Natural convection heat transfer between arrays of horizontal cylinders and their enclosure by Robert Allen Weaver A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mechanical Engineering Montana State University © Copyright by Robert Allen Weaver (1982) Abstract: The natural convection heat transfer between arrays of horizontal, heated cylinders and their isothermal, cooled enclosure was experimentally investigated. Four different cylinder arrays were used: two in-line and two staggered. Four fluids (air, water, 20 cs silicone and 96% glycerine) were used with Prandtl numbers ranging from 0.705 to 13090.0. There was no significant change in the Nusselt number between isothermal and constant heat flux conditions of the cylinder arrays. The average heat transfer coefficient was most affected by the spacing between cylinders and the total surface area of the cylinder arrays. The enclosure reduced the increase in both the average and the local heat transfer coefficients caused by changing the inner body from an in-line arrangement to a staggered arrangement of comparable spacing. An increase in fluid viscosity reduced the influence of the geometric effects. The best empirical equation for all of the experimental data using one correlating parameter was: Nus = 0.214Ra*s^O.260; Ra*s = Ras(L/Ri) for 0.705 ≤ Pr ≤ 1.31xl0^4; 4.45xl0^4 ≤ Ras ≤ 1.17xl0^8 0.602 ≤ L/Ri ≤ 1.041; 4.63x10^4 ≤ Ra*s ≤ 8.15x10^7 with an average percent deviation of 12.00.  STATEMENT OF PERMISSION TO COPY ' In presenting this thesis in partial fulfillment of the requirements' for an advanced degree at Montana State University, I agree that the Library shall make it freely available for inspection. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by my major professor, or, in his absence, by the Director of Libraries. It is understood that any copying or publication of this thesis for financial gain shall not be allowed without my written permission^ Signature (X ___________ Date 11 , 1^ 82.______________ NATURAL CONVECTION HEAT TRANSFER BETWEEN ARRAYS OF HORIZONTAL CYLINDERS AND. THEIR ENCLOSURE by ROBERT ALLEN WEAVER A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mechanical Engineering Approved: Chairperson, Graduate Committee Head, Major JDep^tment Graduate Dean ■MONTANA STATE UNIVERSITY Bozeman, Montana February, 1982 iii ACKNOWLEDGEMENT The author wishes to express his thanks and appreciation to the following for their contribution to this investigation. His advisor, Bob Warrington, for his guidance arid support throughout this investigation. Bill Martindale and Tom Reihman, for serving as committee members and reviewing this thesis. Gordon Williamson, Luther Hartz and Pat Vowell for their helpful assistance in the construction and maintenance of the heat transfer apparatus. The Mechanical Engineering Department of Montana State University, for financial assistance and funding of this investigation. Lastly, yet foremostly, his wife, Cindy, for her never ending encouragement and understanding, and for typing this thesis. TABLE OF CONTENTS Chapter Page VITA................................................ ii ACKNOWLEDGEMENT...........r ....... ................... iii LIST OF TABLES .......................... v LIST OF FIGURES .............................. vi NOMENCLATURE.................................. vii ABSTRACT.............................. X I. INTRODUCTION . . . . . ................................ I II. LITERATURE REVIEW .................................. 3 III. EXPERIMENTAL APPARATUS AND PROCEDURE. .'........... 15 IV. RESULTS................. 30 V. CONCLUSION .............................. . . . . . . 53 APPENDIX I .......................................... 56 APPENDIX I I .......................................... .. . 68 BIBLIOGRAPHY...................................... 73 LIST OF TABLES Table Page 4.1 Range of Dimensionless Parameters . . . ............ 32 4.2 Characteristic Dimensions of Each Inner Body Arrangement...........................................33 4.3 Comparison of the Local Heat Transfer Coefficient of the Bottom Row of Cylinders, hRQwlt to the Upper Rows of Cylinders (hROw2f hROw3, h ROw4). . . . . . . 37 4.4 Correlation Equations for Each Inner Body Arrangement . . . . . . . . . . . . . . . .......... 44 4.5 Correlation Equations for Each Fluid............ 47 4.6 Correlation Equations for Combined In-Line Arrangements, Combined Staggered Arrangements, and All Data Combined...................... 50 4.7 Correlation Results Using the Data From This Study in the Best Correlation Equations of Warrington [12] and Crupper [16].................... 51 V } LIST OF FIGURES Figure Page 3.1 Heat Transfer Apparatus........................ 16 3.2 Schematic of the Heat Transfer Apparatus............. 17 3.3 Nine Cylinder In-Line Arrangement.............. 19 3.4 Sixteen Cylinder In-Line Arrangement. . ............. 20 3.5 Eight Cylinder. Staggered Arrangement................. 21 3.6 Fourteen Cylinder Staggered Arrangement . ........... 22 3.7 Heat Losses from Radiation and Conduction With Air as the Test Fluid ..................................28 4.1 Comparison of the Heat Transfer for Isothermal and Constant Heat Flux Inner Body Conditions. . . . . . . 35 4.2 Geometric and Prandtl Number Effects for All Arrangements Using Air and Water. . . . . . . . . . . 39 4.3 Geometric and Prandtl Number Effects for All Arrangements Using 20 cs Silicone and 96% Glycerine . 40 4.4 Heat Transfer Correlations for the In-Line Data, the Staggered Data, and All of the Data Combined........ 48 vi NOMENCLATURE Description Flow cross-sectional area between cylinders Cross-Sectional area of the copper-constantan thermocouples Cross-sectional area of the heat tape leads Surface area of the inner body Surface area of the outer body Cross-sectional area of the support stems Length of boundary layer on one cylinder, B = ir(d/2) Empirically determined constants Specific heat at constant pressure Diameter of a cylinder Hydraulic diameter, Dj1 = (4ACZ)/AI Acceleration of gravity, 9.81 m/sec^ (32.17 ft/sec2) Grashof number,gp2g (T1-T0)X3/u2, where X is any characteristic length Heat transfer coefficient, h = STAEMNO FOP Vertical pitch between cylinder rows Modified vertical pitch, H 1 = H + d + [d/(number of cylinder rows - I)] Thermal conductivity Horizontal distance between cylinders Thermal conductivity of the copper-constantan thermocouples viii Symbol khtl L A& Nux Pr STAER 0CONV SIOR CPAP r r3X R a \ Ri rO s Tb Tf Tl Description Thermal conductivity of the heat tape leads Thermal conductivity of the support stems Hypothetical gap width, YC - Ri Change in length Nusselt number, hX/k, where X is any characteristic length Prandtl number, Cpy/k. Heat transfer by conduction Heat transfer by convection Heat tranfser by radiation Total amount of heat transfer, SPAP n STAER p STAEM p SIOR Radius of a cylinder Rayleigh n u m b e r , g ( T j - T0)X^cpZyk, where X is any characteristic length ^ Modified Rayleigh number, Ra*% = Ra^(L/R^) Radius of a hypothetical sphere equal in volume to the volume of one cylinder times the number of cylinders in the cylinder array - Radius of a hypothetical sphere equal in volume, to the outer body Characterictic length, S = (R0 - Ri)(A1ZA0) Bulk temperature or average fluid temperature Film temperature, Tf = (Tf, + T1)Z2 Inner Body Temperature Reference temperature, Tj = Tj3 + 0.32(Ts - Tj3) ix Symbol Description Tm Arithmetic mean temperature, Tm = (T1 + T0)/2 Tn Reference temperature, Tn = Tj3 + 0.20(Tg - Tj3) tO Outer body temperature Tg Surface temperature AT Temperature difference, AT = T1 - T0 X Any characteristic length Z Height of the tube bundle 3 Thermal expansion coefficient . U Dynamic viscosity Tr Ratio of circle, circumference to diameter, 3.14159 P Density of the fluid Density of the fluid at atmospheric pressurepatm XABSTRACT The natural convection heat transfer between arrays of horizontal, heated cylinders and their isothermal, cooled enclosure was experimentally investigated. Four different cylinder arrays were used: two in-line and two staggered. Four fluids (air, water, 20 cs silicone and 96% glycerine) were used with Prandtl numbers ranging from 0.705 to 13090.0. There was no significant change in the Nusselt number between isothermal and constant heat flux conditions of the cylinder arrays. The average heat transfer coefficient was most affected by the spacing between cylinders and the total surface area of the cylinder arrays. The enclosure reduced the increase in both the average and the local heat transfer coefficients caused by changing the inner body from an in-line arrangement to a staggered arrangement of comparable spacing. An increase in fluid viscosity reduced the influence of the geometric effects. The best empirical equation for all of the experimental data using one correlating parameter was: Nus = 0.214Ra*g0‘2f>0. Ra*g = Rag(L/R^) for 0.705 I Pr ^ 1.31xl04; 4.45xl04 I Rag < 1.17xl08 0.602 ^ LZRi ^ 1.041; 4.63xl04 I Ra*g I 8.ISxlO7 with an average percent deviation of 12.00. CHAPTER I INTRODUCTION Natural convection heat transfer from a body to an infinite fluid medium has received extensive experimental and analytical study in the past. In recent years there has been a growing demand for an understanding of natural convection heat transfer within enclosures. This phenomenon has important industrial applications in areas such as nuclear reactor technology, electronic instrumentation packaging, aircraft cabin design, crude oil storage tank design, solar collector design, and energy storage systems. This is one of the first studies of multiple body natural convection heat transfer in enclosures. Its purpose is to experimentally investigate the dissipation of heat by natural convection from arrays of heated, horizontal cylinders to a cooled, isothermal, cubical enclosure. The cylinders were subjected to both isothermal and constant heat flux conditions. This study determines the effects of cylinder geometry, and compares the results with the findings of previous studies. Four fluids and four cylinder configurations were utilized. The fluids used were air, water, 96 percent glycerine, and 20cs silicone, with Prandtl numbers ranging from 0.705 to 13090.0. The four cylinder configurations consisted of two in-line arrangements using nine and sixteen cylinders, and two 2 f staggered arrangements using eight and fourteen cylinders. CHAPTER II LITERATURE REVIEW Natural convection heat transfer can be classified in two ways: the external convection problem of flow about a body surrounded by an infinite, fluid medium, and the internal convection problem of flow within an enclosure. The external problem has been investigated extensively in the past while relatively little attention has been given to the internal problem. This is due to the fact that internal natural convection problems are significantly more complex. For external problems the Prandtl boundary layer theory allows one to.assume that the region surrounding the boundary layer is unaffected by the boundary layer. However, in the internal natural convection problem the boundary layer and the region adjacent to the boundary layer interact with each other, making it difficult to obtain analytic solutions to internal problems. The remainder of this chapter is intended to provide a useful background for this particular investigation and is not a complete survey of research in natural convection. The following discussion will be divided into the two categories mentioned above. These are (I) external natural convection to an infinite fluid medium, and (2) internal natural convection in enclosures. However, the discussion is limited to geometries which pertain to this investigation. 