Problems involving mass, momentum and energy transport in one spatial direction in a Cartesian co-ordinate system are considered in this chapter. The concentration, velocity or temperature fields, here denoted field variables, vary along one spatial direction and in time. The ‘forcing’ for the field variables could be due to internal sources of mass, momentum or energy, or due to the fluxes/stresses at boundaries which are planes perpendicular to the spatial co-ordinate. Though the dependence on one spatial co-ordinate and time appears a gross simplification of practical situations, the solution methods developed here are applicable for problems involving transport in multiple directions as well.

There are two steps in the solution procedure. The first step is a ‘shell balance’ to derive a differential equation for the field variables. The procedure, discussed in Section 4.1, is easily extended to multiple dimensions and more complex geometries. The second step is the solution of the differential equation subject to boundary and initial conditions. Steady problems are considered in Section 4.2, where the field variable does not depend on time, and the conservation equation is an ordinary differential equation. For unsteady problems, the equation is a partial differential equation involving one spatial dimension and time. There is no general procedure for solving a partial differential equation; the procedure depends on the configuration and the kind of forcing, and physical insight is necessary to solve the problem. The procedures for different geometries and kinds of forcing are explained in Sections 4.4–4.7.

The conservation equations in Sections 4.2 and 4.4–4.7 are linear differential equations in the field variable—that is, the equations contain the field variable to the first power in addition to inhomogeneous terms independent of the field variable. For the special case of multicomponent diffusion in Section 4.3, the equations are non-linear in the field variable. This is because the diffusion of a molecular species generates a flow velocity, which contributes to the flux of the species. The conservation equation for the simple case of diffusion in a binary mixture is derived in Section 4.3, and some simple applications are discussed.

In Section 4.8, correlations for the average fluxes presented in Chapter 2 are used in the spatial or time evolution equations for the field variables.

Convection can be neglected when the Peclet number is small, and the field variables are determined by solving a Poisson equation ∇‡2Φ fv + S = 0 or a Laplace equation ∇‡2Φ fv = 0, subject to boundary conditions, where Φfv and S are the field variable and the rate of production per unit volume, respectively. It is necessary to specify two boundary conditions in each co-ordinate to solve these equations. The separation of variables procedure is the general procedure to solve these problems in domains where the boundaries are surfaces of constant co-ordinate. This procedure was earlier used in Chapters 4 and 5 for unsteady one-dimensional transport problems.

The procedure for solving the heat conduction equation in Cartesian co-ordinates is illustrated in Section 8.1. The ‘spherical harmonic’ solution for the Laplace equation in spherical co-ordinates is derived using separation of variables in Section 8.2, first for an axisymmetric problem of the heat conduction in a composite, and then for a general three-dimensional configuration. There are two types of solutions, the ‘growing harmonics’ that increase proportional to a positive power of r, and the ‘decaying harmonics’ that decrease as a negative power of r, where r is distance from the origin in the spherical co-ordinate system.

An alternate interpretation of the decaying harmonic solutions of the Laplace equation as superpositions of point sources and sinks of heat is discussed in Section 8.3. It is shown that the each term in the spherical harmonic expansions is equivalent to a term obtained by the superposition of sources and sinks in a ‘multipole expansion’. A physical interpretation of the growing harmonics is also provided.

The solution for a point source is extended to a distributed source in Section 8.4 by dividing the distributed source into a large number of point sources and taking the continuum limit. The Green's function procedure for a finite domain is illustrated by using image sources to satisfy the boundary conditions at planar surfaces.

Cartesian Co-ordinates

Consider the heat conduction in a rectangular block of length L and height H, in which the temperature is T0 at x = 0 and x = L, TA at y = 0 and TB at y = H, as shown in Fig. 8.1.

