THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ANALYSIS OF LAMINAR FILM BOILING IN BOUNDARY-LAYER FLOWS WITH APPRECIABLE RADIATION Hsu-Chieh Yeh Wen-Jei Yang April, 1968 IP-817

ANALYSIS OF LAMINAR FILM BOILING IN BOUNDARY-LAYER FLOWS WITH APPRECIABLE RADIATION Hsu-Chieh Yeh* and Wen-Jei Yang** Department of Mechanical Engineering The University of Michigan, Ann Arbor ABSTRACT This paper presents the theoretical study of the heat transfer and friction characteristics in the natural-convection film boiling on an inclined surface and a sphere, the forced-convection film boiling over a horizontal plate and the stagnation-flow film boiling when radiation is appreciable. The boiling liquid is either at the saturation temperature or subcooled. The two-phase flow and heat transfer problems have been formulated exactly within the framework of boundary-layer theory with the consideration of the shear stress and vapor velocity at the liquid-vapor interface. Through the use of the similarity transformation expressions are obtained to determine the vapor-film thickness, skin friction, and heat transfer rate. It is disclosed that the presence of surface radiation results in an increase in the heat transfer rate and a decrease in the skin friction. * Post-Doctoral Fellow, Institute of Science and Technology. Associate Professor of Mechanical Engineering. Associate Professor of Mechanical Engineering. ii

TABLE OF CONTENTS Page ABSTRACT..................................................... ii LIST OF FIGURES............................................... iv NOMENCLATURE.................................................. v INTRODUCTION.................................................. 1 ANALYSIS..................................................... 4 Natural-Convection Film Boiling............................ 4 Forced-Convection Film Boiling............................. 18 RESULTS....................................................... 24 Heat Transfer.............................................. 24 Skin Friction............................................. 25 NUMERICAL ILLUSTRATIONS....................................... 27 CONCLUDING REMARKS............................................ 35 REFERENCES.................................................... 39 APPENDIX - COMPUTER PROGRAMS.................................. 40 1. For Stagnation Film Boiling in Two-Dimensional Flow.... 41 2. For Forced-Convection Film Boiling Over a Horizontal Plate.................................................. 44 3. For Natural-Convection Film Boiling Over a Vertical (or Inclined) Plate................................... 49 iii

LIST OF FIGURES Figure Page 1 Physical Models and Coordinates for NaturalConvection Film Boiling................................. 2 Physical Models and Coordinates for ForcedConvection Film Boiling................................. 21 3 Functions Associated With Vapor Velocity in NaturalConvection Film Boiling on an Inclined Plate for Ts = T = 212 F......................................... 28 4 Functions Associated With Vapor Temperature in Natural-Convection Film Boiling on an Inclined Flat Plate for Ts = To = 212~F........................ 29 5 Functions Associated With Velocity Profiles in the Vapor and Liquid Boundary Layers for ForcedConvection Film Boiling Over a Horizontal Plate for Ts = T = 212~F.................................... 30 6 Functions Associated With Vapor Temperature in ForcedConvection Film Boiling Over a Horizontal Plate for Ts = T = 212 F......................................... 31 7 Heat Transfer and Skin Friction Characteristics of Stagnation Film Boiling in Two-Dimensional Flow of Water at Atmospheric Pressure........................... 33 iv

NOMENCLATURE a constant defined as U, = ax for stagnation flow A physical parameter, Equation (15-f) B physical parameter, Equation (7-g) B1 physical parameter, Equation (15-f) Cp specific heat D physical parameter, Equation (13-c) E physical parameter, defined as hfg Pr/Cp(Tw-Ts) F temperature variable, Equations (9-c) and (18-d) for naturalconvection film boiling and Equations (25-c) and (28-b) for forced-convection film boiling f velocity variable, Equations (9-b) and (18-c) for natural convection film boiling and Equations (25-b) for the forcedconvection film boiling g gravitational acceleration h local heat transfer coefficient, q/Tw-Ts hfg latent heat of evaporation k thermal conductivity NNu Nusselt number Npr Prandtl number NRe Reynolds number p pressure q local heat flux from wall to vapor R radius of sphere T temperature: Tw = wall temperature; Ts = saturated temperature; T- = free stream temperature UCO free-stream velocity u velocity component of vapor in x-direction v

v velocity component of vapor in y-direction x coordinate measuring distance along the plate from' leading edge y coordinate normal to plate a thermal diffusivity cr absorption coefficient of vapor B ~coefficient of thermal expansion 6 thickness of vapor film, Equations (8) and (18-a) for natural-convection film boiling and Equation (24) for forcedconvection film boiling c emissivity similarity variable, Equation (9-a) and (18-b) for naturalconvection film boiling and Equations (25-a) and (28-a) for forced-convection film boiling All dimensionless vapor film thickness T-Ts 0 dimensionless temperature defined as - for vapor film TL-T wand for liquid layer. s oo v kinematic viscosity p density pr refractivity cr ~ Stefan-Boltzmann constant angle of inclination or x/R stream function, Equations (9-b) and (18-c) for naturalconvection film boiling and Equations (25-b) and (28-b) for forced-convection film boiling Subscripts Unsubscripted quantities —vapor phase L liquid phase r radiation vi

s at saturated state w wall surface oo free stream Superscript' t',, differentiation with respect to vii

INTRODUCTION Film boiling is characterized by a vapor blanketing the entire heated surface. It frequently occurs when the operation of jets or rockets involves the contact of a boiling liquid with high temperature surfaces or in the boiling of mercury especially at high heat fluxes. Film boiling may occur also if cryogenic fluids are used to cool hot surfaces. Since at high temperature differences, the film boiling is the normal type of heat transfer between the heated surface and the liquid, it is therefore of a definite scientific and practical interest. In stable film boiling regime heat is transferred from a heating surface by conduction through the vapor film and by boiling convection from the surface of the film to the surrounding liquid. Superimposed on this heat-flow path is the contribution of radiation to the total heat transfer. There are a few empirical equations being proposed to estimate the total surface conductance for film boiling when radiation is appreciable. However these equations are poor in accuracy and limited in application. This motivates the study of heat transfer and skin friction characteristics in both natural- and forced-convection film boiling through the application of the boundary-layer theory. Natural-convection film boiling over a vertical plate and forced-convection film boiling over a horizontal plate are investigated in Reference 1. This paper is the extension of Reference 1 to include more two-dimensional and axisymmetrical flows and -to demonstrate the generality of the method of analysis for solving laminar film boiling problems. -1

-2 (2-9) Previous studies of film boiling have been concerned with the situation where all motions are induced by gravity forces and where forced convection is absent. Such a process is usually called (2) (3) the natural-convection film boiling. Bromley and Ellion analyzed laminar film boiling on a vertical plate under the assumption of negligibly (4) small inertia forces and convective effects. Hsu and Westwater( studied analytically and experimentally the film boiling in both laminar and turbulent regions. McFadden and Grosh solved the boundary-layer (6) equation for the vapor film and Cess, by means of the integral technique, solved the vapor and liquid boundary-layer equations simultaneously. One feature common to prior analytical work is the assumption of (7) zero interfacial velocity. Koh analyzed the two-phase flow problem with the consideration of the shear stress and vapor velocity at the liquid-vapor interface. The results showed that for water, the effects of the interfacial velocity is small over a wide range of practical interest. Under the assumption of the constancy of vapor properties, the analysis was extended by Sparrow and Cess(8) to include the effects (9) of subcooling and then by Koh and Nilson for the effects of simultaneous action of radiation in saturated film boiling. It is rather unfortunate that the similarity transformation of the conservation equations and the appropriate boundary conditions failed, because the new and old variables coexist in one of the resulting boundary conditions, Equation (26) in Reference 9. In part of the present study, it is attempted to reexamine the problem treated in Reference 9 by introducing a different transformation. Furthermore, consideration is given to the temperature variation of the vapor density.

-3 Cess and Sparrow analyzed the film boiling in forcedconvection boundary-layer flows for the situation in which the liquid is at the saturation temperature or subcooled. Relative to the case of liquid flow, the skin friction is reduced owing to film boiling. The heat transfer is found to increase as (AT)1/. In the present work, attention is focused on the naturalconvection film boiling on an inclined plate and a sphere, the forcedconvection film boiling over a horizontal plate, and tile stagnationflow film boiling. Consideration is given to the convective and radiational exchanges and the associated fluid motions in the vapor film and liquid layer. This is equivalent to solving a two-phase boundary-layer problem. In addition, calculation is carried out for the case of saturated film boiling.

ANALYSIS Natural-Convection Film Boiling The physical model and coordinate system selected for naturalconvection film boiling is shown in Figure 1-a. It consists of an isothermal inclined plate immersed in a large volume of liquid. It is assumed that the vapor forms a stable film over the surface. The liquid has a bulk temperature T which is lower than the saturation temperature Ts prevailing at the liquid-vapor interface y = 6. The temperature of the plate surface is prescribed as Tw and Tw > Ts > To. It is assumed that under a stable film-boiling condition there exists a laminar layer of vapor film adjacent to the plate surface. Since the temperature of the plate and the vapor is relatively high, heat transfer takes place by convection as well as radiation. With the assumption, the application of the conservation laws for mass, momentum, and energy to the vapor film produces the following boundary-layer equations for a gravity-induced flow over the surface. au + 0 (1) ax by oU u 6Lu $2u PL-P u-+ v- = v + V --- g cos6 (2) ox dy oy- p oT T o2T 1 dqr u v + - (3) ax dy ay2 pCp dy where dq,/dy may be determined as follows. -4

-5 Ti L'l LIQUID \\ / BOUNDARY LAYER, TL U t \ I\ LIQUID, T, sV I I g Y/ I / g y^ y (A) NATURAL-CONVECTION FILM BOILING ON AN INCLINED PLA1E VAPOR FILM g (B) NATURAL-CONVECTION FILM BOILING OVER A SPHERE FIGURE 1. Ph,'SICAL MODELS AND COORDINATES FOR NATURAL-CONVECTION FILM BOILING,

-6 The vapor is assumed to be in thermodynamic equilibrium and behaves like a gray gas. If the radiation heat transfer between the vapor and the plate surface and liquid-vapor interface is assumed equivalent to that of a slab of gray gas bounded by two parallel black boundaries, (9) then the local radiation flux may be expressed as T2 aT 4 qr = 2f T4 E2(t-T)dt - 2f T4 E(T-t)dt T O 4 4 + 2 a Ts E3(T2-T) - 2o TW E3(T) For an optically thin vapor for which T2, the product of the absorption coefficient ar of the vapor and the thickness 6 of the vapor film, is much less than unity, the functions E2 and E3 may be approximated by 1-O(t) and 0.5-t+O(t2) respectively. If the temperature gradient of the vapor in y-direction is much larger than that in x-direction, then the net radiation to unit volume of gas becomes dqr 2 ar (Tw + T 4) Equation (3) may, then, be rewritten as dT OT' 32T 4 T 4 l Tw 4 uT'I aT [4 [( ) ( ) ] (4) "I +^ -aH 4ar4 Ts I ~x dy y~ Ts 2 2 Ts The mass and momentum conservation Equations (2) and (3) also apply to the liquid layer, but now a subscript L is employed to identify the physical quantities of the liquid layer. Under assumption that the radiation from the plate surface and vapor to the liquid is completely

