPS AND CONTROLSC THE FLOW OF INFORMATION THROUGH THE SUBROUTINES.CC SUBROUTINE READA READS IN THE INPUT PARAMETERS FROM ANC EXTERIOR DATAFILE CALLED DATA2D. THESE VALUES ARE THENC PRINTED TO AN OUTPUT FILE FOR CONFIRMATION AND INDEXING
kc PURPOSES.cC SUBROIUTINE INITIAL LOADS THE VARIABLE ARRAYS WITH TIE INITIALC CONDITIONS FOR THE BURN.CC SUBROUTINE VARSET RESETS AND UPDATES THE SECOUI)MR VARIABLESC BASED UPON THE REULTS FROM THE PREVIOUS TIME STEP. IT ALSOC PREPARES TE PRIMARY VARIABLES FOR THE NEXT TIME STEP.
C SuBaOUTINE SWEEP PERFORMS THE ACTUAL FINITE DIFFP.RENCEC INTEGRATIONS ON THE PRIMARY VARIABLES. ALL THE CALCULATIONSC ARE PERFORI0 ON A SINGLE NODE BEFORE ADVANCING TO THE NEXTC POSITION. THE SUBROUTINE SWEEPS DOWN THE LENGTH OF THE BELDC BEFORE MOVING MOVING 10 A NMW RADIAL POSITION.
C FUNCTION AVG AVERAGES THE VALUES AT THE INPUT NOW AND HEC PREVIOUS NODE IN THE INPUIT DIRECTICN.C: C FUNCTION OBLAYG AVERAGES THE VALUES OF FOUR NODES-
C SUBROUT INE DUMP CONTROLS THE FORMAT OF THE OUTPUT.CC SUBROUTINE D(JHP2 PRINTS RATIOS OF DRAG AND ?ISW.$, STRESS TEM.CCC
V 79
4b.
.
C THE VARIABLES USED IN THE PROGRAM ARK-:C PRIMARY VARIABLESC RHOl BULK GAS DENSITYC RH02 BULK PARTICLE DENSITYC UG GAS RADIAL VELOCITY COMPONENTC UP PARTICLE RADIAL VELOCITY COMPONENTC WG GAS AXIAL VELOCITY COMPONENTC WP PARTICLE AXIAL VELOCITY COi'ON'ENTC EG GAS TOTAL ENERGYC EP PARTICLE TOTAL ENERGYC SECONDARY VARIABLESC RHOG GAS PHASE DENSITY
. C RHOP PARTICLE PHASE DENSITYC PG GAS PHASE PRESSURZC PP PARTICLE PHASE PRESSUREC PGPHI BULK GAS PRESSUREC PP1MF BULK PARTICLE PRESSUREC TG GAS TEMPERATUREC TP rARTICLE PRESSUREC PHI POROSITYC RADP INSTNTAIEOUS PARTICLE RADIUSC IGN CONDITION OF PARTICLE (INTACT,BURNING,OR BURNED OUT)C MUG GAS VISCOSITYC GAN14AC GAS GENERATION RATE DURING COMBUSTIONC DRAGR RADIAL COMPONENT GAS-PARTICLE INTERACTION DRAGC DRAGZ AXIAL COMPONENT GAS-PARTICLE INTERACTION DRAGC QDOT HEAT TRANSFER RATE FROM GAS TO PARTICLE PHASEC ARVISR ARTIFICIAL VISCOSITY IN THE RADIAL DIRECTIONC ARVISZ ARTIFICIAL VISCOSITY IN THE AXIAL DIRECTIONC OTHER VARIABLESC RLEN ACTUAL RADIAL LENGTH OF BEDC ZLEN ACTUAL AXIAL LENGTH OF BEDC DELR RADIAL DIRECTION DIFFERENCING SPACE STEP
"* C DELZ AXIAL DIRECTION DIFFEREINING SPACE STEPC DELT DIFFERENCING TIME STEPC STEP INCREMENTAL TIME STEP FACTORC EPS TIME SMOOTHER COEFFICIENTC ARV ARTIFICIAL VISCOSIrY COEFFICIENT
* -"C TOL TRUNCATION ERROR 'TOLERANCEC R ACTUAL RADIAL POSITIONC TIME ACTUAL TIMEC CVG GAS SPECIFIC HEAT AT CONSTANT VOLJMEC CVP PARTICLE SPECIFIC HEAT AT CONSTANT VOLUMEC RO GAS CONSTANTC MW MOLECULAR WEIGHT OF GASC B1 MON-IDEAL EQUATION OF STATE CO9.STANTC TOO INITIAL BULK GAS TEMPERATUREC TPO INITIAL BULK PARTICLE TOWPERATUREC TCH INITIAL LOCAL MAXIMJM TEMPERATUREC T!GN TEMPERATURE NECESSARY TO IGNITE PARTICLESC B BLqNING RATE PROPORTIONALITY CONSTANTC BYT BURNING RATE I'DEXC EPT PROPELLANT ENEimYC RHOPO INITIAL PARTICLE DEKSITY
80
C PHIO INITIAL POROSITYC PGO INITIAL BULK GAS PRESSUREC PC0 INITIAL LOCAL MAXIMUM PRESSUREC PWALL MAXIMUM PRESSURE AT THE WALLSC RADPO INITIAL PARTICLE RADIUSC KO PARTICLE BULK MODULUSC HUGO INITIAL GAS VISCOSITYC PR PRANDTL NUMBERC INIT DETERMINES THE TYPE OF INITIATIONC (ONE OR TWO DIMENSIONAL)C TOGGLE TOGGLES GAS VISCOSITY EFFECTS ON AND OFFC ENDR BOUNDARY NODE IN THE RADIAL DIRECTIONC ENDZ BOUNDARY NODE IN THE AXIAL DIRECTION
. C STOPIT WARNING FLAG FOR IMPOSSIBLE CONDITIONS OCCURINGC COUNT INDICATES THE NUMBER OF SWEEPS THROUGH THE BEDC MAX MAXIMUM NUMBER OF INTEGRATION SWEEPS PERMITTEDC RPRINT DETERMINES THE NUMSER OF RADIAL MODES PRINTEDC ZPRINT DETERMINES THE NUMBER OF AXIAL NODES PRINTEDC TPRINT DETERMINES THE NUMBER OF TIME STEPS BETWEEN PRINTINGC NTGR- NUMBER OF NODES AT INITIAL TEMPERATURES/PRESSURESC NPGZ ABOVE THE INITIAL BULK TEMPERATURE/PRESSUREC LOCAL VARIABLESC A SPEED OF SOUND
BLRNED AVERAGE CONDITION OF PARTICLES AT END OF BEDC PHIM 1-POROSITY
c RDOT BURNING RATEC V TOTAL GAS-PARTICLE RELATIVE VELOCITYC RE REYNOLDS NUMBERC FPG DRAG COEFFICIENTC KG GAS THERMAL CONDUCTIVITYC h YC GAS TO PARTICLE HEAT TRANSFER COEFFICIENTC ('I- INTERMEDIATE QUANTITIES INC EP6 CONTINUITY, MOMENTUM, AND ENERGY EQUATIONSC UGRHO1- INTERMEDIATE QUANTITIES IN
4; C EPRH0)2 TIME SMOOTHER EQUATIONSC EGCHEM GAS CHEMICAL ENERGY RELEASED DURING COMBUSTIONC EPCHEM PARTICLE CHEMICAL ENERGY RELEASED DURING COMBUSTIONC RLOC ACTUAL R LOCATION FOR PRINTING
A C ZLOC ACTUAL Z LOCATION FOR PRINTINGC PSI GAS PRESSURE IN CPAC RADPSI PARTICLE RADIUS IN MICRO-METERSC TIMOUT TIME OF PRINTING IN N ."CRO-SECONDSC DTOUT SIZE OF TIME STEP IN IIICRO-SECONDSC PWALOT MAXIMUM WALL PRESSURE IN GPAC I,J,K DO LOOP COUNTERSCCCC LOCAL VARIABLES *
,, .CREAL AINTEGER I, J, BURNED
Cc * COMMON VARIABLES '
