Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data

By R.K. Shah, A. L. London, Thomas F. Irvine and James P. Hartnett

Laminar Flow Forced Convection in Ducts is a sourcebook for compact heat exchanger analytical data. This book describes the analytical solutions for laminar fluid flow and forced convection heat transfer in circular and noncircular pipes, including applicable differential equations and boundary conditions involving velocity and temperature problems of fluid flow. The book also discusses fluid flow—how much power is required to pump fluids through the heat exchanger, as well as the heat transfer—the determination of q" distribution, and the temperature of fluid and walls. The text also analyzes the coolant or heat transfer fluid flows in a nuclear power reactor composed of a bundle of circular section fuel rods located inside a round tube. R.A. Axford addresses fluid flow and heat transfers results for the rod bundle geometry in "Heat Transfer in Rod Bundles." The book also provides an overview and guidelines that can be used for the designer and the applied mathematician. This book is suitable for engineers working in electronics, aerospace, instrumentation, and biomechanics that use cooling or heating exchanges or solar collection systems.

• Language: English
• Publisher: Academic Press
• Release date: Jun 28, 2014
• ISBN: 9781483191300

Book preview

LAMINAR FLOW FORCED CONVECTION IN DUCTS

A Source Book for Compact Heat Exchanger Analytical Data

R.K. Shah

Harrison Radiator Division, General Motors Corporation, Lockport, New York 14094

A.L. London

Mechanical Engineering Department, Stanford University, Stanford, California 94305

Table of Contents

Cover image

Title page

Advances in HEAT TRANSFER

Copyright

Dedication

PREFACE

Chapter 1: Introduction

Publisher Summary

Chapter 2: Differential Equations and Boundary Conditions

Publisher Summary

A Velocity Problem

B Temperature Problem

C Thermal Boundary Conditions

Chapter 3: Dimensionless Groups and Generalized Solutions

Publisher Summary

A Fluid Flow

B Heat Transfer

Chapter 4: General Methods for Solutions

Publisher Summary

A Fully Developed Flow

B Hydrodynamically Developing Flow

C Thermally Developing Flow

Chapter 5: Circular Duct

Publisher Summary

A Fully Developed Flow

B Hydrodynamically Developing Flow

C Thermally Developing and Hydrodynamically Developed Flow

D Simultaneously Developing Flow

Chapter 6: Parallel Plates

Publisher Summary

A Fully Developed Flow

B Hydrodynamically Developing Flow

C Thermally Developing and Hydrodynamically Developed Flow

D Simultaneously Developing Flow

Chapter 7: Rectangular Ducts

Publisher Summary

A Fully Developed Flow

B Hydrodynamically Developing Flow

C Thermally Developing and Hydrodynamically Developed Flow

D Simultaneously Developing Flow

Chapter 8: Triangular Ducts

Publisher Summary

A Fully Developed Flow

B Hydrodynamically Developing Flow

C Thermally Developing Flow

Chapter 9: Elliptical Ducts

Publisher Summary

A Fully Developed Flow

B Thermally Developing Flow

Chapter 10: Other Singly Connected Ducts

Publisher Summary

A Sine Ducts

B Trapezoidal Ducts

C Rhombic Ducts

D Quadrilateral Ducts

E Regular Polygonal Ducts

F Circular Sector Ducts

G Circular Segment Ducts

H Annular Sector Ducts

I Moon-Shaped Ducts

J Circular Ducts with Diametrically Opposite Flat Sides

K Rectangular Ducts with Semicircular Short Sides

L Corrugated Ducts

M Cusped Ducts

N Cardioid Ducts

O Miscellaneous Singly Connected Ducts

Chapter 11: Small Aspect Ratio Ducts

Publisher Summary

Chapter 12: Concentric Annular Ducts

Publisher Summary

A Fully Developed Flow

B Hydrodynamically Developing Flow

C Thermally Developing and Hydrodynamically Developed Flow

D Simultaneously Developing Flow

Chapter 13: Eccentric Annular Ducts

Publisher Summary

A Fully Developed Flow

B Hydrodynamically Developing Flow

C Thermally Developing and Hydrodynamically Developed Flow

D Simultaneously Developing Flow

Chapter 14: Other Doubly Connected Ducts

Publisher Summary

A Confocal Elliptical Ducts

B Regular Polygonal Ducts with Central Circular Cores

C Isosceles Triangular Ducts with Inscribed Circular Cores

D Elliptical Ducts with Central Circular Cores

E Circular Ducts with Central Regular Polygonal Cores

F Miscellaneous Doubly Connected Ducts

Chapter 15: Longitudinal Flow over Circular Cylinders

Publisher Summary

A Fully Developed Flow

B Thermally Developing and Hydrodynamically Developed Flow

Chapter 16: Longitudinal Fins and Twisted Tapes within Ducts

Publisher Summary

A Longitudinal Thin Fins within a Circular Duct

B Longitudinal Thin Fins within Square and Hexagonal Ducts

C Longitudinal Thin Fins from Opposite Walls within Rectangular Ducts

D Longitudinal Thin V-Shaped Fins within a Circular Duct

E Longitudinal Triangular Fins within a Circular Duct

F Circular Duct with a Twisted Tape

Chapter 17: Discussion—An Overview for the Designer and the Applied Mathematician

Publisher Summary

A Comparisons of Solutions

B Heat Exchangers with Multi-Geometry Passages in Parallel

C Influence of Superimposed Free Convection

D Influence of Temperature-Dependent Fluid Properties

E Comments on the Format of Published Papers

F The Complete Solution

G Areas of Future Research

Nomenclature

Appendix

References

Author Index

Subject Index

Advances in HEAT TRANSFER

Edited by

Thomas F. Irvine, Jr.

