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In this list of assumptions, the one which is most likely to be invalid for a practical process is that the plate is an ideal equilibrium stage.
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+ If the efficiency of the plate is not 100 percent, we can introduce an individual tray efficiency of the Murphree type, defined as
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& = xn - Xn+l x:, - Xn+l
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where x z is the concentration of the liquid in equilibrium with gas of composition y n. Notice that for an i&al plate E, = 1 and x,, = xi. In general the efficiency of a plate depends on the design of the plate, the properties of the gas and liquid streams, and the flow rates. We could include efficiency in our mathematical model; however, to do so would greatly increase the complexity of the problem. To account properly for the variation in efficiency with flow rates would require empirical relationships for a specific plate design.
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We begin the analysis of this process by writing an ammonia balance around each plate. A mass balance on ammonia around plate 1 gives H% = L2x2 + Vyo - Llxl - vyI (21.24)
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This last equation states that the accumulation of NH3 on plate 1 is equal to the flow of NH3 into the plate minus the flow of NH3 out of the plate. Notice that V and H do not have subscripts because of assumptions 5 and 6. A mass balance on ammonia around plate 2 gives
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- = vy1 - L2n2 dt
(21.25)
The last equation does not contain a term L3x3, since we have assumed that
x3 = 0 .
For an ideal plate x ,, = xi, and the equilibrium relation of Eq. (21.23) becomes y,, = mx, + b Substituting the equilibrium relationship into (21.24) and (21.25) gives H-dxl L2x2 - Llxl + Vm(xo - xl) = dt and H% = Vm(xl - xq) - L2x2 where xg = (yc - b)lm is the composition of liquid that would be in equilibrium with the entering gas of composition yo. Solving these last two equations for the derivatives gives
dxl - I(L2x2 - LlXl) + X-H
dxz -=
xo - Xl)
F(x, - x2) - ;L2x2
(21.27)
Thus far the analysis has resulted in two nonlinear first-order differential equations. The nonlinear terms in Eqs. (21.26) and (21.27) are L2x2 and Llx 1. The forcing functions in this process, which must be specified as functions of t, are the inlet gas concentration [xe = (ye - b)lm] and the inlet liquid flow rate L3. In order to solve for x l(t) and x z(t), we must have two more equations, ob-
THEORETICAL ANALYSIS OF COMPLEX
FROCESSEiS
mined by considering the liquid-flow dynamics on each plate. Assume that each plate can be considered as a first-order system for which the following equations hold: *
3% = L3-L2
(21.28)
q- = L2-L1 dt
The time constants in these equations (~1 and 72) can be determined experimentally by the methods of Chap. 19. The first-order representation for liquid dynamics was found to be adequate by Nobbe (1961). We now have four differential equations [Eqs. (21.26) to (21.29)], and six variables (X 1, ~2, xa, LI, L2, L3). Since xu and L3 are the forcing functions, which are specified functions of time, these four equations can be solved for xl(t), x2(t), Ll(t), and L2(t) in terms of xc and L3. _ We shall now divide the problem into two cases. The first case requires that we find the response of y2 to a change in the inlet gas concentration only, the liquid flow rate remaining constant. In this case, the problem is linear and only Eqs. (21.26) and (21.27) are needed. In the second case, it is assumed that we want to know the change in outlet concentration y2 for a change in both inlet flow and inlet gas concentration. For this case, four simultaneous differential equations must be solved, two of which contain nonlinear terms. One approach to this problem is to linearize the nonlinear terms as was done in the case of the steam-jacketed kettle of the previous example; however, since this technique has already been illustrated, we shall not repeat it here.
*The assumption that the plate behaves as a first-order system with respect to liquid-flow dynamics would have to be justified experimentally. For the common bubble-cap plate, liquid builds up on the plate and flows over a weir, which may consist of a circular pipe or a vertical plate. The resistance to flow from the plate is therefore a weir, for which flow-head relationships are known (see footnote in Chap. 6). However, these flow-head relationships for weirs have been developed for the flow of liquids that am not aerated. In the case of flow of liquid over a bubble-cap plate, the liquid is very turbulent as a result of the agitation of the bubbles rising through the liquid. For this reason, one cannot expect the flow-head relations developed for quiescent flow to apply to the turbulent conditions present in the liquid on a plate. The true flow-head relation should be determined experimentally. The fact that the flow rate is assumed to vary without change in holdup on the plate (assumption 5) appears to be contradictory. Actually, to increase the flow rate, a slight increase in level (and therefore holdup volume) above the crest of the weir is required. However, for the example under consideration, it will be assumed that the change in level needed to produce a substantial increase in flow is so small that the change in the amount of liquid on the plate is a small fraction of the total liquid holdup.
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