coo-o.-. C. ~‘*oq-'*o«~99~ _--o-- .‘o - HYDRODYNAMIC AND HEAT TRANSFER CHARACTERISTICS j j T ~ OF A SOLAR HEATED AlR DUCT * Thesis for the Degree 0f Ph. D ‘ I p . - MiCHfGAN STATE umveasny _. MUHAMMAD ANWER MALIK 1967 ' ' LIBRAR Y ) Michigm State Unimfiry ‘ . [HESIS n;"‘" This is to certify that the thesis entitled fl/DQ 0 p/va/c, & flé/I 7 7£AM§Z££ C mar/2,4 c 752/5 7/05 0/: A M4 we fig; 75, /4 be flac 7 ~ presented by [WU/r'AM/zmto A‘wiE/Z AiflL/K. has been accepted towards fulfillment of the requirements for Mdegree infiV?‘/‘ . a W Major profesg Dam/IM/VW 0-169 ABSTRACT HYDRODYNAMIC AND HEAT TRANSFER CHARACTERISTICS OF A SOLAR HEATED AIR DUCT by Muhammad Anwer Malik Heated air has many applications in agriculture. The capturing of solar energy is done by means of a col- lector. Many types have been built and tested. For agri- cultural purposes, however, as low as 10 degrees rise of temperature in the outside air is sufficient to, say, speed up the drying of farm crops to moisture contents safe for storage. For low temperature rise, a "no-glass" collector can be used. This consists primarily of a series of rectangular air ducts lying side by side with sheet metal‘covering on the top surface which is exposed to solar radiation. The "no—glass" collector is easily incorporated into the design of a building, as it essentially follows the pattern of a conventional roof. The only addition needed is some type of material to form the bottom of the duct. Hydrodynamic and heat transfer characteristics of the air ducts mentioned above are studied in the present thesis. The problem is essentially that of a fully developed turbulent flow in a rough rectangular duct heated uniformly on the top and having the other three sides essentially adiabatic. The analyses available for the unsymmetrically heated smooth ducts agree only qualitatively. Most of them Muhammad Anwer Malik entail extensive numerical calculations and are of limited use. In the present thesis two analytical solutions are developed for fully-developed turbulent flow in a smooth unsymmetrically heated rectangular duct. Two experimental ducts were built. The bottom for both of them was made of plywood and was insulated. The vertical walls were made of pieces of lumber. The top of one of the ducts was covered by a flat galvanized sheet. The other duct was covered by corrugated galvanized sheet. The second duct was used to estimate the increase in pres— sure drop and heat transfer when a flat top is substituted by a corrugated one. The ducts were used to determine intake losses, intake length and friction coefficients. For both the ducts the friction coefficient was found to depend on Reynolds number and relative roughness. The corrugated roof resulted in an increase in friction coefficient of about #0 per cent. The analytical results for the convective heat transfer coefficient for smooth duct successfully carried over to the rough duct when multiplied by a factor /_f7f6. The increase in the heat transfer coefficient resulting from the use of corrugations was about 20 per cent. Madsen's rule (54) was found to apply to the rough ducts. ’ / . m7 (J 97W HYDRODYNAMIC AND HEAT TRANSFER CHARACTERISTICS 'OF A SOLAR HEATED AIR DUCT By t Muhammad Anwery alik A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1967 ACKNOWLEDGMENTS The author wishes to express his deepest gratitude to Dr. Frederick H. Buelow for his guidance, supervision and inspiration. Thanks are due to my advisor Dr. James S. Boyd, for his supervision, inspiration and graciousness. The author also wishes to thank the other members of the guidance committee: Dr. James V. Beck, Dr. Merle E. Esmay, and Dr. E. A. Nordhaus. The author is grateful to Dr. Ronald C. Hamelink for his willingness to be on the Final Oral Exam Committee. Dr. Carl w. Hall, Chairman of the Department, is thanked for giving financial grant for the project. The assistance offered by Mr. Jim Cawood in the con- struction of test apparatus is gratefully acknowledged. The author is indebted to the Campus Coordinator's Office at Washington State University for coordinating his overall program of study in this country. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . ii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . vii NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . ix Chapter I. INTRODUCTION 1 Flat Plate Solar Energy Collectors 3 Nature of the Problem . . . 6 II. LITERATURE REVIEW . . . . . 9 Hydrodynamic Characteristics of Circular and Noncircular Ducts with a Turbulent Flow 9 Critical Reynolds Number (Beer) . . . . . . 9 Intake Region 9 Friction Coefficient 13 Intake Losses 1“ Rough Pipes and Ducts . . . 16 Heat Transfer Characteristics ér° Circular and Noncircular Ducts with a Turbulent Flow . . . . 20 Fully DeveIOped Turbulent Heat Transfer in. Smooth Ducts . . . . . . . . . . . . 20 Use of Hydraulic Diameter for Noncircular Ducts . . . . . . . 30 Fully Developed Turbulent Flow in Unsym- metrically Heated Ducts . . . . . . . 36 Entrance Region Studies for Symmetrically and Unsymmetrically Heated Circular and Noncircular Smooth Ducts with Turbulent Flow . . AA Fully Developed Turbulent Heat Transfer in Rough Tubes . . . . . . . 52 Theoretical Analysis for Heat Transfer in Rough Tubes . . . . . . . . . . . . . . . 53 iii Chapter III. THEORETICAL ANALYSIS Unsymmetrically Heated Parallel Plate Channels. . . . . . . . . . . . . . Assumptions . . . . . . . .‘. . . . . Temperature Drop in Laminar Sublayer Temperature Drop in the Buffer Layer Temperature Drop in the Turbulent Core Temperature Drop (tw - tB) Symmetrically Heated Parallel Plate Channel . . . . . . . . . . . Temperature Drop Through Laminar Sublayer . . . . . . . . . . . . Temperature Drop Through the Buffer Layer . . . . . . . . . . . . . . . . Temperature Drop Through the Turbulent Core . . . . . . . . . . . . . Temperature Drop (tW - tB) Simplified Analysis for the Unsymmetrical Case . . . . . . . . . . . . . Laminar Sublayer. Turbulent Core . Simplified Analysis for Symmetrically Heated Parallel Plate Channel . Laminar Sublayer . . . . . . Turbulent Layer IV. EXPERIMENTAL INVESTIGATION Construction of Equipment Instrumentation . Measurement of Incident Solar Energy Measurement of Temperature of Upper and Lower Sides of the Ducts. . . . . . Bulk Temperature of Air . . . . . . Measurement of Air Leaving the Ducts . . Measurement of Flow . . . . . Measurement of Pressure DrOp of Air along the Duct . . . . . . Experimental Procedure Data for Hydrodynamic Characteristics Data for Heat Transfer Characteristics of the Ducts . . . . . . . . iv Chapter Page V. ANALYSIS, RESULTS AND DISCUSSION . . . . . . . 91 Analysis of Data . . . . . . . . . 91 Hydrodynamic Characteristics . . . . . . . 91 Heat Transfer Characteristics . . . . . . . 91 Analysis of Sling Psychrometer Data . . . . 9A Results and Discussion . . 102 Hydrodynamic Characteristics of the Ducts . 102 Friction Coefficient for Fully Developed Flow . . . . . . . . . . . . 106 Entrance and Intake Region Losses . . . . . 111 Relative Roughness . . . . . . . 115 Heat Transfer Characteristics of the. Ducts A and B . . . . . . . . . . . . . . 120 VI. CONCLUSIONS . . . . . . . . . . . . . . . . . . 131 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 133 Table \OGDNONAU'I 10. ll. l2. 13. 1A. 15. 16. 17. 18. 19. 20. LIST OF TABLES Page Values of Recrit for Noncircular Ducts as Reported by Various Investigators . . . . . . 7 Tables of the Coefficient fS . . . . . . . . . . 98 Air Flow (CFM) and Corresponding Reynolds Numbers . . . . . . . . . . . . . . . . . . . 102 Intake Length for Duct A . . . . . . . . . . . . 105 Intake Length for Duct B . . . . . . . . . . . . 105 Coefficient of Friction for Duct A . . . . . . . 109 Coefficient of Friction f for Duct B . . . . . . 109 Comparison of Values of f for Ducts A and B . . 113 Entrance Loss Factor (Kl + K2) . . . . . . . . . 113 Entrance Loss Factor (K3 + KM) . . . . . . . . . 116 Factor E for Duct A . . . . . . . . . . . . . . 116 Entrance Loss Factor (Kl + K2) for Duct B . . . 117 Entrance Loss Factor (K3 + K“) for Duct B . . . 117 * Overall Entrance Loss Factor K for Duct B . . . 118 Relative Roughness of Duct A . . . . . . . . . . 118 Relative Roughness of Duct B . . . . . . . . . . 119 Nusselt Numbers for Duct A . . . . . . . . . . . 119 Factor /?£-for Duct A . . . . . . . . . . . . . 122 o Nusselt Number for Duct B . . . . . . . . . . . 122 Factor /%L for Duct B . . . . . . . . . . . . . 124 v 0 vi Figures 1. 10. ll. 12. 13. 1A. 15. 16. LIST OF FIGURES Page Fully Developed Turbulent Flow in an Unsym- metrically Heated Parallel Plate Channel . . . 56 Fully Developed Turbulent Flow in Symmetrically Heated Parallel Plate Channel . . 65 Fully Developed Turbulent Flow in an Unsym- metrically Heated Parallel Plate Channel . . . 68 First Stage in the Construction of the Apparatus . . . . . . . . . . . 76 Second Stage in the Construction of the Apparatus . . . . . . . . . . . . . . . 76 The Apparatus as It Looked after the Completion of the Third Stage of Construction . . . . . 79 Apparatus for Conducting Experiments (trailer is not shown in the Figure) 80 Location of Thermocouples 82 Location of Thermocouples for Measuring Bulk Temperature of Air . . . . . . . . . . . . 85 Location of Pressure Tubes 87 Pressure Manifold 88 Pressure Drop vs. Length of Duct A 92 Pressure Drop vs. Length of Duct B 93 Temperatures twl’ tw2’ and tB vs. Length of Duct A . . . . . . . . . . . 95 Temperatures twl’ tw2’ and tB vs. Length of Duct B . . . . . . .- 96 The Intake Length of Duct A 10A vii \‘ Figure Page 17. The Intake Length of Duct B . . . . . . . . . . 107 18. Coefficient of Friction for Duct A . . . . . . . 110 19. Coefficient of Friction of Duct B . . . . . . . 112 20. Comparison Between Theoretically and Experi— mentally Determined Values of Nusselt Numbers for Duct A . . . . . . . . . . . . . . 123 21. Comparison Between Theoretical and Experi- mental Values of Nusselt Numbers for Duct B . 125 22. Nusselt Numbers for Ducts A and B . . . . . . . 126 23. Verification of "Modified" Madsen's Rule ofr Duct A . . . . . . . . . . . . . . . . . . . . 129 2A. Verification of "Modified" Madsen's Rule for Duct B . . . . . . . . . . . . . . . . . . . . 130 viii a Shorter side of the rectangular duct. a+ : a J?;;6 v A Area of cross-section of the duct. b Longer side of the rectangular duct. C Perimeter of the duct. Cp Specific heat at constant pressure. d Diameter of circular tube. dh Hydraulic diameter of a duct = %é d* Diameter used in rough tubes = //§% f Coefficient of friction. fO Coefficient of friction of hydraulically smooth duct. g Acceleration due to gravity. G Mass velocity. h Convection heat transfer. hgeneral Generalized Madsen's heat transfer coefficient. h* Generalized Madsen's heat transfer coefficient general modified for rough ducts. k Thermal conductivity of fluid. K Absolute roughness. K1 ‘1 K2 Factors for computing entrance losses. K3.— KL: NOMENCLATURE ix K* K/d Le Nu Nuco Re Re cr Re Entrance loss factor defined on page 1A. Relative roughness Length of duct. Intake length. Nusselt Number = hdh/k. Nusselt Number for fully developed flow. Average value of Nusselt Number over a distance. Pressure of air at distance x. Pressure of stationary air ahead of the duct. Pressure at inlet of duct. Pressure at outlet of the duct. Prandtl Number = Cpu/k Turbulent Prandtl Number = em/eq Heat flux at distance y. Intensity of incident solar energy. Heat flux at wall. Volume of air flowing through the duct. Radial distance from the center of the tube. Radius of the tube. Radius of tube. Reynolds Number = umdh/v Critical Reynolds Number. = U*d*/v Hydraulic radius = A/C. Temperature at distance y from the wall. t - t w qw/on/ Tw7o Temperature at the end of buffer layer. Dimensionless Temperature = Bulk temperature of fluid. Temperature at central axis of the tube. Temperature of air entering the duct. Temperature of air leaving the duct. Temperature of the heated wall. Temperature of the wall Opposite to the heated wall. Dimensionless temperature of fluid outside thermal tw_t6 qw/on/Tw7o Velocity at distance y from the wall. boundary layer = Dimensionless velocity = u/VTw7p. Velocity at the outer edge of buffer layer. = ub/V TAN/p, Velocity at center. _ ‘ ,/ - uC/VTw/p. Mean velocity. = um/VTW/p. Freestream velocity outside of the boundary layer either in a flat plate or intake region of a duct. Friction velocity = ume/8. Velocity in fluctuation of dimension = A. xi V Volume of the duct. Vl Volume of air entering the duct. V2 Volume of air as measured by the venturi. x Axial distance measured along the duct. x+ = X/dh’ y Distance from the heated wall of the duct. y+ = y/?;76 xv. yl Thickness of the laminar sublayer. yb Distance between the heated wall of the duct and outer edge of buffer layer. y0 = a/2. + YOVIE75 yo __—5_—— Greek Letters d Thermal diffusivity. 6 Thickness of hydrodynamic boundary layer. 5" = (WT/T) /v. 6t Thickness of thermal boundary layer. u Dynamic viscosity. 0 Kinematic Viscosity. K Energy dissipated per unit mass and time. A0 Local degree of turbulence. 