The simple laws discussed in Chapter 3 involve heat transfer and mass transfer, with and without chemical reactions. The mechanisms of heat and mass transfer that were considered were conduction and diffusion, respectively. Another mechanism for mass and heat transfer is convection.
To consider the convective transport of heat and mass transfer, knowledge of the flow field in the system is required. Lumped parameter models using average velocity are the exception. Estimations of pressure drop, drag forces, viscous dissipation etc., also require the flow field to be known. The subject of transport phenomena was developed with the aim to find mathematical solu-tions for the distribution of velocity, temperature and concentration within a system. The development of numerical techniques to solve the govern-ing equation resulted in the development of computational fluid dynamics (CFD). The governing equations describing transfer processes are derived by applying laws of conservation of mass, momentum and energy. The develop-ment of these equations and their application in modelling various processes are presented in this chapter.
4.1 Laws of Conservation of Momentum, Mass and Energy The subject of transport phenomena is based on the fact that during convection, the momentum, mass and thermal energy are conserved in a system. Changes in any of these (i.e. accumulation of these) are equal to changes due to convection and other phenomena. The momentum is a vector; hence, all of its three components are conserved. Accumulation of the momentum is due to the convective transport, the shear stress in a particular direction and the pressure or gravitational force. Similarly, accumulation of mass of each chemical species is due to convective trans-port, diffusion and chemical reaction. Accumulation of thermal energy is due to convection, conduction, generation of heat due to chemical reaction, viscous dissipation, electrical heating etc. The development of the govern-ing equation is briefly presented here. The detailed derivations of these equations are available in most of the books on transport phenomena (e.g. Bird et al. 1960).
4.1.1 Equation of Continuity
Mass is conserved except in nuclear reactions. It cannot be created or destroyed. It is known as the law of conservation of mass. A substance may consist of various chemical species. If the mass of a particular species is considered, then it may appear or disappear due to a chemical reaction, but the total mass is conserved. Let us apply the law of conservation of total mass over a differential volume element of dimensions Δx, Δy and Δz in Cartesian coordinates (Figure 4.1):
For a fluid of density, ρ, Equation 4.1 can be written as
x y z
After rearranging terms, dividing Equation 4.2 by (Δx Δy Δz) and assuming that the limit of Δx, Δy and Δz approached zero, the following equation is obtained:
Fluid flow through a fixed volume element.
81 Models Based on Laws of Conservation
Equation 4.3 describes the variation of density of the fluid in terms of spatial velocity distribution. It is applicable to both compressible as well as incompressible fluids. It is helpful in simplifying equations of motion, energy and continuity for each species while deriving these (Bird et al.
1960).
For incompressible fluids, ρ is constant. Equation 4.3 simplifies to the following equation:
∂ 0 The spatial distribution of velocity should always follow Equation 4.4 for incompressible fluid. For one-dimensional (1D) fluid flow, vy = vz = 0.
Equation 4.4 reduces to
∂ 0
∂v = x
x (4.5)
The velocity in the direction of flow remains constant. For 2D cases, if vz = 0, Equation 4.4 is reduced to the following equation:
∂ 0 terms of a new variable, ψ, as follows:
= −∂ψ
Equation 4.6 is satisfied, but the governing equations to describe the velocity field will have only one variable, ψ, which is called the stream function.
then Equation 4.6 is again satisfied. The variable φ is known as ‘velocity potential’. From Equations 4.7 and 4.8, we get the Cauchy–Riemenn equations:
An analytical function w(z) of a complex variable, z, may be chosen such that w(z) = ϕ(z) + iψ (z), satisfying Equation 4.8. The velocity components, vy and vz, can be obtained as the real and imaginary parts of the following equation:
= − + dw
dz vx ivy (4.10)
Equation of continuity; stream function, ψ; and velocity potential, φ, in cylindrical and spherical coordinates are given in Tables 4.1, 4.2 and 4.3, respectively. It is obvious that for a steady-state 1D flow problem, the velocity gradient in the direction of flow is always zero irrespective of the coordinate system.
For 2D flows, Equation 4.6 is also used in the following manner:
∫
0 The limits of integration depend upon the problem.TABLE 4.1
Equation of Continuity in Cylindrical and Spherical Coordinates
Coordinate System Equation of Continuity
Cylindrical coordinates
83 Models Based on Laws of Conservation
TABLE 4.2
Stream Function, ψ, and N-S Equation in Cylindrical and Spherical Coordinates Coordinate System Stream Function, ψ N-S Equation
Cylindrical v 1 ;
Note: The operators used are
r r r r r
Velocity potential, φ, and Laplace Equation in Cylindrical and Spherical Coordinates Coordinate System Velocity Potential, φ Laplace Equation
Cylindrical, vz = 0
4.1.2 Laws of Conservation of Momentum, Mass and Energy
Let us consider a differential volume element of dimensions: Δx, Δy and Δz.
