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//I-3 _ ' " I'

• _ t<._

DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS.,_f_,7_ _'- COLLEGE OF ENGINEERING & TECHNOLOGY .....

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA 23508

TRANSIENT RADIATIVE ENERGY TRANSFER IN

INCOMPRESSIBLE LAMINAR FLOWS

By

S. N. Tiwari, Principal Investigator

and

D. J. Singh, Graduate Research Assistant

Progress Report

For the period ended June 30, 1987

Prepared for the

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virginia 23665

Under

Research Grant NAG-1-423

Dr s. A. Kumar and J. P. Drummond, Technical Monitors

HSAD- Computational Methods Branch

(IJISJI-CR-1826C5) ilii_Iglt :lllEiiliIVE E_IEEG][

_BIISIEE Ill IICC_EtES$IBLI LIEIliit ELOW$

IIOg/ISS lepoit, l:eii©d endtig 30 Jun. 1987

({]d ])OliJaiOll OiiV.) 285 _P CSC I- 20D

June 1987

G3t34

H88-18883

Oncla_

0129'981
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DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS

COLLEGE OF ENGINEERING & TECHNOLOGY

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA 23508

TRANSIENT RADIATIVE ENERGY TRMSFER IN

INCOMPRESSIBLE LNIIINAR FLOWS

By

S. N. Tiwari, Principal Investigator

and

D. J. Singh, Graduate Research Assistant

Progress Report

For the period ended June 30, 1987

Prepared for the

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virginia 23665

Under

Research Grant NA8-I-423

Drs. A. Kumar and J. P. Drummond, Technical Monitors

HSAD- Computational Methods Branch

Sublitted by the

Old Dominion University Research Foundation

P.O. Box 6369

NorfoZk, Virginia 23508

June 1987

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FOREWORD

This is a progress report on the research project entitled "Analysis

and Computation of Internal Flow Field in a Scramjet Engine," for the

period ended June 30, 1987. Specific attention has been directed to

investigate "Transient Radiative Energy Transfer in Incompressible Laminar

Flows." This work was supported by the NASA Langley Research Center

(Computational Methods Branch of the High-Speed Aerodynamics Division)

under research grant NAG-I-423. The grant was monitored by Drs. A. Kumar

and J. P. Drummond of the High-Speed Aerodynamics Division.

ii
TRANSIENT RADIATIVE ENERGY TRANSFER

IN INCOMPRESSIBLE LAMINAR FLOWS

S. N. Tiwari I and D. J. Singh 2

Old Dominion University, Norfolk, Virginia 23529-0247

ABSTRACT

Analysis and numerical procedures are presented to investigate the

transient radiative interactions of nongray absorbing-emitting species in

laminar fully-developed flows between two parallel plates. The particular

species considered are OH, CO, C02, and H20 and different mixtures of these

species. Transient and steady-state results are obtained for the temperature

distribution and bulk temperature for different plate spacings, wall

temperatures, and pressures. Results, in general, indicate that the rate of

radiative heating can be quite high during earlier times. This information is

useful in designing thermal protection systems for transient operations.

IEminent Professor, Dept. of Mechanical Engineering and Mechanics.

2Graduate Research Assistant, Dept. of Mechanical Engineering and Mechanics.

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TABLE OF CONTENTS

Page

FOREWORD........................................................... ii

ABSTRACT........................................................... iii

NOMENCLATURE....................................................... vi

INTRODUCTION ....................................................... 1

BASIC FORMULATION.................................................. 3

LIMITING CASES AND SOLUTIONS ....................................... 9

Steady Laminar Flow ........................................... 9

Negligible Radiation .......................................... 10

Optically Thin Limit .......................................... 11

Large Path Length Limit ....................................... 13

METHOD OF SOLUTION ................................................. 14

Steady-State Solutions ........................................ 14

Transient Solutions ........................................... 16

PHYSICAL CONDITIONS AND DATA SOURCE................................ 19

RESULTS AND DISCUSSIONS ............................................ 23

CONCLUDING REMARKS................................................. 58

REFERENCES......................................................... 59

APPENDIX A: SPECTRAL INFORMATION AND CORRELATION FOR

IMPORTANT INFRARED BANDS.............................. A-I

APPENDIX B: RESULTS OF THE THE QUARTIC FORMULATION................ B-1

APPENDIX C: COMPUTER CODE FOR TRANSIENT ENERGY TRANSFER

IN INCOMPRESSIBLE LAMINAR FLOWS....................... C-1

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14a

LIST OF FIGURES

P age

Physical model and coordinate systems for flow of

radiating gases between parallel plates ..................... 4

Comparison of quadratic and quartic results for the

centerline temperature variations with time; Tw = 500 K,

P = 1 atm, and L = 5 cm ..................................... 24

Comparison of quadratic and quartic results for

temperature variations across the duct; P = 1 atm,

L = 10 cm, and Tw = 500 K ................................... 25

Comparison of quadratic and quartic results for

temperature variations across the duct; P = 1 atm,

L = 10 cm, and Tw = 1,000 K ................................. 26

Comparison of quadratic and quartic results for the bulk

temperature; P = 1 atm and Tw = 500 K ....................... 27

Comparison of general and limiting results for the

centerline temperature variations with time; Tw = 500 K,

P = 1 atm, and L = 5 cm .................................... 29

Comparison of centerline temperature variations with

time for Tw = 500 K and 1,000 K; P = 1 atm and L = 5 cm ..... 30

Comparison of temperature variations with time for

= 0.25 and 0.5; Tw = 1,000 K, P = 1 atm, and L = 5 cm ..... 31

Comparison of centerline temperture variations with

time for L = 5 cm and 10 cm; Tw = 1,000 K and P = 1 atm .....

32

Comparison of temperature variations across the duct

for Tw = 500 K, 1,000 K, and 2,000 K; P = I atm and

L-- 10 cm .................................................... 34

Comparison of temperature variations across the duct

for OH; Tw = 500 K, P = 1 atm, and L = 10 cm ................ 35

Comparison of temperature variations across the duct

for CO; Tw= 500 K, P = 1 atm, and L = 10 cm .................. 36

Comparison of temperature variations across the duct

for C02; Tw = 500 K, P = 1 atm, and L = 10 cm ................ 37

Comparison of temperature variations across the duct

for H20; Tw = 500 K, P = 1 atm, and L = 10 cm................ 38
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List of Figures (continued)

Figure Page

14b Comparison of temperature variations across the duct

H20; Tw = 1,000 K, P = 1 atm, and L = 5 cm .................. 39

15a Variation of bulk temperature with plate spacing for OH;

Tw = 500 K and P = 1 atm ..................................... 42

15b Variation of bulk temperature with plate spacing for OH;

Tw = 1,000 K and P = 1 arm ................................... 43

16 Variation of bulk temperature with plate spacing for CO;

Tw = 500 K and P = 1 atm ..................................... 44

17 Variation of bulk temperature with plate spacing for C02;

Tw = 1,000 K and P = 1 atm .................................. 45

18 Variation of bulk temperature with plate spacing for H20;

T w = 1,000 K and P = 1 atm ................................... 46

19a Comparison of bulk temperature results for Tw = 1,000 K

and P = 1 atm ................................................ 47

19b Comparison of bulk temperature results for P = 1 atm ......... 48

20 Bulk temperature results for CO2 + H20 for P = 0.1 atm ....... 50

21 Bulk temperature results for CO2 + H20 for P = 1 atm ......... 51

22 Bulk temperature results for CO2 + H20 for P = 10 atm ........ 52

23 Bulk temperature results for CO2 + H20 for Tw = 2,000 K ...... 53

24 Bulk temperature results for OH and H20 for P = 1 atm ........ 54

25 Bulk temperature results for OH and H20 for P = 10 atm ....... 55

26 Bulk temperature results for OH and H20 for Tw = 2,000 K ..... 56

27 Bulk temperature results for CO2 + H20 + OH for P = 1 atm .... 57

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A

Ao

CO

Cp

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e

mO

el,e 2

Hli ,HI

k

K1

M

N

P

qR

qR,_

q_

S

t

T

T1

Tb

U

Uo

Y

B

e

NOMENCLATURE

band absorptance = A(u,B), cm"1

band width parameter, cm-1

correlation parameter, atm -I - cm-I

specific heat at constant pressure, kJ/kg-k = erg/gm-k

Planck's function, (W-cm'2)/cm -I

Planck's function evaluated at wave number mo

emissive power of surfaces with temperatures T I and T2, W-cm-2

gas property for the large path length limit

thermal conductivity, erg/cm-sec-K

gas property for the optically thin limit distance between plates

large path length parameter, nondimensional

optically thin parameter, nondimensional

pressure, atm

total radiative heat flux, W/cm2

spectral radiation heat flux, (W-cm'2)/cm "I

wall heat flux, W/cm2

integrated intensity of a wide band, atm-l-cm -2

time, sec

temperature, K

wall temperature, K; T I : Tw

bulk temperature, K

nondimensional coordinate : SPy/A o

nondimensional path length = SPL/A o

transverse coordinate, cm

line structures parameter

nondimensional temperature

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eb

K

P

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dimensionless bulk temperature

spectral absorptin coefficient, cm-1

nondimensional coordinate : y/L : u/u o

density, kg/m 3

nondimensional time

wave number, cm"1

wave number at the band center, cm"I

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INTRODUCTION

The radiative energy transfer in participating medium has received

special attention in recent years because of its applications in the areas of

the remote sensing, earth's radiation budget studies and climate modeling,

fire and combustion research, entry

propulsion and defense-oriented research.

mass, momentum, and energy transfer,

and reentry phenomena, hypersonic

In most studies involving combined

however, the radiative transfer

formulation has been coupled mainly with the steady processes [1-20] and the

interaction of radiation in transient processes has received very little

attention. However, the transient approach appears to be the logical way of

formulating a problem in general sense for elegant numerical and computational

solutions. The steady-state solutions can be obtained as limiting solutions

for large times.

The limited number of studies available on the transient radiative

transfer in gaseous systems [21-31] are reviewed critically in [32-34]. The

literature survey reveals that the transient behavior of a physical system can

be influenced significantly in the presence of radiation.

The goal of this research is to include the nongray radiative formulation

in the general governing eauations and provide the step-by-step analysis and

solution procedure for several realistic problems. The basic formulations are

presented in [32] and the specific case of transient radiative exchange in

nongray gases between two parallel plates is investigated in [33, 34]. The

objective of this study is to investigate the interaction of transient radiation in fully-developed laminar flows between two parallel plates. Thus,

special attention is directed to include the nonsteady phenomenon only in the

energy equation. Homogeneous as well as nonhomogeneous mixtures of absorbingI
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emitting species are considered. In subsequent studies, the present analysis

and numerical techniques will be used to investigate the unsteady flow of

compressible and chemically reacting species in one- and multi-dimensional

systems.
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BASIC FORMULATION

The physical system considered is the energy transfer in laminar,

incompressible, constant properties, fully-developed flow of absorbingemitting gases between parallel plates (Fig. 1). The condition of uniform

surface heat flux for each plate is assumed such that the temperature of the

plates varies in the axial direction. Extensive treatment of this problem is

available in the literature under steady state conditions [8, II]. The

primary motivation of this study is to investigate the extent of transient

radiative interaction for high temperature flow conditions.