0 Density of fluid. T Shear stress at distance y. xii Shear stress at the wall. Turbulent diffusivity for momentum. Turbulent diffusivity for heat. Used interchangeably for Em or eq xiii when 6 m CHAPTER I INTRODUCTION Although today's industrialized world acknowledges its dependence on earth for oil, coal and other energy sources, it remains oblivious to the fact that sun is actually the source of all energy. In burning wood, we release the solar energy that was stored in the living tree by chemical action. The fossil fuels-—coal and oil, as well as natural gas--are in effect stored solar energy; they constitute the remains of plant material that flourished millions of years ago in sunlight. The radiation emitted by the sun into the space amounts to 378 x 1018 kilowatts-—an amazing figure indeed. Even the minute portion that is received by the earth-- 170 trillion kilowatts--is astronomically high. Kingston Plant, in the Tennessee Valley Authority, is the largest steam power plant in the world with a capacity of 1.5 million kw. One hundred seventy million Kingston power plants would have to be built to match the amount of solar energy received by the earth. To get an idea of the enormity of solar energy, the following example is cited. If an area of approximately 6,500 square miles of desert land in New Mexico were covered with solar mirrors, enough power could be obtained, assuming only 10 per cent efficiency of conversion, to meet the entire power require- ment of the United States. How long is the stored solar energy going to be available in the form of coal, oil and natural gas? This question has received some appalling answers. Dr. R. G. Jordan and Dr. James L. Threlkeld, both professors at the University of Minnesota, have thoroughly studied the United States fuel reserves and drawn the following conclusions: More than 85 per cent of all fossil fuels ever mined has been consumed since 1900. By "fossil fuels" the authors mean deposits of coal and oil which required about 500 million years for their formation. This high consumption rate is rendered more alarming by the fact that coal and oil cannot be replenished in another 500 million years. Coal supplies of the world are estimated to last not much beyond the twentieth century. So far as the situation of the oil-producing countries is concerned, an even darker picture is painted. The Middle East, the leading supplier of the commodity, is consuming its oil reserves at the rate of 6.6 billion barrels a year, a rate estimated to be tripled by 1975; the 174 billion barrels of oil covered by the soil of the Middle East will last only eight years. Other oil—producing countries have equally alarming figures to quote. Ironically enough as the fossil fuels are rapidly diminishing, the demand for power is increasing in almost geometric proportions. In the face of the present situation, nuclear energy is soon expected to have an impact on industry which will be in every sense as dramatic as those made by the steam engine or electric motor. Nevertheless, only the highly industrialized countries can meet the high cost of invest— ment. Although the underdeveloped and emergent nations are deprived of nuclear energy because of its deterring cost, they are very rich in solar energy. Exploitation of the solar energy, at an ever-increasing scale, has to be under- taken for a prosperous and happier world of the near future. Solar energy costs nothing to generate, transmit or distribute. Inexpensive equipment has been used for several applications of solar power. The heating and dis- tilling of water, desalting the saline and brackish water, operating air conditioning plants and heating homes are a few of the applications to name. Flat Plate Solar Energy Collectors Solar energy collectors, the apparatus used for con— necting solar power, are broadly classified into two cate- gories: the concentrating and the flat type. The concen- trating collectors are used to increase the flux density of incoming solar energy and are used where higher temperatures are desired, as in metallurgical industries. Reflectors and mirrors are frequently employed in this type of collectors to magnify the solar flux density. The flat type collectors are relatively inexpensive and used where moderate or low rise in temperature is needed. Flat type collectors may be subdivided into liquid and gas heaters. The solar water heaters in general consist of copper tubing, with or without fins, laid on a bed of insulation and covered by one or more glass cover plates. The pipes and fins are painted black for maximum absorption of energy. The air heaters are categorized as: the metal plate, the black gauze, and the glass shingle. The metal plate type air heaters, by far the most common, utilize flat metal plate blackened on the side exposed to the sun. If the air passes at the back of the metal plate, an insulation is generally used parallel to the metal plate. The air passes between the metal plate and the insulation. Very little research has been done on the use of solar energy collectors on farms. With the availability of large space for placing collectors, a farm affords attrac— tive possibilities for harnessing solar energy. Moreover the energy requirement for the farm as compared with the energy falling on the available space is low. The temper- ature rise required for drying products and heating farm buildings is low. As little as 10 degrees rise in the temperature of the outside air is sufficient to accelerate drying of farm crops to moisture contents which are safe for storage (1). The solar collector that will heat air for drying crops in summer and fall can be used for heating farm buildings in winter and spring. For small temperature rise, the collector can be of a simple design. The expenditure incurred would as a result be low. Research conducted at Michigan State University has shown that it is possible to heat air by passing it under the galvanized steel sheet roofing (l). The solar energy collector consists primarily of a series of rectangular ducts laid side by side with the sheet metal covering on the top. The sheet metal is exposed to the sun. This collector can be easily incorporated into the construction of a building; as it is essentially the same as a conventional roof with rafters and metal roofing, or a trussed roof with metal roofing and nailing girts. The only additional feature needed is to use some material to constitute the bottom of the duct. Plywood coupled with insulation could be used effectively. The purpose of the present investigation is to study the heat transfer and hydrodynamic characteristics of the rectangular air duct formed by two rafters, constituting the vertical sides; plywood (plus insulation) forming the lower side and the galvanized steel sheet constituting the top. The two rafters constitute the shorter sides of the duct. Nature of the Problem It is a standard practice in the construction of roofs to place successive rafters two feet apart, measured from center to center. The longer side of the duct is, there- fore fixed. The depth between the plywood and galvanized steel is the variable that can be employed to control heat transfer and pressure drop in the duct. Lengths of 50 feet1 or more for the ducts mentioned above will not be uncommon for use on the farm. The flow as a result will be fully developed (both hydrodynamically and thermally) for most part of the duct. Even an air flow as low as 150CFM (which, for a 50 foot long roof, means a flow of 1%CFM per square foot of roof area) will result in a Reynolds number of over 7500 in a 6—inch deep duct. A survey of literature on the critical Reynolds number is presented in Table 1. A typical Reynolds number (Re) of 2300 is generally accepted for tubes and ducts (8). Between Reynolds numbers of 2300 to 7000 the flow is in the transition regime as found in several experimental studies.2 Since air flow in the present investigation is expected to be in excess of lkCFM per square foot of roof area, it may be safely assumed that the flow in the ducts is turbulent. The hydrodynamic and thermal boundary 1For a 2" nominal depth it equals a length of 162.5 hyd. diameter. 2See, for instance, ref. 6. TABLE l.—-Values of Recrit for Noncircular Ducts as Reported by Various Investigators. Investigator Geometry ReGrit Nikuradse, J. (2) Rectangle g = 3.50 2800 Equil. triangle 2800 Right isoceles triangle 2800 Acute triangle 2000—2360 Trapezoid 2000-2360 Washington and b Marks (3) Rectangles, a = 40 and 20 2A00 Schiller, L. (A) Rectangle — = 3.52 1600 Square 2100 Davis and White (5) Rectangles, g = 40 2800 . . g = 5000 Ecxert and Irv1ne (6) Rectangle a 3 6000—-Smooth entrance Isoceles triangle, 23O 1800--Abrupt entrance Cornish, R. J. (7) Rectangles g = 2.92 2800 layers start simultaneously at the entrance of the duct. The duct has an abrupt entrance. The thermal boundary layer develops only on one side as the duct is heated only at the top. The galvanized steel sheet is subjected to uniform heat flux. The bottom of the duct is adiabatic. This gives rise to axially uniform but unsymmetrical boundary conditions. Since plywood, galvanized steel and lumber are used in the construction, the duct is expected to lie in the region of rough tubes. The problem may be summarized as follows: to study fully developed turbulent flow in an unsymmetrically heated rough rectangular air duct, heated from above. Another duct was also built where flat galvanized steel sheet was replaced by a corrugated sheet. The cor- rugations run normal to the direction of flow. This duct will serve to give an estimate of increase in heat transfer and pressure dr0p when corrugated top is used. CHAPTER II LITERATURE REVIEW Hydrodynamic Characteristics of Circular and Noncircular Ducts with a Turbulent Flow Critical Reynolds Number (Recr) A survey of literature on critical Reynolds number is summarized in Table 1. With the conditions that are found in industrial applications flow is usually turbulent (9) when the Reynolds number exceeds the value 3000. By eliminating all disturb- ances, a critical Reynolds number of 500,000 was attained (9)- Intake Region Near the entrance of a duct, the presence of the duct wall makes itself felt only a small distance into the fluid. The influence of the wall is confined to a thin viscous layer. This thin Viscous layer, called boundary layer, grows in the downstream direction until it has finally penetrated through the whole flow. The cross section where this occurs marks the end of the "intake region" and the beginning of the "fully-developed flow region." Actually the flow approaches the developed velocity field asymp- totically and there is always a certain arbitrariness in the definition of "intake length." In this thesis the intake 10 length will be defined as that length at which the pressure drop per unit length has reached a constant value within the accuracy of the measurements. Deissler (10) analytically studied the intake region characteristics for smooth tubes and parallel plate channels. Integral heat transfer and momentum equations were used for calculating thermal and hydrodynamic boundary layers. The flow was assumed to be turbulent at all points along the passage. Deissler has given the following equations relating hydrodynamic boundary layer thickness (0) to distance along the passage. 1. For the tube. + + u r + + + + + + g: f c O 113(57)“]i(2‘§T)uc ‘ —'1T'§ f6 u+(r:‘y+>dy+ C1more) Re r r r 2(r ) 0 —2— O O O O + + II] __j:_§ d f6(u: _ u+) u+(r: _ y+>dy+ 0 2(r ) 0 l\) For parallel plate channel + +a ++ x _ 1 UC 2 6 uc 1 5+ + + + + __ _ f f u dy d(r u ) + d H + 2 + 2 o c h Re (r ) (r ) O 7T O O 1 [j 1 + 5+ + + + + F f + 2 d rO f (uC - u) u dy O (r ) O ll [1 where indicates that the variable of integration should be used as the upper limit. An intake length of less than 10 hyrdaulic diameters (d = A x cross sectional area) was noted. h perimeter Deissler has followed Prandtl's assumption: in the presence of a laminar boundary layer near the entrance, the turbulent portion of the boundary layer behaves as though the boundary layer were turbulent all the way from the entrance (11). Deissler's calculations which assume the flow to be turbulent at all points may be applied to the turbulent portion of the boundary layer even when a laminar boundary layer exists near the entrance. Deissler (12) has investigated the effect of various factors on friction in the entrance region of smooth passages. The influence of Reynolds number, Prandtl number, passage shape and variable fluid properties is predicted. The results based on integral momentum equations indicate that approximately fully developed flow is generally attained in an intake length of less than 10 hydraulic diameters. Deissler's (l2) analysis agrees with the analysis of Pascucci (13), who obtained for turbulent flow in a tube L e _ E- - 3.80 loglORe - 2.1“ 12 Both of these analyses are somewhat higher than the cor- rected predition of Latzko (1A). 2§= 0.623 (Re)an It should be pointed out, however, that all these analyses are considerably lower than the experimental values ob- tained by Schiller and Kirsten (15) who found lengths of 50 diameters and greater to be necessary for the attain- ment of the fully developed flow. It should be mentioned that the long intake lengths found in these experiments refer to the length required for the velocity profile to develop rather than the pressure gradient or shear stress to reach their fully developed values. They apparently did not measure pressure drops. The same result was obtained in Hartnett's (16) experiments where at the tube center the velocity profile was still developing slightly at values of x/d greater than 75 although pressure gradients had long before reached their fully developed values. Hartnett et_al. (17) found eXperimentally that rec- tangular ducts with abrupt entrance have an intake length 2ab m for a of less than 20 hydraulic diameters (dh = rectangular duct). 