Department of Mechanics

State University of New York at Stony Brook

Stony Brook, New York

James P. Hartnett

Energy Resources Center

University of Illinois at Chicago Circle

Chicago, Illinois

Supplement 1

LAMINAR FLOW FORCED CONVECTION IN DUCTS

A Source Book for Compact Heat Exchanger Analytical Data

R. K. Shah and A. L. London

Copyright

Copyright © 1978, by Academic Press, Inc.

all rights reserved.

no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by

ACADEMIC PRESS, INC. (LONDON) LTD.

24/28 Oval Road, London NW1 7DX

Library of Congress Catalog Card Number: 78–62274

ISBN 0-12-020051-1

printed in the united states of america

Dedication

To Rekha Shah

Charlotte London

PREFACE

The purpose of writing this monograph is to present the available analytical solutions for laminar fluid flow and forced convection heat transfer in circular and noncircular pipes. The subject has importance in a large variety of traditional engineering disciplines, such as heating and cooling devices used in electronics, biomechanics, aerospace, instrumentation, and pipelines for oil, water and other fluids, and, in particular, for compact heat exchanger and solar collector designs. Generally, one way to reduce heat exchanger costs is to use more compact surfaces, as both the cost per unit area and heat transfer flux per unit temperature difference are simultaneously improved.

With the advancement of science and technology, a variety of passage geometries are utilized for internal flow forced convection heat transfer applications. Analytical laminar flow solutions for many passage geometries are available in the heat transfer literature. However, these solutions are scattered throughout the world-wide technical journals, reports, and student theses. These solutions, often generated by applied mathematicians, are not always accessible to the practicing engineers and researchers who are generally faced with a limited availability of time. Also, these solutions may be difficult to interpret by the engineer, as they may be presented with all the details of analyses but without the final results which an engineer would like to use. The aim of this monograph is to present all the analytical solutions known to the authors for laminar internal flow, in a readily usable, coordinated, and unified format. In the interest of brevity, derivations of formulas and detailed explanations, such as are appropriate in a textbook, are omitted; only the final results are presented in graphical and tabular forms. The extensive bibliography will provide the detailed information needed by scholars in this field of analysis.

This monograph is the result of the authors’ need for a compendium of solutions for investigating improved surface geometries. It is an outgrowth of an initial compilation of laminar flow solutions prepared five years ago as the dissertation of the senior author. Hopefully, it will also serve as a useful reference to the engineers and researchers in the field.

Since it would take many volumes to do justice to all aspects of internal laminar flows, the scope of this monograph is restricted to laminar flow and forced convection for a Newtonian fluid with constant properties, passing through stationary, straight, nonporous ducts of constant cross section. Except for the twisted-tape flows, all forms of body forces are omitted. Also, magnetohydrodynamic flows, electrical conducting flows, heat radiating flows, and the effects of natural convection, change of phase, mass transfer and chemical reactions are all excluded. Although some of the geometries described may not be visualized for the compact heat exchanger application, they are included for the completeness of available solutions for the laminar flow problem.

Emphasis is given to the summary of analytical solutions. Except for the circular tube, whenever experimental results are available to support or contradict the theory, they are also described in the text. For the circular tube, a vast amount of analytical and experimental results are available for the laminar flow forced convection. These experimental results generally support the analytical results. However, no compilation is provided for these experimental results because of space limitations.

We have not attempted a scientific or technological history. Although an effort was made to compile laminar flow analytical solutions from all available sources, it is quite probable that several important sources may not have come to the authors’ attention.

The first four chapters describe the basic problems, solution techniques, dimensionless groups and generalized solutions. Chapters V through XVI describe the solutions for 39 duct geometries. Chapter XVII provides an overview of these solutions. Table 136 provides a ready reference for locating a particular solution. We recommend reading Chapters I and XVII first.

We are grateful to many researchers who have furnished the information requested. Without their whole-hearted assistance, this monograph would not be of great value. Their assistance is acknowledged as a personal communication in the references. The authors are thankful to Prof. T. F. Irvine, Jr., the editor, Prof. H. C. Perkins and Dr. M. R. Doshi who read the manuscript and made many helpful suggestions. Prof. K. P. Johannsen reviewed Chapter XV. Dr. D. B. Taulbee reviewed sections on the hydrodynamically developing flow. The authors are grateful to these researchers for their constructive suggestions. The institutional support of Stanford University, the Office of Naval Research, and the Harrison Radiator Division of General Motors Corporation is gratefully acknowledged. Lastly, the first author would like to express sincere appreciation to his wife Rekha for her patience, understanding and encouragement during the preparation of this monograph.

