THESIS This is to certify that the thesis entitled THE EFFECT OF CURVATURE AND TORSION ON THE TEMPERATURE DISTRIBUTION IN A HELIX presented by David D. Sayers has been accepted towards fulfillment of the requirements for M.S. degree in Mechanical Engineering W (2.6%: Major professor Date 5 I 83 0—7639 MS U is an Affirmative Action/Equal Opportunity Institution IVIESI_] RETURNING MATERIALS: Place in book drop to LIBRARJES remove this checkout from .—:I-—- your record. FINES will be charged if book is ’returned after the date stamped below. I. "I. . I 7 Q fee .a ti 1‘. .1“, a /<;d— 9%))” 4;) / THE EFFECT OF CURVATURE AND TORSION ON THE TEMPERATURE DISTRIBUTION IN A HELIX By David D. Sayers A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE DEPARTMENT OF MECHANICAL ENGINEERING 1983 ABSTRACT THE EFFECT OF CURVATURE AND TORSION ON THE TEMPERATURE DISTRIBUTION IN A HELIX By David D. Sayers Traditional analysis treats the helix as a straight wire with the effects of nonuniform heating, torsion and large curvature ignored. Using a helical coordinate system the governing partial differential equation including these effects is derived. The equation is then solved numerically using the finite element method. The results indicate a strong dependence of the temperature on the torsion parameter when the curvature parameter is significant. As the curvature parameter increases the temperature distribution becomes skew-symmetric and the maximum temperature in the helix increases. Non- uniform heating influences the temperature distribution independent of the curvature and torsion. ACKNOWLEDGEMENTS The author wishes to express his thanks to his major professor, Dr. Merle C. Potter, Professor of Mechanical Engineering, for his suggestions, guidance and encouragement throughout this study. The author also wishes to thank Dr. L. J. Segerlind and Dr. C. Y. Wang for serving as members of his committee. Also, thanks go to Judy Duncan for her assistance in preparation of the final manuscript. Finally, the author expresses his gratitude to his parents, Vernon and Shirley Sayers, and sister, Chera, without whose support and encouragement this study would not have been possible. ii TABLE OF CONTENTS LIST OF TABLES .................................................. v LIST OF FIGURES ................................................. vi NOMENCLATURE .................................................... viii CHAPTER I - INTRODUCTION ........................................ 1 1.1 REVIEW OF LITERATURE ..................................... 1 1.2 OBJECTIVE OF THE PRESENT STUDY ........................... 2 CHAPTER II - FORMULATION OF THE PROBLEM ......................... 5 2.1 COORDINATE SYSTEM ........................................ 5 2.2 GOVERNING EQUATION ....................................... 7 2.3 TRANSFORMATION OF THE GOVERNING EQUATION ................. 8 2.4 DIMENSIONLESS PARAMETERS ................................. 10 2.5 LIMITS ON PARAMETERS ..................................... 11 CHAPTER III - FINITE ELEMENT METHOD ............................. 13 3.1 THE QUADRILATERAL ELEMENT ................................ 13 3.2 GALERKIN'S METHOD ........................................ 15 3.3 DEVELOPMENT OF EQUATIONS ................................. 16 3.4 NUMERICAL INTEGRATION .................................... 17 3.5 GRIDS .................................................... 17 3.6 COMPUTER PROGRAM ......................................... 20 CHAPTER IV - RESULTS AND CONCLUSIONS ............................ 23 4.1 NUMERICAL RESULTS ........................................ 23 4.2 CONCLUSIONS .............................................. 25 iii APPENDIX ................................................. , ...... 5 4 LIST OF REFERENCES ............................................. 64 iv Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table OGDNOSUT-bwfi) ._a._i_.I.—a._a._a_1._| \IO‘UT-bWN—JO LIST OF TABLES Numerical Integration Points and Weighting Coefficients ....................................... l8 Dimensionless TemperatUres for 8=0.5. x=0.0 ........ 26 Dimensionless Temperatures for e=l.0, x=0.0 ........ 27 Dimensionless Temperatures for B=2.0, A=0.0 ........ 28 Dimensionless Temperatures for =0.5, A=l D ........ 29 Dimensionless Temperatures for B=l.0, A=l 0 ........ 30 Dimensionless Temperatures for 8=2.0, A=l 0 ........ 3l Dimensionless Temperatures for =0.5, A=2 0 ........ 32 Dimensionless Temperatures for B=l.D, A=2.0 ........ 33 Dimensionless Temperatures for 8=2.0, A=2.0 ........ 34 Dimensionless Temperatures for =0.5, A=2.3 ........ 35 Dimensionless Temperatures for 8=l.0, A=2.3 ........ 36 Dimensionless Temperatures for B=2.0, x=2.3 ........ 37 Maximum Dimensionless Temperatures Maximum Dimensionless Temperatures Maximum Dimensionless Temperatures Maximum Dimensionless Temperatures for A=0.0 ....... 38 for X=1.0 ....... 39 for A=2.0 ....... 40 for A=2.3 ....... 4l Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10 11 12 13 14 15 16 LIST OF FIGURES The Helical Coordinate System ....................... The Cylindrical Coordinate System ................... Helical Coils with e=D.l (a) B=3.0, (b) B=0.5, (c) B=0.2 and (d) 8=0.l ............................. A Typical Finite Element with Nodes ................. Numerical Integration Points on an Interior Element ............................................. The Element and Node Numbering Scheme ............... Computer Flowchart .................................. Dimensionless Temperature Profiles for A=0.0, B=0.5 ........................................ Dimensionless Temperature Profiles for A=0.0, B=0.5 ........................................ Dimensionless Temperature Profiles for A=l.0, B=0.5 ........................................ Dimensionless Temperature Profiles for A=l.0, B=D.5 ........................................ Dimensionless Temperature Profiles for A=2.0, B=0.5 ........................................ Dimensionless Temperature Profiles for X=2.0, B=0.5 ........................................ Percent Increase in Dimensionless Temperature for B=0.5 ................................. .......... Percent Increase in Dimensionless Temperature for 8=l.0 ........................................... Percent Increase in Dimensionless Temperature for B=2.0 ........................................... vi 3 9 12 14 18 19 21 42 43 44 45 46 47 48 49 50 Figure l7 Percent Increase in Dimensionless TemperatUre for A=l.0 ........................................... 51 Figure l8 Percent Increase in Dimensionless Temperature ....... 52 Figure 19 Helix Configuration for e=1.0 ....................... 