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Scholars' Mine Masters Theses Student Research & Creative Works 1950 The application of the relaxation method to the solution of steady-state heat transfer through a concentric square insulator William Edward Simpkin Follow this and additional works at: http://scholarsmine.mst.edu/masters_theses Part of the Mechanical Engineering Commons Department: Recommended Citation Simpkin, William Edward, "The application of the relaxation method to the solution of steady-state heat transfer through a concentric square insulator" (1950). Masters Theses. 4896. http://scholarsmine.mst.edu/masters_theses/4896 This Thesis - Open Access is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Masters Theses by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. THE LIC TION OF THE TION THOD TO THE SOLUTION OF STEADY-ST TE HEAT TRANSFER TH 0 TGH C TRIC SQ, I SULATOR. BY vITLLIAM EUNARD SDlPKDJ THESIS submitted to th SCHOOL OF MINE • faculty of the D vIETALLURGY OF 'lm UNIVERSITY OF SSOURI in partial fulfillment of the work required for the Degree of MASTER OF SCIENCE IN MEGHANI CAL ENGINEERING Rolla, Missouri 1950 ii ACKNO OOEMENTS The author is indebted. to Dr. A. J. Miles for the suggestion of and the assistance in the solution of this thesis problem. The author also wishes to take this opportunity to express his appreciation to various members of the faculty for their marginal assistance and to relate his gratitude to his friends and loved ones for their encouragement and sponsorship. 111 TABLE OF COnTENTS Acknowledgements ...............•.. List of Illustrations Page ii iv List of Tables.................... v Introduction....... 1 Statement of Problem...... 7 Review of 1Ji terature. .. . . . .... .. .. 9 Origin of Relaxation Method 10 Di s c u s s ion. . . . . . . . . . . . . . . . . . . . . . . . 12 Initial RssQTIptions 13 Heat-Flow Evaluation 26 Results of Problem Solution......• 30 Conclusions •...................... 31 Discussion of Solution Accuracy .•• 32 Summ.ary. . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Applications for the Results •..•.. 35 Bibliography.................•••.• 36 Vi ta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 -0- iv LIST OF ILLU TRATIONS Fig. 1. Page Sketch of Infinitesimal body in Rectangular Co-ordinates ......•........... 2. 2 A Plot showing the eight symmetrical portions of concentric squares ...•.•..••.. 14 3. One-eighth Section for Case I' ............. 15 4. One-eighth Section for Case II ............ 18 5. One-eighth Section for Case III ........... 20 6. One-eighth Seotion for Case IV ............ 22 7. One-eighth Secti on for Case Plate 1. V ••••••••••••• 24 Page Cs in terms of the ratio of the width of the outside to the inside square .....•. 30 v LI 3 T OF TJJ3L:::S Table I. Tabulation of the Relaxation f 0 l~ I.................... 17 Ca s e Tc:bulation of -etc; ::'(-Jlaxation T"" _ . ...1..- • fOl'" Case II Tabulati on of III. 19 tJH:~ Rel::.x'3tion for Case III IV. Tabulation of for V. Page 'J3se t~e ~elaxat~on TV................... Tabulation of the fo:(' 'Jas"" VI. 21 ~elaxation V.................... Determin:J.tiol] of 23 TJeb 25 t :1'low from Temperature Distribution 27 VII. Determination of Seat Flow through Concentric Conductors ........