For the physical problem considered, the energy equation can be expressed

as [I]

@T @T @2T dp rBu_2

pCp (_tT + u _ + v T_ ) : k _ + BTH _ + _LT_] - divq R (1)

where u and v denote x and y components of velocity, respectively. In

deriving Eq. (1) it has been assumed that the net conduction heat transfer in

the x direction is negligible compared with the net conduction in the y direction. This represents the physical condition of a large value of the Peclet

number. By an analogous reasoning, the radiative heat transfer in the x

direction can be neglected in comparison to that transferred in the y

direction. If, in addition, it is assumed that the Eckert number of the flow

is small, then Eq. (1) reduces to

BT @T @T @2T I BqR (2)

where _ = (k/pCp) represents the thermal diffusivity of the fluid.

For a steady fully-developed flow, v = 0, and u is given by the wellknown parabolic profile as

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IIIIIIIIIIIIIIIIII

U_Zo

(J

I!

O"

corrO

0rr_

U,- O

3_O

u,- O

4-E

u_

_J u_

"O _2.

O_O _J

r.J,--

fO r0

qJ c-- OE

4_

GJ

r033 U•r- t/1

u_ Qj

O- O_

orL:-
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u : 6 um ({ - {2); { : y/L (3)

where um represents the mean fluid velocity. Also, for the flow of a perfect

gas with uniform heat flux, @T/@x is constant and is given by

@T/Bx : (2_qw)/(UmL/k) (4)

Now, by combining Eqs. (2) - (4), the energy equation is expressed in

nondimensional form as

@e + 12 (_ _2) @2e I _qR -- - - (5)

@T @62 qw _

where

T : _t/L 2 ; e = (T - Tl)/(qwL/k)

By assuming that the initial temperature distrihution in the gas is some

uniform value To = TI, the initial and boundary conditions for this problem

can be expressed as

e(_, 0) : 0 (6a)

e(0, T) : e (i, z) : 0 (6b)

e_(_ : I/2) : 0 ; e_(_ : O) : - e{({ : 1) (6c)

It should be noted that all the boundary conditions given in Eqs. (6) are not

independent and any two convenient conditions can be used to obtain solutions.

Also, the initial temperature distribution can be any specified or calculated

value of e(_, O) : f(_).

As discussed in Refs. 32-34, the radiative transfer equations are

formulated for one-dimensional planar systems for many engineering and

astrophysical applications. For diffuse nonreflecting boundaries and in the
radiative flux is given,

!

I absence of scattering, the expression for the total

I for an n-band gaseous system, by [I, 8, 32]

qR(y) = e I - e2

I n y ]dz

I + _ i_'=1 fAmi {fO Fl_i(z)_i exp[-_ K i(y-z)

L 3

I - I y F__mi (z) Kmi exp [-_T_ Kmi (z-y)]dz}dm_ (7) "

i where

FI (z) : e (z) - el ; F_ (z) : e (z) -

I "mi mi 'mi _mi _i e2_i

!

!

Equation (7) is in proper form for obtaining the nongray solutions of

molecular species. In fact, this is an ideal equation for the line-by-line

and narrow-band model formulations. However, in order to be able to use the

wide band models and correlations, Eq. (7) is transformed in terms of the

i correlation quantities as [1, 7-13, 32]

I qR (_) = el - e2

+ _ i A°i u°i{_o Flmi({' ) A_I [_ Uoi ({-{')]d{'

!

! where

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-I F2,.,i({') A_ [_ Uoi ({'-_)]d{'}

(8)

! = u/u ° = y/L; {' : u'/u o = z/L; A = A/Ao;

! u : (S/A o) py', uo = (S/Ao) PL', PS : _ mm dm

Am

!

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It should be noted that Flm i and F2_ i in Eq. (8) represent the values at the

center of the ith band and A'(u) denotes the derivative of A(u) with

respect to u. Upon performing the integration by parts, Eo. (8) can be

expressed in an alternate form as [32]

qR({) : eI - e 2

n

Aoi {f [demi i=1 o

({')/d{'] A i [_ Uoi({-_')]d_'

1

+: [de l ({')/d{'] A i [_ Uoi ({'-{)]d{'}

(9)

A direct differentiation of Eq. (9) provides the expression for the divergence

of radiatve flux as

dqR({) 3 n

d{ - _ i_1Aoi Uoi {_o [demi({')/d{'] x

1

A_ [_ Uoi({-{')]d{' - I [de,,,i({')/d{'] x

A} [_ Uoi(_'-_)]d_'} (I0)

Equations (8) through (10) are the most convenient equations to use when

employing the band-model correlations in radiative transfer analyses.

For the present physical problem, eI = e2 and Flm i = F2m i. Thus, for

the case of linearized radiation, a combination of Eqs. (5) and (8) results in

[8, 32]

n u2 {i{ o{{ - oT - 3No - 12 ({_{2) : (L/k) i:l_ Hli oi o O({', T) x

1

x_, [3 Uoi(___,)]d_,+ : o(_', _) x

A}' [_ Uoi({'-{)]d{ '} (lla)

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n

I N : (pL2/k) KI = (pL2/k) S Si(T) (de /dT)T1 i :1 _i

n

I Hli : Aoi(T) (de i/dT)T 1 , HI = s Hli i=l

i The dimensionless gas property N characterizes the relative importance of

radiation versus conduction within the gas under optically thin conditions.

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Also, by combining Eqs. (5) and (10) another form of the transient energy

equation is obtained as

n

B_C - eT -12 (__ 2):_ (L/k) i:lS Hli u°i {fo (@elB{') x

_,. 3 , , 1 ,

I [_ Uoi({-{ )]d{ - f{ (@01@{) x

I A_ [_ Uoi ({'-{)]d{'} (llb)

1

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Note again that Ea. (11b) can be obtained directly by integrating the righthand side of Eq (lla) by parts. Ouite often, Eq. (llb) is the convenient form

to use in radiative transfer analyses.

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For flow problems, the quantity of primary interest is the bulk

temperature of the gas, which may be expressed as [II]

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: O({,T)(_-_ )d{ (12)

ob (Tb-T1)/(qwL/k) : 6 IO

The heat transfer qw is given by the expression, qw = hc (T1-Tb), where hc is

the convective heat transfer coefficient (W/cm2-K). In general, the heat

transfer results are expressed in terms of the Nusselt number Nu = hc Dh/k.

Here, Dh represents the hydraulic diameter, and for the parallel plate

geometry it equals twice the plate separation, i.e., Dh = 2L. Upon

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eliminating the convective heat transfer coefficient hc from the expressions

for qw and Nu, a relation between the Nusselt number and the bulk temperature

is obtained as

Nu = 2 Lqw/k(T1-Tb) = -2/e b (13)

The heat transfer results, therefore, can be expressed either in terms of Nu

or Bb .

LIMITING CASES AND SOLUTIONS

Before discussing the method of solution for the general case, it is

advisable to explore the various limiting cases. Quite often, closed form

solutions can be obtained for some of these cases. Specifically four limiting

cases are considered here and these are the steady laminar flows, the case of

negligible radiation, the optically thin limit, and the large path length

limit.

Steady Laminar Flow

For steady-state conditions

of the energy equation for this case.

letting @e/BT = 0 in Eq. (5) such that

e" - 12 (___2) = (1/qw) dqR/d_

By integrating Eq.

qR({) and (de/d{)

BelBT : 0 and Eqs. (11) provide two forms

Another convenient form is obtained by

(14)

(14) once and using the conditions that at { = I/2,

are equal to zero, one obtains

6' - 2(3{ 2 - 2{3) + I : qR({)lq w

(15)

A combination of Eqs. (8) and (15) results in

s' - 2(3{ 2 - 2{3) + 1 : _ (L/k)

n

_ H1i Uoi {f

i=1 0

e(c')x

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_, 3 , 1

i [_ Uoi(_-_')]d_ - f_ o(_) x

it 3 ,__ i [_ Uoi(_ )]de'} (16)

It should be pointed out that by combining Eqs. (9) and (15) another useful

form of the energy equation can be obtained for the steady case.

For the case of negligible radiation, Eq. (16) reduces to a very simple

form and utilizing the boundary condition 0(0) = O, the solution of the

resulting equation is found to be

0(_) : _(2 _2 . _3 _ I) (17a)

The result for the bulk temperature is found by combining Eqs. (12) and (17a)

as

The results provided by Eqs.

radiatve contributions.

- 0b : 17170 (17b)

(17) are useful in determining the extent of

Negligible Radiation

For the case of negligible radiation, N = 0 and both forms of Eq. (11)

reduce to

- 0 : 12 ({_{2) (18)

e_ T

By employing the product solution procedure, the solution of Eq. (18) can be

obtained and the result can be expressed in terms of the bulk temperature

through use of Eq. (12). Alternately, the solution of Eq. (18) is assumed to

be of the form

B({,T) : g(5) + h(_,T) (19)

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!

11

From Eqs. (18) and (19), there is obtained two separate equations as

g" = 12({-C 2) (20)

h{{ - hT : 0 (21)

Solutions of Eqs. (20) and (21) are found separately from which the

complete solution for the temperature distribution is obtained as [32]

e({,T) : {(2 _2 _ {3 _ I) + s Cn sin (a _) exp(-a T); a2: n_ (22a)

n=l

where

Cn = (4/a 5 )[(12 - 12a 2 + a4) cos(a) - 24], n = I, 2,...

Finally, the expression for the bulk temperature is found by using Eq. (12) as

0b = -17/70 + 6 s C [(I/a) + (4/a3)] exp(-a2T) (22b)

n:l n

Equations (22) are useful in determining the extent of radiative contributions

for the transient case.

Opticall_ Thin Limit

In the optically thin limit, the expression for the bulk temperature for

the steady case is found to be [ii, 32]

eb = [I/(3N) 3] {576(3N)-I/2(NEXP) - 21.6N 2 + 72N- 288} (23)

where

NEXP : {l-exp [-(3NI/2]}/{I + exp[-(3N)I/2]}

Both forms of the transient energy equation, Eqs. (11a) and (11b), reduce

to a simplified form in the optically thin limit as [32]

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12

e_{ - eT - 3Ne = 12({-{ 2) (24)

Assuming a solution of the form given by Eq. (19), Eq. (24) is written as

h - h - 3Nh = - g + 3Ng + 12(_ - _2) (25)

Consequently,

g" - 3Ng : 12(_ - {2) (26)

and h - h - 3Nh : 0 (27)

From the solution of Eqs. (26) and (27), the solution for Eq. (24) is obtained

as

B(_,T) : (16/3N 2) [sinh (-j'3TI72)/sinh(_r3-FT)]

where

cosh [#3-lq- ({-I/2)] + (4/N)({ 2 - _ + 2/3N)

+ r. Cn sin (as) exp [-(3N + a2) T]; a : n_ (28a)

n=l

Cn 0 , for n even (28b)

: 32 (3N + a2)/(3N 2 a3)

+ 2a/[3N2(3N + a2)] , for n odd (28c)

By combining Eqs. (12) and (28), the expression for the bulk temperature is

obtained as

eb : 6 {(16/3NZ)[sinh(-#3"N/2)/sinh(_TN)] x

[(I/3N) cosh(JTN/2) - (4 + J-3N) (3N) "3/2 x
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|

!

