A large number of models for transfer processes that contain equations are based on laws of conservation of momentum, mass and energy. Some of the models that were developed initially are based on simple laws and were inspired by experimental observations. The laws, though limited in scope, are still used not only as sub-models of various rigorous models but also in various engineering calculations. However, it is essential to understand the simplicity as well as limitations of these models. The simple laws provide us with initial guess values required while using rigorous models. These laws can be used as sub-models or as assumptions if a given process is not very sensitive to a particular parameter. The use of numerical methods is helpful in easily solving some of the problems based on these simple laws. A few commonly used simple laws will be presented in this chapter. They include the ideal gas law, which is used for estimation of thermodynamic proper-ties; the cubic equation of state (EOS), particularly the Peng–Robinson EOS;
Newton’s law of viscosity and its application in flow problems; Fourier’s law for conduction; Fick’s first and second laws for diffusion; kinetic rate expres-sions; the Arrhenius equation and isotherms for adsorption. Definitions of the heat and mass transfer coefficients and resistances to heat and mass transfer are presented. A few applications of the simple laws are presented to demonstrate the importance of these problems.
3.1 Equation of State
The relationship between the pressure, temperature and volume of gases and vapours is called as equation of state. Several thermodynamic properties (e.g. density, enthalpy and vapour–liquid equilibria) can be estimated using methods based on the use of EOSs. These properties determine the perfor-mance of various real processes, such as pressure drop in packed beds and interphase transport in multiphase processes involving compressible fluids.
3.1.1 Ideal Gas Law
The properties of the gases and vapours are used in many engineering calculations related to flow of gases, vapour–liquid equilibria, gas– liquid absorption, adsorption of gases etc. The concentration of a chemical species
in a mixture of gases is generally expressed in terms of volume fraction.
The specific volume of the gas is the reciprocal of density. Good approxi-mations for several properties of real gases are obtained assuming ideal gas laws (Smith et al. 1996). In the ideal gas law, the simplest of the EOS is given as
PV = RT (3.1)
where R is the universal gas constant.
The internal energy for a real gas depends upon temperature and pres-sure. For an ideal gas, the internal energy and enthalpy depend upon tem-perature only. The latter is frequently used while making energy balances.
The enthalpy change for an ideal gas is written as
H C dT
T
∫
T= p
1
2 (3.2)
The correlations for specific heat, Cp, as a function of temperature are known for various gases and can be used for engineering calculation.
At ambient conditions, the compressibility factor for gases up to a few bars is close to 1; hence, the ideal gas law is applicable. For real gases, the ideal gas law gives a good initial guess while using numerical techniques to estimate volume from an EOS.
3.1.2 Cubic Equations of State
At elevated pressures and low temperatures, the ideal gas law is not appli-cable and an EOS for real gas has to be used. Several EOSs are reported in the literature. A class of EOSs for real gases can be expressed in terms of polynomials of the third degree in either volume or compressibility factor, Z.
These EOSs have three roots and hence are called cubic EOSs. These EOSs require less calculation effort and are used while modelling the performance of process equipment. All the three roots of the cubic EOS can be obtained analytically. Van der Waals EOS was the first of these. Although many cubic EOSs have been proposed from time to time, only three of the cubic EOSs are given in Table 3.1. The Peng–Robinson EOS can be written in terms of Z, as given here (Peng and Robinson 1976):
Z3 + (1 − B) Z2 + (A − 3B2− 2B) Z − (AB − B2 − B3) = 0 (3.3)
where A aP
R T B bP
= 2 2, = RT and Z Pv
= RT.
If all the roots are real, then the largest real root corresponds to the gas phase and the smallest corresponds to the liquid phase. The Peng–Robinson EOS provides the roots for both gas and liquid phases more frequently than
55 Models Based on Simple Laws
other cubic EOSs. It is an additional advantage. If one real and two imaginary roots are obtained, then the real root corresponds to the gas phase.
In vapour–liquid equilibrium, at least two components are present in the vapour phase. The Peng–Robinson EOS is used to estimate the properties of the mixture using the following mixing rules:
a x x ai j ij
The binary interaction coefficient, δij, is characteristic of the pair of species, i and j.