R.K. SHAH and A.L. LONDON

Chapter I

Introduction

Publisher Summary

This chapter presents the introduction to laminar flow heat transfer. Interest in heat exchanger surfaces with a high ratio of heat transfer area to core volume is increasing at an accelerated pace. The primary reasons for the use of these more compact surfaces are that smaller, lighter-weight, and lower-cost heat exchangers are the result. These gains are brought about both by the direct geometric advantage of higher area density and also a higher heat transfer coefficient for the smaller flow passages. Because of the smaller flow passage hydraulic radius, with gas flows particularly, the heat exchanger design range for the Reynolds number usually falls well within the laminar flow regime. Thereafter, theoretically derived laminar flow solutions for fluid friction and heat transfer in ducts of various flow cross-section geometries become important.

Interest in heat exchanger surfaces with a high ratio of heat transfer area to core volume is increasing at an accelerated pace. The primary reasons for the use of these more compact surfaces is that smaller, lighter-weight, and lower-cost heat exchangers are the result. These gains are brought about both by the direct geometric advantage of higher area density and also a higher heat transfer coefficient for the smaller flow passages.

Because of the smaller flow passage hydraulic radius, with gas flows particularly, the heat exchanger design range for the Reynolds number usually falls well within the laminar flow regime. It follows then that theoretically derived laminar flow solutions for fluid friction and heat transfer in ducts of various flow cross-section geometries become important. These solutions are the subject matter of this monograph. A direct application of these results may be in the development of new heat exchanger surfaces with improved characteristics. A critical examination of the theoretical solutions may prove to be fruitful because there is a wide range for the heat transfer coefficient at a given friction power for geometries of different cross sections. In addition to compact heat exchangers, applications of laminar flow theory are also of interest in the aerospace, nuclear, biomedical, electronics, and instrumentation fields.

It has long been realized that laminar flow heat transfer is dependent on the duct geometry, flow inlet velocity and temperature profiles, and wall temperature and/or heat flux boundary conditions. These conditions are difficult to produce even in the laboratory; nevertheless, there is substantial ongoing experimental research effort being devoted to this task. A theoretical base is needed in order to interpret the experimental results and extrapolate them for the design of practical heat exchanger systems. However, it is recognized that this theory is founded on idealizations of geometry and boundary conditions that are not necessarily well duplicated either in the application or even in the laboratory. The development of this theoretical base has been a fertile field for applied mathematicians since the early days of the science of heat transfer. Today, by the application of modern computer technology, analysis has exceeded, to some degree, the experimental verification. Since there are a large number of theoretical solutions available in the literature, a compilation and comparison of these solutions, employing a uniform format, should be of value to the designer as well as the researcher.

The study of heat transfer in laminar flow through a closed conduit was first made by Graetz [1] in 1883, and later independently by Nusselt [2] in 1910. Drew [3] in 1931 prepared a compilation of existing theoretical results for heat transfer. Dryden et al. [4] in 1932 compiled fully developed laminar flow solutions for ducts of various geometries. Later, several literature surveys were made for particular duct geometries. Rohsenow and Choi [5] in 1961 presented a limited compilation of solutions for simple cylindrical ducts. Kays and London [6] published a compilation in 1964 pertinent to compact heat exchangers. The theoretical development and the details of analysis for the laminar as well as turbulent flow problems were described in depth by Kays [7] in 1966. Such analyses are also available in other heat transfer text books. Petukhov [8] in 1967 compiled laminar duct flow solutions primarily for liquids. He presented mathematical details for the solutions including the consideration of temperature-dependent viscosity. Only a few solutions are covered by Petukhov compared to the present scope of investigation. Martynenko and Eichhorn [9] in 1968 compiled laminar hydrodynamic and thermal entry length solutions primarily for circular tube and parallel plates. With the cooperation of 30 British industries, Porter [10] in 1971 compiled the laminar flow solutions for Newtonian and non-Newtonian liquids with constant and variable fluid properties. The purpose of Porter’s survey was to identify those areas which presented difficulties in thermal designs of chemical, plastic, and food-processing problems. He suggested the best design equations available and made specific recommendations for future investigation. Kooijman [11] in 1973 compiled laminar thermal entrance solutions for fully developed, constant-property Newtonian fluid flow through circular and noncircular ducts. Kays and Perkins [12] in 1973 presented a compilation of analytical forced-convection solutions for laminar and turbulent flow through circular and noncircular ducts. The heat transfer literature up to 1967 was reviewed by Kays and Perkins, who provided available results in terms of equations, tables, and graphs for design purposes.

Shah and London [13] in 1971 made a literature survey up to December 1970 for laminar forced-convection heat transfer and flow friction through straight and curved ducts. Available analytical solutions for constant-property Newtonian fluids, both liquids and gases, were compiled in contrast to liquids only, as considered by Porter [10]. The area investigated by Shah and London was much more exhaustive for constant-property Newtonian fluids and thus complemented the work of Porter. The present work augments and extends this first effort [13].

The specific objectives of the present work are as follows:s

(1) To provide an up-to-date compilation of analytical laminar internal flow solutions with results in numerical and graphical dimensionless forms.

(2) To clarify and systemize various thermal boundary conditions and dimensionless groups.

(3) To indicate those areas where further contributions may be made.

