Accumulation rate process

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 

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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... 

All terms in the momentum balance have units of momentum per unit time, which is synonymous with the units of force. In this respect, it is necessary to account for external forces (i.e., sources) that act on the fluid within the control volume. In general, these forces are not surface related. They are called body forces because they act volumetrically like the accumulation rate process, which means that each fluid parcel within the system is affected by a body force. The... 

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 ... 

Vector-Tensor Manipulation of the Accumulation Rate Process and Forces Due to Convective Momentum Flux... 

Even though this vector-tensor identity was verified using summation notation in rectangular coordinates, it is valid in any coordinate system. It is extremely tedious to verify vector-tensor identities that involve the del operator in curvilinear coordinate systems because the unit vectors exhibit spatial dependence. Now it is possible to combine terms in the equation of motion due to the accumulation rate process and convective momentum flux. Equations (8-24) and (8-25) yield ... 

Notice that the accumulation rate process and convective forces scale as pV /L, whereas viscous, pressure, and gravity forces scale as pV/L . If one takes the ratio of these two dimensional scaling factors, an important dimensionless number is obtained ... 

This is the integral form of the mass transfer equation within an arbitrary control volume V (f). Notice that there is a term of the form / pi(n Vsurface) dS in the accumulation rate process and in the net rate of input due to mass flux acting across the time-varying surface S(t). These terms are present because the surface that bounds the control volume is in motion. The fact that they cancel provides quantitative support for the claim that the final form of the mass transfer equation is independent of the characteristics of the control volume. All surviving terms in the mass balance. 

Time-Averaged Properties. The unsteady-state macroscopic mass balance for mobile component A is applied to the quiescent liquid, where the rate of interphase mass transfer via equation (11-205) is interpreted as an input term due to diffusion across the gas-liquid interface. There are no output terms, sources, sinks, or contributions from convective mass transfer in the macroscopic mass balance. Hence, the accumulation rate process is balanced by the rate of interphase mass transfer across time-varying surface S t), where both terms have dimensions of moles per time ... 

In every equation of change, one of the terms on the left side represents the accumulation rate process based on a stationary volume element. Hence, one seeks an expression for the accumulation of kinetic energy per unit volume of fluid in a stationary control volume ... 

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