ORNL-TM-13528 Contract No. W-TL05-eng-26 Reactor Division SOLUTION OF THE EQUATION DESCRIBING THE INTERFACE BETWEEN TWO FLUIDS FOR THE VOLUME AND PRESSURE WITHIN ATTACHED, SESSILE SHAPED, BUBBLES AND DROPS J. W. Cocke AUGUST 1971 This report was prepared as an accoun t o sgonsm_'ed by the United States Government %e‘;(})x‘::- é e qutgd States nor the United States Atomi.c Energy th{;l;rrlmlsslon, nor any of their employees, nor any of makesco:tractors, subcontractors, or their employees, mmax l-a y warranty, express or implied, or assumes any eg iability or responsibility for the accuracy, com- pleteness or usefulness of any information app':n‘atus product of process disclosed, or represents ’that its use’ would not infringe ptivately owned rights, OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee Operated by UNION CARBIDE CORPORATION for the U. S. ATOMIC ENERGY COMMISSION iii CONTENTS A.bs-tract . . - . * . a - - - . - L] - » - - . . * Introduction « o« ¢« ¢ & v 4 4 ¢ o o o & o s s e . Derivation of the Interfacial Equation . . . Numerical Solution of the Interfacial Equation . Computer Program . . « « « o & o o o &+ & & Solution of the Differential Equation . . Solution for h/a and V/a® versus B and ¢ . . Solution for h/a and V/a® versus B and r/a . Solution for h/a and V/a® versus r/a . . . . Range, Running Time, and Accuracy . . . Results « ¢ v & ¢ v v v v v v 6 v 0 o 4 o Discussion of Results . « + 4 ¢ « ¢ & « o Estimation of the Accuracy . . « . . . Comparison with Previous Results . . . . . . ConclusionsS v ¢ ¢ « o o e 2 o o o o o o o s o References .« v v v ¢ 4 6 o ¢ o 2 s o o « o o Nomenclature . o o o & o & o o s o s o o o o o 4 AppendiCesS o o ¢« ¢« 4 4 4 s e e s s s s 4 e e e s A, Parametric Crossplots . . . « . « « . . B. Tabulated Results « ¢« « ¢ ¢ « o o « « & C. Computer Program « « « + o « o« o o « « & 10 10 LT 17 21 21 22 25 27 43 SOLUTION OF THE EQUATION DESCRIBING THE INTERFACE BETWEEN TWO FLUIDS FOR THE VOLUME AND PRESSURE WITHIN ATTACHED, SESSILE SHAPED, BUBBLES AND DROPS J. W. Cooke ABSTRACT A numerical computer program was written to solve the equa- tion describing the interface between two immiscible fluids to obtain the shape, size, volume, and pressure of attached bubbles and droplets. These relationships are important to the study of three-phase heat transfer, superheat, critical constants, and interfacial energies. Previous solutions have been obtained with limited accuracy for a restricted number and range of vari- ables. The present results are given in both graphical and tabular form for a wide range and number of parameters, and the computer program is included so that an even broader range and number of variables, as well as specific values, can be obtained as required. In particular, an expression for the maximum- bubble-pressure was derived which is considerably more accurate over a wider range than previous expressions. Keywords. Surface tension, interfacial tension, bubbles, drops, contact angle, maximum-bubble-pressure, filuid-vapor in- terface. INTRODUCTION A knowledge of relationships among volume, pressure, shape, and size of attached bubbles and droplets are important in the study of boiling and condensing heat transfer, superheat and critical point phenomena, mass transfer studies, and in the measurement of contact angles and sur- % These relationships can be obtained from the solution face tension.'- of the second-order, nonlinear differential equation describing the inter- face between two fluids. This equation has been solved by perturbation analysis for very small bubble sizes by Rayleigh® and Schrddinger,® by numerical hand calculations for a wide range of discrete drop shapes by Bashforth and Adams,7 and by numerical computer calculations for droplet volumes and heights by Baumeister and Hamill.® However, the Rayleigh- Schrodinger solution is no longer used extensively because of concern regarding its accuracy; and the Bashforth and Adams (as well as the 1 Baumeister and Hamill) solutions were obtained for a limited number and range of variables. In order to check the accuracy and to extend the usefulness of the above solutions, a numerical computer program was written to solve the interfacial equation. The subject program solves the interfacial equation for any value of the dimensionless shape parameter, B, for positive values of the term 87 (sessile drops or above-attached bufibles)* and extends the previous sclutions for ¢ > 180 degrees. The solution not only provides the dimensionless X, 7, ¢ coordinates describing the profiles of the inter- face, but also gives the bubble volume within and the pressure difference across the interface as a function of profile shape and, in addition, the maximim pressure difference across the interface for a given radius of attachment (which is required for the calculation of the surface tension by the maximum-bubble-pressure technique). This report descrites the derivation of the interfacial equation, its numerical solution, and the computer program employed. The re- sults and an estimation of their accuracy are presented and compared with the previous solutions. DERIVATION OF THE INTERFACIAL EQUATION At the interface separating two immiscible fluids (usually a liquid and its vapor), a force imbalance exists which results in an interfacial tension. The tension is a force per unit length pulling uniformly in all directions tangent to the interface., Neglecting gravitational forces, a force balance on an infinitesimal segment of a free surface is shown in Fig. l1-a. The surface is bowed by a uniform differential pressure, P, where r; and r, are the principal radii of curvature. The force balance would thus be: AZ X P Az =2y | X — + Az —— , (1) 21‘1 21'2 * The case for negative values of the term BZ (pendant drops or below-attached bubbles) will be presented in a later report. ORNL-DWG 7{- 7609 yAX PAZAX LIQUID BUBBLE (8) Fig. 1. Schematic View of a Sessile-Type Drop and Bubble and the Force Balance on an Infinitesimal Surface Element. or P = ‘Y(L + }"—) . (2) If gravitational forces are present, the influence of the dif- ferential fluid density, p, must be considered. For an interface