15 Condenser and Reboiler

Dynamics

15.1 LIQUID-COOLED CONDENSERS WITH NO CONDENSATE HOLDUP

a s mentioned in Chapter 3, we find in the chemical and petroleum industries two principal types of liquid-cooled condensers: (1) the horizontal type with vapor on the shell side and coolant in the tubes, and (2) the vertical design with vapor in the tubes and coolant on the shell side. We can also think of condensers in terms of whether the coolant goes through just once (no axial mixing) or is recirculated to achieve good axial mixing. A condenser with the latter type of cooling is said to have cctempered” cooling.

Condenser with No Axial Mixing of Coolant

Once-through coolant is by far the most common choice. An approximate analysis for a condenser that has a single pass on the coolant side is presented in Chapter 24 of reference 1, and will not be repeated here. It involves the following simplifjring assumptions:

1. Heat storage in the heat exchanger metal is negligible. 2. Subcooling is negligible. 3. Mean temperature difference is arithmetic.

Although these reduce the complexity somewhat, we are still lefi with the job of solving partial differential equations. Perhaps the easiest to read descriptions of their solutions are those by Hempe12 and by G ~ u l d . ~ A much simpler model, largely empirical, has been proposed by Thal-Lar~en.~ Since most liquid-cooled condensers are fairly fast with time constants in the range of 10-60 seconds, we will not pursue their dynamic equations further. For important applications, where subcooling may be of concern, one should probably resort to simulation.

347

348 Gmdenser and Reb& Qparnia

The literature on subcooled condensers is very sparse; one paper has been published by Luyben, Archambault, and Ja~dEet,~ and another by Tyreus.6

If the sensible heat load of subcooling is not too large compared with that of the condensing heat load (and this is usually the case), the following static gains may be derived:

(15.2)

(15.4)

process condensing temperature, O K

coolant inlet temperature, OK

coolant exit temperature, O K

coolant specific heat, pcu/lbm "C pcu/sec "C fcz condenser heat-transfer coefficient,

rate of condensation, Ibm/sec coolant flow rate, Ibm/sec latent heat, pcu/lbm, of process vapor condenser heat-transfer area, fcz

Condenser with Well-Mixed Coolant

Qualitatively the condenser with well-mixed (tempered) coolant is discussed in Chapter 3, Section 9 (see Figure 3.19). A mathematical analysis of its dynamics is given in Chapter 24 of reference 1. We will not repeat it here, but it leads to the following results (simphfjmg assumptions are the same as in the previous section):

15.2 Fhded Co&men-Open-Loop Dynamics

where w, - WA

average bulk temperature, O K , of coolant coolant holdup, Ibm

The subscript OL means open loop. Also:

349

(15.7)

(15.8)

(15.9)

( 15.10)

(15.12)

The first-order dynamics of this type of condenser make it much easier to control than the condenser with once-through coolant.

15.2 FLOODED CONDENSERS-OPEN-LOOP DYNAMICS

Most flooded condensers are of the horizontal type with vapor on the shell side and coolant in the tubes as shown in Figure 15.1 (see also Figure 3.1). As the liquid level in the shell varies, so do heat-transfer area and rate of condensation. We would like to find out how condensing pressure and rate of condensation are affected by rate of vapor flow into the condenser, back pressure downstream of the vent valve, and rate of removal of liquid fiom the condenser shell. We will make several simplifylng assumptions:

350 C&er and Reboiler W m i a

1. The sensible heat load is small enough in comparison with the latent heat load that it may be neglected.

2. Submerged heat-transfer area, A,, is proportional to liquid level above the bottom of the lowest tube. The change in condensing area, A,, is then the negative of the change in A,. This assumption is not bad if there are many tubes and if they are not “layered.” To make this assumption valid may require that in some cases the tube bundle be slightly rotated about its axis.

3. For the time being, we will assume that the vent valve position is fixed. 4. Heat storage in the heat exchanger metal may be neglected.

