MSU LIBRARIES -' RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. HEAT CONDUCTION ACROSS A BENT PLATE AND THE EFFECT OF CURVATURE ON THE TEMPERATURE DISTRIBUTION BY Abbas Arjomandi A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1985 ACKNOWLEDGEMENTS The author wishes to express his thanks to his major professors, C.Y. Wang and Dr. Chuck Bartholomew, both faculty members of mechanical engineering, for their suggestions, guidance and encouragements throughout this study. The author also wishes to thank professors James V. Beck and Dr. John J. Mcgrath for serving as members of his committee. ii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . . . . . NOMENCLATURE . . . . . . . . . . . . . . . . . . CHAPTER 1.1 1.2 CHAPTER 2.1 2.2 2.3 CHAPTER 3.1 3.2 CHAPTER 4.1 I Introduction . . . . . . . . . . . . Review of Literature . . . . . . . . . The Objective of This Study . . . . . . II Formulation of the Problem . . . . . Coordinate System . . . . . . . . . . . Transformation to a Round Corner . . . Parameters.............. III Finite Difference Method . . . . . Boundary Conditions . . . . . . . . . . Formulation for the Interior Nodes . . Formulation for the Boundary Conditions Node Generation . . . . . . . . . . . . Computer Program . . . . . . . . . . . Temperature Distribution Along n-axis . Heat-transfer and Shape Factor . . . . Isotherms . . . . . . . . . . . . . . . Error Estimate . . . . . . . . . . . . IV Results and Conclusions . . . . . . Numerical Results . . . . . . . . . . . iii vii ix 13 13 14 18 20 21 21 25 28 29 31 31 4.2 Conclusion APPENDIX . . . . LIST OF REFERENCES Table Table Table Table Table Table Table Table Table Table Table Table Table Table 10: ll: 12: 13: 14: LIST OF TABLES Dimensionless temperatures when b = 2.0 . . . . . . . Dimensionless temperatures when b = 2.0 . . . . . . . Dimensionless temperatures when 9 = “/4 and 8 varies Dimensionless temperatures when 6 = fl/Z and E varies Dimensionless temperatures when 9 = 3W/4 and E varies Dimensionless temperatures when 9 = fl and e varies . Dimensionless temperatures when 6 = 3u/2 and e varies Dimensionless temperatures when 9 = 2n and e varies . Dimensionless temperatures when 6 = fl/4 and b varies Dimensionless temperatures when 6 = fl/2 and b varies Dimensionless temperatures when 9 = 3fl/4 and b varies Dimensionless temperatures when 9 = W and b varies . Dimensionless temperatures when 6 = 3W/2 and b varies Dimensionless temperatures when 9 = “/4 . . . . . . . along n-axis along along along along along along along along along along along along along s-axis 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Table Table Table Table Table Table Table Table Table Table Table Table Table 15: 16: 17: 18: 19 20: 21: 22: 23: 24: 25: 26: 27: vi Dimensionless temperatures along s-axis When 6 = 1T/2 O O O O O O O O I O O O O 0 O Dimensionless temperatures along s-axis when 6 = 3fl/4 . . . . . . . . . . . . . . Dimensionless temperatures along s-axis When 6 = 1r 0 O O O O O O O O O O O O O O Dimensionless temperatures along s-axis When 6 = 31T/2 O O O O O O O O O O O O O O The dimensionless penetration depth along s-axis to reach the linear values for flat plate . . . . . . . . . . . . . . . . . . The dimensionless temperatures at the indicated nodes when 9 = "/4 . . . . . . . The dimensionless temperatures at the indicated nodes when 9 = “/2 . . : . . . . The dimensionless temperatures at the indicated nodes when 6 = 3n/4 . . . . . . The dimensionless temperatures at the indicated nodes when 6 = w . . . . . . . . The dimensionless temperatures at the indicated nodes when 6 = 3n/2 . . . . . . The shape factors and heat transfer (q/KATO) . . . . . . . . . . . . . . . . . The dimensionless maximum temperature deficiencies and their locations along the n-aXis C O O O O O O O O O O I O O O C Dimensionless temperatures at two loca- tions along the n-axis when h varies and also at s = b/2 along the s-axis when l varies . . . . . . . . . . . . . . . . . 50 51 52 53 54 55 56 57 58 59 60 61 62 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 4: 5: 10: 11: 12: 13: 14: 15: LIST OF FIGURES "90 degree bend with straight boundaries" . . . . . . . . . . . . . . . The Bent Plate and (n,s) Coordinate system 0 O O O O O O O I O O O O O O O O Bent Plates With b=4.0 and (a)e=n, (b)6=3u/2, (c)6=2w, (d)6=n/4, (e)e=w/2 and (f)6=3W/4 . . . . . . . . . . . . . . Bent Plates With 6=2h for a,b,c and d, 9=TT/2 for i,j,k and l 6=TT/4 for e.f.g and h Bent Plates When 6:“ for a,b,c and d, e=3n/2 for e,f,g and h, 6=3n/4 for i,j,k and l . . . . . . . . . . . . . . . . . . Representation of Boundaries and Boundary conditions 0 O O O O O O O O O O O O O 0 Representation of Nodes . . . . . . . . . Computer Flowchart . . . . . . . . . . . Showing the temperature deficiency along n-axis . . . . . . . . . . . . . . . . . The isotherms when e=n/2 and b=4.0 . . . Dimensionless Temperatures along s-axis When b=2 O 0 O O O O O O O O O O O O O O O Dimensionless Temperatures along n-axis When b=2 I 0 O O O O O O O C O O O O I O O Dimensionless Temperatures along s-axis When 6 = “/4 O I O O O O O O O O C O O O O Dimensionless Temperatures along s-axis When 6 =TT/2 O C O O O O O O O O O O O O O Dimensionless Temperatures along s-axis when 9=3N/4 . . . . . . . . . . . . . . . vii 10 12 15 22 23 26 30 63 64 65 66 67 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 16: 17: 18: 20: 21: 22: 24: 25: 26: 27: viii Dimensionless Temperatures along s-axis when 9:“ . . . . . . . . . . . . . . . Dimensionless Temperatures along s—axis When e=3TT/2 O O O O O O O O O O O O O O Dimensionless Temperatures along n-axis When e=n/4 O O O O O O O I O O O O O O Dimensionless Temperatures along n-axis When e=TT/2 O O O O O I C O O O O O O O Dimensionless Temperatures along n-axis When 6:31T/4 O I C O O O O O O O O C O I Dimensionless Temperatures along n-axis When 6:11. C O O O O O O O C O O O I O O Dimensionless Temperatures along n-axis when 6=3W/2 . . . . . . . . . . . . . . Dimensionless Temperatures along n-axis When 8:2" O O O O O O O O O O O O O O 0 Location of the intersection of the different isotherms with n-axis when 9=1T/2 O O O O O O O O O O O O O O I I O The Dimensionless Temperatures versus h at Two Locations Along n-axis . . . . . The Dimensionless Temperatures versus h at Two Locations Along the n-axis . . . The Dimensionless Temperatures at s=b/2 Along the s-axis verses 1 (bottom line) and 12 (top line) . . . . . . . . . . . 