assumed To be symmetrical about an axis of revolution, the force balance equation for the bubble shown in Fig. 1-b will be 1 1 Pg P_y(——+—-—)——(A—Z)=O s (3) ry Ta g c where p = Pp, — Py At the orgin, ry = rz3 = b and thus 2y g P=—+A— . (4) b = Furthermore, with r; = x/sin ¢, Eq. (3) becomes 2y pgz sin ¢ 1 ——+-———y< +-—-—)=o ! (5) b g X To c A similar derivation for the drop shown in Fig. 1-b would also result in Eq. (5). This equation can now be made dimensionless with respect to ©, and after rearranging: 1 sin ¢ - + =2+ Bz , (6) R X where . gpb B = (7) Y&, and To X z R =— " X ==, and Z2 = — . b b b Equation (6) can be transposed into Cartesian coordinates by substituting 1 d*7/dx® R [1+ (az/ax)*]1¥ = and . dz/ax sin = - [1 + (az/ax)® ¥~ to obtain a2z, Az az azel¥ e — + l+(—-—- — = (2 + 82) 1+<—-—-) , (8) axe dx Xdx ax which is a second-order nonlinear differential equation, with boundary conditions: X =0 3 2z =0 X =0 ; 4dz/XaX =1 Although Eq. (8) cannot be solved analytically in terms of ordinary functions, its numerical solution is described in the next section. NUMERICAL SOLUTICN OF THE INTERFACTAI, EQUATION Equation (6) can be rearranged to read 1 R = (9) 2 + BZ — (sin ¢)/X and by definition: ag — =1 dg dX — =Rcos ¢ =F (X, Z, @) , (10) dg dz — B R Sin¢ = G (X, Z, ¢) - (ll) d¢ As long as BZ > 0, R will always be finite,. A numerical technique of fourth-order accuracy developed by Runge- Kutta” was selected to solve the set of simultaneous Egs. (9), (10), and (11); the iterative equations decribing this technique are: 1 X . =X +Z (k, + 2k + 2kp + ka) + O(08)° (12) 1 Zas =Bt 7 (mg + 2my + 2mp + my) + 0(ag)® (13) o I e + 8 “ n+l where O n gt 1 1 1 ky = 8 F(X, + =k, 2 +—m,, ¢ +—=09) , 2 2 2 1 1 1 kg = M@ F(X, +—ky, 2 +—m, ¢ +—208) , 2 2 2 m =A¢G(Xn+;—ko, zn+;mo, q)+-2-a¢) , 1 1 1 mp = AP G(X, +—ky, Dyt —m, ¢ +—08) 2 2 D mg = AP G(Xn + Ka, Zn +my, ¢+ OP) and where the symbol 0(Ag)° represents a term which is small, of the order (Ag)®, when Ap is small. Equations (12) and (13) are of a form that can be readily trans- formed into a computer program. COMPUTER PROGRAM The computer program for the solution of the interfacial equation and for the calculation of the various output parameters is listed in Appendix C. The program consisted of four parts which are discussed below. Solution of the Differential Equation. Two subroutines in both single and double precision and consisting of four iteration loops were written to solve Egs. (12) and (13). The subroutines RHOS and RHOD calculate the values of R and supply the values of the functions F and G to the subroutines RUNGKS and RUNGKD. These latter subroutines cal- culate the values of the coefficients ki and m. and the new values of X, Z, and ¢ for reintroduction in RHOS and RHOD to continue the iterative procedure. The iteration procedure is initiated by equating X, Z, and ¢ to zero and R to one. Solution for h/a and V/a® versus B and ¢. The pressure and volume within the attached bubble are calculated from X, Z, and ¢. The pressure relation is given by Eg. (4) and can be simplified by using Eq. (7) and the definition of the specific cohesion, 2vg e = —= (14) PE to obtain h/a = /2/B + Z.J/B/2 (15) where Z = Z(X = I‘) = A 3 T and h = ch/pg . The volume relation can be obtained from the integration of v d(-—) =X az , (16) b ' where X and dZ can be cobtained from Eq. (6). Integrating the resulting eguation by parts gives the relationship: - - v TX° 2 sin ¢ - =—|2+pz-—] . (17) b° B X Upon substituting Eqs. (6), (14), and (15) into Eq. (17), the final expression for the dimesionless volume 1s obtained: Y T r h ~ == (—)(—) — sin ¢ | , (18) a 8 a a where T - =XV a is the dimensionless radius of attachment. Solution for h/a and V/a® versus § and r/a. To obtain the pres- gure and volume within attached bubbles for a given radius of attachment from the numerical solutions X, Z, ¢, of the interfacial equation, several conditional "IF" statements were required. The interrelation- ship of h/a, 8, r/a, and ¢ are shown schematically in Fig. 2. The iterative solution of the interfacial equation proceeds along constant B lines for given steps of Ap. A conditional check is made to note the intersection of the given r/a curve (denoted by L), and the values at the point of crossing are determined by linear interpolation. ©Since r/a is multi-valued, an additional check is necessary to note when the increasing values of r/a start to decrease at ¢ = nn/2, n =1, 5 ... (denoted by O), or when decreasing values start at ¢ = nn/2, n = 3, 7 .... In this manner, solutions for many values of r/a can be obtained for a given value of 3. Solutions for h/a and V/a® versus r/a. These solutions were some- what more difficult to cbtain than those described above since it was necessary to increment B as well as @¢. Both of the conditional checks described above for crossing a given value of r/a and a change from in- creasing to decreasing r/a values were required as well as a check for the change from increasing to decreasing values of h/a (denoted by A in Fig. 2). ORNL-DWG 71- 7610 h/fa —» B Fig. 2. Between h/a, 'rr/2 ¢ —» Schematic Representation of the Relationships B, r/a, and 3.- 10 For & given value of 8, h/a is a multi-valued function so that the choice of increasing or decreasing B to approach H/a must be carefully considered. Furthermore, the direction of approach to E/a along the given value of r/a must also be carefully considered. To insure the most trouble-free solution over the entire h/a versus ¢ field, a decremental approach from right to left (as shown by the arrows in Fig. 2) was chosen. To reduce the number of iterations required to locate h/a, an esti- mate of the value of g is calculated from equations fitted to a few preliminary results. The initial B value is then decremented along first a coarse grid, and finally along an extra fine grid to obtain the final solution using double precision. Range, Running Time, and Accuracy The ranges of the computer program as presently written are 0 < ¢ < 360° 0.1 €£r/a £2.0, and 0.02 £ B £ 150; however, these can be easily extended at some sacrifice of either the running time or accuracy. The average running time fof