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## Material Information- Title:
- Time-dependent enhanced heat transfer in oscillating pipe flow
- Creator:
- Zhang, Guo-Jie, 1941-
- Publication Date:
- 1988
- Language:
- English
- Physical Description:
- xiii, 162 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Boundary conditions ( jstor )
Heat ( jstor ) Heat flux ( jstor ) Heat transfer ( jstor ) Pipe flow ( jstor ) Pumping ( jstor ) Subroutines ( jstor ) Temperature distribution ( jstor ) Temperature gradients ( jstor ) Velocity ( jstor ) Heat -- Transmission ( lcsh ) Laminar flow ( lcsh ) Pipe -- Fluid dynamics ( lcsh ) Viscous flow ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1988.
- Bibliography:
- Includes bibliographical references.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Guo-Jie Zhang.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 024469817 ( ALEPH )
AFK9835 ( NOTIS ) 19897631 ( OCLC ) AA00004834_00001 ( sobekcm )
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TIME-DEPENDENT ENHANCED HEAT TRANSFER IN OSCILLATING PIPE FLOW By GUO-JIE ZHANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 TIME-DEPENDENT ENHANCED HEAT TRANSFER IN OSCILLATING PIPE FLOW By GUO-JIE ZHANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 To my beloved motherland ACKNOWLEDGEMENTS The author wishes to express his deep gratitude to the chairman of his committee, Dr. Ulrich H. Kurzweg, for the valuable assistance and advice in guiding this research work to its completion and for thoroughly reviewing the entire manuscript leading to the realization of this dissertation. Also he wishes to express his appreciation to Drs. E. Rune Lindgren, Lawrence E. Malvern, Arun K. Varma and David W. Mikolaitis for the many helpful discussions in the formulation of the problem and constructive suggestions for overcoming many difficulties in the solution process. Thanks are expressed here also to Dean Eugene R. Chenette, to the department chairman, Dr. Martin A. Eisenberg, and to Dr. Charles E. Taylor for their support in allowing the author to pursue his educational goals within this lovely country. Part of the work presented here was funded by a grant from the National Science Foundation, under contract number CBT-8611254. This support is gratefully acknowledged. iii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii LIST OF FIGURES vi LIST OF TABLES ix KEY TO SYMBOLS X ABSTRACT xii CHAPTERS I INTRODUCTION 1 II FORMULATION OF THE PROBLEM 18 Governing Equations 18 Boundary Conditions 24 Model 1 24 Model 2 28 Model 3 30 Initial Conditions 33 Calculation of Tidal Displacement 35 Effective Heat Flux 37 III NUMERICAL TECHNIQUES EMPLOYED 38 Transformation 40 Crank-Nicolson Method for Momentum Equation . 46 ADI Method for Axisymmetric Heat Equations. . 48 Convergence Criteria 57 Grid Generation 59 IV NUMERICAL RESULTS AND DISCUSSION 62 Part 1. Oscillatory Pipe Flow Features .... 64 Velocity Profiles 64 Lagrangian Displacements 69 Tidal Displacements 72 Phase Lags 79 Part 2. Enhanced Heat Transfer Investigation 85 Periodic Temperature Build-up in Thermal Pumping Process 87 IV Temperature Distribution in Model 1 95 Temperature Distribution in Model 2 100 Temperature Distribution in Model 3 108 Heat Flux versus Tidal Displacement (Model 2) Ill Influence of Thermodynamic Properties .... 115 Heat Flux versus Tidal Displacement (Model 1) 119 Influence of Wall Thickness 124 Influence of Pipe Diameter 126 Variation of Axial Temperature Gradient In Model 3 130 Comparison of Enhanced Oscillatory Heat Transfer and Heat Conduction 133 Enhanced Feat Flux as a Function of Wormersley Number 135 Tuning Curves 137 V CONCLUDING REMARKS 143 APPENDIX: ETP COMPUTER CODE 149 REFERENCES 157 BIOGRAPHICAL SKETCH 161 V LIST OF FIGURES Figure Page 1-1 F (a) Curve 7 2-1 Thermal Pumping Device 19 2-2 Model 1 Fixed End Temperature Model 25 2-3 Model 2 Periodic Heat and Cold Sources on Insulated Wall 29 2-4 Model 3 Pipe with Extended Conducting Sections 31 3-1 Grid System Used in the Numerical Simulations. 39 3-2 Coordinate Transformation 42 4-1 1-D Velocity Profiles in Oscillating Flow for Wormersley Number a = 1, 10, 100, and 1000 . 65 4-2 Velocity Profiles (a = 5, after Uchida) .... 66 4-3 Magnified View of Velocity Profile near Wall . 67 4-4 Lagrangian Displacement for a =0.1, 1.0 ... 70 4-5 Lagrangian Displacement at a = 10 71 4-6 Dimensionless Cross-Section Averaged Displacement versus Time (a = 0.1 1.0) . 73 4-7 Dimensionless Cross-Section Averaged Displacement versus Time (a = 2 50) 73 4-8 Relationship Between Dimensionless Tidal Displacement AX and Wormersley Number a ... 77 4-9 Relationship Between Tidal Displacement Ax and Exciting Pressure Gradient in Water 78 4-10 Phase Variation Along Radius for Different Wormersley Numbers 84 vi 4-11 Temperature Build-up Process in Oscillating Flow (Model 1, a = 1, Ax = 2cm) 89 4-12 Temperature Build-up Process in Oscillating Flow (Model 1, a = 1, Ax = 5cm) 90 4-13 Temperature Build-up Process in Oscillating Flow (Model 1, a = 1, Ax = 10cm) 91 4-14 Temperature Build-up Process in Steady Flow (Model 2, Uave = 1.5 cm/sec) 93 4-15 Build-up Time versus Tidal Displacement (Model 2, a = 1) 94 4-16 Temperature Distribution in Oscillating Pipe Flow (Model 1, a = 1, Ax = 1cm) 96 4-17 Temperature Distribution in Oscillating Pipe Flow (Model 1, a = 1, Ax = 2cm) 97 4-18 Temperature Distribution in Oscillating Pipe Flow (Model 1, a = 1, Ax = 5cm) 98 4-19 Temperature Distribution in Oscillating Pipe Flow (Model 2, a = 1, Ax = 1cm) 101 4-20 Temperature Distribution in Oscillating Pipe Flow (Model 2, a = 1, Ax = 5cm) 102 4-21 Temperature Distribution in Oscillating Pipe Flow (Model 2, a = 1, Ax = 10cm) 103 4-22 Temperature Distribution in Oscillating Pipe Flow (Model 2, a = 1, Ax = 20cm) 104 4-23 Temperature Distribution in Oscillating Pipe Flow (Model 1, a = 1, Ax = 30cm) 105 4-24 Temperature Distribution in Steady Flow (Model 2, Uave = 0.5 7.5 cm/sec) 106 4-25 Temperature Distribution in Oscillating Pipe Flow (Model 3, a = 1, AX = 10cm) 109 4-26 Magnified View of Temperature in the Central Pipe Section (Model 3, a = 1, Ax = 10cm) . 110 4-27 Heat Flux in Oscillating Flow and Steady Flow (Model 2, a = 1, Water as Working Fluid). . 112 vii 117 4-28 Influence of Thermodynamic Properties of H2O on the Enhanced Heat Flux (Model 2, a = 1, Ax = 10 cm) 4-29 Heat Flux versus Tidal Displacement (Model 1, a = 1, Pr = 7.03) 120 4-30 Heat Flux versus Tidal Displacement (Model 1, a = 3, Pr = 7.03) 121 4-31 Influence of Wall Thickness on Axial Heat Flux (Model 1, Water-Glass, a = 1, Ax = 5cm).... 126 4-32 Influence of Pipe Diameter on Heat Flux for Fixed Frequency (Model 3, Water-glass, a = 3, Ax = 10cm) 128 4-33 Typical Iso-Temperature Contour in Oscillating Pipe Flow (Model 3, Water-Glass, a = 3, Ax = 10cm) 129 4-34 Variation of Temperature T^ and T2 versus Ax (Model 3, Water-Glass, a = 3) 131 4-35 Comparison of Enhanced Heat Transfer and Heat Conduction in Oscillating Pipe Flow (Model 3, Water-Glass, a = 3) 134 4-36 Variation of Axial Heat Flux versus Wormersley Number (Model 3, HoO-Glass, Hg-Steel, Ax = 10cm) 136 4-37 Computed Tuning Curves (Model 3, H20-Steel and Hg-Steel, ax = 10cm) 138 4-38 Tuning Curve versus Wormersley Number (after Kurzweg) 140 4-39 Ratio of Heat Flux due to Conduction to Enhanced Heat Flux versus Wormersley Number (Model 3, H20-Steel, Hg-Steel, Ax = 10cm) . 141 viii LIST OF TABLES Tables Pages 4-1 Dimensionless Tidal Displacement at Different Wormersley Numbers 72 4-2 Phase Lags Along Radius (Working Medium: H2O, AX = 10cm, a 0.1 20) 80 4-3 Comparison of Phase Lags with Different Working Mediums (Ax = 10cm, a = 1, 5) 83 4-4 Comparison of Enhanced Heat Flux Using Numerical Velocity with Heat Flux Using Analytic Velocity (Model 3, H20-Glass, Rl = 0.1cm, R2 = 0.15cm, Pr = 7.03, a = 3) . 86 4-5 Enhanced Axial Heat Flux via Tidal Displacement. 113 4-6 Enhanced Axial Heat Flux in Steady Flow .... 114 4-7 The Influence of Properties of Water on the Enhanced Axial Heat Flux 116 4-8 Variation of the Axial Temperature Gradient versus Wormersley Numbers (Water-Glass, Ax = 10 cm) 132 ix KEY TO SYMBOLS X r t Â£ *? T L R1 r2 w S P P c P K k Ke Pr x coordinate radial coordinate time coordinate (x) in transformed plane coordinate (r) in transformed plane transformed time pipe length pipe inner radius pipe outer radius oscillating frequency boundary layer thickness pressure density specific heat Kinematic viscosity dynamic viscosity thermal diffusivity thermal conductivity coefficient of enhanced heat diffusivity Prandtl number i/p 13p/3x| a measure of the maximum axial pressure gradient (cm/sec2) x a Wormersley number a = Ju>/v T Temperature 7 7 = dT/dx time-averaged axial temperature gradient r f = r/Ri dimensionless radial distance g radial temperature distribution function u velocity u0 representative velocity f velocity shape function X Lagrangian displacement DX dimensionless tidal displacement AX dimensional tidal displacement Qtotal time averaged total enhanced axial heat flow over pipe cross-section axial heat flux Subscript f fluid w wall h hot c cold th thermal eq equivalent adj adjacent XX Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TIME-DEPENDENT ENHANCED HEAT TRANSFER IN OSCILLATING PIPE FLOW By Guo-jie Zhang April 1988 Chairman: Ulrich H. Kurzweg Major Department: Aerospace Engineering, Mechanics, and Engineering Science The problem of time-dependent enhanced heat transfer in an incompressible viscous laminar fluid subjected to sinusoidal oscillations in circular pipes which are connected to a hot reservoir at one end and a cold reservoir at the other end has been examined numerically in detail. Three models were designed for the investigation of such an enhanced thermal pumping process and a computer code (ETP) was developed to implement all the numerical calculations. To increase the understanding of the mechanism of thermal pumping, the periodic velocity profiles and Lagrangian displacements as well as tidal displacements at various Wormersley numbers (from a = 0.1 to 1000) were studied. Some transient problems of enhanced axial heat transfer in oscillating pipe flow such as the periodic final temperature build-up process in oscillating pipe flow were also examined. The time-dependent temperature distribution xii in the different models was numerically studied in detail. The enhanced axial heat flux magnitude versus different tidal displacements with water and mercury as the working fluids bounded by pipe walls of different material were observed and the quadratic coefficients found. The influence of the variation of water properties on the enhanced axial heat flux was numerically examined and the results show that the enhanced axial heat flux can vary about 150 percent even within the temperature range from 0C to 100C. The effects of wall thickness and pipe diameter in enhanced thermal