The momentum flux or the force per unit area on a surface within a fluid can be separated into two components: the pressure and the shear stress. The latter is due to variations in the flow velocity, while the former is present even when there is no flow. Pressure has no analogue in mass and heat transfer, where the fluxes are entirely due to the variations in the concentration/temperature fields. The fluid pressure is the compressive force per unit area exerted on a surface within the fluid in the direction perpendicular to the surface. At a point within the fluid, the pressure is a scalar which is independent of the orientation of the surface; the direction of the force exerted due to the pressure is along the perpendicular to the surface.

There is a distinction between the thermodynamic pressure and the dynamical pressure that drives fluid flow. The thermodynamic pressure is an absolute pressure which is calculated, for example, using the ideal gas equation of state. In contrast, flow is driven by the pressure difference between two locations in an incompressible flow. The velocity field depends on the variations in the dynamical pressure, and the flow field is unchanged if a constant pressure is added everywhere in the domain for an incompressible flow.

A potential flow is a limiting case of a pressure-driven flow where viscous effects are neglected. Some applications of potential flows are first reviewed in Section 6.1. The velocity profile and the friction factor for the laminar flow in a pipe is derived in Section 6.2. As discussed in Chapter 2, there is a transition from a laminar to a turbulent flow when the Reynolds number exceeds a critical value. The salient features of a turbulent flow are discussed in Section 6.3. The oscillatory flow in a pipe due to a sinusoidal pressure variation across the ends is considered in Section 6.4. This flow is used to illustrate the use of complex variables for oscillatory flows, and the approximations and analytical techniques used in the convection-dominated and diffusion-dominated regimes.

Potential Flow: The Bernoulli Equation

At high Reynolds number, viscous effects are neglected in the bulk of the flow, and there is a balance between the pressure, inertial and body forces.

In the previous chapter, a Cartesian co-ordinate system was used to analyse the transport between surfaces of constant co-ordinate, and the boundary conditions were specified at a fixed value of the co-ordinate z. For configurations with curved boundaries, such as a cylindrical pipe or a spherical particle, the boundaries are not surfaces of constant co-ordinate in a Cartesian system. It is necessary to apply boundary conditions at, for example, x2 + y2 + z2 = R2 for the diffusion around a spherical particle of radius R. It is simpler to use a co-ordinate system where one of the co-ordinates is a constant on the boundary, so that the boundary condition can be applied at a fixed value of the co-ordinate. Such co-ordinate systems, where one or more of the co-ordinates is a constant on a curved surface, are called curvilinear co-ordinate systems.

The procedure for deriving balance laws for a Cartesian co-ordinate system can be easily extended to a curvilinear co-ordinate system. First, we identify the differential volume or ‘shell’ between surfaces of constant co-ordinate separated by an infinitesimal distance along the co-ordinate. The balance equation is written for the change in mass/momentum/energy in this differential volume in a small time interval Δt. The balance equation is divided by the volume and Δt to derive the differential equation for the field variable. The balance equations for the cylindrical and spherical co-ordinate system are derived in this chapter, and the solution procedures discussed in Chapter 4 are applied to curvilinear co-ordinate systems.

Cylindrical Co-ordinates

Conservation Equation

A cylindrical surface is characterised by a constant distance from an axis, which is the x axis in Fig. 5.1. It is natural to define one of the co-ordinates r as the distance from the axis, and a second co-ordinate x as the distance along the axis. The third co-ordinate ϕ, which is the angle around the x axis, is considered later in Chapter 7. For unidirectional transport, we consider a variation of concentration, temperature, or velocity only in the r direction and in time, and there is no dependence on ϕ and x.

An anecdote about Prof. P. K. Kelkar, founding director of IIT Kanpur and former director of IIT Bombay, was narrated to me by Prof. M. S. Ananth, my teacher and former director of IIT Madras. A distraught young assistant professor at IIT Kanpur approached the director and complained that ‘the syllabus for the course is too long, and I am will not be able to cover everything’. Prof. Kelkar replied, ‘You do not have to cover everything, you should try to uncover a few things.’ In this book, my objective is to uncover a few things regarding transport processes.