68 69 70 71 72 73 74 75 76 77 78 79 Letters' 7: W :3‘ :3" CD 7% I'-' N OMENC LATU RE Half of the plate thickness (m) Matrix A Half of the curve—length of the bend (dimensionless) Matrix B Convective heat transfer coefficient The mesh length along n-axis (dimensionless) Dimensionless curvature of the centerline Conduction heat transfer coefficient Curvature of the centerline [m]-1 Dimensionless number equal to [l]2/[h]2 Dimensionless radius of curvature of the centerline Dimensionless radius of curvature of the interior boundary Dimensionless radius of curvature of the exterior boundary Dimensionless S coordinate along s-axis Dimensional S' coordinate along S'-axis [m] Dimensional temperature of the interior surface [C°] ix X Tl Dimensional temperature of the exterior surface [C°] T Dimensional ambient temperature [C°] Tm,n Temperature at node (m,n) [C°] Ti,j Temperature at node (i,j) [C°] ATO Overall temperature difference [C°] x', x x coordinate [m] y',y y coordinate [m] z', z z coordinate [m] Greek Symbols A The difference between two quantities 6 The central angle of the bend X Dimensionless temperature n' n'-coordinate along n'-axis [m] n n-coordinate along n-axis [dimensionless] a The subdivision of 6 where a = 6/12 s A small number equal to a/RC 3( )/3( ) Showing partial derivative Vectors N Unit vector along n'-axis R Position vector for points along the center— line i Unit vector along x—axis 3 Unit vector along y-axis R Unit vector along z-axis ABSTRACT HEAT CONDUCTION ACROSS A BENT PLATE AND THE EFFECT OF CURVATURE ON THE TEMPERATURE DISTRIBUTION BY Abbas Arjomandi Due to the complicated geometry, the heat conduction. across a bent plate and the effect of curvature on the tem- perature distribution has never been studied before. The corner of a bent plate is a fundamental structural element which is susceptible to failure. By using the temperature distribution, the thermal-stress can be determined. In this paper, an intrinsic coordination system is utilized in order to obtain a simpler form of the governing differential equation when compared with the cartesian coor- dinate formulation and then the equations are solved numeri- cally. The results indicate the dependence of temperatures on the curvature and angle of the bend. The temperatures are decreased in the vicinity of the bend when compared to the linear distribution for flat plate having the same thickness. A discussion on the influence of these parameters on the shape factor and rate of heat transfer is presented, and the accuracy of the results has a confidence level of three significant digits. CHAPTER I Introduction 1.1 Review of Literature The steady-state heat conduction across a bent plate and the effect of curavture on the temperature dis- tribution is the tOpic of this study. This information can be used in the design of pressure vessels or any sit- uation where two different temperature fluids or gases are separated by a given plate geometry. Since the bend is most susceptible to failure, the results can also be used to determine the limiting thermal stresses. Carslaw [l] was the first to consider the effects of a bent element on heat transfer. He studied "a right angled bend in a wall" where both the interior and exterior angles of the bend were 90°, with no bends or curves involved. Figure 1 shows the details. Carslaw used the Schwarz- Christoffel transformation during the solution of the problem. It was Ozisik [7] who was the first to solve the same problem using a finite difference method. Ozisik showed how to formulate the problem when all the boundaries were straight lines. 2 If the plate is flat or has the shape of a circular cylinder, exact solutions for the problem exist [3]. The present paper studies the case where the plate is neither completely straight nor circular. Separation of variables or conformal mappings are useless in this case. Further- more, there has not been any analytical treatment of the problem until recently. C.Y. Wang [5] has recently applied an intrinsic coordinate system to the problem and solved it analytically for small perturbations. He concluded that in the vicinity of the bend, the temperatures are decreased when compared to their corre- sponding linearly distributed values for a flat plate. The maximum decrease occurs at S=O, n=0 which corresponds to the center of the bend. Wang noticed that as the length of the bend (2b) is decreased, the temperatures tend toward the linear values of a flat plate. The decrease of temperatures also vanished as 2b tended toward zero. He concluded that the local heat transfer changed as the curvature (or 2b) changed, but to the order that problem was solved the total heat transfer is unchanged. 1.2 The Objective of this Study The purpose of this study is to investigate the effect of curvature on the temperature distributions and heat transfer across a bent plate for steady-state heat conduction using the finite difference method. An 3 analytical method based upon a perturbation solution is possible if ratio of the plate width to local radius of curvature is very small. This restriction is not necessary for the finite difference method. (a)-Exterlor corner with convective boundary 2(hAX/k)Tw +(Tm_l'n°Tm'n_R-2((hAX/K)*I)Tm'n=0 m-I n r1010 m.n - ‘\ t ' \ 1 u Ag . | h0,T I ! m,n-l '— — — Ex” — I (b)—lntenor corner with convective boundarg. 2(hAX/K)T +2T *ZT :T T -2(3+nA>2 + (dn')2 + (dz')2 (3) where K' is the curvature and S', n', Z' constitute an orthOgonal intrinsic coordinate system as shown in Figure 2. The boundary is described by n' = i a. Let the temperature on both sides of the plate be maintained at a constant value to produce the following boundary conditions. n' = a, T = To, and n' = -a, T = T1 (4) Figure 2. --Tne Bent Plate and(n.s) Coordinate Sgstem. 7 Introducing the apprOpriate scale factors, Laplace's equation becomes 1 3 1 3T (i-K'n'){35T(1-K'n"as') + g—n.—[(1-K'n')-§%.—]} — o (5) where we will normalize all lengths by '3' and drop the primes. The temperature field is normalized by intro- ducing the following: T = (To - T1)X(n) + T1 (6) By using this representation of the temperature field, we reduce the problem to one where we are solving for the function X(n). Equation (5) thus becomes: 8 1 3X 8 , 3X _ a§(1-Kn as) + §fi[(l‘h”)an] ‘ O (7) If the curvature K is constant (which corresponds to a circular bend), the solution is found to be: 1-Kn 1-K = 1 — X ln(l+K )/‘n(l+K) (9) This result agrees with Reference [3]. 