Primarily, the English language literature up to December 1975 is reviewed. Emphasis is given to the analytical solutions for developed and developing velocity and temperature profiles through axisymmetric and two-dimensional constant cross section straight ducts. Only the forced convection steady-state laminar flow of a constant-property Newtonian fluid through a stationary, nonporous duct is considered. The effects of thermal energy sources, viscous dissipation, flow work, and axial heat conduction within the fluid have been considered in the literature for laminar flow through some duct geometries. This limited available information is included at appropriate places. All forms of body forces are neglected, except for the centrifugal force for the twisted-tape flow. Magnetohydrodynamic flows, electrically conducting flows, high-temperature (heat-radiating) flows, etc., are not considered. Also omitted are the effects of natural convection, change of phase, mass transfer, and chemical reaction. The solutions are applicable for ducts with smooth or rough walls, as long as the surface roughness does not significantly affect the cross section of the duct geometry.

The analytical solutions for the laminar flow problem were obtained from worldwide technical journals (as outlined in the Appendix), reports, and student theses. In reviewing the literature for each geometry, a chronological history is not preserved. Instead, material is classified according to boundary conditions and geometry. No attempt has been made to present the solutions in detail. Only those final results which are useful to a heat transfer designer are presented in tabular and graphical forms. Presented for each geometry is the information available for u, um, f Re, Kd(∞), Ke(∞), Kand Nusselt numbers corresponding to the boundary conditions of Table 2. The originally available tabular and graphical information has been augmented by (1) more information obtained from correspondence with authors, (2) the knowledge of limiting cases of boundary conditions or duct geometries, and (3) results computed by the present authors for a limited number of solutions in more detailed than was heretofore available.

Differential equations and boundary conditions for hydrodynamically and thermally developing and developed laminar flows are described in Chapter II. Definitions of important variables and dimensionless groups associated with the laminar flow problem are presented in Chapter III. General methods used in the heat transfer and fluid mechanics literature to solve the problems formulated in Chapter II are outlined in Chapter IV. The analytical solutions to the aforementioned laminar flow problem are summarized for 40 duct geometries in Chapters V–XVI. For the designer, an overview of the subject is provided in Chapter XVII. This overview includes comparisons of the solutions, approximate solutions for heat exchangers with multigeometry passages in parallel, influence of superimposed free convection, influence of temperature-dependent fluid properties, comments on the format of published papers, the complete solution for the laminar flow problem for constant cross-sectional ducts, and suggested areas of future research.

Chapter II

Differential Equations and Boundary Conditions

Publisher Summary

This chapter presents differential equations and boundary conditions. Flow is laminar when the velocities are free of macroscopic fluctuations at any point in the flow field. For steady-state laminar flow, all velocities at a stationary point in the flow field remain constant with respect to time, but velocities may be different at different points. Laminar flow, also referred to as viscous or streamline flow, is characteristic of a viscous fluid flow at low Reynolds number. Laminar flow in a two-dimensional stationary straight duct is designated as hydrodynamically fully developed when the fluid velocity distribution at a cross section is of an invariant form. The fluid particles move in definite paths called streamlines, and there are no components of fluid velocity normal to the duct axis. In a fully developed laminar flow, the fluid appears to move by sliding laminar of infinitesimal thickness relative to adjacent layers. Depending upon the smoothness of the tube inlet and tube inside wall, fully developed laminar flow persists up to Re ≤ 2300 for a duct length L greater than the hydrodynamic entry length Lhy.

The applicable differential equations and boundary conditions for the velocity and temperature problems are described in this chapter. These are for steady-state laminar flow through constant cross section ducts. Before proceeding to the details, the following terms are defined: laminar developed and developing flows, forced convection heat transfer, and thermally developed and developing flows.

Flow is laminar when the velocities are free of macroscopic fluctuations at any point in the flow field. For steady-state laminar flow, all velocities at a stationary point in the flow field remain constant with respect to time, but velocities may be different at different points. Laminar flow, also referred to as viscous or streamline flow, is characteristic of a viscous fluid flow at low Reynolds number.

Laminar flow in a two-dimensional stationary straight duct is designated as hydrodynamically fully developed (or established) when the fluid velocity distribution at a cross section is of an invariant form, i.e., independent of the axial distance x, as shown in Fig. 1:

FIG. 1 Developed and developing laminar flow.

(1a)

(1b)

2300 for a duct length L greater than the hydrodynamic entry length L2300, the developing flow, as described below, could exist.

The hydrodynamic entrance region of the duct is that region where the velocity boundary layer is developing, for example, from zero thickness at the entrance to a thickness equal to the pipe radius far downstream. In this region, the fluid velocity profile changes from the initial profile at the entrance to an invariant form downstream. The flow in this region, as a result of the viscous fluid behavior, is designated as hydrodynamically developing (or establishing) flow, and is also shown in Fig. 1.

The hydrodynamic entrance length Lhy is defined as the duct length required to achieve a maximum duct section velocity of 99% of that for fully developed flow when the entering fluid velocity profile is uniform. The maximum velocity occurs at the centroid for the ducts symmetrical about two axes (e.g., circular tube and rectangular ducts). The maximum velocity occurs away from the centroid on the axis of symmetry for isosceles triangular, trapezoidal, and sine ducts. For nonsymmetrical ducts, no general statement can be made for the location of umax. There are a number of other definitions used in the literature for Lhy. However, unless specified, the foregoing definition is used throughout this monograph.