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13

sinh(_-3-F]12)] + (41N) [-I130 + II(9N)]

+ _ Cn(I/a + 4/a 3) exp[-(3N + a2)T]} (29)

n:l

Large Path Length Limit

In this limit, the steady-state energy equation, Eq. (16), reduces to [8,

II, 32]

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o' - 2(3{ 2 - 2{ 3) + I : M f o(_') d{'/({ - {') (30)

0

where

n

M : HI L/k : (L/k) E

i:l Aoi (demi/dT)T 1

The nondimensional parameter M constitutes the radiation-conduction

interaction parameter for the large path length limit. Equation (30) does not

appear to possess a closed form solution; a numerical solution, however, can

be obtained easily.

In the large path length limit, the transient energy equations, Eqs.

(11a) and (11b), reduce to

I

+I

- 3NO - 12(_ - {2) : _ M [I

0

e(_',T) d{'/({ - _,)2

e({',T)d{'/({'-{) 2]

I

- - 12 ({ - {2) : M _ (BBIB{')d{'I({-{

e_ eT 0

,)

since I(_-{')21:I(_'-_)21, Eq. (31a) can be written as

I

0{{ - 0 - 3No - 12({ - {2) : _ M I

T 0

e(_' ,3) d{'l({ - {')2

(31a)

(31b)

(31c)

I
14

Through integration by parts, Ea. (31c) can be expressed as

I

- - 3Ne - 12(_ - _2) : r_f (@el@_')d_'/(_-_') (31d)

e_ eT 0

Equations (31a) - (31d) represent different forms of the governing equations

in the large path length limit. With the exception of the term (-3NB) on

the left-hand side, Ea. (31d) is identical to Eq. (31b). Since N represents

the radiation-conduction interaction parameter only in the optically thin

limit [8], it should not appear in the governing equation for the large path

length limit. Thus, Eq. (31b) is the correct equation to use for solution in

the large path length limit; the solution of this equation is obtained by

numerical techniques.

METHOD OF SOLUTION

The solution procedures for both steady and unsteady cases are presented

in this section. In principle, the same numerical procedure applies to both

the general and large path length limit cases.

Stead_-State Solutions

The general solution of Eq. (16) is obtained numerically by employing the

method of variation of parameters. For this, a polynomial form for e({) is

assumed in powers of _ as

n

o({) : _ am _m (32)

m:O

By considering a five term series solution (a quartic solution in

satisfying the boundary conditions e(O) = e'(I/2) : 0

one obtains

e({) : al(_ - 2{ 3 + 4) + a2({2 _ 2{3 + 4)

_) and

and e'(O) : -o'(I),

(33)
I

I

I

!

I

I

I

I

I

I

i

I

I

I

I

!

I

i

15

e'({) : a1(1 - 6{2 + 4{3) + a2(2_ - 6{2 + 4{3)

A substitution of Eq. (34) in Eq. (16) results in

a1(1 - 6{2 + 4{3) + a2(2{ - 6{2 + 4{3)

(34)

n

2(3{ 2 - 2{3) + 1 : _ (L/k) z

n=l

Hli Uoi x

{f

0

e(_') A} [_ Uoi(_ - _')]d_'

- f1 e({ ') A_I [_ Uoi ({' - {)] d_'} (35)

where expressions for 0({') are obtained from Eq. (33).

The two unknown constants aI and a2 in Eq. (35) are evaluated by

satisfying the integral equation at two convenient locations ({=0 and _=1/4

in the present case). The entire procedure for obtaining aI and a2 is

described in [32]. With known values of aI and a2, Eq. (33) provides the

general solution for e({). The expression for the bulk temperature is

obtained by combining Eqs. (12) and (33) as

eb : (1/70) (17a I + 3a2)

(36)

It should be noted that for the case of no radiative interaction a2 = 0 and a1

= -I, and Eq. (36) gives the result of Eq. (17).

The governing equation for the large path length limit is Eq. (30). For

this equation also the solution is given by Eqs. (33) and (36) but the values

of a's are completely different in this case [32].

I
16

Transient Solutions

The governing energy equations for the transient case are Eqs. (lla) and

(11b). General solutions of these equations are obtained also numerically by

employing the method of variation of parameters. For the present problem, a

polynomial form for e({,t) is assumed as

n

e(_,T) = _ am(T ) _m (37)

m:0

For a quadratic temperature distribution in _ (with time dependent

coefficients), Eq. (37) is written as

B(_,T) : a0(T) + at(T) _ + a2(T) _2 (38a)

By using the boundary conditions B(0,T) = 0 and e{({=1/2) : 0, this reduces to

e(_,T) : g(T) ({._2) (38b)

where g(t) represents the time dependent coefficient. Consequently,

B{({,T) : g(T) (I-2{); B{{({,T) : -2g(:); ez(_,T) : ({_ 2) g'(T) (39)

Also, a combination of Eq. (6a) and (38b) yields the initial condition

e(_,0) : g(0) : 0 (4O)

Note that essential boundary conditions are used already in obtaining the

solution represented by Eq. (38b).

By employing Eqs. (38b) and (39), Eqs. (11a) and (11b) are transformed in

alternate forms which are expressed in a compact form as

g'(T) + Jl(_) j2(_ )'] g(T) + 12 = 0 (41)
17

where Jl({) and J2(_) are defined in [32]. The function Jl({) is used for

solution of Eq. (lla) and J2({) is used for solution of EQ. (llb). The

solution of Eq. (41) satisfying the initial conditions of Eq. (40) is given by

g(T) : 12 {exp [-J({)T]-I} (42)

The temperature distribution given by Eq. (38b) can be expressed now as

12

0({,T) : J--(-_T{exp[-J({)T]'l}({-{2) (43)

The expression for the bulk temperature is obtained through use of Eq. (12) as

I

0b = 72 f [({-{2)2/J({)]{exp[-J({)_]-l} (44)

0

Note that in Eqs. (42)-(44), J({) becomes Jl({) for solution of Eq. (11a)

and J2({) for solution of EO. (11b).

For a quartic solution in {, Eq. (37) gives the result similar to Eq.

(33) which for the transient case is expressed as

O(_,T) = g(T) ({ - 2_ 3 + {4) + h(_) ({2 _ 2{3 + {4) (45)

By substituting EQ. (45) into Eq. (11a), one obtains

where

x g'(T) + J3({) g(T) + yh'(T) + J4({) h(T) : - z

x : ({ - 2{3 + {4); Y : ( 2 _ 2{3 + 4);

z = 12({ - {2)

(46)

and functions J3({) and J4({) are defined in [32]. Equation (46)

constitutes one equation in two unknowns, namely g(_) and h(T). However,

since the equation is linear in 3, the principle of superposition can be used

!
18

to split the solution into two equations as

x g'(T) + J3({) g(T) = -z/2

yh'(T) + J4({) h(T) : -z/2

The initial condition for this case can be written as

Consequently,

o({,0) : g(0) ({ - 2{ + 3)

+ h(0) ({2 _ 2{3 + {4) : 0

(47)

(48)

(49a)

(45), the expression for the

g(0) : 0; h(0) : 0 (49b)

The solution of Eqs. (47) and (48) satisfying the appropriate initial

condition of Eq. (49b) is given respectively as

g(T) = [z(5)/2J3({)] {exp[-J3({)T/x(_)]-l} (50)

h(T) : [z({)/2J4({)] {exp[-J4({)T/y({)]-l} (51)

By substituting Eqs. (50) and (51) into Eq.

temperature distribution is obtained as

e(_,T) : [6(_-_2)(_-2_3+_4)/J3(_)] x

{exp[-J3({)T/x({)]-l} + [6(_-{ 2) x

(_2-2_3+_4)/J4(_)] {exp[-d4(_)_/y(_)]-l} (52)

The bulk temperature in this case is given by

I

eb : 36 I [(_-_2)(_'2_3+_4)/J3(_)] x

0
!

I

i

I

I

I

i

19

I 2)

{exp[-j3({)T/x({)]-l}d { + 36 IO [(_-{ x

({2-2{3+{4)/J4({)]{exp[-J4({)T/y({)]-l}d{. (53)

where x and y are defined in Eq. (46).

By substituting Eq. (45) into Eq. (11b), there is obtained

xg' + J5(_) g(T) + yh + J6({) h(T) : -z (54)

where again x,y,z are defined in Eq. (46) and functions J5({) and J6({) are

defined in [32]. The solution procedure for this equation is identical to

that for Eq. (46) and the results for temperature distribution and bulk

temperature are given respectively by Eqs. (52) and (53) with J3 replaced by

J5 and J4 by J6"

In the large path length limit, the two applicable governing equations

are Eqs. (31b) and (31d). The solutions of these equations can be obtained

from the general solutions by evaluating the integrals in J function in the

large path length limit [32].

PHYSICAL CONDITIONS AND DATA SOURCE

As discussed in [32-34], four specific absorbing emitting species were

selected for an extensive study; these are CO, CO2, OH and H20. The species CO

was selected because it contains only one fundamental vibration-rotation (VR)

band and all spectral information are easily available in the literature. It

is a very convenient gas to test the numerical procedure without requiring

excessive computatinal resources. Species OH and H20 are the primary

radiation participating species for the pressure and temperature range

anticipated in the combustor of the scramjet engine. Species CO2, and

combinations of CO2 and H20 are important absorbing-emitting species in many

I
20

other combustion processes. Different mixtures of various species (such as

CO2 + H20, OH + H20, and CO2 + H20 + OH) were selected for parametric

studies. Thermophysical properties of these species are given in [33] for

different temperatures.

In radiative transfer analyses, it is essential to employ a suitable

model to represent the absorption-emission characteristics of specific species

under investigation. Several line-by-line (LBL), narrow-band, and wide-band

models are available to model the absorption of a VR band [7-12]. However, it

is often desirable to use a simple correlation to represent the total

absorption of a wide band. Several such correlations are available in the

literature [7-12]. The relative merits of these correlations are discussed in

[12]. In this study, the correlation proposed by Tien and Lowder [7] is

employed and this is given by

A(u) = In {uf(B) [u u+ +2F(B2 )] + I} (55)

where

f(B) : 2.9411 - exp(- 2.60 B)]

and B represents the line structure parameter.

The spectral information and correlation quantities and thermodynamic and

transport properties needed for different species were obtained from Refs. 7-9

and 33. Spectral information and correlation quantities for important

infrared bands are provided in Appendix A. The specific VR bands considered

for each species are: CO (4.7u fundamental), OH (2.8_ fundamental), CO2 (15u,

4.3u and 2.7_), and H2O (20_, 6.3_, 2.7_, 1.87_, and 1.38_).

In a mixture of several species, spectral lines and bands overlap in

certain spectral regions. The total absorptance in such regions cannot be
I

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l

l

!

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!