2.2 Transformation to a Round Corner For an arbitrary curvature K(S), a perturbation solution is possible when the plate width is small compared to the local radius of curvature, i.e., K'a E K(S) E ek(S) << 1 (10) where e is a small number defined as max [K'a| and k(S) is of order unity. Let us concentrate on the geometry shown in Figure 2. The plate consists of two semi-infinite straight sections joined by a circular section. The curva- ture of the centerline is given as k(S) ={ (11) Substituting Equations (10) and (11) into (7), the final governing equations are: (aZX/anz) + (BZX/BSZ) = o For k(S) = 0, or is] > b (12) (aZX/asz) + e(en-l)(3X/8n) + (l-en)2(32X/3n2) = o For k(S) = l, or |S| g b (13) Note that the first equation describes straight segments while the latter refers to the curved section. 2.3 Parameters The radius of curvature (R') and the thickness of the plate (2a) are dimensional quantities. The dimension- less parameters are 2b and 0, respectively, where 2b is the total length of bend and which is divided into two equal parts b by the n-axis. Facing the arc 2b is the angle 6. See Figure 3f. Equation (14) indicates how 6 is defined and equa- tion (15) is the relation among these parameters. 5 = a/R (l4) 2b€ (15) CD II A value of zero for b indicates that the plate is completely straight without any bend. If e = 0.0, then according to Equation (14), R tends toward infinity which, in turn, indicates that the plate is again straight. This is because "a" can not be equal to zero. Figures 3, 4 and 5 show how the different values of these parameters affect the shape of the bent plate because they are related to one another according to equa- tions (14) and (15). 10 Figure 3.--Bent Plates wun 024.0 and (a)e=n.(o)e=3n/2_ (C)9=2fi.(d)e=n/4.(e)e=11/2 and (r)e=3m4 11 (0)902 (C)£=O.S (d)e=1.o (3)6101 I ~ ./ . I I l ;' \ I I ) / . \/ (9)5‘01 (r)c=o,2 (gie=0.5 (h)£=i.0 // 'I / l, ' / i ' g / 3 ; / i : l I i i I (06:01 (1)5;02 (“£205 (06:10 Figure 4.-- Bent Plates With 9:211 for a,b,c and d 9:11/2 for l,],k and i 9=Tr/4 for e,f,g and h. 12 (b)6=0.2 (c)e=O.S (d)g=i.o @‘él— (2)6301 - . (9)6 -05 (h)e= ‘ I Q {2 Q; Q \ y) \ . (05:0,: (j)e=o.2 MFG-5 il)€='-0 figure S.—-Bent Plates When e=nlor a. he and d ,6=3rr/2 for e.f.q and h, 9:311/4 for i,j,k and l CHAPTER II I Finite Difference Method We will utilize the fact that the plate is symme- tric about the n-axis to reduce our computational effort. This will be used throughout the rest of the formulation. 3.1 Boundary Conditions The inner surface, which corresponds to n = +1, has the temperature of X = 1. On the outer surface, n = -l and X = 0. Along the n-axis (corresponding to S = 0), sym- metry will be used which yields an adiabatic surface with aX/as 2 0. In the segments far away from the bend, the temperature is assumed to approach the flat plate solution. In other words, it changes linearly from X = l to X = 0 because: (i) there is steady-state condition, and (ii) no curvature effects are involved in the geometry. Figure 6 shows these conditions. Thus, two different forms of the governing differ- ential equations are used: one in the curved and the other one in the straight sections of plate. If the curved and straight segments are separated as shown in Figure 6, along the border line, not only the temperature distribution is the same for both segments but the heat 13 14 coming in s—direction from one side must go out through the other side. If at a point (n,s) along this line the temperatures are called X1 and X2 for the curved and straight segments respectively, then the boundary condi- tions along the line separating the two parts would be: x1 = x2 (16) (axl/aS) = (BXZ/BS) (17) Figure 6 shows all the boundary conditions involved. To clearly illustrate the two domains (straight and curved sections), the whole plate is shown as two separate pieces. The cut is at the shared boundary between the two segments. 3.2 Formulation for the Interior Nodes Equations (12) and (13) are converted into finite difference form using a "Central-Difference" representation and taking h and l as mesh lengths along n and S-axis, respectively [2]. x. a- o I 1 _ . . 1+1, 1-1, I§(Xi,j+l-2Xi'j-l+Xl'J_l)-+e(€h1-l) 2h 1 + (l—ehi)2—-l-(x. . - 2x. . + x. .) = o h2 1+1,j 1,3 1-l,j For k = l (18) 15 x is linear f‘ , ’\. x-o.0 XI, 3+1 X36. X," 3.1“- ax,/8s zaxz/as \m 2:— . / ' 2 W J N I l T X- . x--"-\ ’Id 13-1 r I \\. X:o 0 Figure 6.—-Representation of Boundaries and Boundarg Conditions. 16 H l—2(Xioj+l-2Xi,j+xi,j-l) + H For k = 0 (19) The following relations were used in equations (18) and (19): n = hi (20) S = lj (21) axfih1= (Xi+1,j‘xi-1,j)/2h (22) 32X/852 = fi-(xi'jH—zxi’J X1,3-1) (23) aZX/an2 = i2(xi+l,j_zxi,j 1-l,j) (24) If r = 12/h2 (25) then Equations (18) and (19) become 17 125(ehi-l) (-2r(1-ehi)2-2)Xi,j-+(r(l-€hi)2‘* 2h )Xi+l.j + (r(1-ehi)2 ‘ lzeéfihi-l))xi-l,j + xi,j+1 + Xi,j-l = 0 when k = l (26) (-2r-2)Xi,j + rxi+1,j rXi-1,j + Xi,j+l + Xi,j-l = 0 when k = 0 (27) To further simplify the equations, the following substi- tutions are made: -2r(1-ehi)2-2 = 3(1) (28) 12€(€hi-1) r(1-ehi)2 + 2h = F(i) (29) 2 -- r(l-ehi)2 + 1 Eéihl 1’ = G(i) (30) The final form of Equations (12) and (13) for the interior nodes in the curved and straight sections thus take the following form: 18 E(i)xi'j + F(i)Xi+l,j + G(i)Xi_l'j + xi,j-1 + xi,j+l = 0 when k = 1 (31) (-2r-2)Xi'j + rxi+l,j rXi—1,j + Xi,j-1 + Xi,j+l = 0 when k = o (32) Equation (31) holds for interior points when |S| < b, k = l and Equation (32) holds for interior points having [S] > b, k = 0. 3.3 Formulation for the Boundary Conditions a) The first boundary to be discussed is the adiabatic surface corresponding to the symmetry boundary. ax/as z 0 (33) x. . - x. . 1,j+1 1,3-1 - 21 — 0 (34) xi,j+1 = xi,j-l (35) If X is cancelled between Equations (35) and i,j-l (31), the equation which must be used for the adiabatic boundary is: E(i)xi . +F(i)X J i+Lj-+G(1)X. .