The simulation of heat exchanger control with tube-side chemical reaction fouling

Chemtcnl Engineering Science, Vol. 41, No. 9, pp. 2363-7.372, Pnntcd in Great Britain.

THE SIMULATION WITH TUBE-SIDE

1986.

OF HEAT EXCHANGER CHEMICAL REACTION

P. J. FRYER Department

of Chemical

Engineering,

OOO!+ZSO9/S6 S3.00 + 0.00 Pergamon Journals Ltd.

CONTROL FOULING

and N. K. H. SLATERt

University

of Cambridge, U.K. 19 April

(Received

Pembroke

Street,

Cambridge

CB2

3RA,

1985)

Abstract-Various alternative strategies for the control of heat exchangers subject to chemical reaction fouling are simulated, using a numerical technique based on the method of characteristics. The strategies are demonstrated using an experimentally determined local fouling rate model for milk fouling. A performance calculation for four different tubular heat exchangers is presented, showing that the amount ofcontrol action required decreases with increasing exchanger area, and that varying the shell-side flow rate does not guarantee control of the exchanger.

INiRODUCIlON

In a recent article (Fryer and Slater, 1986) we discussed a model computational technique for the dynamic simulation of heat exchanger performance in the presence of tube-side chemical reaction fouling. Deposit accumulation was assumed to obey the familiar Taborek (1972) form of kinetics. The technique employs the method of characteristics to integrate simultaneously the constitutive enthalpy balance equations and the fouling rate equation. Temporal and spatial variations of temperatures through the exchanger are thus explicitly accounted for. For illustration an experimentally determined model for local fouling from milk was employed to demonstrate the technique for four different designs of the exchanger; two constant wall temperature exchangers of different length, a co- and a counter-current exchanger. These exchangers performed the same thermal duty when clean, but owing to fouling their ultimate performance was substantially different. Control strategies for such exchangers were not considered. Industrially it is often necessary to operate heat exchangers in some manner which ensures that process streams have a constant output enthalpy. A primitive solution to this problem was suggested by Kern and Seaton (1959). They developed equations for asymptotic fouling and recommended that extra heat transfer area be included in the exchanger design so that the required output temperature is achieved when fouling has reached an equilibrium level. Such an approach is adequate for mild fouling where an acceptable steadystate performance is attained. However, in certain industries, such as food processing, the severity of deposit accumulation is frequently such that the eventual asymptotic degree of fouling would unacceptably impair performance. In such cases, daily cleaning procedures are necessitated. Food fouling is

+Present address: Research Laboratory, The Netherlands.

Bioprocessing Section, Unilever P.O. Box 114, 3130 AC Vlaardingen, 2343

especially rapid and some form of simple feedback control is often employed to maintain constant process stream output enthalpies. In chemical processes, fouling commonly occurs over a longer time scale but the eventual reduction in thermal performance may well be equally unacceptable. Such an example was considered by Sundaram and Froment (1979), who described a control strategy for furnaces in thermal gasoline cracking plants. In this case, the influence of coke accumulation on the walls of the furnace tubes was mitigated by control of the furnace temperature. The model simulation technique which we described previously can be employed to assist the optimization of heat exchanger design for controlled operation. Without modification, the technique could be used to generate accurate values of the fouling resistance R,, or to allow determination of the extra area required by the Kern-Seaton approach. Unlike the method of Sundaram and Froment, our technique can also be applied to cases where both the shell- and tube-side temperature profiles vary with time. However, if exchangers are to be selected on the basis of their controlled performance, modifications to the procedure must be made. In this paper we first outline the necessary modifications and then demonstrate the procedure for controlled operation of the four exchangers which we previously discussed in the context of fouling from milk. As milk fouling is both rapid and severe, the consideration of milk as a sample fluid permits a thorough testing of the resulting programs; the procedure could, however, be applied to other types of fouling, provided a local rate model is known. The procedure set out in this paper thus constitutes a method of properly integrating local fouling rate models to obtain information on overall controlled exchanger performance. For demonstration purposes the following performance calculation has been considered: A stream of milk (0.25 kg s- ‘) is to be heated from 333 K to a target temperature of approximately 370 K. Utility water is available at 390 K and process steam can be supplied at pressures up to 20 bar. Table 1 gives the specifications

P. J. FRYER

2364

and N. K. H. SLATER

Table 1. Details of heat exchangers used in computer simulations Type of exchanger

Length (m)

TiN

7-P”=

10.0 3.7 7.5 4.5

333.0 333.0 333.0 333.0

370.4 370.5 370.4 370.6

Tgd = constant = 373.0 K

constant = 390.0K Co-current Counter-current TOwd =

wr= 0.25 kg s-l, ws (if applicable) = 0.5 kg s-l, cross-sectional area = 4.90 x 10e4 m2.

of four different concentric tube exchangers which accomplish this duty when clean, and it is desired to estimate their subsequent performance subject to various control strategies. UNCONTROLLED

EXCHANGER SIMULATION

As the simulation procedure for uncontrolled exchangers has been previously set out in detail (Fryer and Slater, 1986), only a brief review is given here to assist the present discussion. Solids deposition was assumed to occur in locally even layers so that the soIid-fluid interfacial temperature, T,, was shown to be given by (see Notation and Fig. 1) T

=

1+#J+si

TpJT

390.0 390.0

373.0 373.0

0.025 m. Shell cross-sectional area = tube

Assuming that the process streams pass along the tube (and shell) sides of any exchanger in plug flow, then the constitutive enthalpy balance equations take the form of ordinary differential equations along certain characteristic lines in the z - t plane. For the tube-side fluid, (3) along characteristic dz z = Similarly,

C+@+WT,

fl

d =

TfN

lines of slope (the a characteristics).