pumping were also studied and the optimum wall thickness was found to be about 20 percent of the pipe radius in the water-glass combination. The tuning effect in the water-steel and the mercury-steel cases was examined and the results show good agreement with analytic predictions. A comparison of the enhanced axial heat flux with the axial heat flow due to heat conduction at various tidal displacement and Wormersley numbers shows that the latter is quite small and negligible provided the tuning condition is satisfied. This study has shown that the enhanced thermal pump is indeed a very effective tool for those problems where large amounts of heat must be transported without an accompanying convective mass exchange. The investigation also indicates that turbulent flow in the reservoirs is preferable to laminar conditions and should receive more attention in future studies. xiii CHAPTER I INTRODUCTION Enhanced heat transport in a viscous laminar fluid subjected to sinusoidal oscillations in a very long pipe which connects a hot fluid reservoir at one end and a cold fluid reservoir at the other end (Fig. 2-1) has been recognized and studied recently by Kurzweg [15, 16, 17, 22], The results obtained show that with this oscillatory pipe flow the heat transferred axially from the hot end to the cold end can be orders of magnitude larger than that obtained by pure molecular conduction in the absence of oscillations. In addition, the more important thing of interest is that this heat transfer process involves no net convective mass transport. Major assumptions made in the above cited studies on enhanced heat diffusion are that a constant time-averaged non-zero axial temperature gradient is always present in the oscillating flow and that the axial molecular conduction along the wall and in the oscillating fluid is negligible. Discovery of this enhanced heat transport phenomenon was made possible by earlier studies on axial dispersion of contaminants within steady laminar flows through capillary tubes by Taylor in 1953 [32], and Aris in 1956 [4]. These 1 2 earlier studies show that when a small quantity of a contaminant is introduced into a circular pipe, the dispersion of the resultant contaminant cloud is greatly enhanced by the flow of the fluid. Bowden's 1965 results show that similar dispersion effects occur in oscillatory flow [8], This enhanced axial dispersion of contaminants in the presence of laminar oscillatory flow within capillary tubes was studied in 1975 by Chatwin who suggested that the assumption of constant time-averaged axial contaminant gradient can be made [10]. Recently, Bohn et al. extended this work to the study of gas component transfer in binary gas mixtures when these are confined to single tubes and a sinusoidal pressure variation is applied [7]. Further studies in 1983 by Watson [38] show that the effective diffusion of contaminants is proportional to the square of the tidal displacement. This has been experimentally verified by Joshi et al. [13] and by Jaeger [12], both in 1983, and most recently by Kurzweg and Jaeger [19], in 1987. All these results show that the contaminant would spread axially in both steady and oscillatory laminar pipe flow at rates as much as five orders of magnitude higher than in the absence of fluid motion. The first significant research work extending these enhanced axially diffusion studies to the heat transfer problem in oscillatory flow within very slender pipes or flat plate channels is due to Kurzweg [15, 16, 22]. In 3 early 1983 Kurzweg suggested that a similar dispersion process should occur in the heat transfer area because of the similarity in both the governing diffusion and heat transfer equations [16], and the first preliminary theory was formulated in 1985 [15, 16], in which, referring to Chatwin's idea, a time averaged constant axial temperature gradient assumption was used. The instantaneous temperature distribution was taken to be of the form [15, 22]. T = 7[x + RX g(r)elwt] (1-1) where 7 = 3T/3x is the time-averaged axial temperature gradient, is the tube radius, x is the axial distance along the capillaries (with x = L/2 and x = -L/2 denoting the ends) L is the pipe length under consideration, w is the oscillating frequency of fluid, g(f) is a radial temperature distribution, and f = r/Ri is the dimensionless radial distance. The theoretical analysis shows that under certain conditions such enhanced axial heat diffusivity can indeed be significantly larger than the axial molecular conduction [15, 16] and this has been verified by some experimental measurements by Kurzweg and Zhao [22]. In order to better understand the physical mechanisms of this interesting and potentially useful heat transfer technique, we shall examine in greater detail the thermal pumping model shown in Fig. 2-1. It is assumed that a bundle of very thin and long tubes is connected to a 4 reservoir which supplies unlimited hot liquid at one end to a second reservoir which supplies unlimited cold liquid at the other end. The liquid in the pipes is oscillating axially with an amplitude such that none of the liquid which is originally in the middle portion of pipes ever runs into either reservoir. That is, there is no net convective mass exchange between two reservoirs. The largest axial fluid dimensionless displacement (when cross-section averaged) is referred to as the nondimensional tidal displacement and is denoted by "AX" (it should not be confused with the dimensional tidal displacement "ax" frequently used in the present study) At time t = 0, the fluid within the pipes is set into axial oscillations at angular frequency w and tidal displacement Ax. After a short transient, this oscillatory motion will lead to very large axial heat flows which can be readily made to exceed those possible with heat pipes. Before exploring the mechanisms of this enhanced heat transport, it is necessary to introduce some new concepts which are commonly used in the study of this type of oscillatory motion. As is well known [36], for high frequency viscous laminar axial oscillations within fluid flow along rigid pipes, the non-slip boundary condition creates a very thin Stokes' viscous boundary layer of thickness S = J2u/o) (1-2) 5 where u is the fluid kinematic viscosity. For room temperature water at a frequency of 10 Hz, this viscous boundary layer is approximately 1.7*102 cm. The corresponding thermal boundary layer thickness is about fith = t/fPr (1-3) where Pr is the Prandtl number. Note that both 5 and 5th decrease in thickness as the oscillating frequency w increases. In the theoretical analysis, it is always assumed that a fully developed velocity profile of the oscillating flow exists within the pipes. At high frequency w, this flow consists essentially of a slug flow over most of the fluid core bounded by a thin boundary layer of width 8 as discussed by Uchida [36], Neglecting end effects, the fully developed oscillating laminar velocity profile in pipes, due to a periodic axial pressure gradient, is found to be [19] U(T ,t) = U0f(r)elwt (1-4) where UQ is a representative velocity, f = r/R^ is again the dimensionless radial distance, f(f) the velocity shape function, and u> the angular velocity of the oscillatory flow. The explicit form of f(f) is 1 - f(r) Jn(7-ir) I j0(/^y J (1-5) 6 where A = Ri213p/5x|/pi/U0 is the nondimensional pressure gradient maximum acting along the capillaries, a = R\J~Jv is the Wormersley number measuring the ratio of inertia to viscous forces, v is the fluid kinematic viscosity of the carrier liguid, and p is the fluid density. This velocity profile will reduce to the familiar Poiseuille parabolic shape as the angular frequency w becomes small, while at moderate frequency, f(r) has the shape demonstrated by Uchida [36]. Another new term commonly used when dealing with oscillating flow is the cross-stream averaged dimensionless tidal displacement AX which can be mathematically defined as AX = 4U0 I Jff (r)dr (1-6) and on integration, yields [18] AX I a p/ax I 1/2 p 1 + 2 a F(a) (1-7) where the complex function F(a) has the form F() . Jn'(Z^) 1 Jo(/=!Â£) with the prime denoting differentiation. Using the definition of Kelvin function: JQ(/-ia) = ber a + i bei a Fig 1-1 F (a) Curve [18] 8 [1], the complex function F(a) (Fig. 1-1) can be further written as: F(a) = Fr (a) + i Fi(a) ber 'a + i bei'a ] ber a + i bei a J (1-8) This dimensionless tidal displacement is related to the maximum of the periodic pressure gradient via [18] AX I 3p/ax| 1/2 1 J2 1- a a 1 (1-9) a 1 Apparently, for any fixed tidal displacement Ax and oscillating frequency w, the axial pressure gradient required is proportional to the inverse square of the Wormersley number when a is small; however, it is independent of a when a is very large. This also implies that the exciting axial pressure gradient |ap/ax| is approximately proportional to the fluid kinematic viscosity v and inversely proportional to the square of the pipe radius when Wormersley number is very small (it happens only at low , small and large v, for example, oil), while it is almost independent of the fluid kinematic viscosity and the pipe radius when the Wormersley is very large (it happens only at very large R^, high w, small v, 9 for instance, a liquid metal). Note, if the tidal displacement AX is fixed, while allowing the oscillating frequency to change, the axial pressure gradient |<3p/dx| can become very large when the oscillating frequency becomes large. This is mainly due to inertial effects and not so much due to viscous drag forces which dominate the oscillatory flow at small Wormersley number. With the definition of the above quantities, we are now in a position to explore the details of the enhanced axial heat transfer in oscillatory flow within pipes. It is assumed that a constant temperature gradient exists along the pipe in the axial direction and that a very large time- dependent radial temperature gradient variation is superimposed. When the fluid moves towards the cooler side (we term this the positive stroke), the hotter fluid within the pipe core which is initially brought into the pipe from the hot reservoir produces a large radial heat flow via conduction to the cooler portions of the fluid within the Stokes' boundary layer and to the cooler pipe wall; while during the negative (or reverse) stroke, i.e., when the fluid moves towards the hotter side, the higher temperature in the boundary layer and the pipe wall conducts the heat back into the cooler fluid core. This coupled radial heat conduction with an axial convective transport leads to an enhanced axial heat flux along the entire length of the pipes. 