4EXTERNAL NATURAL CONVECTION There have been several studies of the heat transferred from single objects (e.g. plates, spheres, cylinders, etc.) to an infinite fluid medium. Morgan [1] gives a very thorough summary of the major correlations for heat transfer from smooth, horizontal, circular cylinders to an infinite fluid medium with Rayleigh numbers ranging from 10--*-® to IO-^. It may be shown by dimensional analysis [2] that the heat transfer from horizontal cylinders varies with the Grashof and Prandtl numbers. The resulting equation is generally of the form ress = A]_ + (GrjjPr)^ l. , B]i, and Ci are constants and X is a characteristic length dimension where Nux = hX/k (Nusselt number), Grx = (g g P2X3ATVu2 (Grashof number), and Pr = yCp/k (Prandtl number.) . The nondimensional grouping (GrxRr) is known as the Rayleigh number, which is rsX = GrxPr = (Cp3gP2X3AT)/ky. Fand, Morris, and Lum [3] presented three different correlations for natural convection heat transfer from horizontal cylinders, based on three different reference temperatures. These are 5Nuf = 0.474 Raf0'25 Prf0*047f NUj = 0.478 Raj0*25 Prj0*050, and Nun = 0.456 Raf0*25 Prf0*057 where the subscripts denote the reference temperature used in evaluating the fluid properties. The reference temperatures were Tf = Tb + 0.5(Tg - Tf,) (film temperature), Tj = Tb + 0.32(Ts - Tb), and Tn = Tb + 0.20(Tg - Tb). Raithby and Hollands [4] have published a correlation equation for laminar and turbulent natural convection from elliptic cylinders of arbitrary eccentricity for the case of constant surface temperature. For the case of horizontal isothermal cylinders their equation takes the form Num JLn {(H-T^ 3Z4)ZIf23Z4C1Ra1/4)}_ + (0.72Ct RaV3) where 3.337 f2 = 2.587 = (2/3)/[I + (0.49/Pr)9/16]4/9 Cb = 0.14 Pr0*004 or 0.15, whichever is smaller. Churchill and Chu [5] suggest that an additive constant is required in the correlation equation for natural convection heat 6transfer from horizontal cylinders. They have published the following correlation equation for heat transfer by natural convection from horizontal cylinders Nud 0.36 + 0.518 ______ ____________ [I + (0.559/Pr)9/16]16/9 0.25 for all Prandtl numbers and for Rayleigh numbers ranging from 10-6 to IO^. They recommend that for large temperature differences, such that the variation of physical properties is signficant, the properties may be evaluated at the average of the bulk and surface temperatures as a first approximation. Kim, Pontikes, and Wollersheim [6] experimentally studied the natural convection heat transfer from a horizontal cylinder with isothermal and constant heat flux surface conditions. The average free convection results were obtained by integrating the local Nusselt, Prandtl, and Grashof numbers over the test section surface area. The average Nusselt numbers for the experimental data were Nur = o.89 Rar0*19 for isothermal surface conditions, and Nur = 0.57 Rar0,20 for constant heat flux surface conditions. Although there has been extensive research performed on natural convection heat transfer from a single horizontal 7cylinder, natural convection heat transfer from multiple cylinders has received little attention. Eckert and Soehngen [7], who performed one of the first studies of natural convection heat transfer from multiple cylinders, investigated the effects that one heated cylinder had on adjacent cylinders in a vertical array of horizontal, isothermal cylinders. They discovered that when one. cylinder was positioned directly over another at a distance of four diameters, there was no change in the Nusselt number of the lowest cylinder as opposed to a single cylinder while the upper cylinder Nusselt number was 87 percent of the value for the lower one. A reduction in the heat transferred from the upper cylinder was said to be caused by the heat ted wake from the lower cylinder striking the upper cylinder. When three cylinders were arranged in a vertical array the Nusselt number of the middle cylinder was 83 percent of the Nusselt number of the bottom cylinder while the Nusselt number of the top cylinder was 65 percent of the value for the bottom cylinder. When the cylinders were staggered such that the middle cylinder was moved laterally out of line by one half of a diameter, the wake of the bottom cylinder missed the middle cylinder and the Nusselt number of the middle cylinder was 103 percent of the value for the bottom cylinder while the Nusselt number of the top cylinder was 87 percent of the value for the bottom cylinder. The increase in the heat transferred from the middle cylinder was a result of the 8higher velocity of fluid movement past this cylinder which was induced by the wake from the bottom cylinder. Liberman and Gebhart [8] investigated the interaction of natural convection wakes between a parallel array of wires. By orienting the array at different angles measured from the vertical and different spacings, they were able to determine the spacing which yielded a maximum Nusselt number for a particular angle of inclination. Marsters [9] performed an experimental study on the natural convection heat transfer from a vertical array of heated horizontal cylinders. He concluded that the heat transfer characteristics exhibited by vertical arrays of heated horizontal cylinders are not predicted by simple superposition of single cylinder behavior. For closely spaced arrays (two diameters between cylinders), individual tube Nusselt numbers were found to be as much as 50 percent lower than for a single cylinder. For wider spacings, individual cylinder Nusselt numbers were as much as 30 percent higher than that of a single cylinder. He concluded that the overall heat transfer characteristics of an array are dependent upon array spacing as well as Rayleigh number. Tillman [10] developed two correlation equations for natural convection heat transfer fran tube bundles: Nuf = 0.057 Raf0-5 9for in-line arrays and Nuf = 0.067 Raf0*5 for staggered arrays. All of the thermal properties except the coefficient of thermal expansion were evaluated at the film temperature. The coefficient of thermal expansion was evaluated at the ambient temperature. The hydraulic diameter for a compact heat exchanger was used for the characteristic length, which was defined as Dh = U A cZ)/A1. Tsubouchi and Saito [11] conducted an experimental study of the natural convection heat transfer from arrays of uniformly heated circular cylinders in air. The cylinders were arranged in various in-line and staggered arrangements. They found that the heat transfer depended on the cylinder spacing, the number of cylinders, and the type of cylinder arrangement. They proposed the following correlation nuM = $2 (0.092) {1 - 0.92 exp[(d-K)/d]} (PrGr11.)0*4 where $2 = 1.00 for in-line banks and $2 = 1*06 for staggered banks. The number of vertical columns ranged from 3 to 5. The number of horizontal rows ranged from 3 to 7 and K was the horizontal distance between the cylinders, which had outer diameters of d. The vertical distance between the cylinders was H. The charactristic length H1 was the modified vertical pitch, defined as 10 H 1 = H + d + d/[(number of horizontal rows) - I]. They concluded by stating that the average heat transfer coefficient was affected more by a variation in the vertical pitch than by a variation in the horizontal pitch. INTERNAL NATURAL CONVECTION Internal natural convection heat transfer utilizes the same dimensionless parameters as are used for external natural convection, however an additional parameter involving a ratio of characteristic dimensions is used in internal problems. Warrington [12] performed an in-depth experimental study of natural convection in enclosures. His work involved the heat transfer between inner bodies such as spheres, cubes, and cylinders to both spherical and cubical enclosures. Several different fluids were utilized with Prandtl numbers ranging from 0.706 to 13,800. The recommended overall correlation from his data was rtl= 0.425 Rali0-234 (LZRi)0*498. For a cylindrical inner body and a cubical enclosure the best correlation was rti = 0.593 Rafi0*240 (LZRi)0*434. Larson, Gartling, and Schimmel [13] used laser interferometry.to experimentally determine the temperature field around a heated horizontal cylinder in an isothermal rectangular 11 box. The purpose of their study was to simulate the possible geometric configurations of a nuclear spent fuel element in a shipping cask and compare the experimental results with numerical results using finite-difference and finite-element techniques. Dutton and Welty [14] conducted an experimental study of natural convection heat transfer in an array of uniformly heated vertical cylinders surrounded by a vertical cylindrical enclosure with mercury as the fluid medium. The cylinders were arranged in an equilateral triangular pattern. Their results indicated that the natural convection heat transfer was strongly dependent on the cylinder spacing and was less dependent on heat flux and circumferential position. In their concluding remarks they suggest that.natural convection heat transfer results in the low Prandtl number range (liquid metals) are well represented by correlations involving the GrxPr2 product, which is independent of viscosity. Van De Sande and Hamer [15] studied the steady and transient natural convection heat transfer between horizontal concentric circular