1
Introduction

1.1 Scope and Objectives of the Book

The two-phase systems covered in this book include boiling, condensation, gas-liquid mixtures, and gas-solid mixtures.

Two-phase heat transfer is involved in numerous applications. These include heat exchangers in refrigeration and air conditioning, conventional and nuclear power generation, solar power plants, aeronautics, chemical processes, petroleum industry, etc. In recent years, there has been increasing use of miniature heat exchangers for computers and other electronic intensive products.

The emphasis in this book is on information that is of practical use. For this reason, theories and methods that do not provide useable and adequately verified solutions are dealt only briefly though sufficient references are provided for more information about them. Effort is made to provide the best available information for the design of a wide variety of heat exchangers in a clear and concise manner. This information includes experimental data, theoretical solutions, and empirical correlations. Accuracy and range of applicability of formulas/correlations presented is stated. Clear recommendations are made for application of the methods presented. A very wide variety of heat exchangers is covered. These include boiling and condensation in tubes and tube bundles, plate heat exchangers of various types, falling film heat exchangers, coils, surfaces cooled by jets, mist cooling, rotating surfaces (tubes, disks, cones, etc.), spheres, etc. Boiling and condensation of metallic fluids is discussed besides those of non-metallic fluids. Also included are heat exchangers with two-component gas-liquid mixtures, fluidized beds, and flowing gas-solid mixtures.

In this chapter, information is provided that is needed for understanding and using the material in other chapters as well as in other publications. This includes explanation of commonly used terms, various models used in solving two-phase flow and heat transfer problems, distinction between minichannels and conventional channels, flow patterns and their prediction, etc. While the focus of this book is on two-phase heat transfer, methods for calculation of single-phase heat transfer, void fraction and pressure drop have also been briefly discussed as these are needed in the design of heat exchangers. References to sources for more information on these topics have been provided.

Only Newtonian fluids are considered in this book. All discussions pertain to non-metallic fluids except where stated otherwise.

1.2 Basic Definitions

Some commonly used terms are explained in the following.

Mass flux or mass velocity is the mass flow rate per unit area. It is usually designated as G. If W be the mass flow rate kg?s-1 in a tube of cross-sectional area Ac (m2), G = W/Ac (kg?m-2?s-1).

Void fraction is the part of the total volume occupied by the gas phase. Consider a gas-liquid mixture flowing in a pipe. If AL is the flow area occupied by liquid and AG is the flow area occupied by gas, void fraction a is

Liquid holdup RL is the part of flow area occupied by liquid phase.

Quality, usually given the symbol x, is mass flow rate of vapor divided by the total flow rate. With WL as the flow rate of liquid and WG that of gas,

Two types of phase velocities are used, actual, and superficial. The actual velocity of gas phase uG is that in the area occupied by the gas phase:

where ?g is the density of gas. The actual liquid velocity is similarly defined and is given by

Superficial gas velocity uGS is the velocity assuming that gas alone is flowing through the entire flow area. In other words, liquid is assumed to be absent. Then,

Similarly, superficial liquid velocity uLS is defined as

The superficial gas and liquid velocities are also called volumetric gas and liquid flux represented by the symbols jG and jL, respectively.

Gas and liquid velocities are often not equal. The difference in phase velocities (uG?-?uL) is called the slip velocity, while uG/uL is known as slip ratio. The latter is expressed by the following relation obtained using Eqs. (1.2.4) and (1.2.5):

The relative velocity between phases uGL can be written as

The drift flux jGL is defined as

where

The drift velocity of gas uGj with respect to a plane moving at a velocity j is defined as

The drift velocity of the liquid phase is

Heat flux, usually represented as q, is defined as the heat applied to a surface per unit area per unit time. If Q Watts are applied to a tube of diameter D and length L,

In boiling systems, quality is usually defined assuming thermodynamic equilibrium between vapor and liquid phases, i.e. all the heat applied is used to evaporate the liquid. Thus, if W?kg?s-1 of saturated liquid enters a tube of length L with heat flux q, quality at exit from tube is

where ifg is the latent heat of vaporization. Equilibrium quality during condensation is defined in a similar way; all heat removed is used to condense the vapor. Unless stated otherwise, the quality used in equations and given in test data is the equilibrium quality.

If Tw be the wall temperature and TSAT the saturation temperature during boiling, (Tw?-?TSAT) = ?TSAT is known as the wall superheat. In condensation, (TSAT?-?Tw) is called wall subcooling. If a liquid is at a temperature T that is lower than the saturation temperature, (TSAT?-?T) = ?TSC is called subcooling.

The term "film temperature" is frequently used. It means the mean of wall and fluid temperature. Unless stated otherwise, it is the arithmetic mean. Thus,

1.3 Various Models

Some basic models used in the analysis of two-phase systems are discussed herein.

1.3.1 Homogeneous Model

It is assumed that gas and liquid are flowing at the same velocity and form a homogeneous mixture. By putting uG = uL in Eq. (1.2.8) and rearranging, the following expression for void fraction a is obtained:

For use in calculation of heat transfer and pressure drop with this model, the properties of the mixture are considered to be the mean of those of gas and liquid. Various methods of calculating the mean values have been proposed, for example, weighted according to the mass fractions of gas and liquid in the mixture.

Homogeneous model works fairly well for bubble flow and mist flow though it has been used in some empirical correlations without regard to the flow pattern.

1.3.2 Separated Flow Models

In the separated flow model, the gas and liquid phases are considered to be separated. Separate equations can then be written for each phase. Additional equations are needed for determining areas occupied by the two phases and interfacial shear. These can be empirical or semi-theoretical correlations or sophisticated analyses such as the two-fluid models in which momentum, energy, and continuity equations are written separately for each phase together with equations for interaction between phases. Closed-form solutions of these equations are rarely possible and hence have to be solved numerically on computers. The two-fluid models are difficult to use and not necessarily more accurate than the simpler models. Empirical and semi-theoretical models are generally used in practical designs.

1.3.3 Flow Pattern-Based Models

In these models, the gas and liquid are considered to be arranged according to the expected flow pattern, and prediction methods are developed specific to particular flow patterns. The prediction methods are most often empirical correlations. Analytical solutions have also been developed notably for stratified, slug, and annular flow patterns. Such analytical solutions use idealized geometry of the flow patterns. For example, annular flow is usually assumed to have uniform liquid layer, no interfacial waves, and no liquid entrainment. These assumptions are usually not correct. Still, the analytical solutions are useful as they provide understanding of the physical phenomena.

The accuracy of flow pattern-based models is further limited by the accuracy of flow pattern...