-+2x = o (36) i-Lj Lj+l b) As discussed earlier, the boundary separating the curved and straight parts uses continuity arguments. Equations (16) and (17) are the boundary conditions (Bxl/Bs) = (BXZ/BS) (37) and by using a central difference, Equation (37) becomes Xi,j+l ' Xi,j-l _ Xi,j+l ' Xi,j-l 21 ‘ 21 -X.. =X Xi,j+l 1,3-1 i,j+l ’ xi,j-l where i and i are two fictitious points as shown in Figure 6. Assuming fictitious points is the common approach used in a finite difference method to handle the nodes located on boundaries. The bars over X are used to make a distinction between the two fictitious nodes. Equations (3l) and (32) along this line become: maxi,j + F(i)Xi+l’j + Gqu 9/3 55 \ "1 \\ 346 I b ( 7 ax \J &9 S a. ( b 19 ‘3 / IO J ’ / . )2 x / 2 3 ' '3 / x ’ " g \i' / Eb " 4 ' \5 ,/ , / Adiaoatm Boundaru 4” S I ‘6 27 l . 6 \‘l / 7 \b 8 9 ll Figure 7.--Reprrzsentation of Nooes 23 Set the values for h,I,e,D and 6 Assemble the matrix A (CALL SKY!) Assemble (he matrix B (CALL SKY2 Catt 'LEOTtF‘ 7 Call OUNDUt 1 Figure 8.1“ Computer flowchart 24 Substituting Equation (6) into (48) 1n(R/Ri) - __ (48) ln(Ro/Ri) which is the dimensionless temperature along the n-axis for a cylinder. The linear distribution of the dimensionless temperature along the n-axis with respect to R is R-R _ O X - 1.2—:1:— (49) O l The difference (dimensionless) between these two distribu- tions is: RO-R 1n(R/Ri) AX=fi-1+W) (50) and this difference is maximum at the point where R -R. O 1 R = W) ‘5“ When the plate thickness is small compared to the radius of the centerline Rc' the maximum temperature deficiency in this case is located at: R = (52) In his work, C.Y. Wang [4] refers to this point as "The worst temperature deficiency" which occurs at S = O, n = 0 along the n-axis only for small values of e. Figure 25 9 shows that the temperature deficiency along n-axis can be closely approximated as a parabolic distribution when the plate thickness is small compared to Rc- Table 26 shows the location and value of maximum temperature deficiency along n-axis for different values of e and 9. 3.7 Heat-transfer and Shape Factor In this section, Figure 7 will be very helpful to and in the understanding of the equations and proofs. From Equation (15), the radius of curvature for the center- line where n = 0 is . _ 2ba RC—-—e— , (53) , _ , _ _ 2ba _ _ _ R i - R c a — —§— a — a((2b 6)/6) (54) where R'i is the radius of curvature for the interior ‘ curved boundary, and "a" is half of the plate thickness. The following shows the calculation of the length of each division along the n-axis and the interior boundary. See Figure 7 for better understanding of the following calcu- lations, which are divided into three parts. 1) The dimensional mesh length along the n-axis N DJ An = 0 = .2a (55) f—J 2) The dimensional mesh length on the interior boundary in the curved segment 0.9 r V Ti 7 0.8 0.7 "1 I 0.61 ‘ 0.3 - 04 L- 0.3 ~ 0.2?— 0.1 L 0.0 0.0 26 Linear diSUlDUIIOH ln(R/R§) K310" (”(Ro/Ri,’ . * : e *f f R! (Ro'Rt)/2 R 1r ‘ ‘ q d Fklure 9.--Showmq the temperature deftcnency alongn axis Ax- Temperature deficiency 27 _ 6/2 _ a — €75 — 6/12 (56) as' = R'ia = 31%;:9) (57) 3) The dimensional mesh length along the interior boundary in the straight segment of the plate as' = 93— (58) The heat conduction equation is [6] 2 -l KA(AT/An') (59) Q ll rmnm From Equation (6), the temperature difference between every two nodes on the same line is AT = (TO-T1)AX = (ATovera11)(AX) = (ATO)(AX) (60) Considering a unit depth in z'-direction and dividing Equation (59) into two parts in order to take care of the fact that the divisions along the interior boundary are not the same in the curved and straight seg- ments of the plate and also substituting Equation (60) for AT, the result is - 6 2b-6 ATOAX 22 ba ATOAX q — i=1 Ka<—I§')(1)(UT§§')*‘§=7K‘676”1“ 0:25) (62) or 2 =7 H'MN 5 b ————)E AX. + IT Axi (63) KATO 2 The shape factor 8 is related to q according to [6] q = KSATO (64) or _fl__.= KAT S (65) O The temperatures along the line next to the inter— ior boundary at the indicated nodes are tabulated in Tables 22 through 26 and the values of heat transfer in the form of which is equal to the shape factor S, are shown in _3__. KAT 0 Table 25. 3.8 Isotherms For each different shape, the isotherms can be found by linear interpolation for the temperatures between adjacent nodes to obtain the desired temperature. As an example, isotherms are shown on Figure 10 when 6 = n/Z and b = 4. For other shapes, isotherms can be found in a sim- ilar way. The location of the intersections of isotherms with the n-axis are also shown in Figure 24 when 6 = n/2. This picture illustrates the deviation of these isotherms from the linear distribution for a flat plate with the same thickness. 29 3.9 Error Estimate By considering two nodes at n +l/3 and n = -l/3 along the n-axis and another one at s b/2 along the 5- axis, different mesh lengths h and l were considered to see how many digits are valid and can be trusted. Results showed that only three digits are valid. Table 27 shows the results and Figure 25 through 27 are the plots of these temperatures verses h. hz, l and 12. These figures show that the temperature varies linearly when they are plotted verses h2 and 12. The error is due to two factors, the first is the truncation error in central-difference formulation in the finite difference method and the second is the computer round off. 30 Figure )0,-- The isotherms when ezn/Zanrt 0:4.0 CHAPTER IV Results and Conclusions 4.1 Numerical Results Temperature distributions were calculated for selected values of 6, b and s using a finite difference method expressed by Equations (31), (32), (36) and (42). Tables 1 through 18 include all the results for different values of these parameters. The results confirm the statements made by C.Y. Wang [4] concerning the analy- tical results. The results indicate that the temperatures at any given (n,S) in the vicinity of the bend are less than their corresponding values for the linear distribution in a flat plate with the same thickness. This is attri- buted to the difference in areas between the inner surface and outer surface. Since the heat transfer rate is prOpor- tional to this area, the area difference results in the isotherms being closer together near the inner surface. See Figure 10. For a given angle of bend and bend length b (or curvature), the temperatures along the centerline are less than the linear value of 0.5, and the minimum temperature along this line is located at S = 0, n = 0 since this 31 32 point is the furthest point from the straight segments. Figures 11 and 13 through 17 show the change of tempera- tures along the S-axis and Tables 2 and 14 through 18 con- tain the numerical