i n the past it has been common practice to treat the response of internal

Transcription

1 13 Tray Dynamics- Material Balance* 13.1 INTRODUCTION i n the past it has been common practice to treat the response of internal reflux to either external reflux or feed changes as the result of a chain of noninteracting, first-order lags. In reality, as we shall see, it is a chain of noninteracting second-order lags, although for many columns the first-order approximation is adequate. It has also been assumed that internal reflux is not affected by vapor flow changes. We now know, however, that, depending on tray design and operating conditions, an increase in vapor flow may cause internal reflux: (1) to increase temporarily, (2) to decrease temporarily, or (3) not to change at all. The first of these is commonly termed inverse response because it causes a momentary increase in low boilers in the column base, which is followed by a long-term decrease in low boiler concentration. This apparently was first noted by Rijnsdorp, although Harriot? observed that in an AIChE report3 a bubble cap tray at low liquid rates had less holdup at higher F factors than at low F factors. In this chapter we will derive the dynamic relationships between internal reflux and both vapor rate and external reflux (or feed) as function of tray and column design. The basic tray hydraulic equations are based on the treatment by Van Winkle4 First we discuss what happens on an individual tray, and then derive an approximate model for a combination of trays. Vapor flow will be assumed to occur without lags, and heat-storage effects will be assumed to be negligible. A discussion of reboiler dynamics will be deferred to Chapter 15. * This chapter is based on reference

2 314 Tray Dynamia-iUawiul Balance 13.2 TRAY HYDRAULlCS (See Figure 13.1) Downcomer Liquid Level The height of liquid level in the downcomer is the integral of the difference between the flow from tray n + 1, w,,+~, and the downcomer outflow, woe. In Laplace transform notation: (13.1) flow, lbm/sec, from downcomer, fromtrayn + 1 overflow, lbm/sec, from tray n + 1, into downcomer liquid density in downcomer, lbm/ft3; liquid usually slightly aerated downcomer cross-sectional area, fi?, assumed to be uniform liquid height, feet, in downcomer Downcomer Pressure Drop and Flow The downcomer pressure drop is determined by the differences between the liquid heads and static heads: (13.2) A POc = downcomer pressure drop, lbf/ft' prn = density of aerated liquid on tray n, Ibm/ft3 Hh = height, feet, of aerated liquid on tray just downstream of inlet weir Pn+ = pressure, lbf/fi?, above liquid on tray n + 1; P,, is pressure on tray R Note that we assume that poc does not vary.

3 13.2 Tray Hya'radh 315 FIGURE 13.1 Distillation tray schematic for flows and liquid evaluations

4 3 16 Tray Dynumia-Mated Balance Equation (13.2) may be Laplace transformed to: AP,(s) - BL - -[PDcKx(~) - jsmhk(s) - KRPTR(J)I (13.3) [Pn(s) - Pn+l(J>l Next, from page 508 of Van Winkle:* 0.03 Q&- hd = (100 Ah)2 hd = liquid head in inches of liquid-flow pressure drop through the downcomer & = downcomer flow, gpm Ah = minimum downcomer flow area, ftz This equation may be rewritten: (13.4) (13.5) or ~& BL - X- PDC BC (13.6) (13.7) and Now since wx i&- 2 hp,c is determined by APx, we may write: (13.9) (13.10) (13.11)

5 13.2 Tray Hydraulics 317 Aerated liquid Holdup and Gradient on Tray With reference to Figure 13.1, we assume a linear gradient across the tray: V, = volume, fi3, of aerated liquid on active area of tray (downcomer volume excluded) A, = active tray area, f? (area bubbling occurs) H,, = height, feet, over outlet weir H, = height, feet, of outlet weir (constant) HA = aerated liquid height, feet, at tray inlet (see Figure 13.1) = H,, + H, or in Laplace transform notation: Inlet Liquid Height Over Weir -- Vm(4 - HW(4 + H%f) (13.13) Am 2 Let us assume that the change in inlet height over weir is the same as the change in outlet height over weir. Then: Volume of Liquid on Tray ~,,(~) = H25) ( 13.14) The volume of liquid on a tray is related to the inventory and the density: (13.15) W, is the active tray holdup, Ibm. By Laplace transforming we obtain: ( 13.16) av, 1 aw, - [From equation (13.15)] Pm

6 318 Tray Dynamia-M&& Balance and avm - -wm- -vm apm P2 Pm [From equation ( 13.15)] Tray Liquid Material Balance On the active part of the tray, the liquid material balance in Laplace transform notation is: ~DC(S) - fp,(s) = wm(s) w, = overflow, Ibm/sec, from tray n. Aerated Liquid Density as a Function of Vapor Velocity S (13.17) In Figure 13.16, page 516, Van Winkle4 presents a p3t that shows that froth density decreases with an increase in vapor rate. From thls we can calculate a slope, apm/af. Then: (13.18) F = F factor u~~ vv -0.5 = UVApy = - p. Am vapor velocity, ft/sec, = corresponding to A~ wv = vapor rate, Ibm/sec Ibm pv = vapor density, - ti3 Liquid Overflow from Tray The Francis weir formula can be written in the form: (13.19) (13.20)

7 13.2 Tray Hydratrlia 3 19 and Q = overflow, ft3/sec, of aerated liquid In Laplace transform notation: b = a constant for any given weir (see Van Winkle: pages ) (13.21) (13.22) (13.23) Tray Pressure Drop In Laplace transform notation: aarm P,(s) - P,,+~(S) = APm(s) = - WV(4 (13.24) at9, For perforated trays Van Wde (page 519 of reference 4) gives the relationship: (13.25) PL = P. = velocity, ft/sec, through the holes WV PAh total hole area per tray, f? dry-plate pressure drop, inches of clear liquid density, lbm/fi3, of clear liquid vapor density, lbm/fi3