-7 absorbed at the interface, the energy equation takes the form 6TL aTL a2T UL ax+ - L. - oL + oy O~y2 (5) The appropriate boundary conditions are y - 0: u = v = O, T = Tw (6-a) (6-b) y -: UL = 0, TL = T0 (for subcooled boiling only) At the liquid-vapor interface, it is required that the continuity of the tangential velocity, the tangential shear, the temperature, the mass-flow crossing interface, and the heat-flow crossing interface be preserved: UL = u (7-a) (7-b) (II au = y Cy TL = Ts (7-c) L (ud L dx T _ V)L= Ts p(u (7-d) (7-e) dS - v) dx (k aT k 4T + B + 2aC ay L oy B T4 dy = p hfg 0 d /J u dy dx o - f u dy 0 (7-t) where 4 4 B (Tw - Ts ) (p/C)L + 1 + (P/E)w (7-g)

-8 is the net radiation flux between the surface and liquid-vapor interface. Equation (7-f) is an energy balance at the interface which states that the sum of the local heat conduction and net radiation gained at the interface is balanced by the heat of vaporization. The expression p d f u d y dx 0 indicates the rate of vaporization per unit area. The contribution of the vapor radiation is included as the fourth term on the left-hand side of Equation (7-f. Now, effects may be directed toward finding solutions. It is assumed that the film thickness 5(x) takes the form 6(x) xl(1 + am m/) (8) m=l where am are the coefficients to be determined. The similarity variable is defined as c=y/x/4 (1 + a xm/4) (9 m=l where r g cv2s L -P14 for subcooled liquid 4v2v' p c =1/4.g cos. PERTs 4v2 cos LRT 1/4 for saturated liquid 1. 4v2 P when the constancy of vapor properties is assumed. The continuity equation can be satisfied by introducing a stream function such that u- = a/y and v = - ar/x. \We introduce the new variables (r) = 4cv x(n+3)/4 (1 + Z am xm/4) fn() (9-b) n=o m=1 (1) = T-Ts = Fo(ra) + x /4 F(2)) + x3 F2 + F3(r) +... (9-c) Tw-Ts

-9 From this and the expansion of /() m/4 1/(1 + Z am xm/4) m=l in a power series according to the binomial theorem, it follows u = - = 4c2v x1/2 (f + xl/4 f + x12 f + x3/4 f+ ) (10-a) toxi 1 l/4(m+n-1) m+n+3 f v = - = - 4cv Z am x +1 n=o m=o (1 + a x1/4 + (2a2 - a2) x/2 + (3a3 - 3ala2 + a) 3/+...)] (10-b) where the primes represent differentiation with respect to the variable. Owing to the employment of the binomial theorem in Equation (10-b), the restriction co ~~ m/4 Z amx / < 1 m=l has been imposed on the solution. The transformations, Equations (8), (9) and (10), also may be applied to the liquid layer provided that CL and GL are defined as 1/4 CL L= 5 J. for subcooled liquid L Tco E PL - P /4 = - - L P for saturated liquid vL P TL - T Ts - To

-10 With the introduction of the transformations into the conservation Equations (1) to (5), followed by collecting similar forms in x one gets the following sets of simultaneous ordinary differential equations for both the vapor film and liquid layer. For the vapor film: IT? " 2 T -T +3ffo - 2(fo) + 1 + - FO (1-a) o Ts,,,,,,, i T,- T2 F af t 1 fl + 3fofl - 5fofl + 4fofl + F1 alfo affo (12-b) Ts s, I, I I IT,, T - T2 F 1 f2 + 3fof2 - 6ff2 + 5 2 + - F2 = alfl - (alfo+ 4fl)fl t 2 it 2' 2 I + 3(1 - alfofl - (3a1 - 2a2)fo + (al - 2a2)ff 2-c) The underlined terms which result from the buoyancy-force term in the momentum equation contribute to the coupling of the energy and momentum equations, i.e. the coupling of the f and F functions. The terms having the coefficient (Tw-Ts)/Ts are absent if the constancy of fluid properties is assumed in the vapor film. For the liquid layer, the underlined terms in Equations (12-a), (12-b) and (12-c) must be replaced by F, F1 and F2, respectively. 1 "I Fp + 3 foo = (13-a) pr 1 " 2al T - F1 + 3foF1 F foF1 = Np F - Fo(alfo - 4fl) (13-b) r r

-11 It I I T T T 4 + 3 fF -2fF2 S F0)4 N F2 oF2 2fF2 - D[(1 + T 2 2 Ts - (5f2 + alfl + 2a2f0 - a f0)F (13-c) T w2a 2t 2 D = 2al - s 2 (T -Ts)pCpvC The underlined terms which result from the radiation term in the energy equation are for the vapor film only. The boundary and matching conditions also may be rephrased in terms of the new variables. It must be noted that since the thickness 5(x) of the vapor film is small, one may take qL 0 at the liquidvapor interface for convenience. With the application of the transformations, there results: Plate surface: I I fn(0) = fn(0) = 0; Fo(O) = 1 and Fn(O) = 0 (14) for all n other than zero Liquid-vapor interface: n() = fLn(O (15-a)

-12-,i, ILPL 1/2 " fo( ) =[Ip ] fo (0) P ~p Lo It it LPL 1/2 it it fl( 5) - alfo(]5) = [I ] [fLl(0) - alfLo (0)] I-O It 2,I - alfl(T1i) - (al - a2)fo(TI) - [PL 1/2 ip ~-P It [fL2(0) (15-b) It - afLl(O) + (al-a)fLO(O)] FLo(O) = 1 and FLn(O) = 0 for all n other than zero (15-c) Fn(Bl) = 0 fn(5) = [ I] fLn(O) ElP (15-d) (15-e) 3E fO(n5) 4E[f1( rQ) = - F o( ) + AFLo(O) + alfo(0 )]l = - [F1(r6) - alFo(S)] + A[FLi(O) - alFLo(O)] + B1 5E[f2(%b) + alfl(i) + a2fo(r)] = - [Ft2(%) - alF (,b) + (a2-a2)F'(%,)] + A[FL(0) - a (0) o 2 F L2() a FLl(O), Tw-T ) + (al-a2)FLo(0)] + DPr j 6 (1 + T Fo)4dR 1 0o Ts15 (15-f) where jkLCL kC A = 0 for _* (T -T \ Tw sT W'Ts/ for subcooled boiling saturated boiling

-13 B1 = B/k(Tw - Ts) C E = p hfg v/k (Tw-Ts) = hfg Npr/Cp (Tw-Ts) For subcooled liquid the functions fLn(O) f (0) and f" (0) are un(O) Ln L replaced by (/ PL T- T,\ 1/2 PL. T T0 \PL-P T o and PL Ts- Too 3/4 f aL-p To( L fLn(O) respectively. PL-P o / In case the vapor density is assumed constant, pL(Ts-To)/(pL-p)To in the matching conditions must be replaced by unity. Free stream: f (oo) = O and Lo f (oo) = 0 for all n other than zero; Ln F (0) = 0. Ln

-14 Figure l(b) shows the physical model for natural-convection film boiling over a sphere with radius R. The governing equations and the boundary and matching conditions are identical with those of the previous case except that the continuity equation now reads 6ur + 6ur o ax Xy (17) and 4 is defined as x/R. Now, the film thickness and the similarity variable are defined as 00 5(x)=l+ a2m42m m=l (18-a) and 00 1 = Cy/(l + ~ a2m 2m), m=l (18-b) respectively, where [g Rv C = / [RRv L Rv2 L PL-P 1/4 p ] p for vapor PL-P ]1/4 for saturated liquid p P PL-P 1/4 ] — for subcooled liquid p The contir luity equation may be satisfied by introducing 00 00 2m + 2n + 1 *(O = cv ~ ~ a2m f2n+l 4 (18-c) m=O n=O such that u = -1 and v = -. The dimensionless r 6y r ax defined as where ao = 1, temperature is 00 G(r) = ~ F2n(q) (2n n=0 (18-d)

-15 When the transformations are introduced into the conservation Equations (2), (3) and (4), there results 2 It IIt f' -2f f -H-f = 0 1 1 1 1 t! It It 1 i It 4f1f3 -2lf3 -4ff+ (- - 2a2)flf + 2a2fl+ J-f = 1 3 1l3 - 1 3 3 1 1 2 1 3 (19-a) 0 (19-b)'2 " " 1 " 1 t" " " 3f3 -4f3f3 ff3 + 1 ff3+ ff3+ flfl- K -2a2ff3-2a2ff3 3 3 2 " t 2 "t I + 2a2f3+ 2a2flf- 4a4flfl (2a4- 3a2)fl- 6f + 2a2 f3 + 2a2flf1- 4a4flfl+ (2a4- 3a2)f1- 6fsf1,, It HtI + 6flf5 - 2ff - f 0 1 5 1-5 5 (19-c) where H= 1, J 1 1 = K =120 6 120 for vapor film H = 0, J = 0, K = 0... for saturated liquid H = FL J =FL2 - FLO L LO Y L2 6T LO Y K = FL4 - FL + 1 F T_____11 4 D[( Ts Fo +1) /= S 0 for liquid k_ for subcooled liquid - (Tw) ]... for vapor Ts 1 Npr Fo + 2f lF l 1o (20-a) 1 F - 2fl2+ 2flF2 + 4f3Fo - fF 2 1 2 2 30 1 0 2 1 NPr a2 Fo + 2 a2flFo TW-T -'s T T-T) Fo + 1)3 TTs + 3 t 2(T")3I s T -T Ts b2]... for vapor 0... for liquid (20-b)

1 Npr 11 t I < I 1', 1, F4 - 4flF4+ 2flF4- 2f3F2+ 4f3F2- 7 flF2+ 6f5Fo- ( 7 - 2a2)f3Fo - 4 + 2a)fF- 1 2 2 a2F 4- (- 4a4 + 2'a2)f_, N(2a4-3a2)FO- ar T -Ts T-T -D[4(W (w Fo+ 1)3 Tws F4 f or liquid 0... for liquid T -t + 6( w Fo + Ts 2 Tw-T 2 ) S(, F4)2] Ts. for vapor (20-c) The boundary and matching conditions are f2n+l O= L(2n+l) (O)= 2~n+l (0 L(2~n+l)(O = 0 (21-a) Fo(O) = FLo(O) = 1, F2(0) = 0 F4(0) = 0,... FL(2m)() = 0 F2m( b6) 0 C 2 f~it (0) = (_) L(2n+l) CL for all m other than zero (21-b) (22-a) V f? (2ntl) (T15), VL (22-a) where ( C)2 v CL VL I 1... for saturated liquid PL-P T 1/2 p Ts-Too... for subcooled liquid f"(0) = G f"(G ) fL (O) a2f(0) + G[fE(r l) - a2fl(rqS)] L3 L fL (0) = a2f3 (0) 2 + (a4- a2)f"l(O) + G[f"(Q5) - a2fi(rlb) 2 - (a4-a2) ft (t]5) 1