CREAL RH01(O:6,O:61,3), R102(O:6,O:61,3), tG(O:6,O:61.3),& UP(O:6,0:61,3), WG(O:6,0:61,3), WP(O:6,0:61,3),& EG(O:6,0:6I,3), EP(O:6,O:61,3)REAL RHOG(0:6,0:61), RHOP(O:6,O:61), PG(O:6,O:61), PP(O:6,O:61),& PGPHI(O:6,O:61), PPIW(0:6,0:61), TG(O:6,0:61), TP(O:6,O:61),& PHI(O:6,O:61), RADP(0:6,O:61), MUG(O:6,O:61),& GANMAC(0:6,0:61). DRAGR(O:6Q0:61), DRAGZ(O:6,O:61),& QDOT(0:6O0:61), ARVISR(0:6,0:61), AIRVISZ(0:6,O:61)INTEGER 1014(0:6,0:61)REAL RLEN, ZLEN, DELR, DELZ, DELT, STEP, EPS, ARV, TOL, R, TIME,& CVG, CVP, RO, MW, B1, TGO, TPO, TCH, TION, B, ENT, EPT, RHOPO,& PHIO, PG0, PCH, PVALL. RADPO, KO, HUGO, PRINTEGER INIT, TOGGLE, ENDR, ENDZ, STOPIT, COUNT, MAX,& RPRINT, ZPRINT, TPRINT, NTGR, N=G, NTVR, NTPZ, NVGR, NP GZCOMMON /PRI/ RHO1, RH02. 110, UP, 110, WP, EG, RPCOMMON /SEC/ RHOG. RHOP, PG, PP, PGPHI, PPlWG, TG, TP, PHI,& RADP, IGN, MUG, GANKAC, DRAGR, DRAGZ, ODOT, ARVISR, ARVISZ
*COMMON // RLEN, ZLEM, DELR, DELZ, DELT, STEP, BPS, ARY, TOL, R,& TIME, CVG, CYP, RO, MV, B1, TOO, TPO, TCN, TIQI, B, BIT, EPT,& RHOPO, PHIO, POO, PCH, PVALL, RADPO, KO, HUGO, PR, 1111?, TOGGLE,& ENDR, ENDZ, STOPIT, COUNT, MAX,& RPRINT, ZPRINT, TPRINT. NTGR, NTGZ, NTPR, NTP Z, NPGR, NP GZ
CC
CALL READA% CALL INITIAL
TIHE=0.0COUNT=0DELT 1 .OE-8CALL DUMPCALL VARSETDO 10 COUNT=1,MAXCALL SWEEP
CC 1*CHECK FOR BURNED OUT CONDITIONS ~C
BtJRMED=ODO 20 I=1,ENDR
AlBURNEDUPRKED.IGN( I,ENDZ-3)20 CONTINUE
IF ((BURNED/EIIDR)OMNR.GE.1) THIENCALL DU14PSTOPEND IF
CIF ((OUNT/TPRINT)TPRINT.Q.CGUNT) CALL DUMPCALL VARSETTIIIE=TIME.DELT
CC COMPUTE NEW TIME STEP
* CDELT=5.OE-7DO 30 I=1,ENDR
82
DO 40O J:1,ENDZ
C & /(1.0+B1ROG(I:J))+RI/M/CVG*(1.0+BIRHEG(I,J))))C ~A=SQRT(PG(J/RH0(I,J)'(1020B1RO(,
DTZ=S 'EP*DELZ/AIF (T.MI.LT.DELT) OELT=DTRIF (DTZ.LT.DELT) DELT=DTZ
40 CONTINUE30 CONTINUE10 'rONTI9UE
Cc IF THIS POINT IS REACHED, THE PROGRAM HAS EXCEEDED *C THE MAXIMUM NU14BER OF TIME STEPS ALLOWED
PRINT 100100 FORMAT(/' THE MAXIMOM NUMBER OF TIME STEPS HAS BEEN REACHED')
STOP4. ~.END
CCC
* SUBROUTINE READACC COMMON VARIABLESC
REAL RH01(O:6,O:61,3), RH02(0:6,0:61,3), IG(0:6,0:61,3),& UP(0:6,O:61,3), VG(O:6,0:61,3), VP(O:6,0:61,3),& EG(0:6,0:61,3), EP(0:6,0:61,3)REAL RHOG(0:6,0:61), RHOP(0:6,0:61), P0(0:6,0:61), PP(O:6,0O:61),& PGPHJ(0:6,0:61), PPlIF(O:6,0:61), TG(0:6,0:61), TP(0:6,0:61),& P111(0:6,0:61), RADP(0:6,0:61), MUG(0:6,O:61),& GAN4AC(0:6,0:61), DRAGR(0:6,0:61), DRAGZ(0:6,0:61),
.1~& QDOT(0:6,0:61), ARVISR(0:6,0:61), ARVISZ(0:6,O:61)INTEGER ICN(0:6,O:61)REAL RLEN, ZLEN, DELII, DELZ, DELT, STEP, EPS, ART, TOL, R, TIME,
& CVG, CVP, RD, MW, B1, TOO, TPO, TON, TICK, B, BUT?, EPT, RHDPO,& PHIO, POO, PCH, PVALL, RADPO, [0, IMXG0, PR
INTEGER INIT, TOGLE, EIIDR, ENDZ. STOPIT, COUNT, MAX,v~. & RPRINT, ZPRIMT, TPRINTr, KTGR, NTGZ, NTPR. UTPZ, NPGR, NPGZ
CObMMON /PRI/ RHOI0, Ra)2, IJO, UP, VO, WP, £0, EPCa*W4~ /SEC/ RHOG, IU"), PG, PP, PGPHI, PPIMF, TG, TP, PHI,
0 & RADP, IGN, WUG, GAUIAC, DRAGH, DHAGZ, QDOT, ARVISR, ARVISZCabGION // RLEN, ZLEN, DELR. DELZ. DELT, STEP, UPS, ART, ?OL, R,& TIME, CVC, ("VP, RO. MV, Bi, TGO, TPO, TCH, TIGN, B, BHT, EPT,& RHOPO, PHIO, PG0, ?CH, PUALL, RADPO, KO, IMtG0, PR, IXIT, TOGGL.& ENOR, ENDZ, STOPIT, COiUNT, MXRMP, PRNG
& RPINT.ZPRNT, PRIT. NGRNTGZ, TRKPZOW N G
CREAD 1, RLENREAD 2, ZLENREAD 3, ENDIRREAD 4,. ENDZ
READ 5, STEPREAD 6, EPSREAD 7, ARVREAD 8, TOLREAD 9, MAXREAD 10, CdOREAD 11, CVPREAD 12, HOREAD 13, P1wREAD 14,BIREAD 15, TGOREAD 16, TPOREAD 17, TCHREAD 18, 117CRREAD 19, NTGZREAD 20, NTPRREAD 21, NTPZ
*READ 22, PGO*READ 23, PCH
READ 241, NPGRREAD 25, NPGZREAD 26, TIGidREAD 27, BREAD 28, BUT?READ 29, KOREAD 30, EPTREAD 31, RHOPOREAD 32, PHIOREAD 33, RADPOREAD 341, MU110READ 35, PRREAD 36, TPRINTREAD 3T, IRFRIMTREAD 38, ZRINTREAD 39, TWITREAD 410, TOGGLE
1 FOJO4AT(//19X,G33. 16)2 FORNAT( 181,033. 16)3 VORMAT(/191,3)41 FORMAT( 181.13)5 FORMAT(29X,G33. 16)6 FORHAT(2'(XG33. 16)7 FORIIAT(331,G33. 16)8 F0OIAT(36,33. 16)9 FORMAT(37X15)10 FORiAT/281.G33.16)11 FORI4AT(33X,G33. 16)12 FOMNT(351,G33. 16)13 F0OMAT(29X!,G33. 16)141 FORMAT(41,G33. 16)15 F0O4AT(I33I,G33. 16)16 FORMET(3$X,G33. 16)17 F0RKAT(3SX,G33. 16)18 FORMAY(/191,13)
84
19 FORKAT( 19, 13)20 FORhAT(/191,13)21 FORAT(181,I3)22 FORMAT(341,G33.16)23 FORMAT(43X,G33.16)24 FORKAT(/19X,13)25 FORKAT(181,13)26 FORMAT(/46XG33.16)27 FORMAT(381,G33.16)28 FORMAT(191,G33.16)29 FORMAT(34X,G33.16)30 FORMAT(24X,G33.16)31 FORMAT(32Z,G33.16)32 FORMAT(17X,G33.16)33 FORKAT(29X,G33.16)34 FORMAT(30X,G33.16)35 FORrAT(19X,G33.16)36 FORMAT(/36,13)37 FORIAT(38X,13)38 FORMAT(37X1,3)39 FORMAT(//41X,11)40 FORAT(45X,I1)
C- C
PRINT 100PRINT 101, RLENPRINT 102, ZLENPRINT 103, ENDRPRINT 104, ENDZPRINT 105, STEPPRINT 106, EPSPRINT 107, ARVPRINT 108, TOLPRINT 109, MAXPRINT 110, CVGFRINT 111, CVPPRINT 112, ROPRINT 113, MVPRINT 114, B1PRINT 115, TODPRINT 116, TPOPRINT 117, TCHPRINT 118, NTGRPRINT 119, NTPR
4 PRINT 120, NTPR_',PRINT 121, WMTPZ