The classic books on transport processes, notably the standard text Transport Phenomena by Bird, Stewart and Lightfoot written about 60 years ago, provided a comprehensive overview of the subject organised into different subject areas. At that time, engineers were required to do design calculations and modeling for different unit operations, and for the sequencing of these operations in process design. This required expertise in laboratory and pilot scale experiments on unit operations and scaling up of these operations using correlations. Proficiency in developing, understanding and using design handbooks and correlations was also needed. In this context, the study of transport processes at the microscopic level, and its implications for design for unit operations, was a pioneering advance that has since become an essential part of the chemical engineering curriculum.

In the last half century, sophisticated computational tools have been developed for detailed flow modeling within unit operations, and for the selection and concatenation of unit operations for achieving the required material transformations. The ease of search for information and data today was inconceivable half a century ago. Routine calculations have been automated, and there is little need for routine tasks such as unit conversion, graphical construction and interpreting engineering tables. There is now a greater need for understanding physical phenomena and processes and their mathematical description.

Using a rigorous understanding of transport processes, an engineer usually contributes to process design in one of two ways. The first is the development and enhancement of models and computational tools for modeling of flows and transformations in unit operations; these result in higher resolution, better representation of the essential physics and inclusion of new phenomena.

The mass/energy conservation laws are derived for two commonly used co-ordinate systems—the Cartesian co-ordinate system in Section 7.1 and the spherical co-ordinate system in Section 7.2. For unidirectional transport, we have seen that the conservation equation has different forms in different co-ordinate systems. Here, conservation equations are first derived using shell balance in three dimensions for the Cartesian and spherical co-ordinate systems. The conservation equations have a common form when expressed in terms of vector differential operators, the gradient, divergence, and Laplacian operators; the expressions for these operators are different in different co-ordinate systems. The conservation equation derived using shell balance is used to identify the differential operators in the the Cartesian and spherical co-ordinate system, and the procedure for deriving these in a general orthogonal co-ordinate system is explained.

Since the conservation equation is universal when expressed using vector differential operators, it is not necessary to go through the shell balance procedure for each individual problem; it is sufficient to substitute the appropriate vector differential operators in the conservation equation expressed in vector form. It is important to note that the derivation here is restricted to orthogonal co-ordinate systems, where the three co-ordinate directions are perpendicular to each other at all locations.

The discussion in Section 7.1 and 7.2 is restricted to mass/energy transfer. The constitutive relation (Newton's law) for momentum transfer for general three-dimensional flows is more complicated than that for mass/heat transfer. Mass and heat are scalars, and the flux of mass/heat is a vector along the direction of decreasing concentration/temperature. Since momentum is a vector, the flux of momentum has two directions associated with it: the direction of the momentum vector and the direction in which the momentum is transported. Due to this, the stress or momentum flux is a ‘second order tensor’ with two physical directions—the direction of momentum and the orientation of the perpendicular to the surface across which momentum is transported.

The number of independent parameters in a problem is reduced when the dependent and independent parameters are expressed in dimensionless form. In the problem of the settling sphere in Section 1.6.1 and the flow through a pipe in Section 1.6.3, the original problem contained one dependent and four independent dimensional quantities. Using dimensional analysis, this was reduced to one independent and one dependent dimensionless groups. The mass transfer problem in Section 1.6.2 contained one dependent and six independent quantities. The problem was reduced to a relationship between one dependent and two independent dimensionless groups, using dimensional analysis and the assumption that the solute mass and total mass can be considered as different dimensions. In the heat transfer problem in Section 1.6.3, there were one dependent and eight independent dimensional quantities. This was reduced to a relationship between one dependent and three independent dimensionless groups, using dimensional analysis and the assumption that the thermal and mechanical energy can be considered as different dimensions. Thus, dimensional analysis has significantly reduced the number of parameters in the problem.

