All physical situations of interest to engineers and scientists are complex enough that a mathematical model of some sort is essential to describe them in sufficient detail for useful analysis and interpretation. Mathematical expressions provide a common language so different disciplines can communicate among each other more effectively. Models are very critical to chemical engineers, chemists, biochemists, and other chemical professionals because most situations of interest are molecular in nature and take place in equipment that does not allow for direct observation. Experiments are needed to extract fundamental knowledge and to obtain critical information for the design and operation of equipment. To do this effectively, one must be able to quantitatively analyze mass, energy, and momentum transfer (transport phenomena) at some level of complexity. In this text we define six levels of complexity, which characterize the level of detail needed in model development. The various levels are summarized in Table 1.1.

Level I, Conservation of Mass and/or Energy. At this level of analysis the control volume is considered a black box. A control volume is some region of space, often a piece of equipment, that is designated for “accounting” purposes in analysis. Only the laws of conservation of mass and/or energy are applied to yield the model equations; there is no consideration of molecular or transport phenomena within the control volume. It is a valuable approach for the analysis of existing manmade or natural systems and is widely employed.

Chemical engineers educated in the undergraduate programs of departments of chemical engineering have received an education that has been proven highly effective. Chemical engineering educational programs have accomplished this by managing to teach a methodology for solving a wide range of problems. They first did so by using case studies from the chemical process industries. They began case studies in the early part of the 20th century by considering the complete processes for the manufacture of certain chemicals and how they were designed, operated, and controlled. This approach was made much more effective when it was recognized that all chemical processes contained elements that had the same characteristics, and the education was then organized around various unit operations. Great progress was made during the 1940s and 1950s in experimental studies that quantified the analysis and design of heat exchangers and equilibrium stage operations such as distillation. The 1960s saw the introduction of reaction and reactor analysis into the curriculum, which emphasized the critical relationship between experiment and mathematical modeling and use of the verified models for practical design. We have built upon this approach, coupled with the tools of transport phenomena, to develop this text.

Our approach to teaching mass and heat transfer has the following goals:

1. Teach students a methodology for rational, engineering analysis of problems in mass and heat transport, i.e., to develop model equations to describe mass and heat transfer based on the relationship between experimental data and model.

2. Using these model equations, teach students to design and interpret laboratory experiments in mass and heat transfer and then to effectively translate this knowledge to the operation and design of mass and heat transfer equipment.

3. […]

A substantial portion of this chapter is taken from Introduction to Chemical Engineering Analysis by Russell and Denn (1972) and is used with permission.

This short chapter is a review of the analysis of simple reacting systems for chemical engineers. It is also designed to teach the fundamentals of analysis to other chemical professionals.

Reactor analysis is the most straightforward issue that chemical professionals encounter because rates of reaction can be obtained experimentally. The analysis of experimental data for reacting systems with mathematical models and the subsequent use of the verified model equations for design provide a template for the analysis of mass and heat transfer. We begin with simple reacting systems because the laboratory-scale experiments enable determination of reaction-rate constants that, with reasonable assumptions, can be used in model equations for design and operation of pilot- or commercial-scale reactors.

The same general principles apply to the design of mass contactors and heat exchangers, although it is more difficult to get the necessary values for mass and heat transfer coefficients from experiment. As we will see, mass transfer analysis is further complicated by the need to determine interfacial areas.

All chemical reaction and reactor analysis begins with experiment. Most experiments are carried out in batch equipment in a laboratory, and efforts are made to ensure the vessel is well mixed. In batch experiments, the concentrations of critical reacting species and products are measured over a given time period, sometimes until equilibrium is reached.

In our study of mass transfer, we noted that both technically feasible analysis and design were complicated by the need to estimate two parameters in the rate expression: the mass transfer coefficient (Km) and the interfacial area (a). In Chapter 6 theoretical and correlative methods for estimating mass transfer coefficients were discussed for those cases in which transfer of some species was limited by a well-defined boundary layer, such as at a solid or liquid surface. Correlations were identified relating the Sherwood number to the Reynolds and Schmidt numbers, with the functionality specific to the geometry and type of flow (i.e., laminar or turbulent). Complications arise in applying these developments when estimating Km in liquid–liquid or gas–liquid mass contactors. We need estimates of bubble or drop size and some knowledge of the fluid motions in the vicinity of the bubbles or drops to calculate the Reynolds and Sherwood numbers. Furthermore, as shown in Section 6.5, resistances to mass transfer may occur in both phases, necessitating knowledge of the fluid motions inside the bubbles or drops. In this chapter we examine methods for estimating these quantities. This is an active research area requiring Level VI two-phase fluid motion modeling and experiment and is not understood nearly as well as its single-phase counterpart. For this reason, most handbooks and textbooks alike provide only empirical correlations of the product Kma specific to particular process equipment and do not address the problem of interfacial area determination independently from the prediction of mass transfer coefficients.

Heat exchanger analysis follows the same procedures as chemical reactor analysis with the added complication that we need to deal with two control volumes separated by a barrier of area a. As shown in Table 3.1 we need to consider both tank type and tubular systems and allow for mixed–mixed, mixed–plug, or plug–plug fluid motions.

A heat exchanger is any device in which energy in the form of heat is transferred. The types of heat exchangers in which we will be most interested are devices in which heat is transferred from a fluid at one temperature to another at a different temperature. This is most commonly done by the confinement of both fluids in some geometry in which they are separated by a conductive material. In such devices the area available for heat transfer is set by the device type and size, and is often the object of a design calculation. If the two fluids are immiscible it is possible to exchange heat by direct contact of one fluid with the other, but such direct contact heat exchange is somewhat unusual. It is, however, the configuration most commonly employed to transfer mass between two fluids, and as such, will be treated in detail in Chapter 4. Note that the area for heat (and mass) transfer is much more difficult to measure or calculate in direct fluid–fluid contacting.