the program to obtain a value of h/a for a given value of r/a is approximately 10 seconds on the IBM 2360-91 Computer system. The average running time for the other programs is considerably shorter per solution. An estimate of the accuracy (to be given later) was obtained by comparing the values of h/a as a function of ¢ for various values of B, ANg, and for single and double precision. The results of the computer solution are discussed in the next section. RESULTS The results in both tabular and graphical form are presented in this section and in the Appendix. The profile of a bubble for B8 = 0.8 is shown in Fig. 3. The pro- file is shown extended to ¢ = 360°, which would be possible if a suitable 11 > ORNL-DWG 74-7641 {. ol LT TN /T IN y ) N \ \ ) / 0.9 0.8 0.6 0.4 N 0.2 0 ——*// {0 08 06 04 02 0 02 04 06 08 10 x/b z/b \ - /\/ Fig. 3. Profile of a Bubble with B = 0.8 for 0° € ¢ < 360° 12 form of attachment were provided, and the center of bouyancy were to remain on the z axis.,. A tabuletion of B, x/b, z/b, h/u, and V/a® for various values of ¢ are presented in Tables B.l through B.6 (in Appendix B). Plots of h/a and V/a® versus ¢ for various r/a are cshown in Figs. 4 and 5, respectively. Figure 4 cleaurly shows the attaimment of a maximum value of h/a (h/a) for a given radius of attachment. (This is the basis of the maximum-bubble- pressure technique® for the measurement of surface tension.) In addition, . .o . 3 there is a minimum value of h/a as well as a maximum value for V/a®. The values of 1i/a and the corresponding values of V/a®, B, ¢, x/b, E/b for various values of r/a are given in Table 1, and various cross- plots of these variables are given in Figs. A.]l through A.4 (Appendix A). These plots show that both Eya and E& approach asymptotic values of Vfig and 180°,* respectively. Thus, the maximum pressure difference (h/a) that a large tube can sustain will be very nearly independent of tube diameter. A least-squares, polynomial fit of the computer solutions for vari- ous formulations of h/a and r/a were made. The forumlation that gave the best fit was a a a r - - = T - - ) (19) h r h a which is of the same form as the perturbation solutions of Rayleigh- Schrbdinger. For this reason, Eq. (19) was fitted to the data of Table 1 by the relationship: £(y) =i + L1y + 1) F(y') = ; = i3 + i4y + isyz + e » (20) Y where io’ i,, and i, are the coefficients of the Rayleigh-Schrddinger solution and i, i,, 1g, «.. were determined by the least-sguares pro- cedure and are listed in Table 2. * These values can be obtained by the solution of Eg. (3) with 1/ry =0 and 1/b = O, 13 ORNL-DWG 71-T76t2 6 r/a =0.2 3o 3 = N 4 \ 0.3 0.2 .fl—-< 33 0.4 \.\\\\ 2 _ \Q\L‘_‘ . — . N\\, 10.0 ° 0 40 80 120 160 200 240 280 320 360 ¢, Fig. 4. Variation of the Dimensionless Pressure Difference, h/a, as a Function of @. for Various Values of 8 and the Dimensionless Radius of Attachment, r/a. 1h ORNL-DWG 74-7613 0 50 100 450 200 250 300 350 #: Fig. 5. Variation of the Dimensionless Volume, V/aa, as a Function of ¢, for Various Values of the Dimensionless Radius of Attachment, r/a. Table 1. 15 Radius of Attachment (r/a) Maximum Values of the Pressure (h/a) and Corresponding Values of the Size, Shape, and Volume of Bubbles as a Function of the r/a ? B x/b 2/b h/a v/a® 0.12 90 .646 0.0290810 0.995156 0.999705 8.413519 0.003651 0.1k 90.795 0.0397216 0.993412 0.998120 7.236L65 0.005806 0.16 91.4ho7 0.0521086 0.991243 1.00397 6.357323 0.008787 0.18 91.906 0.0662774 0.988791 1.00649 5.676507 0.012624 0.20 92. 411 0.082286 0.986012 1.00865 5.134652 0.017439 0.20 92.906 0.100192 0.982929 1.00988 4.693892 0.023404 0.24 93.420 0.120066 0.979526 1.0106k4 L. 328985 0.030656 0.26 93.945 0.141985 0.975814 1.01077 h.opolls 0.0392273 0.28 94,488 0.166037 0.971786 1.01039 3.761790 0.049570 0.30 95 .47k 0.192565 0.966824 1.01574 3.537926 0.062105 0.225 96,472 0.229022 0.960L16 1.01752 3.299450 0.0802973 0.350 97.476 0.269500 0.953463 1.01785 3.097815 0.101911 0.375 98.987 0.315035 0.944859 1.02327 2.,9257k4L 0.128876 0.400 100.225 0.365084 0.936222 1.02319 2.T7TTL3 0.158475 0.4b25 101.985 0.421688 0.925568 1.02746 2.6Lg595 0.197373 0.450 103.483 0.483756 0.914990 1.02640 2.538105 0.239818 0.475 104.821 0.551689 0.904402 1.02173 2. khobes 0.286173 0.50 107.164 0.631617 0.889730 1.02464 2.35527% 0.346254 0.55 111.986 0.822982 0.857398 1.02159 2.214232 0.501889 0.60 117.488 1.072133 0.819487 1.00938 2.104842 0.708170 0.65 123.990 1.407182 0.774913 0.986809 2.01991k 0.987762 0.70 130.993 1.853431 0.727151 0.950582 1.953875 1.347727 0.75 137.489 2.41%075 0.68265L4 0.9002832 1.90210k 1.768822 0.80 1Lko.66k4 2.977900 0.655617 0.853059 1.860k45 2.136163 0.90 152.491 L,841565 0.5784L48 0.742939 1.7986L9 3.270626 1.00 158.hko2 7.100181 0.530738 0.648972 1.753511 L. 356567 1.10 162.989 10.070060 0.490221 0.567348 1.718720 5.521721 1.20 165.994 13.866110 0.L455742 0.4g7922 1.690847 6.726L07 1.30 167.994 18.715220 0.42kg72 0.438376 1.667904 8.005399 1.0 169.991 25.16100 0.394711 0.385323 1.6486L40 9.386485 1.50 171.489 33.48783 0.366575 0.339160 1.632205 10.839000 1.60 172.4904 Lh,16k29 0.340486 0.299033 1.618007 12.356187 1.70 173.488 s58.14618 0.315285 0.263383 1.605608 13.971941 1.80 173.998 75.94175 0.292110 0.232455 1.564682 15.640580 1.90 174.987 99.7358 0.268978 0.204340 1.584k975 17.453834 2.00 175.489 130.1581 0.247919 0.180031 1.576295 19,314144 16 Table 2. Coefficients for the Polynomial Equations Fitted to the Computer Solution Coefficients Eq. (19) Eg. (21) ig 1.00000 -0.00090 iq -0.66667 1.04439 ig -0.66667 -0.47175 ia 0.03230 1.43283 ig -5.52833 -4.59801 ig 61.1913k 5.38228 ig -351.38141 -2.,73720 1 1099.76625 0.51837 ig -1930.93994 ig 1913.36384 110 -1003.22519 111 216.93848 17 The main disadvantage of Eq. (19) is that an iterative procedure is necessary to calculate a/h. A simpler formulation, but less accurate, was also fitted to the computer results: a r - f(-—) . (21) h a The coefficients for the polynomial fit of Egq. (21) are also listed in Table 2. The accuracy of these two polynomial fits is discussed in the next section. DISCUSSION OF RESULTS An estimate of the accuracy of the computer results and compari- sons