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t

I

I

1

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l

21

calculated simply by adding the contributions of different bands and

corrections should be made to account for the partial overlapping. If lineby-line or narrow band models are employed in the general formulation of the

physical problem, then there is no need for such corrections [12, 35, 36].

The solution of LBL formulation, however, requires considerably large

computational resources. Use of narrow band models offers some computational

relief but certain spectral information needed are not available for many

species for temperatures higher than about 600 K. A relatively easier

procedure (called the block method) is suggested by Fdwards [13] and is useful

in calculating the total emissivity of a mixture of several species. Another

method suggested by Penner and Varanasi [37] is probably the most convenient

method to use in the frame work of the radiative flux formulatin expressed in

terms of the wide-band model absorptance and correlations.

For a homogeneous path, the total absorptance of a band is given by

A(y) = f [I - exp(- K y)] d_ (56)

O

where both K and m have units of cm-1. If in a spectral range _i' there

are contributions from bands of different species, then for a homogeneous path

the transmittance is given by

N

TA,.,i : exp(- _ K y) (57) j=l mij

where N represents the number of participating species in the gaseous mixture.

Consequently, Eq. (56) can be expressed as

N

A i = f [I - exp(- _ K y)] d,., (58)

a_ i j=l _ij

!
22

If two bands of different (or same) species are occupying

same spectral range ami, then Eq. (58) reduces to

approximately the

A i = f {1 - expC_mi + _ )y]} dm (59)

Ami I _i 2

By employing the relations for the exponentials, Eqs. (59) can be expressed as

[37]

A i = fA,.,i[I - expC- _mil y]] dm + fami [1 - exp C- m i2) ] dm - AA (60a)

or

A = A 1 + A 2 - AA (6Oh)

where

AA= f {[1 exp(- y)]x

a_ i _iI

[I - exp(- K y)]} dm (60c)

mi2

Use of Eqs. (60) has been made by Felske and Tien [38] and Tien and Lee [39]

to calculate absorptances of homogeneous and nonhomogeneous mixtures of CO2

and H20 in the 2.7 u region for different pressure and temperature

conditions. A similar procedure is used in this study to account for the

overlapping effects of different species.

For the physical problem considered, the dependent variables are e and

(or Nu) and independent variables are T and _. The parameters, in ob

general, are Tw, P, and L. The large path length and optically thin limits

are characterized respectively by parameters M and N. Radiative and thermophysical properties of participating species are evaluated at different

specified pressures and temperatures.
!

I

!

I

23

RESULTS AND DISCUSSIONS

Extensive results have been obtained for variation of o and eb for

different conditions and most of these are available in Appendix B. The

computer program used for numerical solutions is provided in Appendix C.

Selected results are presented here to compare solutions of quadratic and

quartic formulations and demonstrate the variation of o with T and _ and

of ob with L for single component systems and homogeneous mixtures. For all

results presented here, a lower value of o (or ob) in the figures indicates

a higher value of temperature in the medium; this, in turn, implies a relatively higher ability of the gas to transfer radiative energy.

Results of quadratic and quartic formulations are compared in Figs. 2-5

for different species. The centerline temperature variations with nondimensional time are compared in Fig. 2 for P : I atm, Tw : 500 K, and L = 5 cm.

The results show that the steady-state conditions are reached at an eariler

time for H20; and this is followed respectively by C02, CO, and OH.

Significant differences are noted between the quadratic and quartic solutions

for larger times. For the conditions of results presented in the figure, the

difference are found to be greatest for OH and lowest for H20. The results

for o versus { are compared for P : I atm, L : I0 cm, and Tw : 500 K in

Fig. 3 and Tw : 1,000 K in Fig. 4. The results demonstrate that considerable

differences in solutions can occur at different locations in the channel and

that the differences are larger for the lower wall temperature. Results for

ob versus L presented in Fig. 5 show that the differences in two solutions

are relatively larger at lower plate spacings and that quartic results

approach the correct limiting solution for the case of no radiative interaction (Ob = -0.243). From the results presented in Figs. 2-5 and in Refs. 33

and 34 it is concluded that while quadratic and quartic solutions are
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identical for radiative equilibrium and radiation with conduction cases, they

differ significantly for the case of combined conduction, convection and

radiation. As such, all other results for this study were obtained by using

the quartic formulation.

The results for temperature variations with nondimensional time are

presented in Figs. 6-9 for various species and for different physical

conditions. The centerline temperature distribution for general and limiting

cases are illustrated in Fig. 6 for Tw : 500 K, P : I arm, and L : 5 cm. The

results show that the steady-state conditions are reached at about T : 0.5

for all species. As noted earlier, these results also demonstrate that H20 is

a highly radiation participating gas as compared to C02, CO, and OH. In

comparison to other species considered, OH takes relatively longer times to

reach the steady state and is least effective in transferring the radiative

energy. For the specified physical conditions, the large path length solutions are closer to the general solutions and optically thin solutions provide

higher rate of energy transfer. For all species, optically thin solutions

reach the steady state faster than other solutions. Also, in the optically

thin limit, CO2 is more effective in the radiative transfer process than other

species. The reasons for such trends are given in [8, 33, 34]. The results

for Tw : 500 K and 1,000 K are compared in Fig. 7, and they simply indicate

that the rate of energy transfer is higher at the higher temperature. The

results for { : 0.25 and 0.5 are compared in Fig. 8 for Tw : 1,000 K, P : I

atm, and L : 5 cm. It is noted that the rate of energy transfer is higher at

earlier times and at locations closer to the wall. The centerline temperature

variations for L : 5 cm and I0 cm are illustrated in Fig. 9 for Tw : 1,000 K

and P : I arm. As would be expected, the rate of energy transfer is seen to

increase with the increasing path length.
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Figure 14b. Comparison of temperature variations across the duct for H20;

Tw = 1,000 K, P = 1 atm, and L = 5 cm.

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The bulk temperature results as a function of the distance between the

plates are presented in Figs. 15-19 for different times. General as well as

limiting solutions are illustrated in these figures. As mentioned earlier,

the limiting value of eb : - 0.243 corresponds to negligible radiation. For

all species, the results presented in the figures are for a pressure of one

atmosphere. However, for any particular L, the large path length results

essentially represent the limiting solutions for high pressures. The results,

in general, demonstrate that the rate of energy transfer is higher at earlier

times, the effect of radiation increases with increasing plate spacing, and

the radiative transfer is more pronounced at the higher wall temperature.

General as well as limiting solutions for the bulk temperature are

illustrated in Figs. 15-18 for individual species. It is seen that for all

species the general solutions for T : 0.5 and 1.0 are essentially the same

for all plate spacings, and the large path length results are valid for

spacings greater than L = I0 cm for all times. The results for OH are

presented in Figs. 15a and 15b for Tw = 500 K and 1,000 K, respectively. It

is noted that optically thin results provide the correct limiting solutions

for plate spacings upto L = 3 cm for all times. The results for Tw = 500 K

show only slight difference between general and large path length solutions

for T = 0.5, and no significant difference was noted at earlier times (Fig.

15a). This, however, is not the case for the results presented in Fig. 15b at

Tw : 1,000 K. This trend in results for OH was noted also in [33, 34]. The

results for CO are illustrated in Fig. 16 for Tw : 500 K and P : i atm. In

this case, the optically thin solutions are seen to be valid only upto L : 1.5

cm. For Tw : 1,000 K and P : I atm, the results presented in Fig. 17 for CO2

and in Fig. 18 for H20 show the same general trend but the extent of radiative

interaction is entirely different. For C02, there is a considerable
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41

difference in general and optically thin solutions for all times. For H20,

however, the optically thin results are closer to the general solutions for

spacings upto L = 2.5 cm. The results presented in Figs. 15-18 clearly reveal

that for a fixed spacing between the plates the rate of radiative heating will

be considerably higher at earlier times than at the steady state. Thus, in a

particular physical system, the extent of radiative heating can be very

intense during the initial stages of operation. It is also important to note

that both the optically thin and large path length results overestimate the

influence of radiation. Since these solutions can be obtained with less

numerical complications, they can be utilized to assess whether or not, for a

given gas, the interaction of radiation is going to be important.

A comparison of the general band absorptance results for the four gases

is shown in Figs. 19a and 19b for a pressure of one atmosphere. The results

clearly demonstrate the relative ability of the four species for radiative

transfer at different path lengths. For a plate spacing of greater than L =

3.0 cm, the results show the same trend as noted in Figs. 2-10. For lower

plate spacings and relatively higher temperatures, however, CO2 shows a

significantly higher ability for radiative transfer than other species. This

is a typical distinguishing feature of the CO2 under optically thin conditions

[8, 33, 34].

For steady-state conditions, bulk temperature results were obtained for

mixtures of different absorbing-emitting species under various conditions and

some of these are presented in Figs. 20-27. The relative amount of each

species in the mixture and the relative ability of the species for radiative

transfer in a given physical condition determine the extent of radiative

interaction.

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e'_ ¢O

II

.._

,-- (_.

._'_

_.-0

00

•r-- II

SSI (Hl'lMb)l(q.I.-- M.L)=qe-- III
I

1

IIIIIIIIIIIIIIII

/

/

/o

II"

I!

¢/i

I

I III

{_ll-IMb)l[ql-- M1)=qe43

I
I

44

IIiIIIIIIIi

I

X

0 l,i") (0 It') It')

Q

I>ll"lMb)/lql- M1)=qeC.b

S. 0

4-C

ortJ

fO

u'l

¢;

rO

..C

-e"

3=

¢1)

EN

I)J 4-_

4-)

II

c-

_.. n_ 0 V

£--

rO •_ II

rO 3=

S,-

"-I

°pi,

III
IIIII

"r

!

X

I

/

/

/

(HI-IMb)I(q.L- M.L)=q[--

I

C)

45

EU

.J

C,J

C_

_.. 0

_'_

c'-

.r-- U

0.

tO

0.

c'-

4J

]¢

(1) LN

g.--_ •-_ eq.._' 0

C_

O_

•_'- II

t_

,.2

l.

,p.

UI
II

'1IiIIIIIIiIIIIII

(Hi1Mblltql -Ml)=qet_

¢:) P

46

EuJ

O

CM

._- 0

_..

.pU

tD.

e-E

_,_v 0•e,-

_

•P II

c_

°e,-

I
l 47 !!| _ 1

"

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I!!I!!IC_

g_

u'l .._

n_ v...q E I! 0

f,,.

.r.-

(>l/7Mbl/(ql- M1)-.-qe--

!!
48

IIII!i

I--

II

III

I

(Hl7Mblllql--Mll=qe--

III
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i

49

As mentioned before, in many combustion processes involving fossil fuels,

the predominant products of combustion are carbon dioxide and water vapor.

Extensive studies have been conducted in the literature to determine the total

emissivity of CO2 and H20 for homogeneous and nonhomogeneous conditions.

However, only limited studies are available involving mixtures of CO2 and H20

for realistic and important physical conditions. The bulk temperature results

for three different mixtures of CO2 + H20 are presented in Figs. 20-23 for

different temperatures and pressures. As would be expected, the results, in

general, show that the extent of radiative interaction increases with

increasing temperature, pressure, and path length. It is seen that the

radiative interaction is stronger for higher amount of H20 in the mixture.