PRINT 122, POPRINT 123, PCHPRINT 124, NPGRPRINT 125, NPGZ
. PRINT 126, TIGIPRINT 127, BPRINT 128, BUTPRINT 129, KO
85
PRINT 130, EPTPRINT 131, RHOPOPRINT 132, PHIOPRINT 133, RADPOPRINT 134, NUOPRINT 135, PRPRINT 136, TPRINTPRINT 137, RPRINTPRINT 138, ZPRINTIF(INIT.EQ.1) PRINT 139IF(INIT.EQ.0) PRINT 239IF(TOGGLE.EQ.1) PRINT 140IF(TOGGLE.EQ.0) PRINT 240
100 FORMAT('I','INPUT PARAMETERS FOR DDT2DA')101 FORIAT('0','RADIAL LENGTH (CM)=',G12.4)102 FORMAT(' ','AXIAL LENGTH (CMJ:',G12.4)103 FORMAT(' ','NU OF SCALAR MODES--'/
& ' ', RADIAL DIRECTIONz',I3)104 FORMAT( ',' AXIAL DIRECTION:',I3)105 FORMAT( ','INCREENTAL TIME STEP FACTOR=',G12.4)106 FORMAT(' ','TIME SMOOTHING COEn'ICIENT=',G12.j)107 FORMAT(# ','ARTIFICIAL VISCOSITY COEFVICIENT=',G12.4)108 FORMAT( ','TRUNCATION ERROR TOLERANCE (CM/SEC):',G2.4)109 FORMAT( ','MAXIMUM NUMBER OF TIME STEPS ALLOWED-',I5)110 FORMAT('O','GAS SPECIFIC HEAT (ERG/G/i)=,,G12.4)111 FORMAT(' ','PARTICLE SPECIFIC HEAT (ERG/G/I)=',G12.4)112 FORMAT(' ','UNIVERSAL GAS CONSTANT (ERG/NOL/KW)',G12.4)113 FORMAT(' ','GAS MOLECULAR WEIGHT (G/MOL)=',G12.4)114 FORMAT(' ','NOKN-IDEAL EQUATION OF STATE CONSTANT (CC/G)-',G12.4)115 FORHAT('O','INITIAL BULK GAS TEMPERATURE (K)=',G12.4)116 FORMAT( ','INITIAL BULK PARTICLE TEMPERATURE (K)=',G12.4)117 FORMAT(' *,'MAIIMUM LOCAL INITIAL TEMPERATURE (K)=',G12.4)118 FORMAT(' ','NUMBER OF SCALAR NODES ABOVE BULK GAS TEMPERATURE--'/
& ' ',' RADIAL DIRECTION=',13)119 FORMAT(' ',' AXIAL DIRECTIOU:',I3)120 FORMAT(' ',
& 'NUMBER OF SCALAR NODES ABOVE BULK PARTICLE TEIPERATURE--'/& I I,' RADIAL DIRECTION=',13)
121 FORMAT(' ',' AXIAL DIRECTIOK=',I3)122 FORMAT(' ','INITIAL ULZ PRESSURE (DYN/C/C)=',G12.4)123 FORMAT(' ','MAXIMUM LOCAL INITIAL PRESSURE (DYN'CM/CM)= ,G12.4)124 FORMAT(' ','NUMBER OF SCALAR NODES ABOVE BULK PRESSURE--'/
& ' ',' RADIAL DIRECTION:',13)125 FORMAT( ',' AXIAL DIRECTION:',I3)126 FORMAT('O','TEMPERATURE NECESSARY TO IGNITE PARTICLES (K):',G12.4)127 FORMAT(' ','BURNIMG RATE PROPORTIONALITY CONSTANT:',G12.4)128 FORMAT( ','BURNING RATE INDEU=',G12.4)129 FORMAT(' ','PARTICLE BULK MODULUS (DYN/CK/CM)=',G12.4)130 FORMAT( ','PROPELLANT ENERGY (ERG):',G12.4)131 FORMAT( -,'INITIAL PARTICLE DENSITY (G/CC)=',G12.4)132 FORMAT( ','INITIAL POROSITY=',G12.4)133 FORMAT(' ','INITIAL PARTICLE RADIUS (CN)=',G12.14)134 FORMAT( ',-INITIAL GAS VISCOSITY (POISE)W',G12.4)135 FORMAT( ,'GAS PRANDY NU tBER,G12.4)
86
136 FORMAT('O','NUMBER OF TIME STEPS BETWEEN OUTPUT=',I3)137 FORMAT(' ','NUMBER OF RADIAL MODES BETWjEEN OUTPUT=',I3)138 FORMAT(' ','NUMBER OF AXIAL NODES BETWEEN OUTPUT=',I3)139 FORMAT(-'' INITIATION TAKES PLACE TWO-DIMENSIONALLY *'239 FORMAT('-','** INITIATION TAKES PLACE ONE-DIMENSIONALLY *1A40 FORMAT('O,' VISCOUS GAS EFFECTS ARE INCLUDED240 FORMAT('OO,I** VISCOUS GAS EFFECTS ARE NOT INCLUDED )
RETURNEND
CCC
SUBROUTINE INITIAL
C
CINTEGER 1, J, K
cC **COMMON VARIABLESC
REAL RHO1(0:6,O:61,3), R1102(O:6,O:61,3), IG(0:6,0:61,I),& UP(O:6,0:61,3), VG(O:6,O:61,3), WP(O:6,O:61,3),& EG(O:6,O:61,3), EP(O:6,O:61,3)
REAL RHOG(:6,:61), RHOP(:6,:61), P0(0:6,0:61), PP(O:6,0:61),& PGPHI(O:6,O:61), PPlIF(O:6,O:61), TG(0:6,0:61), TP(O:6,0:61),& PHI(O:6,O:61), RADP(0:6,0:61), KUG(O:6,O:61),& GAMMAC(O:6,0:61), DRAGR(0:6,O:61), DRAGZ(O:6,0:61),& QDOT(O:6,O:61), ARVISR(0:6,O:61), ARVISZ(O:6,O:61)INTEGER IGN(O:6,O:61)REAL RLEN, ZLEX, DELR, DELZ, DELT, STEP, EPS, AR, TOL, R, TIME,& CVG, CVP, RO, MW, Bl, TGO, TPO, TCH, TIGN, B, BNT, EPT, RHOPO,& PHIO, PGO, PCH, PWALL, RADPQ, KO, HUGO, PRINTEGER INIT, TOGGLE, ENDR, ENDZ, STOPIT, COUNT, NI'!,& RPRINT, ZPRINT, TPRINT, tITGR, hNTGZ, NTPR, NTPZ, NPGR, NPGZCOMONO /PRI/ RHIl, RHO2, UG, UP, WG, WP, EG, EPCOMMON /SEC/ RHOG, RHOP, PG, PP, PGPHI, PP1MF, TO, 1?, PHI,& RADP, IGN, HUG, GAMIRC, DRAGR, DRAGZ, QDOT, ARVISR, ARVISZ
COMMON // RLEN, ZLEN, DELI, DFLZ, DELT, STEP, EPS, ARY, TOL, R,& TIME, CYG, CVP, RO, MW, BI, TGO, TPO, TCH, TIGN, B, BHT, Efl,& RHOPO, PHIO, PG0, PCH, PWALL, RADPO, KO, HUGO, PR, INIT, TOGGLE,& ENDR, ENDZ, STOPIT, COUNT, MAX,& RPRINT, ZPRIWI, TPRINT, NTGR, NTGZ, NTPR, NTPZ, NPOR, NPGZ
C
EXTERNAL AVGSTOPIT=ODELR: RLEN/ENDRDELZ= ZLEN/ENDZ
* DO 10 I=O,ENDR+10O 20 J=O,ENDZ+i
CC POROSITYC
87
C STEM4PERATURE --
?G(I, J) :TGOIF(INIT.EQ.1.AND.(I.LE.NTGR.AND.J.LE.wroZ)) THEN
C IF(J+(1.O*NrGZ)/NTGR*I.LE NTGZ)C & TG(I,J):(N'rGZ*NTGRTCHNTGZ(TCH-TGO)(1)-NTGR.C & *(TCH-TGO)'(J-1))/(NTaZ*NTGR)
TG( I,J)=TGO+(TcH-TGO)*( (NTGR*1 .O-(I-1) )/NTGR'1 .O)014I.
ELSEIF(J.LE,.NTGZ)
C & TG(I,J)=TCH-(TCH-TGO)*(J-1)/wrcZ& TG(I,J):TGO+(TCH-TGO)((NTGZ*1.0-(J-.1))/NTGZ41.0)**14IO
END IFTP(I,J)=TPOIF(INIT.EQ.1.AND.(I.LE.NTPR.AND.J.LE.NTPZ)) THEN
C IF(J+(1.O*NTPZ)/NTPROI.LE.NTpZ)C & TP(I,J)=(NTPZ*NTPRITCH-NTPZ*(TCH-TPO)*(I-).JTPRC & *(TCH-TPO)*(J-1))/(rrPZ*NrPR)
TP(I,J):TPO+(TCH-TPO)*((NTPR*1 .O-(I-I))/UTPR,1.)*iJL.O& *((NTPZ*1.o-(J1))/NTPZ1.1)1i4.