FIGURE l S . l Horizontal condenser with coolant in tubes and partially flooded on shell side

15.2 F h h d conrlensers-Op~-Loop -b 351

We may now write the following equations, some in the time domain and

(15.13) some in the s domain:

(15.14)

(15.15)

where v = q c =

u, =

A, = T = wc =

wo =

ww = A, = Tc = TA = Pc =

w, =

A T C = Q =

s = =

PR = Rv =

(15.17)

(15.18)

(15.19)

(15.20)

(15.21)

system vapor volume, ft3 rate of heat transfer, pcu/sec

pcu/sec fi? "C

condenser heat-transfer coefficient, ~

condensing heat-transfer area, p "C or OK

Ibm/sec process vapor condensed latent heat of vaporization, pcu/lbm, of condensing process vapor liquid outflow, lbm/sec vapor inflow, lbm/sec vapor outflow, lbm/sec submerged heat-transfer area, fi? condensing temperature, "C or O K

average coolant temperature, "C or O K

vapor space pressure, lbf/fi? pressure, lbf/ftz, downstream of vent valve vent valve resistance, Ibf sec/fi5 pounds of liquid in shell between bottom of lowest tube and top of highest tube total heat-transfer area, ft2, of condenser flow, ft3/sec

352 Chdenser and Reboiler Dynamia

Upon Laplace transforming equations (15.13), (15.14), and (15.16), we

(15.22) obtain:

qc(s) = UcZc AT(@ + Uc hTAc(s)

(15.23)

(15.24)

Response, P,(s), to Various Inputs

To find the response, Pc(s), to various inputs, we combine equations ( 15.13) through (15.24) into the signal flow diagram of Figure 15.2. The first reduction of this is given in Figure 15.3 and the final reduction in Figure 15.4. From this we may write by inspection:

RvlPv -(s + a) a

- V

-Rv 1 + d-+ Kco -2 + s + 1 Fc [: a

PC(S) =

a

TA(s) + &PR(s) + w&) (15.25) 1 - UCAC S I A p

s + a RV where

The first term on the right-hand side of equation (15.25) may be written:

KFC (1 + a) T i s 2 + 2 5 ~ ~ s + 1

For all such condensers we have studied, the denominator has a large damping ratio so that the quadratic may be factored into two terms, one of which has a typical time constant of 1.5-3 minutes, while the other is only a few seconds. The above then reduces to:

KFC (5 + a) TFCS + 1

where

15.2 F

hk

d C

Onrtenret-O

p~-Loop +h

353

il 5 8 B U

C

TI 0

0 e 0

IC

4"

42 E e 3 L

0

m U

0 e

-c

-

2.;;; $e 2

LL

354 G

mh

mcr and R

cbodc~Dynam

ia

2 s 2

- 3 B r

w-

0

E E U

b: I

E F cn w

- 0

G

-2

vis

ZE! -3

Q

gg

15.2 F

W

condnrr~--Optn-Loop w

h

355

i 5 8 B U

C

U

0

0

G

rc

0

4”

e E E L

- P 8

ZT

i

U

G

rnm

r

e

rn s, 3

3

2s

LL

r.

356 Gmdenscr and R e W LIpunia

Case Where R, Approaches Infinity

last equation reduces to: For the case where Rv becomes very large (very little vent gas flow), the

W P V ) (s + 4

s + 1) uczc aTc avpc + -- & P v a r c

PC(S) =

+ - uAd’TA(s) + w&)] s + a

(15.26)

The first term on the right may be written:

Kkc (s + a) S(&S + 1)

where

1 lPv ucxc aTc avpc + -- Ecp P v arc

Kkc =

and

Response, w,(s), to Various Inputs

The signal flow diagram of Figure 15.2 can be redrawn to show wC(s) as an output as shown in Figure 15.5. This can then be reduced to the form of Figure 15.6, fiom which we can write by inspection:

15.3 Reboilrrs-Open-Loop Dynamh 357

where

L($Rvr a AP + 1)

1 P 1 + ( Y - R , + Kco

s + l F c $2 + PC a a

X i c ( g R v s + 1) (15.27a)

Case Where R, Approaches Infinity

Again, as R, becomes very large, the preceding equation becomes simpler:

wc(5) =

L

X 5 1 -

VIR $ + iJ ucAc arc a PIPc + - -

P P h arc

(15.28)

15.3 REBOILERS-OPEN-LOOP DYNAMICS We wish to make an analysis of column bases with associated reboilers

where there is signrficant liquid holdup. The analysis should take into account the temperature of the entering liquid and the sensible heat effect of the liquid mass. Such an analysis also applies to vaporizers with associated separators or knockout drums.