values for different values of parameters. The temperatures in the vicinity of the bend have decreased from their corresponding linear values even for the adjacent straight sections. The temperatures do not relax back to the linear temperature distribution until one reaches a certain penetration depth along S-axis. This is due to the effect of the bend on the neighboring parts of straight sections. Table 19 shows the penetration depth for different values of parameters. As b decreases, the temperatures at any given (n,S) in the vicinity of the bend increase due to the fact that the bend length is being slowly reduced which, in turn, makes the plate behave more like a flat plate. In the limit, as b -+ o the temperature deficiency decreases and temperatures reach their linear values. Along the n-axis or S = 0, the temperatures are less than their corresponding linear values and as dis— cussed in Section 3.6, when the plate thickness is small compared to radius of curvature RC, the temperature defi- ciency is parabolic along this axis. The maximum tempera- ture deficiency in this case occurs at S = 0, n = O. This is an approximation which loses its validity as e 33 increases to larger values, (i.e., the plate thickness is not small compared to the radius of curvature RC). Table 26 shows the maximum temperature deficiency along n-axis and its location. As 8 decreases for a given value of b, the temper- atures along n-axis are increased. This can be seen in Figures 13 to 17. As 0 decreases (which means the angle of bend increases and the plate becomes flatter), the tempera- tures approach the linear values. The temperature distri- butions are shown in Figure 18 through 22. Tables 1 and 3 through 13 also contain additional results. For a given 5, the shape factors and the heat transfer are increased as the angle of the bend increases (6 decreases). For a given angle of the bend (or 6), the shape factors and heat transfer are decreased as 8 increases. See Table 25. 4.2 Conclusion Based on the results obtained and the discussion in the previous section, it can be concluded that: l) The temperatures are decreased in the vicinity of the bend when compared with the linear dis- tribution of a flat plate with the same thickness. 2) For a given 9 (or angle of the bend) and bend length b (or curvature) the temperatures 3) 4) 5) 6) 8) 9) 10) 11) 34 along the S-axis are less than the linear value of .5 for the centerline. The maximum temperature deficiency along the S-axis is observed at n = O, S = 0. The maximum temperature deficiency along any line parallel to S-axis is located at (n,0). Moving along the S-axis, the linear tempera- ture distribution is observed in the straight segments up to a penetration depth that depends on 2b and 6. The temperature deficiency from linear distri- bution along n-axis is parabolic when 6 << 1. As b decreases, the temperatures for any given (n,S) in the vicinity of the bend increase and as b ++ 0 they reach the values of temperature distribution in a flat plate with the same thickness. For a given angle 8, as 6 increases (b decreases) the temperatures at any given point (n,S) in the vicinity of the bend decrease. For a given 8, as 6 increases, the heat trans- fer and shape factors increase. For a given 6 (or angle of the bend) as 8 increases, the heat transfer and shape factors decrease. The results are accurate only to three signifi- cant digits. 35 It is clear that the effect of curvature (bend) cannot be ignored on the temperature distribution along a bend plate. In his work, C.Y. Wang [4] assumed that the plate thickness is very small when compared with the radius of curvature of centerline, (i.e., 6 << 1). The numerical results not only confirm the results of analyti- cal solution, but also relaxes the restriction of being a lot smaller than unity which is imposed upon the analyti- cal solution. 36 Table l: Dimensionless temperatures along n-axis when b = 2.0. n 6=w/4 0=fl/4 8=3n/4 0=n -1 0.000 0.000 0.000 0.000 -4/5 0.084 0.071 0.059 0.048 -3/5 0.172 0.147 0.123 0.101 -2/5 0.262 0.227 0.193 0.159 -1/5 0.356 0.313 0.270 0.224 0.0 0.453 0.405 0.354 0.298 1/5 0.554 0.504 0.448 0.384 2/5 0.658 0.611 0.554 0.484 3/5 0.767 0.727 0.676 0.607 4/5 0.881 0.856 0.821 0.766 1 1.00 1.00 1.00 1.00 37 Table 2: Dimensionless temperatures along s-axis when b = 2.0. 1:1/3 8=n/4 6=n/2 8=3n/4 0=n 0 0.453 0.405 0.354 0.299 1 0.453 0.405 0.355 0.300 2 0.454 0.408 0.359 0.306 3 0.456 0.412 0.366 0.316 4 0.460 0.419 0.377 0.333 5 0.466 0.431 0.396 0.358 6 0.475 0.451 0.425 0.398 7 0.485 0.470 0.454 0.438 8 0.491 0.482 0.472 0.462 9 0.495 0.489 0.483 0.478 10 0.497 0.494 0.490 0.487 11 0.498 0.496 0.494 0.492 12 0.499 0.498 0.496 0.495 13 0.499 0.499 0.448 0.497 14 0.500 0.499 0.499 0.498 15 0.500 0.500 0.499 0.499 16 0.500 0.500 0.500 0.499 17 0.500 0.500 0.500 0.500 18 0.500 0.500 0.500 0.500 19 0.500 0.500 0.500 0.500 20 0.500 0.500 0.500 0.500 21 0.500 0.500 0.500 0.500 38 Table 3: Dimensionless temlperatures along n-axis when 6 n/4 and e varies. n €=0.1 e=0.2 e=0.5 e=1.0 -1 0.000 0.000 0.000 0.000 -4/5 0.091 0.084 0.072 0.066 -3/5 0.185 0.172 0.150 0.139 -2/5 0.280 0.262 0.232 0.219 -1/5 0.376 0.355 0.320 0.307 0 0.475 0.452 0.413 0.402 1/5 0.576 0.553 0.513 0.505 2/5 0.678 0.658 0.620 0.616 3/5 0.783 0.767 0.733 0.736 4/5 0.891 0.881 0.858 0.865 1 1.00 1.00 1.00 1.00 39 Table 4: Dimensionless temperatures along s-axis when 6 = “/2 and e varies. n €=0.1 €=0.2 e=0.5 s=l.0 -1 0.000 0.000 0.000 0.000 -4/5 0.091 0.084 0.066 0.054 -3/5 0.184 0.170 0.137 0.113 -2/5 0.280 0.260 0.213 0.179 -1/5 0.376 0.353 0.295 0.253 0 0.475 0.450 0.384 0.337 1/5 0.576 0.551 0.481 0.431 2/5 0.678 0.6555 0.588 0.530 3/5 0.783 0.765 0.707 0.665 4/5 0.890 0.880 0.842 0.815 1 1.000 1.000 1.00 1.00 40 Table 5: Dimensionless temperatures along n-axis when 8 =3n/4 and e varies. n €=0.1 €=0.2 €=0.5 €=1.0 -1 0.000 0.000 0.000 0.000 -4/5 0.091 0.0836 0.064 0.0456 -3/5 0.185 0.170 0.132 0.097 -2/5 0.280 0.260 0.206 0.154 -l/5 0.376 0.353 0.287 0.219 0 0.475 0.450 0.374 0.292 1/5 0.576 0.550 0.471 0.378 2/5 0.678 0.655 0.578 0.479 3/5 0.783 0.765 0.698 0.602 4/5 0.890 0.880 0.837 0.761 1 1.00 1.00 1.00 1.00 41 Table 6: Dimensionless temperatures along s-axis when 6 = n and e varies. n e=0.1 €=0.2 €=0.5 €=1.0 -1 0.000 0.000 0.000 0.000 -4/5 0.091 0.084 0.063 0.040 -3/5. 