Fluid Mechanics for Chemical Engineering

Chapter 2

Global Theorems of Fluid Mechanics

For many applications, it is not necessary to have a detailed knowledge of the flow at all points in the domain. Global conservation relations enable engineers to determine the main characteristics of a flow. For instance, they provide clues about pressure changes induced in a flowing fluid and enable the calculation of forces on walls in contact with the fluid. They also help in estimating the head losses generated in some flows. Using these principles, some of the relations established in Chapter 1 while solving the local Navier–Stokes equations will be resolved.

It is essential for an engineer to handle global theorems. Generally speaking, this chapter is devoted to the global conservation of mass, momentum, and energy. As far as the notion of conservation is concerned, these three quantities are interpreted differently:

– Mass is necessarily conserved. This notion is articulated in this chapter through flow rate conservation relations.

– The conservation of momentum is more complex. In a steady state, considering a fluid domain, the momentum flux leaving the domain equals that entering the domain, if no force is exerted at the domain boundaries. The momentum theorem enables the evaluation of the forces exerted by a flow on a wall, or the estimation of the global head loss produced in a volume, depending on the circumstances.

– The notion of energy conservation is more subtle. Bernoulli’s theorem embodies an energy conservation ...