Patent Number: 059600500
Section: description

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Referring to FIG. 1, there is shown a schematic illustration of a test holder, generally denoted 10, used in accordance with the method of the present invention to determine fission heat flux for a Uranium 235 fuel-bearing specimen used in a nuclear reactor. The test holder 10 is positioned in a conventional pressurized test loop (not shown) of an ATR reactor to take experimental data used in deriving the absolute value of the thermal fission flux of a prime specimen. Test holder 10 is an SW101-type F 2.times.3 holder and is shown in cross-section as having a series of components positioned in a row. Specifically, the following components are positioned in a row proceeding from left to right: a first bulk water channel (WC) 12; a prime specimen (PRIME) 14; a second WC 12; a first thermocouple test specimen (TC) 16; a third WC 12; a first Zirconium specimen (Zr) 18; a fourth WC 12; a second TC 16; a fifth WC 12; a second Zr specimen 18; and a sixth WC 12. As discussed above, the TC's 16 must be positioned at the same level and this is accomplished by using the test holder 10. FIG. 2 illustrates the composition of each TC 16 and shows in cross section that TC 16 comprises a central backclad (BC) 22. Two outer clads (CLAD) 26 and two fuel fillers (FF) 24 are disposed between the backclad 22 and the two outer clads 26, respectively. Two WC's 12 are shown in dashed lines to help illustrate the positioning of WC's 12 in test holder 10. As mentioned above and as discussed in more detail below, various temperature data is required to ultimately determine the fission flux; FIG. 2 illustrates that the temperature (TM) is measured at the interface between backclad 22 and fuel filler 24. The maximum temperature (T.sub.MAX) of backclad 22 is also measured, as is temperature of the bulk WC (T.sub.W). The temperature (T.sub.C) at the interface between fuel filler 24 and clad 26, and the temperature (T.sub.S) between clad 26 and WC 12 is to be determined later. FIG. 3 illustrates the composition of prime specimen 14 and shows in cross-section that prime specimen 14 comprises a fuel filler 30 disposed adjacent, and between, a pair of clad 32. The first and second WC's 12 are shown in dashed lines to help illustrate the positioning of prime specimen 14 in test holder 10. Since there is no backclad in prime specimen 14, the temperature of the prime specimen 14 is assumed to be the same as the maximum temperature for this type of specimen (i.e., T.sub.M =T.sub.MAX) and the temperature (T.sub.C) is that determined at the interface between clad 32 and fuel filler 30. FIG. 4 illustrates a second embodiment of test holder 10 which is an SE93-type B 2.times.4 holder. The configuration is similar to that illustrated in FIG. 1, except after the fourth WC (proceeding left to right), there is disposed the second Zr specimen 18, the fifth WC 12, the second TC 16, the sixth WC 12, a third Zr specimen 18, and lastly a seventh WC 12. The method of the present invention will be discussed below as part of a mathematical proof which demonstrates that the method does indeed yield a reliable value for the fission heat flux of a prime specimen, e.g., corresponding to specimen 14 based on readily obtainable experimental data. As mentioned above, the method of the present invention requires that two TC's 16 be located at the same elevation. In addition, it is also necessary to measure the temperatures of the WC's 12, and the temperatures of