17 Pressure and AP Control

17.1 INTRODUCTION

C olumn Dressure control. as mentioned earlier. usuallv can be of the I

"averaging" type. Most columns do not need "tight" pressure control, and for some of them it is undesirable; a rapid change in pressure can cause flashing or cessation of boiling in the column. The former might result in flooding and the latter in dumping. For well-damped pressure control, the mathematical models in this chapter, which mostly treat condenser and reboiler dynamics as negligible, are usually adequate.

For tight pressure contol, we should use these models with caution. Most of the tight column pressure controls we have studied have closed-loop resonant fi-equencies in the range of 0.5-2 cpm. For the upper value one should make at least a rough check of condenser and reboiler dynamics. It may be of interest that the only applications of tight pressure control we have found are in heat- recovery schemes where the vapor from one column serves as the heating medium for the reboiler of another column, and perhaps hrnishes heat to other loads. If the vapor flow must be throttled to each load, constant up- and downstream pressures help good flow control.

1 7 . 2 HEAT-STORAGE EFFECT ON COLUMN PRESSURE

The stored heat in the liquid in the column and its base can exercise quite a leveling effect on pressure and differential pressure. If pressure starts to drop, more liquid flashes, which tends to reduce the rate at which pressure falls. If pressure starts to rise, the reverse happens-the rate of boiling is decreased and pressure rises at a slower rate. In this section we will see how vaporization rate, w,,, is affected by heat storage.

A simplified analysis can be made if:

1. Column Al' is small compared with absolute pressure. 2. Latent heats at the top and bottom of the column are nearly the same. 3. And the temperature difference across the column is not too large.

405

406 Pressure and AP Control

These assumptions permit us, at least for annospheric or pressurized columns, to lump the column contents together and to assume an average temperature, Tcp, and an average pressure, Pep, as shown by Figure 17.1. Note that the reboiler is lumped with the column base but the condenser and overhead receiver are not lumped with the column top.

We may now write a heat-balance equation similar to equation (15.30) but with terms for reflux added:

CPTFFwF(S) + CP@FTF(S)+ qdJ) - C p [ ( W c + R) + sJK011 T&)

+ c~T'wR(s) + cp@RTR(s) - ~ p T ~ p f ~ g ( ~ ) - ~p~T~pW~~l(s) (17.1)

= ( A p + C p T p ) %(-r)

The material-balance equation is: 1

wcol(s) = -[wF(s) + wR(s) - wc(s) - w B ( s ) ] (17.2) 5

Reboiler with Steam Flow or Flow-Ratio Controlled

If we treat reboiler dynamics as negligible, and if we assume that steam will be flow or flow-ratio controlled, we may combine the above equations as shown in the preliminary signal flow diagram of Figure 17.2. This, in turn, may be reduced to the form of Figure 17.3.

Reboiler with Direct Throttled Steam

For the case where the steam valve is manipulated by some variable other than flow or flow ratio, we may need to account for reboiler dynamics to calculate qT. Referring to Figure 15.8, we may make a partial signal flow diagram as shown on Figure 17.4 where:

C, = reboiler hot-side acoustic capacitance, fi5/lbf

A R = reboiler heat-transfer area, fi? Pa = reboiler hot-side pressure, lbf/fi?

Q,, = heating-medium flow, actual fi3/sec

pa = heating-medium (vapor) density, lbm/ft3

A, = heating-medium latent heat, pcu/lb

UR = reboiler heat-transfer coefficient, pcu/sec "C fi?

17.2 Heat-Storwe Efect on Column Pressure 407

FIGURE 17.1 Simplified treatment of heat storage effect on column pressure dynamics

408 Pressure and AP Control

This may be reduced to the form of Figure 17.5 where: -

(17.3) URA, aTaIaPa P ( 4 = -

X URAR 1 + - L P a c R s - aQv/apa

For critical flow, note that aQ,,/aP, = 0.

of Figure 17.6. Figures 17.3 and 17.5 may now be combined into the signal flow diagram

17.3 PRESSURE CONTROL VIA VENT AND INERT GAS VALVES

As mentioned in Chapter 3, we often control pressure in a column by a pressure-dividing network such as shown in Figure 3.7. The control-valve input- signal spans are usually the same but one valve opens while the other closes.