10⁵ [14,15]. In this short duct, this laminar boundary layer soon becomes turbulent, and fully developed turbulent flow exists at this high Reynolds number downstream from the developing region.

In convection heat transfer, a combination of mechanisms is active. Pure conduction exists at the wall, but away from the wall internal thermal energy transport takes place by fluid mass motion as well as conduction. Thus the convection heat transfer process requires a knowledge of both heat conduction and fluid flow. If the motion of the fluid arises solely due to external force fields such as gravity, centrifugal, or Coriolis body forces, the process is referred to as natural or free convection. If the fluid motion is induced by some external means such as pump, blower, fan, wind, or vehicle motion, the process is referred to as forced convection. Throughout this monograph, only pure forced convection heat transfer is considered.

Laminar flow in a two-dimensional stationary duct is designated as thermally fully developed (or established) when, according to Seban and Shimazaki [16], the dimensionless fluid temperature distribution, as expressed below in brackets, at a cross section is invariant, i.e., independent of x:

(2)

However, note that t is a function of (y, z) as well as x, unlike u, which is a function of (y, z) only and is independent of x for fully developed flow. Hydrodynamically and thermally developed flow is designated throughout simply as fully developed flow.

The thermal entrance region of the duct is that region where the temperature boundary layer is developing. For this region, the dimensionless temperature profile (tw,m − t)/(tw,m − tm) of the fluid changes from the initial profile at a point where the heating is started to an invariant form downstream. The flow in this region is designated as thermally developing flow. The velocity profile in this region could be either developed or developing. Thermally developing flow with a developing velocity profile is referred to as simultaneously developing flow.

The thermal entrance length Lth is defined, somewhat arbitrarily, as the duct length required to achieve a value of local Nux,bc equal to 1.05 Nubc for fully developed flow, when the entering fluid temperature profile is uniform.

A Velocity Problem

The differential equations and boundary conditions for the velocity problem are presented separately below for developed and developing flows.

1 HYDRODYNAMICALLY DEVELOPED FLOW

Consider a fully developed, steady-state laminar flow in a two-dimensional singly (as in Fig. 2) or multiply connected stationary duct with the boundary Γ. The fluid is idealized as liquid or low-speed gas with the fluid properties ρ, μ, cp, and k constant (independent of fluid temperature). Moreover, body forces such as gravity, centrifugal, Coriolis, and electromagnetic do not exist. The applicable momentum equation is [7]

FIG. 2 A singly connected duct of constant cross-sectional area.

(3)

where x is the axial coordinate along the flow length of the duct and c1 is defined as a pressure drop parameter. ∇² is the two-dimensional Laplacian operator. Note that the right-hand side of Eq. (3) is independent of (y, z) or (r, θ), and so it is designated as a constant c1. Equation (3), in Cartesian coordinates, is

(4)

In cylindrical coordinates, it is

(5)

The boundary condition for the velocity problem is the no-slip condition, namely,

(6)

By the description of fully developed laminar flow of an incompressible fluid, the solution of the continuity equation (conservation of mass) is implicitly given by Eq. (1). Moreover, the continuity equation is utilized in deriving Eq. (3). Consequently, only the solution to the momentum equation† is required for the fully developed laminar fluid flow problem described below.

2 HYDRODYNAMICALLY DEVELOPING FLOW

As described on p. 6, the hydrodynamic entrance region is where the velocity boundary layer is developing. However, the hydrodynamic entrance flow problem is not strictly a boundary layer problem. This is because very near the entrance the axial molecular momentum transport μ(∂²u/∂x²) is not a negligible quantity, and far from the entrance, the boundary layer thickness is not negligible compared to the characteristic dimension of the duct. It is also possible that very close to the entry the transverse pressure gradient across the section may not be negligible. To take these effects into account, a complete set of Navier-Stokes equations needs to be solved. However, except for very close to the entry, μ(∂²u/∂x²) (∂p/∂y), and/or (∂p/∂z) terms are negligible. Even though the physical concept† of a boundary layer introduced by Prandtl is not strictly applicable to the entrance flow problem, the Prandtl boundary layer idealizations

(7a)

(7b)

are good approximations for laminar flow in ducts. As a result, it is found that the terms of the y and z momentum equations are one order of magnitude smaller than the corresponding terms of the x momentum equation and hence may be neglected. In this case, the fluid pressure is a function of x only. Additionally, if all the idealizations made for the fully developed flow are invoked, the governing boundary layer type x momentum equation, for axially symmetric flow, in cylindrical coordinates from [7] is

(8)

and in Cartesian coordinates, it is

(9)

The no-slip boundary condition for this case is

(10)

An initial condition is also required, and usually uniform velocity profile is assumed at the entrance:

(11)

In addition, the continuity equation needs to be solved simultaneously. In cylindrical coordinates [7], it is

(12)

and in Cartesian coordinates [7], it is

(13)

The solution to the hydrodynamic entry length problem is obtained by solving Eqs. (8) and (12) or (9) and (13) simultaneously with the boundary and initial conditions of Eqs. (10) and (11).