53 vii Letters > JUL 1:; {X} {Y} {¢} Dimensionless nonuniform heating parameter- Torsion [m-1] Dimensionless torsion parameter Curvature [m'1] Dimensionless curvature parameter Angle coordinate Christoffel symbol Local element coordinate Local element coordinate Unit binormal vector in the helical system Unit vector in the global system Unit vector in the global system Unit vector in the global system Unit normal in the helical system The eight shape functions Position vector of a point in the helix [m] Radius vector to the helix centerline [m] Unit tangent vector in the helical system The eight nodal coordinates The eight nodal coordinates The eight nodal temperatures ix CHAPTER I INTRODUCTION l.l REVIEW OF LITERATURE The temperature distribution and the maximum temperature in a helix subject to internal heat generation is the topic of this study. A situation in which this information would be necessary would be in the design of an electrical space heater. Traditional analysis often 1/ treats the helix as a straight cylinder with the effects of curvature,—- 2/ torsion,—- and nonuniform heating ignored. Jakob [l] was the first to deal with the effect of variable heat flux on the temperature distri- bution in a solid, straight cylinder. All effects relating to curvature and torsion were ignored because they were believed to be negligible. Jakob's solution to the governing differential equation involves a com- bination of zeroth order Bessel functions of the first kind. It was observed that a temperature of infinity is predicted at the first zero of the zeroth order Bessel function. Physically, heat is being generated so fast near the center of the coil that conduction can no longer carry it away and the coil melts. Jakob also noted that this occurs at a surprisingly low current. l/The radius of curvature is the minimum radius of a segment of the coil; curvature is the inverse of this radius of curvature. -2-/Torsion is related to the distance between the loops measured along the axis of the coil. In his second work, Jakob [2] was the first to discoVer that as much as 40 percent more heat per unit volume is generated at the warmest spot in the coil as compared to a location just below the surface, this is due to the increase in electrical resistance with temperature. His work also showed that nonuniform generation of heat does not significantly affect the shape of the temperature profile in the coil, however, it does affect the maximum temperature generated in the coil. Nicholson [3] was the first to attempt a definition of a helical coordinate system (see Figure l). The coordinate system was erroneously thought to be orthogonal and was subsequently published as such (see reference [4]). The helical coordinate system was first correctly defined by Wang [5]. Hang used this coordinate system to solve for the effect of curvature and nonuniform heat generation on the temperature distribution in a helix for small curvatures. The basic solution was found to be a combination of Bessel functions of the first kind of the zeroth order; the first and second correction terms for curvature and nonuniform heating involved Bessel functions of the first kind of zeroth, first and second order. Hang also correctly noted that torsion was insignificant for small curvatures. l.2 OBJECTIVE OF THE PRESENT STUDY The objective of the present study is to investigate the effect of curvature and torsion on the temperature distribution in a helix. The helix is assumed to be subjected to internal heat generation due to Joule heating (12R). Allowance is made for the variation of the elec- trical resistance with temperature; this leads to nonuniform heat generation. It is assumed that the outer surface of the helix is :N' ‘0 Figure l.--The Helical Coordinate System maintained at a uniform temperature To by forced convection. The dif- ference between this study and previous work is that the effect of torsion is included along with the effects of curvature and nonuniform heating. Previously, nonuniform heating had been modeled, neglecting the effects of torsion and large curvature; Wang [5] did include the effect of small curvature. Nhen torsion is neglected and curvature is assumed small, the solution to the problem can be determined analytically. The present study does not neglect torsion and large curvature and as a result the governing equation becomes more complicated and must be solved numerically. The finite element method was selected as the method of solution. Finite elements were used because grids can be changed rapidly and variable coefficients do not make the solution significantly more difficult. CHAPTER II FORMULATION OF THE PROBLEM The mathematical formulation of the problem is discussed in this Chapter. The partial differential equation which describes the problem is presented in cylindrical coordinates. Since the finite element method is usually formulated in rectangular coordinates, a transformation is made from cylindrical to rectangular coordinates. The dependent and indepen- dent variables are then expressed in dimensionless form in order to identify the governing parameters and to generalize the results. 2.l COORDINATE SYSTEM The radius vector locating a point on the centerline is described by the space curve mn=xon+vofi+7efl h) The helical coordinate system with unit vectors (T, N, B) can be used to describe the radius vector of Equation (l), as shown in Figure l. The vectors T, N and B are mathematically defined by lo. T = ds (2) ~-1fi. N - a dS (3) é=ixfi (4) Here a is the curvature and T, N and B are orthogonal unit vectors. Using the Frenet formulas, the following relationships result: fl - 3% - Té - a? (5) dA A 8.2. = ’TN (6) where T is the torsion. The coordinate system (r, e, 5), shown in Figure l, is constructed such that the Cartesian position vector P can be expressed as P(s) = R(s) + rcose N(s) + rsine 8(5) (7) Note that when 6 is zero, N and r are colinear. Using Equations (2) through (7) there results dP¥dP = (dr)2 + r2(de)2 + [(l-arcose)2 + Izrz] (ds)2 (8) + erzds de The metric tensors gij and 913 and the Christoffel symbols r:j are necessary when expressing the tensorial heat transfer equation in the chosen coordinate system. The metric tensors gij and 913 are expressed, referring to Equation (8), as where 2 2 e (1 - arcose)2 + T r (10) M (l - arcose)2 (11) The nonzero Christoffel symbols are 1- 1.. 1.1% 2.1 P22 ' 'r’ r23 ‘ ‘Tr’ P33 ‘ ‘2 ar’ P21 r 2-19.429. 2.2.92 2-..-G_a_e_ r13 ‘ M (r 2 3r) r23 2M 39’ I‘33 ' 2 so (‘2) 2r M 2 3_-Tr laG 3_i__s_ 3=_-:_g_G_ I"13‘ M +2137" r23-2Mae I‘33 2M 39 These will now be used to write the governing equation in the coordinate system chosen. 2.2 GOVERNING EQUATION The steady-state, tensorial heat transfer equation for conduction in a solid with heat generation [5] is .. 2 -q 13 a T k 3T 0 9 [—~—:— - I'.. ]=——— (13) ax‘axJ ‘3 axIE k where qo accounts for the heat generation per unit volume. Substitut- ing Equations (9) through (l2) into Equation (l3) results in the partial differential equation which describes the temperature field, namely, . 2 2 arcose a51ne 1 r l Trr + (1 ' T - arcose) F'T r (l - arcose) [1 ' (1 _ arcose)2] F'Te 14 Izrz 1 -q0 ( ) + [1 + J"? Tee = ’TT'[1 +'6(T'To)] (l - arcose)2 r where the curvature is expressed as C a . <15) b2+c2 and the torsion is b 1: = (16) b2+C2 The parameter 5 is a measure of the nonuniform heating, a uniform heat flux occurring when 6 = O. The heat flux is assumed to be produced by electrical resistance (Joule) heating. The resistance of a material is not constant but varies with temperature; 5 is a means to provide for this variation. As an example, copper has a value of 5 equal to 0.004°c". Equation (I4) is linear, nonhomogeneous and has variable coeffi- cients. The boundary conditions are T is finite at r = o (17) T = T0 at r = a (l8) This last condition is assumed to be maintained by forced convection. 2.3 TRANSFORMATION OF THE GOVERNING EQUATION Equation (l4) was developed in the helical coordinate system. A rectangular coordinate system (x, y) is generally used with the finite element method; therefore, Equation (l4) was transformed to the rectangular coordinate system. The standard transformation relations, shown in Figure 2, are Figure 2.--The Cylindrical Coordinate System x = rcose y = rsine and 2 2 2 x + y = r Using the chain rule —-1 1 n .1 x x 1 + is: 1 .4 (D n .4 x x CD + ‘5: CD .4 II 2 2 rr Txx(xr) + 2Txyxryr + Tyy(yr) and )2 2 Tee Txx(xe) + 2Txyxeye + Tyy(ye (22) (23) (24) (25) Substituting Equation (l9) through (25) into Equation (14) results in 10 22 22 22 TV 22 Tr TXXIX +y (1 + 2)] + Tnyy +x (1 + 2)] (1-ax) (1-ax) _ 2T XZTZY'Z + T ['GXZ _ M2 (1 _ ‘1'er )_ XTZY‘Z xy(1.0x)2 X (l-ax)2 (l-ax)2 (26) 2 2 + T [xya -r r _ yr r ) y l-ax ((l-ax)2) (l-ax)2 ] q 72 [1 + 6(T-T )](x2+y2) o The quantities in this equation are dimensional; it is more convenient to work with normalized, dimensionless quantities. 