Control volume time-dependent flows

Solution We first select the control volume as shown by the dotted line in the figure, assuming that, at the downstream end of the control volume, the velocity profile in the free jet is flat. Next, we apply the macroscopic momentum balance, Eq. 2.5-3, to the control volume. We need be concerned only with the x component, because this is the only momentum that crosses the control volume boundaries. The flow is steady, and therefore the time-dependent term vanishes, as do the forces, since there are none acting on the control volume. Thus the equation reduces to ... 

In Chapter 2 we developed models based on analyses of systems that had simple inputs. The right-hand side was either a constant or it was simple function of time. In those systems we did not consider the cause of the mass flow—that was literally external to both the control volume and the problem. The case of the flow was left implicit. The pump or driving device was upstream from the control volume, and all we needed to know were the magnitude of the flow the device caused and its time dependence. Given that information we could replace the right-hand side of the balance equation and integrate to the functional description of the system. 

Unfortunately, neither the computer nor the potentiometric recorder measures the primary variable, volume of mobile phase, but does measure the secondary variable, time. This places stringent demands on the LC pump as the necessary accurate and proportional relationship between time and volume flow depends on a constant flow rate. Thus, peak area measurements should never be made unless a good quality pump is used to control the mobile phase flow rate. Furthermore, the pump must be a constant flow pump and not a constant pressure pump. 

In the previous chapter (Section 20.3), we showed the equation describing transport of a non-reacting solute in flowing groundwater (Eqn. 20.24) arises from the divergence principle and the transport laws. By this equation, the time rate of change in the dissolved concentration of a chemical component at any point in the domain depends on the net rate the component accumulates or is depleted by transport. The net rate is the rate the component moves into a control volume, less the rate it moves out. 

Fig. 17.14 Finite-volume, staggered-grid, spatial-difference stencil for the transient compressible stagnation-flow equations. Grid points, which are at control-volume centers, are used to represent all dependent variables except axial velocity, which is represented at the control-volume faces. The grid indexes are shown on the left and the face indexes on the right. The right-facing protuberance on the stencils indicates where the time derivative is evaluated. For the pressure-eigenvalue equation there is no time derivative.

Size reduction that is achieved in a wet mill will depend in part on the residence time of the batch in the mill. Residence time can be controlled by operating in single-pass mode, where the batch is pumped through the milling device from one vessel to another, or in recycle mode. For both, it is important to measure slurry flow rate to confirm that residence times are maintained as a process is scaled up. When the batch is recirculated, one way to quantify this is to convert elapsed time to number of batch turnovers (batch turnovers = elapsed time X flow rate through the mill/total batch volume). It can take several passes or batch turnovers to achieve a steady-state particle size distribution, depending on the mill. 

Summary of Equations of Balance for Open Systems Only the most general equations of mass, energy, and entropy balance appear in the preceding sections. In each case important applications require less general versions. The most common restrictedTcase is for steady flow processes, wherein the mass and thermodynamic properties of the fluid within the control volume are not time-dependent. A further simplification results when there is but one entrance and one exit to the control volume. In this event, m is the same for both streams, and the equations may be divided through by this rate to put them on the basis of a unit amount of fluid flowing through the control volume. Summarized in Table 4-3 are the basic equations of balance and their important restricted forms. 

For the first time we must as a consequence of the plug flow take into account spatial variation as well as time dependence. This means that the concentrations of A and B will have z- and t-dependence and the equations describing them will be made up of partial rather than ordinary differentials. We can derive the equation that describes the plug flow system by first visualizing a zone of reaction (Figure 5) that corresponds to a differential control volume Acr dz. 

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