## Differential Vapor Pressure

Another early effort to compensate for pressure variations in binary distillations resulted in the development of the differential vapor pressure cell.7 The version made by Foxboro is known as the "DVP Cell." As shown in Figure 10.1, a bulb filled with liquid whose composition is the same as that desired on a particular tray is installed on that tray. It is connected to one side of a differential-pressure transmitter. The other side of the AP transmitter is connected directly to the same tray. When the liquid on the tray has the same composition as the liquid in the bulb, pressure in the bulb will be the same as pressure on the tray. For that condition the AP transmitter is normally set to read midscale. Deviations in the tray composition from that in the bulb are then reflected by AP transmitter output signals above or below midscale.

Where applicable, this instrument is capable of great sensitivity. There are, however, some practical problems:

1. Liquid of the desired composition for filling the bulb may not be chemically stable over a long period of time, thus leading to measurement errors.

2. Bulb filling is a job for the factory; reusing an instrument originally bought for another job may not be easy.

3. Response to pressure changes is much faster than the response to composition changes, which causes erroneous readings and sometimes control problems. Some users, therefore, have installed snubbers in the pressure impulse line.

Although this instrument is primarily intended for binary systems, Tivy7 has shown that with a bit of patience and cleverness, it can be used on some multicomponent systems.

To determine the required span of the DVP Cell in inches of water corresponding to a desired composition span, it is necessary to write and solve a few equations. If the reference bulb has been properly filled with a liquid of known composition, then at a reference temperature TR pressure in the bulb is:

where

PR = total pressure, atm abs, in bulb xR = mol fraction low boiler in bulb PL = vapor pressure, atm, of low boiler at TR

PH — vapor pressure, atm, of high boiler at TR J m = low boiler activity coefficient in bulb 7hr = high boiler activity coefficient in bulb At the same temperature, TR, the process pressure is:

where

PT = total pressure in process, atm xL = mol fraction low boiler in process PL = vapor pressure, atm, of low boiler at TR PH = vapor pressure, atm, of high boiler at TR yL = low boiler activity coefficient y h = high boiler activity coefficient

Now the DVP Cell actually measures PT — PR and is normally calibrated in inches of water for such spans as 20-0—20 or 50—0—50. From equations (10.1) and (10.2):

which reduces to:

which is the same as equation (10.2). We can now solve for xL:

Since PR "washes out" of equation (10.3), one might well ask, "Why do we have it?" The answer is that there is only one instrument to calibrate rather than two, and the need for very accurately calibrated temperature and pressure measurements is eliminated.

The activity coefficients may be found from:

where Avl, Bvl are Van Laar coefficients. For other ways of finding activity coefficients, see the various texts on distillation and vapor-liquid equilibrium.

The vapor pressures of the pure components may be found by the Antoine equation:

where

Al,Bl,Cl = Antoine coefficients for the low boiler Ah, Bh,Ch = Antoine coefficients for the high boiler T = temperature, °C For the two-parameter Antoine equation:

If, as we have postulated, the bulb contains the same chemical species as the process, PT — PR will always be zero when the process has die same composition as the bulb. If the chemical species are the same but the compositions are different, PT — PR will vary somewhat with pressure (and, therefore, temperature). That is to say, for xL - xR constant, PT - PR will change slightly as PT changes. To facilitate calculations, an HP-41C program has been written that is valid if the process and bulb contain the same chemical species.

For systems that are not strictly binary but where the nonkey components are present in either relatively constant or very small concentrations, the DVP Cell may often be used by calibrating its output against laboratory analyses. When the desired composition is such that the components would react with one another over a period of time, one may substitute another fill material (if one can be found) that has the same PR as desired at TR. In this event the equations are slightly changed.

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