cylinders with constant heat flux surface conditions. Their experiment showed that a sidewise displacement of the inner cylinder did not affect the heat transfer results. However, the overall heat transfer decreased or increased depending on whether the inner cylinder was above or below the center line of the outer cylinder. An additional dimensionless group was introduced 12 to account for the effect of vertical eccentricity. The results were used to estimate the cooling of buried cable systems surrounded by a water layer. Crupper [16] performed an experimental study of natural convection heat transfer between a set of four isothermal, heated cylinders and an isothermal, cooled, cubical enclosure, to determine the effect of the positioning of the cylinders within the enclosure, and to compare the results with the findings of previous studies on heat transfer from single bodies to an enclosure. Four fluids and inner body positions were utilized. The set of cylinders was oriented in both a horizontal and vertical position. The four fluids had Prandtl numbers ranging from 0.7 to 3.1 x IO4 and Rayleigh numbers, based on gap width, ranging from 6.3 x IO^ to 6.9 x 10®. He found that the best correlations for all of the heat transfer data combined were rti = 0.277 Ras0*274 Pr0'012 . and rti = 0.286 RaB0*275. The best correlations for the cylinders in the horizontal position were rtg = 0.498 RaL°*245 Pr-0*002 NuL - and 0.496 Rali0*245 13 All of the fluid properties were evaluated at the arithmetic mean of the inner and outer body temperatures. Powe [17] investigated the limits of relative gap width for which available correlation equations for natural convection heat transfer in enclosures were applicable. Heat transfer rates for large relative gap widths were shown to be limited by those obtained for free convection to an infinite fluid medium, and this criteria was used to calculate a maximum relative gap width for which the enclosure equations were applicable. A minimum relative gap width for applicability of the enclosure equations was determined by the pure conduction limit. Brown [18] experimentally studied the transfer of heat by natural convection within enclosures at reduced pressures. The best correlation which included a correction for the air density was rtg n halfu Rali1/4 (p/patin) 0 e l 2 9 . The geometries used were cylinder-cube (inner body-outer body) and cube-cube, with the bodies mounted concentrically in both cases. The Rayleigh number ranged from I x IO^ to 2 x 10® and the pressure, ranged from 2670 - 86,180 Pa (20 - 646.4 mm Hg). Powe, Warrington, and Scanlan [19] performed a detailed study of natural convection flow phenomena which occur between a body of relatively arbitrary shape and its spherical enclosure. Resulting trends in the fluid flow data were established to 14 facilitate better predictions of the heat transfer in problems of natural convection in enclosures. As evidenced by this review, the amount of interest in internal natural convection has increased dramatically in the last decade. The intent of this study is to extend the work performed by Warrington and Crupper [16, 20] in the area of natural convection heat transfer between multiple bodies and an enclosure. Chapter III EXPERIMENTAL APPARATUS AND PROCEDURE EXPERIMENTAL APPARATUS The outer body used for this investigation was. a cube 26.67 cm (10.5 in.) along an inner side, constructed from 1.27 cm (0.5 in.) thick, type 6061 aluminum. The assembled outer body and peripheral components are shown in Figure 3.1. A water jacket enclosure, which measured 38.1 cm (15.0 in.) on a side, surrounded the cubical test space. The water jacket consisted of six separate rectangular channels 3.175 cm (1.25 in.) in width, which gave one channel for each face of the cube. Several inlet and outlet ports on each of the channels, fed by a manifold system, ensured a uniform flow over each of the sides. The flow rate of cooling water to each of the channels was separately adjusted to maintain the cube which enclosed the test space at isothermal conditions. The cooling water was collected from the water jacket and pumped through a chiller apparatus, into an insulated storage tank, and from there back into the water jacket. A schematic of the appartus is shown in Figure 3.2. Access to the test chamber was accomplished through a removable rectangular cover on the water jacket and a 25.4 cm (10.0 in.) diameter circular cover on the top face of the enclosing cube. The rectangular cover was sealed with a rubber 16 Figure 3.1 Heat Transfer Apparatus 17 Shunt Panel DC Power Supply I Ohm Resistors Voltmeter J Switch Switch Voltmeter From Heat Tapes To Heat Tapes Inner Body Thermocouple Leads and Switch Variable Resistors Water JacketCubical Test Space Inner Body Cooling Water OutCooling Water Outer Body Thermocouple L Leads and SwitchCooling Water Storage Tank Chiller Figure 3. 2 Schematic of the Heat Transfer Apparatus 18 gasket and the circular cover was flanged and sealed with an O- ring. Four different arrangements of horizontal, heated cylinders were used for inner geometries in this investigation. The four geometries consisted of two in-line and two staggered arrangements. These geometries are shown in Figures 3.3 - 3.6. All of the cylinders were fabricated out of 0.36 cm (0.14 in.) thick copper pipe, 19.46 cm (7.66 in.) long, and 4.22 cm (1.66 in.) outside diameter. Copper end caps 0.25 cm (0.098 in.) thick and 4.22 cm (1.66 in.) in. diameter were mounted flush to both ends of each cylinder and attached with high temperature epoxy. The length to diameter ratio of the cylinders was 4.73. Wooden dowels 0.32 cm (0.125 in.) in diameter were used to provide support and spacing for the cylinders. The ends of the wooden dowels were press fitted into .25 cm (0.1 in.) deep holes in each cylinder. Two adjacent cylinders were connected by one or two wooden dowels, depending on their location in the array of cylinders. Each wooden dowel was covered with shrink tube to preserve the wood and minimize the loss of heat. Each array of cylinders was aligned so that its axis was parallel to the sides of the cube. For each geometry, the axial distance between the ends of the cylinders and the cube was 3.35 cm (1.32 in.). The ratio of cylinder diameter to cylinder spacing (center to center) for the inner geometries was 0.55 for the eight and nine cylinder 19 26 67cm (IQ.SOin) 5.64cm (2.22in) 7.72cm (3.04in) rx r 4.22cm __ ( 1 . 6 6 i n )^ ' 1 Iy Ln Figure 3.3 Nine Cylinder In-Line Arrangement 7, 72 cm L __ __ ^ 5. 64 cm (3 .0 4i n) ” ~ (2 .2 2i n) 26 .6 7c m (I Q. SO in ) 20 4.17cm (1.64in) 4.22cm (1 .66in) Figure 3.4 Sixteen Cylinder In-Line Arrangement 6. 12 cm I _ 4. 17 cm (2 ,A li n) I* I (1 .6 4i n 26 .6 7c m (l O. SO in ) 21 26,67cm (IQ.SOin) 5.64cm 7.72cm (2.22in) 4.22cm (1.66in) 9.50cm (3.74 in) (3.04in) Figure 3.5 Eight Cylinder Staggered Arrangement 22 26.67cm (IQ.SOin) 4,17cm 6.12cm (2.41 i n)(1.64in) 4.22cm (1.66in) 7.21cm (2.84in) 6.12cm (2.41in) Figure 3.6 Fourteen Staggered Cylinder Arrangement 23 arrangements (pitch to diameter was 0.83), and 0.69 for the, fourteen and sixteen cylinder arrangements (pitch to diameter was 0.45). For the staggered arrangements, the cylinder spacing was calculated as the horizontal center to center distance between cylinders. Heat was supplied to each cylinder with electrical resistance heat tape and a direct current power source. The heat tape, 0.05 cm (0.02 in.) thick and .32 cm (.125 in.) wide, consisted of an electrical resistance wire which was rated at 28.87 ohms/m (8.8 ohms/ft.) and a maximum power of 75.46 watts/m (78.53 Btu/hr/ft). The heat tape was applied to the inner surface of each cylinder utilizing two pieces approximately 2.13 m (7 ft.) long with each piece starting at the midpoint of the cylinder and wrapped in a helix fashion toward the end. One end from each piece was connected to a power lead while the other . ends were connected in series. Once the heat tapes were in place, they were coated with a high temperature sealant which kept them securely in place and provided an insulated backing. Input voltages to the tajpes were controlled individually by using Ohmite variable power resistors (0-35 ohms or 0-50 ohms, 150 watts, 2.07 amperes or 1.73 amperes maximum) connected in series with the tapes. The electrical circuitry is shown in Figure 3.2. The inside of each cylinder was filled with a two-part silicone potting compound to protect the electrical connections and 24 minimize any convective currents which might cause an uneven temperature distribution in the cylinders. The inside surface temperature of the enclosing cube was monitored by the use of 25 copper-constantan thermocouples epoxied 0.318 cm (0.125 in.) from the inner surface. The number of thermocouples on each face of the enclosing cube varied from 3 to 7. The thermocouples on any common side were connected in parallel to provide an average temperature for each face. It has been shown by Warrington [12] that the temperature variation on any side was less than .8°C (1.5°F). The temperature of each cylinder was monitored by two copper-constantan thermocouples epoxied from the inside of the cylinder into the wall and flush with the outer surface. The two thermocouples were placed 4.99 cm (1.97 in.) from each end of the cylinder. The lead wires from the thermocouples and heat tape in each cylinder passed through a 0.32 cm (0.125 in.) diameter hole in one of the end caps. The lead wires were directed upward where they exited through a 1.91 cm (0.75 in.) diameter, 12.70 cm (5 in.) long PVC pipe, which was threaded into the circular cover on the top face of the enclosing cube and then passed through a hole in the rectangular cover of the water jacket. The pipe was sealed to the water jacket cover with an O-ring. At the end of the pipe, which protruded above the apparatus, the lead wires 25 were sealed off from the test chamber with silicone rubber cement. EXPERIMENTAL PROCEDURE The array of cylinders was placed and centered in the cubical test space. After attaching and sealing the circular cover and water jacket lid, the lead wires from the inner body were sealed off from the cubical test space with silicone rubber cement and connected as shown in Figure 3.2. With the exception of air as the working fluid, the test space was then filled with a test fluid from a reservoir located above the apparatus. The gravity fed liquid flowed through a fill stem located on the bottom of the apparatus and exited out of a hole in the top when the test space was