• VIII. Tabulati on of OS" • • • • • • • • • • • • • • • 28 29 INTRODUCTION 1 The basic equation for the flow of heat by conduction is given by Fourier's law which states that the instantaneous rate of heat flow, ~, is equal to the product of three factors: the area normal to the path of heat flow, A; the temperature gradient, dt; which is the rate of chan e of tempercirx ature, t, with respect to the length of path, x, in which the heat is flowing; and the dimensional constant, k, the conductivity of the conducting medium. Fourier's law expressed mathematically is as follows: ~. -kill dQ (1) dx where dQ is the amount of heat flowing in a differential time dQ, adjusted to indicate positive heat flow in the direction of decreasing temperature by the minus sign. Z dz x dx Fig. 1. Sketch of infinitesimal body in rectangular co-ordinates. The rate of heat flow, ~ hereafter referred to as dq, dQ into the element in the x direction according to Fourier's law is (see d. ig. 1.) i" =- K" ely J. 'L (~) 2 The heat flowing out of the element in the x direction is r~t ~ (;;>t)J.: t1 cl tx"=-1<..c:Lyclz Lix +~ ~ ~J . The net gain or the amount of heat stored in the element is the difference between the two. J.'t,,:: .,li: - J.f: = - \<." ~ "'J[ f)( ~).,l~ In the same manner it may be shown that the heat stored due to heat flow in the y direction is tr =- k..'( .,luI. z [~ (~ ) tA:t] and in the z direction ;)t')tA. ,1 All. = -""..l)( J.'( [-h, (n ZJ. .A. The rate at which heat is being stored in the element may then be expressed as I :. e>t...:; .l"", + J..1'(+ ~"7. ~ k,.J.'( J.1 ~(I<.~)+~(Ky~)+?i(~ ~ ~)~ . T is is equal to the rate of heat accummulation within the element or • erel' is the density, and c is the specific heat of the material. Therefore we may equate the two expressions with 1 cl"c:lycl.zpc Z9 =,l)(J.ytA.7.[~'c''SXtL+ K~~" + \<.f. 'Sl"J the following e;rttion resul tinge ~~t ;)-t ~'t If the conductivity is the same in all directions, then the above expression can be reduced to ~t I<. (~t ~'a.t J'lt") ~:',oc. where -!Pc ':)(.. +- W +~ (2) is a dimensional constant for each material and is designated by the symbol oC, and is called the thermal dif- 3 fusi vi ty. In the steady state condition of heat transfer, temperature does not vary with time, consequently ~ is zero. There- fore we obtain ~~t ~ ){~ + ~"t: +~. . t" ~ "'(~ ~ 7.. :. 0 (3 ) • which is the general equation for he~t conduction in the steady state, for a homogeneous medium. In the problem at hand, the medium is homogeneous and the flow is two dimensional and steady, hence equation (3) reduces to , Equation (2) is a general quation for all heat-conduc- tion problems in homogeneous media, and is the result of applying Fourier's law·(equ~tion 1). Therefore it may be seen that the solution of all conduction problems re~uires a solu- tion of this differential equation which will meet the boundary conditions. This solution may be simple as in the cas of one dL,ensional hest flow in the steady state, and in two and three dinlensional steady-state heat flow involving circles or spheres. However for most irregular shapes the so- lution of this differential equation has not been made because of the difficulty involved. s a result of this oiffi- culty, recourse has been made where possible to graphical, statistical, experimental, or analogous methods. Graphical analysis was one of the first attempts for the solution of the difficult LaPlacian equation. Early work was done by Binder(l) in the field of electromagnetism. This 4 (1) Binder, L. Munich, 1911 was the forerunner of the Scillrridt(2) graphical analysis meth(2) Schmidt, E., Berlin, 1924 ode The methods as proposed by Moore(3) in the field of elec- (3) Moore, A. D.; Fundamentals of electrical design, McGrawHill Book Co., Inc. 1927 tromagnetism give workable La lacian solutions, and valuable data on thermal properties of heterogeneous systems. The text of Me dams(4) is an indispensable reference in the heat (4) !eAdams, V. H., Heat Transmission, cGraw-Hill Book Co., Inc. 1942 transfer field and presents some results obtained by graphical