-17 [ -O]l/2... for saturated liquid ~ILPL where G [-i 1/2 PL-P T00 3/4 [Up ]1/2 PL-P T ]3/. for subcooled liquid [~LPL p T -Too T -T wD sn 4 AF' (O) - F'(T5) + B1 + - (1 + eo) d = Efl(5) Lo 2 r To d A[F' (O)-a2F' (0)] -Ft(r]) + a2Fo(5) r T T - T 2D 5 w-T.s 3 w s ____ + 2DNr f (l + T F) T F2dTj o s s =3E[a2f1(%) + f3(h5)] FL(2m)(O) - 0' fL(2m)() = 0 (23) It is important to restate that the analysis may be applied to both the saturated and subcooled film boiling. For the latter case, the functions FLn and parameter A become identically zero because the liquid temperature is constant and equal to the saturation temperature, i.e., TL(rL) = To = Tsat ~ Each set of differential equations for fLn' Fn, and FLn requires ten of eleven boundary and matching conditions. The extra one as expressed by Equation (15-f) may be used for the evaluation of the thickness Tb and the coefficients an.

-18 Now, it is desirable to inspect the physical parameters governing the natural-convection film boiling. There is a total of nine: Npr, NprL, Tw/Ts, D, [(pO)L/p] /2, (PL/PL-P)(Ts-T/Too) A, B1 and E. Of these, the first four arise in connection with the differential equations (13) for the vapor film and liquid vapor, while the last six enter through the interface matching conditions. The parameter which appears only in the natural-convection but not in the forced-convection film boiling is (pL/pL-p)(T -TT ). This results from the consideration given to the temperature dependency of the vapor density. In the absence of the vapor and wall radiation, the governing parameters reduce to five: i Pr''PrL 1/2 ^ [(P/)L/Pl] /, A and E. Forced-Convection Film Boiling The physical model and coordinate system are shown in Figure 2(A). The situation is, the laminar boundary-layer flow of a liquid with velocity UO over a flat plate. The liquid has a free-stream temperature To which is lower than;he saturation temperature Ts. The plate is maintained at the temperature Tw, sufficiently higher than Ts that film boiling occurs on the plate. With the same assumption as made in the previous case, the application of the conservation laws for mass, momentum, and energy to both vapor and liquid produces the boundary-layer equations which are identical with Equations (1), (2) with g = 0, (4), and (5). Equation (5) is needed only for the subcooled film boiling since the liquid temperature is essentially constant for the saturated film boiling. The boundary conditions at the surface of the plate (y = ) and the matching conditions at the liquid-vapor interface (y = 5) are identical with Equations (6-a) and (7) respectively. However, far from the plate,

-19 in the bulk of the liquid, the velocity approaches UO and the temperature approaches its bulk temperature T. Therefore, Equation (6-b) may be used as the boundary conditions provided that uL is equated to Uo. Utilizing prior experience with the free-convection film boiling in the previous section, the new dependent and independent variables for flow over a plate are defined: 1/2 c m/2 5(x), x (1 + 1 am x2) (24) m=l U- 1/2 00 / r= Y [] /(1 + m a xm ) (25-a) 2 Vx m m=l 00 (n+l) /2 ~ / ( ) = ~2cv (n+l)/2 (+ am x )f (25-b) n=O m+1 00 (ri) -= x/2Fn() (25-c) n=O where C is defined as (1/2) [UoO/v2] From this it follows that 12, 3/2, u - a = 2c2v[f+ x/f2 f+ +.] (26-a) v d - 2cv Z x { m+nfnf[ (26-b) Again, the restriction im~/2 < x 1 < 1 m=l

-20 has been imposed on the solution owing to the application of the binomial theorem in Equation (26-b). When the transformations defined by Equations (25) and (26) are introduced into the conservation Equations (1), (2), (4) and (5) it yields ttI I t I fo + fofo = 0 (26-a) I ti It I I II tit I f 1+ f 0- fof 1 2f-f1 2alfo a of o (26-b) f + f f - f f + 2f f 2af- 2f alf f 1 ol ol ol lo loo (26-b) tIt!! I It Tit t! f+ff- 2fof2+ 3f f2 = 2alf1- (2f+ alfo)fl' 2 " 2 "' 2 I + (fl) alffl+ (2a2-3a1)fo+ (1 al-a2)fOf (26-c) 1N fF + foo = (27-a) Prpr It' I I II 1 F + f -Fl fF1 - a F - (af - 2f)F (27-b) 1 1 o0 NPr Pr TWTS 4 1 1 Tw )4 + f - 2f, - D[(1 F) - - ) I Ts 2 2 Ts 2__ L.. (2a2-3a1)F0 - [3 + alfl + (2a+ 22 )fF+ f (27-c) With the introduction of the transformations, the appropriate boundary and matching conditions become identical with Equations (13), (14) and (15) provided that the coefficients 3E, 4E, 5E,... of the leftside terms of Equation (14-f) are replaced by 1E, 2E, 3E respectively. The physical model for stagnation film boiling in twodimensional flow is shown in Figure 2(B). The free stream velocity U, can be expressed as ax, where a is a constant.

-21 Uc, Too (A) FORCED-CONVECTION FILM BOILING ON A FLAT PLATE LIQUID x (B) STAGNATION FILM BOILING IN TWO-DIMENSIONAL FLOW FIGURE 2. PHYSICAL MODELS AND COORDINATES FOR FORCED-CONVECTION FILM BOILING,

-22 With the introduction of r = cy t = vcxf(n) T-T (rn) = s Tw-Ts 6x 1W l = X =_ 1 U, f' and aY 2 (28-a) (28-b) (28-c) v = - cf the momentum and energy equations, one obtains differential equations for both the vapor film (29) the following ordinary and liquid layer.. f"'+ ff+ 4 _(f )2 = O (30) 1 " + f' = [(1 + Tw-Ts )4 - 1 1 T )4 Npr 2 Ts 2 2 Ts where 1/2 C = [a/2v] The boundary and matching conditions are: Plate surface: f(0) = f'(0) = 0; 0(0) = 1, (31) (32) L,iquid-vapor interface: - It), f(= 1/2 tt f (f) = fL([), f ( rg) = [ -] fL(O), (L(O) = 1, ('rq5) = 0 tIp f(T5) - [%LPL/'P]fL(O) Ef(Tg) = - G(T5) + A Lt(0) + B1 (33) Free stream: fT( ) = 2, L() = 0 L L (34 (34)

-23 The stagnation film boiling in three-dimensional flow was analyzed in an analogous manner. In the interest of brevity the results are not presented here. It is important to examine the physical parameters which govern the transport phenomena, There is a total of eight in the forced-convection film boiling process. They are: Npr, NrL, Tw/Ts, DO [(PI)L]1/2, L PB1 and E. Of these, Npr and NprL arise in connection with the differential Equation (27) for the vapor film and liquid layer respectively. The next two arise in connection with the differential Equation (27) for the vapor film induced by the vapor radiation. The last four enter through the matching conditions at the liquid-vapor interface. Among the four, B1 is related to the surface radiation processes and A is connected to the subcooling of the free stream. In the absence of subcooling, NprL and A cease to be the governing parameters. When the vapor radiation can be neglected, D must be excluded from consideration as a parameter. When both the vapor and surface radiation can be neglected, Tw/Ts, D, and Bl must be excluded. This leaves five parameters to govern the transport phenomena. The dimensionless vapor film thickness TA and the coefficients am are not considered as independent parameters because there is a unique relation among A, am, A, B1, D, and E as written by Equation (15-f).

RESULTS Heat Transfer The local heat flux contributed by both radiation and convection at the plate surface is aT) 6 4 q - -k (oyyo 2 ar T dy + B 0 When the Reynolds and Nusselt numbers, defined as NRe = Umx/v and Nu hx/k (NNu = hR/k for sphere) respectively, are introduced and the heat flux is rephased into the variables of the analysis, there follows: For natural-convection film boiling one gets cx3/- Fo(0) [F (0) - alF (0) - B -[F/4 (0),, T2-T4s -alFl(O)+(al-a2)Fo(O) + - D Npr f/( l+ F) d]x /+... 0o s (35) for an inclined surface and Nu, NNu 1 I 4 4 3 2 - Fo() + Bl- - DNpr [Bo+ (Boa2+ 4BoB2)~ +...]d) CR- (36) for a sphere, where Bo = (T -Ts)Fo + Ts, B2 = (Tw -Ts)F2 For forced-convection film boiling on a horizontal surface /Nu - 1 Fo(O) - 2 [Fl(O) -alFo(0)-Bl]x1/2 [FO) NRel/ 2 2 - F2 () 2 T-T -alFl(O) + (a O)+ D Npr f (1+ w s Fo)dq]x (37) o Ti

-25 For stagnation film boiling in two-dimensional flow NN - -'(0) + B1 (38) cx where - G'(0) and B1 represent the heat transferred by conduction through the vapor film and by radiation from the surface, respectively. The leading terms of Equations (35) and (37) represent the corresponding heat transfer results in the absence of radiation exchange process. The second terms are the key terms in determining the effects of radiation exchange, since the contribution of the other terms is generally of secondary importance. An investigation of Equations (35) and (37) reveals that their second terms consist of Fo(0), Fi(O), a1, and B1 in which F1(O) and aI are inter-related to the parameter B1 for the surface radiation by Equations (13-b) and (27-b) and the second expression of Equation (15-f). Since the contribution due to the vapor radiation as represented by the parameter D and its associated quantities a2 and F(O) first appears in the third terms of Equations (35) and (37), the radiation effects on heat transfer is contributed mainly by the surface radiation, and the vapor radiation plays a rather unimportant role. Skin Friction The shear stress exerted by the flowing fluid on the surface may be calculated by Newton's shearing formula Tw = - p(au/dy)y=O. A dimensionless representation of the wall shear may be achieved by utilizing 1 2 a friction coefficient defined as Tw/~ p U2. When this is evaluated in 2~ ~ hnti i vlae

-26 terms of the variables of the analysis, there results T N 1/2 Re1/2 = 1 { f"(0) + [f (0) -af(0)] x1/2 1P v 2 = 1 1 o 2 9 f=) -00 + [f2(0) -alf1(0) + (a2-a2)f"(O)] x +.. } (39-a) for forced-convection film boiling over a horizontal plate. = f"(O) (39-b) /2 for stagnation film boiling in two-dimensional flow. The leading term in Equation (39-a) represents the skin friction in the absence of radiation exchange process. The second term is the most important one in determining the radiation effects. Based on the similar arguments for heat transfer, the radiation effects on the skin friction are found to be caused mainly by the surface radiation. The vapor radiation exerts a negligible or secondary effect.