10 Further, from the system point of view, the heated fluid near the cold reservoir will eventually be ejected into the cold reservoir and mixed there with the lower temperature liquid. Contrarily, near the hot reservoir side the fluid within the pipes which has been cooled during each positive stroke is pushed out into the hot reservoir and mixes with the higher temperature liquid. This process of thermal pumping is what leads to a time-averaged heat flow from the hot reservoir into the cold reservoir. It differs essentially from the working principle of a normal pump. For a normal pump one can draw the analogy with transport of a one-way vehicle which transports passengers as well as the carrier from one point to another. For the thermal pump one can draw the analogy with a two-way busline which periodically loads and unloads the passengers (heat) from the hot reservoir to the cold reservoir, heat can be continually transferred, and the carrier, in the time- averaged sense, does not move. This property is particularly important for those systems where a large amount of heat transfer is needed while the working fluid is required to remain in the system (as in nuclear reactors). Note that the axial heat conduction, in general, is assumed to be very small in this thermal pumping process compared with the enhanced axial heat flow [16]. Apparently, the heat transport rate in thermal pumping is governed by both the thermal properties of the 11 working medium and pipe wall and the characteristics of the oscillatory motion of the fluid. The enhanced axial heat flow does increase with increasing oscillating frequency as this thins out the boundary layer and leads to an accompanying increase in the radial temperature gradient. This observation holds only as long as the thermal properties of the liquid and the wall are compatible. If the molecular conduction of the fluid in the radial direction is very small, then even high frequency oscillatory motion will not produce a large increase in the rate of enhanced axial heat flow. This is because such a system fails to supply the "passengers" enough time to be loaded onto the "bus" and to be unloaded from the "bus". Obviously, the system just wastes energy. On the other hand, if the molecular conduction of the fluid in the radial direction is very large, but the frequency of the oscillatory motion is low, once again one can not expect that there will be an efficient enhanced axial heat transfer between two reservoirs because the "bus" is now moving too slowly. The above observation can be confirmed by an analysis of the performance of a water-glass combination (i.e., water is the working fluid medium within a glass tube) and of a mercury-steel combination. For the former, it is necessary to employ rather small diameter tubes and low frequency oscillations with large tidal displacement, for it has 12 relatively poor heat diffusive properties as compared with the mercury-steel case. This small tube diameter-lower oscillation frequency set-up is necessary in order to ensure that there is sufficient time to transfer the excessive heat content of the bulk water core to the tube wall during the positive stroke of each period and to permit the transfer of excess heat from wall to the cooler core fluid during the negative stroke. Otherwise, the water would just carry a portion of its heat content back and forth in the pipe and the condition for achieving optimum enhanced axial heat transfer could not be met. For the mercury-steel case, one can chose relatively large pipe diameters and higher oscillation frequencies with smaller tidal displacement because of the higher thermal conductivity. This assures that only very short times are needed to exchange the heat between the core of the fluid and the wall. It should be pointed out that the suggestion of using smaller tidal displacement is purely due to the mechanical considerations and that one always tries to keep Ax as large as possible in order to produce large axial heat flows. From the above discussion, it can be concluded that the process of enhanced heat transfer via oscillatory pumping requires a precise tuning of parameters governing the enhanced heat transport. Indeed there is an expected "tuning effect" as discussed in references [16, 20, 21]. 13 The tuning effect is a very important concept in the study of the presently considered heat transfer process. It shows that there will be an optimum combination between thermal properties of the working medium and wall and the characteristics of the oscillatory motion. The qualitative aspects of the tuning effect have been observed earlier for both the case of a flat channel and that of the cylindrical pipe [15, 16]. From Fig. 4-38, one can see that an optimum for axial heat transfer occurs only at or near the tuning point which depends on the oscillating frequency and the thermodynamic properties of the fluid and wall. As has been pointed out by Kurzweg [20], in order to obtain the optimum enhanced heat transfer one has to carefully select suitable values for the pipe size, the material of pipe and the working medium as well as the manner of oscillatory motion. The nondimensionalized enhanced heat diffusivity is defined as Ke where Ke = diffusivity, averaged axial temperature gradient, p is the density of the fluid, and c is the specific heat. One can show that this nondimensional enhanced heat diffusivity is a function of both the Wormersley number and the Prandtl number [21] and hence that the dimensional axial heat diffusivity Ke is a 14 function of the tube radius the oscillating frequency w, the kinematic viscosity u, and the square of the dimensionless tidal displacement AX. This can be explained from the fact that the radial heat flow is proportional to the product of the representative radial temperature gradient 7AX/6th an<3 the surface area per unit depth of ttAX available for cross-stream heat transport. The use of large tidal displacement is always beneficial in the enhanced axial heat transfer within oscillating pipe flow. However, in order to avoid the direct convective net mass exchange between the two reservoirs, the tidal displacement must be limited to less than about one half of the pipe length. As has already been predicted by theoretical studies and will be confirmed by the present numerical simulations, the axial heat transfer will be further enhanced if the rigid surface (part of the rigid wall with finite thickness) has a non-zero thermal diffusivity and hence heat storage capability. Note that the existing considerations are restricted to laminar flow. Turbulent flow conditions can occur in oscillating pipe flow at higher values of wAx2/V [26, 27] and apparently would destroy the assumptions of the current analytic model of the thermal pumping process. Fortunately, the condition for optimum enhanced heat transfer in such oscillating pipe flow obtained at the tuning point requires 15 very slender pipes, such that the Reynolds number is usually small enough so that the oscillating motion falls within the laminar range [22]. The theoretical aspects of the oscillatory enhanced axial heat transfer process have been developed much further than its experimental and numerical counterparts. The theoretical predictions are quite limited and consider only cases under certain simplifying assumptions [8], Numerical work is necessary in order to not only to examine the correctness of the theoretical analysis but also to further the development of advanced enhanced thermal pumping devices. Numerical studies are not only fast, economical and accurate, but they also offer a handy way to access complex geometries which can not be handled analytically. It is the purpose of this study to extend the analytic work on thermal pumping by a detailed numerical study. We intend first to examine some transient problems of axial heat transfer in oscillatory pipe flow, such as the development of the velocity profile at various Wormersley numbers in contrast with those of reference [36], where only several special cases with intermediate Wormersley number a were discussed, and to examine numerically the relationship between the tidal displacement and the required corresponding strength of the periodic pressure gradient as a function of Wormersley number a and of tidal displacement Ax. 16 Next, we examine the build-up process of the temperature distribution in a pipe which connects a hot reservoir at one end to a cold reservoir at the other end and see whether there actually exists a constant time- averaged temperature gradient along the pipe axis when the final periodic state is eventually reached. Note that a time-averaged linear temperature distribution along the axial direction is an essential assumption in the existing theoretical studies. The third part of this investigation which forms the main effort, is a computer-aided numerical simulation of the thermal pumping technique, including an investigation of the variation of the enhanced axial heat flux versus the tidal displacement, the variation of enhanced axial heat flux versus different Wormersley numbers, and a study of the variation of heat flux versus different Prandtl numbers. It also includes a study of the influence of wall thickness and pipe diameter as well as the change of the fluid properties on such an enhanced axial heat flux and an examination of the tuning effect in the conducting wall case. Further, we compare axial heat transfer in oscillating flow with that in the steady flow, and compare quantitatively this enhanced heat flux with that induced by the pure axial molecular conduction under various Wormersley numbers and different fluid-solid wall combinations. Finally, the author examines the phase lag phenomenon in oscillatory flow in order to 17 further reveal the mechanism whereby this new heat transfer technique functions. The computational work presented here was completed on the Vax-11/750 electronic computer in the Department of Aerospace Engineering, Mechanics, and Engineering Science, and on Vax-11/780 in the Center for Instructional and Research Computing Activities, University of Florida, Gainesville, Florida. The numerical approach used, for solving the presently considered heat transfer problem, employed a second-order Crank-Nicolson scheme and an Alternating Direction Implicit method (ADI) with Thomas algorithm. A Fortran computer code named ETP (Enhanced Thermal Pumping) was developed to implement all the calculations. CHAPTER II FORMULATION OF THE PROBLEM A single tube with inner radius R^, outer radius R2 and length L (such that L >> R^ and R2) connects a large reservoir of hot fluid at temperature T = Tjj at one end, to another large reservoir of the same fluid at cold temperature T = Tc at the other end (Figs. 2-1 and 2-2) . The tidal displacement Ax and the oscillating frequency u> are adjusted to assure that the turbulent flow does not occur. The tube is oriented in such a manner so that the effect of gravity on the oscillatory motion of the fluid in the pipe is negligible. Water and mercury are employed as the working mediums. They