However, for the optically thin conditions, the radiative contribution is seen

to increase with increasing amount of CO2 at relatively higher temperatures.

The bulk temperatures results for mixtures of OH and H20 are illustrated

in Figs. 24-26 and for mixtures of C02, H20, and OH in Fig. 27 for different

pressures and temperatures. All these results clearly demonstrate that the

radiative ability of a gaseous mixture essentially depends on the amount of

highly radiation participating species in the mixture. For example, a

comparison of results presented in Figs. 24 and 27 for P = 1 atm indicates

that the rate of radiative transfer is significantly higher with the inclusion

of 20% CO2 in the mixture of OH and H20. It is further noted that OH becomes

a highly radiation participating species at higher temperatures and pressures.
iIIIIIIIIIIIII!III

!

X

u')

II II

(>Ill Mb)/(ql-- M.L)=qe-

50

a-)

ro 0

II

o-0

4.. 0+

C_

CO

S.- 0

4.--

E_3

j_

f,-

SQ,. E

Sopt_

I
IIII

0

II II

II _'_ IiI

v

(_I17 Mb)/(ql-- MI): qeII

51

II

CL0

4-O

N

:3: +N0

(.JL0

u,..

E_

u3

j_

t_E3N

opI
iiIII

II II

I

(_l-I Mb Il (ql- MlI:qeI

53 o

II

F--

t_00

C_

"r"+

CO

S.. 0

EU

2;

Q)

$..

Q; E

C:O

C_J

or--

I
i

I

I

/

(>t/7 Mb)/(ql-- Ml):qeO

I!

O..

54

O

qO

"O

c'-

O

O

46.

E

U

a.)

a.)

E

'5

OO

_J

S,-

°l,-

I.JI
I 55

I

i

i

I

!

i

I

I

I

(_t/'1 Mb)/(ql - Ml)=qo--

i
56

I

I c_

,a

0

0

C_l

II

3_

f,-

0

k'-

0

C%1

"I0

SO

4-

C_

c:)

c:; c5 c5

U3 c)

I !O

X

f,-

Q.I

E

4,o

or-,

I (>t/7Mb)l(ql -M1)=qeI

I

I
I (_/7Mb)/(ql--M1)=qe-- III
I

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58

CONCLUDING REMARKS

Analytical formulations and numerical procedures have been developed to

investigate the transient radiative interaction of absorbing-emitting species

in laminar fully-developed flows. Extensive results have been obtained for

OH, CO, CO2 and H20 for different physical conditions, lllustrative results

for the temperature distribution and bulk temperature are presented for

different pressures and wall temperatures. In these results, a lower value of

temperature implies a relatively higher ability of the gas to transfer

radiative energy.

Comparative results of quadratic and quartic formulations confirm the

need to use the quartic formulation in the numerical procedures for the case

of combined conduction, convection, and radiation. The results, in general,

demonstrate that the steady-state conditions are reached at about T = 0.5

for all species, H20 is a highly radiation participating species (as compared

to CO2, CO, and OH), the rate of energy transfer is higher at earlier times

and at locations closer to the wall, differences between the limiting and

general solutions are small at earlier times, the effect of radiation

increases with increasing plate spacing, and the radiative transfer is more

prounounced at higher wall temperature and pressure. Similar conclusions can

be drawn from the results presented for various mixtures. The results clearly

show that for a given physical condition the radiative ability of a gaseous

mixture depends essentially on the amount of highly radiating species in the

mixture. From the results presented in this study, the extent of total

heating can be determined for different times. This information is essential

in designing thermal protection systems for operations during the initial

stages of intense heating.

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59

REFERENCES

1. Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, Brooks/Cole,

Belmont, Calif., 1966 and 1970. New Augmented Edition, Hemisphere

Publishing Corp., Washington, D.C., 1978.

2. Hottel, H. C. and Sarofim, A. F., Radiative Transfer, McGraw-Hill Book

Co., New York, 1967.

3. Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, McGrawHill Book Co., New York, 1971; Second Edition, 1981.

4. Cess, R. D., "The Interaction of Thermal Radiation with Conduction and

Convection Heat Transfer," Advances in Heat Transfer, Vol. 3, Academic

Press, New York, 1966.

5. Sparrow, E. M., "Radiation Heat Transfer between Surfaces," Advances in

Heat Transfer, Vol. 2, Academic Press, New York, 1965.

6. Viskanta, R., "Radiation Transfer and Interaction of Convection with

Radiation Heat Transfer," Advances in Heat Transfer, Vol. 3, Academic

Press, New York, 1966.

7. Tien, C. L., "Thermal Radiation Properties of Gases," Advances in Heat

Transfer, Vol. 5, Academic Press, New York, 1968.

8. Cess, R. D. and Tiwari, S. N., "Infrared Radiative Energy Transfer in

Gases," Advances in Heat Transfer, Vol. 8, Academic Press, New York,

1972.

9. Edwards, D. K., "Molecular Gas Band Radiation," Advances in Heat

Transfer, Vol. 12, Academic Press, New York, 1976.

10. Tiwari, S. N. "Band Models and Correlations for Infrared Radiation,"

Radiative Transfer and Thermal Control (Progress in Astronautics and

Aeronautics), Vol. 49, American Institute of Aeronautics and

Astronautics, New York, 1976.

II. Tiwari, S. N., "Applications of Infrared Band Model Correlations to

Nongray Radiation," International Journal of Heat and _lass Transfer, Vol.

20, No. 7, July 1977, pp. 741-751.

12. Tiwari, S. N., "Models for Atmospheric Radiation," Advances

Geophysics, Vol. 20, Academic Press, New York, 1978.

in

13. Edwards, D. K., Radiation Heat Transfer Notes, Hemisphere Publishing

Corporation, Washington, D.C., 1981.

14. Kuroksi, Y., "Heat Transfer by Simultaneous Radiation and Convention in

an Absorbing and Emitting Medium in a Flow Between Parallel Plates,"

Fourth International Heat Tranfer Conference, Vol. II, Paper R2.5, ParisVersailles, 1970.

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15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

6O

Larsen, P. A. and Lord, H. A., "Convention and Radiative Heat Transfer to

Water Vapor in Uniformly Heated Tubes," Fourth International Heat

Transfer Conference, Vol. III, Paper, R2.6, Paris-Versailles, 1970.

Martin, J. K. and Hwang, C. C. and "Combined Radiant and Convective Heat

Transfer in Laminar Steam Flow Between Gray Parallel Plates with Uniform

Heat Flux," Journal of Ouantitative Spectroscopy and Radiative Transfer,

Vol. 15, December 1975, pp. 1071-1081.

Jeng, D. R., Lee, E. J. and DeWitt, K. J., "A Study of Two Limiting Cases

in Convective and Radiative Heat Transfer with Nongray Gases,"

International Journal of Heat and Mass Transfer, Vol. 19, June 1979, pp.

589-596.

Greif, R., "Laminar Convection with Radiation: Experimental and

Theoretical Results," International Journal of Heat and Mass Transfer,

Vol. 21, April 1978, pp. 477-480.

Kobiyama, M., Taniguchi, H. and Saita, T., "The Numerical Analyses of

Heat Transfer Combined with Radiation and Convection," Bulletin of the

Japanese Society of Mechanical Engineering, Vol. 22, No. 167, May 1979,

pp. 707-714.

Im, K. H. and Ahluwalia, R. K., "Combined Convection and Radiation in

Rectangular Ducts," International Journal of Heat and Mass Transfer, Vol.

27, No. 2, February 1984, pp. 221-231.

Lick, W., "Transient Energy Transfer by Radiation and Conduction,"

International Journal of Heat and Mass Transfer, Vol. 8, 1965, pp. 119-

127.

Chang, Y. P. and Kang, C. S., "Transient and Steady Heat Transfer in a

Conducting and Radiating Medium," AIAA Journal, Vol. 8, No. 4, April

1970, pp. 609-614.

Chang, Y. P. and Smith, R. C., Jr., "Steady and Transient Heat Transfer

by Radiative and Conduction in a Medium Bounded by Two Coaxial

Cylindrical Surfaces," International Journal of Heat and Mass Transfer,

Vol. 13, 1970, pp. 69-80.

Doornink, D. G. and Hering, R. G., "Transient Radiative Heat Transfer in

a Nongray _ledium," Journal of the Quantitative Spectroscopy and Radiative

Transfer, Vol. 12, 1972, pp. 1161-1174.

Larson, D. W. and Vikanta, R., "Transient Combined Laminar Free

Convection and Radiation in a Rectangular Enclosure," Journal of Fluid

Mechanics, Vol. 78, Part I, 1976, pp. 65-85.

Melnikov, V. I. and Sukhovich, E. P., "Transient Heat Exchange Between a

Radiating Plate and a High-Temperature Gas Flow," Heat Transfer-Soviet

Research, Vol. I0, No. 3, May-June 1978, pp. 11-20 (Translation).

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61

27. Heinisch, R. P. and Viskanta, R., "Transient Combined ConductionRadiation in an Optically Thick Semi-lnfinite Medium, AIAA Journal, Vol.

6, 1968, pp. 1409-1411.

28. Hazzah, A. S. and Beck, J. V., "Unsteady Combined Conduction-Radiation

Energy Transfer Using a Rigorous Differential Method," Internatinal

Journal of Heat and Mass Transfer, Vol. 13, March 1970, pp. 517-522.

29. Lii, C. C. and Ozisik, M. N., "Transient Radiation and Conduction in an

Absorbing, Emitting and Scattering Slab with Reflecting Boundaries,"

International Journal of Heat and Mass Transfer, Vol. 15, May 1972, pp.

1175-1179.

30. Weston, K. C. and Hauth, J. L., "Unsteady, Combined Radiation and

Conduction in an Absorbing, Scattering and Emitting Medium," Journal of

Heat Transfer, Vol. 95, August 1973, pp. 357-364.

31. Sutton, W. H., "A Short Time Solution for Coupled Conduction and

Radiation in a Participating Slab Geometry," ASME Paper No. 84-HT-34,

1984.

32. Tiwari, S. N., "Radiative Interactions in Transient Energy Transfer in

Gaseous Systems," Progress Report NAG-I-423, Dept. of Mechanical

Engineering and Mechanics, College Engineering and Technology, Old

Dominion University, Norfolk, VA, December 1985; also NASA-CR-176644 NAS

1.26:176644, December 1985.

33. Tiwari, S. N. and Singh, D. J., "Interaction of Transient Radiation in

Nongray Gaseous Systems," Progress Report NAG-I-423, Dept. of Mechanical

Engineering and Mechanics, College of Engineering and Technology, Old

Dominion University, Norfolk, VA, January 1987; also NASA-CR-181389 NAS

1.26:181389, January 1987.

34. Tiwari, S. N., Singh, D. J. and Kumar, A., "Transient Radiative Energy

Transfer in Nongray Gases," AIAA Paper 87-0323, January 1987.