NUMERICAL ILLUSTRATIONS Equations (12) and (13) for natural-convection film boiling over an inclined plate, Equations (26) and (27) for force-convection film boiling over a horizontal plate and Equations (30) and (31) for stagnation film boiling in two-dimensional flow were numerically integrated (Runge-Kutta method) in conjunction with their appropriate boundary conditions by means of an IBM 7090 digital computer. The first step is to prescribe the dimensionless vapor film thickness 1. The calculation is carried out for the saturated boiling of water under one atmospheric pressure with the neglect of gas radiation in the vapor film. This is justified as long as the vapor film is thin and the vapor pressure is not high. In other words, all radiation terms in the energy equations are neglected. Only the effects of radiation between the plate surface and the fluid interface, which appear in the boundary conditions, are taken into consideration. The range of the surface temperatures was from 280 to 3225 F (corresponding to r, from 0.6 to 1.6) and from 291 to 996 F (corresponding to rd from 0.2 to 0.6) respectively, for the natural and forced-convection film boiling. The emissivities of the wall and liquid-vapor interface are taken to be unity. The typical velocity and temperature profiles are shown in Figures 3 - 6. Figures 3 and 4 show the vapor velocity and temperature profiles in the natural-convection film boiling for the special case of constant vapor property. The terms f' and F' correspond to the velocity and 0 0 temperature profiles respectively in the absence of radiation exchange. Since the magnitudes of f2 and F2 are rather of secondary importance in comparison with those of f and F it is observed from Figures 3 -27

-28 0.18 F4-ly P-.-.o 0.16 0.14 0.12 0.100.08 0.060.04 - 0.02 / -0.02 2 FIGURE 3. 0 2 o 0.4 0.6 0.8o3 1 f2 --- =1.0, Tw = 727 ~F =1.2, =1295 OF u - fI + f'X1/4 + f, X1/2 _fo+f +f +fXI/ o I 2 xv CO > (P-P)/P FUNCTIONS ASSOCIATED WITH VAPOR VELOCITY IN NATURAL-CONVECTION FILMOBOILING ON AN INCLINED PLATE FOR Ts = Too = 12 F.

-29 0 x g.-.N 0o x gLLg 1.0 0.8 0.6 0.4 0.2 0 -0.2 -0.4 0.2 0 F I -I..... =1.0 S8,Tw =727 OF =1.2, =1295 ~F T -T....S F + FX +FX I2 FIGURE 4. FUNCTIONS ASSOCIATED WITH VAPOR TEMPERATURE IN NATURAL-CONVECTION FILM BOILING ON AN INCLINED FLAT PLATE FOR Ts = To = 212 F.

15/ / X -t'- 06 996 10 0o / / -— I\\ ~ —----- I 0.4 541 5 (AJ 2U,/ f / -/ -- 0 0.2 0.4 0.6 0.8 1.0 FIGURE 5. FUNCTIONS ASSOCIATED WITHVELOCITY PROFILES IN THE VAPOR FIGURE 5. FUNCTIONS ASSOCIATED WITH VELOCITY PROFILES IN THE VAPOR AND LIQUID BOUNDARY LAYERS FOR FORCED-CONVECTI FN ILM BOILING OVER A HORIZONTAL PLATE FOR Ts = T, = -1 F.

,L -0.2 /7)'- 0.2 0.4 0.6 0. o IF F" -0.4 FL.0 2 -0.6- TTs -8 -T- =+ 1-Fo/+XF+XF2| Dir) TW, F U|,u ft/sec Tw sS__ _ S _ _ _ _ -0.8 —... 0.6 996 10 -1.0 —- 0.4 541 10 0.4 541 5 FIGURE 0, FUNCTIONS ASSOCIATED WITH VAPOR TEMPERATURE IN FORCED-CONVECTION FILM BOILING OVER A HORIZONTAL PLATE FOR Ts = Too = 212 F

-32 and 4 that the presence of radiation is to increase the velocity profile and decrease the temperature profile. The effects are greater for higher wall temperature or thicker vapor film. For the forced-convection film boiling case, Figure 5 illustrates that the velocity profile f' in the vapor film is practically linear 0 in the absence of radiation. The presence of radiation is to increase the flow velocities in both vapor film and liquid boundary layer, and hence the skin friction is decreased at the plate surface. An increase in the wall temperature (or the vapor film thickness) or a decrease in the freestream velocity results in an increase in the radiation effect. Figure 6 shows that the temperature distribution Fo in the vapor film is practically linear in the absence of radiation process. A simultaneous action of radiation is to decrease the vapor temperature, and hence the heat conduction is increased at the wall surface. As shown in Figure 6, the radiation effects on the temperature profile are larger for thick vapor film (or higher wall temperature) or for lower freestream velocity. Figure 7 shows the heat transfer and skin friction characteristics for the stagnation film boiling in two-dimensional flow of water at a velocity of U0 = 10 x ft/sec under atmospheric pressure. It is seen in the figurethat as the surface temperature increases from 212~F, both skin friction f'(O) and conduction through the vapor film - 9'(0) decrease, while surface radiation Bl increases. When the surface temperature exceeds 1000~R, B is larger than -'(0) indicating that surface radiation becomes more important than conduction through the vapor film.

7 r —-------------------------------- - -- - ~14 ~~~~~~~~6 ~~12 700 800 900 1000 1100 1200 1300 140 10 10 10 10 r~3 2~~~~~~~~~~~~~~~1 r"(O)~~~~~~~~~~~~~2 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 0 WALL TEMPERATURE TW('R) FIGURE 7. HEAT TRANSFER AND SKIN FRICTION CHARACTERISTICS OF STAGNATION FILM BOILING IN TWO-DIMENSIONAL FLOW OF WATER AT ATMOSPHERIC PRESSURE.

-34 Tables 1 to 3 furnish important results for radiation effect on heat transfer performance and shear stress at the wall surface and liquid-vapor interface. For the natural-convection film boiling over a vertical plate, Table 1 indicates that the presence of radiation is to increase the heat transfer from the wall to the vapor and from the vapor to the interface. The radiation effects become greater for higher wall temperature or thicker vapor film. Tables 2 and 3 show that for forced-convection film boiling over a horizontal plate, radiation increases the local Nusselt numbers and decreases the shear stresses at the wall surface and liquid-vapor interface. An increase in the wall temperature or vapor film thickness or a decrease in the free-stream velocity may contribute to an increase in the radiation effects.

CONCLUDING REMARKS To replace the existing empirical equations which have been used for estimating the total surface conductance in film boiling expressions are now obtained for the determination of the heat transfer rate and skin friction in the natural-convection film boiling over an inclined surface and a sphere, the forced-convection film boiling over a horizontal plate, and the stagnation-flow film boiling when radiation is appreciable. The problems have been formulated exactly within the framework of boundary-layer theory with the consideration of the shear stress and vapor velocity at the liquid-vapor interface. The method of analysis may be extended to the natural- and forced-convection film boiling over other surfaces of different geometry. The problems of film boiling on a surface having space-dependent temperature may also be solved by the present method by expanding the surface temperature into an infinite series with respect to the depending space variable. -35

TABLE 1 CERTAIN PROPERTIES ASSOCIATED WITH HEAT TRANSFER PERFORMANCE IN NATURAL CONVECTION FILM BOILING ON A VERTICAL PLATE FOR Ts = To =- 212 F %o, Tw(OF) B1 F'(o) Fi(o) F2(o) F (rl5) F{i(%q) F'(%l) a a2 ba b2 0.6 280 0.487 -1.674 -.00172 -.0014 -1.65.0048.0024.08.00205 -.35 -.oo68 0.8 424 0.640 -1.267 -.00905 -.00108 -1.20).0206.00264.142.00896 -.469 -.0143 1.o 727 1.117 -1.031 -.036 -.0102 -.923.0826.0223.315.0474 -.799 -.053 1.2 1295 2.678 -o.886 -.184 -.0154 -.7095.375.271.948.466 - 1.92 -.104 1.4 2172 6.751 -0.792 -.774 - 1.87 -.541 1.498 2.89 2.93 5.182 - 5.186 - 2.48 1.5 2707 9.806 -0.757 -1.497 - 6.71 -.469 2.635 7.61 4.73 14.97 - 7.623 - 5.92 1.6 3224 14.63 -.729 -2.796 -21.7 -.406 4.62 20.26 8.15 45.6 -11.49 -34.6 Remar:, F - a - B1 I 0\ I Remarks: b1 = F'(O) - alFo(O) - B1 F) - aF() (a a)F b= FI(0) - a1F'(0) + (a2- a2)F_()

TABLE 2 CERTAIN PROPERTIES ASSOCIATED WITH SHEAR STRESS IN FORCED-CONVECTION FILM BOILING OVER A FLAT PLATE FOR Ts = TX = 212~F ft UV(sec) - Tw(~F) f f f'( fo() fi'() f() Cl c2 10 0.6 996 3.409.583.775 3.018 -.469 -.145 -2.503 3.111 10 0.4 541 5.008.196.0093 4.747 -.0096.0198 -1.13.317 10 0.2 291 9.792.0993.0279 9.66.0693.0098 -.67.071 5 0.4 541 5.008.278.172 4.747 -.0134.044 -1.636.782 5 0.3 393 6.61.1856.069 6.416.0635.0346 -1.194.268 5 0.2 291 9.792.139.02-t9 9.66.0968.0204 -.951.1105!<1 Remarks: c = f"(o) - alf"(o) 2 = f,"o) - alf'(o) + (a1 - a2) f

TABLE 3 CERTAIN PROPERTIES ASSOCIATED WITH HEAT TRANSFER PERFORMANCE IN FORCED-CONVECTION FILM BOILING OVER A FLAT PLAT FOR Ts = Too = 212~F U( sec) rl5 Tw( F) B1 Fjo0) Fi(o) F o(o) F ) Fi((6,) F2(65) a1 a b b 10 0.6 996 4.84 -1.708 -.09 -.1203 -1.537.288.301.899 -.026 -3.301 -1.38 10 0.4 541 2.15 -2.527 -.021 -.00979 -2.414.0579.0217.266 -.00115 -1.505 -.181 10 0.2 291 1.28 -5.013 -.0027 -.00064 -4.958.0087.00035.0788.001 -.88 -.026 5 0.4 541 3.04 -2.527 -.0266 -.0179 -2.414.0819.428.376 -.00156 -2.141 -.36 5 0.3 393 2.25 -3.35 -.014 -.00575 -3.27.0355.0141.208.0073 -1.567 -.122 5 0.2 291 1.81 -5.013 -.0039 -.00029 -4.958.0123.00398.111.00201 -1.254 -.045 I 0o CX! Remarks: bl = Fj(o) - alFo(O) - b = Ff(o) - alFi(o) + B1 a - a) Fo(o) 1 2 0