13 Friction Coefficient Pressure drop for fully developed flow in tubes is given by Fanning's equation 2 u dP _ f‘ _m 2 For fully developed turbulent flow in tubes coefficient of friction f is given by Blasius law. Blasius law holds good for smooth ducts of noncircular cross section when Reynolds number is based on hydraulic diameter (18). For isothermal flow in ducts, the integrated flow equation based on perfect gas laws is as follows (19). 2 2 2 4.6Id P. T h 1 where R is gas constant = 15A6 ft lb per (lb. mole) (R0) f and M is molecular weight lbm per (lb. mole). In ducts of appreciable length the last term in the parentheses can be neglected unless the pressure drop is significant. The resulting equation is 2 _ r L G2 _ r L — um P1 - Po _ — _ If— p 2% 2 g p dh h where E is the density of fluid at the average pressure pi + p0 In uniform horizontal ducts, an approximate result is as follows (19) 2 v 2 P. - P = 2.30 11— log —9 + f L G 1 o — 10 v. — s o 1 (dh p)(2g) Intake Losses For design purposes, it is essential to determine the overall pressure drop which fluid would eXperience in flowing through a duct length L. Entrance and intake losses con- stitute an important part of the total pressure drop. This loss is particularly important in short ducts. The following method has been adopted to determine the overall pressure drop. Frictional pressure is calculated as if established flow conditions would exist over the full length L and a second term is added which accounts for the increase in pressure drop due to inlet and intake conditions. 2 2 u u _ L m * m Po'P‘fagp‘é-+Kp‘2‘ 15 where PO is the pressure of stationary fluid ahead of the duct and P is the pressure at distance L from the inlet. The four components of constant K in the above equation account for the following losses: 1. K1: Loss due to acceleration of fluid from rest to the constant velocity um just before it reaches the inlet. 2. K2: Loss due to abrupt contraction at inlet. 3. K3: Loss due to the momentum flux increase in the intake region which is connected with the trans- formation of the velocity fluid. 4. K4: Loss due to an increase in friction in the inlet region. For laminar flow through a circular tube Kl + K2 = 2. Calculations by Schiller (20) show that Kl + K2 + K3 = 2.16. Goldstein (21) calculated the parameter (Kl + K 2 + K3) to be 2.41. * For turbulent flow in a round tube the value of K is found to be 1.058. l6 Rough Pipes and Ducts Most pipes used in engineering practice cannot be regarded as being hydraulically smooth. The resistance to flow offered by rough walls is larger than that given by equations for smooth pipes. Two types of roughness in relation to the resistance formula for rough pipes have been noted. The first kind of roughness causes a resistance which is proportional to the square of the velocity, meaning therby that the coefficient of friction is independent of the Reynolds number. This type of roughness corresponds to relatively coarse and tightly spaced roughness elements, e.g., rough cast iron or cement. In such cases the nature of the roughness can be expressed with the aid of a single roughness parameter K/Rh, the so-called relative roughness, where K is the height of a protrusion and Rh denotes the radius or the hydraulic radius of the cross section. The resistance coefficient is, therefore, a function of relative roughness in the first type of roughness. For geometrically similar roughness Fromm (22) and W. Fritsch (23) found that coefficient of friction is proportional to (K/Rh)0'31u. The second type of roughness occursvdmnwathe protru- sions are more gentle or when a small number of them is distributed over a relatively large area, such as those in wooden ducts and commercial steel pipes. In such cases the l7 coefficient of friction depends both on the Reynolds number and the relative roughness. Extensive and systematic measurements have been carried out on rough pipes by Nikuradse (24) who used circu- lar pipes covered on the inside very tightly with sand of a definite grain size glued on to the wall. By varying the size of the pipes and the grain size, he was able to vary the relative roughness from about 1/1500 to 1/15. Based on Nikuradse's findings, three different regions are to be considered in the case of rough pipes. l. hydraulically smooth regime KVTw7p K O < —————— < 5 f = f(—— , Re) - v - Rh The size of the roughness is so small that all pro- trusions are contained within the laminar sublayer. 2. transition regime K/T /p 5 < ___E__ < 70 f = f(li), Re) — v - Rh Protrusions extend partly outside of the laminar sub- layer and an additional resistance as compared with the smooth pipe, is mainly due to the form drag experienced by the protrusions in the boundary layer. l8 3. Completely rough regime K'Tw:p K —————— > 70 f = f(——) v Rh Here all protrusions reach outside the laminar sub— layer and the largest part of the resistance to flow is due to the form drag which acts on them. Von Karman derived the following formula for com- pletely rough pipes from the similarity laws. Rh -2 f = 2 log (T) + 1.68 Closer agreement is obtained with experiments when the constant 1.68 is replaced with 1.74. The resistance formula for the rough pipes becomes R -2 f = [ 2 log 7? + 1.74 ] An equation which correlates the whole transition regime: was introduced by Colebrook (25). =1.74—21og[§— PAIR h w +4 9° km l______i i9 Moody (26) has reported an extensive experimental work on commercially rough pipes. His graphs of f against Re for different values of relative roughness is in essence identical with Nikuradse's results. Colebrook's function, when solved for relative roughness, gives .15. d = log'1 (0.57 - —1——> - 9'3 h 2/f Re/f Moody's and Nikuradse's graphs are applicable to ducts of noncircular cross sections. Pipe diameter of the duct should be replaced by the hydraulic diameter of the duct. Several analytical and presumptive theories have been advanced about the nature of turbulent flow in rough pipes and the effect of roughness on friction factor. Piggott (27) has suggested that the presence of roughness protru- sions on the pipe wall cause the formation of a stagnant fluid film at the boundary region of the fluid stream. Since the thickness of this stagnant film is proportional to the mean length of the roughness protrusions, it tends to decrease the effective diameter of the fluid stream in the same proportion. The resulting reduction in effective diameter causes a significant increase in the resistance to flow. 20 Heat Transfer Characteristics of Circular and Noncircular Ducts with a Turbulent Flow Fully Developed Turbulent Heat Transfer in Smooth Ducts Reynolds' analogy is eXpressed as qw and assuming that 9 = ?— = const. we obtain w q u —_l’_ .2 = t0 - tw — T c (Pr 1) W p Replacing uc by um and t0 by tB (though not quite correct) and multiplying both sides by the tube wall surface area, there is obtained, for the total heat flow to the fluid c = .2 _ Qw R um (tB tw) 2 where R is the resistance (R = Tw Nd L = A P w %r). This simple relation between the heat flow Qw and the resistance R holds true only for a medium with Pr = 1. Since all gases have Prandtl numbers which deviate only slightly from unity, the expression for Qw is useful in obtaining a first approximation for the heat transfer when the resistance is known. 21 Prandtl (28) and Taylor (29) independently arrived at the following expression for turbulent heat transfer coefficient for a flat plate I cB/u s (P r C'IOCS.‘ CT — 1) s This relation has been modified to apply to fully developed heat transfer in tubes. The resulting expression is St _ 0.038u (Re)-l/u 1 + 1.5 Pr'l/6(Re)-l/8(Pr - 1) This relation is superseded by the following expression which assumes the presence of buffer layer in conjunction with distinct laminar and turbulent layers. , f Nu (Chm/9 B) '8' l + omz’ § {5 (Pr-1) + 5 1n [ (5Pr + 1)/6 ]} where 22 and C _ m ¢m — E— c and for the turbulent velocity profile 0m = 0.82. In reality it changes somewhat with Reynolds number. The friction coefficient f is given for smooth tubes as f = 0.316 Re < 105 (Re)l: For Reynolds number greater than 105, the general resistance law developed by L. Prandtl, T. Von Karman and coworkers (30) is fl; = 2.0 loglO [ (Re) /?'] —0.8 The problem of forced heat convection in turbulent flow through a tube at uniform surface temperature was first analyzed by Latzko in 1921. He gave the following equaiton for temperature distribution in a circular tube when flow is fully developed. 0.. l\) ‘E" r at* 2r 2 3t* 23 where t* = temperature difference between the wall and the given point, and u1/4 (d)3/28 8 m /4 H:— 7 0.199vl Other symbols have the same meanings as given under NOMEN- CLATURE. Latzko arrived at the following solution for the temperature distribution: -P x t E: = 1.29e 1 (0.9544n - 0.02l2n3 + 0.0668n5) w B -P2x 3 5 - 0.1808 (-0.7472n - 4.275n + 6.022n ) ”P3X 3 5 + 0.048e (20.34n - 54.80n + 35.47n ) where 2 1/7 n-E1-<-2a£>3 _ l 4/ v Pn _ Ch 3 um d :1 = 0.1510, :2 = 2.844, :3 = 29.u2 24 Latzko further derived the following equation for a local heat transfer coefficient -Plx -P x —P x + 0.134e -P x -P X —P X 1 + 0.024e + 0.006e 3 _ v 1.078e h — [ 0.0345ump cp(um d)] 0.970e Sparrow and Siegel (31) have examined the effect of wall boundary conditions on Nusselt number for turbulent heat transfer in both the fully develOped and thermal entrance region of a circular tube. The comparisons are made for: 1. Uniform wall temperature 2. Uniform wall heat flux. Computations were carried out for the uniform wall temperature case, [reference (32)] using the same eddy diffusivity and velocity distribution as for uniform wall heat flux in references (33) and (34). The calculations were carried out under the assumption of equal diffusivity of heat and momentum. The results for Pr = 0.7 are presented for four values of Reynolds number (10”, 5x10“, 105, 5x105). The per cent difference was defined as Y = (Nu)uniform flux-(Nu)uniform wall temperature (N u) wall temperature The largest values of'Ywere recorded at low Reynolds numbers, and for each case the values drop at larger distances from 25 the tube entrance. For the ranges of variables considered, the effect of the two different wall boundary conditions was found to be always less than 10 per cent. At higher values of Prandtl number (Pr = 10 and Pr = 100) there was essentially no influence of the two boundary conditions in the range of Reynolds numbers con- sidered. It may be concluded that for turbulent flow the heat transfer mechanism in the thermal entrance and fully developed regions is quite insensitive to the two wall boundary conditions. This conclusion is supported by the findings of Deissler (12) who carried out calculations for the thermal entrance region of a circular tube (using a boundary layer model) for Pr = 0.73. Further support is given by Seban and Shimazaki (35) who considered fully develOped flow in a circular tube for Pr = l. Sparrow, Siegel and Hallman (33) have given the analytical results for the fluid temperature distribution and Nusselt number for tube with a uniform wall heat flux. The fluid temperature distribution is given by 2 1 = 2 x n q d‘ RePr d + G(r) + ? Cn¢n(r)e w 2K n-l where G(r) is the fully developed temperature distribution, 8n and ¢n are the eigenvalues and eigenfunctions, r is the distance from tube axis measured in the radial direction. 26 The Nusselt number given by the authors is 1 Nu = 6(8/2) 1 + An exp(-4BEx/dRe) ll M8 n l where _ d d An - Cn ¢n(2)/G(2) Sparrow and Siegel (32) have considered turbulent heat transfer in tubes with uniform wall temperature. The analytical results are given for fluid temperature distri- bution and heat transfer coefficients. The fluid tempera- ture distribution is given by C 0 (r) exp(-482x/d Re) n n n where 8n and ¢n are the eigenvalues and eigenfunctions, r is the distance from the tube axis measured in a radial direction, and Cn are constants. The Nusselt number is given by Nu _ P nil Cn(_d?2) d/2 exP('”BEX/d Re) ‘ c °2° ii (931) exp (—4B2x/d Re) n=l B2 dr d/2 n n 27 Deissler (12) has given analytical solution for fully developed turbulent flow in a tube having uniform heat flux. According to him Nu = d Pr t+ B and Re = u; d+ where + d+ t+ u+ (%_ _ y+) + f2 + dy t+ _ O l - 8t B _ + + d + d + + 1.71.1 (T-y)dy 0 l-Bt+ and i . u+ a 8 f2 u+(g___y+) dy+ b + 2 0 2 (d) and B is a heat transfer parameter given by 28 The physical prOperties in Deissler's equations are evaluated at wall temperature. Dittus and Boelter (35) recommend the following empirical equations for tubes 0.8 0.4 r) Nu = 0.0243 (Re) (P for heat flow from the wall to the fluid and Nu = 0.0265 (Re)0'8 (Pr)0'3 for heat flow from the fluid to the wall. From a review of the work of various experimenters, McAdams (36) has concluded that a fair correlation of their results for the heating and cooling of various fluids in turbulent flow in horizontal tubes is shown by the equation Nu = 0.023 Reo'8 Pr0°u This equation applies when the Reynolds number is within the range of 10,000 to 12,000, the Prandtl number is between 0.7 and 120. and the physical properties of the fluid are evaluated at the bulk temperature. Martinelli (37) derived an equation based on the following assumptions: 1. Heat and momentum are transferred by molecular action alone in the laminar sublayer. 