Accumulation rate process time derivatives

TABLE 8-1 Appropriate Time Derivative for the Accumulation Rate Process in the Equation of Motion"... 

It should be obvious that this term is volumetric, which means that the accumulation rate process applies to the entire system contained within the control volume. The stipulation that the control volume be stationary simplifies the mathematics to some extent, but the final form of the force balance does not depend on details pertaining to the movement of the control volume. Possibilities for this motion and the appropriate time derivatives are summarized in Table 8-1. The substantial derivative operator... 

The total time derivative in the accumulation rate process can be replaced by the partial time derivative because the control volume is stationary and Vs rface = 0. Furthermore, it is acceptable to reverse the order of integration with respect to V and partial differentiation with respect to time because the coordinates of V are not functions of time. Gauss s law transforms surface integrals to volume integrals as follows ... 

Accumulation of the mass of component i within the control volume is written as a time derivative of a volume integral of the density of component i. In other words, the accumulation rate process is volumetric because it occurs throughout the entire contents of the system. The exact form for the time derivative depends... 

While we laud the virtue of dynamic modeling, we will not duphcate the introduction of basic conservation equations. It is important to recognize that all of the processes that we want to control, e.g. bioieactor, distillation column, flow rate in a pipe, a drag delivery system, etc., are what we have learned in other engineering classes. The so-called model equations are conservation equations in heat, mass, and momentum. We need force balance in mechanical devices, and in electrical engineering, we consider circuits analysis. The difference between what we now use in control and what we are more accustomed to is that control problems are transient in nature. Accordingly, we include the time derivative (also called accumulation) term in our balance (model) equations. 

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