filled. The cooling system and power supply were activated. The cooling water flow rates were adjusted so that the desired isothermal condition was obtained for the outer body. By adjusting the current to each cylinder with the rheostats, approximately seven data points with the inner body at isothermal conditions and four data points with the inner body at constant heat flux conditions were taken for each fIuid/geometry combination. When thermal equilibrium was reached (approximately, two to four hours) the following data were taken: I) The inner body and outer body thermocouple emf readings, 2) The input 26 voltage to each cylinder, and 3) The input amperage to each cylinder. In all, 172 data points were taken utilizing 16 different fluid/geometry combinations. A list of the partially reduced data, along with the type of test fluid, the inner body arrangement, and the inner body boundary condition is shown in Appendix II. The percent temperature variation for either the inner or outer body was defined as Temperature Variation = f^fii0caI, Max---^Local,Min^ iOOf T1 - T0 where Tli0calfMax (TLocal/Min) represents the maximum (minimum) inner or outer body temperature. The outer body had an average temperature variation of 7.31 percent for all of the data. The inner body had an average temperature variation of 12.92 percent for isothermal conditions and 51.47 percent for constant heat flux conditions. The heat transferred by natural convection was obtained by subtracting the heat transfer by radiation between the inner and outer bodies and the conduction through the support system connecting the inner and outer bodies, from the total amount of heat which was transferred. This in equation form is, STAEM n SPAP m SIOR m STAERo ql'dv When air was used as a test fluid the following procedure experimentally measured the amount of heat which was transferred 27 by radiation and conduction. With the complete system assembled as previously explained, a vacuum pump was attached to the fill stem on the bottom of the apparatus which had been used to fill the test space. The test space between the inner and outer bodies was evacuated to a pressure below 6.67Pa (50 microns Hg), which essentially eliminated convection. . Five points, over the range of inner body temperatures used in this experiment, for each inner body arrangement were collected to determine the heat loss by radiation and conduction. These data are shown in Figure 3.7. For the case of air, the average heat loss by radiation and conduction was 77.50 percent of the heat transferred by convection. The three remaining test fluids, water, 96% glycerine and 20cs silicone were opague to radiation while the average heat loss by conduction was 0.26 percent of the heat transferred by convection. Since the supports were insulated with shrink tube, the conduction heat loss from the inner body was calculated using a one-dimensional analysis of the conduction of heat between the bottom row of cylinders and the bottom face of the cubical test space. The conduction of heat through the lead wires from the thermocouples and heat tapes was also included in the analysis. The following equation ^COND = [(ksAs + 1TccaCC + khtlAhtlV(Ajl) ] (Ti-T0) (3.2) was used to calculate the conduction losses. , (W at t) 80.0 60,0 □ - 9 In-Line Cylinders A - 8 Staggered Cylinders 0 - 1 6 In-Line Cylinders v - 14 Staggered Cylinders O V 40,0 " UO UOO 20.0 O 4D O A □ I 20.0 40,0 60.0 T 1 - T0 , (0K) 80.0 __I 100.0 ro CO Figure 3.7 Heat Losses From Radiation and Conduction With Air as the Test Fluid 29 Once ST A E M in equation (3.1) was determined, the average heat transfer coefficient (h) was calculated using h = UCcyr,Lb t(Al)(Tl - T0)]. (3.3) A program was then used to reduce and correlate the data in Appendix II; and existing subroutines [12] were used to calculate the fluid properties of the test fluids. A listing of the program and subroutines is shown in Appendix I. All property values were evaluated at the arithmetic mean fluid temperature, defined as (T0 + T1)/2. (3.4) CHAPTER IV RESULTS This section will discuss the trends and correlations of the experimental data. The chapter is divided into the following sections: (I) Results of inner body boundary condition effects, (2) Results of geometric effects, (3) Results of Prandtl number effects, and (4) Results of all the experimental data combined. Nine characteristic lengths and twenty-four forms of the correlation equations were used to correlate the experimental data. The Rayleigh number, Nusselt number, Prandtl number, and a ratio of characteristic lengths were used as the independent dimensionless parameters. The following equational forms consistently provided the best results using a standard least squares method of curve fitting: Nux = C1RaxcSr (4.1) Nux = C1Rax ^2, (4.2) Nux = C1Raxc^ PrcS, (4.3) Nux = C1RaxcS(LZRi)C3, (4.4) Nux = C1RaxcS(LZRi)c3Prc4. (4.5) Three different characteristic lengths, used to calculate the Nusselt and Rayleigh numbers in the equations above, will be 31 used throughout this chapter. The gap width, L,- is the distance between hypothetical concentric spheres of volumes equal to the actual volumes of the inner and outer bodies. is the radius of a sphere which has a volume equal to the volume of one cylinder times the number of cylinders in the array. B is the distance traveled by the boundary layer on one horizontal cylinder (assuming no flow separation). This distance is defined as one-half of the outer circumference of a cylinder. Both Warrington and Crupper [12,16] found that defining B as the exact boundary layer length, determined from flow visualizations, did not improve the results. S is defined as the ratio of the inner body surface area to the outer body surface area multiplied by the gap width, L. Several other characteristic lengths [10,11] were tried. These characteristic lengths explicitly took into account the cylinder spacing, cylinder diameter and the number of rows of cylinders of the inner body in relation to the cubical outer body. However, the three characteristic lengths, L, B and S consistently yielded the best results, and will be used in the correlation results shown later in the text. The ranges of the independent correlating parameters and the characteristic lengths are shown in Tables 4.1 and 4.2. Reference to parameters which require a characteristic length will be subscripted L, B or S td denote the appropriate 32 TABLE 4.1 RANGE OF DIMENSIONLESS PARAMETERS DIMENSIONLESS PARAMETER MINIMUM MAXIMUM Pr 0.705 13090.0 ffuL 4.040 45.13 nuB 3.663 35.41 Nus 2.657 25.44 GrL 12.29 8.220x10? GrB 9.974 4.196x10? Grs . 3.398 , 2.'143x10? RaL 1.245xl05 3.SOlxlO8 toB . 1.306xl05 1.964xl08 . toS 4.449xl04 1.170xl08 * Ra L 9.772xl04 . ■ 3.958xl08 * Ra B 9.IlSxlO4 1.913xlOS * Ra S 4.633xl04 8.153x10? TABLE 4.2 CHARACTERISTIC DIMENSIONS OF EACH INNER BODY ARRANGEMENT NUMBER OF L B S LZR1 INNER B O D Y . cm cm cm CYLINDERS (in) (in) (in) 9 In-Line 8.115(3.195) 6.624. (2.608) 5.004 (1.970) 0.963 8 Staggered 8.440.(3.323) 6.624 (2.608) 4.625 (1.821) 1.041 16 In-Line 6.332(2.493) 6.624 (2.608) 6.942 (2.733) 0.620 14 Staggered 6.777 6.624 6.500 0.694(2.668) (2.608) (2.559) 34 characteristic length. RESULTS DE 2BE INNER BODY BOUNDARY CONDITION EFFECTS A comparison of isothermal inner body conditions to constant heat flux inner body conditions is shown in Figure 4.1. To remove any geometric effects the graph is comprised of data from only one inner body arrangement, eight staggered cylinders. Figure 4.1 shows that for any one of the four fluid mediums, the constant heat flux data coincide very closely to the isothermal data. The three remaining inner body arrangements also exhibited a negligible difference in the Nusselt number for the different boundary conditions. Overall, the average Nusselt number of the constant heat flux data was only 0.44 percent greater than the average Nusselt number of the isothermal data. The best correlation for all of the isothermal data was rti = 0.229 RaB0,257(L/Ri)0,556Pr0*021 (4.6) which had an average percent deviation of 10.48. The percent deviation at a point is defined as the quantity of the absolute difference between the data value and the equation value divided by the data value. The average percent deviation is the sum of the individual deviations divided by the number of data points. The best correlation for all of the constant heat flux data was 70.0 □ - air O - water A - 20 cs silicone v - 96% glycerine Awr open symbols - isothermal closed symbols - constant heat flux 10.0 I i I Iiiiil I i i i im l I I i I I I ill I I I I 11 ill I I I I 11 ill I I I Iiiiil I I I I I I ill 4 IO5 S 10'r 10 Gr Figure 4.1 Comparison of the Heat Transfer'for Isothermal and Constant Heat Flux Inner Body Conditions Using Data From the 8-Cylinder Arrangement I36 rti = 0.221 RaB0l260(VRi) 0.494pr0.016 (4.7) which had an average percent deviation of 11.03. Although there was no significant difference in the average heat transfer coefficient. Table 4.3 shows a noticeable difference in the local heat transfer coefficients when comparing isothermal and constant heat flux inner body conditions. The local heat transfer coefficient of each cylinder was calculated by using the temperature difference between the cylinder and the mean temperature of the outer body. The local heat transfer coefficient for each row of cylinders was the average of the local heat transfer coefficients of the cylinders in each row. Table 4.3 is a comparison, using all of the fluids, of the local heat transfer coefficients for each row of cylinders above the bottom row to the local heat transfer coefficient of the bottom row. The percentage comparisons were consistently higher for the constant heat flux conditions than they were for the isothermal conditions. As stated earlier, the average percent temperature variation of the inner body was 51.47 for constant heat flux conditions as opposed to 12.92,for isothermal conditions. Under constant heat flux conditions the temperatures of the upper rows of cylinders were forced to be increasingly higher than the temperature of the bottom row. Since the enclosure caused an increase in convective activity in the upper regions of the enclosure, the higher temperature of the upper rows augmented the TABLE 4.3 COMPARISON OF THE LOCAL HEAT TRANSFER COEFFICIENT OF THE BOTTOM ROW OF CYLINDERS, hR O W r T0 THE UPPER R0WS 0F CYLINDERS ( h ^ ' hR0 W S » hR 0 W 4 } NUMBER OF INNER BODY CYLINDERS INNER BODY BOUNDARY CONDITION > W2 x 100 nROWl ^ 0"3 x 100 nROWl ^r0w4 k 100 nROWl 9 In-line 60.115 