2. Heat and momentum are transferred by molecular action and by eddy diffusion in the buffer layer. 29 3. Heat is transferred by thermal conduction and eddy diffusion in the turbulent core, and momentum is transferred by eddy diffusion alone. With these assumptions Martinelli's equation for fully developed turbulent flow in an isothermal tube is 0.04 PrRe7/8 1 Re7/8 Pr + 1n (1 + SPr) + ETITFT In -300— In this equation 8 is a factor which depends on the ratio of molecular conduction to the eddy diffusion heat transfer in the turbulent core. Lyon (38) developed an approximation of the Martinelli equation for values of Pr < 0.1. This equation which gives values within 10 per cent of the Martinelli's values. 0.8 0.8 Nu = 7 + 0.025 (Re) (Pr) In Martinelli's and Lyon's equations the physical properties are evaluated at the bulk temperature if the difference between the bulk temperature and that of the surface the fluid is in contact with, is not large. At higher rates of heat transfer, the temperature which is selected is usually the bulk temperature or the mean of bulk and surface temperatures. 0n the basis of foregoing observation and as a result of tests with viscous oils, Sieder and Tate (39) concluded 30 the surface conductance for both the heating and the cooling of viscous liquids in turbulent flow is well expressed by u 0.14 Nu = 0.027(Re)0'8(Pr)1/3(—§) uw Colburn (40) has given the following relation for turbulent flow h ———— (Pr)2/3 = 0.023(Ae)'0'2 p G c McAdams, Nicolai and Keenan (41) give the following expression for the fully developed turbulent flow (§l%—) (Pr)2/3 = 0.027(Re)'0'23 p Use of Hydraulic Diameter for Noncircular Ducts The predictions of fully developed turbulent pressure drops and heat transfer in noncircular ducts is presently based on the hydraulic diameter concept. It is seen that for a wide variety of duct shapes, the pressure drop and heat transfer correlations which are valid for circular tubes may be applied to noncircular ducts if the hydraulic 4 x cross sectional area perimeter is substituted for the diameter characteristic dimension in the friction factor and the Reynolds number. 31 A large number of investigations which correlate pressure drop data based on hydraulic diameter concept have been summarized by Claireborn (18) and Eckert and Irvine (6). It should be noted, however, that the hydraulic diameter concept does not hold for all geometries; for example, experimental studies (42, 43) have shown that the hydraulic diameter rule can be in error by as much as 20 per cent in small opening angle triangular ducts. For laminar flow the pressure drop is even more sensitive to the shape of duct cross section, and the hydraulic diameter does not satisfactorily correlate the result of circular and noncircular shapes. The hydraulic diameter rule, although it has enjoyed considerable success, has little rational justification for its use and it throws no light on the basic mechanisms involved in noncircular duct flow. Deissler and Taylor (44) have attempted to construct a model of turbulent and noncir- cular duct flow from which calculations of the velocity field and pressure drop may be made. Essentially their procedure consists of the specification that on lines normal to the duct walls the velocity distribution may be charac- terized by the universal velocity profile which has been established for circular tubes. The analysis neglects the secondary flows. The heat transfer characteristics of noncircular geo- metries have been investigated to a lesser extent than the 32 flow. An experimental investigation (42) of turbulent heat transfer in an electrically heated triangular duct with an opening angle of 11.460 has shown that the use of hydraulic diameter overestimates the average heat transfer coefficient by as much as 100 per cent. Another serious limitation of the hydraulic diameter concept is that the local heat transfer conditions cannot be determined. This applies in particular to the circumferential variations of wall temper- ature or specific heat flow at the wall. This is reckoned as a serious limitation, since in certain geometries the local temperatures can be considerably higher than those predicted by the average results. This can result in serious design and metallurgical setbacks. Deissler (12) has given the following equations for a parallel plate channel 4. Nu = 2a Pr t+ B and Re = 2 u+ a+ p where + a /2 t+ u+ + + f0 +dy tB - l - 8t a+/2 + + f0 u dy 33 and 2 + p a+ B 0 1 - Bt+ B = quTW79/g cp Tw t W The physical pr0perties in Deissler's equations noted above should be evaluated at wall temperature. Sparrow and Lin (45) have presented analytical results for the fully developed heat transfer characteristics of turbulent flow in a parallel plate channel with symmetric uniform wall heat flux. The results were obtained by inte- grating the energy equation utilizing a suitably chosen eddy diffusivity for heat. The fully developed Nusselt numbers were presented as a function of Prandtl number for the range Pr = 0.7 to 100. The Reynolds number, which appeared as parameter, ranged from 10,000 to 500,000. The results were compared with McAdam's (36) empirical corre- lation of circular tube data, Nu = 0.023 ReO'8 Pro'u This correlation is usually regarded as the first approxi- mation to noncircular geometries provided that hydraulic diameter is used. The comparison indicated that the use of the hydraulic diameter is reasonably successful in 34 bringing together the results for the two geometries. The analytical results for both geometries suggested that the dependence of the Nusselt number on Prandtl number is more complex than the simple power law appearing in the empirical correlation. The largest deviation between the theory and empirical correlation appeared in the mid-range of Prandtl numbers (10). Barrow (46) has given the following analytical expression for fully developed turbulent flow in a smooth parallel plate channel, both walls of which are subjected to constant and equal heat flux. 0.1986 Re7/8 Pr Nu = 9.7a Pr + 5.03 Re1T8 - 9.74 Barrow used the following expressions for velocity distribution + + . u = 6.2 logloy + 3.6 in turbulent region u+ = y+ for laminar sublayer The presence of buffer layer was neglected in Barrow's analysis. Barrow (47) conducted a theoretical and experimental study of heat transfer in fully developed turbulent flow in a smooth parallel plate channel. In theoretical analysis, 35 surface heat flux at the walls was assumed to be uniform in flow direction. In the experiments, the flow conditions were simulated by ducts of large aspect ratio. The theory is more rigorous than the one given in earlier paper (46) and is adequately supported by the experimental data. The theoretical expression for Nusselt number is: 0.139 Re7/8 (Pr = 0.7) 5.03 Re'178 — 3.06 This equation follows more closely the line repre- senting McAdams' well-established expression Nu = 0.023 Re0°8 Pro'u particularly in the higher range of Reynolds number. In his analysis Barrow assumed a turbulent and a laminar sublayer. Velocity distribution in the channel was taken from Corcoran et a1. (48). + u = 070895 tan h (0.0695 y+) O f y+ f 27 u+ = 5.5 + 2.5 ln y+ y+ > 27 The following expressions were used for eq. 36 [v cosh2 (0.0695 y+) ] - 0 q a+ 2.5 a+ + a+ a+\2 3- 'fif f y E if , Sq = §§—§ - V Pr was assumed to be unity. t Fully Developed Turbulent Flow in Unsymmetrically Heated Ducts Unsymmetrically heated ducts have received attention rather lately. As a first approximation the results given under symmetrically heated tubes can be used by substituting hydraulic diameter of the duct for the tube diameter. It is to be emphasized that full perimeter is used in computing hydraulic diameter, even when only a part is heated or cooled (49). Only in calculating the heat flow from basic heat transfer equations is the heated area alone to be used. Henry Barrow (46) has given a semi-theoretical analysis of unsymmetric heat transfer in fully develOped two- dimensional heat transfer between parallel plates. The boundary conditions consisted of a heat flux qw through one 37 wall and a heat flux of —qu through the other; where y is a factor ranging between —1 and 1. The velocity distribution in the channel as used by Barrow is + + u - 6.2 loglo y + 3.6 for the turbulent layer and for the laminar sublayer. The laminar sublayer with this velocity distribution extends to y+ = 9.74 Barrow gives the following expression for the Nusselt number. 7/8 Pr + 9.74 [Pr - (2 - Y)1 0.1986 Re Nu = 5.03 (2 - v) Rel/8 Barrow (47) presented another theoretical and experi- mental study of unsymmetric heat transfer in fully developed turbulent flow in a parallel plate channel. The heat fluxes at the plate surface were of different magni- tudes and in the theoretical treatment each surface heat flux was assumed to be axially uniform. Ducts with large aspect ratios were used to simulate two-dimensional flow. Heat transfer measurements were made firstly with one wall insulated and then with heat transfer at both walls. The 38 heat transfer coefficients found experimentally for the wall through which heat flowed to the fluid, were seen to be less than the accepted value for the case of symmetrical heat transfer. The heat transfer coefficient was seen to decrease with an increase in the degree of asymmetry. The theory based on an analogy of heat and momentum transfer is more rigorous than the one reported in reference (46) by the same author. According to new analysis, the expressions for Nusselt number are: Case l.—-Wall heat fluxes of equal magnitude but of Opposite sign. Nu _ Re Pr qw c p un(tw - ti) where q Pr t-t= W ___.._ 1.1.1] w B CPD 7;73 [ l 2 3 The integrals I1’ 12, I3 are given by the following expressions: 27 + I =f dy 1 + O [Pr(l - g%r) cosh2 (0.0695y+) - (Pr-1)] a 39 + a /3 dy+ 1 =f + + 27 (1 - 39%?) Pr - (Pr - 1) a and a+ I3 = a+ ___ 6 [(Prfi)-(Pr—l)] Case 2.--0ne wall of the channel insulated. The following expression is given for the Nusselt number. t —‘t Nu = 0.1986 Re 7/8 ( w1 ”2) Il + 12 + 13 + I4 twl - tB where + + I =f a 1 0 2 + 2+ Pr cosh (0.0695y )(1 —-4E—) - (Pr - 1) a L: + a+/3 (l- +)dy I2=f a + 27 (I - @wa 25%";- - (Pr - 1) a 40 a+ I3 _= 3+ 6 [-Pr 22.5 - (Pr - l) ] + y + a+/3 (g?) dy I“ = f + + 27 (1 - arm-213) Pr — (p1- - 1) a These formulas although more accurate need extensive numerical computation and have obvious limitations for practical use. Hatton (50) has considered fully developed turbulent flow in a parallel plate channel with one wall insulated and the other one having a constant axially uniform heat flux. He has solved the energy equation: 1' +z jryog-‘zo /////////////// //// fl 'd + e BZ 8x+ V 3 + 41 The solution results in 423 d0+ Nu = (-—9) 0+ - 9+ dz+ Z+ qw qb 0 where 6+ = t - ti q q(y-yol 2K (t1: the temperature at entrance to heated duct) Z+ f 0+ u+ 6+ d Z+ q + —z e = 0 0 2+ b I i u+ d Z+ -Z c Hatton and Quarmby (51) have presented a solution to the problem of fully developed turbulent flow in a parallel plate channel heated on one side only. A constant eddy diffusivity is assumed for the central region of the channel. In the solution of the energy equation Prt was assumed to be unity. By the principle of superposition, the result is obtained for unequal uniform heat fluxes on each side of the passage. The results of Hatton and Quarmby entail extensive 42 numerical computations and their applicability is limited. Hatton, Quarmby and Grundy in a later paper (52) made a few modifications in the analysis. Em is assumed to be constant over the middle third. em/eq is not taken to be unity. A semi-empirical relation is used for the value em/eq. Sparrow, Lloyd and Hixon (53) have conducted experi- ments on rectangular ducts with unsymmetrical thermal boundary conditions. Results are found to be in qualitative agreement with analytical predicitons for parallel plate channel. Madsen (54) has suggested a characteristic heat transfer coefficient for parallel plates with axially uniform but unsymmetrical heat fluxes. This heat transfer coefficient is independent of heat flux symmetry and is defined as follows: qwl + qw2 (t -t)+(t —t) wl B W2 B hgeneral where suffixes l, 2 refer to the two walls in the parallel plate. Novotny, McComas, Eckert and Sparrow (55) studied fully developed turbulent flow in ducts. The aspect ratios ranged between 1:1 to 1:10. The heating conditions were such that the two longer walls were uniformly heated and the 43 two shorter walls were essentially adiabatic. The results for Nusselt Number (Nu) as a function of Reynolds Number (Re) were compared with the circular tube empirical corre- lation of Colburn (40) as well as with the analytical results of Siegel and Sparrow (34), for the circular tube and Sparrow and Lin (45) for the parallel plate channel. Both analytical studies (45, 34) were carried out for the case of uniform wall heat flux. The analysis for the parallel plate channel yielded Nusselt numbers which were approximately 5 per cent larger than the results of the 5 and circular tube analysis at a Reynolds number of 10 approximately 10 per cent higher at a Reynolds number of 10“. The experimentally-determined Nusselt numbers were, apart from some scatter, bracketed by the results of the two analyses. Additionally it was found that the empir- ically-based Colburn (40) correlation of circular tube data, which is frequently used for engineering calculations, lay somewhat higher than the experimental data, especially at the highest Reynolds numbers. The hydraulic diameter used in the analysis was based on total perimeter. 