2.4 DIMENSIONLESS PARAMETERS The transformed Equation (26), can be placed in a more general form by introducing dimensionless parameters. Choose as reference quantities the helix radius (a), the thermal conductivity (k) and the heat flux (qo). The dimensionless variables are then e = aa (27) B = ta/e = b/C (28) Y = y/a (29) X = x/a (30) _ 2 ¢ - (T-Tolk/qoa (31) X2 = anOB/k and (32) D = Bzezilg+YZJ (33) (l-eX)2 Substituting these dimensionless groups into Equation (26) the partial differential equation that will be solved is 11 ¢XXLX2 + Y2(I+D)] + TYYIYM +x2( 2 eY2 -eX * ¢x[1-ex 1- :x (I 0’ X0] + ¢Y[T l+D)] + ¢XY[— -2XYD] “EXX - 1] YD (34) + tmzn (x2+v2) = with the boundary conditions ¢ = O at X2 + Y2 = l (35) 2 2 = o (36) a is finite at X + Y All of the quantities in Equation (34) are dimensionless and ¢ is of order one. 2.5 LIMITS ON PARAMETERS Several of the dimensionless parameters have bounds which are not apparent from their definitions. For example, 6 is bounded by the fact that loops of the helix can at most touch each other; it is not physically possible for different coil loops to occupy the same physical space. This limits 6 to a maximum of one. A zero value for e corresponds to a straight wire and the solution is independent of 8. Conversely, if 8 goes to infinity the solution also approaches the straight wire case (see Figure 3). Also, at X equal 2.4048 a singularity exists such that X is limited by this value. Figure 3.--Helical coils with :=O.1 (a) 5 (c) E=O.2 and (d) E=O.l CHAPTER III FINITE ELEMENT METHOD The finite element method is a numerical procedure with the follow- ing two primary characteristics: l. The problem is expressed in integral form. 2. The solution is approximated by a piecewise smooth continuous function. It utilizes elements that are not necessarily of the same shape but must contain the same number of nodes. For a given number of elements, as the number of nodes per element is increased the accuracy of the solution increases, as does the time required by the numerical procedure. 3.l THE QUADRILATERAL ELEMENT The two-dimensional, eight node, quadratic, quadrilateral element was used in this study; it is shown in Figure 4. The nodes were numbered starting in the lower left corner and proceeding counterclockwise around the element. The eight shape func- tions are: N1 = - l; (1-£)(1-n)(€+n+1) N2 =%(1-:2)(1-n) (37) N3=%U%H%Mka) N4 = 3,— (1-n2)(1+€) 13 14 (l.l) (4,4) 3 (1,-1) Figure 4.--A Typical Finite Element with Nodes 5 % (1+z)(1+n)(z;+n-1) 2 II 2 I1 6 %(1-g2)(1+n) N7 = -%(1-§)(1+n)(£-n+1) and 2 l1 8 %—(1-n2)(1-:) The shape functions have the following properties: I. They have a value of one at their node. 2. They have a value of zero at every other node in the element. 3. They sum to one at any point. 15 A local (5, n) coordinate system is defined for the element to ease the task of numerical integration. This local coordinate system is curvi- linear and is oriented so that the corners of the element have the values shown in Figure 4. The X and Y coordinates at any point in the element are given by [N]{X} and (38) [N]{Y} (39) where the components of the vector [N] are the eight shape functions X Y evaluated at the desired point (a, n) and the components of the vectors {X} and {Y} are the eight values of the X and Y coordinates of the element nodes. Similarly, the temperature at any point is i =[N]{¢} (40) where the components of the vector {¢} are the eight nodal temperatures. 3.2 GALERKIN'S METHOD Galerkin's method is a weighted residual method used to solve a partial differential equation. When Galerkin's method is used in con- junction with finite elements the formulation of the problem, from [6], demands that I [N]T{ [x2+YZ(l+D)] + [Y2+x2(1+o)] + {-2XYD] A qbxx ivv ¢xv + it [”5"2 - ”2 u-o) - xv] + a; ti"— - 11 YD (41) X 1-eX 1-eX Y l—eX + [1+X2¢](X2+Y2)}dA = o where the shape functions serve as weighting functions. Equation (41) is used to determine the element matrices which are then summed to determine the global matrices. The global matrices are then solved for the nodal values. 16 3.3 DEVELOPMENT OF EQUATIONS The element matrices are calculated from the integral formulation of Galerkin's method using Equation (4T). Integrals involving first order derivatives or no derivatives can be evaluated directly. Integrals in- volving second order derivatives must be expressed in terms of first order terms because the temperature equation does not have a continuous first derivative between elements. This is accomplished by using the product rule, (A g-X-(fmwA = [A § 1) dA + [A f 3% dA (42) For example, the integral T 3 IA 37161 3%] dA (43) can be expressed using Equation (42) as 1 2 a T a _ a|N| a T a - IA 3? [[N] 5%] dA - IA ax f} dA + [A [N] B—xéi - dA (44) Green's theorem can then be used to simplify the formulation to T 2 fL[N]T 33% dL = (AigL §% dA + [A [N]T 3x dA (45) 3 where L is a closed loop surrounding the area A. Deleting the line integral term introduces a negligibly small error when quadratic elements are used since slopes at the intersection of two elements are nearly identical. Thus, the integral on the element boundary is deleted. The form then reduces to 2 1 (A [MT 8"? dA "' ' IA AIL:_)I(_ 3% “A ’ (45) 8X 17 This method, applied to all higher order terms which appear in Equation (4l), produces 3N. 3N. 3N. 3N. IA {[(x2+v2u+m)(7‘x— —,—l) + (Y2+x2(1+o))(——3}— —,i) 3N. 3N. 3N. BN. 2 2 1 J 1 J eX eY ' XYD( ax av + av ax) + (l-eX + 1-ex(1’D) + XD)‘ (47) (N ifi) + vo(1 + EX )(N 311-) + X2(X2+Y2)N N ]{ 1 i 3X 1-5x i aY , j 4 + (X2+Y2)Nj} dA = 0 Equation (47) is integrated numerically for each element to produce the desired solution. 3.4 NUMERICAL INTEGRATION The critical term in Equation (47), when determining the order of numerical integration, is the term containing NiNj' This term is sixth order in g and n. All other terms in Equation (47) are of a lower order. The required sampling points and weighting coefficients, as given in [7], are presented in Table l and shown in Figure 5. 3.5 GRIDS The temperature distribution was calculated for three different grid configurations. The first grid consisted of eight elements. The resulting nodal values were within 2 percent of the analytical solu- tion [8], for the case e = A = 0. The second grid consisted of twelve elements. The resulting nodal values were within l percent of the analytical values. The final grid configuration consists of twelve elements, as shown in Figure 6; this grid results in nodal values 18 Table l.--Numerical Integration Points and Weighting Coefficients 513 ”‘1 WC -0.861136 0.347855 -O.339981 0.652145 0.339981 0.652145 0.861136 0.347855 0 o o o O O O O Figure 5.--Numerical Integration Points on an Interior Element 19 Figure 6.--The Element and Node Numbering Scheme 20 which are accurate to within 0.0l percent. This is the grid used to obtain the results presented in Chapter IV. 3.6 COMPUTER PROGRAM The computer program used is a modification of TDHEAT given in [9]. A schematic of the program is shown in Figure 7. Subroutine VALUE evaluates the shape functions and their first derivatives on g and n at each numerical integration point (a, n). Subroutine JMAT then assembles the Jacobian or J matrix, given by 35 35 [J] = (48) .4): .41 an an b— _11 This is accomplished by differentiating Equation (38) and (39) with respect to g and n. Subroutine PARXY then uses the J matrix to convert from partial derivatives on a and n to X and Y using the chain rule since r r ] ax _] 3E i r = [J] i r (49) k BY) L 3n) Subroutine COEF evaluates the coefficients in Equation (47) at each numerical integration point (5, n). Subroutine BDYVAL incorporates the boundary condition Equation (35) on the global stiffness matrix and force vector using the method of rows and columns. Subroutine LEQTlF is an IMSL (International Mathematical and Statistical Libraries [101] subroutine that was used to solve for the nodal values from the 21 Read title, 1, e, ., nodal coordinates. element node numbers L 4 Call VALUE Call JMAT [Call PARXY__] Call COEF Assemble components into element stiffness matrix and force vectOr Last inteoration n0 point? Add element stiffness matrix and force vector to global stiffness matrix and force vector V ngllBDWE—T [#5911 LEOTlF ] [_ta11 OUTPUTV] Figure 7.