41.722 8 Staggered 62.833 52.318 Isothermal 16 In-line 53.910 36.819 33.925 14 Staggered 59.021 42.428 36.475 9 In-Line 77.515 67.752 ’ 8 Staggered Constant 79.118 70.178 . Heat 16 In-line ■ Flux 75.286 63.917 57.764 14 Staggered 74.409 64.355 57.586 38 driving potential for the heat transfer, resulting in higher local heat transfer coefficients for the upper rows. The remainder of the text will not distinguish between isothermal and constant heat flux inner body conditions when discussing overall heat transfer data correlations. RESULTS OF THE GEOMETRIC EFFECTS There were two major geometric effects evident in the experimental data. First, a staggered inner body arrangement had a slightly higher heat transfer coefficient than ah in-line arrangement of comparable size and spacing. Second, the heat transfer coefficient increased when the spacing of the inner body was increased and the surface area of the inner body was decreased. These geometric effects are shown in Figures 4.2 and 4.3 which are graphs of Nu^i versus Ra^j for all of the experimental data, divided into the separate fluid mediums. These figures^ show that the effects of spacing and total surface area are more | . pronounced than the geometric effect of changing the inner body ,i ',I from an in-line arrangement to a staggered arrangement of; comparable size and spacing. When going from nine in-line cylinders to eight staggered cylinders or from sixteen in-line cylinders to fourteen staggered cylinders the Nusselt number increased only slightly. However, 30.0 10.0 Nul 0 - 4 cylinders. Crupper [lo] o - 9 cylinders 0 - 8 cylinders o - 1 6 cylinders v - 14 cylinders open symbols - in-line arrangements closed symbols - staggered arrangements OOO A AM A 8 A oB I Illll I 80.0 * ° I I Illlll I I___I Ai r (0.7050 i Pr < 0.7121) llllll I i l l 1111 J Nu1 10.0 * A Q ^ CxP A ▼ o % A CD A S T ’’° ° V O I I llllll I Illll Water (4.624 < Pr < 11.03) , , M . lI___I Figure 4.2 Geometric and Prandtl Number Effects for Al I Arrangements Using Air and Water 90.0 rtg 10.0 30.0 Nul L 10.0 Figure 4 0 - 4 cylinders. Crupper [l6] o - 9 cylinders a - 8 cylinders o - 1 6 cylinders open symbols - in-line arrangements closed symbols - staggered arrangements a v - 14 cylinders * * ▼ _o - ' . 0*0* V J O T 8 i Iiiiil j_I Iiiil 20 cs Si Iicone (137.3 < Pr < 307.3) i i I i i i i 11______ J___I * B . “ a o a o° A, ° 0„ ' $ ° a g o ® B 96% Glycerine ^ ° (541.9 < Pr < 1309.0) I l M l I Illll I I Illtll I I Illlll ,3 Geometric and Prandtl Number Effects for Al I Arrangements Using 20 cs Silicone and 96% Glycerine 41 < there was a large increase in the Nusselt number when going from sixteen in-line cylinders and fourteen staggered cylinders (with diameter to spacing ratios of 0.69) to nine in-line cylinders and eight staggered cylinders (with diameter to spacing ratios of 0.55) respectively. Figures 4.2 and 4.3 also show that the data from Crupper [16] for four in-line cylinders (with a diameter to' spacing ratio of (0.33) had a larger Nusselt number than the nine j cylinder in-line arrangment. This supports the premise that ah" increase in the cylinder spacing and a decrease in the inner body ; surface area tends to increase the Nusselt number substantially. \ ,.4 Warrington [12] also discovered that an increase in the inner body surface area reduced its capacity to transfer heat. Crupper [16] found that the average Nusselt number from the top row of cylinders was 79.5 percent of the bottom row with a separation of three diameters between rows. Eckert and Soehngen [7], who performed a similar study using an infinite atmosphere, found that for two horizontal cylinders, placed one above the other and separated by a distance of four diameters, the Nusselt number of the top cylinder decreased to 87 percent of the Nusselt of the bottom cylinder. These same investigators performed the same investigation on three horizontal cylinders placed above each other, the resulting heat transfer from the middle cylinder was 83 percent of the bottom cylinder while the heat transfer from the top cylinder was 65 percent of the bottom cylinder. 42 Since the local heat transfer coefficient of an upper cylinder was decreased when the heated wake from a lower cylinder surrounded it, Eckert and Soehngen [7] investigated the effect of staggering the cylinders so that the wake from the bottom cylinder passed by the side of the middle cylinder. The Nusselt number of the middle cylinder was 103 percent of the bottom cylinder, while the Nusselt number of the top cylinder was 87 percent of the bottom cylinder. When the results of Table 4.3 were compared to the findings of Crupper [16] it was evident that a decrease in the ratio of diameter to spacing, from 0.69 (sixteen in-line and fourteen staggered cylinders) to 0.55 (nine in-line and eight staggered cylinders) to 0.33 (four in-line cylinders [16]), led to a relative increase in the local heat transfer coefficients of the upper rows of cylinders. A similar comparison was made between the results of the eight and nine cylinder arrangements in Table 4.3 and the results of Eckert and Soehngen [7]. Their use of a smaller ratio of diameter to spacing, 0.25, contributed to their higher values for the local heat transfer coefficients of the upper rows. However, the enclosure used in this investigation, as opposed to the infinite atmosphere used by Eckert and Soehngen [7], significantly dampened the increase of the local heat transfer coefficients caused by staggering the cylinders. This was due to the recirculation of the warmer fluid. 43 The best correlations for the nine cylinder and the sixteen cylinder in-line arrangements were Nus = 0.155 Ras0'279 Pr0*0065 (4.8) and Nus=. 0.185 Ras0*252 Pr0'029 (4.9) with average percent deviations of 9.00 and 11.10, respectively. The best correlations for the eight cylinder and the fourteen cylinder staggered arrangements were Nus = 0.227 Ras0*255 Pr0*019 (4.10) and Nus = 0.245 Ras0*239 Pr0*020 (4.11) with average percent deviations of 11.42 and 9.94, respectively. As shown in Table 4.4, equation form 4.3, which is in terms of two correlating parameters, yielded the same average percent . deviation as the more complex equation form 4.5, which is in terms of three correlating parameters. When correlating data from only one geometric arrangement,, the characteristic lengths L, B, and S, each give the same average percent deviations for a particular equation form. Therefore, the correlations in Table 4.4 use only S as the characteristic length. RESULTS OF THE PRANDTL NUMBER EFFECTS As shown in Figures 4.2 and 4.3, the. geometric effect of TABLE 4.4 CORRELATION EQUATIONS FOR EACH INNER BODY ARRANGEMENT EQUATION FORM X EMPIRICAL CONSTANTS AVERAGE % DEVIATION MAXIMUM % DEVIATION ■ ci C 2 . C3 C4 9 In-Line Cylinders 4.1 S .156 .280 9.275 42.029 4.2 S ,158 .280 9.275 42.029 4.3 S .155 .279 .0065 8.995 38.868 4.4 S .049 .280 -30.613 9.275 42.029 4.5 S .049 .279 -30.387 .0065 8,995 37.868 8 Staggered Cylinders 4.1 S .234 .257 11.796 42.107 4.2 S .232 . .257 11.796 42.107 4.3 S .227 .255 .019 11.422 33.053 4.4 S .402 .257 -13.319 11.796 42.107 4.5 S ' .401 .255 -14.103 .019 11.422 33.053 16 In -line Cylinders 4.1 S .176 .263 13.676 33.999 4,2 S .200 .263 13.676 33.999 4.3 S .185 .252 .029 11.105 31.015 4.4 S .091 .263 -1.388 13.676 33.999 4.5 S .101 .252 -1.265 .029 11.105 31.015 ' 14 Staggered Cylinders 4.1 S .264 .240 11.555 31.542 4.2 S .288 .240 11.555 31.542 4.3 S .245 .239 .020 < 9.942 26.323 4.4 S .140 .240 -1.732 11.555 31.542 4.5 S .126 .239 -1.812 .020 9.942 26.323 45 changing the cylinder spacing and inner body surface area became less pronounced with increasing Prandtl number. However, the Prandtl number had no consistent influence when the inner body was changed from an in-line arrangement to a staggered arrangement of equal spacing. Warrington [12] found that the fluid viscosity did not influence the extent,of any geometric effect, while Crupper [16], whose findings are in agreement with those in this study, found that an increase in fluid viscosity tended to damp out geometric effects. Crupper [16] postulated that the difference in the findings between himself and Warrington [12], concerning Prandtl number effects, was due to the fact that his geometric change was more radical than Warrington's [12]. The best correlation equation for the air data of all four inner body arrangements combined was rti = 0.0097 Ra60,207 (LZRi)0*596 Pr-11*171, (4.12) with an average percent deviation of 11.26. The large scatter in the data for air, as shown in Figure 4.2, was possibly caused by the large relative magnitude of the radiation and conduction losses alluded to earlier. The best correlations in terms of three correlating parameters for water, 2Ocs silicone and 96% glycerine were rti = 1.045 RaB0*171 (LZRi) 0 7^12 Pr0*00084, (4.13) .46 Nus = 6.075 Ras0*148 (VRi)0*131 Pr-0*50, (4.14) and Nus = 0.025 Ras0*324 (L/Ri)0,350 Pr0*165, (4.15) with average percent deviations of 5.51, 3.08, and 4.94, respectively. Table 4.5 shows the correlation results for each fluid using equation forms 4.1 through 4.5. RESULTS OF ALL EXPERIMENTAL DATA COMBINED All of the experimental data from this investigation are shown graphically in Figure 4.4, which is a graph of Nug versus Hag*. The data for four horizontal in-line cylinders from. Crupper [16] are also shown in Figure 4.4. In accordance with statements made earlier in the text, there is very.little difference between the correlations for the in-line arrangements and the staggered arrangements, as shown in Figure 4.4. The best correlation for the nine cylinder and the sixteen cylinder in-line arrangements combined was Nus = 0.174Ras0*269 (LZRi)0*331 Pr0*017 (4.16) with an average percent deviation of 10.30* The best correlation for the eight cylinder and the fourteen cylinder staggered arrangements combined was Nus = 0.247Ras0,248(L/Ri)0l369Rr0*020 (4.17) TABLE 4.5 CORRELATION EQUATIONS FOR EACH FLUID EQUATION FORM X EMPIRICAL CONSTANTS AVERAGE % DEVIATION MAXIMUM % DEVIATIONc I C2 C3 C4 Air 4.1 S .502 .185 13.319 51.847 4.2 L .076 .337 13.103 33.820 . 