44 Entrance Region Studies for Symmetrically and Unsymmetrically Heated Circular and Noncircular Smooth Ducts with Turbulent Flow Symmetrically Heated Ducts.--The thin boundary layers and consequently severe temperature gradients at the wall near the entrance produce high heat transfer coefficients in the entrance region. As far back as 1913, Nusselt (56) drew attention to the effect of the length of tube on heat transfer and sug- gested the following equation n m ( Nu = a' Re Pr )c F‘IQ with c = 0.055. This equation gives zero heat transfer as L _+oo. In 1943, H. Hausen (57) gave the following expression for the average Nusselt number in entrance region over length x of a smooth tube “B 0.14 m; = 0.116 [:(Re)2/3 — 125:1 (Pr)l/3 [1+ (%)2/i(r) W where UB is the viscosity at bulk temperature and “w is the viscosity at wall temperature. Local Nusselt number at a distance from the entrance is obtained by differentiation of equation given above. 45 u 0.14 Nu = 0.116[(Re)2/3- 125](Pr)l/3l:L + -§-(%)2/3:l(u—B-) W Hausen's expressions are particularly useful in the case of fluids where the variation of viscosity is the dominant factor. Latzko (58) in 1951 investigated theoretically the effect of what he called "the starting section." He assumed a unit Prandtl number and found that the local heat transfer starts with infinity at the commencement of heating and decreases rapidly along the lengths of the tube until a limit given by p u 0 110° = 0.03814 417732 R e is reached. Heat transfer coefficient in the entrance region as determined by Latzko is given by _ ’ -1/4 h - P um 0 cp (Re) I where P = a function of w and 1.u1 x — 0.0u8 x2 + 0.168 x3 E II and (§)0.8 (Re)-l/5 >< ll 46 In 1930, Eagle and Ferguson (59) heated water in condenser tubes. In one of the experiments, the local heat transfer coefficients at x/d = 2 and x/d = 94 were measured. The former was found to be 50 per cent greater. This was probably the first experimental detection of entrance region thermal effects. In 1952, Davies and Al-Arabi (60) heated water using an electrically-heated pipe and measured the local and average heat transfer coefficients. The following equation was suggested to represent average heat transfer coefficient for x/d > 5 N3 = Nuoo |_l + 2.8 ($01 Hartness (61) heated water and oil in an electrically- heated pipe. The local heat transfer coefficients were measured. A length of 15 diameters brings the local co- efficients within approximately 1 per cent of the value that is reached asymptotically in a long tube. Deissler (12) made a theoretical study of turbulent heat transfer in the entrance region of smooth passages. He considered tubes and parallel plate channels. The results were given in a graphical form for local heat transfer coefficients against length. The results indicate that approximately fully developed heat transfer and friction are attained in an entrance length of less than 10 diameters. 47 Deissler has given the following expressions for entrance region of smooth tubes with turbulent flow Nu = d Pr t+ B Re = d+ u+ p where + d+/2 1+ u+ (%; - y+) + f + dy t+ _ 0 1 — 8t B - + + + + d/2 u (QT-5’) + f + dy 0 1 — 8t + + 5t t+ 11+ (9— - y+) t+ d+/2 + f 2 + c +d + + 0 + 93’ +——'Tf5+u('2"y)dy 1 - 8t 1 - Btc t = + + at u+ (%T ' y+) + 1 d+/2 + d+ + + f + dy+——;f 'U(—2—"Y)dy 0 l-Bt 1-Bt + c 0 t and + + 6 + d + tu(—-y) u; = 8 2 <1 — 31;) f 2 , dy+ + + — <1) 3 1 at + 1 d/2 +d+ + + —"—Tf+ u(-—2—-y)dy l - 8t 0 48 where B = heat transfer parameter The corresponding equations for flow between flat plates are + Nu = 2 a Pr t+ B and Re = 2 u+ a+ p where + + + 6 + + t a /2 f t t u + dy+ + c + f + u+ dy+ 0 l - 8t 1 - 8t 6 + _ c t tB _ + 6 + a /2 f t u + dy+ + l + f + u+ dy+ 0 l — 8t 1 - Btc 6t and + + 6 + t a/2 0+ = i; (1-81;) I t + dy+ + l + f + u+ dy p a 0 l-Bt I—etC 5t Physical properties in the above equations are to be evaluated at the wall temperature. 49 Al—Arabi (62) gives the expression for average Nusselt number for length L of the tube. NH = Nuw (l + §EQ) where 0.3 _ 6000 L Romanenko and Krylova (64) have suggested the following empirical relation for entrance length in smooth ducts. g: = 0.592 (Re)0'u8 h In the discussion on entrance region so far, the flow was hydrodynamically developed before it entered the heating section. ' Limited information is available for the case where hydrodynamic and thermal boundary layers develop simultan- eously. W. Linke and H. Kunze (65) have given experimental results for local heat transfer coefficients in the entrance region of tubes. Entrance length was found to exceed 60 diameters. It appears that, except at the highest Reynolds number (101,600) the flow was laminar throughout the entrance region, for nearly all the friction factors were below the fully developed values. The unusually long length 50 through which a laminar flow was sustained is attributed to a bellmouth entrance. Deissler in his studies (10, 12) on channels and tubes found that fully develOped (hydrodynamically and thermally) flow resulted in less than 10 hydraulic diameters for simul- taneously developing thermal and hydrodynamic boundary layers. The results of heat transfer coefficients are graphically presented against length. The results for tubes in this section are applicable to ducts when hydraulic diameter of the duct is substituted for tube diameter. Unsymmetrically Heated Ducts.--As a first approximation the results of symmetrically heated ducts can be applied. For noncircular cross sections, the tube diameter should be replaced by the hydraulic diameter of the duct. The whole perimeter should be used in calculating the hydraulic diameter even when only a part of the duct is heated or cooled. Hatton (50) has analytically determined the entrance length when two parallel plates are maintained at different temperatures. An entrance length of 20 hydraulic diameters is calculated for Reynolds number = 7096 and Pr = 1. At Re = 73612 and Pr = 1 this length increases to 30 hydraulic diameters. Barrow and Lee (66) considered entrance length for the uniform heat flux on one wall (the other wall being adia- batic). In their analysis they assumed linear heat flux 51 across the thermal boundary layer and an eddy diffusivity distribution as given by Deissler (10). Local Nusselt number in the entrance region is given as: Barrow and Lee in another analysis (67) have assumed a linear variation of e for y+ (y+ > 26). Smaller heat transfer coefficients and entrance lengths are predicted than would be obtained from Hatton and Quarmby's analysis (51). The difference as noted will result from different expressions for a being used. According to Barrow and Lee (67) about 20 hydraulic diameters are required for the flow to develOp thermally for Reynolds number up to 70,000. Hatton and Quarmby (51) predict a much larger entrance length equaling 130 hydraulic diameters. Since thermal boundary layer develops on only one side, unsymmetric boundary conditions will result in longer entrance lengths than would be required by symmetrically heated ducts. Novotny gt_al. (55) conducted experiments with rectan— gular ducts having two heated and two unheated walls. Aspect ratios of the ducts ranged between 1:1 to 1:10. The thermal entrance length was noted to be less than 30 hydraulic 52 diameters. It was further noted that there was no clear influence of the side walls on the thermal development. Fully Developed Turbulent Heat Transfer in Rough Tubes The data available on rough tubes are mainly experi- mental. A. Soennecken (68) noted in 1911 that the value of the coefficient of heat transfer is lower in rough tubes than that in smooth tubes. Stanton (69) obtained a contrary result. This was followed by experimentation by Pohl (70). He concluded that in all the rough tubes the heat transfer coefficient was lower than that in the smooth tube. Pohl did not give a direct measure of roughness. Cope (71) used special knurling procedure to develop roughness in tubes. He found that in turbulent region surface roughness has very little effect on the heat transfer coefficient. Cope further concluded that for a given amount of heat being transferred, the smooth tube gives the lowest pressure drop. Nunner (72) used spring rings which were placed in the tubes to create roughness. The disadvantage as noted by the cited author is that the rings do not form a single body with the tube. This may introduce an error of up to 13 per cent in determining value of h. Nunner noted considerable increase in heat transfer and pressure drop. Nunner gave the following empirical expression 53 (jgi)-l/8 0.68 100 Nu = 0.383 Re (f) Brouillette (73) used triangular grooves to create roughness and found increased heat transfer and pressure drop. The measured values of the heat transfer coefficients for the rough tubes were from 10 to 100 per cent greater than those measured on a plain tube. The results indicated that the height of grooves is the deciding factor in raising the heat transfer. The number of grooves per unit of tube length had relatively little effect. Hobler and Koziol (74) carried out experiments to investigate the influence of alternate and bilateral con- tractions of tubes during the forced flow of air in the range of Reynolds number of 200 to 60,000. Marked increase in heat transfer was noted above Re = 1,000. Theoretical Analysis for Heat Transfer in Rough Tubes For rough tubes the hydrodynamic conditions are expressed by means of lo, the local degree of turbulence. A0 is given by 54 Assuming homogeneous and isotropic turbulence in a tube of diameter d, the following expression is obtained h d* d 2 * 0.5 k = — <-,—> (Pr) 7T 0 and based on experimental results the following expression is obtained: A = 28 2L 0 u* The preceding two equations result in h d* u* d* Nu = k = 0.04 ( )(Pr)0'5 v or Nu = 0.04 Re Pr0.5 T This equation was developed for smooth tubes. As the relation for 10 contains only v and u*, it may be expected to hold also for tubes with rough walls. This was proved experimentally by V. Kolar (75). The value of d* to be used in equations for rough ducts is suggested by most authors to be V d*=/'r'1: where V is the volume of the duct. 55 It was shown by Kolar (76) that Nunner's expression for rough tubes (72) (1.0.0. 1/8 Nu = 0.383 Reo'68 (f) Re is equivalent to N” z 0 05 Re (Pr = 0 72) Pr0.5 ° T ' which is similar to equation cited above as Nu = 0.04 Re PrO'S T CHAPTER III THEORETICAL ANALYSIS Unsymmetrically Heated Parallel Plate Channel Fully developed turbulent flow in a smooth parallel plate channel will be considered. One of the plates is adiabatic whereas the other one transfers heat at a constant rate to the air flowing in the channel. In actual practice a parallel plate channel will be simulated by a duct of large aspect ratio. The problem is schemat- ically presented in Figure l. . W \ , ‘\ 777777777777777/// I/7/////7//T7///// Figure l.--Fully developed turbulent flow in an unsymmetrically heated parallel plate channel. 56 57 The plates are spaced distance "a" apart as shown in Figure l. Hydraulic diameter (dh) for this parallel plate channel will equal 2a. Assumptions l. The velocity distribution in the channel is given by the following: + a. u+=y+ 3’ 55 + + + b u = 5[1 + 1n%71 for buffer layer, 5 5 y 5 30. C 0+ = 5.5 + 2.5 ln y+ y+ > 30. 2. The flow in the laminar sublayer is completely laminar. 3. In the buffer layer the flow changes gradually from laminar to turbulent 4. In the turbulent core, the turbulent diffusivity is much larger than the laminar viscosity or heat conduc— tivity. The laminar terms in shear stress and specific heat flow will be neglected. 5. e = e i.e., Pr = 1 Nusselt number is defined as qw d (l) w - tB) why Nu = (t In order to find temperature drop (tw - t the laminar B)’ sublayer, the buffer layer and the turbulent core will be considered separately. 58 Temperature Drop in Laminar Sublayer The linear temperature drop in the laminar sublayer is given by Introducing the dimensionless value y+(y: = 5) one gets qw p + qw p Atfi - EE—-//;: Pr yl - EE— //;: 5 Pr (2) P W p W Temperature Drop in the Bfiffer Layer Shear stress and the specific heat flow for the buffer layer are given by: l _ du Tb'("+€)pa§ _ 0 dt qb — _ (Pr + e)p Cp dy (Subscript b refers to the buffer layer.) The second equation above gives temperature drop in the buffer layer as: 59 The first equation above gives eddy diffusivity as: as I C _Tb €_-—_ 0 Shear stress and specific heat flow are assumed to be constant in the laminar and buffer layer. -Accordingly we can replace qb and Tb by qw and Tw respectively. Using dimensionless u+ and y+ one gets du+ + Using von Karman's equation for the velocity in the buffer layer: + u = 5 [ l + ln %7'1 one obtains Therefore 60 and _ qw yb dy+ Atb - f Ocpv y 3. y+ l Mei-5") = iii _a. 30 dy+ pc T 5 + D W .32. X..