are understood to be Newtonian and incompressible fluids. The variations of thermal properties of the fluid with temperature during the heat transport process are assumed to be negligible. With these assumptions the problem of enhanced time-dependent heat transfer induced by a simple harmonic oscillatory laminar fluid motion in a very long circular pipe can then readily be formulated. Governing Equations The use of a very slender capillary circular tube with constant cross-sectional area and neglecting end 18 HOT fLUID IN HOTTER FLUID (PRIMARY LOOP) OUT Fig 2-1 Thermal Pumping Device (after Kurzweg, U.S. Patent, 4,590,993 May 1986) 20 effects insures a laminar axisymmetrical one-dimensional time-dependent motion. It is convenient to employ cylindrical coordinates for this problem and we denote the coordinate in the axial direction of the tube by x, and the radial direction by r. The axial velocity can be taken to be independent of x and, in order to satisfy the requirements of continuity the other velocity components must vanish. We shall further assume that the pressure gradient induced by moving the piston (Fig. 2-1) is harmonic and has the form [29] 1 5p ~ A cos t t2-1) where A = l/p|ap/dx| is a constant which measures the maximum pressure gradient existing along the x-axis. Clearly this equation implies that we are now dealing with a time-dependent sinusoidal pressure gradient which is constant over the pipe cross-section at any instant and the pressure varies linearly along the x-axis. The simplified Navier-Stokes equation for this problem is [36] 3U / d r 3U -i = Acos(wt) + 7" 0 < r < Ri (2-2) where U(r,t) is the time and radially dependent axial velocity component. 21 The corresponding temperature T(x,r,t) of the fluid within the pipe is governed by the heat conduction equation [15] 3 T at -u 3T ax + K, f i a aT r ar ar ^ 32T ^ 0 < r < R2 (2-3) where is the inner radius of the tube and *cf the fluid thermal diffusivity which is related the thermal conductivity kf by kf Here p is the fluid density, and c is the specific heat. Note that the viscous heating term has been neglected in equation (2-3) since it is very small for most experimental conditions; this is justified provided one does not deal with very high Prandtl number fluids such as oils. The temperature in the wall can be determined from the solution of 3 T r 1 3 3T a2T 1 3t *w[ r gr (r 3 r) + gx2 J Rl < r < R2 (2-4) where kw is the thermal diffusivity of the conducting wall in Ri < r < R2 and is defined by 22 K w kw P wcw where kw, pw, and cw, are the wall conductivity, density, and the specific heat, respectively. By introducing the following non-dimensional terms, equation (2-2), (2-3) and (2-4) can be treated more easily. t* = t 1/w r * x x S U* u A/ w where 6 = J2v/u> is again the fluid viscous boundary layer thickness. The dimensionless governing system of equations can then be written as air at = cos(t ) + r a2u* ar TT2 + air ar 0 < r < RX (2-5) di atJ = cu aT7 ax r a2T* 2Pr ar Trz~ + aT7 a 2t* ar Tt + ax ^r o < r < R! and (2-6) a t at TT~ = 2S a 2t* r*2 + aT ar a2T* ax Rl < r < R2 (2-7) 23 where and M Cf Pr = K P cw S = *W C -A ^7 This nondimensionalization has some advantages in the computing process to be carried out below. The dimensionless velocity and its distribution over the cross- section found from the momentum equation (2-5) are expected to be universal for any Wormersley number a = R-^Jup/\x and any associated quantities, such as the tidal displacement and Lagrangian displacement. Its final periodic form is of the form given by (1-4). The dimensionless temperature in the pipe is only related to the Prantdl number and the dimensionless velocity, while that in the wall is related to the ratio of wall heat diffusivity to the kinematic viscosity, as seen from equation (2-6) and (2-7). The governing dimensionless equations (2-5), (2-6), and (2-7) are a set of second-order parabolic type of partial differential equations expressed in cylindrical coordinates. To solve this set of simultaneous equations, a corresponding set of boundary conditions and initial conditions are required. Since the velocity is assumed to be a function of r and t only, just two boundary conditions 24 are needed for solving the momentum equation, while for the temperature T(x,r,t) the heat conduction equations require two boundary conditions in both the r and x directions as well as compatibility conditions along the interface between the fluid and the solid wall. It should be pointed out that the initial conditions are less important than the boundary conditions if one seeks only a final periodic state. Boundary Conditions The boundary conditions for the velocity are the usual ones for viscous flow, namely, that the velocity is zero at the inner surface of the wall (r* = R^*) Also by symmetry, the normal derivative of velocity at the axis is zero. That is, U*(R!*, t*) = 0 (2-8) and 3U*(0, t*) The boundary conditions for the temperature depend on the particular model investigated. Model 1 In this model, it is assumed that the temperature at each end of the pipe is equal to that of the connecting fluid filling the reservoirs when the fluid moves into the capillary tube (i.e., during one half of each cycle). The Fig 2-2 Model 1 L=20cm L INSULATING WALL Fixed End Temperature Model R^=0.1cm, R2=0-15cm 26 end boundary temperature at x = 0 and x = L will take the values of temperature of the adjacent pipe fluid elements. With this model, we tried to simulate the real experimental observations where it can be clearly seen that a fluid jet exits the pipe during the outstroke, while the elements of the fluid enter the tube in a funnel pattern during the instroke. This observation shows that there is enough time during each oscillating cycle to allow the fluid elements within the exiting jet to fully mix with the fluid particles which originally are in the reservoir before the next instroke, provided the oscillating frequency is not too high. The end temperature boundary conditions are here taken as (when fluid enters pipe) T*(0, r*, t*) (when fluid leaves pipe) (2-10) (when fluid enters pipe) (when fluid leaves pipe) (2-11) where T*c, T*h are the nondimensional temperatures of cold and hot reservoirs, respectively. While T*acjj is the 27 temperature of the fluid elements which are adjacent to the corresponding ends at a particular instant. The thermal boundary conditions along the outer surface r* = R2 of the pipe wall are taken to satisfy the insulating wall condition, and along the axis of the tube the temperature is assumed to meet the symmetric boundary requirement, i.e., the radial derivatives of temperature along axis are equal to zero. We thus have d T d r r* = R2* = 0 (2-12) and d T dr" r* = 0 = 0 (2-13) The compatibility conditions along the interface between the fluid and the solid wall are that the radial heat flux and temperature are constant across the interface. That is 3T kf dr d T fluid - kw dr* wall at r = R]_ (2-14) and rji ^ ip ^ fluid wall at r = Rp (2-15) Since numerically the same nodes are chosen along the interface, condition (2-15) will be automatically satisfied. 28 Model 2 Here it is assumed that a heat source rim of width 2b is directly mounted on the interface between the solid wall and the fluid at x = L/2, while two small cold rim sources of width b each are mounted at x = 0 and x = L (see Fig. 2- 3) The wall thickness is assumed to be zero. This model is intended to simulate the enhanced heat transfer process of oscillatory fluid in an infinitely long pipe which is heated and cooled by the alternative evenly distributed heat and cold sources along the outer surface of a very thin wall which possesses infinite heat conductivity. Apparently, this geometry can be well simulated by the present model with periodic boundary conditions. Mathematically the boundary conditions are T*(x*, Rl*, t*) = T*h T* (x*, Rl*, t*) = T*c or d T dr 7c = 0 or X2 < x* > X3 * (2-14) 0 < : X* < XI* (2-15) X4* < X* < L* XI* < X* < X2* (2-16) X3 < X* < X4 * X6* are nondimensional coordinates of points which correspond to X = 0, b, L/2-b, L/2+b, L-b, and L, respectively in Fig. 2-3. The periodic end boundary conditions are given as to vo Model 2 Periodic Heat And Cold Sources on Insulated Wall (Except Sources Area) L=20cm, Ri=0.1cm, b=lcm Fig 2-3 30 T*(0, r*, t*) = T*(L*, r*, t*) (2-17) and 3T* i dT* i ^ L* = 0 3X* I x* = L* (2-18) Model 3 In model 1, the boundary conditions for the heat equation at X = 0 and L are somewhat artificial. The temperature distribution in the vicinity of pipe ends may not precisely match the constant temperature end conditions. This is particularly true in the large tidal displacement and/or very high oscillating frequency cases since a temperature drop occurs at the hot end and a rise at the cold end as fluid particles exit the pipe. This leads to a discontinuity of temperature which will lead to unexpected dispersion errors if an even order numerical method is used [3, 24, 30], To improve the end boundary conditions presented in model 1, the following extended model has been considered. It is assumed that an extension pipe which is some 5 times the length of the central pipe is attached to the original pipe used in model 1. The heat and cold source are assumed to be located on the outer surface of the wall in the extended sections of the pipe as well as at both ends of the tube as shown in Fig. 2-4. This model is used to Model 3 Pipe with Conducting Pieces Model L=20cm, Ri=0.1cm Fig 2-4 32 investigate the situation where the heat contained in the jet is exchanged by pure heat conduction with the surrounding fluid elements in each reservoir without any convective mixing. In this case, the boundary conditions along the outer surface of the wall are T* = T*h 0 < x* < XI* (2-19) T* = T*c X2* < X* < X3* (2-20) and d T* = 0 XI* < x* < X2* (2-21) where XI*, X2*, and X3* are nondimensional coordinates of points which correspond to X = 5L, 6L, and 11L, respectively, in Fig. 2-4. At both ends, we have T*(0, r*, t*) = T*h (2-22) and T*(L*, r*, t*) = T*c (2-23) As in the other models, the symmetry condition along the axis of pipe reguires that d T* dr* =0 0 (2-24) 33 and the radial flux and temperature continuity condition along the interface are d T* kf dr* fluid dT* kw wall and (2-25) ip * fluid wall (2-26) Here again kf and kw are the same as described in the previous models. Initial Conditions The initial condition chosen depends on the problem under consideration. If one's goal is to investigate the periodic quasi-steady solution only, the initial conditions chosen should be as close to the quasi-steady state as possible so that a rapid convergence rate at low CPU time cost is achievable. If one intends to study the transient process, then various initial conditions should be supplied according to purpose of the investigating cases selected. For the velocity initial condition we choose for all our studies U*(x*, r*, 0) = 0 (2-27) For the temperature, if the purpose of investigation is to examine the build-up process of the periodic quasi-steady 34 state in the thermal field, the initial condition should be taken as T*(x*, r*, 0) = 0 (2-28) However, for the other cases, in order to gain faster convergence, the initial temperature can be assumed to have a linear distribution in the axial direction, and thus have the form T*(x*, r*, 0) T*h + (T*c T*h)xx* LL* (2-29) where LL* is the dimensionless length within which the axial linear temperature distribution is assumed to hold, and xx* is the dimensionless distance measured from the "origin" which is chosen only for the purpose of this linear temperature initialization. Both the LL* and the origin selected depend on the model considered. For Model 1 and Model 3, LL* is equal to L*, and the origin is taken at the left end (Model 1), or at the left intersection between the central pipe and the left extension pipe (Model 3). However, for Model 2, equation (2-29) is valid only for the right half portion of the pipe as LL* is taken to be L*/2, and the origin is chosen at the middle section of the pipe. The initial temperature of the left half pipe can then be found from the plane symmetric condition about the origin cross section. 