with previous results are presented in this section. Estimation of the Accuracy. Results were obtained for h/a versus ¢ for B = 0.1 and 100.0 with three values of Agp = 1°, 1/2°, and 1/4° using both single and double precision. There was no change in h/a (to the seventh decimal place) as /Ay was decreased from 1° to 1/4° when double precision was used for B = 0.1 and only 0.00008% change for B = 100.0. The single precision results are plotted in Fig. 6, where the percent difference is with reference to the double precision, A¢ = 1/4°, values. As can be seen, an interval less than Ap = 1° decreased the accuracy of the results because of rounding errors when single precision was used. Except for the maximum pressure results listed in Table 1, all the tabulated results were computed using single precision and interval NP = 1°. The values listed in Table 1 were calculated using double pre- cision and are good to at least the sevnth decimal place. The other tabulated results are good to at least the fifth decimal place. Comparison with Previous Results. As anticipated, the careful, tedious hand calculations of Bashforth and Adams (which required a number of years to complete) were in good agreement (to the fifth decimal place) with the present computer sclution. Trouble, however, develops when the “Bashforth and Adams tables are singly and doubly interpolated to apply 18 ORNL-DWG 74— 7644 0.005 A B O |r——— _ .00 04 o 0.25 0. £~ < \\\\\\, ::\\\\\k = -0.005 \ 2 \ \ 100 100 w a W —0.010 N N\ | '8 & \ > 0.50 100 -0.015 < \\ Ll a -0.020 \ 0.25 100 -0.025 0 50 100 450 200 250 300 350 ¢ Fig. 6. Comparison of the Single Precision Results for h/a with the Double Precision, Agp = 1/4, Values as a Function of ¢, Ap, and 8. 19 their results to practical calculations. Sudgen,lo "oy careful inter- polation" of Bashforth's tables, constructed a table for calculating surface tension by the maximum-bubble-pressure method. (This table is the one most often referred to in current literature on surface tension.) The percent difference between our solution and Sudgen's as a function of r/a is shown in Fig. 7. The maximum difference is -0.1%. Although an error of 0.1% can be neglected for some studies at elevated temperatures (where other errors are more significant), this magnitude of error can be significant for many measurements made at room temperature, where theoretical studies of small changes in the molecular structure of the interface are being conducted. Futhermore, this error can be magnified by as much as 20 times when the two tube, differential technique is used to measure surface tension. To be particularly noted in Fig. 7 is that the Rayleigh-Schrddinger solution is in better agreement with the computer solution than Sudgen's results all the way from O € r/a < 0.45. This range of r/a covers a large portion of the surface tension studies that have been conducted in the past. In fact, the Rayleigh-Schrodinger equation is in error by less than 1.05% all the way to r/a = 1.0. Thus, this much simpler analytic solution can be used in many cases where precise surface tension values are not needed. Also shown in Fig. 7 are the deviations of Egs. (19) and (21) from the computer solutions. Equation (19) agrees to within x0.05% all the way to r/a = 1.5. The simpler Eq. (21) agrees to within +0.07% from 0.2 < r/a < 1.5; but Schrddinger's equation is recommended for r/a < 0.2. The main disadvantages of Egs. (19) and (21) is that double precision should be used in their solution, especially at the larger values of r/a. Baumeister and Hamill presented their results as plots of droplet volumes and heights as functions of droplet radii and contact angles. In this form, their results could not be conveniently compared with the present results. In addition, their results were given to only the third significant figure. 20 ORNL-DWG 74-7645 0.5 .4 // \ 410 0.4 / ‘\ 0.9 / ~ = RAYLEIGH-SCHRODINGER . | / \ o SUGDEN - o8 = 03 | \ EQUATION (19) 07 z [ 1] { T ——— EQUATION (21) 1° " \ \ 4 06 9 [ 4 | Z 02 ‘ 0.5 i [} o / \ ~ 0.4 z \ / AT~ ‘ 'y / / L \ ® \ C / \ E 0 v ) ‘5/f\ \‘\"7{, .d\ * A \\ b7 o ° / ® \ / ' \ /'7, * \ / P \ 7 \ *L ° L -~ -0.1 %o -0.2 0 0.2 0.4 0.6 0.8 1.0 .2 1.4 1.6 1.8 r/a Fig. 7. Deviations of Previous Sclutions and Present Polynomial Equations from the Present Computer Solution of the Maximum-Bubble Pressure. 21 CONCLUSIONS A computer program was written to solve the second-order, nonlinear, differential equation describing the axially symmetric interface between two immiscible fluids. The present solution is limited to sessile-shaped drops and bubbles; the solution for pendant-shaped drops and bubbles requires a different approach and will be given in a later report. All of the present results are accurate to at least the fifth decimal place with some results accurate to the seventh decimal place. The results are presented in a form that is useful in the analysis of boiling and condensing heat transfer, superheat, critical constants, and in the measurements of contact angles and surface tension. In addi- tion, these results may be of use in the design of equi-stressed shells for containment vessels (i.e., above-ground water tanks, under-water storage of liquids and gases, lighter-than-air ballons, and submerged marine laboratories). The computer program is listed so that a wider range and number of variables can be obtained as desired. REFERENCES 1. F. Kreith, Principles of Heat Transfer, pp. 308-437, International Textbook Company, Scranton, Pennsylvania, 1lst ed., 1959. 2. J. A. Edvards and H., W. Hoffman, Superheat Correlation for Boiling Alkali Metals, Proceedings of the Fourth International Heat Transfer Conference, Versailles, September 1970 (under the book title '"Heat Transfer 1970"), Elsevier Publishing Ccmpany, Amsterdam, Netherlands, 1970, 3. A. V. Grosse, The Relationship Between the Surface Tension and FEnergies of Liquid Metals and Their Critical Temperatures, J. Inorg. Nucl. Chem., Ph:1l7-156 (1962). 4. D. W. G. White, Theory and Experiment in Methods for the Precision Measurement of Surface Tension, Trans. ASME, 55: 757 (1962). 5. Lord Rayleigh, On the Theory of the Capillary Tube, Proc. Roy. Soc., 92 (Series A): 184-195 (1915). 