Heat-Transfer Dynamics

The combination of thermosyphon reboiler (or any high circulation rate reboiler) and column base or separator may be represented as shown schematically in Figure 15.7. The various equations may then be written as follows.

358 condmrrr and Rcboilcr Dynamurr

a C

al U

C

8 -8 8 37 8 E

E U

G

0

IC

L

- P 8 27ii

E-

=I?! U

6

-c

WP

a*

&&

I

15.3 R

thih~--Opnr-L

oqp QVU

UU

~

359

i 5 U

C

0

0

3 0 0 e O

1c

s 8 E e L

m

I

U

- B

2s W

%!

gs

g

z

e I

C I

w-fn

360 Gmuhcr and R e w mtb

By using perturbation techniques and Laplace transforming this equation we get: cp Ti fpi(s) + cp @i Ti(s) + qT(s) - hrp fPB4.T) - cp @BU TBu(s) - cp T B U wBu(s)

- cp WB TBdS) - cp TBU fPg(s) = cp 5 [TBU W B ( S ) + WB TBdS) 3 (15.30)

Material Balance

(15.31)

(15.32)

j [wi(t) - wBU(t) - w E ( t ) ] dt = WB(t) or

1 [vi($) - wBu(s) - WB (J)] - = W B ( ~ )

5

Steam-Side Dynamics

The following are taken fiom Chapter 25 of reference 1:

(15.33)

(15.34)

Qi(5) = -ps(s) Qv + -pcs(~) aQv + - X v ( s ) Qv (15.35) a p s dPCS Z V

(15.36) qds) 1 Q&) = - - ht Plt qT(5) = URAR [Tcs(s) - T ~ 4 s ) l (15.37)

aTCS 8PCS Qi<s> - Q4s)

C R 5

Tc,(s) = - PCS(5)

Pcs(s) =

(Note that heat storage of the reboiler metal is neglected.)

v FIGURE 15.7 Schematic representation of column base and reboiler holdup

15.3 Reboilnr-Open-Loop LlynamS 361

In these equations:

Tu@) = steam condensing temperature, "C P&) = reboiler shell pressure, lbf/fi?

Q = steam flow rate, ft3/sec qT = heat transfer, pcu/sec

UR = reboiler heat-transfer coefficient, p / " C ft2 sec

A R TBU

= heat-transfer area, fi? = boiling temperature of process fluid, "C = rate of steam condensation, fi3/sec = steam latent heat of condensation, pcu/lbm = steam density Ibm/ft3 at Pa and Ta = acoustic capacitance of reboiler shell, fi5/lbf = VJT', where V, = reboiler shell volume, ft3 = steam supply pressure, Ibf/fi?, upstream of control valve = valve stem position

The terms aQv/aP,, aQv/aP,,, and aQv/dXv may be evaluated by the methods of Chapter 15 of reference 1.

The column-base pressure dynamics may be represented by:

(15.38)

column acoustic impedance, looking up from the base, lbf sec/fi5 density of vapor boilup, lbm/ft3 rate of boilup, lbm/min column-base pressure, lbf/fi?

From the preceding equations we can prepare the signal flow diagram of Figure 15.8. This can be partially reduced as shown on Figure 15.9 where three new functions are defined:

1

1 Psc As2

x -

(15.40)

(15.41)

362 G m h m and Re- Dynumia

For all reboilers examined to date, A(5) has been so small, both statically and dynamically, as to be negligible. It therefore will be omitted in the remainder of this book. It is advisable, however, to calculate A(5) for any new system as a check.

We have also found that the sensible heat effect of the liquid mass in the column base or separator is small. Heat-transfer lags are typically only several seconds; vapor flow finm the separator blows steam flow almost instantaneowly.

FIGURE 15.8 Preliminary signal flow diagram for heat transfer dynamics

15.3 Reboden-Open-Lmp Dynamia 363

Base Level Control Cascaded to Steam flow Control

the signal flow diagram of Figure 15.10. Note that: We assume here that averaging level control is desired. We may then prepare

KCf Gcf(s) = flow controller transfer function K m h

Kch Gc&) = level controller transfer function AB P L = liquid density, lbm/fi3

= liquid-level transmitter gain

= cross-sectional area of column base, fi?

