Introduction to Chemical Engineering Fluid Mechanics

INTRODUCTION

Microscopic analysis, as discussed in Part III, is sometimes impractical. Analytical or even numerical solutions of the differential equations can be very hard to obtain for systems with complex shapes or multiple length scales. The difficulty in predicting velocity and pressure fields tends to increase with the Reynolds number and be greatest with turbulent flow. Macroscopic analysis, which employs integral (control volume) rather than differential (point) balances, provides a way to approach such difficult problems. When using only integral balances, much detail and some precision are sacrificed to obtain useful results. Such analysis is valuable even if only the form of a relationship can be predicted, leaving a coefficient to be evaluated experimentally. The benefits of adopting a more macroscopic viewpoint have been illustrated already by the integral analysis of laminar boundary layers Section 9.4). Conserving momentum only for the boundary layer as a whole, and not necessarily at each point, greatly reduced the computational effort but still yielded acceptable accuracy.

The most direct way to derive a macroscopic balance is to state the volume and surface integrals that represent, respectively, the rate of accumulation of the quantity of interest in a control volume and the net rate at which it enters across the control surface. Any internal formation or loss of the quantity is represented by another volume integral. An alternate method, and sometimes the only option, is to integrate a differential conservation equation over the volume of interest. In this chapter these complementary approaches are used to derive integral balances for mass (Section 11.2), momentum (Section 11.3), and mechanical energy (Section 11.4). The momentum equation, which initially considers only fluid–solid interfaces, is extended then to systems with free surfaces (Section 11.5); no modification of the other integral balances is needed. The use of each kind of macroscopic balance is illustrated by examples.

CONSERVATION OF MASS

General control volume

The continuity equation was derived in Section 5.2 by imagining a small, cubic volume of fluid and equating the rate of increase of its mass with the net rate of mass entry across its surface. Letting the volume approach zero led to the differential equation that expresses conservation of mass at a point. The objective now is to derive a mass balance for a control volume of any size or shape.