- Pr + ,5 l which on integration yields: F )q At = —73- J1 1n (5 Pr + l) ' (3) b pcp Tw Temperature Drop in the Turbulent Core In an unsymmetrically heated parallel plate channel, the temperature profile and the velocity profile are dis- similar as shown in Figure 1. As a result of this the Reynolds analogy is not valid for the turbulent core. For this case a modified Reynolds analogy has been suggested by Mizushina (77). His method will be used as follows: It is assumed that: 1. Velocity is uniform in the turbulent region and equals um. 2. Eddy diffusivity (e) is constant along the turbulent region. 61 Consider the energy balance at distance y from the heated wall for a strip of width Ax, located at distance x from the entrance; consider a unit depth of the strip. The energy balance results in: g at - pcp eq 5; Ax - um 0 cp 5; (a - y) Ax (4) Similarly at distance yb the energy balance yields: q Ax = u p c at w m p 5; (a - yb) Ax ‘ (5) Since the flow is fully developed Bt/Bx = constant. Equations (4) and (5), on division and integration, give: 2 t ' tb ' popeta-yb) {3(y'y6) + (if ‘ 77)} (6) where tb is the temperature at distance yb from the wall. Since the velocity is assumed to be uniform in the turbulent case, the bulk temperature (tB) for fully developed flow in a channel can be approximated by: l a B a - y b yb Substituting for t from equation (6), in equation (7) one gets: 62 q 3 2 _ w a b tb - tB - pc 6 (a-y )5- (II - I?) + (a yb - a p q b 2 yb) (8) Since yb is small compared to the plate spacing a, the equation (8) approximates to: q t - t = w 3 - y:] (9) b B pope:p [3 b which is the temperature drop in the turbulent core. Temperature drop (tw - tB) Since tw - tB = At2 + Atb + (tb - tB) (10) After substituting for component temperature drops from equations (2), (3) and (9) the equation (10) changes to qw o tw'tB-ECT— jr—SPI’+ p W q q .11 J/fli 5 ln(Pr+l) + _JL EL pc I pc 6 p W p q % - yé] (ll) Since 63 and St - u (12) Substitution for (tw - tB) from equation (11) in euqation (12) results in the following expression for Stanton number. -1 _ D 1 /Tw a St - urn/7:; {5P}? + 51n(5Pr+1) + E; '7)- (g - yb)} (13) For turbulent flow in parallel plate channels, the Blasius law satisfactorily correlates the data, when hydraulic diameter dh = 2a is substituted for tube diameter. For a channel, it is, therefore, obtained T —K = 0.038u u2 (Re)-l/u (1“) p m where u (2a) Re=_.I£_—__. v T Substituting the value of I? from equation (14), equation (13) results in: 64 _ 0.1958 Re'l/8 5Pr + 51n(5Pr+1) + 0.0326 E— Re - 30.59 E— q q A constant value of eq across the turbulent core is given by Hatton (78) as: + —l v _ + m E_ - 0.36m (l - —;) - 1 q yo where = B and m+ = l ( + + 26) yo 2 2 yo ' In the ducts that are experimentally studied in the present thesis, a = lft hence y = 1ft and d = 2a = lft ‘I2 0 24 6 ' Since T 7? = 0.0384 u; (Re)'l/u the following expressions result: m+ [0.02A42 Re7/8 + 13] 0.0488 Re7/8 c< ll 65 and .X— l e _ q [0.00u35 Re7/8 - 12u6.65 Re’7/8 + 1] Substituting for g: in equation (15), the following expression q results for Stanton number (St). 0.1958 Re'l/8 0.0326Re7/8 - 30.59 0.00435Re7/8 -12u6.65Re'7/8—1 St 5Pr + 51n(5Pr+l) + (16) It is to be noted that equation (16) will apply to a special case where plate spacing equals one inch. For dif- ferent plate spacings appropriate values of y: and m+ should be substituted in equation (15). Symmetrically Heated Parallel Plate Channel y u it a Y 1 q” j l l 4 " x Figure 2.--Fully developed turbulent flow in symmetri- cally heated parallel plate channel. ‘ls. *ml‘f‘fi'hl “‘1‘ ‘1! W‘M‘u ' ’ ""1"" t' 4 66 The problem is schematically representd in Figure 2. Here the specific heat flow through the two plates is constant and equal. Other conditions remain the same as in the last section. As before temperature drop through laminar sublayer, buffer layer and turbulent core will be determined separately. Temperature Drop Through Laminar Sublayer _ 0 MY. ‘ p—e' }— 5F? (17) Temperature Drop Through the Buffer Layer At = J. E 51n (5Pr + 1) (18) Temperature Drop Through the Turbulent Core Since the temperature and velocity profiles are similar, Reynolds analogy is valid for the turbulent core. Hence q q '— w w p + + t - t = (u - u ) = ——— —— u - u (19) b B CpTw m b pcp Tw [-m 0] Substituting y; = 30 in the von Karman's equation for the buffer zone, namely: 67 the following expression results: u; = 5[:l + 1n 6:] Substitution for u+ in equation (19) gives b t - t - £51 /Jl + 5(1+ln6) (20) b B - pc I um - p v w Temperature Drop (tw - tB) Since tw - tB = At, + Atb + (tb - tB) Substituting for component temperature dr0ps from equations (17), (18) and (20), the following equation is obtained for (tw - tB): C, 2: I C, CD u U (42 ID] I l 5Pr+ 5 ln(5Pr+l) + u; - 5(1+ln6)] Using T 7? = 0.038u u; (Re)-l/u 68 the following expression is obtained for Stanton number: -1/8 St 0.1958 Re (21) 1/8 (5337511) + 5.10 Re 5(Pr—l) + 5 1n Simplified Analysis for the Unsymmetrical Case In this analysis no account will be taken of the buffer or transition zone. The turbulent layer is divided into a laminar sublayer and the turbulent core. Figure 3 is a schematic representation of the problem. ///////////f/{/7////////7' q = 0 Figure 3.--Fully developed turbulent flow in an unsymmetrically heated parallel plate channel. 69 Laminar Sublgyer Here Reynolds analogy applies. Conduction is the dominant mode of heat transfer. Therefore for the laminar sublayer one gets: £1=_1_<. at. T u du Assuming that q 3 = = .3 T constant T w one gets _ k l qw - Tw u E_ (tw - t1) (22) where ul is the velocity at the border of laminar sublayer and the turbulent core. tl is the temperature at the border between the two layers. Turbulent Core Modified Reynolds analogy as suggested by Mizushina (77) will be used here as well. The assumptions are the same as before, namely: 1. Velocity is uniform in the turbulent core 2. sq is constant across the turbulent region. For a strip of width Ax located at distance x from the entrance and having a unit depth normal to the plane of 70 paper, an energy balance at distance y from the heated wall results in: EEAqup @E. cp sq 3y m cp 8x (a-y)Ax (23) - 0 Similarly energy balance at distance yl results in: _ 3t q Ax - pcp um (a-yl) 5; Ax (24) Equations (23) and (24) on division and integration result in: 2 2 qw yl a_y17- a + (7? - A?) (25) l - c e opq( d ll d Since um is assumed to be uniform in the turbulent core, the bulk temperature t is approximated by B _ 1 tB — a _ yl f t dy (26) y1 Substituting for t from equation (25), equation (26) gives 71 For a symmetrically heated parallel plate channel, it is known from Reynolds analogy that symmetrical = ( qw ) (28) (um - u ) c T symmetrical p w symmetrical Furthermore for the case of symmetrical heating it is seen that 1. - pc 6 33 = u oc 33 (3 - y) p q 3y m p 3x 2 2 qw = mocp 3% <3 - Y1) and 3. (tB)symmetrical = ———l———— fa/2 t dy 3 These three equations can be developed into an expression analogous to equation (27). The resulting expression is: 2 (tl-tB) = 2 i3 - E—Zl + a 2 - a 3 )2 l2 2 5’1 3571 q symmetrical w c e a-2 O p q( yl (29) COmbining equations (27), (28) and (29) the temperature drop for ‘the turbulent core for unsymmetrically heated parallel Platme channel is found to be: 72 3 l. 2 i3 - a2 + a 2 — Kl (a-y) 3 y1 y1 3 u-u t -t = l m l 1 B w 2 'c T 3 a y p w 2 a _ 1 + a 2__2 3 2 ‘21 2 3’1 Fl (a—2yl) Since yl is small compared with the plate spacing "a" one gets: u —u = m 1 t1 - tB qu {c T } p w approximately or q = (tl—tB) cpTw (30) w 2(um—uiT Since the film heat transfer coefficient is defined by Equations (22) and (30) can be used to derive the following expression: uul 2(um-ul) l _ h _ T k + c T w p w or u u u l _ m l l H - T C Pr E_ + 2(1 - u ) (31) w p m m 73 For fully developed turbulent flow in a smooth tube we have -1/8 u1 E— = 2.14“ (Re) 3 This expression describes the flow in a channel with sufficient accuracy if hydraulic diameter (dh = 2a) is used instead of tube diameter. u Substituting for U; in (31) and rearranging gives the m following expression for Stanton number: -1/4 St: h_ = 0.0192 R8 (32) pCpum l + 1.22 Re-l/B—(Pr—2) Simplified Analysis for Symmetrically Heated Parallel Plate Channel Here again the buffer layer will be neglected. Laminar Sublayer As for the case of unsymmetric heating in the last section, it is seen that _ R q ‘ T E -— (t - t1) (33) 74 Turbulent Layer As explained before, Reynolds analogy applies for the turbulent boundary layer which gives: l = t —tB = uul + u -ul (35) h qw ka c Tw Using u _$ = 2 AA (Re) ‘1/8 u m and rearranging, the following result is obtained. _ 0.038u Re-l/u St — -1/8* (36) l + 2.44 Re (Pr - l) CHAPTER IV EXPERIMENTAL INVESTIGATION Construction of Equipment Two rectangular ducts (each having a cross-sectional area 1" x 4" and a length of 10') were constructed on a 1/4" thick (Grade A) sheet of plywood. The other two dimensions of the sheet measured 1%' x 10'. The details of the construction are as follows: Three pieces of lumber (1" x 1%" x 10') were fastened on the plywood sheet so that they measured 5%" from center to center leaving a gap of 4" between successive pieces. The pieces of lumber were placed parallel to the longer side of the plywood sheet, with the one-inch sides being normal to the sheet. A flat galwbnized steel sheet (27 guage galvanized 0.02" thick) was used to cover the entire length of the pieces of lumber (i) and (ii) (Figure 4). The sheet measured 5" in width and was symmetrically placed on the pieces of lumber (i) and (ii). The sheet was nailed to the pieces of lumber. This completed the construction of the duct with the flat covering which will hereafter be referred to as Duct A. A corrugated sheet of galvanized steel was used to cover the space over the pieces of lumber (ii) and (iii) 75 76 Pieces of 13;" L b ‘— “ um er 914" . 814‘" %" Plywood -'*4k" Sheet 4 31%| I all/2‘1.— 1n 2Z’2‘!Z’.’.£ / .A ohswflm ma nofiflmppv mucoeflnodxo weaposocoo Lon mapmpmdd .wflm 'qu.0__|\.wm._nl'_‘.l .Nm ILII.NN ILLIJN .IIL Esuoam 81 Measurement of Temperature of Upper and Lower Sides of the Ducts Copper-constantan thermocouples made of 24 cali- bration copper and constantan wires were used. Thermo- couples were soldered to the underside of the flat and corrugated galvanized steel sheets. On the corrugated sheet the thermocouples were soldered to the underside of the crests (Figure 8). Thermocouples were also em- bedded in the lower plywood sheet to measure the temper- ature of the lower sides of the ducts. The plan for the thermocouples for the top and bottom sides of th two ducts is shown in Figure 8. As may be noted in Figure 8, the successive thermo- couples along the top and bottom of Duct A are placed six hydraulic diameters apart. In Duct B, the corresponding distance is approximately 9". An exact distance of 9" could not be maintained for the following reason: the thermocouples were soldered to the underside of the crests of the corrugated sheet; the distance between successive crests was 1%". Slight deviations in the distance of successive crests were, however, noticeable along the length of the sheet. This distance was also affected when the sheet was being nailed to the pieces of lumber during the construction of Duct B. " Kw... ant-M 82 DUCT B DUCT A m WI 2 , l. ’ l T T c... DUCT B DUCT A I o 9" . | 9.6" g 3 .1. 7 Ho 1" t SE 9T5 . 9.6" 8:. Her-1 *1 - 0 I a... .. 1 9* .. . 1 T6 9.6" 0 Thermocouples along I *. the roofing and : 1n ‘1 ‘1' bottom of the duct. | H ‘ {WI ' 9.6 P f 7 1 i. ' . 9.6" 8 3" 0 * OThermocouples for —6 measuring bulk i. 9.6" temperature of air. I 9" ' -t— t + 9.6" 5" ' * f t l 9.6" . . t 71,5,” 4 + 9'6" Hydraulic diameter = 1.6" 8 i f l I 9.6" ll * +1" 1 A 9'6" 8 i 1 1 i... ‘T\ 7" 2‘8 Fig. 8.——Location of thermocouples. 83 Location of Thermocouples along the Roofing and Bottom of the Ducts A and B. Duct A Duct B x Thermo- Thermo- Thermo- Thermo- Distance 7%: couples couples couples couples from the along along along along leading the the the the edge roofing bottom roofing bottom (inches) 6 A1 A13 B1 B14 9 12 A2 A14 B2 B15 181/16 18 A3 A15 Ba B16 271/8 24 A4 A15 B4 B17 363/16 30 A5 A17 85 B18 455/16 36 A6 A18 Be B19 595/16 42 A7 A19 B7 1320 631/. 48 A9 A20 Be 321 721/8 5" A9 A21 Be 322 811/16 60 A10 A22 B10 B23 901/15 66 A11 A23 B11 Bzu 99 72 A12 A24 B12 325 108 313 325 1171/8 Location of Thermocouples Used for Measuring Bulk Temperature of Air. Duct A Duct B Thermocouples Distance from Thermocouples Distance from the leading the leading edge edge A25 B27 H A26 72" B23 72% A27 B29 A29 B30 ’ n A29 96" 331 ngt A30 B32 84 Bulk Temperature of Air The bulk temperature of the air flowing in the ducts was measured at two points inside the ducts. For Duct A, the distances of 6' and 8' from the entrance were chosen. Equivalently these distances were 45 and 60 hydraulic diameters, and the flow here was expected to be fully developed. The corresponding distances for Duct B were 72-1/8" and 96-1/16", which are very nearly the same as in Duct A. The reason for the distances not being exactly 6' and 8' (as is the case of Duct A) is that the bulk temperatures were measured where the crest of the nearest corrugation occurred. For the measurement of bulk temperature at a given point inside the duct, three thermocouples were placed parallel to the flow of air. The thermocouples were mounted on a taut wire and successive thermocouples placed 1/4" apart. Aluminum foil sheets were used to shield the thermocouples from the effects of radiation. The thermo- couples and the aluminum foils alternated. The set-up is shown in Figure 9. The average of the readings of three thermocouples was recorded as the bulk temperature at the given point. The thermocouples were located in the middle of the longer side of the duct. All the thermocouples were connected to a Brown Electronik potentiometer (Model No. 153 x 65 P 12-x-2F) 7:57:31 EFFSIC' 7"! (2T. “3— ‘w--Ji-I:- in ‘2’, ’ ‘4 ' , " “V’- p 85 +++++~+ 1-{1‘1- ‘1): “I —‘l n .