Since Ti varies slowly or not at all, T4s) = 0.

FIGURE 15.9 Partial reduction of figure 15.8

364 condenrer and Reboiler Dynmzia

Noncritical Versus Critical Steam Flow

If steam flow is critical, then:

= 1 (15.42) 1 QV

1 - W) ap,, since

aQ -v = 0 a r c ,

Flow control loop gain and dynamics are determined entirely by the instrument characteristics. Note that:

1 C R 1 -

(15.43) W) = 1 1 a T m

1 + - - (b (4 ap, CR 5 A, Pa

FIGURE 15.10 Signal flow diagram for base level control cascaded to steam flow control

365

If steam flow is noncritical, 1

1 - W) a Q V / e S

has lead characteristics and a static gain of less than unity, permitting higher loop gain, faster flow-control response, and much higher flow controller gain. To avoid problems with flow controller tuning, the system should always be operated in one flow r e p e or the other. In most cases steam flow is noncritical.

Signal Flow Diagram Simplification

Since we have called for averaging level control, the natural frequency of the level control loop will be much lower than that of the flow control loop. Note that:

aQ,. K V GV(4 Kcf Gf (4 ax, B(s)

1 5 - -

1 X

K+G*(s> K+ (15.44)

This leads us to the final signal flow dagram of Figure 15.11.

Base Level Control by Direct Manipulation of Steam Valve

From Figure 15.9 and by assuming A(s) = 0, we can prepare the signal flow diagram of Figure 15.12. Note that if steam flow is critical, aQ,,/dPa = 0.

Further Mathematical Simplification

It has already been indicated that for averaging level control cascaded to steam flow control, one may substitute l/K+for the flow control loop. Use of the mathematical models discussed here on commercial reboilers indicates that typical time constants range from a fraction of a second to 5-10 seconds. Practically speaking ~ ( s ) , 444, and A($) reduce to constants. As we will see in Chapters 16 and 17, it usually will be possible to use much simpler reboiler models than that discussed in this section.

366 CDnrinrer and Rebode-Dynmnia

15.4 PARTIALLY FLOODED REBOILERS

The partially flooded reboiler is similar in many ways to the partially flooded condenser. Usually, although not always, it is a vertical thermosyphon reboiler. As discussed in Chapter 4, Section 2, it is controlled by throttling the steam condensate, which in turn varies the condensate level in the shell and thereby the heat-transfer area for condensation. That area covered by liquid permits only sensible heat transfer h m the condensate; this is a small heat load compared with that of the condensing steam and is treated as neghgble.

The signal flow diagram of Figure 15.5 for a flooded condenser may be used as a starting point. In this case pressure dynamics are essentially negligible. If the steam supply pressure is constant, the steam condensing temperature is also constant.

FIGURE 15.11 Final signal flow diagram for base level control cascaded to steam flow control

15.4 PartiaUy F h U Rebodem 367

The signal flow diagram of Figure 15.13 may be prepared next. As shown by the reduced form of Figure 15.14:

(15.45) WC(S) - 1 --

A* s + l wo(s) - MR UR AT-

awsIf where

wc wo Ast

UR

xi7 aA, awSH ~WSHI- ART WSH

= rate of steam condensation, lbm/min = rate of steam condensate withdrawal, lbm/min = steam latent heat, pcu/lbm

= heat-transfer coefficient, Pmin = average exposed heat-transfer area, f?, for steam condensate - ART

= total heat-transfer area of reboiler, ft2 = condensate on shell side, lbm

p / " C

-

FIGURE 15.12 Signal flow diagram for base level control by direct manipulation of steam valve

368 C

hndenser and Reb&

Dynam

ia

L

a,

0 a

a

n 2

U

a, -0

0

0 E

L

e E E - z g U

G

m c m

- 5; Gi? w

m

c

la-

aa

E5E E

2

369 15.4

Pd

y Fh

hd

Reboden

L n 2

z - - 0

U

0

0

0

L

e E E 3

- F

;g '13

0

G

- U

JZi

U

Ma

l r

5s

S

Z

La

Condmer and Reboiler Dynmtia

weight of condensate that will fill shell side of reboiler, Ibm boilup, Ibm/min laten; - heat, of boiling liquid, pcu/lbm TST - Tsu, "C

Design experience with flooded reboilers is limited but indicates that typical time constants are of the order of 2-5 minutes. Simulation studies show that substantial improvement in response speed may be achieved by lead-lag com- pensations with transfer functions such as:

TD - - s + 1 a

where we let:

(15.46)

Commercial lead-lag compensators commonly have values of a between 6 and 30. Some provide a fixed a and some have adjustable a. The former are usually much less expensive.