35 Calculation of Tidal Displacement An important quantity encountered in the study of the enhanced heat transfer process in oscillatory pipe flow is the tidal displacement, which is usually required to be smaller than one half of the total pipe length in order to avoid any convective mass exchange occurring between the two reservoirs. It has already been defined in the introduction chapter (1-6) and in the present computation the dimensional tidal displacement takes the form (2-30) where is the inner radius of the pipe and x(r,7r/w) is the Lagrangian displacement of the fluid elements located along a radius within the capillary tube at t = w/w. It is assumed these elements are initially lined up at axial position x = L/2 (Model 1 and Model 2) or x = 5.5L (Model 3) half way between the tube ends. Numerically, the dimensional Lagrangian displacement at time t is computed via the equation 0 U(r,t)dt (2-31) It is obvious that the dimensional Lagrangian displacement x(r,t) is a function of both time t and the radial position r. Note that, since the existence of phase lags between the 36 stimulating pressure gradient and the displacements vary for different Wormersley numbers, in actual calculations, the tidal displacement is equal to the sum of the absolute maximum cross-section averaged Lagrangian displacement and the absolute minimum cross-section averaged Lagrangian displacement within a period. With the same non- dimensionalizaton procedure used in the previous section for the governing equations, the dimensionless Lagrangian and tidal displacement can be written as R* (2-32) 0 The nondimensional Lagrangian displacement within the period can then be computed by * / 4- \ x (r ,t ) (2-33) where * x (A/u^) and Ax AX * (A/c2) It is pointed out that this dimensionless tidal displacement Ax* differs from AX defined by equation (1-7) by a constant 1/2. 37 Effective Heat Flux The time averaged total effective axial heat flow within the pipe has the form (2-34) where c is the specific heat, p is the density and again, w is the oscillatory frequency. The per unit area effective heat flow (termed heat flux) can then be written as Equation (2-34) follows from the fact that pcUT is the convective heat flux. The dimensionless total effective heat flow can be written as (2-36) and the dimensionless heat flux by * Q* v total (2-37) CHAPTER III NUMERICAL TECHNIQUES EMPLOYED Equations (2-5) (2-6) and (2-7) which were derived in Chapter II can not generally be solved analytically in any simple manner. Therefore, it is necessary to seek a numerical solution approach to the problem. As is well known, in order to numerically solve a set of simultaneous governing differential equations more accurately and efficiently, an optimized grid system plays a very important role. For the present purpose the grid net works were generated in the following way: within the tube, a non-uniform grid (see Fig. 3-1) is clustered along the radial direction in the vicinity of the inner surface of the pipe where the larger gradients of velocity and temperature are expected to be present, while along the x-axis the grid is distributed uniformly except for those calculations dealing with model 3. The solution process was carried out in the computational plane rather than directly in the physical plane. Thus, a transformation which converts the governing equations as well as the boundary condition plays an essential role. 38 39 1). For Model 1 2). for Model 3 Fig 3-1 Grid Systems Used in the Numerical Simulations 40 The second-order implicit unconditionally stable Crank-Nicolson scheme and the ADI method [2, 31] are employed to break up the transformed equations into finite difference form. A computer code designated ETP (for Enhanced Thermal Pumping) has been developed for obtaining the desired results. Details of this procedure will be described in this chapter. Transformation The numerical solution of a system of partial differential equations can be greatly simplified by a well- constructed grid. It is also true that an improper choice of grid point locations can lead to an apparent instability or lack of convergence. Early work using finite difference methods was restricted to some simple problems which can be numerically solved in the physical domain. As experience was gained, general mappings were used to transform the physical plane into the computational domain as well as the governing differential equations [31]. Such a grid transformation technique brought to the numerical simulation numerous advantages, and the computational work was no longer restricted to a few simple geometries. A body-fitted non-uniform grid in the physical domain could be used, which retains the uniform spaced grid system features in the computational domain [33, 34, 35]. However, several requirements must be placed on the transformation: first, the mapping must be one to one, second, the grid lines must 41 be smooth and have small skewness in the physical domain, third, grid node point spacing should be small where large numerical errors are expected (i.e., large solution gradient regions) in the physical domain (Fig. 3-2). In present study, the grid system in the physical space is numerically generated by solving an algebraic equation which clusters the grid lines in the region where large gradients of the physical quantities are expected so as to gain higher resolution of these physical quantities. Because of the simple pipe geometry, the requirements of grid smoothness and orthogonality are not a serious problem. For the present purpose, the transformation is a simple one which transforms a non-uniform grid network in the physical plane onto a uniform grid system in the computational plane. x* = x*(?) r* = r*(r,) (3-1) t* = t*(r) The inverse transformation can be found as = (x*) n = r, (r*) (3-2) r = r (t*) where x*, r* are the dimensionless coordinates and t* the dimensionless time in the physical domain, Â£, q are the transformed coordinates and r is the transformed time in the w Fig. 3-2 Coordinates Transformation 43 computational domain. With this transformation, the transformed governing equations have the form: au a2u* atr ar f(T) + a^"aV + h{v) dr, (3-3) aT aT3 a2T* aT a r = P as + P2_a?2_ + P3 dr, a2T* + P4^ (3-4) and 3T a r aT a2T* aT wi^r+ + w3~ a2 t* + w4^~ (3-5) where f(r) a(?) b(r?) and PI = P2 = P3 = P4 = at* -cos a r (t*) 1 2 1 at* i ar (ar*/ar?)2 at*r i 1 a 2r 2 dr [ r*(ar*/dr,) (ar*/ar?)J at?2 at* r cu* 1 a2x* ^ a r [ dx*/dÂ£ 2Pr(ax*/a^);j as2 J at* i 5 r 2Pr (ax*/aÂ£2) at* 1 r 1 1 a2r* ^ 5 r 2Pr [r* (dr*/dr,) (ar */dr,)J dr,' J at* 1 dr 2Pr (ar*/ar?)2 'I (3-6) (3-7) (3-8) (3-9) (3-10) (3-11) (3-12) 44 W1 W2 W3 W4 at* i ( . -i d t 2Prw( .(ax*/aOJ ai J at* l i a r 2Prw (ax*/ao2 at* i 1 1 a r 2Prw [r* (dr*/dr,) (dr*/dr,)3 at* 1 1 a r 2Prw (ar */dr,)' d2r*> dr,- (3-13) (3-14) (3-15) (3-16) As mentioned in chapter II, equations (3-3), (3-4) and (3-5) are a set of second-order parabolic partial difference equations in cylindrical coordinates. In addition, since the oscillating flow is considered as incompressible in the present studies, the momentum equation (3-3) can then be independently solved at each time step. As a result, the time-dependent update velocity found can then be substituted into the heat conduction equation (3-4) as a known quantity at the same time step level. Eventually, the coupled heat conduction equations (3-4) and (3-5) are solved simultaneously to obtain the temperature distribution both in the fluid and in the wall at any time. To best solve this set of equations in terms of accuracy and efficiency, the proper choice of numerical technique and grid net-work is dictated by an understanding of the physical aspects of the problem. 45 The same transformation should be also applied on all the boundary conditions proposed in the three different models. For the temperature compatibility conditions along the interface between the fluid and the wall (2-25) one has the following transformed forms: dT* 1 dr] (dr*/drÂ¡) kf fluid dT* 1 dr/ (dr*/di]) kw wall (3-17) or dT dr] = K; flow dTJ d r) wall where kw dr*/dr) |pipe Ka = *Â¡ kf 3r >Hwall (3-18) (3-19) To make the subsequent form of the corresponding finite difference governing equations less cumbersome, the superscript will be dropped from the variables T*, r*, t*, and x*, and in addition, U* will be replaced by V. In the process of deriving the finite difference governing system equations, the second-order central differences in the domain and forward or backward differences along the boundaries or interface of the fluid and wall have been employed at each nodal point. 