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.