4.3 B .456 .255 2.237 16.818 46.964 4.4 B .237 .255 .587 11.484 48.128 4.5 B .0097 .207 .596 -11.171 11.255 50.051 Water 4.1 S 1.182 .151 7.703 21.171 4.2 S .933 .166 5.518 15.951 4.3 ' S 24.164 ■ .040 -.596 5.631 19.755 4.4 B 1.049 .171 .712 5.506 19.097 4.5 B 1.045 .171 .712 .00084 5.507 18.098 20cs Silicone 4.1 S .758 .190 5.538 13.911 4.2 S . .537 .213 3.424 ■ 9.595 4.3 S 39.922 .094 -.438 3.138 10.591 4.4 S .521 .216 .280 3.251 8.743 4.5 S 6.075 .148 • .131 -.250 3.082 9.423 96% Glycerine 4.1 S .382 .215 5.493 37.523 4.2 •s .362 .222 4.839 40.501 4.3 S 1.452 .166 -.084 4.862 40.854 4.4 S .364 .221 .143 4.777 39.108 4.5 S .025 .324 .350 .165 4.938 34,.958 100.0 Nuc- 10.0 Figure 4. open symbols - in-line arrangements closed symbols - staggered arrangements a - 9 cylinders a - 8 cylinders o - 1 6 cylinders v - 14 cylinders 0 - 4 cylinders. Crupper [l6] J__I 1.1 11ill I I I I I I i l l 1. Staggered 2. All I I i i i m l i Iiiii IJ I I 4 Heat Transfer Correlations for the In-Line Data, the Staggered Data, and Al I of the Data Combined 49 with an average percent deviation of 10.70. The best correlation for all of the data combined was Nus = 0.211Ras° *258(LZRi)0.354pr0.019 (4.18) with an average percent deviation of 10.74. The remaining correlation results, using equation forms 4.1 through 4.5, for the in-line data, staggered data, and all of the data combined are shown in Table 4.6. In both Tables 4.5 and 4.6, the correlations employ the characteristic length which yielded the lowest average percent deviation. Since this investigation is a continuation, of the research performed by Warrington [12] and Crupper [16], their best correlations were compared with data from this study. The results of this comparison are shown in Table 4.7. The correlations from Warrington [12], based on the natural convection heat transfer between single bodies and their enclosure, fit the data from this investigation quite well. The large error which occured from the use of Crupper's [16] correlation equations could have been caused by two things. First, the form of Crupper's [16] best correlation equations did not include a ratio of characteristic dimensions as a correlating parameter, as was used by Warrington [12] and the present author. This could reduce the accuracy of Crupper's [16] best correlation equations when applied to investigations with a different TABLE 4.6 CORRELATION EQUATIONS FOR COMBINED IN-LINE ARRANGEMENTS,. COMBINED STAGGERED ARRANGEMENTS, AND ALL DATA COMBINED EQUATION FORM X EMPIRICAL CONSTANTS AVERAGE % DEVIATION MAXIMUM % DEVIATIONci C2 C3 .C4 All Data Combined 4.1 S .222 .254 13.319 . 50.580 4.2 S .214 .260 12.002 52.583 4.3 S .216 .251 .019 12.320 . 40.253 4.4 S .216 .261 .355 11.862 54.609 4.5 B .266 .258 .533 .019 10.714 45.619 Staggered Arrangements Combined 4.1 S .263 .244 12.996 43.702 4.2 S .256 .249 11.789 43.446 4.3 S .250 .243 .020 11.932 37.605 4.4 S .259 .249 .366 ■ 11.543 46.567 4.5 S .247 .248 .369 .020 10.700 36.171 Ili-line Arrangements Combined 4.1 S .179 .266 12.857 45.361 4.2 S .173 .273 11.473 43.555 4.3 S .179 .262 .017 12.686 39.080 4.4 S .174 .274 .336 11.404 45:125 4.5 S .174 .269 .331 .017 10.302 37.459 TABLE 4.7 CORRELATION RESULTS USING THE DATA FROM THIS STUDY IN THE BEST CORRELATION EQUATIONS OF WARRINGTON [12] AND CRUPPER [16] EQUATION AUTHOR AVERAGE % DEVIATION MAXIMUM % DEVIATION rtg = 0.396RaL°1234 (VRi)0'496Pr0'0162 [12] 19.547 78.520 rtg = 0.425RaL0,234(LZRi)0,498 [12] 21.351 92.640 Nufi = O^yyRa130l274Pr0,012 [16] 74.078 165.679 rtg = 0.358RaL0,257Pr0,014 [16] 70.169 156.780 52 hypothetical gap with, L. Second, only one hypothetical gap width L was used during Crupper's [16] investigation. This would also suggest that Crupper's best correlations are limited to a range of hypothetical gap widths in the neighborhood of the hypothetical gap width used in his study. 4 r CHAPTER V 4 CONCLUSIONS This investigation has extended the amount of available data for heat transfer between multiple bodies and an enclosure. In this study isothermal inner body conditions were compared to constant heat flux inner body conditions. There was no appreciable difference in the average heat transfer coefficient between the two conditions, which greatly increases the applicability of the correlations discussed in this study. However, the local heat transfer coefficients of the upper rows of cylinders, when compared to the local heat transfer coefficient of the bottom row of cylinders, were much higher for constant heat flux inner body conditions than they were for isothermal inner body conditions. The constant heat flux condition forced the upper rows to a higher temperature and augmented the driving potential for heat transfer which resulted in higher local heat transfer coefficients for the upper rows. The distance between cylinders and the amount of inner body surface area were the dominate factors influencing the average heat transfer coefficient. The enclosure severly dampened the increase in both the average and the local heat transfer coefficients caused by changing the inner body from an in-line arrangement to a staggered arrangement. It was observed that an increase in Prandtl number had a 54 dampening effect on the geometric efects. The heat transfer data correlated with the characteristic length S, which is the hypothetical gap width L multiplied by the ratio of the inner body surface area to the outer body surface area, generally provided the best correlation results. The following equations, in terms of one, two, or three correlating parameters, are recommended by the present author for the prediction of natural convection heat transfer between arrays of heated horizontal cylinders and their cooled enclosure. These correlation equations are: Nus = 0.2149Ras*0-260 (5.1) Nus = 0.216Ras°*261 (!,/Ri)0 ^355 (5.2) Nus = 0.211Ras0*258(LAi) 0*354Pr0*0189 . (5.3) 0.620 £ (LAi) ^ 1-041; 0.705 X Pr I 1.309 x IO4 4.449 x IO4 Ras < 1.170 x IO8; 4.633 x IO4 I Ra*s £ 8.153 x IO7 which have average percent deviations of 12.00, 11.86, and 10.74, respectively. APPENDICES APPENDIX I HEAT TRANSFER DATA REDUCTION PROGRAM The following is a data reduction program which computes and correlates all of the dimensionless groups. All of the variables, subroutines and function subprograms are defined within the program. 57 C * * * * * H E A T TRANSFER DATA REDUCTION AND CORRELATION PROGRAM C DIMENSION '82f324_Lx8^__fI 0 .82_□'*°H8i<324_< lC8if324_f X ,l8i< 324_ * * C 3 ) * ( P R * * C 4 ) DO 12 1 = 1 , 3 X NM I N ( I ) = R M I N ( I ) = R MM I N ( I ) = G M I N ( I ) =P Rff I N = I O . * * 1 0 12 XNMAX ( I ) = R MA X d ) =RMMAX ( D = GMAX ( I ) =PRMAX = O. K = O P I = 3 . 1 4 1 5 5 2 7 C * * * * * R E A D IN A NEGATIVE INTEGER OR ZERO FOR I DB AFTER THE LAST DATA SE T 5 0 R E A D ( 1 0 5 , 2 5 ) I D B , J J , X C U 8 E , S D , X X X , T A V G I , T A V G 0 , P , P L 25 F O R M A T ( 2 I 5 , 7 F 1 0 . 4 ) I F ( I D B ) I 0 0 , 1 0 0 , 3 0 30 K = K * l ; T AVO ( K ) = TAVGO I T A V I ( K ) = T AVG i ; P P ( K ) = P I P L L ( K ) = P L d I DDBB = IOB R O = ( ( 3 . * X C U B E * * 3 ) / ( 4 . * P I ) ) * * ( 1 . / 3 . ) BLL = PI * S D / 2 . I J I 23 = J J SA0B=XCUBE*XCUBE* 6 . GO TO ( I , 2 , 3 , 4 ) , I DB GO TO 31 c * * * « * 9 CYL I NDERS , I N - L I N E 1 R I = ( 2 7 . * S D * S D * X X X / 1 6 . ) * * ( 1 . / 3 . ) GAP=RO-RI S A l B = 9 . * P I * S D * X X X > 1 8 . * P I * S D * S 0 / 4 . RAOS A= ( SA I B / S A OB ) * GAP GO TO 10 C * * * * * 8 CYL INDERS# STAGGERED 2 R I = ( 2 4 . * S D * S D * X X X / 1 6 . ) * * ( 1 . / 3 . ) GAP=RO-RI S A I B = 8 . * P I * S D * X X X + 1 6 . * P I * S D * S D / 4 . RAOSA= ( SA I B / SAOB ) * GAP GO TO 10 C * * . * * 1 6 CYL I NDERS , I N - L I N E 3 R I = ( 4 8 . * S 0 * S 0 * X X X / 1 6 . ) * * ( 1 . / 3 . ) GAP=RO-RI S A I B = 1 6 . * P I * S D * X X X + 3 2 . * P I * S D * S 0 / 4 . RAOSA= ( SA I B / SAOB ) * GAP GO TO 10 £ * * * ♦ * 1 4 CYL I NDERS , STAGGERED 4 R I = ( 4 2 . * S D * S 0 * X X X / 1 6 . ) * » ( 1 . / 3 . ) GAP=RO-RI S A I 8 = 1 4 . * P I * S D * X X X + 2 8 . * P I * S 0 * S D / 4 . RAOSA=( S A I B / S A O B ) ‘ GAP GO TO 10 £ * * * ♦ * 4 £ Y L I NDERS , I N - L I N E 31 R I = ( 1 2 . * S D * S D * X X X / 1 6 . ) * * ( 1 . / 3 . ) GAP=RO-RI S A I B = 4 . * P I * S D * X X X + 8 . * P I * S D * S D / 4 . RAOSA=( SA I B / SAOB ) * GAP 10 CONTINUE C * » * » * D T I S THE TEMPERATURE DI FFERENCE BETWEEN THE INNER AND OUTER C BODIES ( F ) DT=TAVGI - TAVGO £ . * * * . H IS THE AVERAGE HEAT TRANSFER COEFF I C I ENT ( 8 TU/ HR - FT * * 2 - F ) H = P * 1 4 4 . / ( D T * S A I B ) £ * . ‘ “ CALCULATE THE MEAN TEMPERATURE ( DEG. F ) TM= ( TAVGO+TAVG I > / 2 . £ * * * * . TAVG, MEAN TEMPERATURE (DEG. R ) T AVG = T M + 4 5 9 . 6 9 59 C * * * * * C A L CU L A T E THE FLU I D PROPERTIES C * * * * * V I S C O S I T Y ( L 8 M / HR - F T ) V I S =U ( T AVGz J J > C * * * * * S P E C I F I C HEAT ( B T U / L 8 M - R ) SH=CP ( TAVGz J J ) C * * « * * THERMAL CONDUCT I V I TY ( BTU / H R- FT - F ) COND=CON(T A VG z J J ) C * * * . . D E N S I T Y ( L 8 M / F T . . 3 ) OEN = RHO(TA VGzJ J ) C . » . . . T H E RM A L EXPANSION COEFF I C I ENT ( I Z R ) BET=BETA ( T AV G z J J ) C * . . . . T H E PR A NDTL NUMBER P R ( X ) = V I S ' SH / COND I F ( P R ( K ) . L T . P R M I N ) P R M I N = P R ( K ) I F ( P R ( K ) eGT1 PRv AX ) PRMAX=PR( K ) C . . . . . T H E GR A S HO F NUMBER Z Z d ) = G A P Z Z ( Z ) = B L L ZZ ( J ) =RAOSA Z6 = 3 2 . I 74 . B E d D T * DE N *DE N «3 6 0 0 . ‘ 3 6 0 0 . / ( 1 7 Z 8 . * V I S ‘ V I S ) DO 13 I = 1 z 3 G R ( I z K ) = Z 6 * ( Z Z ( I ) * * 3 ) I F ( G R d z K ) . L T . GM I N ( D ) G M I N ( I ) =G R ( I z K ) I F ( G R d z K ) . G T - G M A X ( D ) G MA X ( I ) = G R ( I z K ) 13 CONTINUE C . . . . . T H E NUSSELT NUMBER Z 7 = H / ( C 0 N D * 1 Z . ) DO 14 1 = 1 , 3 XNUS( I z K ) = Z 7 * Z Z ( I ) I F ( X N U S d z K ) . L T 1 X N M I N ( D ) X N M I N d ) = X NUS ( I z K ) I F ( XNUS ( I z K ) . G T . XNMAX( I ) ) XNMAX( I ) =XNUS( I z K ) 14 CONTINUE C . . . . . T H E RALEIGH NUMBER DO 15 I = I z 3 R A d , K ) =GR ( I zK) «PR ( K ) I F ( R A d z K ) 1L T 1 RM I N ( D ) R M I N ( I ) = R A ( I z K ) I F ( R A d z K ) 1GT1 R M A X ( D ) R MA X ( D = R A d z K ) 15 CONTINUE C * * . . . T H E MODI F I ED RALE IGH NUMBER DO 16 I = I z 3 RAM ( I z K ) = R A ( I z K ) * GA PZRI I F ( R A M ( I z K ) . L T 1 R MM I N ( D ) R MM I N ( I ) = R AM ( I z K ) I F (RAM ( I z K ) 1GT1RMMAX ( D ) RMMAX ( I ) = R AM ( I z K ) 16 CONTINUE SAVG AP ( K ) = GAP ; S A V R I ( K ) = R I GO TO 50 1 0 0 CONTINUE I F ( L L L 1 EQ1 Z)GO TO 999 C . . * . . OUTPUT THE REDUCED DATA AND THE DIMENSIONLESS RESULTS I D B = I I D D B B D J = J I 23 5 0 0 FORMAT C l ' ) 5 05 FORMAT(7Z> 510 FORMAT ( Z Z ) 515 FORMAT( 3 4 X z * 9 CYL INDER I N - L I N E ARRANGEMENT * . » • ) SZO FORMAT( 3 3 X z ' . . . 8 CYL INDER STAGGERED ARRANGEMENT . . . ' ) 52 5 FORMAT( 3 3 X z ' * * * I 6 CYL INDER I N - L I N E ARRANGEMENT . * * • ) 5 3 0 FORMAT( 3 Z X , ' . . . 