For cubic EOSs, some of the properties of the gases can be obtained ana-lytically; for example, for the Peng–Robinson EOS, the fugacity of a pure component is given by
f
Selected Cubic Equation of States for Pure Gases
Investigator Equation
Van der Waals (1873)
( )
0.08664 , 0.48 1.574 0.176
0.5
The fugacity for a mixture of gases may be estimated from
Similarly, the enthalpy departure of a fluid is given by
H H RT Z T da
The codes for the estimation of various properties using the Peng–Robinson EOS and other EOSs are reported in the literature using popular software, including MATLAB® (Nasri and Binous 2009). The built-in function ‘fsolve’
was used to find the roots of the cubic equation. Analytical expressions for other cubic EOSs are available in the literature.
3.2 Henry’s Law
Henry’s law is used to relate the composition of a chemical species in vapour and liquid at equilibrium. The mole fraction or partial pressure of an ith component in the vapour phase, pi, is proportional to the mole fraction of that component in liquid, xi, that is:
pi = Hi xi (3.9)
The proportionality constant, Hi, is known as Henry’s constant. For ideal gases:
pi = xi P (3.10)
where P is the total pressure. From Equations 3.9 and 3.10, a linear equilib-rium relationship is obtained:
y
In case of adsorption of pure gases at pressure, P, the number of moles adsorbed on the surface, n, is given by the following simple relationship known as Henry’s law for adsorption:
n = kP (3.12)
57 Models Based on Simple Laws
3.3 Newton’s Law of Viscosity
One of the recent trends in solving problems of fluid flow is to use computa-tional fluid dynamics, which is based on the law of conservation of momen-tum. This approach requires an expression for shear stress, τyx, in terms of fluid velocity or more precisely shear rate, dv
dy
x. One of the simplest models for this relationship is Newton’s law of viscosity:
dv
yx dyx
τ = − (3.13)
where the proportionality constant, μ, is known as the viscosity of the fluid.
All the gases and liquids and solutions of low-molecular-weight compounds exhibit Newtonian behaviour (Skelland 1967).
Due to the simplicity of the law, it has been possible to obtain analytical solutions for several flow problems. Dimensionless numbers such as Prandtl number, Pr, and Schmidt number, Sc, as defined in Equations 3.14 and 3.15, have been used in several empirical correlations for heat and mass transfer.
C Pr kp
f
= (3.14)
Sc DAB
=ρ (3.15)
where kf is the thermal conductivity of the fluid and DAB is the diffusivity of A into B.
All fluids do not follow Newton’s law of viscosity. If the shear stress is plotted against shear rate, fluids with different behaviour are observed as shown in Figure 3.1. The curve for Newtonian fluid is a straight line passing through the origin. The fluids for which the line does not pass through the origin but is a straight line are known as Bingham plastic fluids. Fluids for which the curve is not a straight line but is described by a power law are known as power law fluids. If the exponent is less than 1, the fluid is known as a pseudoplastic. If the exponent is larger than 1, then it is called a dilatant.
Few fluids exhibit time-dependent behaviour between shear stress and shear rate. These fluids are known as thixotropic or rheopectic fluids depending upon their behaviour with time. Some of the rheological models to describe the non-Newtonian behaviour are given in Table 3.2. Out of various rheologi-cal models for non-Newtonian fluids, the power law fluid and the Bingham plastic model can be used with less difficulty than other models. Solutions for obtaining the velocity field in pipes and channels for these fluids follow-ing these models are frequently available in the literature (Bird et al. 1960;
Skelland 1974). The velocity field is an essential requirement to account for the convective transfer of mass and heat.
TABLE 3.2
Rheological Models for Non-Newtonian Liquids
Investigator Equation
Power law or Ostwald–de
Waele model mdv
dy
If n < 1, pseudoplastic; if n > 1, dilatant; if n = 1, Newtonian.
Eyring model Pseudoplastic
Thixotropic
Rheopectic
FIGURE 3.1
Rheological models for a few non-Newtonian fluids.
59 Models Based on Simple Laws
For non-Newtonian fluids, the viscosity is not defined; instead, an apparent viscosity is defined as
dv dy
yx x a= τ
(3.16)
In absence of a well-defined fluid flow field, mostly encountered in multi-phase systems, the apparent viscosity cannot be conveniently and confidently determined. These empirical correlations using dimensionless numbers involv-ing viscosity cannot be used easily. A proper rheological model has to be used to find out the flow field first. It is possible only if equations based on laws of conservation of momentum are solved.