the TC's 16 and to measure the gamma-scan count ratio of the TC's 16 and the prime specimen 14. The transfer heat coefficient (H.sub.COEF) for the water channel flow is also required, and can be calculated from the measured flow rate of the water channels 12. The thickness of the backclad 22, fuel filler 24, and clad 26 of the TC's 16 must also be determined. Further, the ATR reactor lobe power (POWER) and axial factor for gamma heat (AXF) is used to separate the fission heat from the total heat so that the absolute value of the fission heat fluxes of the TC's 16 can then be determined. Once the absolute value of the fission heat fluxes of the TC's 16 is determined, the absolute value of the fission heat flux of the prime specimen can then be easily determined as shown below. To facilitate the explanation of the method of the invention, Zirconium (Zr) is assumed for the clad and backclad material. The thermal conductivity for the Zr regions can be expressed as a linear function of the average temperature. In addition to the heat generated from fission, there is also gamma heat throughout the specimen. Set forth below is a somewhat detailed derivation for the linearly varying conductivity versus temperature case. For clarification of the derivation, it may be helpful to note the following term abbreviations. Term Abbreviations P=Prime Specimen PA0 TC=Thermocouple Test Specimen PA0 WC=Water Channel PA0 FF=Fuel filler PA0 BC=Backclad PA0 H=Transfer heat coefficient for WC flow PA0 a=6, b=5e-3; ab=a/b=1200 (material constants for Zr) PA0 T.sub.w =Bulk WC temperature PA0 T.sub.s =Temperature at interface between the outer clad and WC in a TC or the temperature between the clad and WC in a prime specimen PA0 T.sub.L =Linear average temperature over fuel PA0 T.sub.M =Temperature at interface between the BC and FF in a TC, or maximum temperature of a prime specimen PA0 k.sub.w =Pseudo WC thermal conductivity PA0 k.sub.s =Pseudo WC film thermal conductivity PA0 k.sub.c =Clad thermal conductivity PA0 k.sub.m =Pseudo thermal conductivity based on T.sub.M PA0 Q.sub.T =Total heat generation for BC, FF, Clad PA0 Q.sub.G =Gamma heat generation PA0 Q.sub.F Fission heat generation PA0 S.sub.C =Clad heat source PA0 S.sub.F =Fuel heat source PA0 t.sub.s =Total specimen thickness PA0 t.sub.c =Thickness of outer clad of TC PA0 t.sub.f =Thickness of fuel filler of TC PA0 t.sub.bc =Thickness of backclad of TC PA0 a.sub.c =t.sub.c /t.sub.s =Clad gamma heat multiplication factor PA0 a.sub.f =(2t.sub.c -t.sub.b)/t.sub.s =Fuel filler gamma heat multiplication factor PA0 a.sub.bc =t.sub.bc /t.sub.s =Backclad gamma heat multiplication factor PA0 A.sub.p =Intercept in conductivity equation for prime specimen PA0 A.sub.i =Intercept in conductivity equation for TC's 1, 2 PA0 B=Slope in conductivity equation for specimens PA0 1,2,P=Subscripts for TC.sub.1, TC.sub.2, prime specimen PA0 AXF=Axial Factor for Gamma Heat PA0 Power=Reactor power PA0 T.sub.MAX =Maximum temperature of backclad in a TC or T.sub.M for prime specimen PA0 T.sub.c =Temperature at interface between FF and outer clad in a TC or temperature at interface between FF and clad in a prime specimen PA0 k=Average thermal conductivity PA0 S.sub.BC =Backclad heat source PA0 T.sub.BC =Backclad temperature PA0 R.sub.TC =Measured scan count ratios of the thermocouple specimens PA0 R.sub.P =Measured scan count ratio of the prime specimen The following statement definitions are also helpful in understanding the derivation which follows: ##EQU1## where: S=heat source term which is