0.185 0.170 0.131 0.085 -2/5 0.280 0.260 0.204 0.136 -1/5 0.376 0.353 0.284 0.193 0 0.475 0.450 0.371 0.260 1/5 0.576 0.550 0.467 0.337 2/5 0.678 0.655 0.574 0.431 3/5 0.783 0.765 0.696 0.549 4/5 0.890 0.880 0.835 0.711 1.0 1.00 1.00 1.00 1.00 42 Table 7: Dimensionless temperatures along s-axis when 0 =3n/2 and e varies. n €=0.l €=0.2 e=0.5 e=l.0 -1 0.000 0.000 0.000 0.000 -4/5 0.091 0.084 0.063 0.033 -3/5 0.185 0.170 0.130 0.071 -2/5 0.280 0.260 0.203 0.112 -l/5 0.376 0.353 0.283 0.160 0 0.475 0.450 0.370 0.216 1/5 0.576 0.550 0.466 0.283 2/5 0.678 0.655 0.573 0.366 3/5 0.783 0.765 0.694 0.474 4/5 0.890 0.880 0.834 0.635 1 1.00 1.00 1.00 1.00 43 Table 8: Dimensionless temperatures along n-axis when 8 = 2N and e varies. n €=0.l €=0.2 e=0.5 -1 0.000 0.000 0.000 -4/5 0.091 0.084 0.063 -3/5 0.185 0.170 0.130 -2/5 0.280 0.260 0.203 -l/5 0.376 0.353 0.282 0 0.475 0.450 0.369 1/5 0.576 0.550 0.465 2/5 0.678 0.655 0.572 3/5 0.783 0.765 0.694 4/5 0.890 0.880 0.834 1 1.00 1.00 1.00 44 Table 9: Dimensionless temperatures along n-axis when 6 =n/4 and b varies. n b=0.5 b=l.0 b=2.0 b=3.0 -1 0.000 0.000 0.000 0.000 -4/5 0.068 0.075 0.084 0.089 -3/5 0.142 0.155 0.172 0.180 -2/5 0.222 0.240 0.262 0.274 -l/5 0.309 0.329 0.356 0.369 0 0.402 0.423 0.453 0.468 1/5 0.502 0.523 0.554 0.568 2/5 0.611 0.629 0.658 0.672 3/5 0.727 0.742 0.767 0.778 4/5 0.854 0.865 0.881 0.887 1 1.00 1.00 1.00 1.00 45 Table 10: Dimensionless temperatures along n-axis when 6==n/2 and b varies. n b=l.0 b=2.0 b=3.0 b=4.0 -1 0.000 0.000 0.000 0.000 -4/5 0.057 0.071 0.079 0.084 -3/5 0.119 0.147 0.162 0.171 -2/5 0.188 0.227 0.249 0.261 -1/5 0.263 0.313 0.339 0.354 0 0.347 0.405 0.435 0.451 1/5 0.440 0.504 0.535 0.551 2/5 0.546 0.611 0.641 0.656 3/5 0.667 0.727 0.753 0.766 4/5 0.811 0.856 0.872 0.880 1 1.00 1.00 1.00 1.00 46 Table 11: Dimensionless temperatures along n-axis when 8 = 3n/4 and b varies. n b=2.0 b=3.0 b=4.0 b=5.0 -1 0.000 0.000 0.000 0.000 -4/5 0.059 0.070 0.077 0.081 -3/5 0.123 0.145 0.157 0.165 -2/5 0.193 0.224 0.242 0.253 -l/5 0.270 0.309 0.331 0.345 0 0.3540 0.401 0.426 0.440 1/5 0.4480 0.499 0.526 0.541 2/5 0.554 0.606 0.632 0.647 3/5 0.676 0.724 0.746 0.758 4/5 0.821 0.854 0.868 0.875 1 1.00 1.00 1.00 1.00 47 Table 12: Dimensionless temperatures along n-axis when 6 = n and b varies. b=2.0 b=4.0 b=6.0 b=8.0 -1 0.000 0.000 0.000 0.000 -4/5 0.048 0.070 0.079 0.084 -3/5 0.101 0.144 0.162 0.171 -2/5 0.159 0.226 0.248 0.261 -1/5 0.224 0.308 0.339 0.354 0 0.299 0.400 0.434 0.451 1/5 0.384 0.498 0.534 0.551 2/5 0.485 0.605 0.640 0.656 3/5 0.607 0.723 0.753 0.766 4/5 0.766 0.853 0.872 0.880 1 1.00 1.00 1.00 1.00 48 Table 13: Dimensionless temperatures along s-axis when 0 = 3w/2 and b varies. n b=3.0 b=4.0 b=5.0 b=6.0 -1 0.000 0.000 0.000 0.000 -4/5 0.045 0.057 0.048 0.070 -3/5 0.095 0.119 0.134 0.144 -2/5 0.150 0.187 0.209 0.223 -1/5 0.212 0.261 0.289 0.308 0 0.283 0.344 0.378 0.399 1/5 0.366 0.437 0.474 0.498 2/5 0.465 0.543 0.582 0.605 3/5 0.588 0.666 0.702 0.723 4/5 0.752 0.815 0.840 0.854 1 1.00 1.00 1.00 1.00 49 Table 14: Dimensionless temperatures along s-axis when 0 = n/4. s/l e=0.l €=0.2 €=0.5 e=1.0 1=n/4.8 1=n/9.6 1=n/2.4 =w/48.0 0 0.475 0.452 0.413 0.402 1 0.475 0.453 0.414 0.403 2 0.475 0.454 0.416 0.404 3 0.476 0.456 0.420 0.408 4 0.477 0.459 0.426 0.412 5 0.480 0.465 0.434 0.418 6 0.488 0.475 0.443 0.425 7 0.495 0.485 0.453 0.433 8 0.498 0.491 0.462 0.440 9 0.499 0.494 0.469 0.450 10 0.500 0.497 0.474 0.452 11 0.500 0.498 0.479 0.458 12 0.500 0.499 0.483 0.463 13 0.500 0.499 0.486 0.468 14 0.500 0.500 0.489 0.472 15 0.500 0.500 0.491 0.476 16 0.500 0.500 0.493 0.480 17 0.500 0.500 0.495 0.484 18 0.500 0.500 0.496 0.487 19 0.500 0.500 0.497 0.490 20 0.500 0.500 0.498 0.494 21 0.500 0.500 0.499 0.497 50 Table 15: Dimensionless temperatures along s-axis when 8 = n/2. 5/1 €=0.1 e=0.2 €=0.5 =1.0 =n/2.4 =n/4.8 l=w/12.0 1=n/24. 0 0.475 0.450 0.384 0.337 1 0.475 0.450 0.385 0.338 2 0.475 0.450 0.388 0.343 3 0.475 0.451 0.394 0.350 4 0.475 0.453 0.403 0.361 5 0.477 0.459 0.417 0.375 6 0.488 0.475 0.437 0.393 7 0.498 0.490 0.457 0.412 8 0.500 0.496 0.471 0.428 9 0.500 0.499 0.481 0.441 10 0.500 0.500 0.487 0.452 11 0.500 0.500 0.492 0.461 12 0.500 0.500 0.494 0.468 13 0.500 0.500 0.496 0.474 14 0.500 0.500 0.498 0.479 15 0.500 0.500 0.498 0.483 16 0.500 0.500 0.499 0.487 17 0.500 0.500 0.499 0.490 18 0.500 0.500 0.500 0.492 19 0.500 0.500 0.500 0.494 20 0.500 0.500 0.500 0.496 21 0.500 0.500 0.500 0.498 51 Table 16: Dimensionless temperatures along s-axis when 6 = 3n/4. s/l e=0.1 e=0.2 e=0.5 e=1.0 l=3n/4.8 1=3n/9.6 l=3n/24.0 1=3n/48 0 0.475 0.450 0.374 0.292 1 0.475 0.450 0.375 0.294 2 0.475 0.450 0.378 0.301 3 0.475 0.450 0.383 0.311 4 0.475 0.451 0.392 0.327 5 0.476 0.456 0.408 0.350 6 0.488 0.475 0.437 0.379 7 0.499 0.494 0.465 0.410 8 0.500 0.498 0.481 0.432 9 0.500 0.500 0.489 0.450 10 0.500 0.500 0.494 0.463 11 0.500 0.500 0.497 0.473 12 0.500 0.500 0.498 0.480 13 0.500 0.500 0.499 0.485 14 0.500 0.500 0.499 0.489 15 0.500 0.500 0.500 0.492 16 0.500 0.500 0.500 0.494 17 0.500 0.500 0.500 0.496 18 0.500 0.500 0.500 0.497 19 0.500 0.500 0.500 0.498 20 0.500 0.500 0.500 0.499 21 0.500 0.500 0.500 0.499 52 Table 17: Dimensionless temperatures along s-axis when 6 = W' s/l e=0.1 €=0.2 €=0.5 e=1.0 l=n/l.2 l=n/2.4 l=n/6.0 =n/12.0 0 0.475 0.450 0.371 0.260 1 0.475 0.450 0.372 0.262 2 0.475 0.450 0.373 0.270 3 0.475 0.450 0.374 0.282 4 0.4750 0.450 0.385 0.302 5 0.476 0.454 0.402 0.331 6 0.488 0.475 0.437 0.372 7 0.499 0.496 0.471 0.413 8 0.500 0.499 0.487 0.412 9 0.500 0.500 0.494 0.461 10 0.500 0.500 0.497 0.474 11 0.500 0.500 0.499 0.482 12 0.500 0.500 0.499 0.488 13 0.500 0.500 0.500 0.492 14 0.500 0.500 0.500 0.495 15 0.500 0.500 0.500 0.497 16 0.500 0.500 0.500 0.498 17 0.500 0.500 0.500 0.499 18 0.500 0.500 0.500 0.499 19 0.500 0.500 0.500 0.499 20 0.500 0.500 0.500 0.500 21 0.500 0.500 0.500 0.500 53 Table 18: Dimensionless temperatures along s-axis when 8 = 3n/2. s/l e=0.l €=0.2 €=0.5 €=1.0 1=3n/2.4 l=3n/4.8 l=3n/12.0 l=3n/24. 0 0.475 0.450 0.370 0.216 1 0.475 0.450 0.370 0.219 2 0.475 0.450 0.370 0.228 3 0.475 0.450 0.372 0.243 4 0.475 0.450 0.377 0.268 5 0.475 0.452 0.393 0.3064 6 0.488 0.475 0.437 0.366 7 0.500 0.498 0.480 0.425 8 0.500 0.500 0.494 0.458 9 0.500 0.500 0.498 0.477 10 0.500 0.500 0.499 0.488 11 0.500 0.500 0.500 0.493 12 0.500 0.500 0.500 0.496 13 0.500 0.500 0.500 0.498 14 0.500 0.500 0.500 0.499 15 0.500 0.500 0.500 0.499 16 0.500 0.500 0.500 0.500 17 0.500 0.500 0.500 0.500 18 0.500 0.500 0.500 0.500 19 0.500 0.500 0.500 0.500 20 0.500 0.500 0.500 0.500 21 0.500 0.500 0.500 0.500 54 Table 19: The dimensionless penetration depth along s-axis to reach the linear values for flat plate. 