It is not possible to further simplify the problem using dimensional analysis. In order to progress further, experiments can be carried out to obtain empirical correlations between the dimensionless groups. Another option, pursued in this text, is to do analytical calculations based on a mathematical description of transport processes. Before proceeding to develop the methodology for the analytical calculations, a physical interpretation of the different dimensionless groups is provided in this chapter.

In dimensional analysis, there is ambiguity in the selection of the dimensional parameters for forming the dimensionless groups. This ambiguity is reduced by a physical understanding of the dimensionless groups as the ratio of different types of forces. Here, a broad framework is established for understanding the different dimensionless groups and the relations between them. The forms of the correlations depend on several factors, such as the flow regime, flow patterns and the boundary conditions.

It is important to note that the correlations listed here are indicative, but not exhaustive. Some commonly used correlations are presented to obtain a physical understanding of the terms in the correlation, and to illustrate their application. More accurate correlations applicable in specific domains can be found in specialised handbooks/technical reports.

Transport of heat/mass is enhanced by an externally generated flow past an object or a surface in ‘forced convection’. Here, the flow is specified, and it is not affected by the change in temperature/concentration due to the heat/mass transfer. The known fluid velocity field is substituted into the convection–diffusion equation in order to determine the temperature/concentration field and the transport rate.

In the previous chapter, we examined the limit of low Peclet number, where transport due to convection is small compared that due to diffusion. There, the approach was to neglect convection altogether, and solve the diffusion equation. In the limit of high Peclet number, an equivalent approach would be to neglect diffusion altogether, and solve the convection equation to obtain the concentration/temperature fields. This approach is not correct for the following mathematical and physical reasons.

Mathematically, when the diffusion term is neglected, the convection–diffusion equation is reduced from a second order to a zeroth order differential equation in the cross-stream co-ordinate. The second order differential for the concentration/temperature field is well posed only if two boundary conditions are specified in each co-ordinate. When diffusion is neglected, the resulting zeroth order equation cannot satisfy both boundary conditions in the cross-stream co-ordinate specified for the original problem. Physically, when diffusion is neglected, there is transport due to convection only along fluid streamlines, and there is no transport across the streamlines. The concentration/temperature is a constant along streamlines in the flow. At bounding surfaces (the pipe surface in a heat exchanger, or particle surfaces in the case of suspended particles), there is no flow perpendicular to the surface. When we neglect diffusion, there is no flux across the surface. Therefore, we obtain the unphysical result that there is no mass/heat transfer across the surface.

A more sophisticated approach is required to obtain solutions for transport in strong convection, based on the following physical picture. In the limit of high Peclet number, mass or heat diffusing from a surface gets rapidly swept downstream due to the strong convection, and so the concentration/temperature variations are restricted to a thin ‘boundary layer’ close to the surface.

In the analysis of transport at high Peclet number in Chapter 9, it was assumed that the fluid velocity field is specified, and is not affected by the concentration or temperature variations. There are situations, especially in the case of heat transfer, where variations in temperature cause small variations in density, which results in flow in a gravitational field due to buoyancy. Examples of these flows range from circulation in the atmosphere to cooking by heating over a flame. In the former, air heated by the earth's surface rises and cold air higher up in the atmosphere descends due to buoyancy; in the latter, hotter and lighter fluid at the bottom rises due to buoyancy and is replaced by colder and heavier fluid at the top, resulting in significantly enhanced heat transfer.

The heat transfer due to natural convection from heated objects is considered here, and correlations are derived for the Nusselt number as a function of the Prandtl number and the Grashof number. The Prandtl number is the ratio of momentum and thermal diffusion. The Grashof number, defined in Section 2.4 (Chapter 2), is the square of the Reynolds number based on the characteristic fluid velocity generated by buoyancy. In order to determine the heat transfer rate, it is necessary to solve the coupled momentum and energy equations, the former for the velocity field due to temperature variations and the latter for the temperature field. The equations are too complex to solve analytically, and attention is restricted to scaling the equations to determine the relative magnitudes of convection, diffusion and buoyancy. We examine how the dimensionless groups emerge when the momentum and energy equations are scaled, and how these lead to correlations for the Nusselt number. The numerical coefficients in these correlations are not calculated here.