ELSEIF(J.LE.NTPZ)
C & TP(I,J)=TCH-(TcH-TPO)*(J-1)/rrPZ& TP(I,J)=TPO+(TCH-TPO)*((NTM *1.O-(J-1))/JTPZI1.0)*14.0
END IFCC PRESSUREC
PG (1,J )=PGOIF(INIT.EQ.1.AND.(I.LE.NPR.AND.J.LE.NpM)) THEN
C IF(J,(1.0'NPGZ)/NPGR*I.LE.NPGZ)C & rG(I,J)=(NPGZ*NPGRPCH-NPGZ.(PCH-PGO)O(I-1)wNPoJC & *(PCH-PCO)*(J-1))/W M *NPcGl)
PG( I,J)=PGQ.(PCH-PGO)*( (NPGR*1 .O-(I-1 ))/NPGR*1 .0)014.O& *((NPGZ*1.O-(J-1))/PGZ1.O)14.0
ELSEIF ( JLE.N'PCZ)
C & PG(I,J)=PCH-(PCH-PGO)*(J-I)fNPGZ& PG(I,J):PGO+(PCH-PGO)*((NPGZ*1 .0-(J-1 ))/NPG*1.O)h1l.O
END IFPP(I,J)=PG(I,J)PGPHI(I,J)=PG(I,J)*PHI(I,J)PP1KF(I,J):PP(I ,J)*(1 .O-PHIU ,J)) *
PWALL=PCH
C lDENSITY '
RHOP(I,J)=RHOPO
C RTGICLE SPI ANGAS RDUCTION "C
88
RADP(I ,J)=RADPOIGi( I,J)=OGAJIAC(I,J):O.O
C*4 DRAG, VISCOSITY, AND HEA TRANSFE RAlT
CDRAOR( I,J)=O.0DRAGZ(I,J)=0.O
QDOT( I,J)=0.OCC PRIMARY VARIABLES
DO 30 K=1.,3
RHO2(I,J,K)=RH0P(I,J)0( 1 .- PHI(I,J))
WG(I ,J,K)0O.0'V WP(I,J,KW=O.O
EG(I,J,K)=(TG(I,J)-TGO)*CVGEP(I ,J,K)=(TP(I ,J)-TPO)*CVP
30 CONTINUE20 CONTINUE10 CONTINUE
RETURNEND
CCC
SUBROUTINE VARSET~4. C
C t LOCAL VARIABLE.SC
REAL PRIM, RDOT, V, RE, FPO, KG, HTCINTEGER I, J, K
CC COMMON VARIABLES *
CREAL RHO1(0:6,O:61,3), RHO2(0:6,O:61,3), UG(O:6,0:61,3),& UP(O:6,O:61,3), WO:6,O:61,3), IP(O:6,O:61,3),& EG(O:6,0:61,3), EP(0:6,O:61,3)REAL RHOG(0:6,O:61), RHOP(O:6,O:61), PG(0:6,O:61), PP(O:6,O:61),
.N1 & 1PGPHI(0:6,O:61), PPlNF(O:6,O:61), TG(0:6,0:61), TP(O:6,O:61),& PHI(0:6,0:61), RADP(0:6,0:61), MUG(0:6,O:61),& GAMOAC(0:6,0:61), DRAGR(0:6,O:61), DRAGZ(O:6,O:61),4 QDOT(O:6,0:61), ARVISR(O:6,O:61), ARV!SZ(O:6,O:6i)INTEGE IGN(O:6,O:6I)REAL RLEN, ZLEN, DELR, DELZ, DEL?, STEP, EPS, ARV, TOL, R, TIME,
I It & CYG, CYP, RO, MW, Bi, TGO, TPO, TOCl, TIGN, B, Bid?, EPT, ROPO,& P1110, PG0, PCH, PWALL, RADPO, KO, KIEO, PR
INTEGER INIT, TOGGLE, ENDR, ENDZ, STOPI?, COUNT, MAX,& RPRINT, ZPRINT, TPRINT, WrOR, NTfGZ, NTPR, NTPZ, NPG, IIPGZ
COMMION J/PRI/ RHOI, RIW2, UG, UP, WG, WP, 9G, EP
* 89
00MON0 /SEC/ Wi F4W PC, Pt 3fl1f "oInW To m,* & ~~RAPPeIGN, WMG GAAC 0404t, OMI W~T, ARVIII, A1113Z
00WbOW /1 RLM MatW ;k.RI J3M4? mWilT RoV us, ANY, T0at it& TIM, BG CV09 ev0l not, " I TqO, "at =11 TIONO 1, am, 9"'&, Rko P1110, PM, PaI PVAt4,j RAMPQ 10, KWtJ, MR 1017, "=J.IA RWM, 9m, S70lTtT? C&IVT MAN#& IlPUIN?, EPRINTt ?PRINTr MR, WTQZ, Will, W9P, Vogt, MM
0EXrRvAL AVGDO 10 I=O,SNDR
DO 20 J=0,MZcc *~POROSITY AP'PR flATTOW 00
PHIRHO112 (IIJ, 1 )/RIIDP( 114)
c TEMPERATURE tiN -
c EQUJATIONS Of STfTYRC
R10P(IJ)0 I(i sit (P (IJ)1,) f0I)@(1/3)
c * OTAI~N WALL4 FAMAMJ~c
c 1F(I'(PU11,I).O PAN ?PA1~p~~R4
09
.5 CC *5GAS VISCOSITY O
-' C NTJG( I,J)=4UG0*((TG( I,J)ITGO) 6 4 0.65)
CC *eREYNOLDS NUMKBER ~C
VZSQRT((UiG(I,J,1)-UP(I,Jll))142.0+(WG(I,J,1)4JF(I,J,1))M42.0)RE:RHOI(I ,J,1)*V2.5 RADP( I,J)/MUG(I ,J)
CC 'DRAGC
FPG=PH1M*42.O/PHI( I J)N42.O'( 150.O+3.89*(RE/PHIM)O*o.87)DRAGR(I,J)=MUG( I,J)/(.L 0 RADP(I,J)*R2.o)*FPp
& *(UG(I,J,2)-UP(I,J,2))DRAGZ(I,J)=KLIG( I,J)/(4.0*RADP(I,J)**2.O)*FpG
& *(WG(I,J,2)-VP(I,J,2))CC HEAT TRANSFER RATE:*C
KG=HUG(I,J)*( (RO/MW)+CVG)/PRHTC=O.58'KG/RADP( I,J)*(RE*U0.7)*(PR**O.3333)IF(IGN(I,J).GT.O) THEN
QDOT(I,J)=0.OELSE
QDOT( I,J)=3.0IPHIM/RADP(I,J)*(TG( I,J)-TP(I,J))*NTCEND IF
CC ARTIFICIAL VISCOSITYC
ARVISR(I,J):ARV*(ABS(AVG(UG,I+1,J,2, 1)+O. 1)+2.liIE5)ARVISZ(I,J)=ARV*(ABS(AVG(WG,I,J.1,2,2)+.. )+2.T4E5)ARI ~lENDR.1 ,J)=ARVISR(ENDR,J)
* - ARVISR(I,ENDZ+Il)=ARVISR(I,EIDZ)ARVISZ(ENDR+1 ,J)=ARVISz(ENDR,J)ARVISZ( I,ENDZ.1 )=ARVISZ(IENDZ)
CC PRIMARY VARIABLE SHIFTC
DO 30 [:3,2,-iRHO1(I, J ,K):RHO1 (IJ ,K- 1)RHO2(I,J,K)=R1102(I,J,K-1)UG(I,JK)=UG(I,J,K-1)WG(I,J,K):WG(I,J,K-1)
EG(I,J,K)=UG(I,J,K-1)
130 CONTINUE20 CONTINUE10 CONTINUE
RETURNEND
C
91
C
SUBROUTINE SWEEP
c LOCAL VARIABLESC
REAL CGI, CG?', CG3, CG4, CPI, CP2, CP3, CP4IREAL MGRI, MGR2, MR3, i=4, NOR5, MOR6, MGR7, HGRd, MmRREAL MOZ1, MUZ2, IIGZ3, 110Z4, MGZ5, MGZ6, MCIZ7, MGZ8, MGZ9REAL HPR1, k4PR2, MPR3, !4PRII, MPR5, MPR6FEAL MPZ1, MPZ2, MPZ3, MPZ4, MPZ5, MPZ6REAL EGi, EG2, EG3, EG4, £05, EG6, EG7, EGS, E09, EG10REAL EP1, EP2, EP3, EP4, EP5, EP6REAL EGCHEH, EPCHEXREAL UGRHO1, WGRHOI, URHO2, WPRHO2, EGRHOI, EPRH02REAL TA[UGGEO:6,O:61), TAUGGZ(0:6,0:61)INTEGER 1, J
C COMONVARIABLES
REAL RHO(:6,o:6i,3), R102(:6,0:61,3), LG(0:6,0:61,3),& UP(0:6,0:61,3), WG(0:6,0:61,3), VP(0:6,O-61,3),& OG(O:6,0:61,3), EP(0:6,0:61,3)REAL RHOG(O:*6,0:61), RHOP(tO:6,O:61), P0(0:6,0:61), PP(0:6,O:61),& PGPHI(0:6,O:61), PPIF(O:6,0:61), TG(O:6,0:61), TP(O:6,0:61),& PHI(0:6,0:61), RADP(O:6,0:61), MUG(0:6,O:61),& GA)4AC(0:6,O:61), DRAGR(0:6,0:61), DRAGZ(0:6,0:61),& QD0T(0:6,0:61), ARVISR(O:6,0:61). ARVISZ(0:6,O:61)INTEGER IGN(0:6,0:61)REAL RLEN, ZLEN, DELR, DELZ, DELT, STEP, EPS, ARV, TOL, R, TIME,
& CVG, CVP, RtO, MWl, Hi, TGO, TPO, TCH, TIGI, B~, MUT, EPT, RIIOPO,& PHIO, PG0, P011, ?WALL, RADPO, KO, HUGO, PRINTEGER INIT, TOGGLE, ENDR, ERIDZ, STOP!?, COUNT, KAX,
& RPRINT, ZPRINT, TPRIUT, NTGR, NTGZ, MTPR, NTPZ, MFR. NPOZCOMMON /PRI/ RMOI, R1402, LJG, UP, VG, WP?, EG, EP
