A rearrangement of equation (18.2) gives:

Substituting for yl from equation (18.3), we get:

Partial differentiation of equation (18.9) leads to:

As discussed in Chapter 17, most evidence available today indicates that there is no significant lag in vapor flow between adjacent trays, so:

(s) + L„+1x„+i(s) + V„Ey„_i(s) + (Jn-l - J„) ^(i) ~ X„SM„(S)

For most trays F„_i = V„ = V, and so the last term cancels out.

Next equations (18.4) and (18.5) may be transformed and combined: Ln(s) 1___1

Harriott1 has shown that:

where

Am - active area of the tray, which is usually the column cross-sectional area minus the area of two downcomers, ft2

dQ = change in flow over the oudet weir 3/f per change in height over the weir, ftVsec ft

By using the approximate form of the Francis weir formula:

we obtain:

Typical values of calculated by the authors for sieve tray columns are in the range of 2.5-8 seconds. But, as mentioned previously, the derivation of t-tr is valid only if the column does not have significant inverse response.

Finally equations (18.8), (18.9), (18.13), and (18.14) may be combined (after some reduction) into the signal flow diagram of Figure 18.2 for a basic tray. The transmissions F(i) and jy„(s) go to the tray above; the transmissions L„{s) and x„(s) go to the tray below. In this way a signal flow diagram for any number of trays can be prepared, although the feed tray and two terminal trays have slighdy modified diagrams.

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