Boussinesq Equations

Consider a heated object with surface temperature T0, in a ambient fluid with temperature T∞ far from the object, as shown in Fig. 10.1 The fluid density is ρ∞ far from the object, but the temperature variation causes a variation in the density near the object. This density variation results in a buoyancy force, which drives the flow.

Significance of Transport Processes

The conversion of raw materials into useful products in a predictable, efficient, economical and environment-friendly manner is an essential part of many branches of engineering. There are two types of transformations: chemical transformations (involving chemical reactions) and physical transformations (melting, evaporation, filtering, mixing, etc.). Both of these transformations involve the motion of constituents relative to each other, and they often involve the transfer of energy in the form of heat. In operations involving fluid flow and mixing, there are forces exerted on the fluid due to pumps, impellers, etc. (input of mechanical energy), in order to overcome the frictional resistance generated by the flow. The subject of this text is the transport of the components in materials relative to each other, the transport of heat energy and the transport of momentum due to applied forces.

This text is limited to operations carried out in the fluid phase. Although solids transport and mixing does form an important part of material transformation processes, fluid-phase operations are the preferred mode for conversion because the transport is enabled by the two fundamental processes: convection and diffusion. Convection is the transport of mass, momentum and energy along with the flowing fluid. Diffusion is transport due to the fluctuating motion of the molecules in a fluid, which takes place even in the absence of fluid flow. Convection does not take place in solids since they do not flow, and diffusion in solids due to vacancy or interstitial migration is a very slow process, which makes it infeasible to effect material transformations over industrial timescales.

Fluids are of two types: liquids and gases. In liquids, the molecules are closely packed, and the distance between molecules is comparable to the molecular diameter. In contrast, in gases, the distance between molecules is about 10 times larger than the molecular diameter under conditions of standard temperature and pressure (STP). Due to this, the density of a liquid is about 103 times that of a gas. In a gas, the molecules interact through discrete collisions, and the period of a collision is much smaller than the average time between collisions.

The two transport mechanism considered in this text are convection and diffusion. Convection is transport due to the flow. It is directional, and takes place only along the flow streamlines. Transport across streamlines, and transport across surfaces (where there is no fluid velocity perpendicular to the surface) necessarily takes place due to diffusion.

Diffusion is the process by which material is transported by the random thermal motion of the molecules within the fluid, even in the absence of fluid flow. The random velocity fluctuations of the molecules are isotropic, and they have no preferred direction. The characteristic velocity and length for the thermal motion are the molecular velocity and the microscopic length scale, which is the molecular size in a liquid or the mean free path (distance between intermolecular collisions) in a gas. While random molecular motion is always present in fluids, when the concentration/temperature/velocity fields are uniform, there is no net transport due to the random motion. Diffusion takes place only when there is a spatial variation, and transport is along direction of variation.

The molecular mechanisms of mass, momentum and thermal diffusion, are discussed in this chapter. Constitutive relations for the fluxes are derived from a molecular description, and the diffusion coefficients are estimated.

The gas diffusivities are estimated using kinetic theory for an ideal gas made of hard spheres, which undergo instantaneous collisions when the surfaces are in contact, but which do not exert any intermolecular force when not in contact. Real gas molecules do not interact like hard spheres—the interaction force between molecules is repulsive at small separations and attractive at larger separations. Diatomic and polyatomic molecules are also not spherically symmetric, and their interaction depends on the relative orientation of the molecules. The diffusion coefficients in the hard sphere model are proportional to √T, where T is the absolute temperature. For molecules with continuous intermolecular potential, the diffusion coefficients are proportional to a power of the temperature which higher than ½. The pressure-density relationship for real gases is also more complicated than that for an ideal gas, and the virial corrections need to be included for dense gases.