46 Crank-Nicolson Method for Momentum Equation The second-order accurate Crank-Nicolson method is quite well known and widely used. It is an unconditional stable, implicit scheme for solving the parabolic types of partial difference equations. When the Crank-Nicolson method is applied to equation (3-3), the finite difference algorithm at a typical node k in the radial direction and at the time step n assumes the simpler form: (Bkvk-i + Dkvk + Akvk+Jn+1 ck k = l, 2, , km (3-20) where - -Hfn+l+ fn) (Bkvk-i+ E,_V,_- A,_V,_ . jn (3-21) Jk k k k+1 Ak " (2a + b ) (3-22) ^ (2 a + b ) (3-23) Dk = 1 + ak (3-24) Ek = 1 ak 1 (3-25) 47 The boundary conditions along the interface (i.e., the inner surface of the wall, k = kmid) and the axis (k = 1) then become V kmid dV dr) (3-26) The initial conditions of the velocity at all nodal points is taken to be zero. n k = 1, 2, 3, kmid (3-27) Equation (3-20) associated with the boundary conditions (3- 26) can then be written in the matrix form. This yields a set of linear system algebraic equations which can be solved in terms of the nodal values of velocity in the capillary tube by using either an iterative method or a Thomas algorithm at each time step. The explicit form is 48 * n ' ' n+1 D, A, V. C. 1 1 1 1 D A C 2 2 2 2 2 B D A_ V, C0 3 3 3 3 - 3 Bkd-lDkd-lAkd-l vkd-i Vkd-1 Bkd Dkd VW J VKd J (3-28) where B 1 kd = kmid-1 ADI Method for Axisvmmetric Heat Equations The governing PDEs (3-3), (3-4) and (3-5) are all of the second-order parabolic type. Thus it might be suggested that the Crank-Nicolson scheme used in solving the momentum equation (3-3) can also be applied to the axisymmetric heat equations (3-4) and (3-5) and one can then take advantage of the tridiagonal matrix form while using this unconditionally stable technique. However, when attempting to use such a formulation, one immediately finds that the resulting system of linear algebraic equations is no longer of the tridiagonal type (3-20) but rather a non-tridiagonal matrix system requiring substantial CPU time to solve. This difficulty can be avoided by applying the unconditionally stable Alternating Direction Implicit method (ADI), 49 developed by Peaceman and Rachford and Douglas in 1955. According to this scheme, the entire solution process at each time step is "split" into two portions, i.e., the first half of solution processes for k^*1 column (radial direction) while the other half processes for the row (axial direction). With the ADI scheme, second-order central differences are used to approximate the values of derivatives at each nodal point in equations (3-4), (3-5). The finite difference algorithm for those equations during the first half of each time step for the jth column are then .BPjj,kTjj,k-l + DPjj,kTjj,k + AP. T jj,k jj,k+l n+1/2 = PXn. . ID,* k = 1, 2 ....kmid (3-29) and BW. ,T. . + . 33,k-l DW. jj,kTjj,k + AW. . T. . ,, 11/k 3D,k+l n+1/2 WX n jj/k k = kmid+1,...kmax (3-30) where the subscript jj is used to emphasize the specific column currently to be computed. It can be seen that the set of difference equations now is in the tridiagonal form 50 since the right-hand-side terms in the equations (3-29), (3- 30) contain only known values from the previous results and the boundary conditions. These values can be computed by PXjj,k (~CPjj,kTjj-l/k + EPjj,kTjj,k FPjj,kTjj+l,k) (3-31) and ( -CW. . T . + EW. . T. . FW ll/k DD-l,k DD,k D},k . T . 1 n (3-32) The computational algorithm is implemented column by column and the unknown value (Tj,^)11 can then be solved by either an iterative or a direct method. In order to do this, equation (3-29) needs to be assembled into the following matrix form. r ' PDX PA1 * n ' T1 n+1/2 ' ' PX1 PS2 pd2 pa2 T2 PX2 pb3 pd3 pa3 T3 = PX3 PBkd-lPDkd-lPAkd-l Tkd-1 PXkd-l PBkd PDkd j T l kd J j PXkd j = 1, 2, 3 jmax (3-33) 51 Similarly, equation (3-30) can be assembled into the matrix form as seen in (3-34). WDkd+l WAkd+l n ' ' T xkd+l n+1/2 WXkd+l WBkd+2 Wkd+2 WAkd+2 T kd+2 WXkd+2 WB, WD. WA. km km km T xkm WXkm-l * WB. WD. kmax kmax j Tkmax j WXkmax j = 1, 2, 3 jmax (3-34) It should be pointed out that the coefficients and/or the right hand side terms in the row marked with symbol "*" in those matrixes needs to be properly modified as well as the limit of either subscript j or subscript k according to the different boundary conditions in the various models considered. For example, along the axis, the symmetry condition requires that dT/dr = 0 and this could be accomplished numerically by equating the temperature Tq and T2 and by employing a new combining coefficient of PA^* = PAi + PBi rather than the original PA^ in the first matrix for evaluating the temperature distribution within the pipe. In addition, if the temperature distributions along certain parts of the outer surface of the wall are given, then the 52 limit of subscript k will be ended with k = kmax-1 rather than k = kmax shown in equation (3-34). The same arguments are also applicable for another subscript j in the axial direction. By using a similar procedures for the second half of each time step, the finite difference algorithm for the kth row then becomes (CPj,kkTj-l,kk + GPj,kkTj,kk + FPj,kkTj+l,kk) n+1 = pYn+l/2 j ,kk j = 1, 2, , 3 max (3-35) H,1:-!,!*+ Gwj,kkTj,kk+ Fwj,kkTj+i,kk)n+1 m"%2. j = 1, 2, j max where the right hand terms can be computed by (3-36) PY1?+l/2= CP. T. + HP T AP T 1 n+1/2 ^ j,kk [ ,kkij,kk-l 3,kk1j,kk ,kkij,kk+lj (3-37) wy"!k2= (-cBj,kkTj,kk-i + Hwj,kkTi,kk- Awj,kkTj,kk+i)n+1/2 (3-38) 53 Similarly, to emphasize that the current calculation is in the kth row, we use the symbol kk rather than k in above formula and the matrix form of equation (3-35) for the k^h row can then be written as PG2 PF2 * n+1/2 r * T2 ' ' PY2 PC3 PG3 PF3 T3 PY3 PC4 pf 4 4 T4 PY4 PC. PG. PF. 3m-1 jm-l jm-l Tjm-1 PXjm-l * i. PC. PG. jm jm J k T . 1 D J k PX. 1 3 n+1/2 k = 2, 3 , kmid (3-39) and the matrix form of equation (3-36) becomes WG2 wf2 * n+1/2 ' 1 T2 N+l wy2 wc3 WG3 WF3 T3 WY3 wc, 4 WG. WF . 4 4 T4 WY 4 WC. WG. jm-l jm-1 WF. jm-l Tjm-1 WX. jm-l * b WC. jm DG. 3 J k T. 1 3 J k WX. 1 3 k = kmid+1, kmid+2 km N+1/2 k (3-40) 54 As indicated above, the symbol shows that the coefficients in that row as well as the right-hand-side terms need to be properly modified corresponding to the different boundary conditions. The matrix terms in equations (3-29), (3-30), (3-31) and (3-32), (3-35), (3-36), (3-37) and (3-38) can be computed by APj,k = "i ^P3 + P4>^ D BPj,k P3 P4>^ J f k CPj,k P1 P2>j,k EPj,k = 1 2*P2>j,k PP j ,k= "iP1 + p2). GPj,k 1 + 2*P2)j,k DPj,k = 1 + 2*P4>j,k and AW. . D,k = W3 + P4) . j ,k BW = ( W3 W4) . j ,k CMj,k < W1 W2>j,k HPj,k t1 2*P4>j,k (3-41) - (1 2*P2>j,k - -<4"wi + w2>j,k = (1 + 2*W2). k DWj f k + 2*M4>j,k HMi,k (3-42) 55 In order to rewrite the equation in a simple form we define the following linear operator : LY [P]j>k = [BP, DP, AP]j,k LY [W]j,k = [BW' DW' AW]j,k LX [P3j/k = [CP, GP, FP]j/k LX [W]j ^ k = [CW, GW, FW]j,k (3-43) and column vectors {TY)j,k {Tj,k-l' Tj,k' Tj,k+1 } {TX}j,k T. {Tj-i,k' xj,k' *j+i,k T. (3-44) Equations (3-29) which are in the radial direction can be simply written as n n+1/2 n LY [P] (TY) = PX j,k j, k j, k (3-45) n n+1/2 LY [W] (TY) j ,k j ,k n (3-46) j = 1, 2, k = 1, 2, 56 Equations (3-33) and (3-34) which are in the axial direction, then assume the following simple forms: n+1/2 n+1 n+1/2 LX [P] (TX) = PY (3-47) j /k j ,k j ,k n+1/2 n+l n+1/2 LX [W] (TX) = WY (3-48) j ,k j / k j/k j = 1, 2, k = 1, 2, where the right side terms can be estimated by PX = (-CP, EP, -FP) (TX) j,k j,k j,k WX = (-CW, EW, -FW) (TX) j,k j,k j, k PY = (-BP, HP, -AP) (TY) j,k j k j k (3-49) WY = (-BW, HW, -AW) (TY) j r k j k j k 57 It is seen, as the result of the ADI "splitting procedure which has been employed in the algorithm associated with different boundary conditions for the various models, that only a tridiagonal system of linear algebraic equations needs to be solved (i.e., during step 1, the coupled tridiagonal matrix (3-33), (3-34) are solved for each j*-*1 column of the grid points, while during step 2, the coupled tridiagonal matrix (3-39) and (3-40) are then solved for each k*-h row of grid points) . Once the periodic steady solution has been obtained, we can calculate both the tidal displacement and the heat flux at different locations within the pipe. The numerical technique used here to integrate equations (2-38), (2-39) and (2-41) for evaluating the tidal displacement and the heat flux at each specified location can be obtained either by the Trapezoidal rule (with end correction) or Simpson's rule. Both numerical approaches are essentially fourth- order methods. Convergence Criteria As is known, once the calculation work has started, the time-matching process will be in the loop forever unless a criterion can be derived that indicates when the goal of the current computing work has been reached and further solution-matching processes do not produce significant increases in accuracy. Such a criterion depends on the purpose of the calculation. If one's goal is to study the 58 transient process, one can indicate a time limit to pause or quit the current computing work. However, if one seeks a final periodic state solution, one has to develop a criterion to test if the solution can be considered acceptable. It seems that the temperature is a good measure of the accuracy of the overall solution process so that the most efficient way is to apply the convergence test on the temperature rather than on the velocity. For the present purpose, two convergence criteria were alternatively used. The choice of the criterion depends on what was the main goal in the particular study case. If the main effort is to observe the distribution of the quasi-steady temperature at any phase angle t within a period, the testing is done by comparing the temperature residual, i.e., by inspecting the averaged temperature difference of each nodal point at thee same t between adjacent periods. This can be written as Resl j s (T11 T22 )? k ]1/2 (JMAX)(KMAX) < j = 1, 2, JMAX k = 1, 2, KMAX (3-50) where Til. is the value of the temperature of node (j,k) at wt in the current period, T22. is the value of the D I K 59 temperature of node (j,k) at the same t in the previous period, and ej is the convergence parameter. If the goal of the investigation is to examine the effective heat transfer, the convergence criterion is established by computing the residual Res2 JSEC * e2 (3-51) j = 1,2, JSEC where JSEC is the number of cross-sections where the axial heat flux was examined, 62 is the convergence parameter, and current period and previous period, respectively. The summation is carried out over all sections (JSEC) where the heat flux is computed. The value of the convergence parameter can be determined by a balance of the acceptable solution accuracy and the cost of CPU time. In the current study both el and e2 were taken between 0.001-0.01 as the residual of the temperature or the heat fluxes for acceptable value. Grid Generation As is well known, the solution accuracy and efficiency in a large degree depends on the grid system used in the numerical calculation. A good grid is characterized by small skewness, high smoothness and capability of high 60 resolution in the large gradient regions in the physical plane. It has been shown that rapid changes of grid size and highly skewed grids can result in undesirable errors [25]. The success of a numerical simulation of a complex thermal fluid dynamics problem does depend strongly upon the grid system used in the computation. In the present study, the grid mesh was generated with the emphasis on the high resolution capability of its simple geometric boundaries. The technique used here is one of the algebraic schemes which cluster the grid lines near the region desired [34], namely, AS. = AS (1 + e) k_1 (3-52) where aSq is the minimum specified grid spacing next to the wall or to some inner interfaces within the tube. The parameter e is determined by a Newton-Raphson iteration process so that the sum of the above increments matches the known arc length between k = 1 and k = kmax. The grid networks used in Model 1 and Model 2 are uniformly distributed along the axial direction in both the fluid and the wall; however, in Model 3, most grid points are clustered near the central portion of the pipe so as to gain higher resolution in the region of interest, while along the radial direction a non-uniform grid was employed in all three models. The minimum specified grid spacing AS0 used 61 depends on the boundary layer thickness 6, namely, the kinematic viscosity of working fluid and the oscillating frequency. In the present study, it was found that a good choice of this value is AS0 = 0.055 for laminar flow cases with a total of about 15-20 nodal points distributed along the radius. CHAPTER IV NUMERICAL RESULTS AND DISCUSSION The problem of time-dependent enhanced heat conduction subjected to sinusoidal oscillations can now be solved numerically for the boundary conditions appropriate to a long capillary tube according to the various models described in previous chapters. The computational tasks fall into two catagories: the first part is a numerical study concerned with the characteristics of the oscillatory pipe flow, which includes an investigation of the velocity profiles, the Lagrangian and the tidal displacement trajectories. The second part of this study includes a thorough investigation of the final periodic state temperature build-up process in oscillating pipe flow, the periodic temperature distribution in the pipe and wall, and a study of the relationship between the enhanced axial heat flux and the tidal displacement. It also includes an investigation of the tuning effect and a comparison of the enhanced axial heat transfer with the corresponding pure molecular axial heat conduction as well as the investigation of the influences of the pipe radius and the wall thickness on the enhanced axial heat transfer. 62 63 Properties of oscillating laminar pipe flow have been analytically discussed by S. Uchida [36] for several different Wormersley numbers, and these results offer a useful reference for comparing with the present numerical studies. However, in the area of thermal fields associated with enhanced thermal pumping there is little information available for comparison, except for some recent results of Kaviany [13]. The present computational work was carried out on the Vax-11/750 computer in the Department of Aerospace Engineering, Mechanics, and Engineering Science, and on the Vax-11/780 in the Center for Instructional and Research Computing Activities, University of Florida, Gainesville, Florida. Note that it is very time-consuming to build-up a final periodic state, for example, if the grid size is 101x22 and the time steps are 2000 per period, it takes 7.5 minutes (CPU) with the VAX-11/780 machine or almost 1 hour (CPU) with the VAX-11/750 machine to run only one period. It usually takes 20-30 periods to reach the final periodic state solution. The process of selecting model size (i.e., total nodal points) is a synthetic balance among the storage requirement, the solution accuracy, and the cost of CPU time for finding an acceptable solution. Such a selection allows each case to be solved with a minimum of expense in computing operation thereby making it possible to do the 64 large number of runs needed to obtain enough data points to plot curves of the desired parameters. The results discussed in this chapter are presented for the purpose of illustrating the effects to be encountered in working with the enhanced thermal pump. These results should be effective in gaining insight into some interesting features of this enhanced heat transfer process. Since the analytic approximations [15, 16, 18, 22] are effective in describing the flow and the heat transfer aspects and to give considerable insight into the problem, they are often used to compare with the numerical results and should give good agreement when applicable. Part 1. Oscillatory Pipe Flow Features In order to better understand the mechanism of the enhanced thermal pump, it is necessary to examine the mechanical features of the oscillating pipe flow. As mentioned in previous chapters, the present interest in the enhanced thermal pumping is confined to an investigation of the central part of the slender pipe, thus the flow field can be well approached by a 1-D time-dependent laminar model which neglects the ends effects. Velocity Profiles Fig. 4-1 shows the numerically computed time- dependent velocity profiles at different phase angles of the exciting pressure when Wormersley numbers are equal to 1, 10, 100, and 1000, respectively. It can be clearly seen 65 a -1000 * * * : 2 U c fa 9 -f r-rn -r n # y M 9 t ise ue ie e jm see Fig 4-1 l-D Velocity Profiles in Oscillating Flow for Wormersley Number a = 1, 10, 100 and 1000 66 that the velocity profile at a = 1 presents a quasi parabolic shape at any instant within a period and is in phase with the stimulating pressure gradient. However, at higher frequency, for example, a = 100 and 1000, the velocity profiles can be clearly separated into two regions: in the vicinity of the wall the flow shows a typical thin boundary layer, while in the region far away from the wall the fluid moves as if it were frictionless slug flow. In fact, within this core region the velocity distribution is independent of the distance from the wall [29]. It can be also seen that the phases of the velocity profiles at higher frequencies cases are shifted about k/2 with respect to the Fig 4-2 Velocity Profile(q=5 ) [29] 67 (Rl-r)/ (Ri~r)/ (Rl~r)/ Fig 4-3 Magnified View of Velocity Profile Near Wall (Wormersley Number a=10, H2O, 6=0.014cm) 68 stimulating pressure gradient. At the intermediate frequency case (a = 10 of Fig. 4-1), the slug flow boundary is not so evident, but one can still note a boundary layer near the wall. The same pattern of the velocity profiles associated with moderate Wormersley numbers can also be found in the reference [36] where the velocity profile at Wormersley number a less than 10 was presented (Fig. 4-2) . In order to better see the variation of the time-dependent velocity profile within the boundary layer, a set of closer view of the velocity profile with respect to Wormersley number at a = 10 during phase intervals wt = 30 is plotted in Fig. 4-3. It should be emphasized that the solutions shown here are under the assumption that the secondary velocity in the radial direction is negligible compared to the axial velocity component, namely, the non-linear inertial terms are not considered in the governing equation (2-1). This is a reasonable approximation for moderate oscillatory frequency w except near the ends of the pipe. However, at high Wormersley numbers, care should also be taken to avoid violating another assumption, namely, that of incompressibility and the concomitant condition that the oscillating phase does not change between the tube ends. High Wormersley number can be obtained either by increasing the oscillating frequency w or by increasing the pipe diameter for a given fluid. An oscillating flow can be 69 considered incompressible if Axw/2 < 0.05C, where C is the speed of sound in the fluid. For mercury, C = 1360 m/sec, one requires a = 27.1, when Ax = 20 cm and = 0.1 cm. To avoid a appreciable phase difference between the tube ends, one requires that L/C 2tt/w. Both the restrictions are met in the examples to be considered below. The Lagrangian Displacements An alternative interesting representation to the oscillatory velocity field are the Lagrangian displacements of the fluid elements at different radii within the pipe. They have been plotted in Figs. 4-4 and 4-5 at time intervals of wt = 30 for Wormersley number a = 0.1, 1.0 and 10. It is noted that since both pipe diameter and the working fluid were fixed in this test, Figs. 4-4 and 4-5 represent the relationship between the Lagrangian displacement and the oscillating frequency. The trajectories plotted in Figs. 4-4 and 4-5 have been normalized by A/w2, where A = l/p|dp/dx| is the amplitude of the sinusoidal pressure gradient as defined in equation (2- 1) and w is the angular velocity. Suffice it here to point out that for the lower frequency case (for example a = 0.1 and 1.0) the Lagrangian displacement trajectory shows a foreseeable parabolic pattern at any moment. Nevertheless, the essential distinction between low frequency oscillatory pipe flow and steady Hagen-Poiseuille flow is that in the former, the Lagrangian displacement trajectories as well as 70 Fig 4-4 Lagrangian Displacement for a = 0.1 and 1.0 71 Fig 4-5 Lagrangian Displacement at a = 10 72 the velocity profiles are periodic so that the fluid particles do not translate axially upon time averaging, while in the later case they will. For intermediate Wormersley number (a = 10), the trajectories of the Lagrangian displacement departs considerably from the standard parabolic shape. This phenomenon can be even more clearly seen in the a = 100 case. Evidently, the higher the oscillating frequency, the thinner the boundary layer (6 = J2v/u>) . Tidal Displacements Figs. 4-6 and 4-7 demonstrate time variation of the cross-section averaged dimensionless Lagrangian displacement at Wormersley numbers in the range from a = 0.1 to 50. The tidal displacement can be obtained by summing the absolute maxima and the absolute minima of these curves. The corresponding non-dimensional tidal displacements with respect to Wormersley number from a = 0.1 to a = 12 are listed in table 4-1 below Table 4-1 Dimensionless Tidal Displacement at Different Wormersley Numbers a o H 1.0 M O 3.0 4.0 5.0 \ AX 0.00246 0.14295 0.67303 1.20213 1.41160 1.49041 a 6.0 7.0 o CO 10.0 12.0 ; ax 1.56294 1.61879 1.66046 1.71900 1.76698 73 Fig 4-6 Dimensionless Cross-section Averaged Displacement Versus Time q = 0.1 1.0 Fig 4-7 Dimensionless Cross-section Averaged Displacement Versus Time q = 2 50 74 It is noticed that for very small Wormersley number cases (a < 0.5), the cross-section averaged displacement varies like a sinusoidal function with respect to time (the ordinate), and as Wormersley number increases, the cross- section averaged displacements are no longer symmetric about the ordinate but rather favor positive values of *. If the Wormersley number is further increased, eventually, the cross-section averaged displacement almost entirely lies on the right half plane. A similar feature can also be seen in Figs. 4-4 and 4-5. In fact, this just again shows the existence of the phase shift in the oscillating pipe flow. When the Wormersley number is small the cross-section averaged displacements as well as