analysis. Closely allied with the graphical methods are the analogous methods. There are several under the classification of electrical analogies. A network type of analogue was devel(5 ) oped by Paschkis and aker (5) Paschkis, V. and aker, H. D., Transactions primarily for transient heat .S.N..E. Vol. 64, No.2, pp 105-112 (Feb. 1942) flow, but could undoubtably be used for determinations of steady-state conditions. In this analogue the heat-flow path, whether one-, two-, or three-dimensional, is represented by a number of electrical resistors arranged in the form of a 5 built-up network to simulate thermal resistance of the original form. The flow of direct current through the network is equivalent to the heat flow conditions. Langmuir developed a geometrical analogue in 1913 which utilizes an electrical conducting bath in a shallow tank for model study of various heat-flow shapes. electrolyte represents the 8i gl ial. conduct'ng liquid isotropic homogeneous mater- With an alternating-current electrical potential est b- lished between the parallel wall electrodes, the equivalent of steady-state heat-flow conditions between constant temperature walls was set up. By means of an electrical probe, isopotential l±nes representing isothermal lines of the 801id body could be established. An electrical geometrical analogue was presented by Kayan(6) in 1945. This method is of the geometric type pri- Kayan, C. F., Transactions .S.M.E. Vol. 67, No.8, pp713-718 Nov. 1945 marily designed for steady-state one- and two-dimensional studies. Conductive sheets of various electrical resistanc- es were used simulating different thermal conductivities of diff rent materials in a memb r to be studied. The basic electrical analogy of this method is the same as the Langmuir method; equivalent temperature conditions f r a heat-flow path can be determined through analysis of an electrical-flow path io which the component resistances shi _ bet'.'Jeen theoselves as the t~ler!c3.1 h~v~ the swne relation- resist,nces. An elec- a~d tricel probe is also used a "contour map" of isopotential lines (vlhich re:,r83cnt isotherr,lal lines) is drawn. It "Jas tor e. :G)Cint8(~ out ::,y :Prantl, in 1903, th9.t the equations l}'J::lo~e11eou~:lelilbrQne are of trlc LaFlacian form. This 6i ~;covor:)T was I113.de as a result of hi s attempts to find a ,solution for the equat~0ns of torsional stress analysis. 1937, Christopherson and Southwell (7) Christopherson, l. ~. ( '1 ) In indicated that every and 30uthwell, .• V., Proceedings os the Royal 30ciety of London, Series A, Vol. 168, pp 317-350 (1938) plane-potential problem has a membrane analogue and that PrantIts analogue is merely a particular example. Southwell(S) (8) Ibid applied this analogy to two-dimensional heat-flow problems. The deflection of the membrane represents, to some scale, the temperature distribution throughout the member under consideration; that is, the hole in the plate over which the membrane is stretched must be geometrically similar to the shape of the member under investigation, and the differences in elevation of the various parts of the model must represent, to some scale, the temperature change between those parts. The boundary conditions must be known and the model constructed accordingly. STATEMENT OF PROBLEM 8 It is the object of this thesis to express the relationship between the heat transfer through a square insulator of variable inside-outside dimension ratios, in terms of a concentric round insulator of the same inside-outside diameter ratio, by means of a continuous curve. athematically speak- ing, this thesis will have a plot of a variable coefficient, cs' to be used in the following relationship, in terms of