modeling in transport phenomena, second edition a conceptual approach

• ISBN: 0444530215 • Publisher: Elsevier Science & Technology Books • Pub. Date: July 2007 PREFACE TO THE SECOND EDITION While the main skeleton of the first edition is preserved, Chapters 10 and 11 have been rewrit- ten and expanded in this new edition. The number of example problems in Chapters 8–11 has been increased to help students to get a better grasp of the basic concepts. Many new prob- lems have been added, showing step-by-step solution procedures. The concept of time scales and their role in attributing a physical significance to dimensionless numbers are introduced in Chapter 3. Several of my colleagues and students helped me in the preparation of this new edition. I thank particularly Dr. Ufuk Bakır, Dr. Ahmet N. Eraslan, Dr. Yusuf Uluda ˘ g, and Meriç Dalgıç for their valuable comments and suggestions. I extend my thanks to Russell Fraser for reading the whole manuscript and improving its English. ˙ ISMA ˙ IL TOSUN (itosun@metu.edu.tr) Ankara, Turkey October 2006 The Solutions Manual is available for instructors who have adopted this book for their course. Please contact the author to receive a copy, or visit http://textbooks.elsevier.com/9780444530219 xvii PREFACE TO THE FIRST EDITION During their undergraduate education, students take various courses on fluid flow, heat trans- fer, mass transfer, chemical reaction engineering, and thermodynamics. Most of them, how- ever, are unable to understand the links between the concepts covered in these courses and have difficulty in formulating equations, even of the simplest nature. This is a typical example of not seeing the forest for the trees. The pathway from the real problem to the mathematical problem has two stages: perception and formulation. The difficulties encountered at both of these stages can be easily resolved if students recognize the forest first. Examination of the trees one by one comes at a later stage. In science and engineering, the forest is represented by the basic concepts, i.e., conserva- tion of chemical species, conservation of mass, conservation of momentum, and conservation of energy. For each one of these conserved quantities, the following inventory rate equation can be written to describe the transformation of the particular conserved quantity ϕ:  Rate of ϕ in  −  Rate of ϕ out  +  Rate of ϕ generation  =  Rate of ϕ accumulation  in which the term ϕ may stand for chemical species, mass, momentum, or energy. My main purpose in writing this textbook is to show students how to translate the inven- tory rate equation into mathematical terms at both the macroscopic and microscopic levels. It is not my intention to exploit various numerical techniques to solve the governing equa- tions in momentum, energy, and mass transport. The emphasis is on obtaining the equation representing a physical phenomenon and its interpretation. I have been using the draft chapters of this text in my third year Mathematical Modelling in Chemical Engineering course for the last two years. It is intended as an undergraduate textbook to be used in an (Introduction to) Transport Phenomena course in the junior year. This book can also be used in unit operations courses in conjunction with standard textbooks. Although it is written for students majoring in chemical engineering, it can also be used as a reference or supplementary text in environmental, mechanical, petroleum, and civil engineer- ing courses. An overview of the manuscript is shown schematically in the figure below. Chapter 1 covers the basic concepts and their characteristics. The terms appearing in the inventory rate equation are discussed qualitatively. Mathematical formulations of the “rate of input” and “rate of output” terms are explained in Chapters 2, 3, and 4. Chapter 2 indicates that the total flux of any quantity is the sum of its molecular and convective fluxes. Chapter 3 deals with the formulation of the inlet and outlet terms when the transfer of matter takes place through the boundaries of the system by making use of the transfer coefficients, i.e., friction factor, heat transfer coefficient, and mass transfer coefficient. The correlations available in the literature to evaluate these transfer coefficients are given in Chapter 4. Chapter 5 briefly talks about the rate of generation in transport of mass, momentum, and energy. xix xx Preface Preface xxi Traditionally, the development of the microscopic balances precedes that of the macro- scopic balances. However, it is my experience that students grasp the ideas better if the reverse pattern is followed. Chapters 6 and 7 deal with the application of the inventory rate equations at the macroscopic level. The last four chapters cover the inventory rate equations at the microscopic level. Once the velocity, temperature, or concentration distributions are determined, the resulting equations are integrated over the volume of the system to obtain the macroscopic equations covered in Chapters 6 and 7. I had the privilege of having Professor Max S. Willis of the University of Akron as my PhD supervisor, who introduced me to the