091N/SEC/ RHOG, RIIOP, PG, P17, PGPHI, PPiMF, MO, TP, PHI,&RADP, IGN, MUG, GAIEAC, DRAGR, DRAGZ, QDOT, ARVISR, ARVISZCO*40N // RLEC, ZLEV, DELR, DELZ, DELT, STEP, EPS, ARV, TOL, R,& TIME, CVGI, CVP, RO, NW, Hi, TGO, TPO, ?CH, TIGN, B, BUlT, CP?,& RHOPO, P1410, PO0, P011, PWALL, RADPO, [0, HUGO, PR, INIT, TOoC!>E,& ENDR, EKDZ, STOPI?, COUNT, MAX,& RPRINT, ZPRINT, TPRIVT, NTGR, NTGZ,, NTPR, NTPZ, NPOR, KMG
C
E XTE RNAL AVG, DBLAVG
EPCHEN=0 .O5*EPTDO 10 I:1,ENDR
DO020 J=1,ENDZCC
,~I. GAS CONTINUITY EQUATIONcC
92
-A4hI % J U
R=DELRI( 1-0.5)CGI=(AVG(RHO1 ,1.1 ,J,2, 1 )(R.DELR/2.0)
& 'UG(1+1,J,2)-AVG(RH01,1,J,2,1)*(R-DELI/2.0)& IUG(IJ,2))/DELR
CCi.2=(AVG(RHlI ,I,J..t,2,2)'II(I,J.1 ,2)
& -AVG(RHO1,I,J,?,2)IIG(i,J.2))/DELZC
CG3:(AVG(ARVISR-I+1,J,I,1)*(RHO1(I+I,J,2)-RHOI(I,4,2))& -AVG(ARVISR,I,J,1, 1)*%RH01(I,J,I)-RN011,J,2)))/DELR
4 CG3=0.0C
&CG4=(AVG(ARVISZ,I,j11,2)4(RHO1(I,J+1,2-HO(1,J,2>)
& -RHO1(I,J.1,2))-AVG(ARVISZ,I,J+1,1 ,2)*(RHO1(1,J+1,2)& -RHO1(1,J,2)))/DELZ
CRHo1uI,J,1)=RH1r,J,3)2.o.DEiT(-.O/R.Cc1-CG2.Cc3.CG4
& +GAIMAC(I,J))CCC *PARTICLE CONTINUITY EQUATIONCC
CPI=(AVG(RH02,I1 1,J,2, 1)*(R+DELIL/2.0)a *UP(I+1,J,2)-AVG(RHO2,I,J,2,1)*(R-DELR2.O)
& *UP(f,J,2))/DELRC
CP2:(AVG(RHO2,I,J,1 ,2,2)*WP(IJ+1,2)& -AVG(RH02,1,J,2,2)*WP(I,J,2))/DELZ
CCP3=(AVG(ARVTSRJ.1,J,1,1)'(RU02(1,1,J,2)-RH02(I,J,2))
& -AVG(ARVISR,I,J,1)*(AIIO2(1,J,2)-RIIO2(I-1,J,2)))/DELRCP3=O.O
C
& -=AG(ARVISZ,I,J11,2(RH0 2( 1, ,)RK2I,-1 2)IJ))& IA(JEQ.'1) CP4=(AG(RISZ,J,2)(R02(,J,2)D
& -RH02(-,J.1,2))-AVG(ARVISZ,I,J+1,1,2)*(RHO2(I&,J+1,2)& -RH02(I,J,2))d/DELZ
CR102(I,J,1)=R112(I,J,3)+2.0*DELT(-./R'CP1-CP2CP3CPI
& -GAIOAC(I,J))CCC *GAS MO(ENTUM EQUATIONC * RADIAL DIRECTIONC-- - - - - -C
R=DELR(I-1)IF(R.EQ.C.O) GO TO 1000
& *(AVG(UG,1+1,J,2.1))*2.0--iI1-1,J,2)
93
& *RDL/.)(V(GIJ21)0.)DLC
WtR2:(DBLAVG(RHO1 .1,4+1,2)& SAVG(UG,I,J+1,22)'AVG(WG,I,J,1,2,1)& -DBLAVG(R1401,i,J,2)'AG(UG,I.J,2,2)& OAVG(WG,I,J,2,1)),DELZ
CIF(TOGGLE.EQ.1) THEN
& ~ -(ROUG(I,J,2)-(R-DELR)*UG(I.1,J,2))/(R-.DELR/2.0))& /(DELRI'2.o)
MGR5:NGR3+(WG(I,J+1 ,2)-WG(I,J,2)-(WGI.1,J.1 2)& -WG(I-1,J,2)))/(DELRODELZ)
CELSE
C M4GR3=O.0
MGR4=0.0
NGR5=0.0
END IFC
MGR6=(GAHKAC(IJ)+GAMAC(I.1 ,J))/2.0*UP(I,J,2)C
HGR7:(PGPHI(I,J)-pGpHI(I-1 ,J))/DELRC
& -AVG(RHO1,!,J,2,1)*Uo(,,2))-ARVISg(I..1,J)
& *UG(I-1,J,2)))/DELR
C GR9=(DBLAVG(ARVIZIJ+11)*(AVG(RJI,I,J+1,2, 1)
& -DBLAVG(ARVISZ,I,J,1)*(AVG(RWO1,1,J,2,1)IUG(I,J,2)& -AVG(RH01,1,J.1,2,1)#UG(I,J.1,2)))/DL
CUG(I,J,1)=(AVG(RIO ,1,J,3,1)*Uo(I,J,3)
& +2.O*DCLT(-1./R*t4GR1GR2+U(I,J)*(34.MR& .1. *G5+G6DARI,)MR+G8MR)& /AVG(R'I7, I, J,1 , 1
CC *~*M~ppujC GAS MOKOEPrw EQUATIONC * AXIAL DIRECTIONc £u:f~a..euenC1000 R=DELR*(I-O.5)
MGZ1:(DBLAVG(RHO1+1 ,4,2)0(R.DELR/2.O)
94
& *AYC(UG,I+I,i,2,2)*AV(C,reI,J,2,t)& -DBLAVG(RHO1,I,J,2)'(R-DELI/2.O)& *AVG(LIGI,J,2,2)*AVG(VG,I,J,2,1))/DeuI
CMGZ2=(RH01(I,J,2)*(AVG(wG,I.J+1 ,22))*2.0
& -RHO1(I.J-1,2)*(AVG(WG,I,J,2,2))"2.O)/DELZc
IF(TOGGLE.EQ.1) THEN
C IF(I.9Q.ENDR) THFEN
C 1 NO-SLIP BOUNDARY CONDITIONC
ttGZ3:( (R+DE'LR/2.O)*(-1.O4 WG(I,J,2)& -1.O/3.O*HG(I-1,J,2))-(R-DELR/2.0)e(Wo(I,J,2)& -WG(I-1,J,2)))/(RDELRI2.0)
CELSE
CHGZ3=( (R+DELR/2.O)*( WG(I+1 ,J,2)-WG(IJ,2))
& -(R-DELR/2.O)'(WG(I,J,2)-vG(I-1,J,2)))/(RDELRj*2.0)
C END IF
C
& +Hcz4C
EL.SE
MGZ3=O.O
?GZ5=O.0
END IFC
HGZ6:(GMAC(I,J).GA(AC(I,J.1 ))/2.0*WP(I,J,2)
C
/ C IF(I.EQ.ENDR) THEN
CHCZ8=(DBLAVG(ARVISR,I,J, 1)*(AVG(RHo ,I,J,2,2)
& *WG(I,J,2)-AVG(RHIh,I-1,J,2,2)WG(I.1,J,2))* & -DBLAVG(ARVISR,I-1,J,1}*(AVG(RHO,I.1,J,2,2)
& *WG(I-1,J,2)-AVG(RHO1,I-2,J,2,2)yG(I.2,J,2)))/DELRC
4 ELSEC
HGZB=(DBLAVG(ARVISR,I+1 ,J, 1)*(AVG(RHI,t+1 4,2,2)
95
& *WG(I.1,J,2)-AVG(RHO1,I,J,2,2)1Kr,(U.J,2))& -BLAVC(ARVISR,I,J,1*(AVG(RN01I,J,2,2)VG(I,J,2)& -AVG(RHO1,I-1,J,2,2)WG(I-1,,2)))/)ER
CEND IF
C IF(J.EQ.1) THEN
WZ9=(ARVISZ( I,J.1 )*(AVG(RH01 , J ,2,2)& *WG(I,J+2,2)..AVC(RHO1,1,I.J ,. ;,2)tWG(I,J+1,2))& -ARIISZ(I,J)/*(AVG(RHO1,I,J,1,2,2)VG(I,J+1.,2)
.4& -AVC(RHOI, , J,2,2)*WG{I,J,2)))/DELZELSE
-~ MGZ9=(ARVISZ(t,J)*(AVG(RHO1 ,I,J.1,2,2)*WG(I,J+1,2)& -AVG(RHO1.I,J,2,2)*VG(I,J,2))-ARVISZ(IJ-1)& '(AVG(RHO1 ,I,J,2,2)*Wo(I,J,2)-AVG(RHO1,I,J-1,2,2)& W(,-,))DL
p END IFWG(I,J,1 )=(AVG(RHIO,I,J,3,2)*G(IJ,3)
& +2.0*0ELT*(-1.0/RMGZMGZ2.4UG(t,j)*(GZ3NGZI& +1.Q/3.o*MGz5)+MGZ6-DRAGz(I,J)-NGZT+Wa844rg))& /AVG(RH01,I,J,1,2)
CCC PARTICLE lNENTU14 EQUATIONC RADIAL DIRECTIONCC
R=DELR*( I-1)IF(ft.EQ.O.O) GO TO 2000MPR=(R02(I,J,2)'(R+DELRi'2.0)
& *(AVG(UP,I+,J,2,1))**2.0-RH2(Im,J,2)& *(R-DELR/2.0)*(AG(UP,I,J,2, 1))"2.0)fDELR
CM4PR2=(DBLAVG(RH102, IJ+1 ,2)
& *AVG(UP,I,J.I,2,2)*AVG(P,I,J.1,2,1)& -DBLAV(RHO2,I,J2)*AVG(IP,I,J,2,2)& OAVG(WP,IJ,2,1))/DELZ
t4PR3z(GAMNAC( I,.)4.GAJOAC(I-1,J))/2.0UP(I,J,2)C
MPR4=(PPlMF(I,J)-PP'NF( I-i ,J))/DELR
& -5AVHISJ2)I,J)*AGRD,,,2,2)-AVIS(t,J2
& *(AVG(RH021,J,2,1)*UP(I,J,2)-AVG(RHO21-1,J,2,1)
& *UP(!-1,J, !)))/DELR
KPR6:(DBLAVG(ARVISZ,I,J11)(AVG(RO2,I,J.1,2,1)& *UPU,J1,2)-AVG(RHO2,I,J,2,)P(I,J,2))&, -DBLAV3(ARVISZI,J,)*(AVG(hI2,I,J,2,1)*UP(I,J,2)
& -AG(RH2,,J-1,2,1)UP(IJ-1,2)))/DELZC
UP(I,J,1):(AVG(RHO2,1,J,3, 1)'UP(I.J,3)
96
ffalY.