the ratio of the width of the outside square to the width of the inside square. REVI:> OF LITERATURE 10 The relaxation method, basically a trial and error procedure, is a means of satisfying a set of given boundary conditions, and internal characteristics for plane- or spacepotential problems. The earliest foundations for the relax- ation method were made in 1932 by cross(9) in the field of (9) Cross, Hardy, "Analysis of continuous frames by distributing fixed end moments." (paper No. 1793) Trans. Am. Soc. Civ. Eng. 96 (1932) stress analysis. Shortly thereafter in England, Southwell(lO) (10) Southwell, R. V., stress-calculation in frameworks by the method of systematic relaxation of constraints. Part III. Froc. Roy. Soc. of Lon. A, Vol. 153 pp 41- 76, (1935) devised a system of 'relaxing' the load from one support on a continuous bema for the solution of the stress in the beam --from which the relaxation method derived its name. Inde- pendently, Cross(ll) used a similar nQ~erical method of 80- (11) Cross, Hardy, Analysis of flow in networks of conduits or conductors. Univ. Ill. Bull. No. 286, (1936) lution for a pipe-line network. Throughout the late thirties application was made in this country and in ·ngland of this method for 'indeterminate' mechanics problems. The first application of the relaxation method to heat transfer was made by Emmons(12) in 1943. In his words, "Ex- 11 (12) Emmons, H. ~., The numerical solution of heat-conduction problems, Trans. A.8.M.E., Vol. 65, No.6, pp 607-615 (1943) cept for the simplest geometric shapes and boundary conditions, the relaxation method is far superior to analytical methods of solution in point of tiffie required to reach a given desired accuracy. The relaxation method has the further adv nta es of permitting the calculator to put into the calculation all the physical intuition he ~ay have about the problem and, at the same time, to know at each step just how seriously his solution still differs fro~ the correct answer. By an exten- sion of the nethoc, one-, two-, and three-dimensional transient-heat-flow problems are easily solved. ~or one-dimension- al problems, the relaxation method is identical to the graphical :nethod of Schmidt. ,,(13) tl5) Schmidt, E. op.cit. P 4 In the words of the writer, the relaxation method is a trial and error guessing for which the right guess can be identified, and then a long arm process of manual summation as analo ous to mathematical integration. However, regardless of the labor involved, it is believed that the relaxation method is an invaluable tool and will solve considerably more practical problems than the claculus except to the esoterica of higher mathematics. DISCUSSION 13 In order to develo a curve for the value of s as d scribed in the statement of the problem, it was decided that C five yoints would be a minimum to determine the curve. To deternline the value of these five points, separate solutio s for each case was necessary. It was arbitrarily decided to give the inside width of each square a value of twelve units, and in the five cases, have a conductor thickness of one, two, three, four, an six units respectively. This gave an out- side-inside ratio of widths of 1.167, 1.333, 1.500, 1.667, and 2.000 respectively. As may be seen from symmetrical portions. mine th ig. 2, a concentric square has eight Therefore it was necessary to deter- temperature distribution in only one of these sections to evaluate the heat flow by the relaxation method. one thousand degree temperature difference between the inside isothermal surface and the outside isothermal surface was arbitrarily chosen for its convenience in calculation. Initial .ssumptions: 1. teady-flow condition. 