35. Soufiani, A., Hartmann, J. M., and Taine, J., "Validity of Band-Model

Calculations for COp and H20 Applied to Radiative Properties and

Conductive-Radiative Transfer, _ Journal of Quantitative Spectroscopy and

Radiative Transfer, Vol. 33, No. 3, March 1985, pp. 243-257.

36. Soufiani, A. and Taine, J., "Application of Statistical Narrow-Band Model

to Coupled Radiation and Convection at High Temperature," International

Journal of Heat and Mass Transfer, Vol. 30, No. 3, March 1_87, pp. 437"-

447.

37. Penner, S. S. and Varanasi, P., "Effect of (Partial) Overlapping of

Spectral Lines on the Total Emissivity of H20-CO 2 Mixtures

(T 9 800 K)," Journal of Quantitative Spectroscopy and Radiative

Transfer, Vol. 6, No. 2, March/April 1966, pp. 181-192.

38. Felske, J. D. and Tien, C. L., "Wide Band Characterization of the Total

Band Absorptance of Overlapping Infrared Gas Bands," Combustion Science

and Technology, Vol. II, 1975, pp. 111-117.

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62

39. Tien, C. L. and Lee, S. C., "Flame Radiation," Progress in Energy and

Combustion Science, Vol. 8, No. I, 1982, pp. 41-59.
APPENDIXA

SPECTRALINFORMATIONANDCORRELATIONQUANTITIES

FORIMPORTANTINFRAREDBANDS

Spectral information and correlation quantities for important infrared

bands are available in [7-9]. The exponential band model correlation

quantities for important bands of CO, C02, H20, and CH 4 are given in Table

A.I. For some species, revised data are now available in [13].

In developing correlation quantities for different bands, it is

important to have information on the integrated band intensity S(To) and the

band width parameter Ao(T ). The band intensities of many molecular bands

are available in the literature [7-9, 13, 33] but this is not the case with

the band width parameter. Different relations for Ao(T ) are proposed by

Edwards et al. in references cited in [9]. From a critical evaluation of

different relations available in the literature for Ao(T) and after personal

communications with Dr. D.K. Edwards, it was decided to adopt the following

two relations:

Ao(T ) : _ r 2 (3_) (4kT._._B)I/2 (A.I)

2 4 hc

Ao(T) : 0.9 r 2 (3_) (2kTB)I/2

4 hc

(A.2)

where B is the rotational constant of lower level, c is the speed of light,

h is the Planck's constant and k is the Bolt_nann's constant. The value of

B is different for different molecules. Equations (A.1) and (A.2) are
A-2

essentially the same except Eq. (A.I) will result in a coefficient of 0.707

instead of 0.9. It is suggested to use Eq. (A.2) in determining approximate

relations for Ao(T ) for all molecules whose values are not available in the

literature. Information on the rotational constants for different

molecules is available in the literature (for example, see Ref. 33). It is

suggested to use the equivalent value for the rotational constant of

polyatomic molecules.

It should be noted that in Eqs. (A.I) and (A.2), B, c, h, and k are

constants and do not depend on the temperature. Thus, Eq. (A.2) may be

expressed as:

where

Ao(T) : CONST (T)I/2 (A.3a)

CONST : 0,9 r 2 (5 (_e-] 1/2 (A.3b)

4 hc

and Be represents the equivalent rotational constant. By evaluating Eq.

(A,3a) at a reference temperature Tref, the value of Ao(Tre f) can be

determined and, therefore, Eq. (A.3a) may be expressed alternately as:

Ao(T ) : Ao(Tre f) (T/Tref)l/2" (A.4)

Equation (A.4) is a convenient form to compare its results with experimental

values.

By noting that r 2 (_) : [r (3/4)] 2 and substituting values for c, h,

and k, Eq, (A.3) can be expressed as:

Ao(T) : 1.59313 (BeT)I/2 (A.5)
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I

t

I

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l

A-3

-1

where Ao and Be have units of cm and T is in degrees Kelvin. For a

particular gas, Ao (Tref) can be obtained from Eq. (A.5) and ther_ Eq. (A.41

can be used to determine Ao(T) at other temperatures. For example, for CO

the rotational constant is 1.931 cm -1 and at a reference temperature of

-1

300 K, Ao(Tre f = 300 K) = 38.344 cm . This compares very well with the

experimental value of 38.1 given if Refs. 7 and 9 and presented in Table

A.I. Similarly, for the 4.3 u band of CO 2, the equivalent rotational

constant is 0.3906 cm °1 and, therefore, Ao(Tre f = 300 K) = 17.246 cm -1", the

-1

experimental value of 19.9 cm given in Table A.1 for this band is slightly

higher.

Spectral Information for OH

For the fundamental band of OH, the following information is obtained

from Ref. 33:

-1

Band center, mc = 3570 cm ,

Band strength at STP, S(T o) = 110 cm-2 atm °1

Also, from Ref. 33 the information on equilibrium rotational constant for OH

is obtained as:

A 2S+ ÷ 17.355 cm"I

X 2Ni ÷ 18.871 cm°1

!
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1

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I

1

1

A-4

Thus, it is suggested to use a value for tne equilibrium rotational constant

-i

for OH as Be = 18 cm Using this, a value of Ao (Tref: 3UU K) is found

as

Ao (Tre f = 300) : 1.59313 (BeTre f)

1/2

= 117.0707. (A.6)

-1

Considering the value of Ao(Tref) = 117 cm , the relation for Ao(T ) for OH

is given as:

Ao(T) = 117 (T/300) I/2. (A.7)

By knowing mc' S (To), and Ao(T), other required spectral information for

OH can be evaluated.
IIIII

0,)

°r",-

,rlI °O

*r,,.

I°

O

I°

o_

I°

¢-

0_O

)<

! ,.,

O,)

I-ro

I--

O

,f,. E!E

q,)

c_E

U _o

r._

0) 02 c"

L.4-I

_0J

_/')EJ

0_ r.._

SO;

c- (-I

N_E

e- 3U

A_

v_

_QO

vv_v__v_vv_v

0____000O00

0O_

CP

ev0

Q;0

A-5
°

o

Correlation quantities are based on the results of reference [9]. The

intensity of the band marked with * was taken from the reference [7].

Notations:

2

KI(T ) : (T/300) I/2 | K2(T) : (300/T) 3/2 D

(T)/KI(T) ] x 10-3 , a2 : ¢3 2(T)/[¢2(T)KI(T)]

h : 6.625 x 10 -27 erg-sec, C : 2.998 x 1010 cm/sec,

-16

k = 1.380 x 10 erg/K , hC/k = 1.44 cm - K

. Temperature range: 300 K < T < T max" For CO, Tmax 1800 K.

For CO2, Tmax = 1400 K. For H20, Tmax = 1100 K. For CH4,

T = 830 K.

max

.

-1

For CO, ,,,: 2143 cm and

¢1(T) : [15.15 + 0.22 (T/To)3/2] [1-exp (-hCm/kT)], TO : 100 K

-1 -1 -1

5. For C02, ml = 1351 cm , m2 = 667 cm ' m3 = 2396 cm

¢2(T) : {I - exp [( - hC/KT ) (ml + m3 )]}x

-i

{[1 - exp ( - hCmllkT)] [I - exp (-hCm31kT)]} , ¢3(T) : 1 +

0.053 (T/IO0) 3/2

A-6
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.

-I

FOF H20, ml = 3652 cm-l' m2 = 1595 cm ' m3 : 3756 cm -i

CvI v 2 v 3 (T) : {1 - exp [ - hC(v I ml + v2 m2 + v3 m3 )/kT]} x

-i

{[l-exp ( -hCml/kT)] [l-exp(-hCm2/kT] [1-exp(-hC(_3/kT]}

¢7 (T) : exp [ - 17.6 (T/IO0) -I/2]

A-7
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APPENDIX B

RESULTS OF THE QUARTIC FORMULATION

Extensive results of the quartic formulation are presented in this

appendix. The order of presentation of results is e vs. T, e vs. _, and

e b vs. L. The order of results for the four species considered is OH, CO,

C02, and H20. Information on other physical conditions is provided on each

figure. The notations used in these figures correspond to the notations

used in the main text according to the following relations:

LARGE PATH = LARGE PATH LENGTH

OPT THIN : OPTICALLY THIN

TBULK : eb : (T w - Tb)/(qwg/K)

THETA : e : (T w - T)/qwL/K)

T :T

TW=T

w

Z=E
IIIIIiIII!IIII

_ iii

II

Av

B-2
II B-3 IIIIIIIIII \ | ", i l',lII l'_'l II _ %% %%

=

I Ii9 """ "-"

II "%%"

._I:JHL5-

t31

o1,--

1.1_

II
IIIIIIIIIIIIiIIiII

B-4

v

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IIIiIIIIIIIIIIIIIII

._J

rln,,"

LIJ ZN

lilll_t_ttJ S

m

• !

C:) ,-t

to 0!

NNI

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II

II

II!

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!I

!!

!!

!!

III111 II1111111tl

B-5

Av

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t.

!q

\-

_. ",,,\-

% -0

(3 -.,,:

Illllllllll illllllllli IIIllllllil IIIIIo

A

"o

qJ

c"

0pc'- 0U

,,x

.pi,
IIIIII I!! I I!I | iI I ! IIIIIIII m Z

li,i,,,il,l,,,,,,,,,l,*,I,,,

II

ii L/"i

II

l--.- ..,.I

Illllllllll lllllllllllllllll

t:t.i.3t.IJ,-

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B-6

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i!. iIIIIIIIIIIIII

"li Z i !I!I t z

II

g

A

ill Q

<,,J

v

,,4

f,.-

Ia=
I B-7 IIII| _ ___. I, i III \ - I , I "", _" , _ .=.I I

A

_.=

°p..

c- Ou

s.-

.r1.1_

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iiLillll

z- Lq_l

i--,-

I--''' il

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•"" II

II ' II IIIIIII11 III111111| I " IlIl l\ \ -

'\\

',,\

",, ,_ ___

_2

II

__J

,i ,,I l,,, I LLmLLL__L_._£1tLJ_._'ii*Ii ' I'ii' Iil LL_

s. n._m

WI,--

sl 0

.?! o

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e- 0o

°l--

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IIIIIIIIIIIIIIIIII

s I

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colw

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,,4

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33

The results for temperature variations within the plates are presented in

Figs. 10-14 for different conditions. Since the temperature profiles are

symmetric, most of these results are illustrated only for _ = 0 to

= 0.5. The stady-state solutions for various species are compared in Fig.

10 for P = 1 atm, L = 10 cm, and different wall temperatures. Results again

demonstrate the relative importance of different species for energy transfer.