3.4 Fourier’s Law of Heat Conduction
When heat transfer takes place due to conduction only, then at steady state the heat transfer rate is proportional to the temperature difference (driving force) and area of heat transfer. It is inversely proportional to the distance between the two faces of the transfer surface. Heat transfer rate q is written as
q kA T T
z kAdT dz
( )
= −
2 1 = − (3.17)
The differential form is known as Fourier’s law of heat conduction. At a steady state, it results in
q k A T T=
( )
δm −
h
2 1 (3.18)
The proportionality constant, km, is the thermal conductivity of the medium. The distance between the two boundaries maintained at constant temperatures T1 and T2 is δh.
One may recall Newton’s law of cooling which is applicable for convective heat transfer at low temperatures. According to this law, the rate of cooling of the body is proportional to the temperature difference between the body and ambient conditions and the heat transfer area. The heat transfer coefficient is defined in a similar fashion:
q hA T T=
(
2− 1)
(3.19)The effect of flow field on the heat transfer rate is included in the heat transfer coefficient, h.
Heat transfer resistance analogous to electrical resistance for convection as well as conduction can be expressed as follows:
1
2 1 h
T T
q hA k Am
(
−)
= = δ(3.20)
The resistance can be added depending upon different heat transfer processes taking place in parallel, in series or a combination of both.
Differentiation of Equation 3.17 gives d dz AdT
dz = 0 (3.21)
This equation may be used if the heat transfer area is a variable.
3.5 Fick’s First Law
Based on experimental observations, Fick proposed that at a steady state, the mass transfer rate remains constant between two points. The concentration pro-file is linear. Hence, the mass transfer rate is assumed to be proportional to the concentration gradient. Since it is also proportional to the area of the mass trans-fer surface, A, the mass transtrans-fer rate, NA, for a steady state may be expressed as
N D A dC
A AB dzA
= − (3.22)
Equation 3.20 is known as Fick’s first law and should be applied only in cases of steady-state diffusion with no convection. The proportionality con-stant, DAB, is known as the diffusivity or diffusion coefficient. Since mass transfer results in some amount of convection (diffusional flux), Equation 3.22 is valid only for dilute solutions or in cases of diffusion in solids. In concen-trated solutions, the diffusion coefficient depends upon the concentration.
In most of the cases, at least two types of molecules are involved. One is that of the medium, and the second is that of the diffusing species. Therefore, the diffusivity is reported for a pair medium and diffusing species. For example, the diffusion of benzene in air will be different from the diffusion of ben-zene in nitrogen. If the types of molecules are different, the diffusivity of A into B may be different than that of B into A. But most of the time, they are considered to be the same. In the case of multicomponent systems, Fick’s law is applicable for each species since each species has a different diffusivity.
If Fick’s law is compared with the definition of mass transfer coefficient, N kA C J A D A dC
A A A AB dzA
= = = − (3.23)
The mass transfer coefficient, k, is equal to the ratio of diffusivity to dis-tance between the points across which the concentration difference exists.
61 Models Based on Simple Laws
N and J are equal only in the absence of convection. At the wall or interface of two phases, the convection is always zero. If there are at least two species diffusing in opposite directions, it is possible that the sum of all the velocities is zero. In a binary system, such a situation is known as equimolal counter- diffusion. However, it does not mean that NA and JA cannot be related otherwise.
The molar flux, NA, and the molar diffusion flux, JA, are relative to stationary coordinates and coordinates moving with bulk flow with velocity, u, respec-tively. Therefore, for a species, i, the two fluxes are given as
Ni = Ciui (3.24)
Ji=C u ui
(
i−)
(3.25)Here, ui is the velocity of each species, and u is the average velocity of all the species.
Fick’s first law may also be written in terms of mass flux, n, and mass diffusion flux, j:
Mass flux, nA, relative to fixed spatial coordinates is given by
AB and molar flux relative to fixed spatial coordinates is given by
AB Fick’s first law does not involve any temporal derivative; hence, it can be applied in steady-state continuous processes. In a batch process, the steady state is never attained. Therefore, understanding the unsteady state is essen-tial for understanding progress in a batch process. It is also helpful for understanding a continuous process during initial periods.