2. Constant conductivity, at k 3. On 1 Btu/hr-ft-oF. foot axial len tho Case I, a conductor ig. 3. * 0 unit thickness is illustrated in The accompanying relaxation is given in Table I. In the other four cases, the descriptive sketch ane its relaxation follow. Since the value of ~ is equal to one, the heat flow through any portion of the network to a given point is equal 14 '. - f----------- - ------4------ T I I Figure 2. A Plot showing the eight symcetr10al portions or oonoentrio squares. 15 to the summation of the temperature differences. Under stead- y-flow conditions, no heat maybe stored or taken from any internal point. Therefore the temperature distribution may be obtained by giving the point temperature~ such values that the net heat flow, Q, at 3ny point is equal to zero. For the purposes of this problem, considering the one thousand degree overall temperature difference, the relaxation was continued until the Q at any point was within tlO or less, thus giving an accuracy of ~O.025% on the points considered. This how- ever can not be considered to be the accuracy of the determination, since an infinite number of points would have to be relaxed to obtain this accuracy. Isothermal 1000· a -. b 1 2 3 4 5 6 Isothermal 0° 789 o 10 11 12 13 14 15 Figure 3. One-eighth Seotion for Case I. ..... oa TABLE I Tabulation ot the Relaxation tor Square with X2 - 1.167, as per Fig. Xl Points b2 Q. t 0 0 0 0 220 215 212 211 t Q. 440 430 424 422 -80 -20 -4 1 b5 b4 b3 t Q. 85 -20 -6 0 460 485 480 479 t Q. 12 16 1 485 490 494 t Q. b8 b7 b6 11 492 3 496 1-498 Q. t 11 494 6 497 1 499 Q. bID b9 t 11 495 3.5 498. 2 499 ~ t Q. 8 496 6 497 1.5 498.5 1.5 499 2 499 2 499 Q. t Points bll Q. b12 t 2.2 499 b13 b14 Q. t Q. t 1.7 499.2 1.2 499.5 -.5 500 Q. t b15 Q. t o 500 .... .....:) Isothermal 1000 o a. _.- - £ 1 1 2 - 3 4 Figure 4. _----- ._- -- -. ---- -- -------- 0 b ------------- _ Isothermal 56 One-ei~hth .. o 7 8 c 9 Section for Case II. ~ CD TAm.E II Tabulation of the Relaxation for Square with !2 • 1.333, as per Fig. Xl Points b2 t Q 0 190 0 205 0 207 b3 Q 95 10 1 t 380 410 414 b4 t Q 65 425 15 445 1 450 b5 Q -22 1 -2 t 485 487 b7 b6 Q 9 -1 1 t 493 495 Q 7 3 t 496 498 b8 t Q 4 498 2 499 + -b9 Q -4 t 500 ..., 1lO Isothermal / 1000 o I ---; a ~--------------_. ,/ .. ~ -.... -- - . b / / o ~ 1 // // I. 2 :3 4 5 Isothermsl 6 O· 7 • 8 "d 9 10 Figure 5. One-eighth Section tor Case III N o I\l Points 02 Q 0 0 -2 -2 0 2 t 80 85 88 90 03 t Q -5 160 30 0 175 5 11 -5 179 -3 0 1 3 4 0 3 -1 0 05 04 Q, t 67 25 -4 3 225 240 255 -4 -1 10 2 -2 -1 258 e 180 0 181 1 2 260 261 3 4 5 1 2 262 Q 20 35 -3 4 7 -5 0 2 4 5 1 3 4 0 1 05 t 272 280 297 Q, 301 302 30 8 3 8 12 15 -5 -2 0 1 2 3 303 0 300 07 t 293 305 315 320 4 321 Q, 20 12 28 -3 1 6 -6 -5 -3 0 1 09 08 t 305 310 313 325 328 Q. -5 3 -4 -2 0 3 -1 0 t 325 Q. t 3 330 332 0 330 331 010 1 2 Q, -5 t 333 Isothermal ~,' /iL__ - _.' / 1000 - ._ _ ~ I I , / : I I i -~----: I I I - I -_. . ~.---,--_.- '--'r--- ~-T---- I //' --lJ " i c I , I I I I • J I i l l 2 j, ---_., --i /------ 1 b t I I I _ "I I ~ I Ii ._- a Ii I : L.t I 1--. 0 Z 4 I 5 Figure 6 ~ I Isothermal 7 ' j 0 8 o J I I J i I : I ,. i 9 10 11 d e 12 One-eighth Section tor Case IV l\) {\) TABLE IV i 12 • Tabulation of the Relaxation tor Square with Points b4 Q, -162 -40 0 70 40 60 0 5 10 6 -? -3 1 -2 t 503 480 480 440 455 475 480 482 485 483 481 480 Points 04 Q. -23 -47 33 43 -1 14 1 6 2 -12 -8 -2 -3 t 320 300 310 315 317 318 315 312 311 60 71 100 46 23 0 1 0 -2 -3 -1 t 80 70 85 95 100 104 103 t 655 620 Q, -14 -44 53 83 ?8 -42 8 5 -I? -2 -3 --1 610 630 660 650 652 656 650 649 648 Q 140 90 _ 100 50 40 22 10 4 -1 -73 -13 -6 -3 -2 112 72 52 30 17 5 1 -1 -2 -3 0 t 95 235 250 130 