14 CYL INDER STAGGERED ARRANGEMENT . * . • ) 5 35 FORMAT( 3 4 X z * » . * 4 CYL INDER I N - L I N E ARRANGEMENT . . » • ) 60 c 540 F O RMA T ( 3 5 X , ' » * * THE TEST SPACE 545 FORMAT C35X z ' * * * THE TEST SPACE 550 FORMAT C 3 0 X , ' * * * THE TEST SPACE 555 FORMAT C 3 1 X , ' * * * THE TEST SPACE 560 FORM AT ( I X , 1 0 6 ' * ' , Z z 1 X z ' * ' , 3 1 X, 31X,'* ' ) z ' * ' z 2 8 X , 'WHERE TH E CHARACTERIST IC LENGTH "GAP " = Z ' * 'z28x. DL I i P i THE CHARACTERIST IC LENGTH " A L L ” = z ' * ' ,27X, 'WHERE THE CHARACTERIST IC LENGTH "RA OS A 20 CS S I L I CONE * * * ' , / / / ) 96% GLYCERINE * * * ' , / / / > S2 8X , ) 5 70 FORMAT( I X S E 1 0 . 4 / 2 8 X 575 FORMAT( I X $ , E 1 0 . A z 2 7 X , ' * • ) 5 90 F O R M A T ( 1 X , 1 0 6 ' * ' , / , 1 X , ' * ' , 2 ( 3 X , ' A V E R A G E ' , 2 X , ' * ' ) , 2 ( 1 1 X , ' . ' ) , 3 ( 1 0 X , I**')/I X Z ' MODI F I ED *z l x Z 1 * ’ z i o x Z** *z / z l X z * * OUTER 90PY * I NNER BODY $ * QCONV * QLOSS * • , ' PRANDTL » GRASHOF * , z 2 * RALEIGH * * $ z * NUSSELT oo. b . dx. u 'j TEMP CF) • , 2 ' * ( BTUZHR) ' , $ 5 * * . NUMBER * z * * ' z / z l X z 1 0 6 ' * ’ ) 595 F O R M A T ( 1 X , ' * ' z 2 X z F 8 . 4 z A X z F 9 . 4 z 2 X z 2 ( 1 X z F 1 0 . A z 1 X ) z 5 ( E T 0 . t z 1 X ) , ' z ' ) 6 0 0 FORMAT C I X z 1 0 6 ' * * ZZz I X z ' * THE MINIMUM PRANDTL NUMBER = ' z E I 0 . 4 z I 7Xz I ' TH E MAXIMUM PR ANDTL NUMBER = ' z E 1O. 4 z 11 Xz ' * ' ) 60 5 FORMAT( I X z ' * THE MINIMUM GRASHOF NUMBER= ' z E 1 3 . 4 , 1 7 X , $ ' THE MAXIMUM GRASHOF NUMf lER = ' z E 1 0 . 4 z 11 Xz ' * ' ) 6 1 0 FORMAT( I X z ' « THE MINIMUM NUSSELT NUMBER= ' z E I O. 4 , 1 7X , $ ' THE MAXIMUM NUSSELT NUMBER= ' z E 1 0 . 4 , 1 1 X z ' * ' ) 6 1 5 FORMAT C l X , ' * THE MINIMUM RALEIGH NUMBER= ' , £ 1 3 . 4 , 1 7X , $ ' THE MAXIMUM RALEIGH NUMBER= ' z E I O. 4 , 11 Xz ' * ' ) 6 20 FORMAT C I X , ' * THE MINIMUM MODI F I ED RALE IGH NUv 3 ER= ' z E I 0 . 4 , 8 X, $ ' THE MAXIMUM MOD I F I ED RALEIGH NUMB ER = ' z E I O. 4 , 2 X , ' * ' , Zz I X , I 0 6 1 * ' ) WR I TEC I 0 8 , 5 0 0 ) W R I T E ( 1 0 8 , 5 1 0 ) GO TO ( 6 2 5 , 6 3 0 , 6 3 5 , 6 4 0 ) , I D B W R I T E ( 1 0 S z 5 3 5 ) ; G 0 TO 6 4 5 6 2 5 URI TEC I 0 8 , 5 1 5 ) FGO TO 6 4 5 6 3 0 WRITEC I 0 8 , 5 2 0 ) FGO TO 64 5 635 WRI TEC I 0 8 , 5 2 5 ) FGO TO 64 5 6 4 0 WR I TEC I 0 8 , 5 3 0 ) 6 4 5 CONTINUE GO TO ( 6 5 0 , 6 5 5 , 6 6 0 , 1 0 0 0 , 6 6 5 ) , J J 6 50 WRITECI 0 8 , 5 4 0 ) F GO TO 6 7 0 655 W R I T E ( 1 0 8 , 5 4 5 ) FGO TO 6 7 0 6 6 0 WR I TEC I 0 8 , 5 5 0 ) FGO TO 6 7 0 665 W R I T E ( 1 0 8 , 5 5 5 ) 6 7 0 CONTINUE DO 17 1 = 1 , 3 W R I T E ( 1 0 8 z 5 6 0 ) GO TO ( 6 8 0 , 6 8 5 , 6 9 0 ) , ! 6 8 0 WR I TECI 0 8 , 5 6 5 ) ZZC I ) FGO TO 705 6 8 5 W R I T E C 1 0 8 , 5 7 0 ) Z Z ( 2 ) ; G 0 TO 705 6 9 0 WR I TEC I 0 8 , 5 7 5 ) Z Z ( 3 ) 7 0 5 WR I TEC I 0 8 , 5 9 0 ) DO 18 I I = I z K 18 W R I T E ( 1 0 8 z 5 9 5 ) T A V O ( I I ) , T A V I ( I I ) z P P C I I ) , P L L ( I I ) z P R C I I ) z G R ( I z I I ) z IRAC I , I I ) ,R AMC I z I I ) , XNUS C I z I I ) WRITECI 0 8 , 6 0 0 ) PR MINzPRMAX W R I T E C 1 0 8 , 6 0 5 ) G M I N ( I ) , G M A X ( I > W R I T E ( 1 0 8 , 6 1 0 ) X N M I N ( I ) , XNMAX ( I ) W R I T E ( 1 0 8 z 6 1 5 ) R M I N ( I ) , R M A X ( I ) 61 WRITEC1 0 8 , 6 2 0 ) RMMIN ( I ) , RMMAX( I ) WR I T EC I 0 8 , 5 0 0 ) 17 WR I T EC I 0 8 , 5 0 5 ) _ I G M M MS i $ S i F X . ( 3 9 9 9 CONTINUE CORRELATE THE D IMENSIONLESS RESULTS 1010 FORMAT C * I • , / / , 4 0 X , ' * * * CORRELATION NO. ' , 1 2 , ' * * * ' ) 1 0 15 FORMATC46X , ' NU = C I * C R A * * C 2 ) ' ) 1 0 20 FORMAT C 4 6 X , * NU = C I * CRAM* »C2 ) ' ) 1025 F 0 R M A T C 3 3 X , ' N U = C 1 * C R A * * C 2 ) * C C G A P / R I ) * * C 3 ) ' ) 1 0 30 F 0 R M A T C 3 4 X , ' N U = C 1 * C R A * * C 2 ) * C C G A P / R I ) * « C 3 ) * C P R * * C 4 ) ' ) 104 0 FORMAT C 4 I X , ' NU = C I * C R A * * C 2 ) * C P R * * C 3 ) ' ) 1 0 6 0 F ORMATC 3 2 X , ' * * * THE CHARACTERIST IC LENGTH IS ' ' GA= " » . . ' / / ) 1 0 65 FORMAT C 3 2 X , * * * * THE CHARACTERIST IC LENGTH IS " 3 L L " » * * ' / / ) 1 0 7 0 FORMAT C3 1 X , ' * * * THE CHARACTERIST IC LENGTH IS " RAOS A" * * * ' / / ) 1085 FORMAT C 35X , ' THE COE F F I C I E N T , " C ' # I 1 , " ' = ' , E l 4 . 8) DO I 800 I C N = I , 5 DO I 300 MN = I , 3 WRITEC I 0 8 , 1 0 1 0 ) I CN GO TO C I 2 0 0 , I 2 0 5 , I 2 2 5 , I 2 1 0 , 1 2 1 5 ) , ! CN 1 2 0 0 WR I T ECI 0 8 , 1 0 1 5 ) ; GO TO I 250 1205 WRI TECI 0 8 , 1 0 2 0 ) 1 6 0 TO I 250 1 2 1 0 WRITEC 1 0 8 , 1 0 2 5 ) IGO TO I 250 1215 WRI TECI 0 8 , 1 0 3 0 ) IGO TO I 250 1225 WR I TEC1 0 8 , 1 0 4C> 1 2 5 0 CONTINUE GO TO Cl 2 5 5 , 1 2 6 0 , 1 265 ) , MN 1255 W R I T EC 1 0 8 , 1 0 6 0 ) 1 6 0 TO I 300 1260 WRI TECI 0 8 , 1 0 6 5 ) IGO TO I 300 1265 WR I TEC I 0 8 , 1 0 7 0 ) 1 3 0 0 CONTINUE GO TO C I 3 5 0 , 1 4 0 0 , 1 6 0 0 , 1 4 5 0 , 1 5 0 0 ) , I CN C * * * »NU=C1 *CRA * * C2> 1 3 5 0 NV=2 ! N0B=K DO 1355 1 1 = 1 , K X C l , 1 1 ) = 1 . X C 2 , I I ) = ALOGCRACMN , ! I ) ) 1 355 x C 3 , i i ) = ) M ( z G V 2 5 X U» , K BBF F CALL CURFT C N V , N 0 9 , X , C ) DO I 360 L O X = I , 2 1 3 6 0 DDCLOX) =CCLOX) DDCD= EXPCDDC I ) ) DO I 36 5 L O W = I , 2 1365 WRI TEC I 0 8 , 1 0 8 5 ) LOW , OD CLOW) CALL E RRORCNV , N 0 9 , X , C , SS 0 ) GO TO I 8 00 C * * * * * N U = C 1 * C R AM * * C 2 ) 1 4 0 0 NV= 2 INOB=K DO I 40 5 11 = 1 , K X C 1 , I I ) = 1 . XC2 , I I ) =ALOGCRAM CMN, I I ) ) 1405 X C 3 , I D = AL OG CXNUSC M N , I D ) CALL CURFTCNV ,NOB , X , C ) DO I 410 LOX = I , 2 1410 DDCLOX)=CCLOX) DDCD= EXPCDDC I ) ) DO 1415 LOW=I , 2 1415 WRI TECI 0 8 , 1 0 8 5 ) L OW , DD CLOW) o o r > o 62 CALL ERROR ( NV , NOO , X , C , SSQ ) GO TO I 8 00 C * * * * * N U = C 1 * ( R A * » C 2 ) * ( ( G A P / R I ) * * C 3 ) B J : - 2 9 6 % Y2 ( 8 6 / OO 1455 I I = I z K X ( I z I I ) = I . X ( 2 z I I ) = AL O G ( R A ( M N z I I ) ) X ( 3 z I I ) = A L 0 G ( S A V G A P ( I I ) / S A V R I ( I I ) ) 1 4 55 X ( 4 z I l ) =ALOG( XNUS (MNz I I ) ) CALL CURFT ( NVzNOQzXzC) DO 1 460 LOX = T z 3 1 4 6 0 DD ( LOX ) =C ( LOX ) D D d )= E X P ( DD( I ) ) DO 1 465 L OW = I z 3 I 4 65 UR I TE ( I 0 8 z I 0 8 5 ) LOWzOD( LOU) CALL ERROR(NVzNOBzXzCzSSO) GO TO I 800 C * * * * * N U = C 1 * ( R A * * C 2 ) * ( ( G A P / R I ) * * C 3 ) » ( P R * * C 4 ) 1 5 0 0 NV=4 ; N0 8 =K DO I 50 5 I I = I z K X ( I z I I ) = I . X ( 2 z I I ) = A LOG ( RA (MNz 1 1 ) ) X ( S z I I ) = A L OG ( S A V G A P d D / S A V R K I D ) X ( 4 , I I ) = A LOGCPRd I ) ) 1 5 0 5 X ( S z I I ) = AL O G ( X N U S ( M N z I I ) ) CALL CURFT (NVzNOBzXzC ) DO 1510 LO X = I z 4 1 5 1 0 DO ( LOX ) =C ( LOX ) D D ( I ) = E X P ( D O d ) ) DO I 515 LOW= I z 4 I 5 15 W R I T E d 0 8 z 1 0 8 5 ) L 0 WzDD(LOW) CALL ERROR(NVzNOBzXzCzSSO) GO TO I 8 0 0 C * * * * * N U = C 1 * ( R A * * C 2 ) * ( P R * * C 3 ) 1 6 0 0 NV=SINOB=K DO 1605 I I = I z K X ( I z I I ) = T . X ( 2 z I I ) =ALOG( RA (MN , I I ) ) X ( 3 z I I ) = A L O G ( P R ( I I ) ) 1 605 X ( 4 , I I ) = A L OG( XNUS( MNz I I ) ) CALL CURFT ( NVzNOBzXzC) DO I 610 LOX = I z 3 1 6 1 0 DD ( LOX ) =C ( LOX ) D D d ) = EXP(DD ( I ) ) DO 1615 LOW = I z 3 1615 WR I T E dO S z 10 8 5 ) LOWz DD ( LOW) CALL ERROR (NVzNOBzXzCzSSO) 1 8 0 0 CONTINUE 1 0 0 0 END * * * * * C U R V E F I T SUBROUTINE USING THE METHOD OF LEAST SQUARES SUBROUTINE CURFT ( NVzNOBzXzC ) DIMENSION X ( 5 z 2 5 0 ) z C ( 4 ) z S ( 5 z 6 ) DOUBLE PREC I S ION SzD C * * * « * N V I S THE NUMBER OF INDEPENDENT VAR I ABLES C * * * * * N O B IS THE NUMBER OF OBSERVATIONS C * * * * * X ( I , K ) ARE EXPRESSIONS COMPRISED OF THE D IMENSIONLESS RESULTS 63 C * * » * * C ( I ) ARE THE COEFF I C I ENTS OF X ( I , K > C * * * « * Y = X ( N V * 1 , K ) , THE DEPENDENT VARIABLE M=NV+1 MP = M-H DO I I = I z M DO 1 J = I z M P 1 S ( I z J ) = O . DO DO 2 I = I z N OB DO 2 J = I z M DO 2 K = I zM 2 S ( J z K ) = S ( J zK) + X rt o 64 $ / , 9 X , 9 1 • * • ) 13 FORMAT( 9 X , 9 1 • « ' ) 00 2 1 * 1 * N OB YE = X (M, I ) YC=O . DO DO 20 I J = 1 , N V 2 0 YC = YC + C I I J > * X ( I J , I ) YC=EXP(YC) YE=EXP (YE ) E=YC-YE E P = I O O . * E / YE EPA=ABS( EP > I F ( EPA - EMX ) 6 , 6 , 7 7 EMX=EPA 6 DO 8 J = 1 , 5 ACD=J=S I F ( EPA - ACD ) 9 , 9 , 8 9 A N P ( J ) = A N P ( J ) = I . 8 CONTINUE TS=TS=E=E TE=TE=ABS ( EP ) 2 - W R I T E d O 8 , 3 ) 1 , Y E , Y C , E , EP 3 F 0 R M A T ( 9 X , ' * ' , 5 X , I 3 , 3 X , 4 ( 5 X , E 1 4 . 8 ) , 2 X , ' = ' ) U R I T E ( 1 0 8 , 1 3 ) A = NOB SSO= ( T S / A ) • * . 5 TE=TEZA WRITE ( 1 0 8 , 4 ) S S Q, TE 4 FORMAT( ' I * , 1 0 / , 3 0 X , 4 9 * * * , / , 3 0 X , ' * THE STANDARD D E V I A T I O N = ' , E 1 4 . 8 , 9 S X , • * ' , / , 3 0 X , ' * THE AVERAGE PER CENT D EV I A T I ON = ' , E l 4 . 8 , ' * ' ) WRITE ( 1 0 8 , 1 0 ) EMX 10 FORMAT( lhO 'D* THE MAXIMUM PER CENT DEVI ATION = ' , E l 4 . 8 , ' = ' , / , I O X , 90 $ ' * ' ) 14 FORMAT( I O X , 9 0 ' * ' ) DO 11 1 = 1 , 5 XNP = ANP ( I ) XNP= IOO. =XNPZA ACD=I =S 11 WRITE ( 1 0 8 , 1 2 ) XNPz ACO 12 FORMAT( I O X , ' * ' , E 1 4 . 8 , ' PER CENT OF THE DATA ARE WI TH IN ' , F 3 . 0 , S ' PER CENT OF THE CORRELATION EQUATION • ' ) WRI T E ( 1 0 8 , 14 ) RETURN END FUNCTION U ( T , J J ) GO TO ( I , 2 , 3 , 4 , 5 ) JJ C* = * * * ABSOLU TE V I SCOS I TY OF A I R I CO = I 3 4 . 3 7 5 C 1 = 6 . 0 1 3 3 8 3 4 C 2 = 1 . 8 4 3 2 2 9 9 C 3 = 1 . 3 3 4 7 0 5 0 U = T * * C 2 Z ( E X P ( C 1 ) * ( T = C O ) * * C 3 ) GO TO 50 C * * * * « ABSOLUTE V I SCOS I TY OF WATER 65 2 T P = T - 5 9 3 . 3 3 2 0 3 C l = . 0 0 7 1 6 9 5 1 4 9 C2 = . 0 1 1 751 302 C 3 = . 0 0 8 7 7 9 1 9 4 2 C 4 * . 3 1 6 5 4 7 0 4 V 1 S = C 1 * T P + C 2 » ( 1 . + C 3 * ( I P * * 2 ) ) * * . 5 + C 4 U = I . ZVI S GO TO 50 C * . * . . ABSOLUTE V I SCOS I T Y OF 20CS DOW 200 S I L I CONE 3 V = . O 3 8 7 5 * ( 4 . 6 * 1 0 * * 5 ) / ( T - 3 5 9 . 6 9 ) * * 1 . 9 1 2 C 1 = 5 2 . 7 5 4 6 8 4 C 2 = . 0 4 5 4 3 7 5 3 3 C 3 = 5 . 1 8 3 2 3 3 6 / ( 1 0 . * * 5 ) RHO = Cl + ( C 2 - C 3 * D * T U=RHO* . 9 4 9 * V GO TO 50 4 STOP C * * * * * V I S C O S I TY OF 96% GLYCERIN 5 C 1 = 1 0 3 . 5 1 1 9 9 C 2 = - . 5 4 0 4 0 7 3 6 C 3 = . 1 1 1 2 8 5 9 5 E - 0 2 C 4 = - . 1 0 5 2 6 0 6 4 E - 0 5 CS = . 38 2 O 2064 E - 0 9 U = C 1 + C 2 * T + C 3 * T * T + C 4 * T * * 3 + C 5 * T * * 4 U = I O . * * U 5 0 RETURN END C C C FUNCTION C P ( T y J J ) GO TO (1 , 2 , 3 , 4 , 5 ) JJ C * * * * * S P E C IF I C HEAT OF AIR 1 C0 = 2 . 2 3 6 7 7 5 / ( 1 0 . * * 5 ) C l = . 2 2 7 9 7 7 4 9 CP=C1+C0*T GO TO 10 C * * » ♦ *SPEC IF I C HEAT OF WATER 2 C I = I . 3 7 5 7 0 9 5 C2 = . O01 2 9 6 8 9 6 5 C3 = 1 . 1 1 1 05 3 3 / ( 1 0 . * * 6 ) C P = C 1 - ( C 2 - C 3 * T ) * T GO TO 10 C * * * * . SPEC I F I C HEAT OF 20CS DOW 200 S I L I CONE 3 T P = S . * ( T - 4 9 1 . 6 9 ) / 9 . C l = . 3 4 4 8 3 3 4 C2 = 7 . 