‘ -\.A_-oml m . ‘ 3 1} FIGURE 9.-—Location of Thermcouples for measuring bulk temperature of air. which recorded the temperature automatically. Temperature could be read accurately to 0.50F. Measurement of Air Leaving the Ducts Mercury-in—glass thermometers were used for measuring the temperature of the air leaving the ducts. The ther- mometers were inserted in the insulated exit chambers (Figure 7). Measurement of Humidity Ratio of Air Entering the Duct A sling psychrometer was used for measuring the dry bUlb and the wet bulb temperature of air. Both thermometers were mercury-in-glass type with a bare bulb diameter of 0.25". Both thermometers were unshielded. __ 1‘“. d 86 The temperature of the air entering the duct was measured by a mercury-in-glass thermometer. It was assumed that the entering air temperature is the same as that of the outside air. Measurement of Flow The venturi shown in Figure 7 was used to measure air flow. The venturi had a throat diameter of 2" and a pipe diameter of 4". Calibration for the venturi was taken from a thesis.* An inclined tube manometer was used to measure the drop of pressure in the converging section of the venturi. The manometer could accurately record pressure drops of up to 0.005 inches of water. An adjustable opening was used at the fan outlet to vary the flow. The following fans were used; 1. Aerovent Fan no. 124 Type DF Ser. 58351 Model SKC BB-4l D HP 1/2 RPM 3450. 2. Frame E.85 No. 1715000 A PH 1 ILG Electric Ventilating Co. The first fan (Aerovent No. 124) was used alone until the flow requirement exceeded its capacity. The second fan was then turned on so that the two together would supply the required volume of air. *Stephen Weller, "Air flow characteristics of a scale mOdel chamber," Michigan State University, Dept. of Ag. Eug., M.S., 1966. 87 Z .31 a: we I. - L. . Lew, .wobzp oLSmmoLd mo GOHmeoqll.oa .wflm khan“ been mm ea .%As mocmhpcm EOLM 2mg :u : : : QOCGDwHQ ~N ON mu awn h.— o o b o o ooze MOCML cm EOCH ..0m 2m: :mm ::N :Nfi :oH :m :n :o :m :: :m :uWN :WH :tm :wfl woflmpmflm w“ mm .7." ma N.— ua ofi m o h o m J m N H o o o o o b b b o o o b e b b b oboe mocmh cm EOCH Emmuzoo~ :mm :36 :NB 9 .H :mu moflmpwflfl and own and mum saw @958 $0.8m...” Cm SOL“ em e: on e~ Na m e s e m e m km W" b MP p m : : : : : : : : : : : : :~ :~ : m : H mofimpmHQ mu m +— m Nm 0 m 3 N m an so am w Aim am m on an em as m mm m in mode - m h m m a Ql/Q/ . // uN ON mu mm hm ma m~ a P p b p o o . o m. . W. 1 e 1 t d .1 .1 q illldHWAKRlii m BODQ NH Om a m +. N b o o o b o . a_m an em mm m. H // I/ F; ANN ONM.v mud. mud ham. mud arm .m . . - Hi 1 4 1 l t I 1| 1 EN: < BUDQ Nam cum mm mm :M Nd 88 Measurement of Pressure Drop of Air along the Duct Eighth-inch copper tubes were made flush with the plywood. The pressure tubes were located along the center lines of ducts A and B. The plan for the tubing is given in Figure 10. Rubber hoses connected the copper tubes to a pressure manifold (Figure 11), which in turn was connected to an inclined tube manometer. Pressure of air passing a given tube in the duct could be measured by releasing the pinch clip on the corresponding hose. COpper tubes,/~”' fl,” 1..4 Connected (connected :;\:M~ ' 71— I ' ' I II I to the to the l l I l I I l ‘ manometer pressure tubes in the ducts) Figure ll.—-Pressure manifold. 'The inclined tube manometer could record pressure correctly ‘to 0.005 inches of water. In the vertical position the mano- merter could read correctly to 0.025 inches of water. The madiometer could be used in an inclined position for measuring prflessure up to 1.2 inches of water. With the vertical DOESition, the range increased to six inches of water. 89 Experimental Procedure Data for Hydrodynamic Characteristics The data for hydrodynamic characteristics of the ducts were collected indoors. When Duct A was being tested the entrance to Duct B was closed. Later on Duct A would be closed to study Duct B. The tests were run under essentially isothermal conditions. The range of the Reynolds number studied was between 10,000 and 50,000. For each rate of flow (which was controlled by the adjustable opening at the fan outlet) pressure was recorded for the pressure tubes by the manometer. The following procedure was followed for both ducts. 1. Starting with Re = 10,000, the pressure drop was recorded as a function of the length of duct. Pressure recorded at a given point along the length was the Weighted average of three readings. This was affected to nquimize the error, particularly at low pressure drops. 2. The procedure was repeated for the next Reynolds Inunber under investigation (which was 15,000). The entire Paruge of Reynolds numbers (10,000 to 15,000) was thus COVnered. This constitutes a set of readings, which here- afteer will be referred to as Set I. This was followed by SGVan sets of readings, giving a total of eight sets. These setss of readings will hereafter be referred to as the Sets 1 tl'lrough 8. \x...mtu§m .11 mi“ gt‘...£ . 90 Data for Heat Transfer Characteristics of the Ducts The data for heat transfer characteristics of the ducts were collected on days when the sky was almost clear. This ensured ea steady incoming solar radiation which was necessary for valid test data. The trailer carrying the apparatus was rotated about every twenty minutes so that the ducts faced the sun at all times. Before any tests were conducted, the ducts were allowed to warm up for at least half an hour. No data were taken until all temperatures remained stabilized for at least 10 minutes. While the temperatures were approaching constant values, the air flow, incident solar energy, the dry and wet bulb temperatures, and the temperatures of air entering and leaving the duct were noted. The barometric pressure was noted once a day at noon; a mercury-in-glass barometer was used. For a particular Reynolds number the Ducts A and B would be opened one at a time, keeping the entry to the other one closed. After all temperatures had remained constant for ten minutes the thermocouple temperatures were recorded. Air flow was changed and the procedure repeated. CHAPTER V ANALYSIS, RESULTS AND DISCUSSION Analysis of Data Hydrodynamic Characteristics The pressure readings from the tubes are plotted against the duct length for each Reynolds number. This is done for both the ducts A and B. Figures 12 and 13 represent typical graphs showing pressure drop versus duct length for the ducts A and B. These graphs are used to determine intake length, intake losses and coefficients of friction, as will be explained under the section "Results and Discussion." Heat Transfer Characteristics The following procedure is used for both the ducts A and B for each Reynolds number under investigation. 1. The bulk temperature of air is plotted against the duct length. Three points are plotted on the graph: a. Temperature of air leaving the duct b. Temperature of air at 6 feet from the entrance for duct A and 72-1/2 inches for duct B. This 91 92 mm mm .< seam co gpmcma 00 mm 0m m: mm .m> gopc whammopmll.ma .mflm m x _ r: HH 0mm In ooo.ma H mm A I: a Boss 0 rl In 0 _ 0 _ _ _ _ _ _ _ _ _ 0 _ 00m.0 05m.0 0:m.0 0HN.0 00H.0 0ma.0 0mH.0 000.0 000.0 0m0.0 000.0 Jaqem JO saqou: doaq aanssaa 93 .m peso uo apmcoa .m> dopp whammopmll.MH ooo.om H pom "mm m 8000 0mw.0 msm.o 000.0 mmm.0 om:.o mum.o 00m.0 mmm.0 0mH.0 mwo.o good .mfim 5:2.77nn'r FIHJ'KT 2-7710023? Tn 9“ temperature is mean of three thermocouple readings. The thermocouples were placed parallel to the direction of flow. c. Temperature of air at 8 feet for duct A and 96-1/16 inches for duct B from the entrance. This temperature is the mean of three thermo— ~x couple readings Just as above. a 2. The temperatures of the top and bottom horizontal walls are plotted against the duct length. Figures 1“ and 15 represent typical temperature A versus length graphs for the two ducts. Analysis of Sling Psychrometer Data The following symbols will be used in this part of the analysis: w Humidity ratio of moist air. Lbs. water per lb. dry air. C Specific heat of moist air = 0.24 + 0.45W BTU per (lbs. dry air) (F). w Humidity ratio at saturation. Lbs. water per lb. dry air. Humidity ratio at saturation. Calculated at wet bulb temperature. Lbs. water per lb. dry air. t =t Dry bulb temperature-~FO. t Wet bulb temperature--FO. t Temperature of surfaces surrounding psychro- meter--FO. 95 .< Boga no npwemfi .m> ms was .mzs .H 3p mopzpmpooaoeln.:a .wfim am N mm 05 mm 00 mm 0m m: 0: mm 0m mm 0m ma 0H m _ _ _ _ _ 0 _ 0 a _ _ 0 _ _ O\\\\AW\\ II map Q\\\\\Q\\\\\ mp C O \\\ omm.m H mm .ll .0\ .\C\\\\\o\\\\\ H pmm H3 0 a song \.\_\i _ _ _ a; _ _ a _ i __ 0m 00 on 00 00 00H 30 aanqeaedma; 96 I‘ ALAN- Hfllfllé . .. . n _ . I. alliaqfi All .I .l‘ . like}? 0 .m3 .0 p050 mo Spwcoa .m> p and .Hz 9 p mmpzpmmeEoBII.mH .wflm go eanqeaadmeg mm M ms 0» mm 00 mm om m: 0: mm om mm om ma 0H m om _ _ L 0 _ _ _ _ _ _ _ _ _ _ I 1 mm I No l1 00 II as I 0m 0 l \\ I mm \\ II IIL mm H pmm o mm I 03$ H mm .1 o\O\ 3» I m 925 I i: _ _ _ _ _ _ _ _ _ r L _ _ P _ 1. Q: 97 Enthalpy of moist air = 0.2ut+W(106l+0.45t). BTU per lb. dry air. Volume of moist air entering the duct. Cu. ft. per minute. Barometric pressure--Psia or inches Hg. Partial pressure of water vapor; Psia or inches Hg. P for saturated water vapor at t; w,s P for saturated water vapor at tW ws,wb b' Enthalpy of saturated liquid water. BTU per lb. Enthalpy of saturated water vapor. BTU per lb. Latent heat of vaporization of water = hg-hf. BTU per lb; h evaluated at wet bulb fg,wb temperature. Convection heat transfer coefficient. BTU per (hr.) (sq. ft.) (F). Convection mass transfer coefficient. Lbs. water per (hr.) (Sq. ft.) (lb. water per lb. dry air). Lewis number--h/h --dimensionless. DCP,a Radiation heat transfer coefficient. BTU per (hr.) (sq. ft.) (F). Coefficient used in Equation (37)--dimensionless. Mass of moist air. lb./hr. Mass of dry air. lb./hr. xv.‘———x— — - -‘ o—.-'- 1.“.2'.‘ , n A x “x 98 k Thermal conductivity of air. BTU per (hr.) (F) (ft.). At Change in temperature of moist air—-FO. Ah Change in enthalpy of moist air=At(0.240+ 0.45W). BTU per lb. dry air. The dry and wet bulb temperatures recorded by the sling psychrometer are analyzed as follows: 1. Calculate WS wb from the following equation: 3 f P = s ws,wb ws,wb 0.622 P—f P (37) s ws,wb where the magnitudes of fS have been given by Goff (81). Its values are listed in Table 2. TABLE 2.--Va1ues of the Coefficient fs. __ 0 Temp. F f8 40 1.0044 50 1.0044 60 1.0044 70 1.0045 80 1.0047 90 1.0048 100 1.0050 {Hue values of Pws wb to be used in equation (37) are taken 3 frwom reference (82). 2. w was calculated from the following equation: ; w = wS wb - K (tdb — twb) (38) 99 where , LeC a hR K=h 22. 1+? (39) fg,wb assuming that tS = t 4 T 4 (1‘80) - (388') where ewb is the emissivity of wet bulb. For the flow of air normal to single wires or cylinders, McAdams (83) has suggested the following relations: 0.466 %? = 0.615 (Eva) for 40 < Re < 4.000 (41) 0.618 g? = 0.174 (9%9) 4000 < Re < 40,000 (42) where d is the bulb diameter and V the velocity of air. Threlkeld (84) has presented the ratio ER in the forms of h graphs. The graphs are based on equations (40), (41), and (42). A value of Ewb = 0.9 was used in equation (40). Figures (10.5), (10.6) and (10.8) of reference (84) h were used to determine Le and T?’ An air velocity of 800 ft. per min. was used. 100 Values of h were taken from reference (82). fg,wb 3. Mass of dry air flowing through the duct was calculated as follows: The volume of air flowing through the venturi was corrected for pressure drop and temeprature rise using perfect gas relations. This gave the volume of air cor- responding to the temperature and pressure at entrance to the duct. Let this volume be indicated by Vl Mass of V1 dry air = m8 = 77 where = 53-35 (460 + t)/(P - PW) the mass of dry air flowing through the duct being known, Reynolds number is given by: ma.(1+w) dh Re=—’_A—'—'T where A = 1/36 sq. ft. for either duct A or B, dh = 1.6 in. The prOperty values are inserted at mean bulk temper- 0.14 ature and then multiplied by (fig) where “B and “w are w evaluated at mean bulk and mean wall temperature respectively. 4. In fully developed flow %% = constant and on the graphs of temperature against duct length, this condition is indicated by the wall and bulk temperatures increasing linearly along the duct length. 101 The experiments showed that fully developed flow was attained in less than 3 feet from the duct entrance. The enthalpy increase for air in passing through a length 3 feet (between 5 feet and 8 feet from the entrance) for either duct was calculated as: maAh' = ma (t'-t") (0.24 + 0.45w) where t' and t” are the bulk air temperatures at two points located at distances 8 feet and 5 feet from the entrance. Since the width of the duct is 4 inches, a length of 3 feet along the duct will give an area of heated wall equal to 1 sq. ft. 5. The Nusselt number for heat transfer between the heated wall and air is then defined as: maAh' dh Nu = ——-——— - —— k tw -tB l where twl-tB is the mean of the temperature difference: t - t , in fully developed region. wl B Property values are inserted at mean bulk temperature and then corrected for variation in physical properties by 0.14 0 using a multiplication factor (32) . As before “B and “w W are evaluated at mean bulk and mean wall temperature respec- tively. 