8 320 Tray Dynanak-Material Balance C, = discharge coefficient that is a function of both the tray-thickness/hole-diameter ratio, and the hole-area/active-area ratio. This is presented in Figure 13.18, page 5 19, of Van Winkle.4 Thistlethwaite6 has developed a correlation for C, that facilitates computations: T, = tray thickness and Dh = hole diameter. From the above: so (13.26) (13.27) = 2(%) (13.28) For valve trays, if the caps are not M y lifted: Once the caps have been fully lifted, the equation for perforated trays applies DERIVATION OF OVERALL TRAY EQUATION As a first step in deriving an overall tray equation, let us construct a signal flowdiagramfiomequations (13.1), (13.3), (13.11), (13.13), (13.14), (13.16), (13.17), (13.18), (13.21), and (13.24). This is shown in Figure By successive reductions this signal flow diagram can be reduced until the following equation is derived: (13.29)

9 13.3 DeriPatiOn.foPerall Tray Eqrcation 321 \ t For all columns examined so far, the denominator quadratic has factored into two terms with substantially unequal time constants. The smaller, in most cases, has been roughly equal to the numerator time constant in the multiplier for w&). Then equation (13.29) reduces to: (13.30) (13.31) Note that K, + [ - v,- af ap, +--- af ap, 2 - How(A, has the dimensions aw9 af at9, a3 3 lbm/sec lbm/sec x sec +A,) ] (13.32)

10 322 Tray Dy~~--Mated Balance " U z m C s m a - m ti c, E c, P L s E 5 P : 0 m - I C m N" $2 W- C SE us Gk

11 13.4 Mathematical Moa!.el fw Combined Trays 323 We can now make three generalizations about KTR and w,: 1. If KTR is positive, an increase in w,, causes a temporary increase in internal reflux. This results in an inverse response, since low boiler concentration is increased temporarily in the base of the column. 2. If Km is zero, a change in w, causes no change in internal reflux. This we term neutral response. 3. If KTR is nedative, an increase in w, causes a temporary decrease in internal reflux. This we have termed direct)) response; it temporarily augments the long-term decrease in base low boilers caused by increasing w,,. The term that seems to affect the sign of KTR most strongly is dapm/&,,. This term is almost zero for valve trays, and for all valve-tray columns we have checked so far, the calculations predict inverse response over the entire range of normal operation. For sieve trays, dap,/dw, is small at low boilup rates and the calculations predict inverse response. This term increases rapidly with boilup, however [see equation (13.27)] and the calculations indicate that as w, increases, the column shows next neutral response, and finally, for large values of w,, direct response MATHEMATICAL MODEL FOR COMBINED TRAYS A column with a number of trays may be represented by a mathematical model such as shown in Figure This may be simulated readily on a large digital computer. For a mathematical analysis, however, or for hand calculations, we need a simpler model. A system with a large number of identical first-order lags may be represented by a dead time a, which is equal to nt n is the number of lags. Therefore, the response of reflux from the lowest tray, wl, to a change in external reflux n trays away is: (13.33) To = condensing temperature, C TR = external reflux temperature, C wr = external reflux, Ibm/sec A simplified model for response to w, may be arrived at by a somewhat intuitive method. Consider the overflow response of an individual tray to a

12 324 Tray Llymwaia-Mated Balance FIGURE 13.3 Material balance coupling with vapor and liquid flow

13 325 step change in w,: The integrated outflow is: At t = a: is : Km -t/rm w(t) = Aw, - e Tm Ibm/sec (13.34) W(t) = Aw,Km(l - (13.35) W(w) = Atp,,Km (1 3.36) For n trays the total amount of liquid that is either displaced or held up CW(m) = nawkm (13.37) Since we are assuming that Aw, is felt by all of the trays simultaneously, equations (13.29) and (13.30) apply to all trays at the same time. But the outflow fiom each tray must flow through all lower trays. The time required for the bulk of the liquid displaced or held up to flow down is approximately ntm. Therefore, the average outflow rate is: (13.38) Overall, then, we obtain the approximate transfer hction for an n-tray column: (13.39) It was previously indicated that if the sign of Km is positive, the columnbase composition will exhibit an inverse response to a change in boilup. Equation (13.39) indicates the possibility of another kind of inverse response, that of column-base level. If Km is positive and if Km/~m is greater than unity, an iwease in boilup will result in a temporary iweuse in base level. As will be discussed in Chapter 16, this can cause great difficulties base level is controlled by boilup. If Km/~m is positive and close to unity, a change in boilup causes no change in level for a period of time equal to ntm. The control system seems af liaed with dead time, but in reality, as shown by equation (13.39), it is not. Thistlethwaite6 has carried out a more extensive analysis of inverse response in distillation columns. REFERENCES 1. Rijnsdorp, J. E., Birmiqgbam Univ. 2. Harriott, P., Process Conml, McGraw- Chm. EM. 12:5-14 (1961). Hill, New York, 1964.

14 326 Tray DynamM -Mat& Balance 3. Williams, B., J. W. Begley, and C. Wu, Rollins, Inverse Response in Tray Efficiencies in Distillation a Distillation Column, CEP, Columns, final report of AIChE 71(6):83-84 (July, 1975). Research Committee, 1960, page 6. Thistlethwaite, E. A., Analysis of In- 13. verse Response Behavior in Distil- 4. Van Winkle, M., Distifhtk, McGraw- lation Columns, M.S. Thesis, De- Hill, New York, partment of Chemical Engineering, 5. Buckley, P. S., R. K. Cox, and D. L. Louisiana State University, 1980.