the Lagrangian displacements show a sinusoidal variation with respect to time and this has ?r/2 phase lag with respect to the stimulating pressure gradient which is assumed to be a cosine function of wt with zero initial phase angle. However, for higher Wormersley number, the phase lags increase to almost n with respect to the phase of the exciting pressure gradient. Note that the Lagrangian displacements in Figs. 4-4 and 4-5 were computed by lining up all the fluid particles on the plane x = 0 at wt = 0, however, by using non-zero velocity at wt = 0 as shown in Fig. 4-1. This implies that if the phase of the exciting pressure gradient is taken as the base of the measurement, it is generally not possible to assure the same phase to the 75 velocity and Lagrangian displacement as well as the tidal displacement. For example, at phase of the stimulating pressure gradient wt = 0 the corresponding velocity phase may be 7r/2 and the Lagrangian displacement phase may be n. In fact, this is just as true for very high Wormersley numbers; however, the phase difference with respect to the exciting pressure gradient phase is less than the value shown above for low Wormersley number. One can well see that if the phase lags of the Lagrangian displacement or the tidal displacement is given, one can certainly re-draw the diagram shown in Figs. 4-6, 4-7 by lining up the phase with itself, and then an exact symmetric pattern of the tidal displacement curve similar to that in the very small Wormersley number case can be obtained. Unfortunately, the phase lags are a function of the Wormersley number and they are not known in advance. In order to verify the numerical method used in this study and to check the ETP code developed, a comparison of the computed dimensionless tidal displacement to the one using the analytic equations (1-7) and (1-8) given by Kurzweg [18] for Wormersley number varying from 0.1 to 100 has been calculated and plotted in Fig. 4-8. The solid line shows the analytic solution obtained by use of equations (1- 7) and (1-8), while the dashed line shows the results with the ETP code developed in this study. The agreement is quite good, particularly when the Wormersley is less than 76 3.0. However, at high Wormersley number the numerical solution shows a very slight deviation from the analytic solution. This deviation is believed due to an inaccurate numerical integration over the cross-section using relatively large time steps (our time steps per period in the calculation were between 1000-2500). A comparison with using 104 time steps per period for Wormersley number a = 10 was studied and shows some improvement. However, using such small time steps in the present investigation is beyond the capacity of the current VAX computer facility used. The numerical error becomes particularly serious as the oscillating frequency becomes large where the extremely thin boundary layer requires more grid nodal points to resolve the flow variables in the vicinity of the wall. Fig. 4-8 shows that as the Wormersley number gets large, the dimensionless tidal displacement tends to the limit of 2.0, which agrees with the limit of 1.0 in the analytical solution given by Kurzweg [18] for the reason that the normalization parameter used in [18] is twice as large as that in the present numerical simulation. Fig. 4-9 shows the required stimulating axial pressure gradient used in the present study for a pipe radius = 0.1 cm and water (u = 0.01 cm2/sec) taken as the working medium versus the dimensional tidal displacement Ax in cm for various Wormersley numbers (namely, oscillating frequency). It is evident from these results that for fixed Dimensionless Tidal Displacement AX 77 id"' 1U 101 102 Wormersley Number a Fig 4-8 Relationship Between Dimensionless Tidal Displacement AX and Wormersley Number a (Uio) Relationship Between Tidal Displacement ax and Exciting Pressure Gradient in Water 03 Fig 4-9 79 tidal displacement, the required axial pressure gradient in the large Wormersley number case is orders of magnitude higher than that in the small Wormersley number case. This may eventually put some constraint on the use of very high frequency in the enhanced heat transfer technique. Fortunately, to meet the tuning condition, the required Wormersley number in this case is of order 1. As already discussed in the introduction, in order to gain the benefit of axial heat transfer, the use of large tidal displacement is always preferred. However, such an increase must be limited by the requirement of no convective net mass transfer occurring between two reservoirs and may also be constrained by the ability of the device to withstand the increase in the exciting pressure gradient. Phase Lags We are now in the position to study the phase lags of the Lagrangian displacement in the oscillatory pipe flow. Fig. 4-10 shows the phase variations (in degrees) along radius for Wormersley number varying from a = 0.1 to 4. Some numerical results are also shown in Tables 4-2 and 4-3. All of the data shown in Fig. 4-10 and the tables have the phase angle measured relative to the exciting pressure gradient. Two features can be seen in Fig. 4-10; first, in the core portion, the phase lags are almost equal to n/2 when the Wormersley number is small, while the lags are almost n when the Wormersley number is large, and second, 80 the phase lags vary along radius, especially in the boundary layer. It is such phase lags that allow the existing temperature gradient in the very thin boundary layer of the oscillating pipe flow to act as region of temporary heat storage. It absorbs heat when the temperature of the core Table 4-2 Phase Lags Along Radius (Working Medium: H2O, Ax = 10 cm) Nodal point K a = 0.1 a = 1. a = 2. ?=r/Rl phase rÂ¡= r/Rl phase r?=r/Rl phase 1 0.0000 90.63 0.0000 102.06 0.0000 134.28 2 0.1667 90.63 0.0366 102.06 0.1095 134.10 3 0.3333 90.63 0.0957 102.06 0.2164 133.38 4 0.5000 90.63 0.1913 101.88 0.2836 132.66 5 0.5833 90.63 0.2593 101.70 0.3620 131.76 6 0.6667 90.63 0.3458 101.52 0.4535 130.50 7 0.7500 90.63 0.4559 101.16 0.5603 128.70 8 0.8333 90.63 0.5958 100.62 0.6849 126.00 9 0.9167 90.63 0.7737 99.72 0.8303 122.58 10 1.0000 0.00 1.0000 0.00 1.0000 0.00 81 Table 4-2 continued Nodal point K a = 3 a = 4 a = 7 7=r/Rl phase r?=r/Rl phase r?=r/Rl phase 1 0.0000 170.83 0.0000 179.82 0.0000 181.62 2 0.0917 170.47 0.1242 179.28 0.0970 181.62 3 0.1818 169.75 0.2370 178.02 0.1892 181.62 4 0.2703 168.31 0.3395 176.04 0.2769 181.44 5 0.3573 166.51 0.4324 173.70 0.3604 180.72 6 0.4427 164.00 0.5168 171.00 0.4397 179.28 7 0.5267 161.12 0.5935 168.12 0.5152 177.12 8 0.6091 157.88 0.6630 165.24 0.5869 174.06 9 0.6901 154.64 0.7262 162.18 0.6552 170.28 10 0.7696 150.69 0.7835 159.12 0.7201 165.78 11 0.8478 146.73 0.8356 156.06 0.7818 160.74 12 0.9246 142.42 0.8828 153.18 0.8406 155.16 13 1.0000 0.00 0.9257 150.30 0.8964 149.04 14 0.9646 147.60 0.9495 142.56 15 1.0000 0.00 1.0000 0.00 82 Table 4-2 continued Nodal Point K a = 10 H II a a = 20 7 =r/Rl phase 7=r/Rl phase 7=r/Rl phase 1 0.0000 180.54 0.0000 179.46 0.0000 179.10 2 0.0884 180.72 0.0834 179.10 0.1635 180.54 3 0.1740 181.08 0.1647 180.18 0.3032 185.93 4 0.2566 181.80 0.2440 184.50 0.4227 182.34 5 0.3364 182.52 0.3215 190.25 0.5247 186.65 6 0.4135 182.70 0.3970 192.41 0.6120 184.14 7 0.4880 181.80 0.4709 189.53 0.6865 184.14 8 0.5599 179.64 0.5428 183.06 0.7502 181.98 9 0.6295 175.68 0.6130 174.07 0.8046 178.38 10 0.6966 170.46 0.6815 164.00 0.8512 173.71 11 0.7616 163.80 0.7484 153.20 0.8909 167.95 12 0.8243 156.06 0.8136 141.70 0.9249 161.12 13 0.8849 147.24 0.8773 130.19 0.9540 153.57 14 0.9434 137.52 0.9394 118.32 0.9788 146.01 15 1.0000 0.00 1.0000 0.00 1.0000 0.00 83 Table 4-3 Comparison of Phase Lags With Different Working Mediums (Ax = 10) Nodal Point a = 1. a = 5. K r) =r/Rl Phase Lags t) =r/Rl Phase Lags Water Mercury Water Mercury 1 0.0000 102.06 102.14 0.0000 181.62 181.62 2 0.0161 102.06 102.14 0.0714 181.44 181.26 3 0.0366 102.06 102.14 0.1429 180.90 180.54 4 0.0626 102.06 102.14 0.2143 179.64 179.46 5 0.0957 102.06 102.14 0.2857 178.02 178.02 6 0.1378 101.88 102.14 0.3571 176.04 175.86 7 0.1912 101.88 102.14 0.4286 173.34 173.37 8 0.2593 101.70 102.14 0.5000 170.28 170.11 9 0.3458 101.52 101.78 0.5714 166.50 166.51 10 0.4559 101.16 101.42 0.6429 162.18 162.20 11 0.5957 100.62 101.06 0.7143 157.50 157.52 12 0.7737 99.72 99.98 0.7857 152.10 152.13 13 0.0000 0.00 0.00 0.8571 146.16 146.37 14 0.9286 139.86 139.90 15 1.0000 0.00 0.00 Dimensionless Radius Fig 4-10 Phase Variation Along Radius for Different Wormersley Numbers 85 slug flow is higher than that of the boundary layer, and it releases heat as the core temperature is relatively lower. This large temperature gradient enhanced by the existing velocity phase lags allows a large amount heat to be conductively transferred radially within a very short time and subsequently to be transferred axially by a convective coupling. Part 2. The Enhanced Heat Transfer Investigation We have examined some mechanical characteristics in oscillating pipe flow, and compared the computed solution of the velocity field with Uchida's solution. The results for the velocity profiles are in good agreement. Nevertheless, before a detailed examination of the thermal field, it is first necessary to test the current developed ETP code when applying the temperature equations (2-6),(2-7). It is seen that the energy equations strongly depend on the velocity distribution and its build-up process, so that one can use analytic periodic velocity state (Eqs. 1-4 and 1-5, with no build-up process), and the computed velocity (with build-up process) to verify the correctness of the resulting thermal variables. The enhanced heat flux is a function of both velocity and temperature (Eqs. 2-34 and 2-35) and was chosen for a comparison of the analytic and numerical results of the problem. Part A of table 4-4 shows the results of the computed enhanced axial heat flux as well as the axial conduction heat flux when using the analytic velocity 86 Table 4-4 The Comparison of Enhanced Heat Flux Using Numerical Velocity with Heat Flux Using Analytical Velocity (Model 3, Water-Glass, Pr = 7.03, a = 3) A. Heat Flux Using Analytic Velocity AX (cm) (w/cm2 0 K) 4>f (w/cm2 K) (w/cm2 K) PA (w/cm4 K) 0.9839 0.0230 0.0149 0.0811 0.0238 2.9519 0.1855 0.0132 0.0724 0.0213 4.9194 0.3864 0.0098 0.0541 0.0159 6.8868 0.5810 0.0075 0.0415 0.0122 7.8847 0.6775 0.0067 0.0369 0.0109 9.8390 0.8638 0.0055 0.0302 0.0089 B. Heat Flux Using Computed Velocity 1.0033 0.0234 0.0149 0.0811 0.0232 3.0101 0.1870 0.0133 0.0728 0.0206 5.0164 0.3899 0.0097 0.0533 0.0155 7.0226 0.5790 0.0076 0.0420 0.0117 8.0402 0.6746 0.0068 0.0374 0.0104 10.0328 0.8597 0.0055 0.0302 0.0085 * Model 3, Water-Glass * 4> enhanced axial heat flux * * 4>\j axial heat flux by conduction in pipe wall * p = |

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