& +2.O4DELT(-I .O/RNPRI-PR2-PI3+DRAGR( I J)-NPR4+i4PR5& +MPR6))/AVG(RH02,I,J,1,1)
Cc *3UUIfhhI*#4*C *PARTICLE MOMENTM EQUATIONC AXIAL DIRECTIONC -------------I3 IBO B3C2000 R:DELR*(I-0.5)
IPZ1=(DBLAVG(RH02,.1 1,J,2)*(R+DELR/2.0)4 & *AVG(rJF,I+.1J,2,2)'AVG(WP,I+11J,2,1)
& -DELAVG(RH102,1,J,2)*(R-DELR/2.0)& *AVG(UP,I,J,2,2)*AVG(WP,I,J,2,1))/DELR
C
& -RH02(I,J-1,2)*(AVG(VfP,I,J,2,2))**2.0)/DELZC
MPZ3=(GAIO4AC(I,J).GAM(AC( I-i ,J-l))/2.O*V(I,J,2)C
MPZ4=(PPiF( I,J)-PPleF(I,J-1) )/DELZC
& MPZ5=(DBLAVG(ARVISR,IsI,J,1)*(AVG(RH02,I+1,J,2,2)
& -DBLAVG(ARVISR,I,J,1)*(AVG(RHO2,I,J.2,2)W(I,J,2)* & -AVG(RJ$2,1-1,J,2,2)*WP(I-1,J,2)))/OELR
CIF(J.EQ.1) TREK
MPZ6=(ARVISZ( I,J)+1)*(AVG(RiO2,IJ+2,2,2)& *W(I,J.2,2)-AVG(1U132,I,J+1,2,2)*W(I,J+l,2))& -ARVISZ(I,J)*(AVG(RHO2,1,J.1,2,2)*VP(I,J.1,2)& -AVG(RiI2,1,J,2,2)*VP(I,J,2)))/DELZ
r~l.. &-AVG(RHOD2,I,J,2,2)VP(I,J,2))-ARVISZ(I,J-l)& *(AVG(RIE12,IJ,2,2)*vP(I,J,2)-ATG(RW02,I,J-1,2,2)
& idP(I,J-1,2)))/DELZEND IF
CWP(%I,J, 1)=(AVG(RHO2,I,J,3,2)OVP(I,J,3)
* & +2 .ODELT'(-1 .O/RHMPZI-MPZ2-MPZ3.DRAGZ( I,J)-MPZA41MPZ5& 44PZ6))/AVG(RIK2,I,J,1,2)
C
EG1=(AVG(RH01I,I+1 ,.,2, 1 )UG(I+1 ,J,2)*(Ri.DELR/2.0)& *(AVG(EG,I.1,J,2,1).((PGPJII(IeI,J).PGPHI(I,J))/2.O)& /AVG(RHI1,I.1,J,2,1))-AVG(1~,I,J,2,I)& *UG(I,J,2)*(R-DELR/2.0)'(AVG(EG,I,J,2,I)& +((PGPHI(I,J)+PGPHIl(I-1,J))/2.O)& /AVG(RBO1,I,J,2,1)))/DELR
-IC
- -* G(H11,+,.2"(.J12
& *(AVG(EG,11J+1,2,2)+t(PGPMI(I.J1)PGP4I(I,J))/2.O)& /AVC(RIUQ1, 1,J..1,22))-AVC(RN1, I, J,2,2)
# *G(I,J,2)*(AVG(EG,I,J,2,2)
& +((PGPHI(I,J)ePGPH1(IJ-1))/2.0)& /AVG(RHOI,I,J,2,2)))/DELZ
CIF(TOGGLE.EQ.1) THEN
CEG3:(UG(i+1 ,J,2)-tJC(I,J,2))/DELR
CEG4=AVG(UG,I+1,J,2,1)/R
C
E'f5=(WG(I,J.1 ,2)-WG(I,J,2))/DELZ
EG6=(DBLAVG(WG, 1+1 ,J,1,2)& -DBLAVG(WC,I,Je1,2))/DELR
& +(DBLAVG(UG,I+1,J*1,2)4' & -DBLAVG(UG,I+1,J,2))/DELZ
cELSE
CE-G3=0.O
EG4:0 .0
EG5zO .0
EG6=O.o
END IFC
EC7=(DRAGR(t.1,J)ORAGR(IJ))(UP(II,J,2)UP(I,J,2))/I.O4' & .(DRAGZ(I,J+1 )+DRACZ(I,J))*(WIP(I,J.1,2).MF(I,J,2))/4.O
EG8=GANKAC( I,J)'(EGCHEN.AVG(UP, 1+1,J 12,1I)2.0/2.O
& .AVG(WP,I,J.1,2,2)"2.O/2.0)
EG9:(AVG(ARVISR,I+1,J,1,1)*(RIO1(I+1,J,?)OEG(I.1,J,2)& -RHO1(I,J,2)*EG(I,J,2))-AVG(ARVISRI,J,1, 1)0(RHIW(.J,2)& *Et3(I,J,2)-RJ40(I-1,J,2)UEG(I-1,J,2)))/flE.
lF(I.EQ.1)& EG9=(AVGCARVISRI+2,J,1,1)(RI(I+2,Jp2)*EG(I,2,J,2)& -RHOI(I+1,J,2)'EG(1*1,J,2))-AVG(ARVISR,I+1,J,1,1)& *(RHOI(I.I ,J,2)*EG(I.1 ,J,2)-HH]Dl(I,J,2)*EG(I,J,2)))/DELR
* & -RHJIO(I-,J,2)EG(I-1,J,2))-AVG(ARVISR,I-1,J,1,1)* .& *(RH01(I-1,J,2)'EG(I-1,J,2)-REOI(I-2,J,2)*EG(I-2,J,2)))/DELR
C
4'-, & -RlfI(I,J,2)*EG(I,J,2))-AVG(AJIVISZI,J,1,2)'(RHII)(I,J,2)& *EG(I,J,2)-RHOI(I,J-1,2)*EG(I,J-1,2)))/DILZ
& *EG(I.J.2,2)-RIDI(IJ.1,2)*EG(I,J.1,2))
98
& -RHO1(1,J,2)*EG(I,J,2)))/DELZ
& +MUG(I,J)*(2.04(EG3**2.0+EG#*2.0.E5*'2.0)& -2.cw3.O*(EG3.EG4+EG5)"2.O+EC,6"*2.0+AVG(UG,I+1,J,2, 1)& *(I4Qi3+GR+1./3.Ot4GR5)+AVG(WG,I,J+1,2,2)& *(MGZ3+GZI+1 .O/3.O*NGZ5))& -QDOT(I,J)-EG?+EG8+EG9+EGIO))/RiO1(I,J,l)
C4 C
C PAR~TICLE ENERGY EQUATION
A CEP1=(AVG(RHO2,1+1,J,2, 1 )*P(I+1 ,J,2)*(R+DELR/2.0)
& *(AVG(EP,I41,J,2,1)*((PPlKF(TI,J)4PPlNF(I:J))/2.O)& /AVGt'R02,11,J,2,1))-AVG(RHO2,I,J,2,!)& *UP(I,J,2)*(R-DELR/2.O)*(AVG(EP,I,J,2,I)& +((PP'MF(I,J).PPlMF(I-1,J))/2.O)& /AVG(H2,1,J,2,1)))/DU~R
& EP2=(AVG(RI*02,I,J*1 ,2,2)'WP(IJ+e1,2)
& /AVG(RHO02,I,J+1,2,2))-AVG(R102,I.J,2,2)& *WP(I,J,2)*(AVG(EP,I,J,2,2)& +((P1F(1,J)+PPIKF(I,J-1))/2.O)& /AVG(RHO2,I,J,2.2)))/DELZ
CEP3=EG7
CEP4:GAMOA(I,J)*(EPCHENAVG(UPI+1,J,2,1)**2.0/2.o
& -AVG(WP,I,J+1,.-,2)**2.0/2.0,C
& -RHO2(I,J,2)'EP(I,4,2))-AVG(ARVISR,I,J, 1.1)'(RHO2(I,J,2)& *EP(I,J,2)-RJIO2(1-1,J,2)*EP(T-1,J,2)))/DIiR
IF(I.EQ.1)& EPS=(AVG(ARVISR,I.2,J?1,1)*(RH02(I+2,J,2)*EP(1+2,J,2)& -RHI2(I+1,J,2)'EP(1.1,J,2))-AVG(ARVISR,I.1,J,1,1)& *(R102(I.1,J,2)*EP(I.1,J,2)-RH02(I,J,2)*EP(I,J,2)))/DELR
IF(1.EQI.ENDR)& EP5=(AVC(ARVISR,I,J,1,1)*(RH02(I,J,2)*EP(I,J,2)
*& -RHO2(I-1,J,2)*EP(I-1,J,2)),AVC(ARVISR,I-1,J,1,1)& *(H2I1J2*PI1J2-H0(-,,)E(-,,))DL
CEP6=(AVG(AP.VISZ,I,J.1,1,2)O(RH2(,J+e1,2)*EP(I,J.1,2)
& -RH02(I,J,2)'EP(I,J,2))-AVG(ARVISZ,I,J,1,2)*(RHO2(I,J,2)& *EP(I,J,2).-RHO2(1,J-1,2)'EP(I,J-1,2)))/DELZ
IF(J.EQ.1) EP6=(AVG(ARVISZ,I,J+2,1,2)'(RH02(1,J+2,2)& *EP(I,J+2,2)-RH02(I1J.1,2)EP(I,J.1,2))& -AVG(ARVISZ,I,J.1,1,2) 4(RH2(I,J1,2)*EP(I,J+1,2)& -R102(I,J,2)*EP(I,J,2)))/DELZ
a 99
& +QDOT(I,J)+EP3+EPW,.EP5+EP6))/RHO2(1,J,1)CC
IF(C0UNT.CE.00.AND.COUNT.LE.10)1 fTENWRITE(6,100) I,JWRITE(6,200) CGI,CG2,C03,CGM,(RHOI(I,J,r),K: ,3)WRITE(6,200) CP1,CP2,CP3,CP4,(RH2(ItJIC),K=1,3)WRII'E(6,300) HGRt,NCR2,HUG(I,J,40R3,4GR4,NGR5,MGR6,
& DRAGR(I,J),4oR7,4GR8,?4R9,(Uo(I,J,K),x:1,3)& WRITE(6,300) HGZ1,HCZ2,MUG(I,J),MGz3,4GZ14,NGZ5,4GZ6,& DRAGZ(I,J),MGZ7,MGZ8,MGz9,(W(I,J,K),c:1,3)
I bWRlTE(6,1400) MPRI,MPR2,KPR3,MPRJ4,KPR5,(UP(I,J,K),K=1,3)WRITE(6,1400) MPZ1,FZ2,PZ3,PZ4,MPz5,(IWP(I,J,K),K=1,3)WRITE(6,500) EG1,EG2,EG3,EG-4,EG5,EG6,EG7,EG8,EG9,EG1O,