—-Computer Flowchart 22 global stiffness matrix and force vector. Subroutine OUTPUT uses the nodal values to determine the maximum temperature and its location. OUTPUT then uses the nodal values to calculate the temperatures at various locations on the X- and Y- axis. A program listing and sample output is found in the Appendix. CHAPTER IV RESULTS AND CONCLUSIONS 4.1 NUMERICAL RESULTS Temperature distributions and maximum temperatures were calculated for selected values of e, 8 and A using the finite element method, expressed by Equation (47). Tables 2 through 13 present the values of the dimensionless tempera- ture at selected positions for various values of e, B and A. It is evident that increasing 2, B or X results in an increase in the dimensionless temperature at any point. This is shown graphically in Figures 8 to 13. Note that as 5 increases, the temperature distribution is no longer symmetric on the X-axis but remains symmetric with respect to the Y-axis. The actual temperature distribution, however, remains nearly parabolic across either the X- or Y-axis. The maximum temperature increases when- ever a, B or X increases. The maximum temperature always occurs on the X- axis. It is apparent that B has a very minimal effect on the temperature distribution when 6 is small (less than approximately 0.2). Tables 14 to 17 contain the location and value of the maximum dimensionless temperature for various values of e, B and X. As expected, an increase in the maximum temperature is realized whenever a, B or X is increased. As 6 approaches one, the mathematical model breaks down. This is particularly evident for data presented at A equal 2.3. This is not unexpected and was first predicted by Jakob [l], a singularity due to 23 24 the first zero of the zeroth order Bessel function of the first kind exists at X equal 2.4048. Thus, at A equal 2.3 the solution is strongly influenced by this singularity. Figures 14 through 17 show the effect of changing B and X on the percent increase in the maximum dimensionless temperature. Increasing B or X intensifies the effect of e on the maximum temperature. Figure lBis from the paper by Wang [5]; note the similarity to Figures 14 to 17. Wang assumed small values for e and as a result underestimates the effect of increasing 8 since for small a the solution is essentially independent of B. Wang let a attain values greater than one when it is limited phy- sically to a maximum of one. At 6 = 1.0 the coil is wrapped so tightly that it is continually in contact with itself (see Figure 19). Figure 18 does show that as the first zero of the zeroth order Bessel function is approached the maximum temperature becomes unbounded, independent of the value of e or B. It is apparent from Tables 2 to 13 that as e approaches one the numerical scheme used to solve the problem breaks down. This is due to the terms involving l/(l-eX) in the governing equations becoming very large, i.e., 1-eX approaches zero. Just before this occurs the con- dition of a double maximum occurs. Of these two maxima the larger maximum always occurs at the larger value of X. Since c has a value near one the isothermal boundary condition should be replaced by a convection boundary condition. Forced convection is ineffective where loops of the coil are very close and as a result the temperature of the fluid surrounding the coil increases above To. This could result in the need to use a different numerical method to solve the problem. 25 4.2 CONCLUSIONS Based on the results obtained and discussion of the previous section it can be concluded that: 1. As 8 increases ¢max increases and ¢max occurs at an ever increasing value of X. For 5 = 0, ¢max = .2506 at X = 0; for e = 0.5, ¢max = .2644 at X = 0.070 (B = 1.0 and X = 0). As 8 increases Pmax increases and the effect of e is amplified. For B = 0.5, ¢max = 0.3163; for B = 1.0, Omax = 0.3288 (8 = 0.5 BHd A = 1.0). Increasing 1 results in an increase in ¢max- For A = 1.0, ¢max = 0.3099; for A = 2.0, ¢max = 0.8824 (c = 0.2 and B = 1.0). The temperature profile is not affected by changes in e, B or X to the extent that ¢max is affected. For small values of a (less than approximately 0.2) 8 does not have a significant influence on the maximum temperature,¢max,or the temperature distribution. For B = 1.0, ¢max = 0.3099; for B = 2.0, ¢max = 0.3163, a change of less than 2 percent (a = 0.2 and X = 1.0). Maximum temperatures always lie on the X—axis. The temperature distribution is always symmetrical about the Y-axis. It is evident that the torsion parameter 8 cannot be neglected if e > 0.2. Also, when 6 > 0.2 the higher order terms in the asymptotic expansion technique used by Wang [5] become significant and must be included. 26 Table 2.--Dimensionless Temperatures for 3:0.5, )=0.0 'X c=0 c=.2 c=.4 e=.5 e=.7 c=.8 c=.9 ~1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0308 0.0296 0.0288 0.0285 0.0281 0.0280 0.0281 -0.8750 0.0595 0.0573 0.0559 0.0553 0.0546 0.0545 0.0546 -0.8125 0.0860 0.0830 0.0811 0.0804 0.0794 0.0793 0.0797 —0.7500 0.1103 0.1068 0.1045 0.1036 0.1026 0.1026 0.1032 -0.6B75 0.1325 0.1287 0.1261 0.1252 0.1242 0.1242 0.1253 -0.6250 0.1526 0.1486 0.1459 0.1450 0.1441 0.1443 0.1458 -0.5625 0.1705 0.1665 0.1639 0.1630 0.1624 0.1628 0.1647 -0.5000 0.1862 0.1825 0.1801 0.1793 0.1790 0.1797 0.1822 -0.4375 0.2011 0.1976 0.1955 0.1949 0.1950 0.1961 0.1993 -0.3750 0.2141 0.2109 0.2091 0.2087 0.2093 0.2109 0.2151 -0.3125 0.2251 0.2224 0.2210 0.2208 0.2221 0.2242 0.2294 -0.2500 0.2341 0.2319 0.2311 0.2312 0.2333 0.2360 0.2425 -0.1875 0.2412 0.2396 0.2393 0.2400 0.2428 0.2462 0.2542 -0.1250 0.2463 0.2454 0.2459 0.2468 0.2508 0.2550 0.2645 —0.0625 0.2495 0.2493 0.2506 0.2520 0.2571 0.2623 0.2735 0.0000 0.2506 0.2513 0.2535 0.2554 0.2619 0.2680 0.2811 0.0625 0.2495 0.2510 0.2541 0.2565 0.2642 0.2711 0.2851 0.1250 0.2463 0.2486 0.2526 0.2555 0.2645 0.2721 0.2870 0.1875 0.2412 0.2442 0.2491 0.2525 0.2626 0.2711 0.2869 0.2500 0.2341 0.2378 0.2434 0.2473 0.2587 0.2680 0.2847 0.3125 0.2251 0.2293 0.2357 0.2400 0.2527 0.2629 0.2805 0.3750 0.2141 0.2189 0.2259 0.2306 0.2446 0.2557 0.2742 0.4375 0.2011 0.2063 0.2139 0.2191 0.2344 0.2465 0.2659 0 5000 0.1862 0.1918 0.1999 0.2055 0.2221 0.2352 0.2555 0.5625 0.1705 0.1765 0.1856 0.1922 0.2135 0.2338 0.2797 0.6250 0.1526 0.1587 0.1682 0.1753 0.1995 0.2243 0.2879 0.6875 0.1326 0.1384 0.1478 0.1549 0.1800 0.2069 0.2800 0.7500 0.1104 0.1157 0.1243 0.1309 0.1549 0.1815 0.2561 0.8125 0.0860 0.0905 0.0978 0.1135 0.1244 0.1481 0.2161 0.8750 0.0595 0.0628 0.0683 0.0725 0.0884 0.1067 0.1601 0.9375 0.0308 0.0326 0.0357 0.0380 0.0470 0.0573 0.0881 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0000 27 Table 3.