115 40 10 0 -5 -5 -2 -3 680 650630 660 680 700 705 708 ?10 712 710 709 t 380 400 405 412 418 421 423 Q -2 33 31 2 -3 5 -1 -2 ?10 730 700 710 715 ?30 732 53 47 47 35 35 0 -5 -2 9 7 5 3 1 731 t 450 455 453 454 453 Q, 140 100 40 30 20 10 3 -2 -16 -9 -1 d5 195 198 200 198 196 194 Q. 22 0 3 -2 9 1 -1 735 740 741 b10 t 742 745 Q 0 1 -1 4 1 2 t Q, 749 -6 -3 -2 t 430 440 460 465 470 4?5 478 480 t 488 Q, 2 10 6 4 6 -3 -2 490 t 10 20 40 25 18 7 2 0 -2 0 480 t 498 Q. -1 0 t Q -8 500 330 180 190 195 200 207 210 211 210 209 20? d2 Q -40 -20 30 10 6 0 -2 485 488 492 494 495 t Q. ell 010 Q 03 ?50 -560 40 20 30 24 8 2 +2 -2 0 -2 -3 0 -4 746 09 ~ t Q. ?42 08 - b11 '7.32 t 50 40 35 45 47 52 478 t 215 220 222 220 222 221 Q. -23 2 0 2 -3 -2 t 240 -235 236 d9 d8 d7 d6 t 192 b9 t 733 737 Q 8 15 -15 3 8 3 0 2 0 07 410 405 403 402 Q t 120 -10 0 130 3 140 10 147 3 152 -1 154 8 153 -3 1 0 152 -1 b8 t Q, 06 d4 Q, b? Q, 05 Points d3 Q, b6 b5 1.66?, as per Fig. Q 4 -1 2 iTl4 0 t 242 243 Q. 8 1 2 dl0 t 244 246 Q 4 2 t 247 248 dl1 Q. -8 t 250 1'> u. /:I , . : 1000 0 Isothermal I I I' , . I , I r-j ; -.-r----.------ -r --. a --. - ~ b ; I , ; i . I I I I i - - i i ---" - ~. I I I - - I - - - - --- ---'-1- - I c ______: . ----r----"----II d - , . I " /11 --- -r---------- - ----. // 1 I I t : .. /r--~II -ji--T-t--r--r--~ I I I 2 3 4 5 6 7 I I l I . ! I I I I , I ; I I 1 e - - - . -..--.- I _. . I ;t 8 ' , I Isothermal 1 --- - "-------- .-----J-_ _. i I .J -- --- -- -- - I , - I I I, _ _..i.-,_ _ " ----._- I I .r _ --- +_. -- - - I ...._~-~ 'I i I 1 __ t g 00 9 10 11 12 13 Figure 7. One-eithth Section for Case V. l\) ~ TABLE V Tabulation ot the Relaxation tor Square with !2 • 2.000, as per Fig.? X1 Points B6 t 590 610 615 Q. -80 20 10 5 -4 -4 613 611 b7 t 480 490 Q. 45 15 -35 -5 -10 -9 -4 -5 -7 485 483 480 477 t 740 Q. -5 20 0 5 0 -7 -1 -3 Points 06 745 t 790 795 Q. 17 -3 -7 3 0 -4 07 793 bl0 t Q. 1 2 0 -2 -4 815 b12 bll t Q. 1 -2 -1 -3 t Q, 575 -13 -23 -28 -7 -12 -17 -2 -7 -8 -6 -8 09 08 570 565 563 561 t 622 Q. -5 -11 -16 0 -5 -10 -15 -10 -2 -2 -6 825 t Q. 7 828 3 .829 2 618 615 612 611 410 a 349 640 638 -9 -3 -5 406 t 662 Q, -2 -5 -4 -3 80 81 85 10 0 6 Points e12 t 330 Q, 3 t 120 Q. 140 60 5 7 132 80 14 9 4 6 3 1 2 Q -5 t 455 450 446 t Q 478 6 -7 -12 2 -3 -8 -4 -4 -5 475 472 471 e7 t Q 240 5 -45 -25 -5 -5 235 ~30 227 t3 t 20 333 -10 0 2 Q -15 -10 -2 -5 -4 -5 -6 t 10 265 10 3 263 2 1 Q, t 492 490 488 486 Q, Q 20 0 5 36 19 4 3 140 145 150 152 Q -8 10 -2 -1 t 156 159 Q 2 -1 t 300 240 220 210 208 206 204 495 0 -2 -4 -3 -4 t 35 40 41 t 60 Q -5 6 5 64 t13 t Q t 162 163 :3 164 165 0 Q -5 t 167 Q, 20 6 1 :3 d13 t 498 Q -2 -3 t 500 Q -4 494 ----- Q 8 13 -7 -2 --- ell el0 t 308 310 309 t Q. -2 -6 1 322 t 328 327 Q. -3 -1 320 292 291 t6 t5 t4 t12 tIl tl0 t t Q e9 t 300 295 Q. -25 -12 -7 -12 -6 -2 -3 d12 d11 e8 Points t9 -5 661 dl0 181 Q. d4 Q t 667 -260 -120 -20 0 -10 -2 -4 6 Q 653 d9 t 472 465 460 Q. -43 -25 -15 -20 -25 -29 -12 -17 -2 -1 -4 t2 e13 388 385 381 380 U' e6 t 180 Q, 380 390 IV a5 e4 t Q. 160 20 -10 -6 -10 -4 -2 -8 013 t 665 Q. -1 -2 - e3 t 832 05 t 833 Q. -1 012 Points Q, b13 t Q. -2 -1 011 t 658 655 Q. -7 -1 -3 -5 -1 -2 d8 t 445 425 420 115 Q, t 420 -48 360 -35 -20 -10 -15 355 5 010 t 646 642 Q. -11 -2 -5 -10 -5 0 -1 -2 d7 d6 Q, t 360 -155 300 5 290 0 -5 -10 0 285 279 -5 -16 -2 -5 -7 -160 -30 -10 -15 -5 -13 -2 b9 742 Points d5 Q, b8 t 80 86 88 Q 35 1 -1 0 f8 f7 t 100 110 Q. 35 15 -4 o. t 115 125 130 129 t Q 35 130 10 140 0 143 25 The heat flow may be determined from the temperature distribution by multiplying the difference in temperature between the series of points next to the outside surface, and the points along the outside surface by the kA (=1) facx tor. One-half of the