6. FErwin Schrddinger, Notizilber den Kapillardruck in Gasblasen, Ann. Physik., 46: 413-418 (1915) T T 9. 10. 0 @] | 22 F. Bashforth and J. C. Adams, An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluids, University Press, Cambridge, Massachusetts, 1833. K. J. Baumeister and T. D. Hamill, Liquid Drops: Numerical and Asymptotic Solutions of their Shapes, NASA-TN-D-4779, National Advisory Committee for Aeronautics, September 1968. F. B. Hildebrand, Introduction to Numerical Analysis, pp. 236-239, McGraw-Hill, New York, 1956. S. Sugden, The Determination of Surface Tension from Maximum Pressure in Bubbles, J. Chem. Soc., 1: 858-866 (1922). NOMENCLATURE distance from the origin to the plane of attachment, cm specific cohesion 2ygc/pg, enf radius of curvature at the vertex of the bubble, cm local acceleration due to gravity, cm/sec” dimensional constant, dyne.sec®/g.cm pressure differential across interface, cm of fluid with density p meximum value of h for given value of r/a, cm of fluid with density p coefficients of polynomial equations coefficients of numerical eguations coefficients of numerical equations pressure differential across interface, dyne/cm? radius of circle of attachment, cm dimensionless radius of attachment r; Pprincipal radii of curvature of the interface, cm dimensionless value of ry, rg/b volume enclosed by the interface, em® < 23 volume enclosed by the interface for a given value of r/a at h/a = h/a, cm® horizontal coordinate, cm value of x for a given value of r/a at h/a = h/a, cm dimensionless x, x/b value of X at x =r independent variable in polynomial equation vertical coordinate, cm value of z for a given value of r/a at h/a = h/a, cm dimensionless z, z/b value of Zat x =1 Greek Letters al &6 BB < dimensionless parameter = gpbs/gcy = 2(b/a)? value of 8 for a given r/a at h/a = h/a interfacial tension, dyne/cm angle between axis of revolution and the normal to the surface value of ¢ at the radius of attachment value of ¢ for a given r/a at h/a = h/a positive density difference between the two fluids, g/cm3 APPENDICES A. PARAMETRIC CROSSPLOTS Various crossplots of the parameters shown in Figs. 4 and 5 are given in Figs. A.l, A.2, and A.3. The maximum bubble radius as a function of the angle qbr for various radii of attachment is shown plotted in Fig. A.4. TFor 3., < 0%, the maximum bubble radius would be the radius of attachment. Also shown in Fig. A.4 is the locil of the maximum bubble radius where the maximum bubble pressure 1is reached. 27 28 ORNL-DWG 7{— 7646 7 N 2 —~— O 02 04 06 08 10 12 44 16 8 2.0 f/a Fig. A.l. Dimensionless Maximum-Pressure-Difference Across the Interface Separating Two Fluids as a Function of the Dimensionless Radius of Attachment. 29 ORNL-DWG 74-T7617 80 70 60 50 30 / 20 m — T 90 {00 #10 120 130_ 440 1150 160 4170 480 $r 4__——/ Fig. A.2. The Value of the Parameter B at h/a = h/a as a Function of ¢ . r 30 ORNL—DWG T7i-T7648 20 r 1.5 ' /// o 1.0 // 05 / / 90 100 HO 120 {70 180 0o 130 140 {50 160 ér Fig. A.3. The Dimensionless Radius of Attachment as a Function of ¢. at h/a = h/a. 31 ORNL-DWG 71-7619 2.2 2.0 Ofi:::f \\\\ \LOCI OF (%/0)yax {.8 SN 7 1.0 4 / N\ / | ‘\\ x 1.2 s /—\ = 0.6 3 ‘///,f" “\\\\\ 1.0 N TN 0.8 ’ l e \ / /03 N \ ’—\ 0.4 |= // // " / 0.2 Hm=] 0 90 120 150 180 210 240 270 300 330 360 ¢ Fig. A.4. Maximum Radius of a Bubble (x/a at ¢ = 90°) as a Function of ¢_ for Various Radii of Attachment (r/a). TFor ¢_ < 90°, (x/a) = r/a. T T ? max B. TABULATED RESULTS The computer results for the size, shape, pressure, and volume of attached bubbles and drops for given radii of attachment are listed in Table B.l through B.6. The Results are arranged in ascending values of ¢r. 33 34 Table B.1l. ©Size, Shape, Pressure, and Volume of Bubbles and Drops with a Given Radius of Attachment r/a = 0.2 3 ?.. x/b z/b h/a V/a 30.00 2.99 0.05168 0.00135 0.26341 0.00030 18.00 3.86 0.06667 0.00225 0.34009 0.00042 8.00 5.80 0.10000 0.00505 0.51009 0.00063 5.50 7.00 0.12061 0.00734 0.61519 0.00077 k.00 8.21 0.14142 0.01013 0.72143 0.00090 3.50 8.7 0.15119 0.01158 0.77125 0.00096 2.90 9.66 0.16609 0.01400 0.84731 0.00106 2.30 10.86 0.18650 0.01765 0.95144 0.00119 2.00 11.66 0.20000 0.02317 1.02032 0.00128 1.80 12.30 0.21082 0.02262 1.07556 0.00135 1.50 13.49 0.23094 0.02717 1.17823 0.00149 1.10 15.81 0.26968 0.03727 1.37604 0.00175 0.95 17.05 0.29019 0.04327 1.48078 0.00189 ¢.80 18.63 0.31623 0.05161 1.61378 0.00207 0.66 20.59 0.34816 0.06295 1.77694 0.00229 0.58 22.0k 0.37139 0.07194 1.89569 0.00247 0.50 23.84 0.40000 0.08399 2.04200 0.00268 0.46 2L .92 0.41703 0.09166 2.12910 0.00281 0.Lh2 26.17 0.4364Y 0.10089 2.22841 0.00296 0.38 27.62 0.45883 0.11220 2.34306 0.003173 0.34 29.35 0.48507 0.12636 247745 0.00334 0.30 31.46 0.51640 0.14467 2.63801 0.00361 0.25 34.87 0.56569 0.17662 2.89087 0.00h05 0.23 36.59 0.58977 0.19386 2.01458 0.00k27 0.21 38.60 0.61721 0.21486 3.15569 0.00455 0.19 40.99 0.64889 0.24102 3.31872 0.00489 0.17 L3.01 0.68599 0.274 7k 3.51007 0.00531 0.15 47.60 0.73030 0.31993 3.73910 0.00585 0.14 49.86 0.75593 0.34889 3.87195 0.00622 0.13 52.51 0.78446 0.38410 L.02025 0.00664 0.12 55.69 0.81650 0.42796 4.,18731 0.00719 0.11 59.65 0.85280 0.48479 L.37771 0.00792 0.10 6L.87 0.89443 0.56306 4,5980L 0.00898 0.09 72.70 0.94281 0.6861L 4.85960 0.01079 0.0828 84.85 0.98295 0.88466 5.09473 0.01h4k 35 Table B.1 (Continued) 8 ¢ x/b z/b h/a v/a® 0.0828 95.17 0.98295 1.05331 5.12905 0.01878 0.09 107.58 0.94281 1.24471 4,97809 0.02660 0.10 115.80 0.89443 1.35781 L.77575 0.03445 0.11 121.43 0.85280 1.42604 L .59845 0.04173 0.12 125.82 0.81650 1.47280 b.hhy30kh 0.04890 0.13 129.46 0. 78446 1.50654 4.30642 0.05606 0.14 132.59 0.75593 1.53162 4.18487 0.06328 0.15 135434 0.73030 1.55053 L.07611 0.07058 0.17 140.07 0.68599 1.57565 3.88935 0.08542 0.19 144,07 0.64889 1.58951 3.73435 0.,10060 0.21 147.60 0.61721 1.59609 3.60326 0.11610 0.23 150.78 0.58977 1.59777 3.49067 0.13189 0.25 153.70 0.56569 1.59596 3.39269 0.14794 0.34 164 .94 0.48507 1.56461 3.07046 0.22259 0.38 169. 