For axisymmetric ducts, two unknowns u and v are obtained by solving Eqs. (8) and (12). Since Eq. (8) is nonlinear, various approximate methods have been used to obtain the solution. These methods are described briefly in Chapter IV. The axial pressure distribution is then obtained from another physical constraint such as from the solution of the mechanical energy integral equation† or integral continuity equation. For a two-dimensional duct, in which case both v and w components exist in the entrance region, a third equation [in addition to Eqs. (9) and (13)] is essential for a rigorous solution for u, v, and w. Such an equation for the square and eccentric annular ducts is briefly described on pp. 211 and 333.

B Temperature Problem

The solution to the temperature problem involves the determination of the fluid and wall temperature distributions and/or the heat transfer rate between the wall and the fluid. Such a problem has mathematically complex features, such as heat conduction in normal, peripheral, and axial directions; variable heat transfer coefficients along the periphery and in the axial direction; and invariant or changing dimensionless velocity and temperature profiles along the flow length. The theoretical solutions of such problems provide the quantitative design information on the controlling dimensionless groups and indicate when to neglect certain effects. To illustrate this interplay, some idealized problems are formulated in this section for laminar internal flow forced convection heat transfer.

Attention is focused primarily on the forced convection heat transfer rate from the wall to the fluid (or vice versa). The determination of this heat transfer involves the solution of either (a) the conventional convection problem or (b) the conjugated problem.

In the conventional convection problem, heat transfer through the wall is characterized by a thermal boundary condition directly or indirectly specified at the wall-fluid interface. The velocity and forced convection temperature problems are solved only for the fluid region. The solution to the temperature problem for the solid wall is not needed; but in its application it is implicitly assumed that the duct has a uniform wall thickness and there does not exist simultaneous heat conduction in the wall in axial, peripheral, and normal directions. The heat transfer rate through the wall, normal to the flow, is subsequently determined from the solution of the fluid temperature problem for the specified wall-fluid interface boundary conditions. The dimensionless temperature profile of the fluid may be either fully developed or developing along the flow length for this problem class.

However, in a broader view, the heat transfer through the duct wall by conduction may have significant normal and/or peripheral as well as axial components; or the fluid heating (or cooling) flux may be nonuniform around the duct periphery; or the duct wall may be of nonuniform thickness. As a result, the temperature problem for the solid wall needs to be analyzed simultaneously with that for the fluid in order to establish the actual wall-fluid interface heat transfer flux distribution. This combination is referred to as the conjugated problem. The simultaneous solutions of the energy equations for the solid and fluid media are obtained by considering temperature and heat flux as continuous functions at the solid wall-fluid interface. The velocity distribution for the fluid medium must first be found by solving the applicable continuity and momentum equations. The dimensionless temperature profile is always variant (never fully developed) for this class of problems.

Some idealized conjugated and conventional convection problems are now formulated.

1 CONJUGATED PROBLEM

As described above, the formulation of the conjugated problem involves the application of the energy equations for both the fluid and solid regions. The temperature and heat fluxes at the solid-fluid interface are considered continuous.

As an illustration, the conjugated problem is formulated for a thick-walled circular tube and the fluid on the inside. The heat transfer from the other fluid side is represented by a thermal boundary condition. Consider a steady-state, laminar, constant-properties flow of a Newtonian fluid in the duct of constant cross-sectional area. Thermal energy sources, viscous dissipation effects, and flow work within the fluid are neglected. In the absence of free convection, mass diffusion, chemical reaction, change of phase, and electromagnetic effects, the governing differential equations and boundary conditions are as follows [17] (refer to Fig. 3 for the system coordinates).

FIG. 3 Coordinate system for the circular tube conjugated problem.

Fluid

(14)

(15)

(16)

(17)

Solid

(18)

(19)

(20)

Introduce x* = x/Dh Pe, r* = r/a, X = x/L, u* = u/um, and R* = R/a’ = (r – a)/a’. Equations (14), (16), and (18) reduce to the forms

(21)

(22)

(23)

where Rw = kf/Uw/Dh and Uw = kw/a’.

As can be seen, the energy equations for the solid and fluid media are coupled by the boundary conditions, Eqs. (15) and (16).

The conjugated problem for the circular tube [18–20] and parallel plates [18,19,21–23] has been analyzed by employing highly sophisticated mathematical techniques. The solution to the conjugated problem for a circular tube by Mori et al. [20] is the most comprehensive.

2 CONVENTIONAL CONVECTION PROBLEM

As described earlier, the formulation of this class of temperature problem involves the specification of the applicable energy equation and the thermal boundary condition at the duct wall. The differential energy equations are presented in the following text for two kinds of flows: (a) fully developed flow [both hydrodynamically and thermally, Eqs. (1) and (2)], and (b) thermally developing flows (with developed or developing velocity profiles). In contrast, note that the flow is always thermally developing for a conjugated problem.

a Thermally Developed Flow

Consider a steady-state, laminar, constant-properties (μ, cp, k) flow in a duct of constant cross-sectional area. In the absence of free convection, mass diffusion, chemical reaction, change of phase, and electromagnetic effects, the governing differential energy equation for a perfect gas is [7]

(24)

The third term on the right-hand side of Eq. (24) represents part of the work done by the fluid on adjacent layers due to action of shear forces and is usually referred to as viscous dissipation in the English language literature and as internal friction in the Russian literature. The fourth term on the right is referred to as the flow work and work done by pressure forces by English and Russian authors, respectively. Sometimes flow work is also referred to as gas compression work. This appears in the energy equation when the equations for energy conservation and momentum are manipulated so as to eliminate the kinetic energy term [7].