a function of fission and/or gamma heat plus the specimen material thicknesses. k=Average thermal conductivity EQU k=a+bT EQU T=(T.sub.2 +T.sub.1)/2=AVERAGE TEMPERATURE BETWEEN SURFACES 2 AND 1 a, b=average material constants where the material may contain Zr, U235, etc. As a result of an adiabatic heat surface in the backclad, the backclad heat source, denoted S.sub.BC, depends only on the material gamma heat and the backclad thickness (t.sub.bc). Since the thermocouple well is contained in the backclad (e.g., corresponding to backclad 22 of FIG. 2), it follows that T.sub.BC =(T.sub.MAX +T.sub.M)/2, and, therefore, the temperature drop across the backclad can be considered constant throughout the derivation for a particular TC. Using the fact that the average thermal conductivity is assumed to vary similarly with temperature for both TC's (1 and 2), then EQU (dk/dT.sub.L).sub.2 =(dk/dT.sub.L).sub.1 =(dk/dT.sub.L).sub.P wherein, in this last equation, the P refers to a prime specimen. As mentioned above, test holders such as the types F (2.times.3) and B (2.times.4), shown in FIGS. 1 and 4, generally contain two fueled thermocouple specimens (TC's 16) and one or more prime specimens (specimens 14). The derivation or proof set forth below is for simplest case whereby the conductivity in the fuel region is assumed to vary linearly with temperature. Conductivities for the clad and backclad are also assumed to vary linearly with temperature which is a very good assumption for Zr material. Moreover, the heat transfer is assumed as being one-dimensional. The composition of the prime specimen 14 and the TC's 16 was discussed previously. For purposes of the derivation which follows, it is assumed that all specimens have the same clad and backclad material and that the backclad 22 and clad 26 are Zr material with a linear conductivity temperature dependence. A further assumption is that dk/dT is the same for all fuel fillers. In the derivation or proof which follows, equations for the clad and fuel regions are set up in turn. It should be noted that a thermal adiabatic surface occurs in the backclad region 22. Consequently, since the conductivity temperature dependence is known, the temperature drop to the fuel surface fuel filler 24 depends only on the backclad thickness (t.sub.bc), gamma heat rate (Q.sub.G), and the TC temperature reading in the backclad. Some of the variables in the equations are symbolic to facilitate the derivation. Clad Region Let ##EQU2## It follows that EQU k.sub.c.sup.2 -k.sub.s.sup.2 =2bS.sub.c Eq (1) By differentiation of Eq (1) and using the equation for S.sub.c, then ##EQU3## Fuel Region Let ##EQU4## By differentiation of the last equation and using Eq (2) ##EQU5## where T.sub.M is fixed and Eq (2) has been used. It follows that ##EQU6## Let k.sub.w =a+bT.sub.w, then ##EQU7## From rules of differentiation and noting that T.sub.L =(T.sub.C +T.sub.M)/2 ##EQU8## By definition ##EQU9## Canceling constants it follows that ##EQU10## Next, we need to determine the variables k.sub.s, k.sub.c. From Eq (1) and Eq (3) ##EQU11## Solving the quadratic in k.sub.s yields ##EQU12## Let k.sub.s +t.sub.c H/12=X, k.sub.c =a+bT.sub.c and k.sub.m =a+bT.sub.M. Note that the only unknown variable on the right side of Eq (7) is k.sub.c. Eq (6) may be written in terms of X, k.sub.m and k.sub.c as follows ##EQU13## The total heat generation is Q.sub.T =Q.sub.F +Q.sub.G. Since ##EQU14## Then it follows that ##EQU15## Here R.sub.TC is the measured gamma scan ratio of the TCs. Equations (7), (8), (9) are