3n/2 35.3 63495 11.8 e €=0.l €=0.2 €=0.5 e=1.0 n/4 9.16 7.53 3.27 1.83 n/Z 14.4 9.82 6.54 3.66 3n/4 19.6 12.8 8.25 5.50 n 26.2 14.4 8.90 7.33 9.82 55 Table 20: The dimensionless temperatures at the indicated nodes when 6 = n/4. Node e=0.1 e=0.2 e=0.5 €=1.0 l .890 .881 .858 .865 10 .890 .881 .859 .865 19 .890 .881 .860 .866 28 .891 .882 .862 .867 37 .891 .883 .866 .869 46 .892 .886 .870 .872 55 .895 .890 .876 .875 64 .898 .895 .882 .878 73 .899 .897 .886 .880 82 .900 .898 .889 .882 91 .900 .899 .891 .884 100 .900 .899 .893 .886 109 .900 .900 .894 .888 118 .900 .900 .896 .890 127 .900 .900 .896 .891 136 .900 .900 .897 .892 145 .900 .900 .898 .894 154 .900 .900 .898 .895 163 .900 .900 .899 .896 172 .900 .900 .899 .897 181 .900 .900 .899 .898 190 .900 .900 .900 .899 56 Table 21: The dimensionless temperatures at the indicated nodes when 9 = n/2. Node e=0.1 e=0.2 €=0.5 e=1.0 1 0.890 0.880 0.842 0.815 10 0.890 0.880 0.843 0.816 19 0.890 0.880 0.845 0.819 28 0.890 0.880 0.848 0.825 37 0.891 0.881 0.853 0.833 46 0.891 0.883 0.861 0.843 55 0.895 0.891 0.874 0.855 64 0.899 0.897 0.884 0.866 73 0.900 0.899 0.890 0.874 82 0.900 0.900 0.894 0.879 91 0.900 0.900 0.896 0.884 100 0.900 0.900 0.897 0.887 109 0.900 0.900 0.898 0.890 118 0.900 0.900 0.899 0.892 127 0.900 0.900 0.899 0.893 136 0.900 0.900 0.900 0.895 145 0.900 0.900 0.900 0.896 154 0.900 0.900 0.900 0.897 163 0.900 0.900 0.900 0.898 172 0.900 0.900 0.900 0.898 181 ‘ 0.900 0.900 0.900 0.899 190 0.900 0.900 0.900 0.900 57 Table 22: The dimensionless temperatures at the indicated nodes when 0 = 3fl/4. Node e=0.1 e=0.2 e=0.5 e=l.0 1 0.890 0.880 0.837 0.761 10 0.890 0.880 0.837 0.763 19 0.890 0.880 0.839 0.770 28 0.890 0.880 0.842 0.781 37 0.890 0.880 0.847 0.797 46 0.891 0.882 0.856 0.818 55 0.895 0.891 0.874 0.844 64 0.900 0.898 0.888 0.864 73 0.900 0.900 0.894 0.875 82 0.900 0.900 0.897 0.883 91 0.900 0.900 0.898 0.888 100 0.900 0.900 0.899 0.891 109 0.900 0.900 0.899 0.894 118 0.900 0.900 0.900 0.895 127 0.900 0.900 0.900 0.897 136 0.900 0.900 0.900 0.898 145 0.900 0.900 0.900 0.898 154 0.900 0.900 0.900 0.899 163 0.900 0.900 0.900 0.899 172 0.900 0.900 0.900 0.899 181 0.900 0.900 0.900 0.900 190 0.900 0.900 0.900 0.900 58 Table 23: The dimensionless temperatures at the indicated nodes when 0 = n. Node €=0.1 €=0.2 e=0.5 €=1.0 1 0.890 0.880 0.835 0.711 10 0.890 0.880 0.836 0.714 19 0.890 0.880 0.836 0.724 28 0.890 0.880 0.839 0.741 37 0.890 0.880 0.843 0.766 46 9.891 0.881 0.853 0.798 55 0.895 0.891 0.874 0.838 64 0.900 0.899 0.890 0.865 73 0.900 0.900 0.896 0.879 82 0.900 0.900 0.898 0.887 91 0.900 0.900 0.899 0.891 100 0.900 0.900 0.900 0.894 109 0.900 0.900 0.900 0.896 118 0.900 0.900 0.900 0.898 127 0.900 0.900 0.900 0.899 136 0.900 0.900 0.900 0.899 145 0.900 0.900 0.900 0.900 154 0.900 0.900 0.900 0.900 163 0.900 0.900 0.900 0.900 172 0.900 0.900 0.900 0.900 181 0.900 0.900 0.900 0.900 190 0.900 0.900 0.900 0.900 59 Table 24: The dimensionless temperatures at the indicated nodes when 6 = 3n/2. Node e=0.1 €=0.2 e=0.5 €=1.0 1 0.890 0.880 0.834 0.635 10 0.890 0.880 0.834 0.640 19 0.890 0.880 0.835 0.654 28 0.890 0.880 0.836 0.679 37 0.890 0.880 0.839 0.715 46 0.890 0.881 0.848 0.767 55 0.891 0.891 0.875 0.835 64 0.895 0.900 0.893 0.871 73 0.900 0.900 0.898 0.886 82 0.900 0.900 0.899 0.892 91 0.900 0.900 0.900 0.896 100 0.900 0.900 0.900 0.898 109 0.900 0.900 0.900 0.899 118 0.900 0.900 0.900 0.899 127 0.900 0.900 0.900 0.900 136 0.900 0.900 0.900 0.900 145 0.900 0.900 0.900 0.900 154 0.900 0.900 0.900 0.900 163 0.900 0.900 0.900 0.900 172 0.900 0.900 0.900 0.900 181 0.900 0.900 0.900 0.900 190 0.900 0.900 0.900 0.900 60 Table 25: The shape factors and heat transfer (q/KATO). 6 e=0.1 €=0.2 €=0.5 e=1.0 n/4 7.19 3.58 1.39 0.581 w/Z 14.4 7.16 2.78 1.18 3n/4 21.6 10.7 4.17 1.75 n 28.7 14.3 5.56 2.30 3n/2 43.3 21.5 8.33 3.38 61 Table 26: The dimensionless maximum temperature deficien- cies and their locations along the n-axis. 0 e n AX 0.1 0 0.025 “/4 0.2 0 0.048 0.5 1/5 0.087 1.0 0 0.098 0.1 0 0.025 ”/2 0.2 0 0.050 0.5 1/5 0.119 1.0 1/5 0.169 0.1 o 0.025 0.2 0 0.050 3"/4 0.5 1/5 0.129 1.0 1/5 0.222 0.1 0 0.025 n 0.2 0 0.050 0.5 1/5 0.133 1.0 2/5 0.269 0.1 0 0.025 0.2 0 0.050 3"/2 0.5 1/5 0.134 1.0 2/5 0.334 62 Table 27: Dimensionless temperatures at two locations along n-axis when h varies and at s = b/2 along s-axis when l varies. n h = 1/3 h = 1/6 h = 1/12 +1/3 0.62235 0.62226 0.62224 -1/3 0.29263 - 0.29256 0.29255 S 1 = b/4 1 = b/8 1 = b/l6 b/2 0.45588 0.45571 0.45568 63 3.3 :2: 949m 953 35.2853 828.2239--. : 3:91 _\m -N ON 0. m_ m. A: . brb r — p — n — h h yak-FWHHMu FWp—p-n— pflth ppmpm OPLM W hut-N» .P- O l 11...1_.di.l.1_411....3..+..atqtq.... on: 1, :HQ rmo 55$ .. whim two : 3:8 1. 0.0 64 1 1 fi ,L : 1 f § 7L 5 . v - 5 1.0 4/5 3/5 2/5 V5 9” 4’5 J 1 v Figure :2.--0imensionless Temperatures Along n-axis wnen 0:2.0 1 . 1 1 1 . -2/5 -3/5 «M; -t 65 3: no :23 m_xm-m 933 3.5.8383 www.cemcmeathfl use: _\m .mommzftfeinm.:o_omhmmvmmlo bu-pb.hLPIbnpvn-i—th-PL-b-p-nbhrhn~.npbrL- . 4(1):)«ddd1d h 1| -.LI—1... —lLbIdIF d1)...«11.4)..44..7.n14.d1414...114114.tfiia- 111L mmd 69 .N Om 0. mp m. 35% 8;; 988% 933 motz.mt......_.E p $255555 1.: 3%: E . 8. m. w. n. m. : 9 +1. omeo mVMm.n TillliiilTill+T+l+1lil1+iTli+11|l1|TTTii+4i11 70 L 1 'l g ‘4/5 '1 1 1 4 1 L l—A L1 1‘44; 0.05%1f,fi1 . 1' 1 I 1 4/5 3/5 2/5 1/5 01) 1 1 -1/5 -2/5 -3/5 Hour: :8." DimenSionicss Temperatures along n—axis When 6:77/4 0.91 1 0.7“ 0.6" l 0.5) 0.41‘ 0.31)- 0.2+ 0.11" 71 5:10 \ 6=o.5 '\ \x \. \ 1 #1 l 0.0 1 , . v V , fiv— 1L § ‘1‘ Ar '- L ; l 1 41/5 3/5 2/5 1/5 0 -1/5 -2/5 -3/5 _4/5 _1 71 Figure ‘9." Dimensionless Temperatures along maxus when 9:77/2 72 e=0. 0 0.5‘L’ 6:0.) 6:02 0.41, .1 6:10 0.3)” 027' 01]) 0.0:5e$:3;1:*1:’r:‘54‘ 1 4/5 3/5 2/5 1/5 0 -)/5 -2/5 -3/5 4.1/5 -1 11 Figure 20.--01men‘3=0ntess Tsmreratures along maps wnpn 9:3.7/2: ab- .0 91 T 0.14- 0.0 1 . 1 .475. 73 /—_~ $00 630.1 6:02 5:05 -2 \ \ ‘ t ‘1 ; 4 : Jr 4 .L at * J. : .L t r 3/5 2/5 1/5 0.0 -1/5 -2/5 ‘3/5 -4/5 -1 71 v figure 21 _--Dimen51001ess Tamperefures along man's when 9:71 74 r 0.8) 0.7‘ 0.416 ‘ . ’0 0.3) 0.24” 4* ‘ ‘ ‘ : 1 t t 1.0 4/5 ' 375 ' 275 175 . 0 ~1/5 ~2/s -3/5 -4/5 ‘1 Figure 22 ." Dimensionless Temperatures atonq n-axis when 9:311/2 0.0 . 