3k 0.45883 1.54451 2.96739 0.25668 0.h2 173.53 0.43644 1.52274 2.87998 0.29109 0.46 177.57 0.41703 1.50000 2.80452 0.3257h 0.58 189.15 0.37139 1.42956 2.62679 0.43005 0.66 196.74 0.34816 1.38256 2.53500 0.49949 0.70 200.55 0.33806 1.35938 2.40453 0.53405 0.80 210.35 0.31623 1.30143 2.404h23 0.61955 0.90 £220.90 0.2981h 1.24333 2.32476 0.70350 1.00 233.12 0.28284 1.18282 2.25059 0.78543 1.10 250.16 0.26968 1.11197 2.17306 0.86409 1.10 289.74 0.26068 1.01358 2.10009 0.85531 1.00 306.55 0.28284 1.00345 2.12376 O.77164 0.90 318.54 0.29814 1.01038 2.16850 0.68854 0.80 328.88 0.31623 1.02769 2.23111 0.60513 36 Table B.2. Size, Shape, Pressure, and Volume of Bubbles and Drops with Given Radius of Attachment r/a = 0.3 B ® x /b 2 /b h/a v/a® 30.0 4.55 0.07746 0.00307 0.27011 0.00165 18.0 5.87 0.10000 0.00509 0.34859 0.00215 12.0 7.20 0.12247 0.00764 0.42700 0.00264 8.0 8.83 0.15000 0.01145 0.52201 0.00325 5.5 10.66 0.18091 0.01672 0.63076 0.00393 4.0 12.53 0.21213 0.02303 0.73967 0.0046k4 3.5 13.41 0.22678 0.02640 0.79085 0.00496 2.9 14,77 0.2491k 0.03194 0.86891 0.00547 2.3 16.63 0.27975 0.04045 0.97588 0.00618 2.0 17.87 0. 30000 0.04663 1.04663 0.00667 1.8 18.88 0.31623 0.05197 1.10340 0.00745 1.7 19.45 0.32540 0.05512 1.13547 0.00727 1.5 20.76 0.34641 0.06271 1.20901 0.00779 1.25 22.85 0.37947 0.07579 1.32483 0.00861 1.2 234+35 0.38730 0.07908 1.35225 0.00881 1.1 24 46 0.40k452 0.08662 1.41264 0.00925 1.0 25 .7k 0.42426 0.0957h4 1.48191 0.00977 0.95 26.46 0.43529 0.10110 1.52063 0.01005 0.9 27.24 0.44721 0.1070k 1.56252 0.01038 0.8 29.05 0.47434 0.12135 1.65789 0.01113 0.7 31.28 0.50709 0.14017 1.77323 0.01206 0.66 32.32 0.52223 C.14943 1.82662 0.01250 0.58 3L.79 0.55709 0.17222 1.94970 0.01358 0.5 37.92 0.60000 0.20334 2.10167 0.01499 0.46 39.86 0.62554 0.22365 2.19240 0.01588 0.42 42,13 0.65465 0.24857 2.29609 0.01695 0.38 h4.87 0.68825 0.27994 2.41618 0.01829 0.34 48,25 0.72761 0.32082 2,55764 0.02000 0.3 52.62 0.77460 0.37669 2.72788 0.02236 0.25 60.61 0.84853 0.48659 3.00046 0.02715 0.23 65.36 0.88465 0455577 3.13731 0.03036 0.21 72 .20 0.92582 0.65884 3.29956 0.03559 0.195 81.53 0.96077 0.80327 3.45338 0.04423 37 Table B.2 (Continued) B ¢ x/o z /b h/a v/a® 0.1925 95.40 0.96694 1.01465 3.53793 0.06203 0.21 108.40 0.92582 1.19288 3.47260 0.08758 0.23 115.90 0.88465 1.27945 3.38272 0.1086k% 0.25 121 .34 0.84853 1.33222 3.2994L 0.12796 0.3 131.17 0.77460 1.4ho212 3.12503 0.1741k4 0.34 137.09 0.7276 1.42704 3.0137h 0.21045 0.38 142.08 0.68825 1.43800 2.92095 0.24666 0.42 146,46 0.65465 1.44035 2.84223 0.28285 0.46 150.41 0.62554 1.43727 2. 77443 0.31905 0.50 154,05 0.60000 1.43052 2.71526 0.35524 0.58 160.65 0.55709 1.41014 2.61634 0.42743 0.66 166.63 0.52223 1.38465 2.53620 0.49919 0.7 169.45 0.50709 1.37088 2.50133 0.53471 0.8 176.18 0. 47434 1.33487 2.42538 0.62303 0.9 182.56 0.44721 1.29805 2.36147 0.7098% 1.0 188,74 0.42426 1.26124 2.30604 0.79517 1.1 194.79 0.4ok52 1.22489 2.25680 0.87863 1.2 200.82 0.38730 1.18910 2.21206 0.96038 1.5 219.70 0.34641 1.08403 2.09350 1.19399 1.7 234.58 0.32540 1.01218 2.01784 1.33856 1.7 304.88 0.32540 0.83893 1.85810 1.29849 1.2 337.48 0.38730 0.87818 1.97123 0.91837 1.1 343,30 0.40452 0.89730 2.01386 0.84032 1.0 349,15 0.h2k26 0.92136 2.06571 0.761k2 0.9 355.15 0.4k721 0.,95155 2.12903 0.68168 Table B.3. 38 Size, Shape, Pressure, and Volume of Bubbles and Drops with Given Radius of Attachment r/a = O.h ¢ x/b z/b h/a v/a® 30.0 6.17 0.10328 0.00548 0.27941 0.00533 18.0 7.98 0.13333 0.00912 0.36068 0.00692 12.0 9.79 0.16330 0.01372 0.44187 0.00850 8.0 12.02 0.20000 0.02063 0.54126 0.01047 5.5 14.54 0.24121 0.03019 0.65309 0.01270 4.0 17.13 0.28284 0.04173 0.76612 0.01504 3.5 18.35 0.30237 0.04787 0.81926 0.01613 2.9 20 .2k 0.33218 0.05808 0.90040 0.01786 2.3 22.86 0.37300 0.07385 1.01170 0.02030 2.0 2L ,63 0.40000 0.08546 1.08546 0.02196 1.8 26.06 0.42164 0.09548 1.14468 0.02332 1.5 28.78 0.46188 0.11589 1.25507 0.02594 1.25 21.84 0.50596 0.14108 1.3764kL 0.02897 1.1 3k.23 0.53936 0.16228 1.46875 0.03139 1.0 36.17 0.56569 0.18038 1.54176 0.03341 0.8 41.32 0.62246 0.23257 1.72823 0.03897 0.66 W6.69 0.69631 0.29255 1.90883 0.04512 0.58 50.97 0.7h278 0.34404 2.04222 0.05041 0.5 56.89 0.80000 0.41997 2.20998 0.05828 0.46 60.93 0.83L406 0. 47434 2.31263 0.06L4173 0.42 66.32 0.87287 0.54540 2.43395 0.07262 0.38 74.69 0.91766 0.66956 2.58601 0.08785 0.36 97.25 0.94281 0.98565 2.77520 0.14837 0.38 105.99 0.91766 1.09093 2.76968 0.18416 0.k2 115.44 0.87287 1.18463 2.72505 0.23502 0.46 121.96 0.83406 1.23412 2.67701 0.27950 0.5 127.16 0.80000 1.26371 2.63185 0.32143 0.58 135.45 0.74278 1.12924 2.55291 0.40169 0.66 142.15 0.69631 1.29953 2.48730 0.47925 0.7 145.12 0.67612 1.29854 2.45854 0.51723 0.8 151.82 0.63246 1.2880L4 2.39577 0.61073 0.9 157.75 0.59628 1.27074 2.34315 0.70194 1.0 163.17 0.56569 1.24977 2.2979k4 0.79124 1.1 168.21 0.53936 1.22692 2.25831 0.87839 39 Table B.3 (Continued) ¢ x/b z /b h/a V/a® 1.2 172.98 0.51640 1.20312 2.22293 0.96372 1.2 175.29 0.50596 1.19105 2.20652 1.00587 1.5 186.20 0.46188 1.13075 2.13396 1.208L46 1.7 194 .51 0.43386 1.08367 2.08375 1.36230 .0 206,81 0.40000 1.01557 2.01557 1.57998 .5 229,01 0.35777 0.90542 1.90671 1.90695 2.5 310.00 0.35777 0.714k42 1.69317 1.81377 2.0 331.06 0.40000 0.7340L 1.73k401 1.47971 1.7 32,67 0.43386 0.76316 1.78825 1.27326 1.5 250.54 0.46188 0.79225 1.84081 1.13175 Table B.k. Lo Size, Shape, Pressure, and Volume of Bubbles and Drops with Given Radius of Attachment r/a = 0.6 B 3. x/b z /b h/a. V/a® 60.0 6.98 0.11094 0.00647 0.21801 0.01748 30.0 9.75 0.15492 0.01266 0.3072k 0.02813 18.0 12,64 0.20000 0.02120 0.39692 0.03656 12.0 15.54 0.24495 0.03198 0.48658 0.04515 8.0 19.17 0. 