The first application of the control volume equations will be to develop a suitable analytical model for the long-term steady operation of devices such as turbines, compressors, nozzles, boilers, condensers—a very large class of problems of interest in thermodynamic analysis. This model will not include the short-term transient start-up or shutdown of such devices, but only the steady operating period of time.
Consider a certain set of assumptions that lead to a reasonable model for the steady-state process:
• The control volume does not move relative to the coordinate frame
• The state of mass at each point in the control volume does not vary with time
• As for the mass that flows across the control surface, the mass flux and the state of this mass at each discrete area of flow on the control surface do not vary with time. The rates at which heat and work cross the control surface remain constant.
As an example of a steady-state process consider a centrifugal air compressor that operates with constant mass rate of flow into and out of the compressor, constant properties at each point across the inlet and exit ducts, a constant rate of heat transfer to the surroundings, and a constant power input. At each point in the compressor the properties are constant with time, even though the properties of a given elemental mass of air vary as it flows through the compressor. Often, such a process is referred to as a steady-flow process, since concern is primarily with the properties of the fluid entering and leaving the control volume. However, in the analysis of certain heat transfer problems in which the same assumptions apply, primary interest is in the spatial distribution of properties, particularly temperature, and such a process is referred to as a steady-state process. The term steady-state process will be used for both. It should be realized that the terms steady-state process and steady-flow process are both used extensively in engineering.
Now, consider the significance of each of these assumptions for the steady-state process.
• The assumption that the control volume does not move relative to the coordinate frame means that all velocities measured relative to the coordinate frame are also velocities relative to the control surface, and there is no work associated with the acceleration of the control volume.
• The assumption that the state of the mass at each point in the control volume does not vary with time requires that:
and also:
Therefore, for the steady-state process it can be written