temperature difference is used for the center line calculation, an axis of s~nmetry, since half of the heat flow along that line must be contributed to each of the sides. The summation and tabulation of the data for each of the five cases is given in Table VI. In order to express the heat-flow through the hollow square conductor in terms of the heat-flow through a conoentrio circular conductor, of the same outside-inside dimensional ratio, it was necessar] to evaluate the radial heat flow through the circular conductors, using the s~~e diameter ra- tios, and considering the conductivity to be one, an one thousillld degree temperature difference, and a one foot in length section, as before. This data is calculated and tabulated in Table VII. The value of c s ' the ratio of the heat-flow through the square to the heat-flow through the ciroular conductors, is calculated and tabulated in Table VIII for each of ttle five cases. A oontinuous curve showin.g the value of Os in ter~IlS of the ratlo of the v'Jidths of the outside to the inside square (~) is shown on Platt I. 27 TABLE VI Determination ot Heat Flow trom Temperature D1stribution Caee I, from Table I q • 8(211+422+&?9+494+498+499+499+499+499+499+499.2+499.5+ 500+ix500) • 50,?74 Btu/hr-tt Caee II, trom Table II q • 8(20?+414+450+48?+495+498+499+!x500) • 25,408 Btu/hr-tt Caee III. trom Table III q • 8(90+181+262+303+321+328+331+332+ix333) • • 18,520 Btu/hr-tt Case IV, trom Table IV q • 8(52+103+152+194+221+236+243+246+248+1x250) • • 14,560 Btu/hr-tt Case V. trom Table V q • 8(20+41+64+88+110+129+143+152+159+163+165+ ix167) • 10,544 Btu/hr-tt 28 TABLE VII Determination of Heat Flow through Concentrio Conduotors Case I, with r2/rl m 1.167 q. 2nk(tZ -t,} • 6.28(1000} _ 40,700 Btu/hr-rt In 1.167 In (r2!rl) Case II, with r2/rl - 1,333 q - 6,28(1000} • 21,840 Btu/hr-rt In 1,333 Case III, with r2/rl - 1.500 q - 6i~8i~gggl - 15,490 Btu/hr-ft Case IV, with r2/r, • 1.667 q - 6.28 1000) • 12,300 Btu/hr-rt In 1.667 Case V. with r2/rl • 2.000 q • 6.28(10001 a 9070 Btu/hr-rt In 2.000 29 TABLE VIII Tabulation of Case I C s· q a 50,774 1 246 q 0 • ~O,'oO· ~a • 26,408 • 1.210 21,840 · Case II Os • q • Case III ° • 18,520 8 1~,490 • 1.196 Case IV 0 14,560 8 • 12,300 a 1.18' Oa8. V Os • 10,544 9,070 • 1.162 as 30 •: I t ~ d; II l~~ 11 '1 I ....... " t It,. Ii J . CO CLUSIONS 3 From the plot on Plate I, it may be seen that the heatflow through the square conductor approaches that through the circular conductor as the outside-inside es. ~idth ratio increas- This is what one would expect, since the geometry of the thick square more closely approximates the concentric thick walled circles. The only available indication as to the accuracy of the heat-flow throueh the square conductor that could be found was in ar- empirical formula as given by Jakob. (14) (19) Jakob, M~x, The agree- and Hawkins, George A., Elements of heat transfer and insulation, 2nd edition 1950, John VTiley P 39 ment with this form was within the order of +3%. The relaxation of the various internal points was carried on until the Q for that point was within ~lO, which would mean the temperature was within three degrees, or 0.3%. In most cases the Q was in the order of from 0 to ~4, which would. give an error of one degree, or 0.1%. This gave an ac- curacy in the heat-flow calculation of from 0.2 to 0.8%, since the temperature of the first inside row of points varied from 80 to 500°F. However this accuracy could not be attained un- less a great nwnber of points had been used. With the number of points used, it is thought that the accuracy was within one percent. The relaxation method is quite applicable to the solution -of difficult heat transfer