30000 0.04838 0.59676 0.05604 5.5 23.35 0.36181 0.07133 0.72131 0.06879 4.0 27.72 0.42426 0.09970 0.84811 0.08255 3.5 29.83 0.45356 0.11501 0.90807 0.08935 2.9 33.16 0.49827 0.14101 1.00025 0.10029 2.3 37.95 0.55950 0.18242 1.12813 0.11668 2.0 41.31 0.60000 0.21408 1.21407 0.12865 1.8 L4 .15 0.63246 0.24231 1.28397 0.13910 1.7 45.82 0.65079 0.25954 1.32394 0.1454, 1.5 49.88 0.69282 0.30298 1.41709 0.16135 1.2 57.15 0.75895 0.38608 1.57014 0.19222 1.2 59.11 0.7T460 0.40935 1.60808 0.20112 1.1 63.92 0.80904 0.46757 1.69516 0.22420 1.0 70.86 0.84853 0.55331 1.80546 0.26115 1.0 110.89 0.84853 0.97076 2.10065 0.61473 1.1 119.62 0.80904 1.01944 2,104k 0.7h1L7 1.2 126.22 0.77460 1.04278 2.,09873 0.85283 1.2 129.07 0.75895 1.04934 2.09449 0.90534 1.5 1L0.75 0.69282 1.05581 2.06906 1.14754 1.7 148.28 0.65079 1.04488 2.04798 1.32506 1.8 151.67 0.63246 1.03679 2.37679 1.40998 2.0 157.90 0.60000 1.01784 2.01784 1.57280 2.3 166.26 0.55950 0.98579 1.98964 1.80255 3.0 182.08 0.48990 0.90845 1.92912 2.28289 3.5 193.87 0.45356 0.8557h 1.88796 2,58703 .0 204,25 0.42426 0.80597 1.84692 2.86294 6.0 258.64 0.34641 0.60326 1.62222 3.68271 6.0 281.23 0.34641 0.55639 1.54104 3.59171 4,0 332.13 0.42426 0.55032 1.48537 2.56096 2.0 251.28 0.48990 0.59863 1.54966 2.03835 b1 Table B.5. Size, Shape, Pressure, and Volume of Bubbles and Drops with Given Radius of Attachment r/a = 1.0 8 3. x/b z /b h/a v/a® 60.0 13.46 0.18257 0.01914 0.28739 0.17182 30.0 19.254 0.25820 0.03877 0.40835 0.24755 18.0 25,2k 0.33333 0.06585 0.53089 0.32807 16.0 26.91 0.35355 0.0745k4 0.56L438 0.35097 12,0 31.60 0.40825 0.10137 0.65655 0.41635 8.0 40.18 0.50000 0.15888 0.81776 0.54200 6.5 45,96 0.55470 0.20262 0.91998 0.63182 5.5 51.73 0.60302 0.24929 1.01642 0.72663 4.5 61.02 0.66667 0.32813 1.15886 0.89237 ) 69.09 0.70711 0.39734 1.26903 1.05200 3.7 77.81 0.73522 0.46919 1.37338 1.24390 3.7 103.15 0.73522 0.63176 1.59450 1.95007 b,o 113.93 0.70711 0.67052 1.65536 2.32888 4.5 125.33 0.66667 0.69002 1.T70L70 2.78302 5.5 140.90 0.60302 0.68546 1.73973 3.48414 6.5 152.52 0.55470 0.66411 1.75195 L.oskhe 8.0 166.51 0.50000 0,62550 1.75100 L, 76824 10.0 181.82 0. 44721 0.57493 1.73279 5,54361 16.0 220 .40 0.35355 0. 44901 1.62354 7.13667 20.0 253,01 0.31623 0.36935 1.48421 7.66733 20.0 286.54 0.31623 0.32147 1.33281 7.19870 16.0 316.08 0.35355 0.31169 1.23514 6.05930 10.0 348,47 0.kh721 0.35006 1.22997 b o208 42 Teble B.6. Size, Shape, Pressure, and Volume of Bubbles and Drops with Given Radius of Attachment rfa = 1.5 B 2, x /b z/b h/a v/a® 100.0 21.00 0.21213 0.03058 0.35762 0.83884 80.0 23.69 0.23717 0.03859 0.40220 0.94928 60.0 27471 0.27386 0.05210 0.46793 1.11632 40.0 34.97 0.33541 0.08054 0.58381 1.42593 30.0 41.80 0.38730 0.11119 0.68884 1.72815 2L.0 48.69 0.43301 0.14478 0.79022 2.04596 20.0 56.13 0. 47434 0.18271 0.894k02 2.40o6k49 18.0 61.81 0.50000 0.21185 0.96887 2.69503 16.0 70.85 0.53033 0.25677 1.07982 3.18127 18.0 124.73 0.50000 0439672 1.5234%9 6.89629 20.0 133.98 0.4743k 0.3944k 1.56354 7.6608L 24,0 148.02 0.43301 0.38033 1.60617 8.85724 20.0 163.84 0.38730 0.35409 1.62959 10.20717 50.0 200.61 0.30000 0.2791k4 1.59568 12.93770 6L .0 222,48 0.26517 0.23868 1.52697 13.97587 80.0 255.04 0.23717 0.19292 1.37824 14 .29483 80.0 284 .50 0.23717 0.16827 1.22238 13.20281 6k.0 313.1k 0.26517 0.16132 1.08935 11.13856 50.0 330.62 0.30000 0.16778 1.03892 9.65580 32.0 354,65 0.37500 0.19654 1.03617 7.76366 C. COMPUTER PROGRAM The Fortran listing of the computer program for solving the inter- facial equation and computing the various parameters presented in this report is given in Table C.1. The ranges of ¢, B, and r/a can be easlly extended at some sacrifice of either the running time or the accuracy. Other parameters, such as maximum drop height and radius as a function of drop volume and contact angle, can be obtained either by modifying one of the MAIN subroutines or by adding another routine. 43 Ly Table C.1., Computer Program O OO SOLUTION DF INTERFACIAL EQ. FOR THE PRESS. AND VOL. WITHIN ATTACHED BUBBLE. CASE 1-ATTACHED ABOVE (POSITIVE BETA) IMPLICIT REAL*B (D) 4REAL*4(A-Cy E-H,40-2) DIMENSIONA{(4) sB{4)3Cl4)sD(4)4E(8)R{4),G1(10}),4,P(4) DIMENSIONDA(4) 4 DB(4)4DC(4)4DZ(4)4DE(8)4DP(4),DR(4) 9 FORMAT (I1,12) 11 READ 9 ,41,4N3 IF (I-1)401,510,501 SOLUTION AT MAX ,PRESS FOR GIVEN VALUES OF R/A(=G) 300 FORMAT(F15.7) 10 CONTINUE DO 201 N&=1,N3 READ300,+6 ESTIMATE OF INITIAL BETA IF(GeGE«0.099 .AND.G.LT.0.4)G0T01001 IF(GeGE«0+39.AND.G«LT.0.8)G0TO1002 IF(GCeGEW0e79 e ANDG oL T«3.01)G0T0O1003 1001 BETA=411175-,92434%G+3.92521%G%% 2 GOTO301 1002 BETA=2,87267-12.78143%GC+16.40021%G**x2, GOTO301 1003 BETA=16 43044349 443188%G+40454628 %G 32, GOTO301 C COURSE BETA GRID 301 BETA=BETA+O041%BETA N20=0 P(2)=0 58 BETA=BETA-.O01=*BETA CALL MAINS(A4BsCsZ 9EsH sJ4sP +BETA,G4R) IF(P{1)-P(21))59,459,60 60 P(2)=P (1) N20=N20+1 GOT0O58 53 TF(N20-1)301,301,57 57 PRINT 50,+BETAsH PRINT 100 PRINT 2004,E(8)4E(6)4E(TIWR{1)4P(1),V BETA=BETA+.,02*BETA P(2)=0 C FINE BETA GRID 65 BETA=BETA-.O001*BETA CALL MAINS(AyBsCyZsEsHyJyP+BETAGHR) IF(P(1)-P (21161461462 62 P{2)=P (1) GOTO65 C OO0 OO, 45 Teble C.1 (Continued) 61 PRINT 504+BETA,H PRINT 100 PRINT 2004E(8)yE(6)sE(T)sR{1)4P (1)4V BETA=BETA+.002%BETA bP(2)=0 EXTRA FINE BETA GRID WITH DOUBLE PRECI SION DBETA=BETA DG =G 66 DBETA=DBETA-.0001*DBETA CALL MAIND(DA+DB+DC+DZ +DEsDH,J,DP,DBETA,DG,DR) TF{DP (1) -DP (2)1634634,64 64 DP{2)=DP (1) GOTO66 63 DV=3414159265%DG #*{DP (1) =*DG =DSIN(DE(5)})) DE(4)=DE(1)%57 ,29577957 DE(8)=DE(5)=57 .29577957 PRINT 50,DBETA,DH 50 FORMAT(IHZ2 46HPETA =43E15,7 46X y3HH =,E15,7///) 100 FORMAT(1HO 95X 9 3HPHI #8X ¢ 3HX /By 13X 3 3HZ /By 13X, 3HR/ Ay 13X, 3HH/ A, 113X 44HV/AZ) 200 FORMAT(1IH ,0P1lF943,1P5E16 .7} PRINT100 PRINTZ200,DE(8B),DE(6)}¢DE(7)+sDGC,DP (1) ,DV 201 CONTINUE GOTO 11 SOLUTION FOR GIVEN VALUES CF BETA AND R/A(=G1) 400 FORMAT(B8F10.7) 401 CONTINUE DO 202 N4=1,N3 READ 4004BETA,(G1(1),1=1,8) C,’—\LL ”AINIS(A,B,C 72 7E 7H 9J ’P ,BETA,G].,R ) 202 COCNTIMUE GOTO 11 SOLUTION FOR GIVEN VALUES OF BETA AT INTERVALS (N1) OF PHI UP TO N2 501 CONTINUE DO 203 N4=1,4N3 READ 5004+BETAyNLI,N2 500 FORMAT(F1074214) CALL MATINZ2S(A4BosCyZ yEsH 9 J 4P +sBETASZG4R yN1,N2) 203 CONTINUE GOTO 11 END L6 Table C.1 (Continued) O D O~ 150 SUBROUTINE MAINS{A4ByCosZ yEgH 43 J9eP ¢BETA,G4R) IMPLICIT REAL*8 (D) 4REAL*4(A-Cy E=Hy0~7) DIMENSTONA(4)yB(4)yC{4) 2 {4)yE(8) 4P (4)4R(4) H=0.1745329E-01 E(6)=0. J=3 E{(1)=0,. E(2)=0. E(32)=0, Nl=1 R(2)=0. DO150M=14360 CALL RUNGKS(AyRsCosZ yEyH s JyBETA4G,P) R{1)=E(2):SORT(BETA/2.,0) GOTO(445)4N1 IFIGR(1))543,3 TF{(R{1)-R{2))8B,48,10 IF(GR(1))146,7 N1=2 R{2)=R (1) E(5)=E(1) E{6)=FE(2) E(T)=E(3) CONTINUE F={G-R({(2)})Y/(R(1)-R(2)) E(B)=F==(E(1)-E(5))+E(5) (6 )=F3({Z{2)~FE(6))+E(6) LT ) =Fa(E(23)~E(7)Y)+E(T7) P(1)=SCRTI(?