& (EG(I,J,K),K=1,3)WRITE(6,600) EP1,EP2,EP3,EP4,EP5,EP6,CEP(I,J,K),K=1,3)
100 FORMATUf,'PQSITION ',lI,2(I3))200 FORI4AT(lX,T(2X,G1O.4))300 FORI4AT(lX,10(2X,GI0.14)/lX,14(21,Glo.4))400 FORMAT(lX,8(2X,GIO.4))500 FORI4AT(lX,1O(2X,G1O.4)/lX,3(2z,GlO.4))600 FORKAT(lX,9(2X,G1.4)//)
END IFCC TAUGGR(I,J):MUG(I,J)*(NGR3+NGRJI+1.O/3.0OMCR5)C TAUGGZ(I,J)=MUG(I,J)*(mGz3IGZ4+1.0/3.0*MGZ5)C20 CONTINUE
.- f ~10 CONTINUE
CC CALL DUMP2CCC BOUNDARY CONDITIONSCCC AT Z=0 END WALL SYMMETRY CONDITIONSC
DO 30 I--0,ENDR+1RHO1(I,0, 1)=RIIO1(I, 1,1)
UG(I,0,1 )=UG(I,1,1)
WG(i,0,1 )=WG(I,2,1)UP(I,0, 1 )UP( , 1 *1)
V. WP(I, 1,1 )=0.OWP(I,0, 1 )WP(I,2, 1)EG(I,0,1 )=EG(I, 1,1)EP(I,0,1 )=EP(I,1,1)
30 CONTINUECC 'IAT R=O CENFTER LINE SYMMETRY CONDITIONS
DO 40 J=1,ENDZ
100
iRH02(0,J, 1)=HH02(1, j, 1)UG(1,J 1)=:0.0UG(O,J, 1)=UG(2,J, 1)WG(0,J,1)=WC(1,J,1)
UP(1,J, 1)=P2j 1).UP(O,J, i)=UP(2~, 1)
EG(O,J,1)zEG(1 ,J,1)EP(0,.J,I)=EP(l,J,1)
40 CONTINUE
C 0 0IOED+
C' AT Z=ENDZ END WALL RADIATIVE CONDITIONS *
RHO1(I,ENDZ.1, 1 )(5.ORHI(I,EIDZ, 1)-4.O'RHOI(I,ENDZ-1, I)& +RHO1(I,ENDZ-2,1))/2.O
RHO2(1,ENDZ.1,1):(5.0*RH2(,ENDZ,)-4.0R02(I,ENDZ.1,1)& +RHO2(I,ENDZ-2,1))/2.0
& G(IENDZ-,1))/.0GIED,)40U(ENZ11
& +UG(I,ENDZ-2,1))/2.O
& *U(I,ENDZ-,1))/.O*GIED,)40WIEM11
& +WP(I,ENDZ-2,1))/2.OEG(I,ENDZa.1,1)=(5.OECP(IENDZ,1)-4.0oU(I,DZ.1, 1)
& .EG(I,ENDZ-2,1))/2.OWP(I,ENDZ.1,1)=(5.O*EP(I,ENDZ1)-$.O'EPCI,E Mz-1, 1)
& +EP(I,ENIYL-2,1))/2.O
50 CONTINUECC **AT R=ENDR CIRCUNFERENTIAL WALL NO-SLIP CONDITIONSC CRACK OPENING CONDITIONSCC NOTE: NO-SLIP CONDITIONS ARE APPLIED ONLY IFC THE VISCOUS GAS EFFECTS ARE TO BE INCLUDED.C CONSEQUENITLY, ACTUAL BOUNDARY CONDITIONSC ARE APPLIED VIA ONE SIDED DIFFERENCING INC THE MAIN INTEGRATION LOOP IF TOGLEm1.C CONDITIONS LISTED HERE ARE ONLY UJSED TO KC FALSE EXTERIOR NODES FILLED AND HENCE THE VALUESC MAOR MAY NOT BE CORRECT.C
DO 60 J=1,ENDZRHO1(ENDR+1,J,1)=RM~1(ENDR,J, 1)1W02(ENDR+1 ,J, 1 )RH02(ENDR,J, 1)UG(ENDR+l .J, 1 )O.0MG(ENDR.1 ,J, 1)=WG(ENDR,J, 1)UP(ENDR+1 ,J, 1)=O.OWP(EIIDR.1 ,J, 1 )WG(ENDR,J, 1)EG(ENDR+t ,J, 1 )EG(ENDR,J, 1)9P(ENDlR+1 ,J,l1)xEP(ENDRJ, 1)
101
K Ki 7
60 CONTINUECC h------ ---h------h----ue~uC *REDU5CTION OF TRUNCATION ERRORS-*c iu.eiiueuu**mw:u*uC
DO 70 I=O,ENDR+lDO 75 J=O,ENDZ+l
IF(AB$(UG(I,J,1)).LE.TOL) IG(I.J,1):0.0
IF(ABS(WP(I,J,1)).LE.TOL) 4P(I,J,1)=O.O
75 CONTINUET0 CONTINUE
C
C *TIME SMOOTHERCC
DO 80 I=0,ENDR.1DO 85 J=O,ENDZ.1EGfO:EG(I,J,2)*RHJI(I,J,2)+EPS(EG(I,J,1)ORHO1(I,J, 1)
& -2.O*EG(I,J,2)*RQI(I,J,2)+EG(I,J3)RIO(I,4,3))EPRH02=EP(I,J,2)*RH2(,J,2).,P*(EP(I,J,1)*RR02(1,J,1)
& -2.OOEP(I,J,2)*RRO2(IJ,2)e.EP(I,J,3)*iWO2(1,J,3))
& .-Z.O*UG(IJ2)RED(I,J,2)E.UC(J ,JIO1IJR1,3)& RHO1:G(I,J,2)RHO1(I,J,2)+EFV(IJ,1)*RhI(IJ1)
& -2.O#VC(I,J,2)'RH01(I,J,2)+VG(I,J,3)*RH01(I,J,3))
& -2.0.IJ(I,J,2)*RH02(I,J,2).4JP(I,J,3)*RJV2(I,J,3))WR102=VP(I,J,2)'RHD2(1,J,2)+EPS*(VP(I.J,1)*IIO2(I,4,1)
& -2.0iWP(I,J,2)*RHD2(I,J,2).VP(I,J.3)RI2(I,J,3))
& .4W01(1,J,3))
& 4RHO2(1,J,3))UG(I,J,2)=UGRHO/RHO(I,J,2)WGI,J,2)WGRHDI/RHO(I,J,2)UF(i,J,2):UPRBO2/REIO2(I.J,2)WP(I,J,2)=WPRH2/RH2W1J,2)
EP(I,J,2)=EPRHO2/RH02(I,J,2)IF(COUJIT.GE.0.AND!.COUNT.LE. 10) THEN
WRITE(6,700) I,J,RHO1(I,J,2),RB0D2(1,J,2),uG(r,J,2),& WG(I,J,2) ,UP(I,J,2),WP(I,J,2),EG(r,J,2),EF(I,J,2)
TOO FORlMAT(1X,2(13),8(GlO.l))END IF
85 CONTINUE80 CONTINUE
RETURNEND
c
102
FUNCTION AVG (VALU, SITER, SITEZ, TIME, DIR)
C LOCAL VARIABLESC
REAL VALU(O:6,O:61,3)INTEGER SITER, SITEZ, TIME, DIR
cC
IF (DIR.EQ.1) THENAVG:(VALU(SITER,SITEZ,TINE)+VALU(SITER-1,SITEZ,TIME))/2.0
ELSE IF (DIR.EQ.2) THENAVG:(VALU(SITER,SITEZ,TIME)+VALU(SITER,SITEZ- 1,TIME) )/2.OEND IF
RETURNEND
CCC
FUNCTION DBLAVG (VALU, SITER, SITEZ, TIME)CC LOCAL VARIABLES ~C
REAL VALU(O:6,o:61,3)INTEGER SITER, SITEZ, TIME
.1 C
DBLAVG=(VALU(SITER,SITEZ,TIME).VALU(SITER-1 ,SITEZ,TIME)& +VALU(SITERSITEZ-1 ,TIME)+VALU(SITER-1 ,SITEZ-1 ,TIME))/14.0RETURNEND
CCC
SUBROUTINE DUMPCC LOCAL VARIABLESC
REAL RLOC, ZLOC, PGST, RADPSI, TIIEXJT, DYQUT, PWALOTINTEGER 1, J
CC **COMMON VARIABLES
B CREAL RHO1(O:6,0:61,3), RHO2(O:6,0:61,3), (0(:60o:61,3),& IP(0:6,O:61,3), WG(O:6,O:61,3), VP(0:6,o:6i,3),& EC(O:6,O:61.,3), EP(0:6,O:61,3)REAL RHOG(O:6,0:61), RHOP(O:6,0:61), P0(0:6,0:61), PP(0:6,0:61),& PGPHI(0:6,0:61), PPlNF(O:6,O:61), TG(O:6,O:61), TP(0:6,0:61),& PHI(O:6,O:61), RADP(0:6,0:61), ?VG(O:6,0:61),& GAJ4AC(0:6,0:61), DRAGR(O.±6,0:61), DRAGZ(0:6,0:61),& QDOT(0:6,O:61), ARVISR(O:6,0:61), ARtVISZ(0:6,O:61)INTEGER IGN(0:6,O:61)REAL RLEK, MLEN, DELR, DELZ, DELT, STEP, EP3, ARV, TOL, R, TIME,
& CVG, CVP, RD, MW, B1, TGO, TPO, TCH, TIGI, B, BUT, EPT, RHOPO,
103
& P1110, PG0, PCH, PWALL, RADPO, KO, HU=O, PRINTEGER INIT, TOGGLE, ENDR, SNDZ, STOPIT, COUNT, MAX.