Let Q be any of the quantity of momentum, mass or energy. The laws of con-servation of Q can be written for this element.
Q
Usually, the molecular transport of momentum, mass and heat is described by Newton’s law of viscosity, Fick’s first law of diffusion and Fourier’s law, respectively. The appropriate substitution of various terms in Equation 4.12 is followed by division by the volume, ΔxΔyΔz. As Δt→0, Δx→0, Δy→0 and Δz→0 Equation 4.12 gives the desired differential equa-tions for conservation of Q. Simplifying the equaequa-tions with the help of the equation of continuity, the equation of change for momentum, mass and thermal energy may be obtained. It is not an easy task to use the shell balance method to model a process in curvilinear coordinates. Bird et al.
(1960) suggests starting the problem formulation directly from the equation of motion.
The following equations for momentum transport in Cartesian coor-dinates are obtained for all types of fluids, whether Newtonian or non-Newtonian, and compressible or incompressible. An absence of derivatives of ρ does not mean that these equations are valid for only incompressible fluids. The derivatives of ρ were eliminated with the help of the equation of continuity.
Equations 4.13 through 4.15 are the x, y and z components of the equa-tion of change, respectively. The stress is a tensor having nine components.
85 Models Based on Laws of Conservation
From these equations, the following equations, called ‘Navier–Stokes (N-S) equations’, are obtained for Newtonian incompressible fluids:
2
Analytical solutions of these equations may be obtained for equations with one variable and, in a few cases, with two variables. Numerical solutions to equations with two variables can be obtained using various types of soft-ware, including MATLAB®. To solve a problem with three or four variables, CFD software is required.
In cases of mass transfer with a chemical reaction in incompressible fluid, the following equation is applicable for each of the chemical species.
For example, if there are three chemical species, then three equations are obtained. The diffusivity is assumed to be constant which is true only in the case of dilute solutions:
C In cases of heat transfer, only one equation is obtained:
C T In Equations 4.19 and 4.20, the generation terms (i.e. the last term on the right hand side) are due to chemical reactions only. In the absence of any reaction, −rA = 0 and ∆Hrxn = 0.
Several simple models are based on special cases of these equations.
The left side has a temporal derivative which is zero in steady-state models.
The temporal derivative is considered in dynamic models. The other three terms on the left-hand side of these equations are convective terms. In the absence of convective transport, these terms will be zero. One-dimensional momentum balance for an unsteady-state model gives
2
The equation of continuity for 1D flow is given by Equation 4.5. Using it in Equation 4.21 gives
2
ρ∂ 2
∂ = −∂
∂ + ∂
∂ + ρ v
t p x
v
y g
x x
x (4.22)
A few simple 1D model equations in Cartesian, cylindrical and spherical coordinate systems are presented in Table 4.4. The solution of these equations depends upon the boundary conditions. The equations in Table 4.4 have been used to model various processes involving multiphase systems. Solutions in Cartesian coordinates are useful in describing the processes occurring at the flat walls and large surfaces. Solutions in cylindrical coordinates are used to describe the processes in tubes, capillaries and cylindrical vessels.
Solutions in spherical coordinates are used to describe the processes around the dispersed systems such as bubbles, drops and particles. Often, the non-spherical shapes of the dispersed objects have been approximated by spheres with correction in terms of sphericity.
The well-defined velocities in N-S equations exist in cases of laminar flows only. The random nature of the velocity fluctuations observed in turbulent flows is not described in terms of velocities only.
N-S equations in cylindrical and spherical coordinates are given in Appendix A.
4.1.3 Boundary Conditions
The solution of the differential equations depends upon the boundary con-ditions (i.e. the concon-ditions at the boundaries of the system). The equations of change based on the laws of conservation are second order in the spatial coordinates and first order in time. The solution requires that two boundary conditions, and one initial condition must be specified to describe the sys-tem. Three different types of boundary conditions are commonly used while solving heat and mass transfer problems (Figure 4.2).
4.1.3.1 Boundary Conditions of the First Kind
For heat transfer, the temperature at the boundaries is specified. For mass transfer problems, the concentration of chemical species at the boundary or surface is specified. This type of boundary condition is very frequently used;
it is known as a Dirichlet condition, or a boundary condition of the first kind.
However, the temperature or concentration may be a function of time and location at the boundary. Such situations are very frequently observed while studying the dynamic response of a system to ramp or sinusoidal input.