A simple heat exchanger, easily constructed, is one in which the fluids are pumped through two pipes, one inside the other.

As in the case of viscous fluids, very good approximations to exact results for viscoelastic fluids can be obtained from purely irrotational studies of stability. Here we consider RT instability (§21.1) and capillary instability (§21.2) of an Oldroyd B fluid. Viscoelastic effects enter into the irrotational analysis of RT instability through the normal stress at the free surface. For capillary instability, the short waves are stabilized by surface tension, and an irrotational viscoelastic pressure must be added to achieve excellent agreements with the exact solution. The extra pressure gives the same result as the dissipation method as is true in viscous fluids where VPF works for short waves and VCVPF and DM give the same results for capillary instability.

Rayleigh–Taylor instability of viscoelastic drops at high Weber numbers

Movies of the breakup of viscous and viscoelastic drops in the high-speed airstream behind a shock wave in a shock tube have been reported by Joseph, Belanger, and Beavers (1999; hereafter JBB). They performed a RT stability analysis for the initial breakup of a drop of Newtonian liquid and found that the most unstable RT wave fits nearly perfectly with waves measured on enhanced images of drops from the movies, but the effects of viscosity cannot be neglected. Snapshots from these movies are displayed in figures 21.1 to 21.4 and 21.9; data for the experiments are shown in table 21.1.

We study the force on a 2D cylinder near a wall in two potential flows: the flow that is due to the circulation 2πκ about the cylinder and the uniform streaming flow with velocity V past the cylinder. The pressure is computed with Bernoulli's equation, and the viscous normal stress is calculated with VPF; the shear stress is ignored. The forces on the cylinder are computed by integration of the normal stress over the surface of the cylinder. In both of the two cases, the force perpendicular to the wall (lift) is due to only the pressure and the force parallel to the wall (drag) is due to only the viscous normal stress. Our results show that the drag on a cylinder near a wall is larger than on a cylinder in an unbounded domain. In the flow induced by circulation or in the streaming flow, the lift force is always pushing the cylinder toward the wall. However, when the two flows are combined, the lift force can be pushing the cylinder away from the wall or toward the wall.

The flow that is due to the circulation about the cylinder

Figure 10.1 shows a cylinder with radius a near the wall x = 0.

When a vessel containing liquid is made to vibrate vertically with constant frequency and amplitude, a pattern of standing waves on the gas–liquid surface can appear. For some combinations of frequency and amplitude, waves appear; for other combinations the free surface remains flat. These waves were first studied in the experiments of Faraday (1831), who noticed that the frequency of the liquid vibrations was only half that of the vessel. Nowadays, this would be described as a symmetry-breaking vibration of a type that characterized the motion of a simple pendulum subjected to a vertical oscillation of its purpose.

The first mathematical study of Faraday waves are due to Rayleigh (1883a, 1883b) but the first definitive study is due to Benjamin and Ursell (1954; hereafter BU) who remark that “The present work has been made possible by the development of the theory of Mathieu functions.”

Faraday's problem is a rich source of problems in pattern formation, bifurcation, chaos, and other topics within the framework of fluid mechanics applications in the modern theory of dynamical system. Under the excitation of different parameters governing the Faraday system, different patterns, stripes, squares, hexagons, and time-dependent states can be observed. These features have spawned a large recent literature on Faraday waves. The experiments of Ciliberto and Gollub (1985) and Simonelli and Gollub (1989) on chaos, symmetry, and mode interactions are often cited.

In this chapter we present the form of the Navier–Stokes equations implied by the Helmholtz decomposition in which the relation of the irrotational and rotational velocity fields is made explicit. The idea of self-equilibration of irrotational viscous stresses is introduced. The decomposition is constructed first by selection of the irrotational flow compatible with the flow boundaries and other prescribed conditions. The rotational component of velocity is then the difference between the solution of the Navier–Stokes equations and the selected irrotational flow. To satisfy the boundary conditions, the irrotational field is required, and it depends on the viscosity. Five unknown fields are determined by the decomposed form of the Navier–Stokes equations for an incompressible fluid: the three rotational components of velocity, the pressure, and the harmonic potential. These five fields may be readily identified in analytic solutions available in the literature. It is clear from these exact solutions that potential flow of a viscous fluid is required for satisfying prescribed conditions, such as the no-slip condition at the boundary of a solid or continuity conditions across a two-fluid boundary. The decomposed form of the Navier–Stokes equations may be suitable for boundary layers because the target irrotational flow that is expected to appear in the limit, say at large Reynolds numbers, is an explicit to-be-determined field.

In this chapter we carry out an analysis of the stability of a liquid jet into a gas or another fluid by using VPF. This instability may be driven by KH instability that is due to a velocity difference and a neck-down that is due to capillary instability. KH instabilities are driven by pressures generated by a dynamically active ambient flow, gas or liquid. On the other hand, capillary instability can occur in a vacuum; the ambient can be neglected. KH instability is included by a discontinuity of the velocity at a two-fluid interface. This discontinuity is inconsistent with the no-slip condition for Navier–Stokes studies of viscous fluids, but is consistent with the theory of potential flow of a viscous fluid. We start our study with an analysis of capillary instability.

Capillary instability of a liquid cylinder in another fluid

The study of this problem is especially valuable because it can be solved exactly and was solved by Tomotika (1935). This solution allows one to compute the effects of vorticity generated by the no-slip condition. The ES can be compared with irrotational solutions of the same problem. One effect of viscosity on the irrotational motion may be introduced by evaluation of the viscous normal stress at the liquid–liquid interface on the irrotational motions.