--Dimensionless Temperatures for B=l.0, )=0.0 X c=0 e=.2 c=.4 e=.5 c=.7 e=.8 c=.9 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0308 0.0298 0.0294 0.0294 0.0297 0.0304 0.0346 -0.8750 0.0595 0.0576 0.0569 0.0569 0.0577 0.0591 0.0679 -0.8125 0.0860 0.0835 0.0826 0.0826 0.0840 0.0862 0.0999 -0.7500 0.1103 0.1074 0.1064 0.1065 0.1086 0.1116 0.1307 -0.6875 0.1325 0.1293 0.1283 0.1284 0.1314 0.1355 0.1603 -0.6250 0.1526 ’0.1492 0.1484 0.1489 0.1526 0.1576 0.1885 -0.5625 0.1705 0.1672 0.1667 0.1674 0.1720 0.1781 0.2156 -015000 0.1862 0.1832 0.1830 0.1840 0.1896 0.1970 0.2413 -0.4375 0.2011 0.1984 0.1987 0.2000 0.2068 0.2157 0.2715 -0.3750 0.2141 0.2118 0.2125 0.2142 0.2224 0.2331 0.3013 -0.3125 0.2251 0.2232 0.2245 0.2266 0.2364 0.2490 0.3306 -0.2500 0.2341 0.2328 0.2348 0.2374 0.2488 0.2635 0.3595 -0.1875 0.2412 0.2405 0.2433 0.2464 0.2597 0.2767 0.3879 -0.1250 0.2463 0.2463 0.2500 0.2537 0.2690 0.2884 0.4159 -0.0625 0.2495 0.2502 0.2549 0.2592 0.2768 0.2988 0.4433 0.0000 0.2506 0.2523 0.2580 0.2631 0.2834 0.3078 0.4704 0.0625 0.2495 0.2520 0.2587 0.2644 0.2859 0.3116 0.4746 0.1250 0.2463 0.2496 0.2573 0.2636 0.2867 0.3133 0.4726 0.1875 0.2412 0.2452 0.2538 0.2607 0.2854 0.3130 0.4643 0.2500 0.2341 0.2388 0.2482 0.2557 0.2821 0.3106 0.4496 0.3125 0.2251 0.2304 0.2406 0.2486 0.2768 0.3061 0.4287 0.3750 0.2141 0.2199 0.2308 0.2394 0.2694 0 2995 0.4015 0.4375 0.2011 0.2073 0.2190 0.2281 0.2600 0.2909 0.3679 0.5000 0.1862 0.1928 0.2050 0.2148 0.2485 0.2801 0.3281 0.5625 0.1705 0.1775 0.1911 0.2026 0.2483 0.3069 0.7056 0.6250 0.1526 0.1597 0.1739 0.1862 0.2392 0.3161 0.9636 0.6875 0.1326 0.1394 0.1533 0.1656 0.2214 0.3075 1.1019 0.7500 0.1104 0.1166 0.1293 0.1409 0.1947 0.2813 1.1207 0.8125 0.0860 0.0912 0.1020 0.1119 0.1592 0.2375 1.0199 0.8750 010595 0.0633 0.0714 0.0788 0.1150 0.1760 0.7995 0.9375 0.0308 0.0329 0.0374 0.0415 0.0619 0.0968 0.4595 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 28 table 4.--Dimensionless Temperatures for 3:2.0. =0.0 X e=0 c=.2 e=.4 e=.5 c=.7 c=.8 ~-1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0308 0.0305 0.0318 0.0333 0.0410 0.0670 -0.8750 0.0595 0.0589 0.0616 0.0645 0.0797 0.1307 -0.8125 0.0860 0.0853 0.0893 0.0935 0.1161 0.1912 -0.7500 0.1103 0.1096 0.1148 0.1203 0.1502 0.2485 -0.6875 0.1325 0.1319 0.1383 0.1451 0.1821 0.3025 -0.6250 0.1526 0.1521 0.1597 0.1677 0.2116 0.3533 -0.5625 0.1705 0.1703 0.1789 0.1881 0.2389 0.4009 -0.5000 0.1862 0.1864 0.1961 0.2064 0.2638 0.4453 -0.4375 0.2011 0.2017 0.2126 0.2241 0.2900 0.5089 -0.3750 0.2141 0.2152 0.2273 0.2401 0.3147 0.5709 -0.3125 0.2251 0.2268 0.2402 0.2543 0.3378 0.6316 -0.2500 0.2341 0.2365 0.2513 0.2668 0.3594 0.6908 -0.1875 0.2412 0.2443 0.2606 0.2776 0.3795 0.7485 -0.1250 0.2463 0.2502 0.2681 0.2866 0.3980 0.8048 -0.0625 0.2495 0.2543 0.2738 0.2939 0.4150 0.8597 0.0000 0.2506 0.2564 0.2777 0.2995 0.4305 0.9131 0.0625 0.2495 0.2561 0.2786 0.3019 0.4316 0.8839 0.1250 0.2463 0.2538 0.2775 0.3009 0.4307 0.8416 0.1875 0.2412 0.2494 0.2743 0.2985 0 4279 0.7861 0.2500 0.2341 0.2430 0.2690 0.2941 0.4232 0.7175 0.3125 0.2251 0.2345 0.2616 0.2877 0.4166 0.6356 0.3750 0.2141 0.2240 0.2521 0.2792 0.4081 0.5406 0.4375 0.2011 0.2115 0.2406 0.2687 0.3977 0.4325 0.5000 0.1862 0.1969 0.2270 0.2562 0.3855 0.3111 0.5625 0.1705 0.1818 0.2152 0.2506 0.4688 1.2009 0.6250 0.1526 0.1639 0.1987 0.2375 0.5145 1.8253 0.6875 0.1326 0.1433 0.1774 0.2168 0.5227 2.1844 0.7500 0.1104 0.1201 0.1514 0.1886 0.4933 2.2782 0.8125 0.0860 0 0941 0.1207 0 1528 0.4263 2.1067 0.8750 0.0595 0.0654 0.0852 0.1094 0.3128 1.6698 0.9375 0.0308 0.0341 0.0450 0.0585 0.1797 0.9676 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 29 Table 5.--Dimensionless Temperatures for 6=0.5, X=l.0 X c=0 c=.2 c=.4 6:.5 £=.7 c=.8 €=-9 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0358 0.0343 0.0333 0.0329 0.0325 0.0324 0-0327 -0.8750 0.0693 0.0666 0.0648 0.0641 0.0634 0.0634 0-0641 -0.8125 0.1008 0.0970 0.0945 0.0937 0.0928 0.0930 0-0941 -0.7500 0.1300 0.1255 0.1226 0.1261 0.1207 0.1210 0-1227 -0.6875 0.1570 0.1520 0.1489 0.1478 0.1470 0.1476 0.1501 -0.6250 0.1819 0.1767 0.1734 0.1724 0.1718 0.1727 0.1760 -0.5625 0.2046 0.1994 0.1962 0.1953 0.1951 0.1964 0-2007 -0.5000 0.2251 0.2202 0.2172 0.2165 0.2169 0.2187 0.2240 -0.4375 0.2442 0.2395 0.2370 0.2364 0.2375 0.2399 0.2455 -0.3750 0.2607 0.2565 0.2544 0.2542 0.2560 0.2591 0-2673 -0.3125 0.2747 0.2711 0.2696 0.2697 0.2725 0.2765 0-2354 -0.2500 0.2862 0.2833 0.2825 0.2831 0.2870 0.2119 0-3038 -0.1875 0.2952 0.2931 0.2932 0.2942 0.2995 0.3055 0-3195 -0.1250 0.3016 0.3005 0.3016 0.3032 0.3100 0.3171 0-3335 -0.0625 0.3056 0.3055 0.3077 0.3100 0.3184 0.3269 0-3459 0.0000 0.3071 0.3081 0.3116 0.3145 0.3248 0.3347 0.3566 0.0625 0.3056 0.3079 0.3126 0.3163 0.3283 0.3393 0-3627 0.1250 0.3017 0.3049 0.3109 0.3153 0.3290 0.3411 0-3551 0.1875 0.2952 0.2994 0.3065 0.3115 0.3270 0.3402 0-3657 0.2500 0.2862 0.2913 0.2994 0.3051 0.3222 0.3366 0-3645 0.3125 0.2747 0.2806 0.2896 0.2958 0.3146 0.3303 0-3595 0.3750 0.2607 0.2672 0.2771 0.2839 0.3044 0.3213 0.3518 0.4375 0.2442 0.2512 0.2619 0.2692 0.2913 0.3095 0-3413 0.5000 0.2252 0.2326 0.2439 0.2518 0.2755 0.2950 0-3280 0.5625 0.2046 0.2126 0.2250 0.2339 0.2634 0.2917 0.3579 0.6250 0.1819 0.1900 0.2027 0.2122 0.2449 0.2789 0-3575 0.6875 0.1571 0.1647 0.1771 0.1865 0.2200 0.2564 0.3570 0.7500 0.1300 0.1370 0.1483 0.1570 0.1888 0.2243 0.3261 0.8125 0.1008 0.1066 0.1162 0.1236 0.1512 0.1826 0-2750 0.8750 0.0694 0.0736 0.0807 0.0863 0.1071 0.1313 0.2036 0.9375 0.0358 0.0381 0.0420 0.0451 0.0568 0.0705 0.1119 1.0000 0 0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3O Table 6.--Dimensionless Temperatures for 5=1.0, X=l.0 X e=0 e=.2 e=.4 e=.5 c=.7 c=.8 e=.9 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0358 0.0345 0.0340 0.0341 0.0349 0.0363 0.0493 -0.8750 0.0693 0.0670 0.0663 0.0664 0.0682 0.0711 0.0981 -0.8125 0.1008 0.0976 0.0967 0.0970 0.1000 0.1045 0.1466 -0.7500 0.1300 0.1262 0.1254 0.1259 0.1301 0.1363 0.1947 -0.6875 0.1570 0.1529 0.1522 0.1530 0.1586 0.1668 0.2424 -0.6250 0.1819 0.1777 0.1772 0.1784 0.1855 0.1957 0.2897 -0.5625 0.2046 0.2005 0.2004 0.2020 0.2108 0.2232 0.3367 -0.5000 0.2251 0.2214 0.2218 0.2239 0.2345 0.2493 0.3832 -0.4375 0.2442 0.2408 0.2419 0.2445 0.2572 0.2747 0.4381 -0.3750 0.2607 0.2578 0.2597 0.2628 0.2778 0.2984 0.4930 -0.3125 0.2747 0.2725 0.2752 0.2790 0.2964 0.3202 0.5478 -0.2500 0.2862 0.2847 0.2884 0.2929 0.3130 0.3403 0.6026 -0.1875 0.2952 0.2945 0.2994 0.3047 0.3276 0.3586 0.6574 -0.1250 0.3016 0.3020 0.3081 0.3142 0.3402 0.3751 0.7122 -0.0625 0.3056 0.3070 0.3145 0.3216 0.3508 0.3898 0.7669 0.0000 0.3071 0.3097 0.3186 0.3267 0.3594 0.4028 0.8216 0.0625 0.3056 0.3094 0.3198 0.3287 0.3639 0.4090 0.8339 0.1250 0.3017 0.3065 0.3182 0.3280 0.3655 0.4124 0.8345 0.1875 0.2952 0.3010 0.3139 0.3245 0.3644 0.4129 0.8235 0.2500 0.2862 0.2929 0.3069 0.3182 0.3605 0.4104 0.8009 0.3125 0.2747 0.2821 0.2971 0.3092 0.3538 0.4051 0.7666 0.3750 0.2607 0.2688 0.2847 0.2975 0.3443 0.3968 0.7207 0.4375 0.2442 0.2528 0.2695 0.2829 0.3321 0.3855 0.6632 0.5000 0.2252 0.2341 0.2515 0.2656 0.3171 0.3714 0.5940 0.5625 0.2046 0.2141 0.2330 0.2490 0 3152 0.4056 1.3087 0.6250 0.1819 0.1914 0.2107 0.2277 0.3026 0.4168 1.7980 0.6875 0.1571 0.1661 0.1848 0.2016 0.2791 0.4049 2.0619 0.7500 0.1300 0.1381 0.1552 0.1708 0.2449 0.3700 2.1003 0.8125 0.1008 0.1076 0.1219 0.1352 0.1999 0.3120 1.9134 0 8750 0.0694 0.0743 0.0850 0.0949 0.1440 0.2310 1.5010 0.9375 0.0358 0.0385 0.0443 0.0498 0.0774 0.1270 0.8632 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 31 Table 7.—-Dimensionless Temperatures for 5=2.0, A=l.0 X e=0 c-.2 e-.4 e-.5 c—.7 c-.8 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0358 0.0354 0.0376 0.0401 0.0565 0.3495 -0.8750 0.0693 0.0688 0.0732 0.0780 0.1108 0.6916 -0.8125 0.1008 0.1001 0.1066 0.1138 0.1628 1.0263 -0.7500 0.1300 0.1294 0.1380 0.1475 0.2126 1.3535 -0.6875 0.1570 0.1567 0.1673 0.1790 0.2601 1.6733 -0.6250 0.1819 0.1819 0.1945 0.2084 0.3053 1.9856 -0.5625 0.2046 0.2050 0.2196 0.2357 0.3483 2.2905 -0.5000 0.2251 0.2262 0.2426 0.2609 0.3891 2.5879 -0.4375 0.2442 0.2459 0.2643 0.2847 0.4312 3.0143 -0.3750 0.2607 0.2631 0.2836 0.3063 0.4710 3.4343 -0.3125 0.2747 0.2780 0.3006 0.3256 0.5085 3.8482 -0.2500 0.2862 0.2905 0.3152 0.3425 0.5438 4.2557 -0.1875 0.2952 0.3005 0.3275 0.3572 0.5768 4.6570 -0.1250 0.3016 0.3081 0.3375 0.3695 0.6051 5.0521 -0.0625 0.3056 0.3133 0.3451 0.3796 0.6359 5.4409 0.0000 0.3071 0.3161 0.3504 0.3874 0.6621 5.8234 0.0625 0.3056 0.3159 0.3520 0.3903 0.6659 5.6441 0.1250 0.3017 0.3130 0.3508 0.3904 0.6666 5.3776 0.1875 0.2952 0.3075 0.3469 0.3877 . 