/EETAY+E(T7)*=SQRT(BETA/2.) GOTOS P(1)=1.0 P{(2)1=2.0 R ETUR N END SUBROUTIME MAIND(DA,DByDCyDZ 4DEsDH +JsDPy DBETA, DG4 DR) IMPLICIT REAL*B (D) REAL*4(A~-CyE-H,0-2) DIMENSTONDA(4) ,DB(4),DC(4),DZ (4),DE(B)4sDP(4)43DR(4) DH=0.2726646D=02 DE(6)=0. J=3 DE(1)=0. DE(2)=0. DE(3)=0. N1=1 b7 Table C.1 (Continued) OO, DO150M=1,720 CALL RUNGKD(DAWDByDCyDZ4DEyDHy JyDBEETALZ DG4 DP) DR{1I=DE(2)*DSART(DBETA/2.0) GOTO(445)4N1 IF(DG-DR({1}))5,150,150 TF(DG=DR (1)) 146,7 DR{Z)=DR (1) DE(5)=DE(1) DE(6)Y=DEL(2) DELT7)Y=DE(3) N1=2 CONTINUE DF=(DG-DR (2))/(DR(1)-DR (2)) DE(5)=DF*{DE{1)-DE(5))+DE(5) DE(6)Y=DF*={DE(2)-DE(6))+DE(6) DE(7)=DF*{DE(3)-DE(7))+DE(T7) DP(1)=DSQRT(2./DBETA)+DE(T)==DSQRTI(DBETA/2.) RETURN END SUBROUTINE MAINIS{A,BsCyZ yE4H J4PBETA,G1,R) IMPLICIT REAL=8 (D) REAL*4{A-CyE-H,0-7Z) DIMENSIONA(G) ,B(4),C(4),Z(4),E(8)4P(4)4R(4),G1(10) H=(6726646E-02 E{&6)=0. OO0 OO0 . 4 9 e n o #n Zmmm>Pc = b — I=1 DO 150 M=1,720 CALL RUNGKS(AsByCsZ+EsHyJyBETALZG1,4P) R{1)=E{z2)*SQRT(BETA/2.0) GOTO(445422)4M1 ITF(GI(IY-R{1))T7 46,3 IF(RT1)-R{2))10,10,414C I=I-1 M2=-1 hl=2 IF(GIIII-RI(1)}20 46,7 TF(RI1}Y-R(21)140421,21 I=1+1 L3 Table C.1 (Continued) Oy Oy O O 272 N2=1 N1=3 IF(GI(I)-R (1117 464140 T F={(GI(I)-R(2))/(R{1)R(2)) E(L)=F={E(L)-E(5))}+E(5) E(2)=F*x{E(2)-E(6))+E(6) E(3)=F=(E(3)}-E(T7)})+E(T) 6 P{1)=SQRTI{2./BETA)+E(3)*SQRT(BETA/2.) V=3,14159265%G 1(I )= (P (1}*G1{I)=SIN(E(1))) El4a)= E(1)%57 429577957 50 FORMAT(1HZ y6HBETA =4yE15.7 46X,y 3HH =yE15.7///) PRINT 504BETAH PRINT 100 PRINT 200,E(4)yE(2),E(3)sG1(I)4P(1),V 200 FORMAT(1H 40P1F9.3,1P5E16,7) 100 FORMAT(1HO,5Xy3HPHI 38X ¢3HX /By 13X y3HZ /By 13X43HR/A,13X43HH/A, 113X 44HV/A3) I=1+NZ2 IF(I-8}130,130,3 130 TF(I-11)941404140C 140 R{2)=R (1) E(5)=E(1) E(6)=E(2) E(7)=E(3) 150 COMTINUE S RETURN END SUBROUTINE MAIN2S(As4ByCoeZ sEgH 9 J9yP +BETA,G4RyN1,N2) IMPLICIT REAL®B (D) yREAL*4{A-CyEH ,0-2) DIMENSTIONA(L) 3B (4) 3C(4)9Z(4),E(8)4P(4) 4R (4) 101 FORMAT(1HQ 35X y3HPHI 48X 33HX /By 13X y3HZ /By 13Xy 3HR/ Ay 13X 3HH/ A, 112X 44HV/A3) 50 FORMAT(1IH]1 sSHBETA=9E15.7 96X 92HH=yEL5.7///) 200 FORMAT(1H s0P1F9.341P5E16.7) H=0s1745229E-01 PRINT 504BETA,H PRINT 101 E(6)=0. J=3 N2=N2/N1 M3=,1746E-01/H M1=N2%N1 E(1)=0. E(2)=0. E(3)=0. DO 150 HM=14N2 OO O OO0 100 150 20 50 30 60 49 Teble C.1 (Continued) DO 100 N=1,N1 CALL RUNGKS(A4ByCoZsEsHyJyBETALG,P) CONTINUE G=E(2)*SQRT(BETA/2.,0) E(6)=E(2) E(7)=E(3) P(1)=SQRT(2./BETAY+E(T)*SQRT(BETA/2.) V=3e1415933G*(P(1)*G-SIN(E(1))) E(4)=E(1)%57,29577957 PRINT 2004+E(4)4E(6)4E(T)4G4P(1),V CONTINUE RETURN END SUBROUTINERUNGKD(DA,DB,DC,DZ ,DE,DH,J,DBETA,DG,DP) IMPLICIT REAL*B (D) sREAL*4 (A-CyE-H,0-Z) DIMENSIONDA(4)4DB(4)9DC(4),DZ(4)4DE(8)4DP(4),DR(4) DA(1)=DH*0.16666666667 DA(2)=DH*0433333333333 DA(3)=DA(2) DA(4)=DA(]1) DB(4)=0. DB(1)=DH*>0e5 DB(2)=DB(1) DB (3}=DH DO2CI=1,J DC(I)=0. DZ(I1)=DE(I) DO30K=144 CALL RHOD{(DA,DB,DC,DZ yDE,DH,J,DBETA,DG,DP) DZ{li=1. DO501 =1,J DC(I }=DC(I)+DA(K)=*=DZ (I ) DZ(I)=DE(I)+DB(K)=*DZ (I} CONTINUE DC(1)=DH DO60I =144 DE(I)=DE(I}+DCI(I) RETURN END SUBRGUTINE RHOD (DA,DByDC+DZ4DE,DH,J+DBETA,DG,DP) IMPLICIT REAL*8B(D)ysREAL*4 (A=CyE-H,0-Z) DIMENSTONDA(4) 4DB(4)4DC(4)yDZ(4)4DE(8),DP(4),DR(4) 50 Table C.1 (Continued) OO OO OO ™ 70 75 76 20 10 75 16 IF(DZ(1))70475470 DRH=140/{2.0+DBETA=DZ (3)~DSIN(DZ(1))/DZ(2)) GOTOT76 DRH=140 DZ(2)=DRH=*DCOS(DZ (1)) DZ (3)=DRH*DSIN(DZ (1)) RETURN END SUBROUTINERUNGKS(AsByCsZ yEsH 9y J+BETA,G,P) IMPLICIT REAL®B (D) yREAL*4(A-Cy E-H,0-2) DIMENSIONA(4) 9B(&) o Cl4) 9Z {4) 3 E(8)4P (4) 4R (4) All)=H*0.16666666667 A(2)=H*0433333333333 A(3)=A(2) Al4)=A(1) B{4)=0. B(1l)=H*0.5 B(2)=B(1) B{3)=H DOZ201=14J C(I}Y=0. Z({I)=E(I) DO30K=1,+4 CALLRHOS (A 4B 4CeZ yE9H s J9BETALG,4P) Z(1l)=1. DO50I=1,4 CII)=CIlI }Y+A(K)=Z{ Z(I)=E(I)+B(K)=Z | CONTINUE Cl1)=H DO60CI =1,4J E(IY=E(I)+C(I) RETURN END I) 1) SUBROUTINERHOS({AyBaCyZ sEsH s J4BETA,G4P) IMPLICIT REAL*8 (D} sREAL*4(A-CyE-H,0-2) DIMENSIONA(4) 9yB{4)3Cl4) 32 (4),E{8) 4P (4) 4R (4) IF(Z(1))70475,470 RH=1.0/(2.,0+BETAXZ(3)-SIN(Z (1)) /Z(2)) GATO76 RH=1.0 Z(2)=RH=*=COS{(Z (1) Z(3)=RH=SIN(Z (1) RETURN END ) ) \O CO—3 O\ B o o e W N H O 1k, - - ?99o@@gwypgywzbg?smOzmzwbb>quzgmmwmmmmzqommm2002@ 51 ORNL-TM-3528 INTERNAL DISTRIBUTION Alexander . Anderson . Baes . Barton . Bautista . Beall Bettis . Blankenship . Boyd HEOEHGLOGHGQQ . Braustein . A. Bredig . B. Briggs . R. Bronstein . D. Bundy, K-25 . Cantor . Chapman . Claiborne, Jr. . Compere . Cooke . Cottrell . Crowley . Culler . DeVan Dworkin Eissenberg Fraas Ferguson . Fontana Freidman . Furlong . Gabbard Gallsher Gambill Gilpatrick . Grimes Grindell . Harms Haubenreich . Hitch . Hoffman Hurtt Kasten . Kedl . Keyes, Jr. Kidd, Jr., K-25 . Kirslis Klepper . Koger . Kolb Korsmeyer 9:502meCqC-l':UQE!'ijZOQ'JUO'JUUijbmm*d!gmmt“t“bflflt"&im 61. 62. 63, 6k, 65. 66. 67. 68. 69, 70. 1. 2. T3 h. 5. 76. T 78, 79 80. 81. % 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. Krakoviak . Kress . Krewson Lackey . LaVerne TLawson . Llewellyn Lloyd . Lundin Lyon . MacPherson . Mauney . McCurdy . McElroy . Mcliain . McNeese . McWherter . Miller Mixon . Moore Parsley . Perry 1dkow1cz Redmon Richardson Robbins Romberger . Rosenthal Ross * ':.’)€'J>UZU‘TJZ"!:IL—‘WQWMPHOEHZHtflmQMFfizmH QUWERQUGGPHI R PO DNUNAYYNRUQAR GG 1 o o B D = n J P. Sanders W. K. Sartory H. C., bavage Dunlap Scott J. H. Shaffer Myrtleen Sheldon J. D. Sheppard M. J. Skinhner . Spiewak . Sunberg . Thoma . Thomas . Trauger Tobias . Wantland . Watson . Weaver . Whitman . Wichner . M Yarosh EHQ QGO EU0HNYH WO D E Qe 111. 112. 113-11kh, 115. 116-117. 118. 119-121. 122-123. 124h-125, 126. 127. 128, 129. 130. 131. 132-133. 13k, 135-139. 52 J. P. Young H. C. Young Central Research Library Y-12 Document Research Section Iaboratory Records Laboratory Records - Record Copy EXTERNAL DISTRIBUTION Director, Division of Reactor Licensing, AEC, Washington Director, Division of Reactor Standards, AEC, Washington Division of Technical Information Extension (DTIE) Laboratory and University Division, AEC, ORO D. F. Cope, RDT Site Office, ORNL A. R. DeGrazia, AEC, Washington Ronald Feit, AEC, Washington Norton Haberman, ARFC, Washington Kermit Laughon, RDT Site Office, ORNL T. W. McIntosh, AEC, Washington R. M. Scroggins, AEC, Washington Executive Secretary, Advisory Committee on Reactor Safeguards, AEC, Washington