If the surface is heated or cooled by a fluid going through a phase change, the surface temperature is constant. This kind of boundary condition is applied to heat transfer equipment heated by steam.
87 Models Based on Laws of Conservation
TABLE 4.4
A Few Simple Models Based on Laws of Conservation
Mass Transfer Heat Transfer
Cartesian Coordinates Conditions
Steady state; no reaction; no Steady state; with
reaction; no
Unsteady state; no reaction; no Unsteady state; no
reaction; diffusion or conduction in radial direction
Unsteady state; no reaction; diffusion or conduction in axial direction
Unsteady state; with reaction; diffusion or conduction in radial direction or conduction in radial direction;
convection in axial direction or conduction in axial direction;
convection in axial direction or conduction in radial direction
4.1.3.2 Boundary Conditions of the Second Kind
A large number of problems require that, at the surface, the heat transfer or mass transfer flux is constant. The flux is specified by Fourier’s law or Fick’s first law in cases of heat and mass transfer, respectively. This type of bound-ary condition is specified as given here:
q k T This boundary condition is known as a boundary condition of the second kind or Neumann condition. It is observed in electrically heated heat trans-fer equipment or solar collectors. Mass transtrans-fer across two diftrans-ferent phases, with one of them being pure, also has this kind of boundary condition.
In cases of adiabatic process, q =0.
4.1.3.3 Boundary Conditions of the Third Kind
In the case of convective heat and mass transfer across the surface, the boundary condition of the third kind is observed. It is usually described in terms of heat or mass transfer coefficient.
k T
The temperature at the surface, Ts, or concentration of A, CA∞, need not be constant. It may be a function of time.
(a) (b) (c)
T2 T1 and T2= constants q is not constant
T1
Types of boundary conditions: (a) Dirichlet, or first kind, (b) Neumann, or second kind and (c) third kind.
89 Models Based on Laws of Conservation
4.2 Laminar Flow
In laminar flow, the fluid flows in an orderly manner. No macroscopic inter-mixing of the fluid elements with its neighbour is observed. Laminar flow takes place at low flow rates. From the modelling point of view, a laminar flow can be considered as a well-defined flow which can be expressed in terms of time and space variables.
4.2.1 Velocity Field in Laminar Flow
The laws of conservation provide governing equations which, when solved, give the distribution of velocity, temperature and concentration in a system.
These are therefore useful in developing a distributed parameter model.
The first terms on the left-hand side of Equations 4.18 and 4.19 are tempo-ral derivative terms. These terms are zero in steady-state models. The other three terms correspond to convection terms.
If the boundary conditions are a function of time or the velocity is a func-tion of time, then the temporal term should be used. However, sometimes the process is so slow that accumulation is negligible, and the temporal term is dropped. This is called a ‘quasi-steady-state assumption’.
If the velocity components, vx, vy and vz, are considered to be independent of temperature or concentration, then the equation of motion may be solved independently to get the velocity profile.
The equation of thermal energy and equation of continuity for a chemical species may be solved after substitution of the expression for the velocity profile. This approach is applicable in cases of forced convection. Natural convection is caused by the density difference due to the temperature or concentration gradient. In such cases, the equation of motion and equation of change for thermal energy or the equation of continuity for a chemical species should be solved simultaneously. The models for natural convection are thus somewhat more complex than those for forced convection problems.
4.2.2 Velocity Profile in Simple Geometries
In process industries, fluids flow in circular tubes and cylindrical vessels.
The jacketed wall used to exchange heat in a cylindrical vessel is similar to annulus. The fluid flow in many separation devices is similar to flow between two parallel plates and slits. The flow around dispersed systems resemble to flow around spheres. The flow across cylindrical tubes is encountered in heat exchangers. Thus, the most frequently observed geometries are circular tubes, plates, annulus, cylinders and spheres. The flow around these objects has been studied extensively and is available in books of fluid mechanics and transport phenomena. Only a few cases will be discussed to illustrate the use of laws of conservation to obtain the velocity, temperature or concentration profile.
Let us consider the steady-state fully developed flow of Newtonian incompressible fluid between two parallel plates (Figure 4.3a) separated by a distance 2B. The width of the plate W is very large as compared to the gap 2B. Fully developed flow means that there are no end effects. This assumption is true only when the length of plate, L, in the direction of flow is very large. All the convective terms in N-S equations are absent for 1D flow when the equation of continuity is substituted. The gravity has no effect in a horizontal tube or in the case of forced convection. At steady state, the time derivative is zero. After these simplifications and taking
p x
P L
∂
∂ = , we get
2
= 2
P L
d v dy
z (4.27)
The boundary conditions are as follows:
At y = B, vz = 0 At y = 0, dv =0
dy
z
(a)
(b)
(c) vz r z
R
L vz r
z
vz Ro
Ri
L
B vz
y z
L W B
FIGURE 4.3
Flow of Newtonian incompressible fluid: (a) between two parallel plates, (b) in a circular tube and (c) in annulus.