0.6641 5.0239 0.2500 0.2862 0.2994 0.3401 0.3821 0.6584 4.5832 0.3125 0.2747 0.2886 0.3306 0.3738 0.6495 4.0553 0.3750 0.2607 0.2751 0.3183 0.3626 0.6374 3.4402 0.4375 0.2442 0.2590 0.3031 0.3486 0.6221 2.7380 0.5000 0.2252 0.2403 0.2852 0.3317 0.6036 1.9487 0.5625 0.2046 0.2203 0.2688 0.3228 0.7331 8.0091 0.6250 0.1819 0.1974 0.2469 0.3046 0.8041 12.268 0.6875 0.1571 0.1717 0.2195 0.2770 0.8164 14.726 0.7500 0.1300 0.1431 0.1866 0.2402 0.7703 15.383 0.8125 0.1008 0.1116 0.1482 0.1941 0.6655 14.239 0.8750 0.0694 0.0773 0.1043 0.1387 0.5022 11.294 0.9375 0.0358 0.0401 0.0549 0.0740 0.2804 6.5475 1.0000 0.0000 0 0000 0.0000 0.0000 0.0000 0.0000 32 Table 8.--Dimensionless Temperatures for 8=0.5, A=2.0 X c=0 c=.2 c=.4 e=.5 c-.7 e=.8 c=.9 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0821 0.0775 0.0754 0.0753 0.0779 0.0825 0.0993 -0.8750 0.1622 0.1538 0.1502 0.1502 0.1560 0.1658 0.2010 -0.8125 0.2402 0.2288 0.2243 0.2247 0.2344 0.2499 0.3050 -0.7500 0.3163 0.3025 0.2977 0.2988 0.3131 0.3349 0.4114 -0.6875 0.3903 0.3750 0.3704 0.3725 0.3921 0.4207 0.5202 -0.6250 0.4622 0.4462 0.4424 0.4458 0.4713 0.5073 0.6314 -0.5625 0.5322 0.5161 0.5138 0.5187 0.5508 0.5948 0.7449 -0.5000 0.6001 0.5848 0.5844 0.5912 0.6306 0.6831 0.8608 -0.4375 0.6602 0.6463 0.6485 0.6573 0.7042 0.7653 0.9708 -0.3750 0.7123 0.7004 0.7055 0.7164 0.7712 0.8410 1.0745 -0.3125 0.7565 0.7469 0.7553 0.7687 0.8316 0.9102 1.1718 -0.2500 0.7928 0.7860 0.7981 0.8140 0.8854 0.9731 1.2629 -0.1875 0.8210 0.8175 0.8338 0.8524 0.9326 1.0295 1.3477 -0.1250 0.8414 0.8416 0.8624 0.8839 0.9732 1.0795 1.4262 -0.0625 0.8537 0.8582 0.8839 0.9085 1.0073 1.1230 1.4984 0.0000 0.8581 0.8673 0.8982 0.9261 1.0347 1.1601 1.5643 0.0625 0.8537 0.8676 0.9040 0.9352 1.0531 1.1866 1.6116 0.1250 0.8414 0.8594 0.9006 0.9347 1.0608 1.2015 1.6439 0.1875 0.8211 0.8427 0.8880 0.9246 1.0579 1.2047 1.6610 0.2500 0.7928 0.8175 0.8663 0.9050 1.0444 1.1964 1.6631 0.3125 0.7566 0.7838 0.8354 0.8758 1.0202 1.1764 1.6500 0.3750 0.7124 0.7417 0.7953 0.8371 0.9853 1.1448 1.6219 0.4375 0.6602 0.6910 0.7461 0.7887 0.9399 1.1016 1.5787 0.5000 0.6001 0.6318 0.6877 0.7309 0.8838 1.0467 1.5204 0.5625 0.5322 0.5641 0.6208 0.6653 0.8306 1.0202 1.6457 0.6250 0.4623 0.4932 0.5484 0.5924 0.7610 0.9639 1.6809 0.6875 0.3903 0.4190 0.4706 0.5121 0.6751 0.8778 1.6260 0.7500 0.3163 0.3417 0.3873 0.4245 0.5728 0.7618 1.4810 0.8125 0.2403 0.2611 0.2986 0.3294 0.4542 0.6161 1.2459 0.8750 0.1622 0.1773 0.2045 0.2270 0.3191 0.4405 0.9207 0 9375 0.0821 0.0903 0.1050 0.1172 0.1677 0.2352 0.5054 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 33 Table 9.--Dimensionless Temperatures for B=l.0, 1=2.0 X c=0 c=.2 c=.4 e=.5 c=.7 e=.8 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0821 0.0789 0.0810 0.0849 0.1122 0.2260 -0.8750 0.1622 0.1564 0.1613 0.1695 0.2257 0.4593 -0.8125 0.2402 0.2327 0.2408 0.2536 0.3404 0.7001 -O.7500 0.3163 0.3076 0.3195 0.3373 0.4564 0.9482 -0.6875 0.3903 0.3812 0.3975 0.4206 0.5735 1.2037 -0.6250 0.4622 0.4535 0.4748 0.5035 0.6920 1.4667 -0.5625 0.5322 0.5245 0.5513 0.5860 0.8116 1.7370 -0.5000 0.6001 0.5942 0.6270 0.6681 0.9325 2.0147 -0.4375 0.6602 0.6566 0.6957 0.7431 1.0452 2.2822 -0.3750 0.7123 0.7115 0.7569 0.8104 1.1493 2.5357 -0.3125 0.7565 0.7587 0.8105 0.8701 1.2446 2.7754 -0.2500 0.7928 0.7984 0.8567 0.9222 1.3312 3.0011 -0.1875 0.8210 0.8305 0.8954 0.9667 1.4091 3.2129 -0.1250 0.8414 0.8549 0.9265 1.0036 1.4783 3.4109 -0.0625 0.8537 0.8718 0.9502 1.0330 1.5388 3.5949 0.0000 0.8581 0.8811 0.9663 1.0547 1.5906 3.7651 0.0625 0.8537 0.8815 0.9732 1.0665 1.6253 3.8762 0.1250 0.8414 0.8733 0.9703 1.0676 1.6646 3.9525 0.1875 0.8211 0.8564 0.9577 1.0581 1.6486 3.9941 0.2500 0.7928 0.8310 0.9353 1.0378 1.6372 4.0010 0.3125 0.7566 0.7969 0.9032 1.0069 1.6104 3.9730 0.3750 0.7124 0.7542 0.8612 0.9653 1.5682 3.9104 0.4375 0.6602 0.7029 0.8096 0.9129 1.5106 3.8130 0.5000 0.6001 0.6430 0.7481 0.8499 1.4377 3.6808 0.5625 0.5322 0.5745 0.6785 0.7815 1.4091 3.9879 0.6250 0.4623 0.5026 0.6020 0.7022 1.3373 4.0757 0.6875 0.3903 0.4273 0.5188 0.6122 1.2223 3.9444 0.7500 0.3163 0.3487 0.4287 0.5113 1.0642 3.5939 0.8125 0.2403 0.2666 0.3317 0.3997 0.8629 3.0242 0.8750 0.1622 0.1811 0.2280 0.2772 0.6184 2.2354 0.9375 0.0821 0.0923 0.1174 0.1440 0.3308 1.2273 1.0000 0.0000 0.0000 0 0000 0.0000 0.0000 0.0000 34 Table 10.--Dimensionless Temperatures for e=2.0, X=2.0 X c=0 e-.2 e=.4 e=.5 -1.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.0821 0.0848 0.1160 0.1896 -0.8750 0.1622 0.1680 0.2308 0.3787 -0.8125 0.2402 0.2497 0.3444 0.5674 -0.7500 0.3163 0.3299 0.4568 0.7558 -0.6875 0.3903 0.4085 0.5679 0.9436 -0.6250 0.4622 0.4856 0.6778 1.1311 -0.5625 0.5322 0.5611 0.7865 1.3182 -0.5000 0.6001 0.6351 0.8940 1.5048 -0.4375 0.6602 0.7014 0.9916 1.6764 -0.3750 0.7123 0.7597 1.0792 1.8328 -0.3125 0.7565 0.8100 1.1566 1.9738 -0.2500 0.7928 0.8523 1.2240 2.0996 -0.1875 0.8210 0.8865 1.2813 2.2101 -0.1250 0.8414 0.9127 1.3284 2.3053 -0.0625 0.8537 0.9309 1.3655 2.3852 0.0000 0.8581 0.9411 1.3926 2.4498 0.0625 0.8537 0.9419 1.4058 2.4867 0.1250 0.8414 0.9335 1.4057 2.5012 0.1875 0.8211 0.9161 1.3921 2.4933 0.2500 0.7928 0.8894 1.3651 2.4631 0.3125 0.7566 0.8537 1.3247 2.4105 0.3750 0.7124 0.8088 1.2709 2.3355 0.4375 0.6602 0.7548 1.2036 2.2382 0.5000 0.6001 0.6916 1.1229 2.1185 0.5625 0.5322 0.6197 1.0380 2.0279 0.6250 0.4623 0.5436 0.9372 1.8876 0.6875 0.3903 0.4634 0.8205 1.6974 0.7500 0.3163 0.3790 0.6881 1.4575 0.8125 0.2403 0.2904 0.5398 1.1678 0.8750 0.1622 0.1978 0.3757 0.8283 0.9375 0.0821 0.1010 0.1958 0.4390 1.0000 0.0000 0.0000 0.0000 0.0000 35 X e=0 e=.2 e=.4 e-.5 e-.7 -1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.2659 0.2545 0.2713 0.2991 0.5511 -0.8750 0.5310 0.5113 0.5480 0.6059 1.1245 -0.8125 0.7954 0.7704 0.8301 0.9203 1.7201 -0.7500 1.0591 1.0317 1.1176 1.2424 2.3381 -0.6875 1.3221 1.2954 1.4104 1.5721 2.9782 -O.6250 1.5843 1.5613 1.7086 1.9095 3.6407 -0.5625 1.8458 1.8294 2.0121 2.2545 4.3255 -0.5000 2.1065 2.0998 2.3210 2.6071 5.0325 -0.4375 2.3347 2.3403 2.5997 2.9274 5.6842 -0.3750 2.5326 2.5518 2.8482 3.2153 6.2812 -0.3125 2.7003 2.7343 3.0666 3.4708 6.8236 -0.2500 2.8379 2.8879 3.2549 3.6938 7.3114 -0.1875 2.9452 3.0124 3.4131 3.8845 7.7444 -0.1250 3.0223 3.1080 3.5412 4.0427 8.1228 -0.0625 3.0692 3.1746 3.6391 4.1682 8.4466 0.0000 3.0858 3.2122 3.7069 4.2619 8.7156 0.0625 3.0692 3.2158 3.7378 4.3147 8.9084 0.1250 3.0224 3.1859 3.7280 4.3199 9.0030 0.1875 2.9453 3.1224 3.6774 4.2774 8.9994 0.2500 2.8380 3.0252 3.5861 4.1873 8.8977 0.3125 2.7005 2.8945 3.4541 4.0496 8.6978 0.3750 2.5328 2.7302 3.2814 3.8642 8.3997 0.4375 2.3349 2.5324 3.0679 3.6312 8.0035 0.5000 2.1068 2.3009 2.8137 3.3506 7.5091 0.5625 1.8460 2.0318 2.5148 3.0224 7.0067 0.6250 1.5845 1.7574 2.2009 2.6684 6.3798 0.6875 1.3223 1.4778 1.8719 2.2884 5.6281 0.7500 1.0593 1.1928 1.5277 1.8826 4.7518 0.8125 0.7956 0.9025 1.1684 1.4508 3.7508 0.8750 0.5311 0.6070 0.7941 0.9931 2.6252 0.9375 0.2659 0.3061 0.4046 0.5095 1.3749 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 36 Table 12.--Dimensionless Temperatures for B=l.0, X=2.3 X e=0 e=.2 e=.4 e=.5 -1.0000 0.0000 0.0000 0.0000 0.0000 -0.9375 0.2659 0.2713 0.3884 0.7165 -0.8750 0.5310 0.5450 0.7845 1.4527 -0.8125 0.7954 0.8209 1.1884 2.2086 -0.7500 1.0591 1.0992 1.6000 2.9841 -0.6875 1.3221 1.3798 2.0194 3.7792 -0.6250 1.5843 1.6628 2.4464 4.5939 -0.5625 1.8458 1.9480 2.8813 5.4283 -0.5000 2.1065 2.2355 3.3238 6.2823 -0.4375 2.3347 2.4912 3.7231 7.0593 -0.3750 2.5326 2.7161 4.0800 7.7604 -0.3125 2.7003 2.9102 4.3943 8.3856 -0.2500 2.8379 3.0736 4.6661 8.9350 -0.1875 2.9452 3.2062 4.8954 9.4085 -0.1250 3.0223 3.3080 5.0822 9.8063 -0.0625 3.0692 3.3791 5.2265 10.128 0.0000 3.0858 3.4194 5.3283 10.374 0.0625 3.0692 3.4237 5.3778 10.522 0.1250 3.0224 3.3923 5.3691 10.555 0.1875 2.9453 3.3252 5.3023 10.474 0.2500 2.8380 3.2224 5.1773 10.279 0.3125 2.7005 3.0839 4.9941 9.9691 0.3750 2.5328 2.9098 4.7529 9.5449 0.4375 2.3349 2.6999 4.4534 9.0064 0.5000 2.1068 2.4543 4.0959 8.3535 0.5625 1.8460 2.1688 3.6793 7.6146 0.6250 1.5845 1.8773 3.2353 6.7884 0.6875 1.3223 1.5796 2.7642 5.8751 0.7500 1.0593 1.2759 2.2658 4.8745 0.8125 0.7956 0.9661 1.7402 3.7867 0.8750 0.5311 0.6501 1.1874 2.6117 0.9375 0.2659 0.3281 0.6073 1.3495 1.0000 0.0000 0.0000 0.0000 0.0000 37 Table 13.