It is noted that the results for CO2 at Tw = 500 K are exactly the same as for

OH at Tw = 1,000 K, and the results for CO and H20 are about the same at Tw =

1,000 K and 2,000 K. This obviously is a coincidence for the physical case

considered. The general and limiting solutions for OH, CO, CO2, and H20 are

presented, respectively, in Figs. 11 through 14a for Tw = 500 K, P = 1 atm,

and L = 10 cm. These results also show that the steady-state conditions reach

earlier for H20 and CO2 than for CO and OH. In general, the differences

between limiting and general solutions are found to be small at earlier times;

the maximum difference occurs at the steady-state conditions. In each case,

the large path length solutions are closer to the general solutions, hut the

optically thin solutions are found to deviate considerably. This is because

for P = I atm and L = 10 cm, the pressure path length is sufficiently high and

the optically thin limit is not the correct limit for the physical case

considered. It should be noted that for CO, CO2, and H20, the optically thin

solutions are identical for _ ) 0.05. Additional results given in Appendix B

reveal that the differences between general and large path length solutions

are insignificant at higher temperatures for all species. The general

solutions for the temperature variation across the entire duct are illustrated

in Fig. 14b for H20 for the physical conditions of Tw = 1,000 K, P = I atm,

and L = 5 cm. The parabolic nature of the transient profiles is clearly

evident and, in this case, the steady-state is reached at T _ 0.5.
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APPENDIX C

COMPUTER CODE FOR TRANSIENT ENERGY TRANSFER IN

INCOMPRESSIBLE LAMINAR FLOWS

Following the numerical procedure discussed under the section "Method

of Solution," a general computer program is developed to calculate the

temperature variation and bulk temperature for a fully-developed laminar

flow between two parallel plates having uniform wall heat flux. The

program provides the general solution as well as the limiting (optically

thin and large path length) solutions. The program is written for the

three-band CO2 gas, but can be used for any gas by employing appropriate

thermophysical and radiative properties.
I

C-2

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PROGRAMTRAN(OUTPUT:65,TAPE6,TAPEII,TAPEIS,TAPE10,TAPE1,TAPE2,

&TAPE3)

C____________

C

C THIS PROGRAMCALCULATES THE TRANSIENT RADIATIVE

C INTERACTION IN FULLY DEVELOPED LAMINAR FLOW OF

C C02 BETWEEN TWO PARALLEL BLACK WALLS WITH CONSTANT

C HEAT FLUX.

C IT IS BASED ON REPORT BY DR. S.N. TIWARI(REPORT NO

C

I-NAG 432)

C

C

C L DISTANCE BETWEEN PLATES (IN CM)

C TW WALL TEMPRATURE ( K)

C P PRESSURE (ATM)

C TT TIME (NON DIMENSIONAL)

C Z NON DIMENSIONAL DISTANCE BETWEEN THE PLATES

C Z : 0.0 AT LOWER WALL

C Z : 1.0 AT UPPER WALL

C

C

C

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C

C

108

109

EXTERNAL F211,F212,F213,F221,F222,F223,F511,F512,F513,F611,F612,

&F613,F521,F522,F523,F621,F622,F623

REAL L,KFB

DIMENSION U(60),EPS(3),PRES(IO),TEMP(8),ZZ(60),TT(60),TB(65)

&,TBO(65),TBL(65),DZZ(20)

COMMONFI,F2,F3,BI,B2,B3,Z

DATA U/0.1,0.2,0.4,0.5,1.0,2.,3.,4.,6.,7.,8.,I0.,20.,40.,60.,80.

&,I00.,200.,500.,I000./

DATA PRES/O.OI,O.I,I.0,10.O/

DATA TEMP/500.O,IO00.O,2000.O,3000.O/

DATA TT/O.05,0.1,O.5,1.O/

DATA DZZ/O.2,0.3,0.4,.5,.6,.7,.8/

WRITE(6,110)

WRITE(IO,120)

WRITE(15,130)

NP=20

NT=4

TIME MARCHING

DO 22 ITT=I,NT

T=TT(ITT)

IB=I

SET IB=O FOR CALCULTING BULK

IM=I

SET IM=I FOR QUARTIC

IF(IB.EQ.1) GOTO 108

WRITE(6,150)

WRITE(IO,150)

WRITE(15,150)

GOTO 109

WRITE(6,118)

WRITE(IO,II8)

WRITE(15,118)

TEMPRATURE,OTHERWISE IB=O

AND IM=2 FOR QUADRATIC

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WRITE(6,140)T

WRITE(15,140) T

WRITE(IO,140) T

DO 44 IT=I,3

TW=TEMP(IT)

DO 55 KK=3,3

P=PRES(KK)

DO 66 I=I,NP

L:U(1)

N11=7

DZ:O. i

DO 77 IZ=I,NII

Z:DZZ(IZ)

CALCULATION OF PLANCK FUNCTION AND ITS DERIVATIVE

WNB BAND CENTER (1/CM)

HCK CONSTANT (K CM)

CCC CI*C2 (ERG-K-CM**3/SEC)

PFDBI PLANCK FUNCTION DERIVATIVE FOR I BAND

CESS=TW**2

STU=TW**O.5

SNT=TW/273.0

HCK=1.439257246

CCC=0.000053847734

SPECTROSCOPIC PROPERTIES OF C02

HERE WE HAVE CONSIDERED ONLY THREE BANDS(15,4.3 & 2.7 MICRONS)

WNBI BAND CENTER (1/CM)

WNB1=667.0

WNB2=2347.0

WNB3=3716.0

C2BI=HCK*WNB1

C2B2=HCK*WNB2

C2B3=HCK*WNB3

CCC=0.000053847734

CCBI=CCC*(WNBI**4)

CCB2=CCC*(WNB2**4)

CCB3=CCC*(WNB3**4)

TBI=C2BI/TW

TB2=C2B2/TW

TB3=C2B3/TW

TEBI=EXP(TBI)

TEB2=EXP(TB2)

TEB3=EXP(TB3)

SNTBI=CESS*((TEBI-I.O)**2.0)

SNTB2=CESS*((TEB2-1.O)**2.0)

SNTB3=CESS*((TEB3-1.0)**2.0)

PFDBI=(CCBI*TEB1)/SNTB1

PFDB2=(CCB2*TEB2)/SNTB2

PFDB3=(CCB3*TEB3)/SNTB3

BAND MODEL CORRELATIONS (TIEN & LOWDER WIDE BAND MODEL)

!
IIIIiIIIIIIIIIIII

1i CCCCCCCCCCCCCCC

AZI AOI (I/CM)

CZSI COl**2 (1/(ATM-CM))

BSI B**2 (NON DIMENSIONAL)

OMGI WAVE NUMBER(1/CM)

SI INTEGRATED BAND INTENSITY (1/(ATM CM**2))

AKI=(TW/300.O)**O.5

AK2=(3OO.O/TW)**I.5

AZ1=1.29*STU

AZ2=1.15*STU

AZ3=2.4*STU

KFB THERMAL CONDUCTIVITY (ERG/CM-SEC-K)

KFB=(1488.365171)*(SNT**I.23)

0MG1=1351.0

0MG2=667.0

0MG3=2396.0

TX=-(HCK/TW)

TXI=TX*OMG1

TX2=TX*OMG2

TX3=TX*OMG3

ETXI=EXP(TX1)

ETX2=EXP(TX2)

ETX3=EXP(TX3)

CI=I.0-ETX1

C2=I.0-ETX2

C3=I.0-ETX3

BRKT=TX*(OMGI+OMG3)

PHI2=(I.O-EXP(BRKT))/(CI*C3)

TS=3OO.O/TW

SI=339.685"TS

$2=2702.7"TS

S3=71.497*TS*PHI2

CZSI=SI/AZI

CZS2=S2/AZ2

CZS3=S3/AZ3

PL PRESSURE PATH LENGTH (ATM-CM)

PL=P*L

UZI=CZSI*PL

UZ2=CZS2*PL

UZ3=CZS3*PL

BS1=O.O841*AK1

BS2=O.32895*AK1

PHI3=I.0+O.O53*((TW/IO0.O)**1.5)

DEL2=(PHI3**2.0)/(PHI2*AKI)

BS3=O.1112*DEL2

PEI EFFECTIVE PRESSURE FOR EACH BAND (NON

PEI=(I.3*P)**0.7

PE2=(I.3*P)**O.8

PE3=(I.3*P)**O.65

DIMENSIONAL)

C-4

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678

BETAI LINE STRUCTURE PARAMETER

BETAI=BSI*PEI

BETA2=BS2*PE2

BETA3=BS3*PE3

CORRELATION FOR EACH BAND

FI:2.94*(I.O-EXP(-(2.60*BETAI)))

F2:2.94*(I.O-EXP(-(2.60*BETA2)))

F3:2.94*(I.O-EXP(-(2.60*BETA3)))

BI=I.5*UZl

B2:I.5*UZ2

B3:I.5*UZ3

HI=AZI*PFDBI

H2:AZ2*PFDB2

H3:AZ3*PFDB3

Z2:Z*Z

CZ:I./(Z-Z2)

NUMERICAL INTEGRATION

FOR DETAILS OF INTEGRATION REFER

AT NASA LANGLEY (DOCUMENT N-3)

MATHEMATICAL

EPS(1):IE-12

EPS(2):IE-12

X=O.O

IF(IM.EQ.I)

CALL

CALL

CALL

CALL

CALL

CALL

GOTO 678

CADRE(X,Z,F211,EPS,ITEXT,G211,1ERR)

CADRE(X,Z,F212,EPS,ITEXT,G212,1ERR)

CADRE(X,Z,F213,EPS,ITEXT,G213,1ERR)

CADRE(Z,I.,F221,EPS,ITEXT,G221,1ERR)

CADRE(Z,I.,F222,EPS,ITEXT,G222,1ERR)

CADRE(Z,I.,F223,EPS,ITEXT,G223,1ERR)

SUM21:(G211-G221)

SUM22:(G212-G222)

SUM23=(G213-G223)

BRI=H1*UZI*SUM21

BR2=H2*UZ2*SUM22

BR3=H3*UZ3*SUM23

BR=BRI+BR2+BR3

AJ2=CZ*(2+I.5*L/KFB*BR)

GT2=I2*(EXP(-AJ2*T)-1)/AJ2

THETA=GT2*(Z-Z2)

GOTO 679

CONTINUE

CALL CADRE(X,Z,F511,EPS,ITEXT,G511,1ERR)

CALL CADRE(X,Z,F512,EPS,ITEXT,G512,1ERR)

CALL

CALL

CALL

CALL

CALL

CALL

CADRE(X,Z,F513,EPS,ITEXT,G513,1ERR)

CADRE(Z,I.,F521,EPS,ITEXT,G521,1ERR)

CADRE(Z,I.,F522,EPS,ITEXT,G522,1ERR)

CADRE(Z,I.,F523,EPS,ITEXT,G523,1ERR)

CADRE(X,Z,F611,EPS,ITEXT,G611,1ERR)

CADRE(X,Z,F612,EPS,ITEXT,G612,1ERR)

LIBRARY

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679

207 CCC

801

CALL

CALL

CALL CADRE(Z,I.

CALL CADRE(Z,1.

SUM51=G511-G521

SUM52=G512-G522

SUM53=G513-G523

SUM61=G611-G621

SUM62=G612-G622

SUM63=G613-G623

BR51=H1*UZ1*SUM51

BR52=H2*UZ2*SUM52

BR53=H3*UZ3*SUM53

BR61=H1*UZ1*SUM61

BR62=H2*UZ2*SUM62

BR63=H3*UZ3*SUM63

XZ=Z-2*Z**3.+Z**4.