REFERENCES 1. Hsu-Chieh Yeh and Wen-Jei Yang. "Radiation Effects on Film Boiling in Natural- and Forced-Convection Boundary-Layer Flows," ASME paper No. 66-WA/HT-6 (1966). 2. L. A. Bromley. "Heat Transfer in Stable Film Boiling," Chemical Engineering Progress, Vol. 46, 1950, p. 221. 3. M. E. Ellion. "A Study of the Mechanism of Boiling Heat Transfer," Memo No. 20-88, JPL, CIT, 1954. 4. Y. Y. Hsu and J. W. Westwater. "Approximate Theory for Film Boiling on Vertical Surfaces," Chemical Engineering Progress Symposium Series, Vol. 56, No. 30, Heat Transfer-Storrs, 1960, pp. 15-24. 5. P. W. McFadden and R. J. Grosh. "An Analysis of Laminar Film Boiling with Variable Properties," International Journal of Heat and Mass Transfer, Vol. 1, No. 4, 1961, pp. 325-335. 6. R. D. Cess. "Analysis of Laminar Film Boiling from a Vertical Flat Plate," Trans. ASME, Series C, Vol. 81, 1959, p. 1. 7. S. C. Y. Koh. "Analysis of Film Boiling on Vertical Surfaces," Journal of Heat Transfer, Trans. ASME, Series C, Vol. 84, 1962, p. 52. 8. E. M. Sparrow and R. D. Cess. "The Effect of Subcooled Liquid on Laminar Film Boiling," Journal of Heat Transfer, Trans. ASME, Series C, Vol. 84, 1962, pp. 149-156. 9. J. C. Y. Koh and T. W. Nilson. "Simultaneous Convection and Radiation in Laminar Film Boiling on Vertical Surfaces," ASME Paper No. 63-HT-2, 1963. 10. R. D. Cess and E. M. Sparrow. "Film Boiling in a Forced-Convection Boundary Layer Flow," Journal of Heat Transfer, Trans. ASME, Series C, Vol. 84, 1962, pp. 370-376. 11. R. D. Cess and E. M. Sparrow. "Subcooled Forced-Convection Film Boiling on a Flat Plate," Journal of Heat Transfer, Trans. ASME, Series C, Vol. 84, 1962, pp. 377-379. -39

APPENDIX COMPUTER PROGRAMS -40

1. For Stagnation Film Boiling in Two-Dimensional Flow $COMPILE MAD, EXECUTE, PRINT OBJECT, DUMP, PUNCH OBJECT PRI I CONiM::ENI $FILM BOILING, STAGNATION FLOW$ PRINT CUVMMENT $UNIT IN FT, HOUR, DEGREE R, LBM, BTlj -D..YElTTOsNT Y ( 3 ), FCT,~ —OT3),P(-T00-, FG ( 100), FGP 100), FGPP (100), FGPPP (100), PL (100), FL (100),FLP (100 ),FLPP(100),'2FL-P-P(P0"-);-TT —O- TT - T, —TR( 10)., G(10 ), TWG( 10 ) 3LP(100), R(100),RP(100),RPP(100), GR(10) VECIUR VALUES RESULI = $1H,5F18.6*$ VECTOR VALUES OBTAIN = $1H,4F18.6*$ -TNiTE-GE J.Rv~-j-Ji-A'X-, — T, —N-,; Z START READ AND PRINT DATA S3 J=O tbXELU I E S i RKD. (3D,Y(1 ),( 1),,X, S EP) X=0. YTIT=TT. Y(2)=0. Y3T3F —-FGP-TTT SAVE J=J+1 "TT(-JT(X FG(J)=Y(1) -FG-P —JT=yT)-( F(;PP(J)=Y(3) F-GTPP FrJT) —FT) H= ( J-1. )/DALJ Z=( J-1 ) / DALJ WIHENEVER H.E. Z P-R I FST — F-0R-rS ATR — JSUL'T, ----—;-PT J T-,-F-PJFP T PT ), FPP F G P P Pi (-'END OF CiONDITIONiAL -:WEl-.EVE-R-" —— E.-F- S.. iG'lA X-, T R-ATS-F-R-TJ-S CALL S=RK)EO. ( 0 ) WlHEiTEVER S.E. 1.0 F(1)=Y(2) TF T-2) =Y ( 3 ) F ( 3 ) =-FG ( J ):FG;PP ( J )-4. +FGP (J ) FGP (J ) TRAiNSFER TO CALL ENlD O F C )'OND I T I 0 AL TRANSFER TO SAVE SS J=0 EXECUTE SETRK).(3,Y( 1),F( 1),0,X,LSTEP) X=O. Y ( 1 ) =L*L'FG ( GJMAX ) Y (2)=FGP (GJ MAX) Y(3)= L*FGPP (GJivAX) LSAVE J=J+l PL(J)=X FL(J)=Y(1) "F L P ( J)"= Y-(-2 FLPP(J)=Y(3) FLPPP ( J ) =F (3 H= (J-1. )/)ALJ 7-'= rJ-:i )-7)D A L J WHENEVER H.E. Z -PRTNT-iTFT-ORr"AT RESULT, PL( J ), FL( J),FLP ( ), FLPP ( J ),FLPPP (J ) END OF CONDIITIONAL GG=FLP ( J )-2. WHENEVER.ABS. FLPP(J).G.. EPFLPP, TRANSFER TO SSS WHENEVER.ABS. GG.L. EPFLP, TRANSFER TO S8 SSS WHENEVER J.E. LJMIAX, TRANSFER TO S4 LCALL S=RKDEO. (O)

;. HEi!,EV VER S.. 1. F(1)=Y(2) F ( 2 Y 3 ) F(2)=Y(3) F( 3 )'=-LF(-': —FL —-'P(,:) —4+Fi- LP(J)' FL( J ) T k A [\. S F [E T il L C L L Hi ri.i) lJPF CG ir...i I1 l' IA t..,,~ A L 1" R A i. S F E T L S \! F-.%'4: I=I + l 1=1+1 WHEtiEVER I.E. 1u, IASFE TAiJSFk START 1 I ( )= PP ( 1) G( I)=( GG Ti-T —iES LR T S — T-( 1) ~ GTIT WHE.; F,iEVEI:,E I.L. 2, TRAhSFER TO S5 UL=- ( 1 )'AL/ ( G (- -G 1-1 ) ) S5 FGPP(1) =i(F PP (1. ) +D1AL TRASi'F-TS-ER.T.- S-3-.. S H8 I,, E' E \F J. E. L J I AX, TRAN S F E R T S 9 J3-3 1 — LP(J )=LP( J- ) +LST-LEP — FL( J )=Y ( ) +1-LP ( J ), LSTEPFLP(J )=Y(2) T —LPP' —C )I'-. -LPP ( J ) =.. -T-,]Ai- S-F —E'..i- TS'' ~1-vA;HSjP!-F.' Hi SF ) S9 Pk IiN\l-T CU(MMEiT $Ii:X, X F(,, P(P, FPP, GFG PPP$ T H R 0 U G A S, F (.JR i= 1,1, I. G. G J i X LAST PRI IT FULIhRiN'A RESUL T, P(",) FG(i ",),FGP(N), FGPP (I,) T FGPPP(N) "P It'T —C('.'iii',-i.-s-XL -'L' L F P FLPP, L' PPP TI HR UH. J- L L S T, F ( R =l I 1, ri. (. LJ1,,AX L L AS'T -i.T F-T"RT T R.SU L -P"('i'~-, (i I ), \ FLP(l) F L P P (I-,) FL-PP P-( t EXECUTE SEITRK)_. (2,Y(1),F( 1 ), ), X,STEP) I = S23 J=( Y(1)=1. YT z-)R —P'r -() GET J=J+1 P ( J ) =X P(J)=X R(J)=Y(1) -R-P T-)..=Y (2 — T1 RPP(J)=F(2) -H= —TJ — I-T 7/TIA- LTJ Z= (J-1 ) / ALJ WHEItlhVtR K H... L PRINIiT FURMAT U TAIi, P ( J ) R ( J ) TRP ( J ) RPP (J) E-ND ~J F —CUL 1 I -T[I- UilNAL - WHENEVER J.E. GJIMAX, TRANSFER TO SSSS C ONTI S=-RK [)E i( —.(. ) WHErNE\iER S E. 1.0 F(1) =Y(2) F( 2 )=-PR: FG (; J ): RP( J ) "T'R,A'-i< SF E-"......T 0"-Cui'-TTT'....T I- --- ErHJD'-lF COTil:) IT'TICiNA, TRAi.' S-Ft ER -T' — SET.. --- - SSSS'lHNEVER.OAS, R(J).L. EPR, TRANSFER TO S29 T= I+1 G;R(I) =R(J) TR-( = —R- PT-(PRINiT RESULTS (GR(I), TR(I) W'FtHE' NEVE R...-I.-L. -- 2-, - TR ANi'<SFE R T- T — S25

-43 RDAL=-GR (I )R [)AL/ ( GR (I )-GR ( I- ) ) S75 -TP ( 1 )=RP( 1 +RDALTRANSFE'R T1- S23 -S2 9 P R II' I C I i, AENT S X, R, R P,- R P P — THROU3.GH RLAST, FOR N=1, N.,G. GJMlAX__ -R L A-ST — P- I-T, -FT-It:'- T T --'.l- I, - -( -) - R(i ),R. P ( ),T R P P ( N) I =0 TT.-3 I=I1+1 E = RO!N'J UH. F G / K< / ( T 1.I- T S )_....'T-'-'-SI-'R-'-(-T i4 —S-) " 4 ('T:.'-' T'4 -TN S,' T S ) / (( RE F LL / E i'RI T L 1+1 + R E F L W'/ E' I T, ) 4: K) ( 2. ^:NU /A ) P. 0.5 -TY-'T'.r-,T"'I-E —F - (' Gf'i3-i-/A;-XT "-)-+ —F-R rG-J-7'T,'X-%- - B 1 PRINT RESULTS -TNG( I ), TiJ, 81, E,,THENEVER- I.E. 10, TRANSFER TO STARTWHENiEVER.AB5S. TWG(I).L, EPTW, TRANSFER TO START — EB tEFER-r —-.- L-. -2TR-'SFE-T'D TWSS5 TWt DAI=-TIG ( I )T:: DA )L/(T G( TG I )-TWG( I-1) ) TWSS'-F-Tr,.VTT.TAT, TRANSFE 1 T- t' S3 END OF P l..) (G AriRA $DATA -D AL=1 — 0 —-. 7. —T"EP- L —'-..EP-0.OOt1, RP 1T ) -=-0 (.5, HFG=970.3, TS=672., SIGMA=0.172E-08, REFLL=O.1, EMITL=0.9, REFLW=0., EMITW=1., -ERPTW'-1,R ) A L 0.O1, T DALO., L=.00 515, GJ MIAX=21, D)ALJ=5, LJvItAX=21, EPFLPP=().01, EPFLP=0.1, LJMAX=16, PR=.94, L=. 0-05-16, A=36000., TW=850., STEP=,022, FGPP(1)=5.33, ROW=.0328, i\lUl=1.09, K= 0171*: TW=1-05-0., TEP-3, FPP 1)=4.106, R tW=.0288, NJ=1 42, K=.02TW=1450., STEP=.065, FGPP(1)=3.592, ROW=.0233, NU=2.2, K=.0257* TW=1 —800.7, —- STEP=.1- FGPP- 1) =3. 542, R OW =. 196, NU=3.08, K=. Q32 1