--Dimensionless Temperatures for B=2.0. X=2.3 X e=0 6= 2 -1.0000 0.0000 0.0000 -0.9375 0.2659 0.3679 -0.8750 0.5310 0.7384 -0.8125 0.7954 1.1115 -0.7500 1.0591 1.4871 -0.6875 1.3221 1.8654 -0.6250 1.5843 2.2461 -0.5625 1.8458 2.6295 -0.5000 2.1065 3.0154 -0.4375 2.3347 3.3584 -0.3750 2.5326 3.6603 -0.3125 2.7003 3.9212 -0.2500 2.8379 4.1410 -0.1875 2.9452 4.3199 -0.1250 3.0223 4.4577 -0.0625 3.0692 4.5544 0.0000 3.0858 4-5102 0.0625 3.0692 4.6182 0.1250 3,0224 4.5785 0.1875 2.9453 4.4909 0.2500 2.8380 4.3555 0.3125 2.7005 4.1722 0.3750 2.5328 3-9411 0.4375 2.3349 3.6621 0.5000 2.1068 3.3353 0.5625 1.8460 2.9560 0.6250 1.5845 2.5659 0.6875 1.3223 2.1651 0.7500 1.0593 1.7536 0.8125 0.7956 1.3313 0.8750 0.5311 0.8983 0.9375 0.2659 0.4545 1.0000 0.0000 0.0000 38 Table T4.--Maximum DimensionTess Temperatures for A=0.0 E B ¢max X 0.0 -- 0.2506 0.000 0.2 0.5 0.2514 0.020 0.4 0.5 0.2542 0.050 0.5 0.5 0.2565 0.065 0.7 0.5 0.2646 0.100 0.8 0.5 0.2721 0.125 0.9 0.5 0.2872 0.150 1.0 0.5 0.2968 0.500 0.2 1.5 0.2524 0.020 0.4 1.0 0.2587 0.050 0.5 1.0 0.2644 0.070 0.7 1.0 0.2867 0.120 0.8 1.0 0.3134 0.145 0.9 1.0 0.4747 0.075 1.0 1.0 0.5579 0.500 0.2 2.0 0.2565 0.020 0.4 2.0 0.2786 0.060 0.5 2.0 0.3013 0.085 0.7 2.0 0.4316 0.065 39 Table l5.--Maximum Dimensionless Temperatures for A=l.0 5 3 ¢max X 0.0 -- 0.3071 0.000 0.2 0.5 0.3083 0.025 0.4 0.5 0.3126 0.055 0.5 0.5 0.3163 0.070 0.7 0.5 0.3291 0.110 0.8 0.5 0.3412 0.135 0.9 0.5 0.3668 0.170 0.2 1.0 0.3099 0.025 0.4 1.0 0.3198 0.060 0.5 1.0 0.3288 0.075 0.7 1.0 0.3656 0.130 0.8 1.0 0.4131 0.165 0.9 1.0 0.8356 0.095 1.0 1.0 0.7668 0.500 0.2 2.0 0.3163 0.025 0.4 2.0 0.3520 0.070 0.5 2.0 0.3907 0.095 0.7 2.0 0.6667 0.105 40 Table l6.--Maximum Dimensionless Temperatures for A 2.0 5 B ¢max X 0.0 -- 0.8581 0.000 0.2 0.5 0.8685 0.035 0.4 0.5 0.9041 0.070 0.5 0.5 0.9361 0.090 0.7 0.5 1.0611 0.140 0.8 0.5 1.2050 0.175 0.9 0.5 1.6640 0.225 0.2 1.0 0.8824 0.035 0.4 1.0 0.9734 0.075 0.5 1.0 1.0685 0.100 0.7 1.0 1.6490 0.170 0.8 1.0 4.0026 0.230 0.2 2.0 0.9426 0.035 0.4 2.0 1.4074 0.095 0.5 2.0 2.5014 0.135 41 Table l7.--Maximum Dimensionless Temperatures for A=2.3 e 8 45max X 0.0 -- 3.0858 0.000 0.2 0.5 3.2184 0.040 0.4 0.5 3.7392 0.080 0.5 0.5 4.3235 0.100 0.7 0.5 9.0135 0.155 0.2 1.0 3.4263 0.040 0.4 1.0 5.3814 0.085 0.5 1.0 10.5580 0.110 0.2 2.0 4.6208 0.040 42 m.onm .o.ou4 Low mwpwmoga mgaumgmnsmh mmmpcowmcmswo--.m mgzmwa 0.0» 0.0.: 5.0. u «.o. . 43 mdum .o.oux .8» 85.8.2“. 832353 33:03:25-... mduw 5.0.: \l O mgzm?u 1 «.0 1 r2 44 dem 6.7,“ .8» 82.3.5 9.32353 325353923 953“. 0.0 ru #6 45 m.o-m .o._u« 20$ moppmoga mgaumgmasmh mmmrcowmcmswo--._F mgzmwu 9?: ho". h-O 1 0.0 46 m.o|m .o.~uK Lo» mmpwwogm wgzumgmasmh mmmpcowmcwema--.mp wgsmwu 47 m.oum .o.~n« Lo» mwpwwoga mgaumgmnsm» mmmpcowmcwswo--.mp mesmwu 48 2(196‘ 2.3 2 I A=0 H396- Figure l4.--Percent Increase in Dimensionless Temperature for 8=0.5 (3max/¢max|e=0.0) 49 ”*1 2K39S- 2.3 2 I A30 HDQS“ (J .2 .4’ 6 8 Figure l5.--Percent Increase in Dimensionless Temperature for B=l.0 (¢max/¢maxle=0.0) 50 3096- 2096- ELS 2 l A=() |O%d I I l l 0 2 .4 6 B c Figure le.--Percent Increase in Dimensionless Temperature for 8=2.0 (¢max/¢max1e=0.0) 51 3(VIr 2096- 2 I IO % a B 8 0.5 O .2 .4 .6 .3 Figure l7.--Percent Increase in Dimensionless Temperature for A=l.0 (¢max/¢max1e=0.0) 52 2CN*:~ 2.4048 23 2 I A80 "395- Figure 18.--Percent Increase in Dimensionless Temperature 1¢max/¢max|e=0.0) 53 Figure l9.--Helix Configuration for c=1.0 APPENDIX 10 15 2O 25 30 35 40 45 50 55 60 65 7O 75 54 PROCRAR ISOPAR (INF; OUTP T TAPEoo-INPUT. TAPED1-OUTPUT) DIHENS ON VN 1 Pm E a1” DINENSION VO 2 g).xa 'v D woe DINENSION AC a? we 5 1 O FA?‘ OEFB: DINENSION S SN l.EF(D1. 165.65 (DDI.DIDOI.ID(DO).VN(1OO) ‘DMMON/TLE T TLE 20 130 REAL xxx K v.xxv.xx.xv N DATA COEFA/-.ae1136.-.D:IRDD1é .aaRRo1 D11ao/ DATA COEFa/.aATesa. 6571175172)‘5 57 D55 DATA 1’ 60é.IO/61 L 1 1610 NL/D c DATA A 422 calm/6500.; C DEFINITION OF THE INPUT PARAN ETERS 130 c TITLE-A OESCRIPTIVE STATEN ENT F THE PRODLEN BEING SOLVED c NP-NUNDER OF NOOES AND EOUATIDNS c NE-NUNDER OF ELENENTS 210 c xxx-HATERIAL PROPERTv IN A DIRECTION 230 c xvv-NATERIAL PROPERTV IN v DIRECTION 240 c O-COEFFICIENT NULTIPLTINC PHI IN THE DIFF E0 250 c -SOURCE TERN 260 c -x COORDINATES OF THE NOOE; 210 C V-v COORDINATES OF THE NOOE ROD c NEL-ELEHENT NUNaER 290 c Ns-NODE NUNDERS FOR ELENENT NODES 1 TO D c VN CONTAINS THE SHAPE FUNCTION VALUES AT E AND N C E Is THE VALUE OF SIGH c Is THE VALUE OF NETA C PNE CONTAINS THE SHAPE FUNCTION PART}AL§ g“ E C CONTAINS THE SHAPE FUNCTION PART AL N C CONTAINS THE PARTIAL: ON x c PNv CONTAINS THE PARTIAL ON T c VU CONTAINS THE UACOBIAN OR a RATRIx c VUI CONTAINS THE INVERSE OF 8 c VD I THE VALUE OF THE OETE RINENT OF U C A Is HE CLDDA STIFFNESS HATR x c F Is THE CLODAL FORCE VECTOR 8 ES C INPUT OF THE T TLE CARD CONTROL PARANETERS AND NDOAL COORDINATES 330 R5AD(IN SIT TLE 340 a F RNAT(§O A :50 READ IN.o E.DE. BL READ IN.- .N READ IN.o {Ac “1 I-1..NP 310 c READ IN.- VC -1 NP 333 g OUTPUT OF TITLE. PARANETER VALUES AND NDDAL COORDINATES 338 OVRITE TLE. NP N E. A FORNA 11H1/u§}é}x;.20142)1x‘§§NE) -ID/1A.SHNE -ID/1x.DHEE -E1s.s/1 so 11 FORNA 1:17HNOD:&CY M?D$NATES‘gx.4HNODE.4x.IHX.I4X.1HY) 300 HRIT 3161'Y1 I-1 010 12 FORNA IA. 2 1 D20 Cocoa-:0 o. .30 c ASSENDLVINC OF THE DLDDAL STIFFNESS NATRIx AND CLOOAL FORCE NATRIAOAO Cocoon-Do 650 C 660 8 INPUT AND ECHO PRINT OF ELENENT DATA :38 HRITE$12 ,) 690 O 5327‘“ I 1x.12HELENENT DATA/1L 11HNEL NOOE NUNDERS) 118 READEW120 NRIT 16 32 ENELs Ns 130 c 23 FORNA 1L I 2L £14) 750 g CALCULATION OF THE ELENENT STIFFNESS NATRIx AND FORCE VECTOR 193 DODI-z I -Ns I TOO x I - c II; 000 9 v I -vc II D10 N -1.D Kd-;.O ESN NI .NU -o. BO 85 90 95 100 105 110 125 55 RD CONTINUE EF(NI)-O. DD CONTINUE 00 so 0'1 A E-COEFA(J1 DO 29 x-1 A N-COEFAiKé CALL VALU (E.N.VN PNN PN ) CALL JMAT(V¥.X.V P P CALL PAva VDU.PNE. PNN. Nx. PNv. CALL COEF(X.Y VN xxx. va. xxv. NL 'NV .0. C. EE. DE. BL) Uc-COEFB‘JgOCOEF8( ) Do 95 L- . Do 94 H-1 D SHIL.H)- SN( L. :1 xxxoPNxI )OPNX§ HM)+KVYOPNY‘ oPNv C H;-xxoVN L)- PN N -KY0VN(L oPNv( )+Nxvo(PNxL oPNv c H 1-.sg-Voum 94 ON NU EF(L -EF(L) 1 VN(L)‘O°UCOVDJ 95 CONT NUE 29 CONTINUE c 30 CONTINUE g ASSEHOLE A AND F FRON ESN AND EF DO 41 Ic- .O 00 40 uc-1. II -st II sdd -A(II. ad)+ESN(IC. ac) -F III‘EF(!C) c CONTINUE g MODIFICATION AND SOLUTION OF THE STSTEN OF EOUATIONS CALL OOVVAL(A.F.NP.O.ID) ID CA LL E3T1F A.H.NP.NP.F.IOCT.UN.IER) NRITE 1 }go 100 FORHA ”{Z‘ “9x.- NODAL VALUEs-I ERITE $10,261) I.F(I) 20°C 201 IA.E .5) CALL OU PUT(F.XC.YC Cocooooca; 1:1;Pa¥?(k 2;VN menu 40"! CC“) 1O 15 20 25 1O 1O 56 SUBROUTINE VAL E (E N VN. PNN PNE) DIMENSION VN(8 .PNNlB). PNE(85 REAL N VN 1 =-.251(1. -E)t 1. -N): E+N+1.) VN 2 . .51(1. -Et-2 -(1. VN 3 - .25:(1. +E)* 1.-N)1 E- N- 1. ) VN 4 . .5*(1.-N'*2 c(1.+ VN 5 - .251(1.+E)t 1. +N)- E+N- 1. ) VN 6 - .Sté1.-E**2 1(1. +N VN 7 . -.2 (1 .-E (1. +N 1(E- N+1. ) VN e - .51(1. -N:- PNN 1 - .25: -Ett + -2. - -N+2 1N) PNN 2 . -.5: 1,-Ett2 PNN 3 -.251-Eé-r2+2.1N+2.-N1E) PNN 4 --N1( PNN 5 - .25'(E*E +E+21N+2. EE'N) PNN 6 - .5-(1. PNN 7 -.251(E:-2.:N+2.-N1E) PNN e - -N-(1. PNE 1 - .25~ 2. :N- 2.-E-N+N-N--2) PNE 2 -E* 1. PNE 3 - .25: -N+2.1 -2. *E*N+N“2) PNE 4 - .51(1.- m12 PNE 5 - 2512+~ ~ +2. 