91 Models Based on Laws of Conservation
The velocity profile can be obtained by solving Equation 4.27.
2 2 1
For axial flow in a circular tube (Figure 4.3b), we again obtain the similar equation. The variable y is replaced by r. The following boundary conditions are used:
At r = R: vz = 0 At r = 0: dv =0
dr
z
The velocity profile in this case also is parabolic.
4 2 1
A fully developed flow of incompressible non-Newtonian fluid in an annulus with Ro and Ri as the radii of the outer and inner tubes, respec-tively, results in the same equation as in the case of flow in a circular pipe (Figure 4.3c). The boundary conditions are as follows:
At r = Ro: vz = 0 At r = Ri: vz = 0
The velocity profile is as follows:
( )
Detailed proofs of these problems are not given here as they are available in several books on transport phenomena (Bird et al. 2002). Equation 4.30 reduces to Equation 4.28 for values of
o
R R
i close to 1. Under this condition, the flow in an annular flow can be treated as the flow in a slit (Bird et al. 2002). Similar simpli-fications are used to avoid the equation of change in curvilinear coordinates.
The use of N-S equations in rectilinear coordinates is much easier.
In all three problems, the convective terms are zero. The velocity profile still exists as it is related to shear stress.
For flow of non-Newtonian fluids, the starting equations are Equations 4.13 through 4.15. Let us consider flow of a power law fluid in a circular pipe.
For axial flow, the z component of the equation of motion is simplified to p
Equation 4.31 requires a rheological model to proceed. The correctness of the velocity profile depends on the correct choice of the rheological model (Skelland 1964). For a power law fluid substituting the value of τrz in Equation 4.31, we get
The boundary conditions are as follows:
At r = R: vz = 0 At r = 0: dv =0
dr
z
Equation 4.32 can now be integrated twice to get the velocity profile.
v P
4.2.3 Convective Heat and Mass Transfer in Simple Geometries
Let us consider unsteady-state mass transfer in a flat plane in the absence of convection with no reaction. The convection and generation terms are zero, and Equation 4.19 reduces to Equation 3.31. An analogous equation for heat transfer is also obtained. Fick’s second law and Fourier’s law were not obtained from the law of conservation, but they are used in the equations of continuity, momentum and thermal energy. The problems based on Fick’s and Fourier’s laws were discussed in Chapter 3 and were in the absence of convection. Heat and mass transfer problems in the presence of convection are discussed in this chapter.
Let us consider that a cold fluid at temperature T0 is falling down a vertical wall, as shown in Figure 4.4. The hot surface is maintained at a constant temperature, Ts. If the fluid is incompressible non-Newtonian, the velocity profile is given by Equation 4.29. The equation of change of thermal energy for a steady-state heat transfer is obtained by simplifying Equation 4.20:
The boundary conditions are as follows:
T = T0 at z = 0 and for y > 0 T = T0 at y = δ and for finite z T = Ts at y = 0 and for z > 0
93 Models Based on Laws of Conservation
To obtain an analytical solution, the second boundary condition, y = δ, is replaced by the condition y→∞. For a short contact time, the parabolic velocity profile is approximated by linear velocity in the vicinity of the surface, that is,
=ρ δ
vz g y (4.35)
If the heat transfer flux is to be determined, then the temperature gradient at the wall is required. Therefore, the above assumption seems appropriate.
The analytical solution of Equation 4.34 is as follows (Bird et al. 1960):
∫ ( )
( )
( )
( )−
− =
Γ −η η
η
1 ∞ 0 exp
s 0
3 4 / 3
T T
T T d (4.36)
where
η =
ρ δ
9
2 p 1 3
y k C g x
Ts
Ts z
0 yT0
vz
T0
T0 Velocity profile
Temperature profile Transfer
surface
Liquid film
δ
FIGURE 4.4
Heat transfer for fluid falling down a vertical wall.
If the velocity profile is given by Equation 4.29, then Equation 4.34 can be solved numerically. The ‘pdepe’ solver of MATLAB can be used by replacing the time derivative by spatial derivative. This solver can be used for para-bolic partial differential equations for various types of boundary conditions for slabs, cylinders and spheres, and it can consider the generation term also.
While using the ‘pdepe’ solver, use of variables ‘t’ and ‘x’ only is permitted.
The codes for the function, initial condition and boundary conditions are
The codes for the function, initial condition and boundary conditions are