real nature of transport phenomena. All that I pro- fess to know about transport phenomena is based on the discussions with him as a student, a colleague, a friend, and a mentor. His inuence is clear throughout this book. Two of my col- leagues, Gỹniz Gỹrỹz and Zeynep Hiỗásaásmaz Katnaás, kindly read the entire manuscript and made many helpful suggestions. My thanks are also extended to the members of the Chemical Engineering Department for their many discussions with me and especially to Timur Do gu, Tỹrker Gỹrkan, Gỹrkan Karakaás, ệnder ệzbelge, Canan ệzgen, Deniz ĩner, Levent Ylmaz, and Hayrettin Yỹcel. I appreciate the help provided by my students, Gỹlden Camỗ, Yeásim Gỹỗbilmez, and ệzge O guzer, for proofreading and checking the numerical calculations. Finally, without the continuous understanding, encouragement and tolerance shown by my wife Ayáse and our children ầi gdem and Burcu, this book could not have been completed and I am particularly grateful to them. Suggestions and criticisms from instructors and students using this book will be appreci- ated. ISMA IL TOSUN (itosun@metu.edu.tr) Ankara, Turkey March 2002 Table of Contents Preface 1 Introduction 1 2 Molecular and Convective Transport 15 3 Interphase Transport and Transfer Coefficients 41 4 Evaluation of Transfer Coefficients: Engineering Correlations 65 5 Rate of Generation in Momentum, Energy and Mass Transfer 133 6 Steady-State Macroscopic Balances 149 7 Unsteady-State Macroscopic Balances 181 8 Steady-State Microscopic Balances Without Generation 237 9 Steady-State Microscopic Balances With Generation 325 10 Unsteady-State Microscopic Balances Without Generation 429 11 Unsteady-State Microscopic Balances With Generation 473 A Mathematical Preliminaries 491 B Solutions of Differential Equations 531 C Flux Expressions 567 D Physical Properties 575 E Constants and Conversion Factors 583 Index 586 1 INTRODUCTION 1.1 BASIC CONCEPTS A concept is a unit of thought. Any part of experience that we can organize into an idea is a concept. For example, man’s concept of cancer is changing all the time as new medical information is gained as a result of experiments. Concepts or ideas that are the basis of science and engineering are chemical species, mass, momentum,andenergy. These are all conserved quantities. A conserved quantity is one that can be transformed. However, transformation does not alter the total amount of the quantity. For example, money can be transferred from a checking account to a savings account but the transfer does not affect the total assets. For any quantity that is conserved, an inventory rate equation can be written to describe the transformation of the conserved quantity. Inventory of the conserved quantity is based on a specified unit of time, which is reflected in the term rate. In words, this rate equation for any conserved quantity ϕ takes the form  Rate of input of ϕ  −  Rate of output of ϕ  +  Rate of generation of ϕ  =  Rate of accumulation of ϕ  (1.1-1) Basic concepts upon which the technique for solving engineering problems is based are the rate equations for the • Conservation of chemical species, • Conservation of mass, • Conservation of momentum, • Conservation of energy. The entropy inequality is also a basic concept but it only indicates the feasibility of a process and, as such, is not expressed as an inventory rate equation. A rate equation based on the conservation of the value of money can also be considered as a basic concept, i.e., economics. Economics, however, is outside the scope of this text. 1.1.1 Characteristics of the Basic Concepts The basic concepts have certain characteristics that are always taken for granted but seldom stated explicitly. The basic concepts are • Independent of the level of application, • Independent of the coordinate system to which they are applied, • Independent of the substance to which they are applied. 1 2 1. Introduction Table 1.1. Levels of application of the basic concepts Level Theory Experiment Microscopic Equations of Change Constitutive Equations Macroscopic Design Equations Process Correlations The basic concepts are applied at both the microscopic and the macroscopic levels as shown in Table 1.1. At the microscopic level, the basic concepts appear as partial differential equations in three independent space variables and time. Basic concepts at the microscopic level are called the equations of change, i.e., conservation of chemical species, mass, momentum, and energy. Any mathematical description of the response of a material to spatial gradients is called a constitutive equation. Just as the reaction of different people to the same joke may vary, the response of materials to the variable condition in a process differs. Constitutive equations are postulated and cannot be derived from the fundamental principles 1 . The coefficients appearing in the constitutive equations are obtained from experiments. Integration of the equations of change over an arbitrary engineering volume exchanging mass and energy with the surroundings gives the basic concepts at the macroscopic level. The resulting equations appear as ordinary differential