1. Conservation Laws—Macroscopic Forms
In this section of the course, we will obtain equations suitable for “macroscopic” balances for conservation of mass, momentum, and energy. On occasion, it will be advantageous to apply conservation laws to an entire unit. We may not know the details at points in space within the unit operation. The conservation laws in a macroscopic form can be valuable to analyze problems of this type. The term “macroscopic” suggests that the balance equation can be applied to a large “black box” without necessarily knowing what’s happening at distinct points in the interior.
We choose a boundary to demarcate the region over which we will do our “accounting.” This should be an enclosed surface defining our control volume. Matter, forces and energy can flow or be transmitted across the control surface of the control volume. Experience helps us to make a smart choice for the control volume. By evaluating what’s happening at the surface with particular attention to entrances and exits, we can determine information about changes in the amounts of mass, energy or momentum with time. One of the challenges is how to represent the various quantities of interest in convenient mathematical forms suitable for obtaining information after analysis.
While we aspire to apply mathematical equations to these systems, keep the physical picture in mind. Understanding the physical meaning of terms allows you to decide which terms are important and which are not.
Later in the course, we’ll see that there are microscopic forms of these equations can be applied to points in space. These yield the continuity equation, equation of motion or the Navier-Stokes equations, but don’t worry about them for now. We will learn how these equations can be used to get velocity profiles, volumetric flowrates, stresses and other information.