& RPRINT, ZPRIRT, TPRINT, NTGR, NTGZ, KTPR. NTPZ, NPGR, NPGZCOtfHON /PRI/ RHO1, RHIO2, UG, UP, WG, VP. EG, EPCOMMON /SEC/ RHOG, RHlOP, PG, PP, PGPHI, PPIKF, TG, TP, PHI,& RADP, IGN, HUG, GAMlKAC, DRAGR, DRAGZ. QDOT, ARVISR, ARVISZ
COMMON // RLEN, ZLEN, DELR, DELZ, DEL?, STEP, EPS, ARV, TOL, R,& TIME, CYG, CyP, RO, MW. Bl, TGO, TPO, TCli, TIGN, B, BNT, EPT.& RHOPO, PHIO, PG0, PCH, PYALL, RADPO, [0, HUGO, PR, INIT, TOGGLE,& ENDR, ENDZ, STOPIT, COUNT, MAX,& RPRINT, ZPRIMT, TPRINT, NTGR, NTGZ, NTPR, NTPZ, NPGR, NPGZ
CC
EXTERNAL AVGPRINT 100DO 10 I=1,ENDR
RLOC: I*DELR-DELRf2 .0DO 20 J=1,EMIDZ
ZLOC=J*DELZ-DELZ/2 .0PGSI=PG( I,J)*1 .OE-10RADPSI:RADP(I,J)*I .0E1IJGPTAVG(UG,IJ,2, 1)WGPRT=AVG(UG,I,J,2,2)
-' UPPRT=AVG(UP,I,J,2, 1)WPPRT=AVG( HP, 1,J,2,2)IF (((J.EQ.0).OR.(J.EQ.ENDZ).OR.((J/ZPRINT)'ZPRINT.EQ.J))
* & .AND.((I.EQ.O) .OR.(I.EQ.ENDR).OR.((I/RPRINT)*RPRINT.EQ.I)))& PRINT 200,RLOC,ZLOC,UGPRT,UPPRT,VGPRT,WPPRT,& PGSI,TG(I,J) ,TP(I,J),RHOG(I,J),RHDP(I,J),PHI(I,J),RADPSI.& GAIIAC(I,J),QDOT(I,J),DRAGR(I,J),DRAGZ(I,J)
20 CONTINUEPRINT 40
10 CONTINUEPWALOT:PVALL1. OE- 10TIMOUTTIHE*1 .E6DTOUT=DELTl.E6PRINT 300 ,TINOVT,COUNT,DTOUT,PW AL0T'
CC100 FORIAT('1',lX,'RLOC',3X,'ZLOC',5X,'UG',6X,'UPI,61,'VGO,61,'WP',
& 3X,'GAS PRESS',3X,'TG',51,'TP',4X,-RH0G',3X,REP',3X,'PHI',& 2X,'RADIUS',2X,'GAMA',51,'QDOT',5X,'DRAGR',3Z,'DRAGZ'/& 3X,2('GM',5X),AI('CN/S',i4X),'GPA',5X,2('DEG K',31),'G/cC't& 3X,-G/C',1OX,'UM/134('-'))
200 FORI4AT(' ',2(FS.3,lX),14(F7.O,11),F8.5,1X,2(F6.O,1X),2(F6.I,lX),& F5.3,lX,F6.2,4(lXG8.2))
300 FOFJ4AT(/ DATA IS RECORDED AT 1,F8.4,- M~ICRO-SECODS,6X,& 15,' INTEGRATIONS AT THIS TIMEV// CURRENT TIME STEP IS '
& F8.4i,- MICRO-SECONDS' ,6X,kAXI4UM WALL PRESSURE IS,*1 & F8.5,' GPA')
4 400 FORIKAT( ')RETURNEND
C
104
SUBROUTINE OUMP2(TAUGGII,TAUGGZ,DRACR, DRAGZ , EDR, EIDZ , D~R, DEZ)REAL TAUGGR(0:6,O:61),TAUGGZ(0:6,0:61),DRAGR(0:6,.:61),
& DRAGZ(0:6,0:61),DELR,tELZ,LAR,LAZ,RLOC,ZLOCINTEGER I,J,EYDR,EJIZWRITE(7, 100)DO 10 I=0,ENWRRLOC= IDELRD0 20 J=O,ENDZ
ZLOC=J*DELZLAMR:TAUGGR( I,J)/DRAGR( I,J)LAHZ=TAUGGZ( I,J)/DRAGZ( I,J)WRITE(7,200) RL.OC,ZLOC,TAUGGR(I,J),TAUGGZ(I,J),DRAGR(I,J),
& DRAGZCI,J),LAMR,LAMZ20 CONTINUE
WRITE( 7,300)10 CONTINUE100 FORMAT('1',lx,'RtL0C'I3X,'ZL0C',3X,'TAUGGRI,3X,'TAUGGZ',3X,
& tDRAGR',l4X,'DRAGZ',X,'LA4 R',4X,'LAM Z'/3X,'CHt,51,'Cx',& 4X,4('DYN/CC',3X)/68(t-I))
200 -FORKAT(2(1X,F6.3),6(lX,G8.2))300 FORI4AT( I
* RETURNENID
105
LIST OF SYMBOLS
Symbol Definition Units
a speed of sound in gas cm/s
b burning rate coefficient (cm/s)/(dyn/cm2 )n
hI non-ideal gas equation of state co-volume cm3/g
gas specific heat at constant volume erg/g'K
Cvp particle specific heat at constant volume erg/g*K
Dr gas-particle interaction drag,radial component dyn/cm 3
D gas-particle interaction drag,
axial component dyn/cm
V Eg gas total energy erg/g
Ep particle total energy erg/gE che m gas chemical energy releasedg during combustion
erg/g
E pchem particle chemical energy releasedduring combustion erg/g
ign particle ignition energy erg/g
fpg interphase friction factor
hp Pconvective heat transfer coefficient erg/scm2.K
kg gas thermal conductivity erg/scm*K
K0 bulk modulus of solid g/cms 2
L total length of bed cm
n burning rate index
Pg gas phase pressure dyn/cm2
P p particle phase pressure dyn/cm 2
106
LIST OF SYMBOLS (continued)
Symbol Definition Units
P go initial gas phase pressure dyn/cm2
P Po initial particle phase pressure dyn/cm2
Pr gas Prandtl number
0 interphase convective heat transfer erg/cm 3 "s
r radial coordinate
rp particle radius Pm
,::ro initial particle radius Pm
surface burning rate cm/s
R total bed radius cm
R' gas constant erg/K"gmol
Re Reynolds number
t time s
Tg 9dS temperature K
Tp particle temperature K
Tgo initial gas temperature K
T initial particle temperature K
Ug radial component of gas velocity cm/s
1j b ,jp radial component of particle velocity (M/!;
Wg axial component of gas velocity Cmls
axial component of particle velocity cm/s
z axial coordinate
zc location of opening in outer wall cm
rc gas production rate during combustion q/cm3"s
107
-. ...I ...
LIST OF SYi4BOLS (concluded)
Symbols Def0tion Units
6 finite differencing operator
ar radial distance between nodes cm
AZ axial distince bftwen nodes cm
Vzc length of opening in outer wall cm
time filter coefficient
Mg gas viscosity poise
U initial gas viscosity poise
Pg gas phase density g/cm 3
particle phise density g/cm 3
J initial gas phase density g/cm 3
initial particle phase density g/cm3
p1 bulk gas density g/cm3
p? bulk particle density g/cm3
shear stress components dyn/cm2
porosity
108
< _ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~p. A ........ ,1 I
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