2. For Forced-Convection Film Boiling Over a Horizontal Plate $COMPILE MAD, EXECUTE, PRINT OBJECT, DUMiP, PUNCH OBJECT PRINT COMMENT $FILM BOILING, HORIZONTAL$ DIMENSION Y(3),F(3),0(3), P(100), 1 Y1 (300,WA),Y2(300,lB),Y3(300,WC),F3(300,WD), 2R(300,WE),RR(300WF ),RRR(300,WG),T(30,WH),G(30,WI), 3LY1(300,WtJ),LY2(300,WK),LY3(300,WL), LF3(300,WM), 4LR(300,WN), LRR(300,WP),LRRR(300,WO), LP(100), 5A(30,WR),H(30,WS) VECTOR VALUES WA =2,1,100 — vTCTO.'V Ai-UES WB — =2-T,1, 100 VECTOR VALUES WC =2,1,100 — VEfVCTORVA- — L U S WD -,T,2 VECTOR VALUES WE =2,1,100 VECIOR VALUES WF =2,1,100 VECTOR VALUES WG =2,1,100 -VECT —R VALUES WH -=2,1,10 VECTOR VALUES WI =2,1,10 -V'E-CTO..VA-LUE- WJ, -i, 1 = 00 VECTOR VALUES WK =2,1,100 VEClOR VALUES WL =2,.1,100 VECTOR VALUES WM =2,1,100 -TECT-R VA LF-TES-W N =2,1,100 VECTOR VALUES WP =2,1,100 -VECT'V-AOLUES- WO =2,1,100 VECTOR VALUES WR =2,1,10 VECTOR VALUES WLS =2,1,10 - VECTOR VALUES RESULT = $1H,5F18.6*$ I- — E, N 1, U, TS- ~ - INTEGER I,J,N,U,IM-i START READ AND PRINT DATA M=0 S1 I=0 M=M+1 WHENEVER M.E.2 U=O A1=B1/(2"E"Y1( 1, JMvlAX) -RR(1, JMi1AX)) END OF CONDITIONAL WHENEVER M.E.3 U)=0 A2=(A1*RR (2, JMAX)-AliAl'"RR( 1, JMAX)-3*E*A1*Y1 (2, JMAX) )/ 1(3' E Y1( 1, J AX )-RR ( 1t,J AX) ) END OF CONDITIONAL S3 J=O EXECUTE SETRKD. (3,Y(1),F(l),QX,STEP) X=O. Y(1)=O. rYT2iYO-. --- Y(3)=Y3(M,1) SAVEF _J. — _ P(J)=X Yl(, J)=Y()1) Y2(M,J ) =Y(2) -y3T,-3A- T=Y3).. F3(M, J)=F(3) WHE N EVR JE M A X — T'A NSFETT-TO-SCALL S=RKDEQ.(0) WHENEVER S.E.1.O F( 1)=Y(2) -F(T2)-=Y 3) WHENEVER M. E.1 - FT3 -YT1-T-YT3-1 END OF CONDITIONAL WHENEVER M.E.2 F(3)=- Y1(1,J)*Y(3)+ Y2( 1,J)*Y(2)-2*Y3(1,J)*Y(1)+2 IPAlt t^ I,J 1-AI Yl i J t i, J Y) t 1,J END OF CONDITIONAL W-HEN EVER-M E-J.3

F(3)=-Yl(1,J)*Y(3)+2''Y2(1,J)*Y(2)-3*Y3(1,J)*Y(1)+A1*2*F3(2,J) 1-(A1Y1( 1,J)+2*Y1(2,J) )*Y3(2,J)+ Y2(2,J)*Y2(2,J)-Al*Yl(2,J) 2*Y3( 1J)-(3*A1*A 1-2*A2)*F3(1,J)+( 0. 5'A*A1 -A2)*' -3Y-Tr-, J-T.VT,T-,f -3- - END OF CONDITIONAL -TR-AN7S'FER' — T -C-A END OF CONDITIONAL IRANSHFR 1LU SAVE SSS J=O -EXECUBTE.S ET R K D.3T7Y T-T7F IT';,'T CSTEIP] X=o. -Y1 1 1= -0-. Y(2)=Y2(M, JMAX) WFIENEVER M.E.1 Y(3)=L,Y3( 1,JMAX)'E ND.(TF — C- I T I O A L WH-iENE VER MlE. 2 YT3 ) — A-LY3 +L ( Y3( 2,JAX ) -A1 —3 ( 1, JAX ) ) END OF CONDITIONAL W HENEVE R M.E.3 Y( 3)=41 *LY3 ( 2, 1 )-( A1'A1-A2 ), LY31(, _ ) 1 +L 3 ( ( 3, iAX ) -A1 Y3 ( 2, JAX ) - ( A 1 A 1-A2 ) 3 ( 1, JMAX) ) END, OF CONDITIONIAL LS A V E J= J+ - - A — —. LP ( J)-'X.. LY1T ( i', J )=Y ( 1) L Y 3 ( ii, J ) = Y ( 2 ) -E-VE- TT J"J —Y —( —STLF3(Mi, J)=F(3) T-A; H FRI' EV'F T 7.E — GG=Y(2)-2. I HltER iV I SE GG=Y ( 2)'E- rT — -FK C o N IT -I T —O'AL WVHENEVER.ABS.Y(3).G.EPSIH, TRAPiSFER TO SS F:ET E'V-EP-7-7-.-S —.-GG-.'~L —. —-i- i -T-SVT R AN- F E R T S 8 SS WHENE\EVER J.E.LJ'IMAX, 1TRANSFER TO S4 LCALL S=R KDEO.(0) WHENEVER S.E.1.0 - FCT-=-VYT.Z'F(2)=Y(3)'-FTEv EVr T';I-..-' F ( 3)=-Y ( 1 ):Y ( 3 ) END) OF COLi'l I TI UAL WH E E V E,R. i.4. E. 2 -' —LYTT- Y-3)+LY2(1, J ) Y ( 2)-2 LY3 ( 1, J ):Y ( 2 )+2"' 1A1':'LF3(1,J)- Al'LY1( 1,J)' LY3( 1,J) WHENEVER r.Ei. E.3 F (3= ) =-LY.( 1, J )':Y ( 3 ) +2*LY2( J )Y 2 -3LY3 ( 1, ) Y LY3( J ) Y ( 1 ) +2*A1 1LF3(2,J )-(A1;LY. ( 1,J)+2"LY 1 (2,J) )'LY3(2 J)+L Y 2(2,J)* E..D.OFi-N..~F [TI —C- FT-OBf L TRANSFER TO LCALL'-T1D FFC- r0T)-TT-ITON —L TRANSFER TO LSAVE ^SZI = I+1 T(, I )=Y3(MI11) (7- (' I )=-GG PRINT RESULTS G(ii, I), T(M, I) WHEEVER I-.L.2, TRANSFER TOC) S5 WHENEVER G(i,I).G.O, TRANSFER TO S6 WHENEVER G(M,I-1).G.O., TRANSFER TO S7

-46 TRANiSFER'TO S15 DAL=DALTA Y3(i, I1 ) Y3 (,1:, 1 )+ )AL S6 WiHENEVER G((iN, I- ).L.0., TRANSFER TO S7 TR Aii SF R T S15 S57.AL -('( ii, I )' I)AL/ ( G ( v, I )-G( h', I-1 ) )'Y 3 ( 1;,Y-' - 3 ( li, 1 I) + D) AL TRAlFISFER Tl O S3 S 15. [AL =- G( ",'T ) O D'-('(-, I i' —IGT'-',1 -1i)) Y3(i, )=Y3 (i,1) + )AL TRA"l"SFER Ti) S3 S8 Ii-WEL EVEP J.E.LJiMAX, TRAliISFER TO S9,= J+ LP(J)=LP(J-1 )+LSTEP L-?Y1Ti-('~q,'') —— YTTT LY2 ( J ) =Y ( 2 ) — LY3 (, J ) =(). LF3(I, J )=0. f'i:A;1S F -ER TO' SS8 S9 PRI\NT.i RESULTS M - H -.'c-i-(-^.....'.'..i..-_ —....i_ —.".... _ _....' T H RUO GH LAST H FO 1 l 1. 11. G. J AX. LAST PRIINT FORiAT RESULTP(N),YI( tN),Y2(v1MtN), Y3(MiN)F3(iV'N) THRUU(GH LLAST, FOR N=1, 1,N.G LJMiAX LLAST PR.INT FURPiHAT RESULT'LP(iN)LY1( (,nr),LY2(MiN) LY3(riiN),LF3(WMN) tEXECTUTE. SETRKO (2,Y(1-), F ( 1), O, X,STEP ) X=O ",.~2-'3' J =0 X = (_) WiHE L-VER.i.l 1 Y ( ) = 10 OTytR-:,i I SE Y( )=(O. T'- -[TF- i ['.) T ) r'T -F I A L Y ( 2 ) = R R ( iN, 1 ).(TET' J + 1 Pt ( J )=J+ P(J)=X wuin^JTiTTT - RR (iJ ) =Y(2) — KR,-'(-',"i —~-C2-I WHENEVER J.E.JIAX, TTRA\NSFER T) SSSS CLNTI S=RKLDE).( 0) IHENEVER S.E.1.0 T'-I'- (-ZT WHENEVER ri.E.i -F-T2T —=P —-,'T7-1J) ]Y( 2 ) END OF CONDITIONAL WHENEVER i.E.2 F(2)=PR-R(-Y1(1,J) —'Y(2)+Y2(1,J)*Y(1)+(2/PR)*Al-RRR(1,J)-( I- rYT-rr-T2 avTiTt-27J) "WRR f-i' END OF CO)i,)DITIONAL F(2)=PR-'(-Y1(1,J)-Y(2)+2'Y2( 1, J';:Y(1)+(2/PR) *AlRRR (2,J) 1- ( A1,lY 1 ( 1, J )-2"'Y 1 (2,J))*-RR(2,J)-Y2(2,J) *R(2,J)+(1/PR.)* 2( 2.A2 —3%'Al' Al)*RRR (1, J )-(Y2( 3,J )+6*Y(3, J)+A1* -3-YT —' —~~ 2'A -... —:A1 —A-)-Y 7f —T11J )J)*RR (1,J ) ) END OF CONDITIONAL — Tir -SFER- TOT-' CNTT — END OF CONDITIONAL -TRANSFER Il ( I SSSS WHENEVER.ABS. Y(1).L.EPSID, TRANSFER TO S29 — TRN-A —SF-E-TT O —'S-2-4