It is interesting to look at the response of wc to a change in AT. From Figure 15.14:

(15.47)

If w, were held constant, we would intuitively expect a step increase in AT to cause an initial increase in heat transfer, and therefore in wc. Condensate level, however, would eventually increase (thereby decreasing&) and eventually wc would have to equal w,. Solving equation (15.47) for a step increase in AT shows this to be true:

so that:

(15.49)

In equation (15.47) wc ( t ) is a perturbation variable (deviation from steady state). As shown by equation (15.49), it decays to zero as time goes to infinity.

15.5 Partially Flooakd Rebodem fm Low-Boiling Materials 371

PARTIALLY FLOODED REBOILERS FOR LOW-BOILING 15.5 MATERIALS

In Chapter 4, Section 2, we discussed a variation in flooded reboiler design for low-boiling materials. As shown by Figure 4.4, we throttle the steam instead of the condensate. The static force-balance relationship is given by equation (4.1) :

Hs pLBL + P, + APLine ( 4 4 BL = HL PL - BC BC

where HL = loop seal or standpipe height, feet Hs = condensate level in the shell, feet Pc, = steam pressure in shell, lbf/f? abs pL = condensate density, lbm/ft3

The vented loop seal maintains P, just a little above atmospheric pressure. We can now write the following equations:

~ 4 5 ) - wc(5) (15.50)

Z ( 5 ) = ---&) aT, (15.51)

From equation (4.1), if we assume that the line loss Mhne is negligible,

5 P Y (VRJ

ap,

PcS(s) =

BL 8;

and that HL pL - is constant, then:

(15.52)

a R -=- A R T ( 15.53)

- P,(5>

P L (BL/Bc) H45) =

aHr [Hrlrnax

&(5) = a ~ , U R Hr(5) (15.54)

(15.55)

(15.56)

wc = @ (15.57)

WBW -41. - (15.58)

q ~ ( t ) = UR A R ( ~ ) AT(t)

AT = T, - TB,

A,

ABW

3 72 Cbnhser and Reboiler Dpunia

FIGURE 15.15 Signal flow diagram for flooded reboiler for low boiling point materials

FIGURE 15.16 Reduced signal flow diagram for flooded reboiler for low boiling point materials

373

where

ws = steam flow, Ibm/min wc = rate of steam condensation, Ibm/min p I = steam density in shell, Ibm/ft3 V = average free volume in shell above liquid level, ft3 Tu = condensing temperature, "C pL = condensate density, Ibm/ft3

Other terms are as defined earlier in this chapter. To get equation (15.55) into the s domain, write:

-

where

(15.59)

(15.60)

(15.61)

We may now prepare the signal flow diagram of Figure 15.15, which may then be reduced to the form of Figure 15.16. From the latter we see that:

where

Also:

(15.62)

(15.63)

(15.64)

As in the previous section, if there were a step change in Tsv, there would be an immediate spike in wc, which would slowly decay to zero.

REFERENCES 2. Hempel, A., TramASME, 244 (1961). 3. Gould, L. A., Chemiurl PromControl,

1. Buckley, P. S., Techniques of Pmcea Control, Wiley, New York, 1964.

374 W e r and Reboiler Qynamaa

Addison-Welsey, Reading, Mass., . 923-928 (1973). 1969. 6. Tyreus, B. D. “Modeling and Sim-

4. Thal-Larsen, H., ASME paper 59-A- dation of Vertical Subcooling 117. Condensers,” presented at ACC

5. Luyben, W. L., J. P. Archambault, meeting, San Francisco, Calif., June and J. P. JaUffiett,AIChEJ, 19 (5): 22, 1983.