The corresponding differential energy equation for an incompressible liquid is the same as Eq. (24) with the omission of the u(dp/dx)/J term, because in the derivation this term cancels with the pressure component of enthalpy for an incompressible liquid [7]. For some liquids, under special circumstances, the flow work term may not be negligible, as discussed on p. 81.

Note that if Eq. (24) is operated on by ∇², the right-hand side of Eq. (24) will contain ∇²u, which equals c1 from Eq. (3). The resulting equation will be a fourth-order differential equation for the dependent variable t.

Equation (24) in the following development is made dimensionless, except for t, in a particular manner so that when the effects of axial heat conduction, thermal energy sources, and viscous dissipation are negligible, the resulting energy equation is parameter-free. For thermally developed or developing flow, introduce the dimensionless x* (= x/Dh Pe), y* (= y/Dh), z* (= z/Dh), and u* (= u/um) in Eq. (24). After some rearrangements, it reduces to the form

(25)

where for hydrodynamically developed flow

(26)

is a constant, dependent on the duct geometry. For hydrodynamically developing flow, the right-hand side of Eq. (26) is first obtained from the solution of the velocity problem and then is used in Eq. (25) in place of f Re.

For constant axial wall temperature boundary conditions, the dimensionless temperature θ is defined as

(27)

Then Eq. (25) is reduced to the dimensionless form

(28)

where the dissipation number Br is defined by Eq. (139), and ψ designates the bracketed term of Eq. (25) which includes viscous dissipation and flow work terms.

For constant axial wall heat flux boundary conditions, the dimensionless temperature Θ may be defined as

(29)

Substitution of this into Eq. (25) results in a dimensionless equation with the dissipation number Br’ defined by Eq. (140) as a parameter for the viscous dissipation (and for gases also flow work) effects.

The thermal boundary conditions associated with Eq. (24) will be discussed separately later. It should be emphasized that only the fluid medium needs to be analyzed for the solution of the posed (fully developed) temperature problem and the associated thermal boundary conditions. The solution to the temperature problem for the solid medium (duct wall) is not required for the analysis of fully developed flows regardless of thick or thin duct walls.

b Thermally Developing Flow

All the idealizations made in the fully developed case are still applicable except that thermal energy sources, viscous dissipation, and flow work within the fluid are neglected. Also, the boundary layer type idealizations, Eq. (7), and

(30)

are invoked. (Refer to the footnote on p. 9 and associated discussion.) The governing boundary layer type energy equation for the developing laminar temperature profile of a perfect gas or an incompressible liquid is [7]

(31)

in cylindrical coordinates, and

(32)

in Cartesian coordinates. Equation (31) includes the idealization that the heating is axially symmetrical.

The effects of thermal energy sources, viscous dissipation and flow work can be included in the energy equation by adding corresponding terms on the right-hand side of Eq. (31) or (32). These terms for the boundary layer type idealizations, when multiplied by ρcp, are presented in Eq. (24).

In addition to the thermal boundary conditions to be described in the following section, an initial condition is also required and is normally employed as the uniform temperature at the point where heating (or cooling) is started:

(33)

For the exact solution to the thermal entry length problem, the continuity and momentum equations need to be solved first to determine u, v, and w.

Thermal entry length problems are of three categories: (1) the velocity profile is fully developed and remains fixed while the temperature profile develops, (2) the velocity and temperature profiles develop simultaneously, and (3) the temperature profile starts developing at some point in the hydrodynamic entry region.

The thermal entry length problem of the second category, simultaneously developing velocity and temperature profiles, is also referred to as the combined entry length problem. The rate of developments of velocity and temperature profiles in the entrance region depends upon the fluid Prandtl number. For Pr = 1, the velocity and temperature profiles develop at the same rate, if both are uniform at the entrance. For Pr > 1, the velocity profile develops more rapidly than the temperature profile. For Pr < 1, the temperature profile develops more rapidly than the velocity profile. For the limiting case of Pr = ∞, the velocity profile is developed before the temperature profile starts developing. For the other limiting case of Pr = 0, the velocity profile never develops (remains uniform) while the temperature profile is developing. The idealized Pr = ∞ and 0 cases are good approximations for highly viscous fluids and liquid metals, respectively.

It may be noted that the thermal entry length problem of the first category, which is valid for any fluid (any Pr), is identical to the combined entry length problem for Pr = ∞. Thus, the combined entry length solution for Pr = ∞ is also a solution for any fluid with a developed velocity profile and a developing temperature profile. Similarly, the combined entry length solution for Pr = 0 also refers to the solution for any fluid with uniform velocity profile (slug flow) and developing temperature profile. One way of presenting the combined entry length solution is to present the Nusselt number versus x* = x/(Dh Re Pr) with Pr as a parameter. In such a plot, however, the parameters Pr = ∞ and 0 designate the nature of the velocity profile in the thermal entrance region as mentioned above, and should not be confused with Pr in x*.

C Thermal Boundary Conditions

The thermal boundary condition is the set of specifications describing temperature and/or heat flux conditions at the inside 1. Hence, the following classification of thermal boundary conditions, though applicable also to turbulent flows, is useful mainly for laminar flows.