524 I=I+1 (.(!.T I )=Y( IT (, I') =RR ( I.i, 1 ) PKRli' I R SLL I S I ( i, I ), T ( vi,T WJHEi"E\VER I.L.2, TRAliSFER TO S25 E-I \,71 E i' VE-fR. 2' —— G — I —-(7.-'(, —'-P, -/-: i'S'F'E-'F-T S 2 6 WiJi-IEiEVER (.(, 1 -1 ).G.O., T-RA'lSFER TO S27 I TRASFERiI-SAN S F ER S25 D A L =)AL TA RR (,',1 ) =RR. ( it ) +U.AL TRAS,,SFER TI S23 _ T R A i; V S F E TT T I:3t S 3T -S 2-6 -i i': EY "' G (-i'y'- I -— )- OT,-' -RI-'sS F R- -Tr S 27 TRANiSFER TO) S35 S 2-7' f L- — G (-i.;-I- T-'L)AL-7 -GT-( —T,-I ) —- ( —- 1- )) — RR ( i'i, 1 ) =RR ( M, 1 ) +i)AL TRANiSHFER il.) S23 S35 )AL=-( ( i I )' AL/ ( G( i, I )- ( 1 ) ) Ri' (-'', 1' ) =-R —R-(,1 i)+- AL:.)' TRAilSFER t:J S23 S2 9!-HENEV E'i.' E. E=-RR. ( 1, J iAX ) / Y ( 1, J iAX) I II I... (i,; H (. I I / ( K A' -: ) + T S B1=S I GMA, ( T T! + T S )' ( T I T + T S- T S ) / ( ( REF L/ E I TL F-FP..-TR-E,7-.IT-T-;aiTYK -A-'IN-'' F-F P.':"5-)N t -) —. 5 T H R G..lH AA A, F (R =1, 1,1. G.J Z-Y2(2," TJ —A —L:-Y — (, J -' — Y3,-v3 ) -( 3',:*AlA1-2,A2 ) *' LF3 ( 1, J ) + ( Al 3:. 5 A'1-A2 )'LY1 ( 1, J ) LY3( 1, ) A P R i kI I R bE S tL I S P ( R' ), R ( M iT, ) K ( IM r ) T R ( I- t, ) P R I T RE SL TS E, T^, i E ID F C I" T I S L ET HE TF E X. --— E'-2 -...... AA1= ( b1-RR ( 2,JiAX )-2 E*':Y 1 ( 2, JMAX ) ) / ( 2*EY 1 ( 1 JMi\X ) I-KK ( t JilIAX ) J IiFl.Eiv EVER-), AB-S. (A1-A- A1).L.EPS I A,TRANiSSFER TO S50 H (, t J ) = A A 1-A 1 4-(7'TFt-T1- ~ —-I PRINT RESU.JL TS H(Hlt ), A(Mt),A 1,Hbtii:tVtR Oi.L.2, TIRAlNSFER TO S65 WiHEi!EVER k i(M 1 ).(.0. O TRAl\NJSFERK Tl) S66?A_ E'T TJ-_ -1I T,' ^ — S67 T. Fi A Ni S F t P, T A S - E7 TRANI.SFER TO S75 56 5 L A =U L I AA A].=AL1+D)A I RAxiNSFbR T —.) S3 S66 -WHEiN'JEVER H(lvii,.J-1 ).L.0.,TRANSFER TO S67 TR'A-iTFER. T — S75- -. —-- - S67 1)A =-H ( MU ): ^*A / (H (, )-H ( iJ-1) ) TRAN\SFER TO S3,S"-'T5 [ )DA =-H(,.i )A )A / ( H( v,U)-H (,-1)) A1=A1+DA END OCF CC:NDI)TIONAL W f-E, i EV' E- iWe5'-'E -'.-.-3 —. ---- AA2=(-RR ( 3 t JivAX +A lRR ( 2 JAX )-A 1A.A 1*RR ( 1 JIvIAX) 1-3, E'.Y 1 ( 3, JiiAX ) -3* E* A1* Y 1 ( 2 t JlAi.X ) ) / ( 3*t:Y 1( 1. t J MAX) 2-RR(1,JVlAX))

WJHEN'\E\VER.ABS. (A2-AA2).L. EPSI, TR AN'ISFER TO S51 U=U+ 1 — ti7;t J) — AA-2 — i. A(l, I )=A2 P R I N T R E S U L T S H ( t UlJ ), A- ( U y 1. ) I A 2 UlHElF\/ER ). J.L 2, TRAN'SFER T(:) S85 HF R H (9,U)VF G U UU "T R AS hR T U S ~6 H E t! e'V t. Ii- (,.'i!,. -i ). o0 o. (. T r A i"\i S F I PR T 0 S87 WHENEVER H( (i1,U-1 ).U.0., TR/NSFER. TU S87 T RA i', S F E R 9 — 1. S9 - S85 i. = i )A I_ A2=A2+UA TRAt'!SFER TI1 S32 S8-6 -N-NEi.fEVEER -H'J(-'~iY, ). L().., TF.:RANSFER TCi S87 TRAN'SHFER'O'' S95 587 DU =-A ('!' / ( h ( lH I V) -'I, U-1 ) 2 = A 2 + l) A TRANSFER TU S3 S9 _5 DA = — (,T tJ ) )A / ( H, ( i,! ) —H ( M, lJ-1 ) ) A2=A2+ilA 1"RA iNSFER. T S3 E t' l.i D 0' F C i,. i.) Il T" I ( ji'i L S5') PRI,!"'l"' RESI.LTS Al. THR 0UGI14 ASS, FEJI, l.i= 1, 1, f.,..J ASS PRIIt'T RESL.LTS P ( \ ), R (2,i l) RR (2, l ),R R(3,') S51 PRINT RESULTS A2 THR[JUGH ASSS, FUR i'I=llN.G.J'T H Ri:) UJ (>rh A- S S S, F U iR l'l = 1, 1, Ni. G J ASSS PRIi\T RESULTS P( i\),R(3,N),RR (3,N),,RRR( 3, ) TRANSFER TU STAK I ENU OF PROGRAi'i Y3(l 1)=0 5,Y3(2,l)=0.5,Y3(3,1)=0.5,RR( 1,)=0.5,RR(2,1)=0.5,RR(31l)=0.5, -DALTA —T'().2- P=O.9EPS- - E P S I:B= ).01, P S EPSIID=0.001, REFL=0. 1, ELMI TL=O. 9,REL=0.,ET EIT=., EPSIH=O.01(,EPSI I=.001,LJ IAX =40,L=O.00556, l)ALTAA=-100.,LSTEP=0.1, RUW lG=0. 2,HFG=97 0.3, NU=3. 60() KA=(. 035, T S=672,SI (IIA=o. 172E-8, JH-AX-I- I T0 U I NF=3 6000, S TEP=. 00 6:. -UTINF 1 3-O-(-',STEP-=.O-C4O-, UINF=18000,STEP=.004'

3. For Natural-Convection Film Boiling Over a Vertical (or Inclined) Plate IClOIMPV!ILL MAD), EXECtC.I-t, PRIEiT O-RJECT, D.viP, Plli\CH t JEtCT PRINT Cni]imEN'JT $EILii HB(ILII',,1G, VERITICAL $ T- 1A i E \IS I0~..-.Y ( 3), F (3) (-3), - P —' -0)' 1 Y1 ( 600,A ),Y2( 600, W ),Y3(60()(), C),F3(600,!.1), - -7T-6(-T 7,-7' E-,) RT-60-'7-WF1], RR R —(-60b-O-'G -—,-Wf-G-) h,., ), G ( 6,, 7 I ), 3A ( 60, iJ ),H( 6(), iK) Vt LI UR VALUES H4A =2, 1 I t 2 VECT.)R \/ALIJES WrN =2, 1 2.)0 -VECT OR —VA-l[J E S- -v -iC-'.,- 2 —1 —20'VECT'I-[. VALUES W) =2, ], 200 -VE-C-i'O'R —'-'V-UAUJE S- - I-:,iF -. —2 i —,- 200VECT() VALUES.EF =2 1, 200 VEL I UI VALUbJb I.', =Z 1 i,2 0)O VECTOR VALUES l'H-1 =2,1, 20 TECT \TR —~ATiL-TE....., I — = 2-,1, -2-0V\ECTI)R \/ALt ES "'(J =2, 1 20 VECTlOR VALGLUES -' =2 2 0 1 2 ) VECTI)R VALUES RESUJLT =.1H,5E18.6': INTEGER I,J, i Ui,l,I, START HE AI.) A il) R I 1\1NT O)A —\ S1 I=O Sl~ ~ t 1=( WHENEVER.,E.2 A1=B1/ ( 4- E Yl ( 1 tJ AX) -R R( 1, JMAXX ) ) -FNiD F ) T — F-.C-iT-TTi C'i AL WI N I i \1 V E VER 1. E.3 — =0 A 2=( Ai -' i R ( 2, J i AX )-A 1 A 1 k R ( 1, J Ml A X ) 5 E * A 1 Y.1 ( 2, JM A X ) ) / 1 ( Ib' t;e Yl ( 1,JriiA X )-Rk R ( 1, J. IlAX ) ) E l) L F C i.N.!) I TI I.'\AL S3 Jli( EXELCIJTE SEfTRKtIK. 3,Y( 1 ) F,( 1),,X, STEP) X =). Y( 1)=0. Y ( 3 ) =J. Y( 3) =Y3 (i, 1 ) SA \lF J=J+1 P(J)=X Yl(ll,J)=Y ( 1 ) Y2 ( l, J)=Y(2) Y (I,,J. ) =Y ( 3 ) F 3 ( vi, J ) =F ( 3 ) F3(0hJ)=E(3) -I.FFFi-EVER- J'.T.-Ji ii,AX-, T'RAISF-ER T SS S CALL S=RKL)E(-J. (()) WHFEFxf-EV-EFRS. E -— I - F(1)=Y(2) - (2)=Y ( 3)!.HElIJE\/ER l. E. 1 F (3'=- 3Y)IY ( ) + 2Y ( 2 ): Y( 2)-i1 tENL ) IJE CONI\) I T IJOIAL -- JIiTH.iN EVER" W.-E-. 2F( 3)=-3",Y1 ( 1,J ),Y ( 34 +.-"Y2( 1, J ).Y(2 )- 4,Y3( 1,J ) Y( 1) 1+A1 E l3 ( 1, J ) -Al*Yl ( 1, J ) -'Y ( t J ) EN\1i) IF CONI)I.I I I;NAL. - WHTE TEVE -TF- 7TF. E 3 F( 3)=-3", Y1( 1,J )-Y(3)+6",Y22 (, 1,J J ) )*2-5*,J)Y( )+Al F3(2,J) -I-A —I-,.lT — +4- YI -27,J')"-'"Y-,T2,-(J- +3 Y-2- ( d - 2:-Y 2-, ) -A- I -y ( 2, J ) 2 Y Y3( 1, J )-( 3 A1A1- 2 ) F3 (t, J ) + ( A1 ]A1- 2 A2 )' Y ( 1, J ) Y3 ( 1, J ) trENL) LU) C(ON!DIT I UNAL TRAiNSFEtR TI) CALL'-B'! D-' —0TF C- FYI)- TT iI'A' —LTRANSFER Tl SAVE S S S WH —i-'ENV'E'-R —-;'ABS TY( 2T -t'-EP-S-T-C';-TR-IAi TRANiSFER TO S4 5-4 1=1+1 T( i, I ) =Y3 ( l, 1) (,1T i1-T) -=Y 2 ") PRINT RESULTS G(l'i,I), T(ivi,I) -FTFrEV-EVE-R- L'-. —72, — TR-PFNSTFTER —T.. 5WHENiEVER G(M,I). G. (), TRAN\ISFER TO S6 HEitNiEVER G( mi, I-1) G.)., TRAiNSFER TI3 S7 TRANSFER TlO S15