problems. This numerical 33 analysis has the added advantage of the incorporation of variable conductivity, and of directional conductivity. The solution by this method requires only a nominal amount of time. The solution of Case V, having fifty points to relax, took approximately seven hours. This length of time was con- siderably shorter than what would have been normally necessary, because of previous practice. In conclusion, it may be stated that it is believed that the results presented in this thesis are accurate to well within the normal accuracy of heat-flow calculations. 35 A brief review is made of the solution of heat conduction problems by means of electrical an and by graphical methods. membrane analogies, The origin of and various applica- tions of the relaxation method of solution is discussed. The solution of a continuous concentric square heat conduction problem is made by numerical analysis. The relation- ship between the heat conducted through the concentric square conductor and that cnducted through a concentric circular \ conductor is determined and indicated in one continuous curve. The accuracy of this determination is surveyed and coneluded to be within plus or minus one percent. In summary, the author would like to bring to your attention the applications for the above work in the fields of heat-flow determinations in furnace, kiln, tunnel, and duct 36 BIBLIOGRAPHY 1. Books: Dusinberre, G. M., The Numerical Analysis of Heat ~low. McGraw-Hill Book Co. pp 53-59 (1949) Jakob, ~ax, Heat Transfer, Vol. 1, John Wiley pp 365- 373 (1949) Jakob, Max and Hawkins, G. A., Elements of Heat Transfer and Insulation, 2nd Edition, John Wiley, P 39 (1950) l:oore, A. D., Fundamentals of Electrical Design, McGrawHill, p 73-77 (1927) McAdams, W. H., Heat Transmission, 2nd Edition, Mc rawHill, P 1-25 (1942) Southwell, R. V., Relaxation Uethods in "l"":ngineering Science, Oxford University Press pp 1-4 (1949) 2. Periodicals: Christopherson, B. A., and uouthwell, R• • , Methods Applied to ngineering Problems. of the Royal Society of London, elaxation Proceedings eries A, Vol. 168, pp 317-350 (1938) Emmons, H. " robl .ms 0_ The Numerical Solution of Heat-Conduction Trans. of the A. S. M.E. Vol. 63, No.6, pp 607-615 (1943) Kayan, 0 F., An lectrical Geometrical Analogue for Complex Heat Flow. Trans. of the A.S.~oE. Vol. 67, 37 No.8, pp 713-718 (1945) Paschkis, IT. and Baker, H. D., A Method for Determining Unsteady-state Heat Transfer by means of Analogy. n Electrical Trans. of the A.S.N;.E. Vol. 64, No.2, pp 105-112 (1942) Unpublished Material: Mann, H. T., Dete~mination of the Economic Feasibi1- i ty of Producing Marginal Gas '.'lJells. Thesis, 1issouri chool of Mines and Metallurgy, Rolla, Missouri (1950) Tilson, L. H., The pplication of the embrane to the Solution of Heat Oonduction Problems. issouri School of alogy Thesis, ines and .etallurgy, Rolla, s- souri. (1948) zwierzchowski, ethod to the ., The Application of the Relaxation olution of Problems Involving t e Fluids through Porous Media. of lines and low Thesis, Missouri School etallurgy, Rolla, issouri. (1949) 38 VITA The author was born July 26, 1926, at Hannibal, Missouri. His high-school education was obtained at the Hannibal High chool, and was completed in June 1944. Pre-engineering work was taken at the Hannibal-LaGrange College in Hannibal, Nissouri. First entrance to the ~issouri allurgy was made in September 1946. ~chool of Mines and Met- He was graduated in June 1948 with the degree of Bachelor of Science in Mechanical Engineering from this institution. concurrent wit f e- ntrance for graduate work, an instructorship in the Mechanical Engineering Department, was made in September, 1948.