7 4 9 9 / ( 1 0 . * * 5 ) C 3 = 4 . 1 6 7 / ( 1 0 . * * 8 ) CP = C1 + ( C2+C3 * T P ) * TP GO TO 10 4 STOP C * * * SPEC I F I C HEAT OF 100% GLYCERIN 5 I F ( T . L T . 5 9 9 . 6 9 ) GO TO 6 C l = . 2 2 4 2 0651 C 2 = . 6 6 6 6 6 7 0 1 E - 0 3 CP = C 1 + C 2 * T GO TO 10 6 C l = . 1 5 4 8 1 5 1 4 66 C 2 = . 7 8 5 7 1 4 0 2 E - 0 3 CP = C I + C 2 * T 10 RETURN END C C C FUNCTION C O N ( T , J J ) GO TO C l , 2 , 3 # 4 # 5 > JJ C * * * * * T H E RN 4 L CONDUCT I V I TY OF A IR 1 X P = . I C O = - S . 5 9 6 4 9 6 5 0 = 3 4 4 9 0 . 8 9 C 2 = 8 6 8 . 2 3 8 37 C 3 = 8 0 5 6 5 8 3 . 8 7 X = XP F = C 0 + C 1 * X + C 2 * X * X + C 3 * X * X * X - T F P = C U 2 . * C 2 * X + 3 . » X » X » C3 XP=X - FZFP I F ( A3S ( ( X P - X ) / X > - 0 . 0 0 0 1 ) 6 , 6 , 7 6 CON=XP GO TO 10 C * * * » * TH ERMA l CONDUCTIVI TY OF WATER 2 C l = . 2 3 7 0 5 4 1 7 C 2 = . 0 0 1 7 1 5 6 7 9 7 C3 = 1 . 1 5 6 3 7 7 0 / 0 0 . * * 6 ) CON= - C I + ( C 2 - C 3 * T ) * I GO TO 10 C * * » * * THERMAL CONDUCTIVI TY OF 20CS DOW 2 0 0 S I L I CONE 3 CON= . 0 0 0 3 4 / 0 . 0 0 4 1 3 4 GO TO 10 « STOP [ * * * * . THERMAL CONDUCTI V I TY OF 96% GLYCERIN 5 C l = . 1 4 3 4 0 1 2 7 C 2 = . 4 7 2 2 2 2 7 5 E - 0 4 CON = C I + C 2 * T 10 RETURN END C C C FUNCTION R HO ( T , J J ) GO TO ( 1 , 2 , 3 , 4 , 5 ) JJ C * * * * * D E N S I T Y OF AIR A l ATMOSPHERIC PRESS. L BM / FT 3 1 A= I 2 . 5 R H O = A * 1 4 4 . / ( 5 3 . 3 6 * T ) GO TO 10 C * * * « * D E N S I T Y OF WATER 2 C 1 = 5 2 . 7 5 4 6 8 4 C 2 = . 0 4 5 4 3 7 5 3 3 C3 = 5 . 1 8 3 2 3 3 6 / ( 1 0 . * * 5 ) RHO = CI + ( C2-C 3« T ) » T GO TO 10 c * * . . +DENS I TY OF 20CS DOW 200 S I L I CONE 3 C 1 = 5 2 . 7 5 4 684 C 2 = . 0 4 5 4 3 7 5 3 3 C 3 = 5 . 1 8 3 2 3 3 6 / ( 1 0 . * + 5 ) R H 0 = . 9 4 9 * ( C 1 + ( C 2 - C 3 + T ) * T ) 67 CO TO 1 0 4 STOP c * . * * . DENSITY OF 96X GLYCERIN 5 [ 1 = 8 9 . 7 8 9 9 3 2 C 2 = - . 2 2 2 1 1 63E - 0 1 RHO=C I +C2 * T I O RETURN END C C C FUNCTION 9ETA ( T , J J ) GO TO ( I , 2 , 3 , 4 , 5 ) JJ C * * * * * THERMA L EXPANSION COEFF' . OF A I R 1 BETA = I . Z T GO TO 10 C * * * * * TH ERMA L EXPANSION COE F F . OF WATER 2 TP = T / I 0 0 . I F ( T - 5 4 9 . 5 9 ) 6 , 6 , 7 6 [ 1 = 6 0 3 . 1 1 8 4 1 C 2 = - 3 5 3 . 0 3 8 8 2 [ 3 = 6 8 . 2 9 7 0 1 2 C 4 = - 4 . 3 6 1 1 460 B P = C 1 + ( C 2 + ( C 3 + C 4 * T P ) ‘ T P ) * T P GO TO 8 7 C 1 = - 1 2 8 . 4 4 9 2 0 C 2 = 6 8 . 8 2 7 9 2 7 C 3 = - 1 3 . 8 5 8 4 8 9 C4 = 1 . 2 6 0 8 5 8 5 CS = - . 0 4 2 4 9 5 2 3 6 B P = C 1 + C C 2 + ( C 3 + ( C 4 + C 5 * I P ) * T P ) * T P ) * T P 8 BETA = B P Z ( 1 0 . « * 4 ) GO TO 10 C * * * * * TH ERMA L EXPANSION COE F F . OF 20CS DOW 200 S I L I CONE 3 B E TA = . 0 0 1 0 7 Z 1 . 8 GO TO 10 4 STOP [ • ♦ • ♦ • T H E RM A L EXPANSION COEFF I C I ENT OF 96% GLYCERIN 5 [ 1 = 8 9 . 7 8 9 9 3 2 C 2 = . 2 2 2 1 1 6 3 E - 0 1 8 E T A = C 2 Z ( C 1 - C 2 * T ) 1 0 RETURN END APPENDIX II PARTIALLY REDUCED DATA The following is a listing of all the data taken in this investigation, in partially reduced form. The column headings are: IDG is the inner body indentifier 1 = 9 In-Line Cylinders 2 = 8 Staggered Cylinders 3 = 16 In-Line Cylinders 4 = 14 Staggered Cylinders IDBC is the inner body boundary condition identifier . 1 = Isothermal Conditions 2 = Constant Heat Flux Conditions IDF is the fluid identifier 1 = Air 2 = Distilled Water 3 = Dow Corning Dimethylpolysiloxane (20 centipoise at 25°C) 4 = 96% Aqueous Glycerine by Weight TAVGO is the average outer body temperature in Kelvin TAVGI is the average inner body temperature in Kelvin QCONV is the heat transferred by natural convection in Watts QLOSS is the heat loss due to conduction and radiation in Watts 69 IDG IDBC IDF TAVGO TAVGI QCONV QLOSS i I I 281.312 290.597 3.380 2.724 i 1 I 279,736 295.114 8.861 5.34 9 i I 1 281,038 305.262 15.408 9,160 i I I 282,824 323.066 29.178 16.061 i I I 280.227 330.035 38.638 2 0 . I 82 i I I 281.232 350.672 59.886 28.640 i I I 281.940 361.679 71.197 33.077 i 2 I 281.097 292.109 6.437 3.468 i 2 1 281.029 305.105 17.698 9.096 i 2 I 280.645 328.977 38.837 I 9.546 1 2 I 280.810 353.280 66.892 29.945 i I 2 277.198 282.367 156.293 . I 76 i I 2 .284,799 292.020 312.992 ,246 i I 2 281,367 290.176 382.805 . 300 i I 2 287,475 298.104 534.934 .361 i 1 2 285.105 297.766 624.889 .431 i I 2 292.258 307.748 980.162 .527 i 1 2 298.586 319.462 1379.387 .710 i 2 2 280.113 284.419 147.325 .146 i 2 2 282.950 291.431 377.156 .288 i 2 2 288.282 302.268 795.726 .476 I 2 2 297.605 315.172 I I 76.177 ,597 1 I 3 281.282 290.064 82.479 .299 I I 3 283.722 303.427 234.215 .670 i I 3 283,928 31 1.562 364.986 .940 I I 3 285.937 323.560 543.572 1.279 i I 3 288.070 333.519 696.462 1.545 1 I. 3 290.334 • 341.587 837.199 1.743 i I 3 294.032. 350.329 952.991 1.914 i 2 3 279.448 287.972 84.747 .290 i 2 3 280,945 301.7 35 259.969 .707 i 2 3 286.879 324.798 570.588 1.289 i 2 3 293.761 350.751 985,538 1.938 i I 4 279.410 291.091 67.498 .397 i I 4 278.935 299.061 I 57.223 .684 I I 4 282.171 311.951 304.023 1,013 i I 4 286.208 324.251 459.548 1.294 1 I 4 289.617 333.411 587.662 1.489 i I 4 292.704 341.199 709.315 1.649 1 I 4 295.733 353.163 966.895 1.953 i 2 4 283.265 295.760 65.160 .425 i 2 . 4 . 283.071 307.606 238.811 . 834 i 2 4 289.775 329.167 516.796 1.339 i 2 4 297.605 352.466 973.225 I . 865 ix ir xi rv ix ir xi ru fx jr u fx ji xi P u ix jr vr xj p jr xi ru ix ii vi ru iX Jt xi rv jr xi ru ix ir vi ix jr u rx ii xi rv ix ir M ru ix ir xj ru ru ix if N Ji xi iv 70 IDBC IDF TAVGO TAVGI QCONV . QLOSS I 1 279.495 284,991 1.825 1.957 I 1 277.147 290.886 8.506 7.480 I I 277.066 302.149 18.576 15.080 I I 277.717 317.060 32.918 24.635 I I 278.431 330.108 45.987 32.899 1 I 279.321 3 4 0 . I 04 54.789 38.999 1 I 280.041 347.624 61.183 43.556 2 I 276.674 283.932 2.900 3.138 2 I 277.670 295.278 10.313 10.072 2 1 278.494 320.339 27.468 26.311 2 I 279.503 346.549 59.573 43.195 I 2 279.567 284.989 164.819 .169 I 2 282.588 291.768 351.008 .286 I . 2 287.158 300.426 578.711 .414 I 2 292.559 309.597 796.122 . .532 I 2 298.335 316.011 1090.437 .551 I 2 293.403 309.528 914.615 .503 1 2 300.051 319.153 1185.275 .“596 2. 2 281.291 286.258 157.602 .155 2 2 285.715 295.116 389.211 . 293 2 2 292.300 305.846 701.227 .423 2 2 300.485 318.167 1087,938 . 5.52 I 3 279.177 287.667 83.764 .265 I 3 281.413 300.954 233.796 .610 I 3 283.403 307.582 304.556 .754 I 3 286.041 319.706 462.703 1.050 I 3 289.771 333.166 656.871 1.3 54 I 3 293.999 346.009 878.013 1.623 I 3 297.139 355.597 1040,040 . 1.824 2 3 279,190 286.9)4 71.524 .241 2 3 283.655 303.763 230.210 .627 2 3 288.427 324.21 I 497.598 1.116 2 3 295.692 346:279 856.343 1.578 I 4 276.998 288.601 68.260 . 362 I 4 278.163 298.404 I 52.249 .631 I 4 280.801 308.706 275.903 .871 I 4 282.588 319.4 76 434.133 1.151 I 4 285.907 331.750 628.645 1.430 1 4 290.429 345.388 883.639 1.715 I 4 296.187 358.373 I 1 35.827 1.940 2 4 281.502- 299.920 135.513 .575 2 4 287.887. 31 9.820 354.299 .996 2 4 294.353 338.682 658.876 1.383 71 IDG IDBC IDF TAVGO TAVGI . QCONV' 2 2 4 298.656 355.718 1046.355 3 I I 279.516 287.858 7.880 3 I I 278.872 297.846 I 5.2 54 3 I I 278.159 309.658 29.056 3 I I 278.244 323. 332 40.515 3 I I 280.544 339.745 61.229 3 I I 281.076 353.584 81.977 3 I I 280.020 358.336 92.106 3 2 I 278.766 286.469 7.724 3 2 I 278.027 303.078 19.038 3 2 I 279.736 325.819 42.025 3 2 I 282.037 365.308 110.805 3 I 2 282.176 286.696 219.992 3 I 2 286.821 293.952 412.581 3 I 2 287.233 295.839 527.228 3 I 2 291.463 301.054 623.250 3 I 2 288.693 301.543 789.595 3 2 2 281.666 288.319 348.427 3 2 2 287.762 297.319 632.830 3 2 2 294.959 309.241 I 116.902 3 I 3 280.299 287.101 91 .428 3 I 3 284.045 295.846 I 78.808 3 I 3 286.058 304.166 307.621 3 I 3 288.780 313.104 448.474 3 I 3 290.698 322.878 635.853 3 I 3 293.386 333.210 878.316 3 I 3 297.494 343.908 1132.037 3 2 3 281.401 292.234 I 66.665 3 2 3 283.693 302.517 342.034 3 2 3 287.636 316.026 591.105 3 2 3 293.900 336.968 1012.306 3 ■ I 4 280.632 290.320 103.230 3 I 4 282.660 300.898 220.113 3 I 4 284.565 309.339 369^934 3 I 4 287.995 320.063 566.140 3 I 4 289.788 325.956 682.421 3 I 4 293.855 337.131 958.382 3 I 4 298.041 346.089 1135.092 3 2 4 290.144 315.959 419.581 3 2 4 291.173 323.556 608.203 3 2 4 : 295.810 336.533 928.255 3 2 4 302.728 350.836 I 281.974 4 I I 279.601 289.270 5.872 QLOSS 1.7&0 3.6-16 14.714 27.787 41.971 56.704 70.593 76.657 2. 949 21.057 43.011 81.828 .303 .478 .577 .643 .862 .-4 4 6 .641 . 958 . .456 . 792 1.215 1.631 2.158 2.671 3.113 .727 1.263 1.904 2.889 .650 1.223 1.662 2.151 2.426 2.903 3.223 1.732 2.172 2.731 3.227 5.233 72 < IDG IDBC IDF TAVGO TAVGI 0C0NV QLOSS 4 I I 281.569 298.399 17.103 12.948 4 1 I 281.502 312.059 28.844 27. 739 4 I I 279.897 321.412 38.894 39 . 5 4 7 4 I I 281.389 333.768 55.824 50.713 4 I I 282.748 352.136 80.434 69.580 4 I I 284.309 363.680 99.634 80.336 4 2 I 281.358 288.532 6.256 2.543 4 2 I 281.147 300.846 17.663 16.039 4 2 1 231.999 321.410 37.261 .37.279 4 2 I 284.133 345.670 69.625 61.120 4 1 . 2 284.498 288.896 212.688 .272 4 1 2 285.673 291.605 315.462 . 366 4 I 2 286.971 294.165 412.796 . 444 4 1 2 288.930 297.953 541.556 .557 4 1 2 290.537 301.332 682.380 .664 4 I 2 293.284 307.133 843.062 .855 4 I 2 295.648 310.801 1022.397 .936 4 2 2 284.971 288.968 160.379 .“247 4 2 2 287.267 293.613 . 304.453 .392 4 2 2 289.983 298.993 499.159 .557 4 2 2 293.284 305.027 738.901 . 725 4 I 3 279.456 287.843 107.805 .518 4 I ■ 3 281.615 295.046 I 88.364 .830 4 1 3 283.374 301.716 279.468 1.133 4 I 3. - 284.866 308.982 401 .756 I .490 4 I 3 286.712 315.942 529.155 1.805 4 I 3 289.559 327.440 690.546 2.340 4 2 3 282.113 291.096 123.722 .555 4 2 3 284.640 303.233 309.055 1.148 4 2 3 288.448 317.477 562.422 1.793 4 2. 3 295.999 339.449 900.080 2.684 4 1 4 280.907 289.909 80.268 .556 4 I 4 283.126 296.962 145.109 .855 4 I 4 284.862 304.639 237.031 1.222 4 1 4 287.042 313.183 359.723 1.615 4 1 4 289.182 320.607 500.630 ■ 1.941 4 1 4 291.127 329.253 671 .090 2.355 4 I 4 293.633 337.900 864.529 2.734 4 2 4 281.316 292.468 104.340 .689 4 2 4 283.143 303.937 262.373 I .284 4 2 4 285.844,/ 317.365 500.549 1.947 4 2 4 291.779 333.840 817.442 2.598 BIBLIOGRAPHY BIBLIOGRAPHY i [1] Morgan, Vincent T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, Advances in Heat Transfer. 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O., and Crupper7 G., Natural Convection Heat Transfer Between Cylindrical Tube Bundles and a Cubical Enclosure, Transactions of ASME. Journal of Heat Transfer. Vol. 103, pp. 103-107, 1981. MONTANA STATE UNIVERSITY LIBRARIES stks N 3 7 8 . W 3 7 8 5 @ T h e s e s R L Natural convection heat transfer between 3 1762 00113127 3 -WN Li&