102 Results and Discussion Hydrodynamic Characteristics of the Ducts Air velocity, air flow and corresponding Reynolds numbers studied for each duct are given in Table 3. TABLE 3.--Air Flow (CFM) and Corresponding Reynolds Numbers. Um-—ft./sec. Q--CFM Re 12.084 20.14 10,000 18.126 30.21 15,000 24.168 40.28 20,000 30.210 50.35 25,000 36.252 60.42 30,000 48.336 80.56 40,000 60.420 100.70 50,000 Under each section, the results for duct A will be presented first. This will be followed by the results for duct B. Intake Length.--As explained earlier, the flow in a duct attains the fully develOped velocity profile asymp- totically. In the present thesis the intake length will be defined as that length where pressure gradient attains a constant value, commensurate with the accuracy of the measuring instruments. The graphs showing pressure against duct length for each Reynold number are used to locate the duct cross section where the pressure gradient assumes a constant value. The g I ¢,——.— n-mu— —— y-‘n—l—u- ~ . 1 k-q ‘ 103 distance of this cross section from the entrance of the duct is, by definition, the intake length. The results for the intake length are presented in Table 4. Results for Latzko (14) are also tabulated for comparison. £3 is plotted against Reynolds number in Figure 16. (:1h The present analysis approximates the intake length by the formula: Le 0.166 (43) dh = 1.022 Re The present results deviate from Latzko's results by as low as 2.3 per cent and as high as 11.7 per cent in the range of Reynolds numbers studied. This deviation as noticed in Figure 16 increases with a rise in the magnitude of Reynolds number. Latzko (14) assumed that the boundary layer is turbulent starting at the entrance. The tube was further assumed to be smooth. In the present analysis, the presence of laminar boundary layer was observed for a very short distance of about two hydraulic diameters at lower Reynolds numbers. This distance became exceedingly small reducing to less than one hydraulic diameter at higher Reynolds numbers. It may safely be assumed that at higher Reynolds numbers the boundary layer is turbulent right at the entrance. In other Words, at higher Reynolds numbers the duct A fulfills the Intake Length Le/dh 10 104 I 1 T LATZKO Present Analysis 0 Present Analysis I l l -w 2 3 4 Re x 10_“ Fig. 16.-—The intake length of Duct A. U1 vmv‘m 1' A'E'M'.f.maé muiwarm' Q -. ' 105 TABLE 4.--Intake Length for Duct A. Le/dh Re Latzko* Set Set Set Set‘ Set Set Set Set I II III IV V VI VII VIII 10,000 6.45 6.40 6.50 6.45 6.40 6.50 6.45 6.40 6.23 15,000 6.55 6.90 6.95 6.95 6.90 7.00 6.90 6.60 6.92 20,000 7.25 7.25 7.50 7.20 7.25 7.25 6.90 6.95 7.48 25,000 7.50 7.45 7.55 7.45 7.45 7.50 7.70 7.50 7.92 30,000 7.75 7.70 7.75 7.70 7.75 7.75 7.70 7.70 8.28 40,000 8.20 8.25 8.30 8.30 8.25 8.25 8.00 8.30 8.92 50,000 8.40 8.40 8.35 8.35 8.40 8.40 8.60 8.35 9.40 *Latzko awn—Taf— = 0.623O°25 . h TABLE 5.--Intake Length for Duct B. Le/dh Re Set Set Set Set Set Set Set Set I II III IV V VI VII VIII 10,000 9.05 9.30 9.25 9.30 10.05 9.30 9.30 9.25 15,000 10.05 10.00 10.05 9.60 10.25 10.50 10.00 10.05 20,000 11.00 11.10 11.05 11.15 11.00 10.25 11.00 10.75 25,000 11.50 11.55 11.55 11.50 11.45 12.50 11.45 11.55 30,000 12.10 12.05 12.05 12.20 12.25 12.20 12.05 12.10 40,000 12.80 13.35 12.00 12.75 12.70 12.65 13.00 13.00 50,000 13.75 13.55 13.70 13.75 13.50 13.45 13.75 13.50 106 assumptions made by Latzko (14) except that it is rough and not hydraulically smooth. The additional roughness causes the turbulent boundary layer to develop faster, thereby resulting in a shorter intake length than what will be predicted by Latzko (14). At lower Reynolds number, on the other hand, the presence of laminar boundary layer results in a longer intake length than in the case of Latzko's analysis (14). It is because turbulent layer develops faster than the laminar one. The increase in the intake length due to the presence of laminar layer, it seems, is not offset by the roughness of the tube which causes the turbulent layer to develop faster. The corresponding values of intake length (Le/dh) for duct B are given in Table 5. The results are graphically presented in Figure 17. The intake length (Le/dh) is approximated by the relation: Le 0 24 —— = 0.918 Re ' <44) dh Friction Coefficient for Fully Developed Flow For a noncircular cross section, the fully developed friction coefficient is given by: dP -dh ( ~—) f = _____§£_ (05) Intake Length Le/dh 30 20 CD 107 _. 0 Present Analysis -1 - '1 L. .— I I I I 1 2 3 4 5 Fig. l7.--The intake length of Duct B. ' . MW: ‘2}! 108 For both ducts A and B, the hydraulic diameter is given by: ——— = 1.6" Pressure gradients for various Reynolds numbers are determined from the graphs of pressure against duct length. The resulting values of f as calculated from Equation (45) are given in Table 6. These 8 sets of values are graphically presented in Figure 18. Friction factors for a smooth tube as given by Blasius Law are also plotted. The coefficient of friction for the duct A is seen to depend on both Reynolds number and the relative roughness of the duct. Comparison with Nikuradse's (24) and Moody's (26) charts shows that the duct A very closely follows the pattern of rough pipes in the transition regime < 70 i.e., the protrusions extend partly outside the laminar sub- layer and additional resistance is introduced by the form drag experienced by the protrusions in the boundary layer. The type of roughness which is experienced by the duct by having a galvanized steel roof, two pieces of lumber as the vertical sides and plywood as the bottom, may be (as mentioned in Chapter II) classified as being of the second . 2..» A . fl. .. 109 a. 1:25.111 1.0.8? m©m0.0 0%m0.0 mwm0.0 0mm0.0 0000.0 mmm0.0 Nom0-0 mwm0.0 000n0m 2000.0 2®m0.0 me0.0 mmm0.0 20m0.0 H©m0.0 me0.0 0wm0.0 000n02 mwm0.0 NNm0.0 me0.0 2500.0 05m0.0 20m0.0 Nwm0.0 ©2m0.0 000n0m mwm0.0 20m0.0 2200.0 0500.0 N0m0.0 Nwmo.0 w©m0.0 2Nm0.0 000amm mbm0.0 2Nm0.0 0Nm0.0 w©m0.0 0000.0 mmm0.0 05m0.0 0wm0.0 000n0m 2Nm0.0 Nmm0.0 Nwmo.o H0m0.0 N000.0 0wm0.0 NNm0.0 mwm0.0 000an 5000.0 @200.0 wm00.0 Nm©0.0 2000.0 2000.0 H000.0 0m©0.0 000n0H HHH> HH> H> > >H HHH HH H pmm pmm pwm umm umm pmm pmm pmm M .m poem sou u ceasefism no pcmfioanmmoouu.w mqm HH> H> > >H HHH HH H pmm pmm 00m 00m 00m pom 00m 00m mm ..< 0.050 .HOrH COHUOHLMRH wo pCmHOHwM®00II.0 mqm<9 Coefficient of fraction f .08 .07 .06 .05 .04 .03 .02 .01 110 9““--__..£L_ a o DUCT A Q 10 v s w 0— u Blasius Law for smooth ducts f = 0.316/(Re)% Fig. 2 _ 3 4 Re x 10 “ 18.—-Coefficient of friction for Duct A. 111 type. In this kind of roughness, the friction coefficient depends on both the Reynolds number and the relative roughness. The corresponding values of f for duct B are given in Table 7. The results are graphically presented in Figure 19. Friction coefficient for hydraulically smooth pipe based on Blasius Law are also given in the Figure for comparison. The coefficient of friction for duct B, too, is seen to be dependent on both the Reynolds number and the relative roughness in the range of Reynolds number: 10,000 to 50,000. Duct B also lies in the transition regime of roughness. The roughness of a rectangular duct with a corrugated top, a plywood bottom and vertical walls made of lumber, may be (as mentioned in Chapter II) classified as being of the second type. The friction coefficients for the two ducts are compared in Table 8. The values of f are taken from Figures 18 and 19. Entrance and Intake Region Losses As explained earlier, frictional pressure drop in a duct is calculated..as if fully developed flow existed over the entire length L, and a second term is added which accounts for the increase in pressure drop due to entrance and intake region. The resulting equation for pressure drop is: Coefficient of fraction f Fig. l9.——Coefficient of friction of Duct B. 112 .08 1 1 .07 T" -" DUCT B .06 }' O & p, Q, Q —E v r W 0 c O 0 05 _' b— .04 r— —‘ 03 —- ‘” Blasius Law .024— — .01 I 1 2 _ 4 Re x 10 “ Eh ‘I. 113 TABLE 8.--Comparison of Values of f for Ducts A and B. Reynolds Smooth Duct Duct Per cent Per cent Per cent number duct A B increase increase increase Blasius in f of in f of in f of Law duct a duct B duct B over over over smooth smooth duct A duct duct 10,000 0.0316 0.0440 0.0650 39.23 105.69 47.72 15,000 0.0285 0.0420 0.0585 47.36 105.36 39.28 20,000 0.0265 0.0414 0.0573 56.22 116.22 38.40 25,000 0.0250 0.0412 0.0573 64.80 129.20 39.07 30,000 0.0240 0.0410 0.0573 70.83 138.75 39.75 40,000 0.0223 0.0410 0.0573 83.85 156.95 39.75 50,000 0.0211 0.0410 0.0573 94.31 171.56 39.75 TABLE 9.--Entrance Loss Factor (Kl + K2). Reynolds a number Factor (Kl + K2) 10,000 0.924 15,000 0.887 20,000 0.885 25,000 0.874 30,000 0.898 40,000 0.885 50,000 0.850 aWeighted average of all 8 sets of readings. Average value of K1 + K2 = 0.8861. where P i o duct and P * Also K = K The losses: K1 In t found from The between th by the pre The factor * K where K 114 02 U2 * P-P=f—--p—2fl+1 For an asymmetrically heated channel, with one wall essentially adiabatic, the relationship reduces to: _ qw h _ general (Ew::EB) + (55;:50) where the bar refers tO‘URamean values of the temperature difference. The general heat transfer coefficient suggested by Madsen (54) is for smooth ducts. An attempt was made in this thesis to see if Madsen's Rule will carry over to the rough ducts. A certain modi- fication is obviously necessary and the following procedure was used. The heat transfer coefficients for symmetrically heated smooth channels obtained from different analyses were multiplied by the factor /%L . o This gives a modified h for rough ducts. Let general * it be denoted as hgeneral' it The value of hgeneral as found from experiments for the rough ducts can be defined as: 128 I * ma.Ah h _ general ———:— _ (tW tB) + (tw t ) 1 2 B x The experimental and theoretical values of hgeneral (after changing them to corresponding Nusselt numbers) are plotted in Figures 23 and 24 for ducts A and B respectively. The numbers in Figures 23 and 24 refer to the references from which analyses for smooth symmetrically heated channels are taken. Nu 129 300 T 1 l | I 2T DUCT A 200 _ Eqn. (36) Present 6 Analysis 100 - 90 - Eqn. (21) 8O Present 70 Analysis 60 50 00 d Experimental values 9 Ref. (85) 1 3O _ [:1 Ref. (45) 0-8 0.4 f 0.023 Re Pr \f/fo All these are shown as 20 _. discreet points. No curves were drawn through them. 10 I I I I I I 1 2 3 4 5 6 7 Re x 10-” Fig. 23.—-Verification of "modified" Madsen's Rule for Duct A. 130 30 I I r I II 20 L Equation (36) Present Analysis 6 d 10 b d 9 — 8 P 7 ‘ Equation (21) 6 Present Analysis Nu x 10‘1 d 5 4 __ d Experimental Results A Ref (85) 3 _ D REf (145) O 0 023 Re°°8 Pr°'“ (890 2 _. All these are shown as discreet points. No curves drawn through them. 1 I I I I I l 2 3 4 5 6 7 Re x 10-“ Fig. 24.—-Verification of "modified" Madsen's Rule for Duct B. CHAPTER VI CONCLUSIONS Two dimensional forced convection for a constant property, incompressible fluid in an unsymmetrically heated rough duct was studied in this thesis. The effects of conduction and radiation were neglected. For the range of the Reynolds numbers studied, the findings may be summarized as follows: 1. In fully developed turbulent flow, the friction coefficient for duct B is about 40 per cent higher than that for duct A. 2. The friction coefficient for both ducts is a function of the Reynolds number and relative roughness. 3. Duct B has a longer intake length, the boundary layer developing on one side only. The flow was fully developed in less than 15 hydraulic diameters. 4. The value of the factor K used in determining the inlet losses roughly equals unityfkuiboth the ducts. 5. The absolute roughness of duct A is about 0.011 and that of duct B equals 0.029. 6. The convection heat transfer coefficient for an unsymmetrically heated duct is less than that of the symmetrically heated duct. The analytical expressions deve10ped in this thesis for unsymmetrically heated smooth channel apply satisfactorily to the case of an unsymmetri- cally heated rough duct when heat transfer coefficient is multiplied by g; . O 131 132 7. Hydraulic diameter and Reynolds number based on inlet cross section of the corrugated-top—duct give satisfactory correlation. 8. The generalized heat transfer coefficient sug- gested by Madsen for smooth tubes, carried over to the rough ducts studied in this thesis. The heat ransfer coefficient, however, had to be corrected according to item 6 given above. 9. An obvious increase in heat transfer coefficient results when rough duct replaces a hydraulically smooth one. However, this increase has to be paid for by an increase in pressure drop; whereas the heat transfer coefficient is proportional to g; , the pressure varies o f as the ratio —— . f0 10. The thermal entrance length appears to be less than 22 hydraulic diameters. Because of the asymptotic nature of the thermal development and insufficient points in the entrance region, it was not possible to assign a more precise value to the thermal entrance length. 11. In the analytic expressions developed in this thesis, equality of Em and sq for all y and Re was assumed. 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