FIGURE 17.2 Preliminary signal flow diagram for column heat storage dynamics

17.3 Pressure Control via Vent and Inert G

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410 Pressure and AI? Control

FIGURE 17.6 Combined signal flow diagram for figures 17.3 and 17.5

17.3 Pressure Control via Vent and Inert Gas Valves

The equation for the inert gas valve is:

Similarly, the equation for the vent valve is:

411

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(17.5)

where WIG = inert gas flow, lbm/sec

we = vent flow, lbm/sec

Pq = column pressure, Ibf/ft2

PIG = inert gas supply pressure, lbf/fi?

PR = pressure downstream of vent valve, lbf/@

6, = controller output signal

These equations may be combined into the partial signal flow diagram of Figure 17.7. Note that the valve gains, dwIG/aO, and aw,/aO,, are both assumed to be positive; reverse action of the inert gas valve is obtained by the - 1 term. If we can assume that PIG and PR are sufficiently constant, we may reduce Figure 17.7 to the form of Figure 17.8. The term C, is the acoustic capacitance of the column, vapor line to condenser, and the condenser.

For the case where reboiler steam is flow or flow-ratio controlled we can now combine Figures 17.8 and 17.3 into the signal flow diagram of Figure 17.9. Note the pressure feedback on w, through dT,/aP,, and the addition of the pressure measurement, K,G,(s), and the controller, K,G,(s). Figure 17.9 can be reduced to the form of Figure 17.10 where:

(17.6)

From equation (17.6) we can see that open-loop pressure dynamics are essentially first order. Since in most cases the inert gas bleed and the vent flow are fairly small, the valve gains, awIG/d6, and awe/aO,, tend to be small. Together with the first-order dynamics, this commonly leads to large controller gains (small proportional bands) and control valve saturation for fairly small disturbances.

412 Pressure and AI’ Control

FIGURE 17.7 Partial signal flow diagram for column pressure control via manipulation of iner gas and vent valves

FIGURE 17.8 Reduction of signal flow diagram of figure 17.7

17.3 Pressure C

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17.4 PRESSURE CONTROL VIA FLOODED CONDENSER

Here we consider four cases:

1. Flow- or flow-ratio-controlled steam to reboiler, signhcant inerts. 2. Flow- or flow-ratio-controlled steam to reboiler, negligible inerts. 3. Steam to reboiler not flow or flow-ratio controlled, signdicant inerts. 4. Steam to reboiler not flow or flow-ratio controlled, negligible inerts.

Flow- or Flow-Ratio-Controlled Reboiler Steam, Significant lnerts

For this case we combine Figures 15.4 and 17.3, which leads to the signal flow diagram of Figure 17.11. This reduces to the form of Figure 17.12 where:

KFC6 + a)

This will clear to an expression that is first order in the numerator and second order in the denominator.

Flow- or Flow-Ratio-Controlled Steam, Negligible Inerts

This system may also be represented by a signal flow diagram similar to that of Figure 17.12 except that Kp3Gp3(s) replaces KpzGpz(s) and has a slightly different definition [see equation (15.26)]:

G C ( S + a) S ( 4 J + 1)

x -[(we + WB) + sW,,,]- S ( 4 J + 1) Ap apLp

(17.8) Kp3Gp3(s) = Kf-,(s + a) cp - aT, 1 +

Steam to Reboiler N o t Flow or Flow-Ratio Controlled, Significant lnerts

Pressure controls via a flooded condenser with signhcant inerts and where reboiler steam is not flow or flow-ratio controlled may be represented by a signal flow diagram (Figure 17.13) formed by the combination of Figures 17.6 and 15.4. This diagram may be reduced to the form of Figure 17.14 where: KP4GP4(S)

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that Kp5Gp5(s) is defined differently [see equation (13.26)]: The signal flow diagram for this case is the same as Figure 17.14 except

KXs + a\

(17.10)

17.5 PRESSURE CONTROL VIA CONDENSER COOLING WATER

This method of controlling pressure, although once popular, has fallen into some disfavor in recent years. This is particularly true for once-through coolant. Since its flow rate cannot be allowed to go too low-which would lead to fouling as well as excessive exit coolant temperatures, which, in turn, contribute to corrosion-it permits only limited control of pressure. Tempered coolant, which avoids these problems, is a better choice for column pressure control via coolant flow manipulation.