independent. The T.sub.M as defined here is not the measured T.sub.MAX since there is a slight temperature rise to the middle of the backclad. The temperature T.sub.M as a function of T.sub.MAX is determined as follows. ##EQU16## and therefore ##EQU17## Solving the resulting quadratic equation yields for T.sub.M ##EQU18## Since T.sub.MAX is measured, and all other quantities are known, the T.sub.M for the TC is determined as is k.sub.m =a+bT.sub.M. Noting that k.sub.c from Eq (7) may be written in terms of X, then ##EQU19## for either TC. Therefore, it follows Eq (8) that ##EQU20## After applying Eq (11) to Eq (12), and using Eq (9) and (13) then the X value for either TC is determined. From the value of X and Eq (7), k.sub.s, k.sub.c and hence T.sub.S, T.sub.C are determined. Since, by definition EQU Q.sub.T.sbsb.1 =H.sub.1 (T.sub.S.sbsb.1 -T.sub.W.sbsb.1); Q.sub.F.sbsb.1 =Q.sub.T.sbsb.1 -Q.sub.G.sbsb.1 Eq (14) and similarly for TC.sub.2. The Q.sub.F values are therefore determined. From the TC value (T.sub.MAX), the delta T across the filler, T.sub.M -T.sub.C, is known. Since ##EQU21## the value of k is hence determined for either TC. The additional required equation, namely, Q.sub.G, may be written simply as EQU Q.sub.G =25773.times.POWER.times.0.3.times.AXF.times.t.sub.SEq (16) The above set of equations have been derived for the TC's. The following gives equations for the prime specimens. The gamma scan ratio of the prime specimen, R.sub.P is a measured quantity, i.e. ##EQU22## Since R.sub.p, Q.sub.F.sbsb.1 are known, the Q.sub.F.sbsb.P is determined. From the specimen dimensions, Q.sub.G.sbsb.P is found and hence Q.sub.T.sbsb.P. By analogy from the preceding equations for the CLAD region and FUEL region, the T.sub.C.sbsb.P, T.sub.S.sbsb.P are readily found. Since there is no backclad for the prime specimen, T.sub.M.sbsb.P is the maximum value. Therefore, for a linear temperature dependence of the thermal conductivity in the fuel filler, and noting that ##EQU23## we have found from Eq (8) ##EQU24## Since Y.sub.1 for TC.sub.1 is known, and X.sub.p, k.sub.c.sbsb.P is determined from T.sub.C.sbsb.P, T.sub.S.sbsb.P, Q.sub.T.sbsb.P then the only remaining variable in Eq (18) is k.sub.m.sbsb.P. Solving for k.sub.m.sbsb.P which is the pseudo filler thermal conductivity for the prime specimen, and then using the following ##EQU25## and k.sub.p, B, T.sub.M.sbsb.P, T.sub.C.sbsb.P are known, then the intercept A.sub.p for the prime specimen can be determined. Substituting the appropriate variables in Eq (21), the A.sub.i values for the TC's are similarly determined. A computer program was written in MS pro-basic to run on an IBM type 486 computer. The program listing plus the input for the ATR SW101 and SE93 test trains are attached. A comparison of results with the measured results is given in Table 1. Following Table 1 is a listing of the prompts for input and the values entered for both of the disclosed embodiments. Lastly, there is a listing of the computer program itself. TABLE 1 __________________________________________________________________________ COMPARISON OF INVENTION WITH THE MEASURED RESULTS Specimen T.sub.W T.sub.S T.sub.C T.sub.W T.sub.WAX Q.sub.F .times. 10.sup.