YZ=Z**2-Z

ZZ=I2*(Z-Z**2)

AJS=ZZ+1.5*L/KFB*(BR51+BR52+BR53)

AJ6=-2.+1.5*L/KFB*(BR61+BR62+BR63)

DAL=AJS*T/XZ

DALI=AJ6*T/YZ

DS=EXP(-DAL)

DSI=EXP(-DAL1)

GT=ZZ/(2.*AJ5)*(DS-I)

HT=ZZ/(2.*AJ6)*(DSI-I)

THETA=GT*XZ+HT*YZ

TB(IZ)=THETA*(Z-Z2)

IF(IB.EQ.I) GOTO 207

WRITE(6,160) TW,P,L,THETA,Z

CONTINUE

CADRE(X,Z,F613,EPS,ITEXT,G613,1ERR)

CADRE(Z,I.,F621,EPS,ITEXT,G621,1ERR)

,F622,EPS,ITEXT,G622,1ERR)

,F623,EPS,ITEXT,G623,1ERR)

LARGE PATH LENGTH

IF(IM.EQ.I) GOTO 801

DS=ABS(Z/(Z-1))

AJ=ALOG(DS)+ALOG(-2/Z+I/Z**2+I)*Z+2

H=HI+H2+H3

AJL=CZ*(2+H*L*AJ/KFB)

GL=12*(EXP(-AjL*T)-I)/AJL

THEL=GL*(Z-Z2)

GOTO 802

CONTINUE

ZI=Z-1

ALI=ZI**3./3.-3.*Z*ZI**2./2.+3.*Z*Z*ZI-Z**3.*ALOG(I-Z)-Z**3.

&*(II./6.-ALOG(Z))

AL2=-(ZI**2./2.-2.*Z*ZI+Z**2*ALOG(I-Z)+Z*Z*(1.5-ALOG(Z)))

AL3=-(ALOG(I-Z)-ALOG(Z))

AL=4.*AL1-6.*AL2+AL3

AJS=ZZ+L/KFB*AL*(HI+H2+H3)

ALLI=-I.-Z*ALOG(I.-Z)+Z*ALOG(Z)

ALL2=AL3

ALL=2.*ALL1-ALL2

AJ6=-2+L/KFB*ALL*(HI+H2+H3)

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DAL=AJ5*T/XZ

DALI=AJ6*T/YZ

DS=EXP(-DAL)

DSI=EXP(-DALI)

GT=ZZ/(2.*AJ5)*(DS-I)

HT=ZZ/(2.*AJ6)*(DSI-I)

THEL=GT*XZ+HT*YZ

TBL(IZ)=THEL*(Z-Z2)

IF(IB.EQ.I) GOTO 208

WRITE(IO,160) TW,P,L,THEL,Z

CONTINUE

OPTICALLY THIN

IF(IM.EQ.I) GOTO 701

ANI=HI*UZ1

AN2=H2*UZ2

AN3=H3*UZ3

AN=L/KFB*(ANI+AN2+AN3)

AJO=2*CZ+3*AN

GO=12*(EXP(-AJO*T)-1)/AJO

THEO=GO*(Z-Z2)

GOTO 702

CONTINUE

ANI=P*L*L/KFB*(SI*PFDBI+S2*PFDB2+S3+PFDB3)

EE=Z**4-2*Z**3+Z

FF=Z**2-Z

AJS=ZZ+3.*ANI*EE

AJ6=-2.+3.*ANI*FF

DAL=AJ5*T/XZ

DALI=AJ6*T/YZ

DS=EXP(-DAL)

DSI=EXP(-DAL1)

GT=ZZ/(2.*AJ5)*(DS-1)

HT=ZZ/(2.*AJ6)*(DSI-1)

THEO=GT*XZ+HT*YZ

TBO(IZ)=THEO*(Z-Z2)

IF(IB.EQ.I) GOTO 209

TBO(IZ)=THEO*(Z-Z2)

IF(IB.EQ.1) GOTO 209

WRITE(15,160) TW,P,L,THEO,Z

CONTINUE

CONTINUE

IF(IB.EQ.O) GOTO 234

TBUL=O.

TBULL=O.

TBULO=O.

DO 10 JB=1,N11

TBUL=TBUL+TB(JB)

TBULL=TBULL+TBL(JB)

TBULO=TBULO+TBO(JB)

BULK=6.*DZ*TBUL

BULKL=6.*DZ*TBULL

BULKO=6.*DZ*TBULO

WRITE(6,160) TW,P,L,BULK
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WRITE(10,160) TW,P,L,BULKL

WRITE(15,160) TW,P,L,BULKO

CONTINUE

CONTINUE

CONTINUE

CONTINUE

CONTINUE

CONTINUE

FORMAT(7X,*GENERAL SOLUTION FOR C02 (CONSTANT FLUX CASE)*)

FORMAT(4X,*TW*,TI8,*P*,T28,*L*,T38,*TBULK*)

FORMAT(7X,*LARGE PATH LENGTH FOR C02 (CONSTANT FLUX CASE)*)

FORMAT(7X,*OPTICALLY THIN FOR C02 (CONSTANT FLUX CASE)*)

FORMAT(IOX,*TIME=*,F8.3)

FORMAT(4X,*TW*,T18,*P*,T28,*L*,T38,*THETA*,T48,*Z*)

FORMAT(7(IX, F9.4))

STOP

END

FUNCTION F211(ZB)

COMMON FI,F2,F3,BI,B2,B3,Z

XX=BI*(Z-ZB)

DENI=(FI*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F1)

AUDI=(FI*((XX**2.0)+(4.0*XX*FI)+(4.0*F1)))/DENI

F211=(1-2*ZB)*AUD1

RETURN

END

FUNCTION F212(ZB)

COMMON FI,F2,F3,B1,B2,B3,Z

XX=B2*(Z-ZB)

DEN2=(F2*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F2)

AUD2=(F2*((XX**2.0)+(4.0*XX*F2)+(4.0*F2)))/DEN2

F212=(1-2*ZB)*AUD2

RETURN

END

FUNCTION F213(ZB)

COMMON FI,F2,F3,B1,B2,B3,Z

XX=B3*(Z-ZB)

DEN3=(F3*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F3)

AUD3=(F3*((XX**2.0)+(4.0*XX*F3)+(4.0*F3)))/DEN3

F213=(1-2*ZB)*AUD3

RETURN

END

FUNCTION F221(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=BI*(ZB-Z)

DENI=(FI*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F1)

AUDI=(FI*((XX**2.0)+(4.0*XX*FI)+(4.0*FI)))/DENI

F221=(1-2*ZB)*AUD1

RETURN

END

FUNCTION F222(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=B2*(ZB-Z)

DEN2=(F2*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F2)

AUD2=(F2*((XX**2.0)+(4.0*XX*F2)+(4.0*F2)))/DEN2

F222=(1-2*ZB)*AUD2

C-8

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! C-9

IIIIIIIIIi

RETURN

END

FUNCTION F223(ZB)

COMMON F1,F2,F3,BI,B2,B3,Z

XX=B3*(ZB-Z)

DEN3=(F3*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F3)

AUD3=(F3*((XX**2.0)+(4.0*XX*F3)+(4.0*F3)))/DEN3

F223=(I-2*ZB)*AUD3

RETURN

END

FUNCTION F511(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=BI*(Z-ZB)

DENI=(FI*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F1)

AUDI=(FI*((XX**2.0)+(4.0*XX*F1)+(4.0*FI)))/DEN1

F511=(I-6*ZB*ZB+4*ZB**3)*AUDI

RETURN

END

FUNCTION F512(ZB)

COMMON F1,F2,F3,BI,B2,B3,Z

XX=B2*(Z-ZB)

DEN2=(F2*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F2)

AUD2=(F2*((XX**2.0)+(4.0*XX*F2)+(4.0*F2)))/DEN2

F512=(1-6*ZB*ZB+4*ZB**3)*AUD2

RETURN

END

FUNCTION FSI3(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=B3*(Z-ZB)

DEN3=(F3*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F3)

AUD3=(F3*((XX**2.0)+(4.0*XX*F3)+(n.O*F3)))/DEN3

F513=(1-6*ZB*ZB+4*ZB**3)*AUD3

RETURN

END

FUNCTION F521(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=BI*(ZB-Z)

DENI=(FI*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F1)

AUDI=(FI*((XX**2.0)+(4.0*XX*FI)+(4.0*FI)))/DEN1

F521=(1-6*ZB*ZB+4*ZB**3)*AUD1

RETURN

END

FUNCTION F522(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=B2*(ZB-Z)

DEN2=(F2*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F2)

AUD2=(F2*((XX**2.0)+(4.0*XX*F2)+(4.0*F2)))/DEN2

F522=(1-6*ZB*ZB+4*ZB**3)*AUD2

RETURN

END

FUNCTION F523(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=B3*(ZB-Z)

DEN3=(FB*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F3)

AUD3=(F3*((XX**2.0)+(4.0*XX*F3)+(4.0*F3)))/DEN3
! C-IO IIIIIIIIIIIIIIII

F523=(1-6*ZB*ZB+4*ZB**3)*AUD3

RETURN

END

FUNCTION FG11(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=BI*(Z-ZB)

DENI=(FI*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*FI)

AUDI=(FI*((XX**2.0)+(4.0*XX*FI)+(4.0*FI)))/DENI

F611=(2*ZB-1)*AUD1

RETURN

END

FUNCTION F612(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=B2*(Z-ZB)

DEN2=(F2*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F2)

AUD2=(F2*((XX**2.0)+(4.0*XX*F2)+(4.0*F2)))/DEN2

F612=(2*ZB-1)*AUD2

RETURN

END

FUNCTION F613(ZB)

COMMON FI,F2,F3,B1,B2,B3,Z

XX=B3*(Z-ZB)

DEN3=(F3*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F3)

AUD3=(F3*((XX**2.0)+(4.0*XX*F3)+(4.0*F3)))/DEN3

F613=(2*ZB-1)*AUD3

RETURN

END

FUNCTION F621(ZB)

COMMON FI,F2,F3,BI,B2,B3,Z

XX=BI*(ZB-Z)

DENI=(FI*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*FI)

AUDI=(FI*((XX**2.0)+(4.0*XX*FI)+(n.O*F1)))/DEN1

F621=(2*ZB-I)*AUD1

RETURN

END

FUNCTION F622(ZB)

COMMON F1,F2,F3,B1,B2,B3,Z

XX=B2*(ZB-Z)

DEN2=(F2*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F2)

AUD2=(F2*((XX**2.0)+(4.0*XX*F2)+(4.0*F2)))/DEN2

F622=(2*ZB-1)*AUD2

RETURN

END

FUNCTION F623(ZB)

COMMON FI,F2,F3,B1,B2,B3,Z

XX=B3*(ZB-Z)

DEN3=(F3*((XX**2.0)+(2.0*XX)+2.0)+XX)*(XX+2.0*F3)

AUD3=(F3*((XX**2.0)+(4.0*XX*F3)+(4.0*F3)))/DEN3

F623=(2*ZB-1)*AUD3

RETURN

END

!!