equations, with time as the only inde- pendent variable. The basic concepts at this level are called the design equations or macro- scopic balances. For example, when the microscopic level mechanical energy balance is in- tegrated over an arbitrary engineering volume, the result is the macroscopic level engineering Bernoulli equation. Constitutive equations, when combined with the equations of change, may or may not comprise a determinate mathematical system. For a determinate mathematical system, i.e., the number of unknowns is equal to the number of independent equations, the solutions of the equations of change together with the constitutive equations result in the velocity, tem- perature, pressure, and concentration profiles within the system of interest. These profiles are called theoretical (or analytical) solutions. A theoretical solution enables one to design and operate a process without resorting to experiments or scale-up. Unfortunately, the number of such theoretical solutions is small relative to the number of engineering problems that must be solved. If the required number of constitutive equations is not available, i.e., the number of un- knowns is greater than the number of independent equations, then the mathematical descrip- tion at the microscopic level is indeterminate. In this case, the design procedure appeals to an experimental information called process correlation to replace the theoretical solution. All process correlations are limited to a specific geometry, equipment configuration, boundary conditions, and substance. 1.2 DEFINITIONS The functional notation ϕ =ϕ(t,x,y,z) (1.2-1) 1 The mathematical form of a constitutive equation is constrained by the second law of thermodynamics so as to yield a positive entropy generation. 1.2 Definitions 3 indicates that there are three independent space variables, x, y, z, and one independent time variable, t.Theϕ on the right side of Eq. (1.2-1) represents the functional form, and the ϕ on the left side represents the value of the dependent variable, ϕ. 1.2.1 Steady-State The term steady-state means that at a particular location in space the dependent variable does not change as a function of time. If the dependent variable is ϕ,then  ∂ϕ ∂t  x,y,z =0 (1.2-2) The partial derivative notation indicates that the dependent variable is a function of more than one independent variable. In this particular case, the independent variables are (x, y, z) and t. The specified location in space is indicated by the subscripts (x, y, z), and Eq. (1.2-2) implies that ϕ is not a function of time, t. When an ordinary derivative is used, i.e., dϕ/dt =0, then this implies that ϕ is a constant. It is important to distinguish between partial and ordinary derivatives because the conclusions are very different. Example 1.1 A Newtonian fluid with constant viscosity μ and density ρ is initially at rest in a very long horizontal pipe of length L and radius R.Att =0, a pressure gradient, |P |/L, is imposed on the system and the volumetric flow rate, Q, is expressed as Q= πR 4 | P | 8μL  1 −32 ∞  n=1 exp(−λ 2 n τ) λ 4 n  where τ is the dimensionless time defined by τ = μt ρR 2 and λ 1 = 2.405, λ 2 = 5.520, λ 3 = 8.654, etc. Determine the volumetric flow rate under steady conditions. Solution Steady-state solutions are independent of time. To eliminate time from the unsteady-state solution, we have to let t →∞. In that case, the exponential term approaches zero and the resulting steady-state solution is given by Q= πR 4 | P | 8μL which is known as the Hagen-Poiseuille law. Comment: If time appears in the exponential term, then the term must have a negative sign to ensure that the solution does not blow as t →∞. 4 1. Introduction Example 1.2 A cylindrical tank is initially half full with water. The water is fed into the tank from the top and it leaves the tank from the bottom. The inlet and outlet volumetric flow rates are different from each other. The differential equation describing the time rate of change of water height is given by dh dt =6 −8 √ h where h is the height of water in meters. Calculate the height of water in the tank under steady conditions. Solution Under steady conditions dh/dt must be zero. Then 0 =6 −8 √ h or, h =0.56 m 1.2.2 Uniform The term uniform means that at a particular instant in time, the dependent variable is not a function of position. This requires that all three of the partial derivatives with respect to position be zero, i.e.,  ∂ϕ ∂x  y,z,t =  ∂ϕ ∂y  x,z,t =  ∂ϕ ∂z  x,y,t =0 (1.2-3) The variation of a physical quantity with respect to position is called gradient. Therefore, the gradient of a quantity must be zero for a uniform condition to exist with respect to that quantity. 1.2.3 Equilibrium Asystemisinequilibrium if both steady-state and uniform conditions are met simultane- ously. An equilibrium system does not exhibit any variation with respect to position or time. The state of an equilibrium system is specified completely by the non-Euclidean coordinates 2 (P,V,T). The response of a material under equilibrium conditions is called property corre- lation. The ideal gas law is an example of a thermodynamic property correlation that is called an equation of state. 