A large variety of thermal boundary conditions can be specified for the temperature problem. Generally, these boundary conditions are not clearly and consistently defined in the literature, and therefore these highly sophisticated results are difficult to interpret by a designer. Shah and London [24] attempted to systemize the thermal boundary conditions. These results are now summarized for singly, doubly, and multiply connected ducts.

1 THERMAL BOUNDARY CONDITIONS FOR SINGLY CONNECTED DUCTS

A region Ω is called singly connected if any simple closed curve that lies in Ω can be shrunk to a point without leaving Ω. A region Ω that is not singly connected is called multiply connected. The duct cross sections considered in Chapters V–XI and again in Chapter XVI are singly connected.

Numerous thermal boundary conditions can be specified for a singly connected duct of constant cross-sectional area (Fig. 2). These are categorized in three groups in the first column of Table 1: (1) a specified axial wall temperature distribution tw, (2) a specified axial wall heat flux distribution q’, and (3) a specified combination of axial wall temperature and heat flux distributions. Around the periphery of the duct Γ, any combination of tw, q", or (q" ∞ twn) may be specified. These boundary conditions may be applicable to thermally developed and/or thermally developing flows. A general classification of thermal boundary conditions and their application to thermally developed or developing flows is presented in Table 1. Those thermal boundary and initial conditions with finite axial heat conduction within the fluid are described in Figs. 17 and 23 and the associated discussion. Axial heat conduction in a thin wall could be important for axially constant heat flux boundary conditions. The resulting thermal boundary condition is described on p. 30.

TABLE 1

GENERAL CLASSIFICATION OF THERMAL BOUNDARY CONDITIONS

†Some specific solutions, using superposition techniques, are available for thermally developing and hydrodynamically developing flows. These are described in the text at appropriate places.

For the case of specified arbitrary variations in wall temperature or wall heat flux, the solution to the energy equation may be obtained by superposition methods for constant property flow. Literature sources describing the superposition methods for arbitrary variations in tw or q" axially or peripherally are indicated in Table 1. Since so many boundary conditions can be generated by the superposition techniques, specific names are not recommended for axially variable thermal boundary conditions.

In the Soviet literature [18,33], conjugated problems and boundary conditions of four kinds are widely used for unsteady heat conduction problems. These are also applied to the duct flow forced convection problems. In a most general form, these boundary conditions are

(34)

(35)

(36)

(37)

(38)

Boundary conditions of the first, second, and third kinds are, respectively, the cases 1d, 2d, and the restricted 3d of Table 1 [corresponding to the linear functional relationship of Eq. (41)], except that in steady laminar flow no time dependence is involved.

Since the above boundary conditions do not make any distinction between axial and peripheral conditions, and since the solutions appear in the literature with this distinction, some specific thermal boundary conditions are proposed and systemized for thermally developed and developing flows. They are summarized in Table 2 and described in the following section in further detail for the singly connected duct of Fig. 2. From the solutions for these boundary conditions, a solution may be obtained for arbitrary variations in corresponding parameters by superposition methods. These basic boundary conditions are relatively simple to analyze mathematically, are realized approximately in practical systems, and have been analyzed to a varying degree for ducts having different cross sections, as described in Chapters V–XVI. Thermal boundary conditions that yield thermally fully developed flow for a long duct are limited in number. According to the authors’ knowledge, Table 2 appears to be a complete list of such thermal boundary conditions of practical interest.

TABLE 2

THERMAL BOUNDARY CONDITIONS FOR DEVELOPED AND DEVELOPING FLOWS THROUGH SINGLY CONNECTED DUCTS†

boundary condition (refer to text).

boundary condition. It has been analyzed for a circular tube [25,34], and rectangular and elliptical ducts [35].

The following nomenclature scheme is used for these boundary conditions. Generally, two characters are enclosed in a circle; the first character (T or H) represents axially constant wall temperature or heat transfer rate, respectively; the second character (1, 2, or 3) indicates, respectively, the foregoing boundary conditions of the first, second, or third kind in the peripheral direction.represents a constant axial heat transfer rate q’ with a constant peripheral surface temperature. The numerals 4 and 5 designate other specialized axial or peripheral boundary conditions.

In reporting the forced convection heat transfer results, it is recommended that the dimensionless variables be presented with either one or two sets of subscripts, as shown here in a specific example for the Nusselt number:

The subscripts x and m represent the local (peripheral average) and mean (flow length average) values, respectively, in the thermal entrance region; p represents the peripheral local value; Nubc denotes the peripheral average Nusselt number; Nup,bc and Nubc are for the fully developed region; the subscript bc designates the thermal boundary condition for which the solution has been obtained. Since the peripheral local Nusselt number in the thermal entrance region is not usually required in the design of a heat exchanger (except for the hot or cold spots), and since this information is currently not available in the literature, no specific nomenclature is recommended for it.

The thermal boundary conditions, which are summarized in Table 2, are described below. In all these boundary conditions, axial heat conduction in the duct wall is considered as zero.

a Constant Axial Wall Temperature, , , and

have linear and nonlinear combinations of wall heat flux and temperature on the duct periphery as described below.

Boundary Condition.