PROGRAM GRAETZ C... C... The Graetz problem stated by Jakob (1949) is C... C... 2*vavg*v(r)*ua = D*(ua + (1/r)*ua ) (1) C... z rr r C... C... v(r) = 1 - r**(2) C... C... where ua = (T - Ts)/(To - Ts) C... C... For the numerical solution, u = (T - To)/(Ts - To). u can be C... expressed in terms of ua. Thus, C... C... T = Ts + ua*(To - Ts) C... C... u = (Ts + ua*(To - Ts) - To)/(Ts - To) = 1 - ua C... C... The analytical (series) solution given by Jakob, eq. (22-45), C... is C... C... ua = 1.477*exp(-3.658*(1/Pe)*z)*R0(r) C... C... - 0.810*exp(22.178*(1/Pe)*z)*R1(r) C... C... + 0.385*exp(-53.05*(1/Pe)*z)*R2(r) C... C... where z = z/ro, r = r/ro, (the LHS z and r are dimensionaless), C... R0(0) = R1(0) = R2(0) = 1 (from Jakob, Table,22-1, page 455), C... Pe = ro*vavg/D (D = k/(rho*Cp)) C... C... Jakob, M., Heat Transfer, John Wiley and Sons, Inc. New York, C... Vol. 1, 1949 C... C... The output from this program is C... C Series solution = 60.000 30.000 0.7628 C Terms = 0.2372 0.0000 0.0000 C C Series solution = 60.000 90.000 0.9939 C Terms = 0.0061 0.0000 0.0000 C C Series solution = 120.000 30.000 0.4113 C Terms = 0.5919 -0.0032 0.0000 C C Series solution = 120.000 90.000 0.9050 C Terms = 0.0950 0.0000 0.0000 C C Series solution = 180.000 30.000 0.2172 C Terms = 0.8028 -0.0201 0.0001 C C Series solution = 180.000 90.000 0.7628 C Terms = 0.2372 0.0000 0.0000 C... IMPLICIT DOUBLE PRECISION (A-H,O-Z) C... C... OUTPUT FILE OPEN(1,FILE='output') C... C... PECLET NUMBERS DO 3 NPE=1,3 C... C... PE IF(NPE.EQ.1)PE= 60.0D0 IF(NPE.EQ.2)PE=120.0D0 IF(NPE.EQ.3)PE=180.0D0 C... C... TWO LENGTHS DO 2 NLEN=1,2 C... C... Z IF(NLEN.EQ.1)Z=30.D0 IF(NLEN.EQ.2)Z=90.D0 ZPE=Z/PE C... C... UA(0,Z) TERM1= 1.477D0*DEXP( -3.658D0*ZPE) TERM2=-0.810D0*DEXP(-22.178D0*ZPE) TERM3= 0.385D0*DEXP( -53.05D0*ZPE) UA=TERM1+TERM2+TERM3 C... C... FOR COMPARISON WITH THE NUMERICAL SOLUTION, U = 1 - UA U=1.0D0-UA C... C... WRITE SOLUTION WRITE(1,1)PE,Z,U,TERM1,TERM2,TERM3 1 FORMAT(' Series solution = ',2F10.3,F10.4,/, + 10X,' Terms = ',3F10.4,/) C... C... NEXT LENGTH 2 CONTINUE C... C... NEXT PE 3 CONTINUE C... C... END OF CALCULATION STOP END