3.6 Fick’s Second Law
Let us consider diffusion through a semi-infinite slab. It is assumed that during a short time interval, the system behaves as a steady-state process.
Such an assumption is called ‘quasi–steady state’. Fick’s first law may be
differentiated during this period. The rate of accumulation of a chemical species in a volume is equal to the difference between the rate of species entering at z due to diffusion and the rate of species leaving.
(Change in conc.)(volume) rate of A entering
– rate of A leaving (area)(time)
= (3.30)
Substitution of terms in Equation 3.30 gives an equation for unsteady-state diffusion:
Equation 3.31 is known as Fick’s second law and is applicable only in the absence of convection.
Several processes involve cylindrical and spherical objects (e.g. equipment and dispersed phases). In these situations, an appropriate coordinate system should be chosen. Fick’s second law in cylindrical coordinates and spherical coordinates, respectively, is given in Equations 3.32 and 3.33:
C
Integration of Fick’s first law with the boundary condition as a function of time and the solution of Fick’s second law are different.
3.7 Film Model
Mass and heat transfer take place between a fluid and a transfer surface. The transfer surface may be a rigid wall or interface of two different phases.
The region of interest is near the transfer surface. Mass transfer takes place by diffusion and convection mechanisms. Heat transfer takes place by conduc-tion, convection and radiaconduc-tion, the latter being significant at high temperature.
While diffusion and conduction may be estimated using Fick’s first law and
63 Models Based on Simple Laws
Fourier’s law, respectively, the convective transport of mass and heat transfer requires knowledge of the convection (i.e. the flow field). The fluid velocity depends upon space variables as well as time. Problems related to diffusion or conduction can be solved much easier than those involving convective trans-port, as the fluid velocity is absent. At low fluid velocity, the flow is lami-nar and randomness of the turbulence is absent. However, due to the strong dependence of the flow upon viscosity, the effect of flow field upon transfer rates cannot be ignored and is to be used to estimate mass or heat transfer rates. The flow field adjacent to the transfer surface strongly depends upon the flow field in the bulk of the fluid. It happens in the case of laminar flow.
When the fluid velocities are large, the flow becomes turbulent. The con-vective transport is higher than the transport due to diffusion or conduction.
Therefore, the flow field away from the interface does not seem to have a significant effect on the transfer rates.
Many engineering problems are related to transfer processes at the inter-face (e.g. fluid–fluid and fluid–solid interinter-faces). In the bulk of the fluid, the distribution of the temperature or concentration is studied. One of the simplest models to describe mass or heat transfer is the film model or film theory. This model is based on the following assumptions:
1. The entire variation of concentration or temperature takes place within a thin fluid film adjacent to the interface.
2. Within the film, the mass or heat transfer takes place by steady-state one-dimensional diffusion or conduction, respectively.
The first assumption is close to a situation where a high degree of mix-ing in the bulk fluid is present so that the concentration or temperature is almost uniform in the bulk fluid. Therefore, the effect of fluid flow field on mass or heat transfer is not significant. It also indicates that the film model should not be used in cases of laminar flow. The second assumption is very bold. It ensures that flow field is not considered for the estimation of mass or heat transfer rates. It does not say that there is no flow in the film. But the fluid velocity normal to the interface is absent. The fluid velocity in the vicinity of the interface is laminar and parallel to the interface. Therefore, it is neglected, as the convective transport in the direction of diffusional or thermal transport is normal in regard to the direction of fluid flow.
With these assumptions, mass transfer flux for species A (i.e. NA, or mass transfer rate per unit area) can be written using Fick’s first law:
N D C C
A AB A2 A1
m
( )
= −
δ (3.34)
where CA1 and CA2 are the concentrations of the species A at the interface and on the other face of the fictitious film, respectively. The latter is equal to the concentration of A in the bulk fluid.
The diffusional film thickness, δm, cannot be measured. It may, however, be estimated. The diffusional flux may be neglected in the case of dilute solutions. In the case of equimolal counter-diffusion, the diffusional flux is zero. In the case of concentrated solutions, the diffusivity depends upon the concentration, and it can vary in the vicinity of the interface.
The mass transfer flux can also be written in terms of mass transfer
The mass transfer flux can also be written in terms of mass transfer