Daniel Hershey

Finite Difference Calculus

In this chapter we introduce the calculus of finite differences, with applications in difference equations, interpolation and extrapolation and solutions to simple difference equations. Finite difference integration methods are discussed. More complex finite difference equations are then solved by operator techniques. Finally simultaneous difference equations and nonlinear equations are solved.

Models for Simultaneous Diffusion and Chemical Reaction of Oxygen within the Intact Red Cell of Whole Blood

A great amount of work has been done in the area of the oxygenation of blood. Studies have been devoted to the diffusion of ligand gases through plasma, hemoglobin solutions, and whole blood. Chemical reaction studies have been made testing various kinetic models. Much work has been done with hemoglobin solutions and millipore membranes to simulate the environment within the red cell. Fleischman (7) has presented a survey of some of the pertinent work done in this field. Of more interest to this discussion are the studies that have involved the simultaneous considerations of oxygen diffusion and chemical reaction with hemoglobin in the intact red cell of whole blood.

Some Mathematical Concepts

In this chapter we shall introduce a few basic concepts involving units, Newton’s second law of motion, and some “common sense” statements of basic transport relationships. After a brief calculus review, there will be presented statements of continuity, differentiability, and differentials, L’Ho-pital’s rule, and the Leibnitz rule for differentiating integrals. We conclude with some vector and tensor operations, and a discussion of matrix algebra.

Transport Analysis in Mass Transport Phenomena

In this chapter we utilize the equations and principles presented in Chapter 5 to analyze mass transport situations involving diffusion, diffusion plus chemical kinetics, and diffusion plus chemical kinetics and convection.

Transport Analysis in Cascaded Systems

In this chapter we illustrate the finite difference approach of Chapter 7 in stagewise transport operations such as extraction, absorption, stirred tank reactors, and distillation columns. Startup and unsteady-state operations are included. Finite difference techniques are applied with the emphasis on closed form analytical solutions.

Transport Analysis in Fluid Flow Phenomena

In this chapter velocity profiles and volumetric flow rate equations are developed from the equations of motion introduced in Chapter 3. From these equations a discussion of macroscopic flow phenomena is presented, including an analysis of friction factors in laminar and turbulent flow. Then non-Newtonian fluid behavior is discussed, from the point of view of Reynolds number-friction factor plots. Some examples of viscoelastic properties are then briefly presented, along with two-phase solid-liquid suspension flow. The chapter concludes with an analysis of some fluid flow phenomena by boundary layer theory.

Derivation of the Momentum Transport Equations

In the analysis of fluid flow problems, the equation of continuity (material balance) is usually the initial principle invoked. The information gleaned from the equation of continuity is then applied to the equation of motion. From the equation of motion and the fluid properties, the velocity profiles, average velocities, and energy loss can be calculated.

Membrane Resistance and Diffusion Coefficients: The Intact Red Cell

The transport of oxygen through the interior of the red cell is characterized by a diffusion coefficient and oxygen transport through the membrane is described in terms of a mass transfer coefficient. In order to determine these coefficients, a flow apparatus is used in which blood is mixed with isotonic saline solution and flows down a glass tube. The basic data obtained is extracellular oxygen partial pressure versus time. The results are independent of hemoglobin reaction or diffusion since at all timers the hemoglobin is kept fully saturated with oxygen. Therefore, there is no need for hemoglobin-oxygen rate constants and hemoglobin diffusivity. Also, the controlling diffusion equation is linear because of the absence of reaction terms and a straight forward analytical solution is possible.

The Effects of Inert Gases, Preservatives and Storage on Oxygen Transport in Blood

Research in the area of membrane transport has concentrated on the determination of ionic fluxes across biological barriers (1, 2, 11). Erythrocyte membranes are frequent subjects of such studies since they are readily available (1, 11). The transport of oxygen through erythrocyte membranes in particular has received much attention recently. Some previous oxygenation studies have relied on a lumped model approach in which all resistances to oxygen transport are combined into one parameter (8, 12, 13, 14). Differences between predicted and experimental results for these models were explained in terms of possible membrane resistance. This study provides a method for the direct determination of membrane resistance from experimental data (4).

In Vitro Measurement of Uncombined Oxygen Concentration in Intact Red Cells

Oxygen transfer in blood is receiving increasing attention because of current activities in the areas of heart transplants and oxygenator development. In the course of our work on simultaneous diffusion and chemical reaction (1) of oxygen within red cells, it became necessary to measure the concentration of uncombined oxygen in intact cells. Since no method was available, a technique using polarographic PO2 electrodes was developed and is described in this paper. The data thus obtained was used in the governing equations for intracellular oxygen transport to evaluate the overall diffusion coefficient of oxygen in the red cell at 25°C.