1 1 1 1 1 +4 . 1 ‘Y I L -1 75 0.9‘ 0.81 0.7‘ 0.6) 0.51 0.4) 0.3") 0.2: 00.4]11114:111.111gg?\1 . ,‘ 1 , , v T 1 v 1 7 I - 1 4/5 3/5 5/5 1/5 21“";S -2/5 -3/5 -4/5 1 F‘QUT‘? 33¢- otmensiontess Temperatures along n-firis when 6:7.1T 76 ’1‘)— MUS-4 T *3/54 T D=1.0 I *2/5‘ :i/Sit- Linear distribution 06 415(— 1 - 2/51 1)- b=2.0 1 -3/5] I "“5: t :41 t i L f :4171 1 t 14; ,1; 1 0.2 0.3 0.4 0.5 0.6 0.7 018 0.9 1.0 X “gun? 24_—«LOC31100 or mp intersection oi the diffreni isotherms with 11-31115 when 9:0/2 A 1 Yr. .1 0.0 0 77 {p :1» W. Jr Read 1’ ,——D u .. 3t ‘L T) “l/ ' 4» n *3 -r 3 - LU . 1.. 4L 10 . .( )- . Jr 4)- IIF «b ‘P 1‘ - 'J 033‘- ) «r291 1 1 1J 1/13 1,”. 1”” :1 Tigure :5 ——The Dimensionless Temperatures verses h at Two I. Q Locacions Alone "1-:ri: 753 r"; \ A) (J '4- 1L- " Q—J F + Read 0 " - 7.‘ ') 05.. "' q— _';2 L l 1 r v :3 ‘ Iluu 1/36 L") It ~r ’ -\ >1' ‘\ ' ' ‘. v. «M . .. ~ ~~w~-.~ “ ylgure -b..-..e 31m9n31on¢-;~ .~mc~rqturn. .e‘ua: n it TWO '.— a ’7‘, — . 4~ J,caticns A-3ng -39 ,.-ax-u 79 J PM U -0\\ I) \J’l C dn'j‘ Tr L' C' \ -1 t: L4 Read 1 Y 0 'J 0/16 b/U b/ll ‘ W. K o h J v P‘J - W"! ‘ " ~ ‘ ‘m '71, . . ,. a -_‘ ". " A Figure ;r.—-fhe Jimenuioniebu .cmonratdrug 4t ”-n/- ALsnR The . \ 2 ~ s-axis verses l ;b0ttom Line) and l nton Llne) APPENDIX 10 15 20 25 30 35 000000000000 APPENDIX THE COMPUTER PROGRAM PROGRAM ATHENA REAL A(198,l98),B(198,l),E(9),G(9),U(5),WKAREA(198), Z(9) INTEGER L(ll),M,N,MA,IB,IDGT,IER OPEN(20,FILE='OUTPUT') IN THIS PROGRAM, FIRST THE VALUES OF 5 , l,h,b AND 9 ARE CALCULATED AND THEN THE SUBROUTINES SKYl AND SKYZ ARE CALLED TO FORM THE ELEMENTS OF A AND B MATRICES. THEN THE SUBROUTINE 'LEQTlF' IS CALLED TO SOLVE THE EQUATION AX=B. VARIABLES:L(1) IS THE NUMBER OF DIVISIONS ALONG n-axis L(2) IS THE NUMBER OF DIVISIONS ALONG s-axis IN THE CURVED SEGMENT. L(3) IS THE NUMBER OF DIVISIONS ALONG s-axis ONLY IN THE STRAIGHT SEGMENTS OF THE PLATE. C REPRESENTS l WHICH IS THE LENGTH OF MESH ALONG s—axis. H REPRESENTS h WHICH IS THE LENGTH OF MESH ALONG n-axis. W STANDS FOR e=a/R. L(l)=9 L(2)=5 L(3)=15 L(4)=L(l)*L(2)+L(l) L(5)=L(l)*L(2)+L(l)*L(3)+2*L(1) L(6)=L(1)+l L(8)=L(l)*L(2)+L(l)+l L(9)=L(l)*L(2)+2*L(l) L(10)=L(5)-L(l) L(ll)=L(2)+1 U(1)=L(1l) HERE THE MESH LENGTHS ARE CALCULATED. C IS EQUAL TO SMALL 1 C=B/U(l) U(2)=L(6) H=2.0/U(2) W=0.5 NOW THE ELEMENTS OF THE A MATRIX ARE BEING FORMED. R=(C**2)/(H**2) U(3)=-(2.0+2.0*R) D=((L(l)-l)/2)+l DO 2 I=1,L(l) 8O 4O 45 50 55 60 65 70 75 80 85 90 00k) 81 D=D-l.0 E(I)=-(2.0+R*2.0*((l.0-W*H*D)**2)) F(I)=(R*((l.0-W*H*D)**2)+w*(W*D*H-l.0)*(C**2)/(2.0*H)) G(I)=R*((l.0-W*H*D)**2)-W*(W*H*D—l.0)*((C**2)/(2.0*H) Z(I)=E(l)-R*2.0-2.0 IF(I.EQ.1)THEN 0(4)=F(I) U(5)=F(I)+R ELSE END IF CONTINUE HERE IT MAKES!\CALL1I)THE SUBROUTINE SKYl WHICH FORMS ELEMENTS OF MATRIX A CALL SKYl(A,L,E,G,F,Z,U) SUBROUTINE SKYI REAL A(198,l98),E(9),F(9),G(9),U(9),Z(9) INTEGER L(11) DO 3 I=l,L(ll) D) 4 L=1,L(5) IF(I.EQ.J)THEN A(I,J)=E(l) ELSE IF(I.EQ.L(I).AND.J.EQ.L(6))THEN A(I,J)=0.0 ELSE IF(J.EQ.I+1)THEN A(I,J)=G(i) ELSE IF(J.EQ.L(1)+I)THEN A(I,J)=2.0 ELSE IF(J.EQ.I-1)THEN A(I,J)=F(I) ELSE A(I,J)=0.0 END IF CONTINUE CONTINUE DO 8 I=L(6),L(4) DO 9 J=1,L(5) K=I/L(l) M=l-K*L(l) L(7)=K*L(l)+l IF(I.EQ.J.AND.I.NE.K*L(1))THEN A(K,J)=E(M) ELSE IF (I.EQ.J.AND.I.EQ.K*L(I))THEN N=L(l) A(I,J)=E(N) ELSE IF(I.EQ.L(7).AND.J.EQ.I-I)THEN A(I,J)=0.0 ELSE IF(I.EQ.K*L(1).AND.J.EQ.I+1)THEN A(I,J)=0.0 ELSE IF(J.EQ.I~1.ANDI.NE.K*L(1))THEN A(I.J)=F(M) ELSE IF(J.EQ.I-1.AND.I.EQ.K*L(1))THEN N=L(l) A(I,J)=E(N) 100 105 110 115 120 125 130 135 140 145 150 13 12 82 ELSE IF(I.EQ.L(7).AND.J.EQ.I-1)THEN A(I,J)=0.0 ELSE IF(I.EQ.K*L(1).AND.J.EQ.I+1)THEN A(I,J)=0.0 ELSE IF(J.EQ.I-1.AND.I.NE.K*L(l))THEN A(I,J)=F(M) ELSE IF(J.EQ.I-1.AND.I.EQ.K*L(1))THEN N=L(l) A(I,J)=F(N) ELSE IF(J.EQ.I+1)THEN A(I,J)=G(M) ELSE IF(J.EQ.I-L(1))THEN A(I,J)=l.0 ELSE IF(J.EQ.I+L(1))THEN A(I,J)=l.0 ELSE A(I,J)=0.0 END IF CONTINUE CONTINUE DO 12 I=L(8),L(9) DO 13 J=1,L(5) K=I-L(4) IF(I.EQ.J)THEN A(I,J)=Z(K) ELSE IF(I.EQ.L(8).AND.J.EQ.I-1)THEN A(I,J)=0.0 ELSE IF(I.EQ.L(9).AND.J.EQ.I+1)THEN A(I,J)=0.0 ELSE IF(J.EQ.I+1)THEN A(I,J)=G(K)+R ELSE IF(J.EQ.I-1)THEN A(I,J)=F(K)+R ELSE IF(J.EQ.I-L(l))THEN A(I,J)=2.0 ELSE IF(J.EQ.I+L(1))THEN A(I,J)=2.0 ELSE A(I,J)=0.0 END IF CONTINUE CONTINUE DO 17 I=L(9(+1,L(S) DO 16 J=1,L(5) K=I/L(1) N=L(l) IF(I.EQ.J)THEN A(I,J)=U(3) ELSE IF(I.EQ.N*K.AND.J.EQ.I+1)THEN A(I,J)=0.0 ELSE IF(I.EQ.N*K+1.AND.J.EQ.I-1)THEN A(I,J)=0.0 ELSE IF(J.EQ.I+1)THEN 83 A(I,J)=R ELSE IF(J.EQ.I-1)THEN A(I,J)=R ELSE IF(I.LT.L(10)+1.AND.J.EQ.I+L(1))THEN 155 A(I,J)=1.0 ELSE A(I,J)=0.0 END IF 160 16 CONTINUE 17 CONTINUE C HERE IT MAKES}\CALL1K)THE SUBROUTINE SKY2 WHICH FORMS C THE ELEMENTS OF MATRIX B CALL SKYZ 165 SUBROUTINE SKY2 REAL B(198,1),U(5) INTEGER L(11) DO 25 I=1,L(11) IF(I.EQ.J)THEN 170 D0 25 I=1,L(1) IF(I.EQ.1)THEN ELSE B(I,1)=0.0 175 END IF 25 CONTINUE DO 26 I=L(6),L(4) M=I/L(1) IF(I.EQ.M*L(1)+1)THEN 180 B(I,1)=-U(4) ELSE B(I,1)=0.0 END IF 27 CONTINUE 185 D0 28 I=L(9)+1,L(10) M=I/L(1) IF(I.EQ.M*L(1)+1)THEN B(I,1)=-R ELSE 190 B(I,1)=0.0 END IF 28 CONTINUE D=L(6) DO 29 I=L(10)+1,L(5) 195 D=D-l.0 T=D/U(2) IF(I.EQ.L(10)+1)THEN B(I,1)=-(R+T) ELSE 200 B(I,l)=-T END IF 29 CONTINUE C HERE THE DATA NEEDED FOR THE CALL FROM "IMSL" ARE SET UP N=L(5) 84 205 IDGT=3 IB=L(5) IA=L(5) M=1 C HERE THE CALL IS MADE TO THE ROUTINE "LEQTlF". 210 CALL LEQTlF(A,M,N,EA,B,IDGT,WKAREA,IER) C NOW THE OUTPUT IS CALLED TO PRINT THE RESULT OF C SOLUTION. DO 31 J=1,L(5) WRITE(20,30)B(J,1) 215 30 FORMAT(60X,F8.7) 31 CONTINUE STOP 218 END LIST OF REFERENCES LI ST OF REFERENCES Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, University Press, Oxford, 2nd Edition, 1959, pp. 445-446. Ozisik, M.N., Heat Conduction, Wiley, New York, 1980, pp. 471-516. Carslaw, H.S., Introduction to the Mathematical Theory of the Conduction of Heat in Solids, Dover, New York, 1945. Wang, C.Y., "A New Analytical Method for the Heat Con- duction Across a Bent Plate," Letters in Heat and Mass Transfer, Vol. 9, 1982, pp. 199-207. Wang, C.Y., "Flow in Narrow Curved Channels," Journal of Applied Mechanics, Vol. 47, 1980, pp. 7-10. Holman, J.R., Heat Transfer, McGraw Hill, 4th Edition, 1976, PP. 63-86. Ozisik, M.N., Boundary Value Problems of Heat Conduc- tion, International Text Book, Pa., 1968. Sokoinikoff, 1.8., Tensor Analysis--Theory and Applica- tionstx>Geometry and Mechanics of Continua, Chapter 3, Wiley, New York, 1964. 85