The signal flow diagram for pressure control via coolant flow manipulation is given in Figure 17.15. Once-through coolant and steam flow or flow-ratio control are assumed. The lefi-hand side of this diagram comes tiom Figure 17.3. Note that wc is the rate of condensation, Ibm/sec.

17.6 COLUMN AP CONTROL VIA HEAT TO REBOILER

The control of column AP by throttling steam to the reboiler was once very popular in the chemical industry, particularly for small columns. The usual practice was to run at a boilup that would give considerably more reflux than called for by design. This would usually provide a product purer than specification. In an era when it was common practice to overdesign columns (lowffactors, bubble-cap trays, and extra trays) and there was little concern about saving energy, this approach to control did have the advantage of usually providing a good-quality product with simple insmentation. For today's tightly designed columns, it is technically less satisfactory, and with the rapidly rising energy costs its wastage of steam is economically unattractive. Nevertheless we still have an interest in this control technique for override purposes; an override controller is now commonly used to keep column AI' from exceeding the maximum value speciiied by the column designer or determined by plant tests.

Before looking at the overall control scheme, let us discuss what is meant by the column impedance, Z&).

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Relationship Between Boilup and Column Pressure Drop

So far we have assumed that there is negligible lag in vapor flow between adjacent trays. This means that we treat the acoustic impedance Z , , between the bottom and top trays as a pure resistance approximately equal to

2 (2 = + _ - z;) where the subscripts s and Y refer to the stripping and rectification sections of the column. The validity of this assumption has been shown by tests run by Stanton and Bremer' on a 90-tray column and by the computer studies of Williams, Hamett, and Rose2

If, however, we are interested in the high-frequency behavior of the column, then we must treat the impedance as that of an RC chain as shown in Figure 17.16. Here each RC section represents one tray, the resistance is that of the tray and layer of liquid to vapor flow, and the capacitance is the acoustic capacitance of the space between the trays. The terminal impedance, Z&), is simply P&)/Q,(s). Mathematically the entire network may be studied by the methods of transmission-line analysis.

If the individual RC sections are equivalent, or nearly so, then the impedance looking up from the reboiler is approximately:

where

and

Z+) + Z&) tanh nl Z,(s) + ZT(s) tanh nl

(17.11)

(17.12)

( 17.13)

(17.14)

where R and C are the resistance and capacitance, respectively, for each tray and vapor space, and 1 is the total number of trays. Two cases are now of primary interest.

If ZT = 0, as would be true for an atmospheric column or for a column with tight overhead pressure control, then:

As can be seen, the impedance becomes IR

( 17.15)

] (17.16) 2 4 2 2 2 - 1 R C r - . * . 15 at low fiequencies.

17.6 Column AI? Control Pia Heat to Reb& 423

FIGURE 17.16 Equivalent network for vapor flow and pressures in column

424 Pressure and AI? C

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The other case of interest is that of a tall column (many trays). Then at high fiequencies:

(17.17)

A more detailed, more rigorous treatment of this subject will be found in Day.3

Af Control Cascaded to Steam Flow Control

Let us assume that column AP control is cascaded to steam flow control and that the latter, the secondary or slave loop, is tuned to be much faster than the primary loop. As in the case of level control cascaded to flow control, the flow loop must have a linear flow meter or an orifice meter followed by a square root extractor.

We may now simpMy and rearrange Figure 15.8 and add the AP controller as shown in Figure 17.17. Note that for constant top pressure control, column- base pressure control and AP control are equivalent.

Af Control Via Direct Steam Valve Manipulation

If, as is usually the case, the AP controller is connected directly to the steam valve positioner, we need a somewhat different analysis. This calls for another rearrangement of Figure 15.8, as shown by Figure 17.18.

Note that P(s) is defined by equation (17.3).

REFERENCES

Paper W-2-58, Wilmington, Del., 3. Day, R. L., “Plant and ProcessDynamic 1958. Characteri&qJJ edited by A. J.

2. Williams, T. J., B. T. Hamett, and A. Young, Academic Press, New Yo& Rose, I d . Eqg. Chem., 48(6): 1957.

1. Stanton, B. D., and A. Bremer, ISA 1008-1019 (1956).