-50 S5 DAL=DALTA Y-3 (, 1 ) =Y3 ( v, 1 )+DAL TR-A-SiSE Rf TO S3..... S6 WHEr'iEVER G(M,lI-1).L.O., TRANSFER TO S7 TRANSFER TO S15 S7 DAL=- ( (, I )'DAL/ ( G( 1l, I )-G ( Fi, I -1 )) Y3( rI' ) =Y3 (' -i, 1 ) +) AL TRANSFER TU S3 S 15'-iAl — G-(-i; i Ti, -A L -- /' CG ( i;,-, ( -) —-I( —l-, —i-I T )Y3 ( I 1 )=Y3 (I, 1 ) +)AL TRAN'SFER T'(')F S3 S9 PRINT RESULTS, il T h (JkO-UGH t AST, F'R- -N —- -, 1 N.G,.J' LAST PRIiT FU'.RRMAT RESULT,P(I\I),Y1(M,VN),Y2(M,lN),Y3(M,N)F3(M,N) E X E X EJ T E S t R K O. —( 2, )-Y,F (i- -, -XST-EPI =0() S23 J = ) X=( Y ( 1 )=. 0 Yi.(l)T 1 R i —S Ut Y( I )=0(.). ENI) UiF CUlJi:) I TI O Y ( 2 ) =R, 1 ) cG E J.j= J + P(J )=X R(i,J)j Y(1) RR(i(M,J)=Y(2) RRR (, J ) =F ( 2) WHEINEVER J.E.JiAX, TRAI\ISFER TO SSSS CONl I S=RK)EO. ( 0 ) WHENEVER S.E.1.0 F (1) — =Y ( 2) IHfl\fiEVER Ivi E.1 F(2) =-PR' 3-Y1 ( 1, J)*Y(2) El'\) OF COINDITIONIAL F(2) =PR*(-3*Y1( l,J)*Y(2)+Y2(1,J)*Y( 1)+2*Al*RRR(1,J)/PR, —A7I.-TT( )RR)T7T — YT1(- 7- J ). * R R- ( 1, ) ) ENiDJ) UF C'OND IT I ONAL WHtENEVER M. it.3 F(2) =PR*(-3*Y1(1,J)*'Y(2)+2*2Y2(1,J)*-Y( 1)+2*A1*RRR(2,J)/PR 1T-T A-IIF-( T1 Y; —JY+-T2' — 2';-JTT —RRT-2TTJ + Y 2 2 ) *R ( 2, J ) -I *A* A -2A2 ) 2*RRR( 1,J)/PR-(5*Y1( 3,J)+A1*Y1(2,J)+2*A2*Y1 (,J)-A1-*AI 37Y^IT- I,J-TYT')R I, I'J 3-)Y END IF COND IT I UONAL IRANS-ER O1 CONlII EIND OF CONDITIONAL TR-Ai -SFEIR —-T -GET ---- SSSS WHENEVER.ABS.Y(1).L.EPSID, TRANSFER TO S29 TR-ATN S-FFR —T —-S-24 S24 I=I+1 GTMFT1)=Y ( ) T(M,I)=RR(M, 1 )'PRTNTRESU1LT S GT - ) —T-TTTTi,T-Y WHENEVER I.L.2, TRANSFER TO S25 W RET\ -EVE-R'-(7 -TT -G0.- T -A1T-S1F PE- T- -~2 t., WHENEVER G(tf, I-1).G.O., TRANSFER TO S27 IRANS-ER TO S35 S25 DAL =DALTA RR-T'=T-,- R RiTi( FT+D-ATRANSFER TO S23 S 2-6 WFTENEiVER' - --,-I" -- )-.- L.-0, —TRkA'-SFE R-TO-S2-T TRANSFER TO S35 S2T DAL=-G(M, I)*DAL/(G M,I )-G(M,I-1) ) RR(, 1 )=RR(M, 1 )+ AL TRA-MlSF ER-TT —"' S2-3 S35 DAL=-G(M,I)*DAL/(G(M,I)-G(M,I-1)) TRANSFER TO S23 S79 WHENEVER M.E.1 E=-RR(1, I JMAX ) / ( 3*Y ( 1, JMAX) ) TW = R-O W-F GT TU)7-(1-KAE1T+ S B1=SIGMA*(TW+TS)'(TW*TW+TS*TS)/((REFL/ EMITL T+i + R EL'W/-E'i- TF- ) *K —-C) THROUGH AAA, FOR N=1,1,N.G.J

-51 3AAA PR I N I RESUL TS P(N),R (, N),RR(M, N ),RRR(M,'N) PRINT RESULTS E,TW,B1 -— TR A N SF tER"TO- T1 — S END OF CONDITIONAL -...WH-Et'q-EVEi~T\ —;.-. 2AAi=(Bl-RR(2,JMiAX)-4*E'Yl(2,JJMAX ))/(4*E*Yl(l,JMAX) 1-RI l, JMAX) ) WHENEVER.AHS.(Al-AAl).L.EPSIA,TRANSFER TO S50 ---— U+-L - I -------- -- - - - - - ----- - - H(, U )=AA1-Al A(MO)-A1............ PRINT RESoLTS H(M,U), A(M,(J),Al WHENEVER l.L.2, TRANSFER TO S65 WHENEVER H(M, I).G.O., TRANSFER TO S66 — WNHEql-EVETr- Ti1TT~-J-IT-.G.S-; RA, —-TR.A|S F E R.- TO S6 7 TRANSFER TO S75 -S 65 -'D-A.- D —=A —A —...A 1=A1+ L)DA TRANSFER TO S3 S66 WHENEVER H(M,lJ-1 ).L.0.,TRANSFER TO S67 S67 DA =-H ( M,0J ),' -DA /(H ( ivi, U )-H(M, U-1) )'-A 1-=-A' I1-'TY-/-.. - -~-A1 r -A-+ 7 ~- ~~-~ -~~-~-~ — -~ —~ ----. —. --- ----- ~- -— ~ — TRAN\SFER TO S3 ST5 A) = -H ( f' ) )A / ( H i, J)-H(M,-1) ) Al=A1+DA -TR-AWxTS E R,-Tfn- S3 - - END (OF C..ONt_.)I T I LiNAL -'WFTEN-VER —-E..3. 3 —-. 1-b _Y 1 ( 3, Jr,AX ) b,'t -Al:'(Y I(2, JAX ) ) / (. 5*E'Y 1 ( 1, JMviAX ) 2-RR( l, JiiAX ) ) -. TW Er'EVERA —-_A-S. ( A 2 -.L —E7PS-. TBST'_TRAINSFFER TO S51 - 0=11+1 A( v, J )=A2 PR I IT RESULTS H (',0), A ( 1i, J ), A2 WHEN\EVER O.L.2, TRANSFER TO S85 WHE'tEV -RE H (V, L).G. (0., TRFANSFR TO3 SShWHENEVER H(M,J- ).G.O., TRANSFER TO S87.TRAIN!SFER TTl S95-........... S85 IA =DALTAA A2=A2+DA TRANSFER TO S3 SS6 W HE E H N ( FI, -1i) r L.0., TRA\ S F ER S87 TRANSFER TO S95 S87 -- ----—'-H FU YO —*'f-A —7 I (Fit- -H (1, - yl ) - A2=A2+)DA TRANSFER TO S3 S95 DA =-H(M, U) *A /(H( iv, ) -H(MlU- )) -A-2'2-=Z+D-A......... —---- --...___ _. TRANSFER TO S3 — EN-l)-OF'-CO;' D T-TTTOI-A- S50 PRINT RESULTS Al THROUGH ASS, FOR N=11,,N.G.J ASS PRINT RESOLTS'P(N),R(2, N),RR(2,N),RRR(3,N) -T R- AWS-F-R~ T - -S 1S51 PRINT'RESULTS A2 -_ TH-RHOUH- A — SS —-E-DR N =-1,1, N.G.J ASSS PRINT RESULTS P(N)R(3,N),RR(3,N),RRR'(3,N) TRANSFER TO START END OF PROGRAM $ D'A T-..... — ---'Y3(1,1)=0.5,Y3(2,1)=0.5,Y3(3,1)=0.5,RR(1,1)=0.5,RR(2,1)=0.5,RR(3,1)=0.5, D-ALTAT=0. —2,- EP-S IAO.OO1, EPSI B=0.O01, EPS IC=0.001, EPS I.D=O.O01, HFG=970.3,.TS=672,SIGMA=0.172E-8,DALTAA=-100., REFL=O0.1,EMITL=0.9,RELW=O.,EMITW=l., STEP=O.01,PR=0.91,ROW.G=0.016-9,NU=4. 170KA=0.0388,C= 144, JMAX-141* STEP=-O. OO-,P-R=0.89,ROWG=O.0145,NU=5.94,KA=0.0508,C=125,JMAX=151* STEP=0.01,PR=0.87,ROWG=0.0120,NU=7.710,KA=0.0610,C=106,JMAX=161* 659 LINES PRINTED 00162 6 02-12- 68 001626 02-12-68 001626 02-12-68 001626 02-12