1.2.4 Flux The flux of a certain quantity is defined by Flux = Flow of a quantity/Time Area = Flow rate Area (1.2-4) where area is normal to the direction of flow. The units of momentum, energy, mass, and molar fluxes are Pa (N/m 2 ,orkg/m·s 2 ), W/m 2 (J/m 2 ·s), kg/m 2 ·s, and kmol/m 2 ·s, respectively. 2 A Euclidean coordinate system is one in which length can be defined. The coordinate system (P,V,T)is non-Euclidean. [...]... First Law of Diffusion Consider two large parallel plates of area A The lower one is coated with a material, A, which has a very low solubility in the stagnant fluid B filling the space between the plates Suppose that the saturation concentration of A is ρAo and A undergoes a rapid chemical reaction at the surface of the upper plate and its concentration is zero at that surface At t = 0 the lower plate... temperature To As time proceeds, the temperature profile in the slab changes, and ultimately a linear steady-state temperature is attained as shown in Figure 2.3 Experimental measurements made at steady-state indicate that the rate of heat flow per unit area is proportional to the temperature gradient, i.e., ˙ Q = A Energy flux k T1 − To Y Transport property Temperature gradient (2.1-3) 16 2 Molecular and... Fourier’s Law of Heat Conduction Consider a slab of solid material of area A between two large parallel plates of a distance Y apart Initially the solid material is at temperature To throughout Then the lower plate is suddenly brought to a slightly higher temperature, T1 , and maintained at that temperature The second law of thermodynamics states that heat flows spontaneously from the higher temperature... the inventory rate equation for money as Change in amount Service Dollars Checks = (Interest) − + − of dollars charge deposited written Identify the terms in the above equation 1.2 Determine whether steady- or unsteady-state conditions prevail for the following cases: a) The height of water in a dam during heavy rain, b) The weight of an athlete during a marathon, c) The temperature of an ice cube as...1.3 Mathematical Formulation of the Basic Concepts 5 1.3 MATHEMATICAL FORMULATION OF THE BASIC CONCEPTS In order to obtain the mathematical description of a process, the general inventory rate equation given by Eq (1.1-1) should be translated into mathematical terms 1.3.1 Inlet and Outlet Terms A quantity may enter or leave the system by two means: (i) by inlet and/or outlet streams, (ii) by exchange... Assumptions 1 The total molar concentration, c, is constant 2 Naphthalene plate is also at a temperature of 95 ◦ C Analysis The molar flux of naphthalene transferred from the plate surface to the flowing stream is determined from ∗ JAx x=0 = −DAB dcA dx (1) x=0 It is possible to calculate the concentration gradient on the surface of the plate by using one of the several methods explained in Section A. 5... through a pipe, it is considered a single phase and a single component system In this case, there is no ambiguity in defining the characteristic velocity However, if the oxygen in the air were reacting, then the fact that air is composed predominantly of two species, O2 and N2 , would have to be taken into account Hence, air should be considered a single phase, binary component system For a single phase... is kept stationary a) Calculate the steady force applied to the upper plate b) The fluid in part (a) is replaced with another Newtonian fluid of viscosity 5 cP If the steady force applied to the upper plate is the same as that of part (a) , calculate the velocity of the upper plate (Answer: a) 5 N b) 4 m/s) 2.3 Three parallel flat plates are separated by two fluids as shown in the figure below What should... are called transport properties 2.1.1 Newton’s Law of Viscosity Consider a fluid contained between two large parallel plates of area A, separated by a very small distance Y The system is initially at rest but at time t = 0 the lower plate is set in motion in the x-direction at a constant velocity V by applying a force F in the x-direction while the upper plate is kept stationary The resulting velocity... direction of decreasing temperature Example 2.2 One side of a copper slab receives a net heat input at a rate of 5000 W due to radiation The other face is held at a temperature of 35 ◦ C If steady-state conditions prevail, calculate the surface temperature of the side receiving radiant energy The surface area of each face is 0.05 m2 , and the slab thickness is 4 cm Solution Physical Properties For . thought. Any part of experience that we can organize into an idea is a concept. For example, man’s concept of cancer is changing all the time as new medical information is gained as a result. this text in my third year Mathematical Modelling in Chemical Engineering course for the last two years. It is intended as an undergraduate textbook to be used in an (Introduction to) Transport. below. Chapter 1 covers the basic concepts and their characteristics. The terms appearing in the inventory rate equation are discussed qualitatively. Mathematical formulations of the “rate of input” and