1E1N+N+Nt-2) PNE 5 e -E*é PNE 7 = - 2 13- 2. -E+N- 2. .EcN1N112) PNE e = - .51( -N1:2) RETURN END SUBROUTINE JMAT NN gNEI DIMENSION vu(2.2 M1 1W81pa).p~s1a) vu 1.1 :0. VO 1.2 :0. VJ 2.1 -0. VJ 2.2 -0. DO 1 1:1.8 vu 1.1 -vu 1.1 +PNE I 1x I vu 1.2 -vu 1.2 +PNE I v I vu 2.1 -vu 2.1 +PNN I x I vu 2 2 svu 2.2 +pNN I v I CONTINUE RETURN END SUBROUTINE PARXY svofé PNE. PNN PNX. PN vu) DIMENSION PNX(85. Nv LIP N'§(a) RNNSOL VId(2.2).Vd(2.2) VDJFVJ(1.1)*VJ( .2)-vu 1vul2 1 VDu-Aes vouz VIJ 1.1 -vu 2.2)lvou VIU 1.2 F-Vd(1.2;/VDJ VId 2.1 --VUL2.1 cvoa 35012N21'gd .1)/ DJ PNX xgc bNszgtv1u‘1 11+ PNN u; VIJ€1.2; PNY K - PNE K -v10 2 1 + PNN Vlu 2.2 CON INUE RETURN END 1O 15 20 1O 15 2O 25 3O 35 GOO 000 200 dd 0 d 208 207 57 SUBROUTINE coer1xk v. VN. KXX.KYY.KXY.KX.KY.O.G.EE.BE.BL) “5 xxx va v DIMENSION x(é). xv18). VN(8) YY:O. OO xx-xx+x i tVN‘I ; EB§YY+Y I 'VN O'BEt‘2tEEt‘2t(XX¢t2+YY*‘2)/((1.-EEtxx)tt2) R-XX*#2+YYt Kxx- xxtt2+YY;*2*‘1.+D;;% va- YVct2+xx:n2t 1 +0 :iI: ?‘§§;;t-D i/(1.-E£cxx)+(§sovv6;2) (1. -O)/(1. -EEtXX) +XXtD)/R E gas-xx) 1 ~EEtxx) +1 tvv SUBROUTINE ?°YV AL F.NP. a 13; DIMENSION NP NP #3 p). 81N .!B(NP) COMMON TLE TIfLE éo DATA 1 /60/ 104?; NR! E FORMA111H1. /////.1X.20A4) INPUT OF THE PRESCRIBED NODAL VALUES . ‘??NI:"?S(I 13111.: . 1 GO TO 101 IN.* B?I? $0.92 G? TO 1000 $ éézo 063:3‘E22E5C31?EDW (NRDAL VALgES‘) NOD. OF THE MATR. USING THE IETHOD 0F ROUS AND COLNS. tuna Inrmno D A N ”(H02 anonmo HA2 343424+OAO ’M’Wh EAR M ”H” H F 00 ZMOAAOAAAOO UdZHHZXX H822: p ) '0. . B B J ):8K 1 (4)) * ( ) ~nvcnvNLII w-0 m ”LIV-fig 15(gz))-1. Mnfiflbnbfi) C-flDW-P vii-t” ”HMO-CH I m ZECGZW‘AXL 58 zflAT x-E15.s) AEY‘BDJNOBLNSU).61.).T(I).PNX“).PNY(O) E. I 3 7 I. 5 ”WK 2 E1 . m. o o s ) \I \I a. nae? v m m 2 11- W O c O u I “ O O O ‘ X) . W .5 . E 0 F.) 5 w T ”H“ a" c x . T T . T T . W .V .0 .. W W )1 T W .1).- T ” W )1 w ) I a, v 0 LL 0 O O 0 LL I O I 0 LL a).q;u1me) N .11u u .L N .11? v L N .31.) .1 .7 .5 m . TX K . TYY V m m. TX T . E o O E o o o o T W”. glen-M-Myml‘ . ‘1“ l. 0).!) 1114 l in. l\ .1; \I 2 U .1 11 .1! . m 1111 a) O In! 0 E oLL‘ ‘2 ‘1 o \l... o E uLL‘ ‘ ‘r o In. ‘- OLL “ mm in all/E101 I811 u 1(1 o o 1101 7881 u 0‘ o o 1.1 Tel. 1” 1(1 I ./1///2- Ell.- L IWWOE 00- .01. L uWWOE O - .oIlll. nL nWW . TIT. “ENJJFCK I L I ”(1...."lele I L I I. . 81ng KA L 2V... SSNCI TTTT I ( x )V ++( )V 1+1 (00 sx 7V +§OAT EAAAAZS- .7 K GTXEI E59”.85- .7 K.CTYE1 . .E- 39“....)- - 31X .5- REEL“TN~"M1D1N1111L0.01TXTT1OT7202N241LQ027YT 1°TVV303N11LMLQ°3TX+TV NIAHHIMMWWE .dd ”wuh. .ulmhhllE .JJ .:u no! .ulAA A udd.. Mb” thHA ‘01.: N00 FFffWIWIwaNECTumTXWCXTWWWRwITY NCTW TV CTTWSSWIWHN1NE CTXWHXEIT 1234 I .70 I .73 u 1 111 2 a; u 1 5 10 15 20 25 30 35 40 45 $0 55 00 65 70 5.3) TSAv.xsav v-xx 8&11‘?¥ IETUSN END X ii 75 59 THESIS PROGRAM NP 3 65 NE ' 16 EE ' .50000E*00 BE ' .50000E+OO BL = .10000E+01 NODAL COORDINATES NOOE X 1 -.10000E+01 2 -.96825E+00 3 -.96825E+00 4 -.86603E+00 5 '.86603E+OO 6 -.75000E+00 7 *.75000E+00 8 -.75000E+00 9 -.75000E+00 10 -.75000E+00 11 -.70711E+00 12 -.70711E+00 13 -.66144E+00 14 -.66144E+OO 15 -.50000E+00 16 -.50000E+OO 17 -.50000E+00 18 -.50000E+00 19 -.50000E+00 20 -.50000E+OO 21 -.50000E+00 22 ~.50000E+00 23 -.50000E+00 24 -.25000E+00 25 -.25000E+00 26 '.25000E+OO 27 -.25000E+OO 28 -.25000E+OO 29 0. 3O 0. 31 0. 32 O. 33 0. 34 0. 35 0. 36 0. 37 0. 38 .25000E+OO 39 .25000E+00 40 .25000E+OO 41 .25000E+OO 42 .25000E+00 43 .50000E+00 44 .50000E+00 45 .50000E+OO 46 .50000E+00 Y .25000E+00 .25000E*00 .50000E+00 .50000E+00 .66144E+00 .50000E+00 .50000E+00 .66144E+00 .70711E+00 .70711E+00 .75000E+00 .75000E+00 .86603E+00 .75000E+00 .50000E+00 -.25000E+00 0 .25000E+00 .50000E+00 .7SOOOE+00 .86603E+OO .9682SE+00 -.50000E+00 0 .50000E+00 .96825E+00 .10000E+01 .75000E+00 -.50000E+00 .25000E+00 .25000E+00 .50000E+00 .75000E+00 .10000E+01 .96825E+00 -.50000E+00 1O .50000E+00 .96825E+00 .86603E+00 .75000E+00 .50000E+00 .25000E+00 .50000E+00 .50000E+00 .50000E+00 .50000E+00 .50000E+00 .66144E+OO .66144E+00 .70711E+OO .70711E+00 .75000E+OO .75000E+00 .75000E+00 .75000E+00 .75000E*00 .86603E+00 .86603E+00 .96825E+00 .96852E+00 .10000E+01 ELEMENT DATA NEL NOOE NUMBERS @OQOUbUM-b 11 13 15 60 0. .25000E+00 .50000E+00 .75000E+00 .86603E+00 -.75000E+00 .75000E+OO -.70711E+00 .70711E+00 -.66144E+00 -.50000E+00 0. .50000E+OO .66144E+00 -.50000E+00 .50000E+00 -.25000E+00 .25000E+00 0. THESIS PROORAI 1 0. 1O 0. 23 0. 42 O. 55 O. 34 0. NODAL VALUES 1 O. 2 0. 3 0. 4 0. 3 0. 3 O. 7 .43527E'01 3 .12131E100 3 .43327E-01 10 0. 11 O. 12 O. 13 O. 14 0. 1S 0. 1O .30233E-01 17 .14394E100 13 .13337E‘OO 19 .21OS1E000 20 .13337E100 21 .14334E000 22 .30233E-01 23 O. 24 0. 25 .20477EOOO 23 .23303E000 27 .20477E900 23 0. 23 O. 30 .13333E0OO 31 .23004E‘OO 32 .23233E1OO 33 .314OSE‘OO 34 .23233E‘OO 33 .23004E‘OO 33 .13333E1OO 37 O. 33 O. PRESCRIOEO NODAL VALUES 2 11 24 43 33 03 999999 61 3 0. 4 O 12 0. 13 O 23 O 23 O 31 0. 32 O 30 O. 31 0 99999 34 33 99999 .87500E+00 39 .22214E+00 40 .30506E+00 41 .22215E+OO 42 0. 43 0. 44 .58478E-01 45 .16968E+OO 46 .23359E+00 47 .25181E+00 48 .23359E+OO 49 .16970E+00 50 .58483E-01 51 0. 52 0. 53 0. 54 0. 55 0. 56 0. 57 .61290E‘01 58 .15700E+00 59 .61299E-01 60 0. 61 0. 62 0. 63 0. 64 0. 65 0. X’COORD -.10000E+01 0 .93750E+00 .87500E+00 .81250E+00 .75000E+00 .68750E+00 .62500E+00 .56250E+OO .50000E+OO .43750E+OO .37500E+00 .31250E+00 .2SOOOE+OO .18750E*OO .12500E+00 .62500E-01 0. .62500E-01 .12500E+OO .18750E+00 .25000E+00 .31250E+00 .37500E+00 .43750E+OO .50000E+00 .56250E+00 .62500E+00 .68750E+00 .75000E+OO .81250E+00 62 TEMP .329056-01 .64140E-01 .93707E-O1 .12161E+OO .14783E+OO .17239E+OO .19529E+OO .21651E+OO .23644E+OO .25418E+OO .26973E+00 .28308E+OO .29424E+OO .3032OE+OO .30997E+00 .314555+OO .31628E+00 .31528E+oo .31153E+OO .30506E+OO .2958$E+00 .28390E+OO .26922E+OO .25181E+OO .23394E+OO .21218E+OO .18653E+OO .157005+OO .12358E+OO .86273E-O1 .93750E+00 .1000OE+O1 Y-COORD .10000E+01 .93750E+OO .87500E+00 .81250E+00 .75000E+OO .68750E+OO .62500E+00 .56250E+OO .5000OE+00 .43750E+00 .37500E+OO .31250E+OO .25000E+OO .18750E+OO .12500E+OO .62500E-01 .62500E'01 .12500E+00 .18750E+OO .25000E+00 .31250E+00 .37500E+OO .43750E+00 .50000E+OO .56250E+00 .62500E+OO .68750E+00 .75000E+00 .81250E+OO .87500E400 .93750E+OO .10000E+01 MAX TEMP3 63 .45080E-01 0. 0 TEMP .36865E-01 .71434E'01 .10371E+OO .13368E+00 .16137E+OO .18675E+00 .20984E+00 .23064E+OO .25000E+00 .26683E+OO .28112E+00 .29288E+00 .30210E+OO .30879E+OO .31294E+00 .31455E+OO .31294E+OO .30879E+00 .30210E+OO .29288E+OO .28112E+00 .26683E+OO .25000E+OO .23064E+00 .20984E+00 .18675E+OO .16137E+00 .13369E+OO .10371E+OO .71435E-01 .36865E-01 0. .31630E+00 AT XI .70000E-01 LIST OF REFERENCES 10. LIST OF REFERENCES Jakob, M., Heat Transfer, Wiley, New York, Vol. 1, 1949, pp. 173-194. Jakob, M., "Influence Of Nonuniform Development of Heat Upon the Temperature Distribution in Electrical Coils and Similar Heat Sources of Simple Form," Trans. ASME, Vol. 65, 1943, pp. 593-605. Nicholson, J. M., "The Effective Resistance and Inductance of a Helical Coil," Philosophical Magazine, Vol. 19, 1910, pp. 77-91. Smithsonian Mathematical Formulae and Tables of Elliptic Functions, compiled by E. P. Adams and R. L. Hippisley, The Smithsonian Institution, Washington, D.C., 1922, p. 106. Wang, C. Y., "The Helical Coordinate System and the Temperature Distribution Inside a Helical Coil," Journal of Applied Mathematics, Vol. 47, 1980. pp. 951—953. Zienkiewicz, 0. C., The Finite Element Method, McGraw Hill, 3rd Edition, 1977. pp. 49-58. Conte, S. D. and deBoor, C., Elementary Numerical Analysis An Algorithmic Approach, McGraw Hill,31980, 3rd Edition. pp. 303-319. Holman, J. R., Heat Transfer, McGraw Hill, 1976, 4th Edition, pp. 33-35. Segerlind, L. J., Applied Finite Element Analysis, Wiley, lst Edition, 1976. pp. 387-391. International Mathematical and Statistical Libraries, Inc., Vol. 2, Houston, Texas, 1980. 64 "111111111111111111111“