-6 Percent ID M I M I M I M I M I M I Diff __________________________________________________________________________ SW101 Prime 224 224 467 472 828 840 1167 1142 1167 1142 2.01 2.05 +2.0 TC.sub.2 221 221 447 454 781 794 -- 1073 1084 1084 1.77 1.81 +2.3 TC.sub.1 215 215 364 369 599 607 -- 840 854 853 1.15 1.17 +1.7 SE93 Prime 492 492 558 559 764 767 -- 935 936 935 .836 .846 +1.2 TC.sub.2 492 492 558 559 714 718 -- 901 909 910 .784 .794 +1.3 TC.sub.1 489 489 543 543 672 674 -- 837 845 846 .633 .641 +1.3 __________________________________________________________________________ Invention I Measured Results M __________________________________________________________________________ PROMPTS FOR INPUT AND ENTERED VALUES INPUT TRAIN ID, SPECIMEN ID'S PRIME TC, ALT TC SW101, XX, YY INPUT POWER, GAMMA AXF, FISS RAT PRIME TO ALT TC, AND SIDE TO ALT TC 41, .939, 1.545, 1.751 INPUT BW, TMAX, HCOEF, THICC, THICF, THICBC, SWEL FOR PRIME TC 221, 1084, 8141, .0198, .064, .1453, 0 INPUT BW, TMAX, HCOEF, THICC, THICF, THICBC, SWEL FOR ALT TC 215, 853, 8141, .0195, .0637, .1458, .0048 INPUT SPEC ID, BW, HCOEF, THICC, THICF, THICBC, SWEL FOR SIDE SPECIMEN ZZ, 224, 8387, .0197, .0685, 0, .0334 INPUT TRAIN ID, SPECIMEN ID'S PRIME TC, ALT TC SE93, XX, YY INPUT POWER, GAMMA AXF, FISS RAT PRIME TO ALT TC, AND SIDE TO ALT TC 31.8, .951, 1.2385, 1.3207 INPUT BW, TMAX, HCOEF, THICC, THICF, THICBC, SWEL FOR PRIME TC 492, 910, 12820, .0206, .0682, .1404, .02284 INPUT BW, TMAX, HCOEF, THICC, THICF, THICBC, SWEL FOR ALT TC 489, 846, 12900, .0204, .0682, .1403, .02728 INPUT SPEC ID, BW, HCOEF, THICC, THICF, THICBC, SWEL FOR SIDE SPECIMEN ZZ, 492, 13100, .0268, .0673, 0, .04053 PROGRAM LISTING 9 OPEN "LPT1" FOR OUTPUT AS #1 10 PRINT "INPUT TRAIN ID, SPECIMEN ID'S PRIME TC, ALT TC" 20 INPUT Idt$, Id2$, Id1$ 30 PRINT "INPUT POWER, GAMMA AXF, FISS RAT PRIME TO ALT TC, AND SIDE TO ALT TC" 40 INPUT P, Axf, R, Rs 50 IF Axf &gt; 1 THEN Axf = COS(.05775 * Axf) 60 PRINT "INPUT BW, TMAX, HCOEF, THICC, THICF, THICBC, SWEL FOR PRIME TC" 70 INPUT Tw2, Tmax2, H2, Tc2, Tf2, Tbc2, Sw2 80 INPUT "INPUT BW, TMAX, HCOEF, THICC, THICF, THICBC, SWEL FOR ALT TC" 90 INPUT Tw1, Tmax1, H1, Tc1, Tf1, Tbc1, Sw1: REM Sw1 = 0.273: SW2 = .0228 100 A = 6: B = .005; Ab = A/B: Tf2s = Tf2*(1 + Sw2): Tf1s = TF1*(1 + Sw1) 110 Ts2 = 2*Tc2 + Tf2 + Tbc2: Qg2 = 25773*P*.3*Axf*Ts2: Kw2 = A + B*Tw2: Kch2 = Tc2*H2/12 120 Ts1 = 2*Tc1 + Tf1 + Tbc1: Qg1 = 25773*P.3*Axf*Ts1: Kw1 = A + B*Tw1: Kch1 = Tc1*H1/12 130 Kfh2 = Tf2*H2/48: Kfh1 = Tf1*H1/48: Kgh2 = B*Qg2/H2 Kgh1 = B*Qg1/H1: Kwb1 = Kw1 + Kgh1 140 Ac2 = Tc2/Ts2: Af2 = (2*Tc2 - Tbc2)/Ts2: Kwb2 = Kw2 + Kgh2 150 Ac1 = Tc1/Ts1: Af1 = (2*Tc1 - Tbc1)/Ts1 160 Kwc2 = Kw2 + Ac2*Kgh2: Kwc1 = Kw1 + Ac1*Kgh1: Kwf2 = Kw2 + Af2*Kgh2: Kwf1 = Kw1 + Af1*Kgh1 170 Al1 = Kfh1*KWf1: G1 = Kch1 + Kwf1: G12 = Kch2 + Kwf2: G3 = Kch1 2 + 2*Kch1*Kwc1: G31 = G3 180 G32 = Kch2 2 + 2*Kch2*Kwc2: : P1 = Kwb1 + Kch1: P2 = Kwb2 + Kch2: Pw1 = Kwc1/B: Pw2 = Kwc2/B 190 Ph1 = (Kwc1 + .5*Kch1)/B: Ph2 = (Kwc2 + .5*Kch2)/B 200 IF Rs = 0 THEN 260 210 PRINT "INPUT SPEC ID, BW, HCOEF, THICC, THICF, THICBC, SWEL FOR SIDE SPECIMEN" 220 INPUT Ids$, Tws, Hs, Tcs, Tfs, Tbcs, Sws: REM Sws = .0405: Tfss = Tfs*(1 + Sws) 230 Tst = 2*Tcs + Tfs + Tbcs: Qgs = 25773*P*.3*Axf*Tst: Acs = Tcs/Tst: Afs = (2*Tcs - Tbcs)/Tst 240 Kchs = Tcs*Hs/12: Kfhs = Tfs*Hs/48: Kghs = B*Qgs/Hs: Kws = A + B*Tws: Kwcs = Kws + Acs*Kghs 250 Kwbs = Kws + Kghs 260 PRINT #1, : PRINT #1, "*******THE TRAIN ID IS"; Idt$; "*******" 270 PRINT #1, : PRINT #1, "SPECIMEN ID FOR PRIME ALT TC'S ARE": Id2$; "AND"; Id1$ 280 IF Rs &lt;&gt; 0 THEN PRINT #1, "SPECIMEN ID FOR SIDE SPECIMEN IS"; Ids$ 290 REM*******DETERMINED BCLAD DT AND FUEL/BCLAD INTERFACE TEMPERATURE******** 300 Tm2 = (-A + SQR((A + B*Tmax2) 2-B*Qg2*Tbc2 2/(24*T12)))/B: Dtb2 = Tmax2 - Tm2 310 Tm1 = (-A + SQR((A + B*Tmax1) 2-B*Qg1*Tbc1 2/(24*Ts1)))/B: Dtb1 = Tmaxl - Tm1 320 PRINT #1, : PRINT #1, "AXF ="; Axf; "tm2 ="; Tm2: "dtb2 ="; Dtb2; "tm1 ="; Tm1; "dtb1 ="; Dtb1 330 Kc1 = A + .002*Tm1: Dk = .001: Km2 = A + B*Tm2: Km1 = A + B*Tm1 340 C3 = Kwf2 2 + Kch2*Kgh2*(Af2 - Ac2) - Km2 2: C4 = Kwf1 2 + Kch1*Kgh1*(Af1 - Ac1) - Km1 2 350 FOR M = 1 TO 1000 360 Tcn1 = Kc1/B - Ab: Ks1 = -Kch1 + SQR(Kch1 2 + 2*Kch1*(Kw1 + Ac1*Kgh1) + Kc1 2) 370 Tsxl = Ks1/B - Ab: Ks2 = Kw2 + (Ks1 - Kw1 - Kgh1)*R*H1/H2 + Kgh2: Tsx2 = Ks2/B - Ab 380 Qt1 = H1*(Tsx1 - Tw1): Qf1 = Qt1 - Qg1: Qt2 = H2*(Tsx2 - Tw2): Qf2 = Qt2 - Qg2 390 Qct1 = Qt1 - AC1*Qg1: Qct2 = Qt2 - Ac2*Qg2: Qft1 = Qt1 - Af1*Qg1: Qft2 = Qf2 - Af2*Qg2 400 Dtl = Tm1 - Tcn1: S2 = Tf2*Qft2/48: S1 = Tf1*Qft1/48: K1 = S1/Dt1 410 Kc2 + SQR(Ks2 2 + 2*Kch2*(Ks2 - Kwc2)): Tcn2 = Kc2/B - Ab: Dt2 = Tm2 - Tcn2: K2 = S2/Dt2 420 Y2 = B*Kfh2*((Ks2 - Kwf2)/(Km2 - Kc2) 2 + Kc2/((Km2 - Kc2)*(Ks2 + Kch2))) 430 Y1 = B*Kfh1*((Ks1 - Kwf1(/(Km1 - Kc1) 2 + Kc1/((Km1 - Kc1)*(Ks1 + Kch1))):Ra = Y2/Y1 440 IF ABS(Ra - 1) &lt; .001 THEN 480 450 IF M = 1 THEN 470 460 S1 = (1 - Ra)/(Ra - Rao): Kc1 = Kc1 + .5*S1*Dk: Me = M + S1: REM PRINT M; Me; Ra; Kc1; K2; K1 470 Kc1 = Kc1 + Dk: Y1o = Y1: Y2o = Y2: Rao = Y2o/Y1o: NEXT M 480 IF Rs = 0 THEN 560 490 Kss = Kws + (Ks1 - Kw1 - Kgh1)*Rs*H1/Hs + Hghs: Tsxs = Kss/B - Ab 500 Kcs = SQR(Kss 2 + 2*Kchs*(Kss - Kws - Acs*Kghs)) Tcns = Kcs/B - Ab 510 Ps = B*Kfhs*Kcs/(Kss + Kchs) 520 Kms = Kcs + .5*(Ps + SQR(Ps 2 + 4*Y1*B*Kfhs*(Kss - Kws - Afs*Kghs)))/Y1 530 Tms = Kms/B - Ab: Qfs = Rs*Qf1: Qts = Qfs + Qgs: Qfts = Qts - Afs*Qgs 540 Tmaxs = (-A + SQR((A + B*Tms) 2 + B *Qgs*Tbcs 2/(24*Tst)))/B: Dtbs = Tmaxs - Tms 550 Ks = Tfs*Qfts/(48*(Tms - Tcns)): PRINT #1, "TINTS ="; Tcns; "TMAXS ="; Tmaxs 560 PRINT #1, ".sub.-------------------- " 570 PRINT #1, "THE FOLLOWING VALUES WERE FOUND AFTER" M; "ITERATIONS" 580 PRINT #1, "me, RAT"; Me; Ra; "K2 ,K1, Ks"; K2; K1; Ks; "QF2, QF1, QFS"; Qf2, Qf1; Qfs 590 PRINT #1, "TSX1, TSX2, TSXS, TCN1, TCN2, TCNS"; Tsx1; Tsx2; Tsxs; Tcn1; Tcn2; Tcns 600 T1 = .5*(Tm1 + Tcn1): T2 = .5*(Tm2 + Tcn2): Q = Tf2*Qft2*(Tm1 - T1)/(Tf1*Qft1*(Tm2 - T2)) 610 S1 = (K2 - K1)/(T2 - T1): PRINT "SLOPE ="; S1; "INT ="; (T2 - Q*T1)*S1/(Q - 1) 620 Rh = H1*R/H2: Bb1 = Kch1*(Kch1 + 2*Kwc1): Ff = Kch2 + Kwb2 - Rh*(Kch1 + Kwb1) 630 Bb2 = Kch2*(Kch2 + 2*Kwc2): Kc1 = A + B*Tcn1: U1 = (Kc1 2 + Bb1)*Rh 2: Kc2o = Kc2 640 U2 = (Ff + SQR(U1)) 2: Kc2 = SQR(U2 - Bb2): Tcn2n = Kc2/B - Ab: K2ra = K2/K1 650 Cc = Tf2*Qft2/(Tf1*K2ra*Qft1): Sx = (Cc/Rh) 2 - 1: Ax = Sx 2: Kmd2 = (Km2 - Cc*Km1) 2 660 Hh = Ff 2 - Kmd2 + Bb1*Cc 2 - Bb2: Cx = Hh 2 + 4*Bb1*Kmd2*Cc 2 670 Bx = -4*Kmd2*(Cc/Rh) 2 - 2*Hh*Sx = 4*Ff 2: G4 = Ax 2: G3 = 2*Ax*Bx - 16*(Sx*Ff) 2 680 G2 = Bx 2 + 2*Cx*Ax + 32*Hh*Sx*Ff 2: G1 = 2*Bx*Cx - 16*(Hh*Ff) 2: G0 = Cx 2 681 Gx = SQR(SQR(G0)): Y1 = U1/Gx: G3 = G3/Gx: G2 = G2/Gx 2: G1 = G1/Gx 3: GO = 1: Y1 = .9086 690 Eq = G4*Y1 4 + G3*Y1 3 + G2*Y1 2 + G1*Y1 + GO: PRINT Eq; G4; G3; G2; G1; GO 700 PRINT #1, Kc2o; Kc2; Tcn2; Tcn2n; Tcn2n - Tcn2 701 PRINT #1, Kc1; SQR(U1/Rh 2 - Bb1); Tcn1; SQR(U1/Rh 2 - Bb1)/B - Ab 702 PRINT #1, : PRINT #1, "THE ABOVE FLUX ESTIMATES ARE THE LEAST REQUIRED CALCULATION" 703 PRINT #1, : "********************************************" 704 PRINT 710 REM*******FIND INITIAL INTERCEPT AND SLOPE FOR CONDUCTIVITY EQUATION***** 720 REM NOTE K2 VERSUS K1 IS GENERAL; T2 - T1 DEPENDS ON LINEAR AVERAGE 730 REM PRINT "INPUT INITIAL KBAR1 ESTIMATE": INPUT K1 740 L = L + 1: PRINT #1, : PRINT #1, "-------FOR L ="; L; "-------" 750 F1 = .5*Kch1 + Kfh1*(Km1 + A11/K1)/K1 760 Term = (1 - (Kfh1/K1) 2)*(Kch1*Kwc1 + (Km1 + A11/K1) 2) 770 G1 = (-F1 + SQR(F1 2 + Term))/(1 - (Kfh1/K1) 2) - Kwb1 780 Krgh = R*G1*H1/H2: S = Kwb2 + Krgh 790 D1 = SQR((S + .5*Kch2) 2 - .5*Kch2*(.5*Kch2 + 2*Kwc2)) 800 K2t = Kfh2*(Krgh + Kwb2 - Kwf2)/(Km2 - D1) 810 B3 = (Kwf2 + .5*Kch2 + Kfh2*Km2/K2)*2*B*K2/Kfh2 820 B4 = (Kwf1 + .5*Kch1 + Kfh1*Km1/K1)*2*B*K1/Kfh1 830 A3 = 4*B 2*((K2/Kfh2) 2 - 1): A4 = 4*B 2*((K1/Kfh1) 2 - 1) 840 IF L = 1 THEN 860 850 Sloo = Slo: T2o = T2: T1o = T1: Tcno2 = Tcn2: Tcno1 = Tcn1 860 T2 = Tm2 + B3/A3 + SQR((B3/A3) 2 - C3/A3): T1 = Tm1 + B4/A4 + SQR((B4/A4) 2 - C4/A4) 870 Slo = (K2 - K1)/(T2 - T1) 880 Q = Tf2*Qft2*(Tm1 - T1)/(Tf1*Qft1*(Tm2 - T2)): Nt = (T2 - Q*T1)*Slo/(Q - 1) 890 PRINT #1, : PRINT #1, "*******SLOPE ="; Slo; "INT ="; Nt; "******": PRINT #1, 900 PRINT #1, : Qt2 = Qft2 + Af2*Qg2: Qtl = Qft1 + Af1*Qg1: Qf2 = Qt2 - Qg2: Qf1 = Qt1 - Qg1 910 S2 = Tf2*Qft2/48: S1 = Tf1*Qft1/48 920 PRINT #1, "K2 ="; K2; "K1 ="; K1; ".backslash.QF2"; Qf2; "QF1 ="; Qf1 930 PRINT #1, "QT2 ="; Qt2; "Qt1 ="; Qt1; ".backslash.QFT2 ="; Qft2; "QFT1 ="; Qft1 940 PRINT #1, "R ="; R; "R EST ="; Qf2/Qf1; "R RATIO ="; R*Qf1/Qf2 950 PRINT #1, "S2 ="; S2; "S1 ="; S1; ".backslash.TBR2 ="; T2; "TBR1 ="; T1 960 Cc = 1: Qfol = Qf1: FOR M = 1 TO 200 970 IF M &gt; = 2 THEN Cc = .9999*Cc 980 Qf1 = Cc*Qfo1: Qf2 = R*Qf1 990 Qt1 = Qf1 + Qg1: Qt2 = Qf2 + Qg2: Qct1 = Qt1 - Ac1*Qg1: Qct2 = Qt2 - Ac2*Qg2 1000 Tsx1 = Tw1 + Qt1/H1: Tcx1 = -Ab + SQR(Ab 2 + Tsx1 2 + 2*Ab*Tsx1 + 2*Tc1*Qct1/(12*B)) 1010 Tsx2 = Tw2 + Qt2/H2: Tcx2 = -Ab + SQR(Ab 2 + Tsx2 2 + 2*Ab*Tsx2 + 2*Tc2*Qct2/(12*B)) 1020 Ps2 = Qf1 *R/H2 + (Kwb2 + .5 *Kch2)/B: Tcn2 = -Ab + SQR(Ps2 2 - Ph2 2 + Pw2 2) 1030 Ps1 = Qf1/H1 + (Kwb1 + .5*Kch1)/B: Tcn1 = -Ab + SQR(Ps1 2 - Ph1 2 + Pw1 2) 1040 S2 = Tf2*(R*Qf1 + (1 - Af2)*Qg2)/48: S1 = Tf1*(Qf1 + (1 - Af1)*Qg1)/48 1050 Nu = (Tcn2 2 - Tm2 2 + 2*S2/B)*(Tm1 - Tcn1): Den = (Tcn1 2 - Tm1 2 + 2*S1/B)*(Tm2 - Tcn2) 1060 Ratio = Nu/Den: K2 = Tf2*(Qf2 + (1 - Af2)*Qg2)/(48*(Tm2 - Tcx2)) 1070 K1 = Tf1*(Qf1 + (1 - Af1)*Qg1)/(48*(Tm1 - Tcx1)) 1080 K2 = Tf2*(Qf2 + (1 - Af2)*Qg2)/(48*(Tm2 - Tcx2)) 1090 IF M &gt; = 2 THEN Cc = Cc + .5*(1 - Ratio)*(Cc - Cco)/(Ratio - Rato) 1100 Rato = Ratio: Cco = Cc 1110 IF ABS(Ratio - 1) &lt; = .0001 THEN 1130 1120 NEXT M 1130 PRINT #1, "--------------------------------------------" 1140 PRINT #1, "THE FOLLOWING VALUES WERE FOUND FOR L ="; L, "AFTER"; M; "ITERATIONS" 1150 PRINT #1, "C, Rat"; Cc; Ratio; "K2, K1"; K2; K1; "QF2, QF1"; Qf2; Qf1; "K2R"; K2t/K2 1160 Qft1 = Qf1 + (1 - Af1)*Qg1: Qft2 = Qf2 + (1 - Af2)*Qg2: Tcxo2 = Tcx2: Tcxo1 = Tcx1 1170 IF L = 1 THEN 1500 1180 Qf2 = Qt1 + Qg2: Qt1 = Qf1 + Qg1 1190 C2 = B*Tf2*H2/(K2t*48): D2 = Km2 + Af2*C2*Qg2/H2 1200 Term2 = C*D2 + B*Kw2 + 4*Tc2*C2*K2t/Tf2 1210 Term22 = D2 2 - Kw2 2 + 8*Tc2*K2t*C2*Ac2*Qg2/H2 1220 Qf2 = H2*(Term2 - SQR(Term2 2 - (C2 2 - B 2)*Term22))/(C2 2 - B 2) - Qg2:Qf1 = Qf2/R 1230 Qt1 = Qf1 + Qg1: Tcx1 = SQR((Kw1 + B*Qt1/H1) 2 + 2*B*Tc1*(Qt1 - Ac1*Qg1)/12)/B - Ab 1240 Qt2 = Qf2 + Qg2: Tcx2 = Tm2 - Tf2*(Qt2 - Af2*Qg2)/(48*K2t) 1250 K1t = Tf1*(Qt1 - Af1*Qg1)/(48*(Tm1 - Tcx1)): K1 = K1t 1260 K2 = Tf2*(Qt2 - A12*Qg2)/(48*(Tm2 - Tcx2)) 1270 Qft2 = Qt2 - Af2*Qg2: Qft1 = Qt1 - Af1*Qg1: Qct2 = Qt2 - Ac2*Qg2: Qct1 = Qt1 - Ac1*Qg1 1280 Dd = S2*K1/(S1*K2):Tcx2c = Tm2 - Tm1*Dd + Dd*Tcx1 1290 Gf = (Tm2*Tcx2 - Tm1*Tcx1)/(T2 - T1): Gfo = (Tm2*Tcxo2 - Tm1*Tcxo1)/(T2o - T1o) 1300 Ks1 = A + B*(Tw1 + Qt1/H1): Dt1 = Tm1 - Tcx1: Y1 = Tf1*H1*(Ks1 - Kw1 - B*Af1*Qg1/H1)/Dt1 2 1310 Kcx1 = A + B*Tcx1: Y1 = Y1 + Tf1*B*Kcx1/(Dt1*(Ks1/H1 + Tc1/12)): Kcx2 = A + B*Tcx2 1320 Ks2 = A + B*(Tw2 + Qt2/H2): Dt2 = Tm2 - Tcx2: Y2 = Tf2*H2*(Ks2 - Kw2 - B*Af2*Qg2/H2)/Dt2 2 1330 Y2 = Y2 + Tf2*B*(A + B*Tcx2)/(Dt2*(Ks2/H2 + Tc2/12)): PRINT "Y2, Y1"; Y2; Y1; Y2/Y1 1340 X1 = SQR(G31 + Kcx1 2): Z1 = 2*X1 1350 Kcx = SQR((P2 + (X1 - P1)*H1*R/H2) 2 - G32): X2 = SQR(G32 + Kcx 2): Z2 = 2*X2 1360 Trad = Km2 - SQR((2*G32 + Km2 2 + Kcx 2 - Z2*G12)/(1 + Z2*Y1/(B 2*Tf2*H2))): Trat = Kcx/Trad 1370 PRINT #1, "TRAT1 ="; Trat 1380 1F Rs = 0 THEN 1500 1390 Qfs = Rs*Qf1:Qts = Qfs + Qgs: Qfcs = Qts - Acs*Qgs: Qffs = Qts - Afs*Qgs 1400 Tss = Tws + Qts/Hs: Tes = -Ab + SQR(Tss + Ab) 2 + 2*Tcs*Qfcs/(12*B)): Tsum1 = Tm1 + Tcx1 1410 Tcb = K1/Slo: Tps = Tcb - Tsum1/2: Tms = -Tps + SQR((Tps + Tes) 2 + 2*Tfs*Qffs/(48*Slo)) 1420 F1 = .5*Kch1 + Kfh1*(Km1 + A11/K1)/K1 1430 Term = (1 - (Kfh1/K1) 2)*(Kch1*Kwc1 + (Km1 + A11/K1) 2) 1440 G1 = (-F1 + SQR(F1 2 + Term))/(1 - (Kfh1/K1) 2) - Kwb1:Krghs = Rs*G1*H1/Hs 1450 Fs = Tfs*Hs*(Krgs + (1 - Afs)*Kghs): Kws = A + B*Tws 1460 Ss = Kwbs + Krghs: Srs = Tfs*Qffs/48 1470 Ds = SQR(Ss + .5*Kchs) 2 - .5*Kchs*(.5*Kchs + 2*Kwcs)) 1480 Tmss = ((Ds - A)*Tfs*Qfffs - Fs*Tes)/(B*Tfs*Qffs - Fs): K = Tfs*Qffs/(48*(Tmss - Tes)) 1490 PRINT #1, "TRUE TMAX ="; Tmss; "TRUE KBARS ="; K; K2: K2t 1500 IF ABS(K2 - K2t) &lt; .001 THEN 1521 1510 IF L = 2 THEN 1521 1520 GOTO 740 1521 LPRINT CHR$(27) + "E" 1530 PRINT #1, "--------------------------------------------" 1540 PRINT #1, "********TEMPERATURES FOR PRIME TC*******" 1550 PRINT #1, "BW2 ="; Tw2; "TSURF2 ="; Tsx2; "TINT2 ="; Tcx2; "TM2 ="; Tm2; TMAX2 ="; Tmax2 1560 PRINT #1, "DTS2 ="; Tsx2 - Tw2; "DTC2 ="; Tcx2 - Tsx2; "DTF2 ="; Tm2 - Tcx2; "DTBC2 ="; Dtb2 1570 PRINT #1, : PRINT #1, "*******TEMPERATURES FOR ALTERNATE TC*******" 1580 PRINT #1, "BW1 ="; Tw1; "TSURF ="; Tsx1; "TINT1 ="; Tcx1; "TM1 ="; Tm1; "TMAX ="; Tmax1 1590 PRINT #1, "DTS1 ="; Tsx1 - Tw1; "DTC1 ="; Tcx1 - Tsx1; "DTF1 ="; Tm1 - Tcx1; "DTBC1 ="; Dbt1 1600 PRINT #1, : PRINT #1, "*******FLUXES FOR BOTH TC'S*******" 1610 PRINT #1, "QT2 ="; Qt2; "QT1 ="; Qt1; "QF2 ="; Qf2; "QF1 ="; Qf1; "QG2 ="; Qg2; "QG1 ="; Qg1 1620 PRINT #1, : PRINT #1, "*******CONDUCTIVITIES AND TBAR'S FOR BOTH TC'S" 1630 PRINT #1, "KBAR2 ="; K2; "KBAR1 ="; K1: Tb12 = Tcx2 + .5*S2/K2:Tb11 = Tcx1 + .5*S1/K1 1640 PRINT #1, "TBLIN2 ="; Tbl2; Tm2 - Dd*Tm1 + Dd*Tbl1; "TBLIN ="; Tbl1 1650 Tt2 = .5*(Tm2 + Tcx2): Tt1 = .5*(Tm1 + Tcx1) 1660 IF Rs = 0 THEN 1780 1670 IF Tbcs = 0 THEN 1690 1680 Kms = A + B*Tms: Tmaxs = (-A + SQR(Kms 2 + 2*B*Qgs*Tbcs 2/(24*Tst)))/ B:Bs = Tmaxs - Tms 1690 PRINT #1, "--------------------------------------------" 1700 PRINT #1, "PRINT #1, "*******TEMPERATURES FOR SIDE SPECIMEN*******" 1710 PRINT #1, "BWS ="; Tws; "TSURFS ="; Tss; "TINTS ="; Tes; "TMS ="; Tms: "TMAXS ="; Tms + Bs 1720 PRINT #1, "DTSS ="; Tss. - Tws; "DTCS ="; Tes - Tss; "DFTS ="; Tms - Tes; "DTBCS ="; Bs 1730 PRINT #1, : PRINT #1, "*******FLUXES FOR SIDE SPECIMEN**************" 1740 PRINT #1, "QTS ="; Qts; "QFS ="; Qfs; "QGS ="; Qgs: Tbar1 = .5*(Tes + Tms): Zqs = Tbar1*Dq 1750 PRINT #1, : PRINT #1, "*******CONDUCITIVITIES AND TBAR'S FOR SIDE SPECIMEN*******" 1760 Ks = Tfs*Qffs/(48*(Tms - Tes)): PRINT #1, "KBARS ="; Ks: Sqs = 1 + (Tms/Tbar1 - 1) 2/3 1770 PRINT #1, "TBLINS ="; Tbar1; "TBARS ="; Tbar1*(1 + Zqs*Sqs)/(1 + Zqs) 1780 STOP __________________________________________________________________________ Although the present invention has been described to specific exemplary embodiments thereof, it will be understood by those skilled in the art that variations and modifications can be effected in these exemplary embodiments without departing from the scope and spirit of the invention.