In the past 5 years, membrane ultrafiltration has gained increasing prominence as a simple and convenient process for concentrating, purifying, and fractionating solutions of moderate-to-high molecular weight solutes and colloids, and for purifying water and other solvents containing such solutes. The emergence of this new molecular separation technique for both laboratory and industrial applications is almost entirely attributable to the development of a family of uniquely structured polymeric membranes which display extraordinarily high hydraulic permeabilities coupled with the capacity to retain even quite small solute molecules.

Solute polarization and cake formation in membrane ultrafiltration: causes, consequences, and control techniques

A derivation of the equations for multi-phase transport

Membrane separation processes

An exact solution to the unsteady convective diffusion equation for miscible displacement in fully developed laminar flow in tubes is obtained. The most interesting result of this work is that it shows the generalized one-dimensional dispersion model describes the average concentration distribution exactly for all values of the independent variables if the dispersion coefficients are defined properly as functions of time. The analysis given reveals the precise structure of the dispersion coefficients necessary for the dispersion model to be exact. Also, it is shown that an exact solution for the local distribution can be constructed in which the dispersion coefficients play the role of eigenvalues which are functions of $\tau $.

Exact analysis of unsteady convective diffusion

The transport equations for multi-phase systems

An orthogonal expansion technique for solving a new class of counterflow heat transfer problems is developed and applied to the detailed study of laminar flow concentric tube heat exchangers. The exchanger problem is solved for fully developed laminar velocity profiles, negligible longitudinal conduction in the fluid streams and in the exchanger walls, and with fluid properties which are independent of the temperature.

A description of the variation of the local Nusselt numbers and the temperature at the wall between the two streams is given. Also reported are bulk temperature changes in the two streams and mean overall Nusselt numbers. It is shown that for long exchangers, which are of some industrial importance, asymptotic Nusselt numbers exist in counterflow as in single-phase and cocurrent systems. Numerical values of asymptotic Nusselt numbers are reported for a wide range of parameters. Comparisons are made with single-stream solutions such as the Graetz problem, with empirical correlations of experimental data, and with cocurrent flow exchangers.

To solve this problem it was necessary to derive new orthogonality relations, and also expressions for determining positive and negative sets of eigenvalues and eigenvectors. Satisfaction of inlet boundary conditions at both ends of counterflow exchangers requires a complete set of eigenfunctions and thus one must use both the positive and negative sets.

An analytical study of laminar counterflow double‐pipe heat exchangers

The effects of axial conduction in the wall on heat transfer with laminar flow

An exact analytical theory of the local behavior of finite slugs which are dispersed in three-dimensional velocity fields is developed. It is shown that the dispersion coefficient enters the problem as an eigenvalue. For the important special case of fully developed straight pipe flow, considered first by Taylor and later generalized by Aris, the dispersion coefficient can be found very easily by the present method, which also provides the basis for studying a large class of more complicated problems.

A detailed numerical analysis is carried out to determine the limits of applicability of the analytical theory. Fortunately, it is found that dispersion of slug stimuli in fully developed laminar tube flow is described by Equation (15) when τ exceeds certain minimum values which depend only weakly on Peclet number and are independent of slug length in the region considered here. Since peak mean concentrations are predicated accurately by Equation (19) for X > 0.1, this is a convenient quantity to use to determine dispersion coefficients experimentally.

On comparing the numerical and analytical results for local point concentrations, good agreement is found, provided the average concentration is predicted accurately by Equation (15). This is significant because it implies that a broad class of difficult transient heat and mass transfer problems, with aperiodic inlet disturbances, can be solved by using superposition in conjunction with the present approach.

Laminar dispersion in capillaries: Part IV. The slug stimulus

Mass transfer with chemical reaction from spherical one or two component bubbles or drops

The effect of inlet boundary conditions on the transient approach to the asymptotic Taylor-Aris theory was studied for three different cases. It was found that for values of τ of practical interest the solutions for these cases were different only if NPe < 100. The asymptotic solution was independent of inlet conditions in the three cases studied once τ exceeded the necessary minima.

A rather complete numerical solution to the equation that describes laminar flow in tubes with a single continuous stagnant pocket of arbitrary depth along the entire length of the pore has been obtained, and results for point, average, and bulk concentration distributions are presented for a wide range of parameters. The range of applicability is extended to infinity for both τ and NPe by the asymptotic theory of Aris.

Higher capacitance systems require longer times before their dispersion behavior can be described by error-function solutions. This is important, since only then is the concept of an equivalent axial dispersion coefficient useful because then it allows the calculation of the complete average concentration distribution from a minimum of data. This suggests that in work on high capacitance systems such as packed beds, one should ensure that experimental systems are long enough for the asymptotic theory to apply if results are to be interpreted in terms of dispersion coefficients.

Numerical results for the Turner model agree very well with the modified effective dispersion coefficient, which was suggested by Aris and is given in Equation (15), as long as τ is sufficiently large. These lower limits for τ are given in Table 3. When the lower limits on τ are exceeded, the new solution developed for the point distribution given by Equations (31) and (32) is also valid. This is important for at least two reasons. First, it implies that the dispersion model can be used to describe the local as well as the average behavior of complex transient systems if only the velocity distribution across the system is known. More work is needed to ascertain the limitations and possibilities of the dispersion model in this regard. Second, it shows that one can find simple and accurate approximate local solutions without neglecting axial molecular diffusion.

William N. Gill

Papers

An analytical study of laminar counterflow double‐pipe heat exchangers

The effects of axial conduction in the wall on heat transfer with laminar flow

Laminar dispersion in capillaries: Part IV. The slug stimulus

Mass transfer with chemical reaction from spherical one or two component bubbles or drops

Laminar dispersion in capillaries: Part II. Effect of inlet boundary conditions and turner type of system capacitance