Uf

for the shell-side fluid (where

relevant),

-

(5)

where along characteristic gi (= h&3,/1,), a Biot number, is a dimensionless local fouling factor which was shown by experiment to vary according to the generalized form similar to that suggested by Kern and Seaton (1959), and modified by Taborek et a[. (1972). dBi = k,(r,)exp dt

-kk,Bi.

(2)

For the fouling from milk investigated by Fryer and Slater, it was found that E, the activation energy for deposition of milk solids, = 89 + 6 kJ mol - ‘; kd(rw), a deposition rate constant, = (9.9 +- 1.0) x lO’O/r, s- ‘; and k,, a deposit removal rate constant, = (1.3 k 0.4) x lo-3s-*.

SHELL

(4)

TUBE

TS

walI

ho s Fig. 1. Idealized distribution of fluid, wall and deposit temperatures at a point on the heat transfer surface of a generalized concentric tube heat exchanger.

dz -_=v dt

s

lines of slope (the y characteristic).

(6)

Closure is permitted by eq. (2), which gives the temporal variation of Bi along lines of constant 2 /?characteristics. (infinite slope) in the z--t plane-the A discussion of the construction of the characteristic mesh for each exchanger type is not repeated here. Solution of eqs (2), (3) and (5) by direct integration along the appropriate characteristics was performed by a modified Euler predictor-corrector algorithm similar to that described by Acrivos (1956). Boundary conditions were set by specifying the shell and tube fluid inlet temperatures, and, where appropriate, the tube wall temperature. Finally, to maximize the thermal effects of fouling, the overall heat-transfer resistance was arbitrarily assumed to be concentrated on the tube side (4 = 0). The method of characteristics, is, however, equally applicable to the more general case of non-zero 4, and could be extended to the case where fouling occurs on both the shell and tube sides.

SIMULATION

OF CONTROL STRATEGIES

Three types of control might be attempted to mitigate the effects of fouling and maintain the tubeside process stream outlet temperature constant. The advantages and disadvantages of these strategies, together with any modifications to our procedure which are necessary to implement each of them, are considered in turn.

2365

Simulation of heat exchanger control

Decreasing

the tube-side flow rate

Generally such a strategy would be inappropriate as any decrease in tube-side flow rate may accelerate local fouling rates for two reasons. Firstly, because it has been found that the rate of solids accumulation is inversely proportional to the surface shear stress [cf. eq. (2)] (Fryer and Slater, 1986), and secondly, because a reduction in ur would diminish local tube-side heattransfer coefficients causing Tfi to approach T,; this again results in enhanced fouling rates [eqs (1) and (2)]. For these reasons no simulation has been attempted using this strategy. In practice it is vital to maintain the tube-side flow rate as high as possible. For example, in milk processing, problems occur within the tube-banks of multiple-stage falling film evaporators; when flow down one tube is reduced due to an initially small amount of deposit the rate of fouling is increased and the tube eventually blocks completely. The use of flow bypasses for control in milk plants is additionally restricted to the shell side since all milk must undergo the requisite amount of heating, i.e. for pasteurization.

Increasing

the shell-side

temperature

Unlike method (i), this strategy will produce a stable steady state, albeit at a higher value of g (the tube average Bi) than in the uncontrolled state since the mean shell-side temperature will be higher. Although a higher Twd initially enhances the solids accumulation rate by increasing T, [see eq. (2)], the rate will not increase indefinitely, since in the limit of infinite deposit thickness Ts will approach Tr. As the tube- and shell-side flow rates do not change, it is unnecessary to adjust the characteristic mesh for the co- and counter-current systems. It is convenient, however, to introduce an extra characteristic in the case of the uniform wall temperature exchanger as changes in T,, occur simultaneously at all points along the tube whereas the tube-side fluid takes a finite time to cross the exchanger. Each fluid a characteristic may thus encounter a range of Twd. An additional y characteristic of zero slope (t = constant) was introduced along which Twd = constant. This resulting characteristic mesh is shown in Fig. 2, together with the appropriate directions of integration. The inset to Fig. 2 shows the calculation of mode (i + 1, j, k + l), for

which Twd(i + 1, j, k+ 1) = T&i, j+ 1, k-c 1). This value of Twd is then used to calculate T,, and hence the rate of deposit accumulation can be found.

Increasing

the shell-side flow rate

This strategy does not necessarily guarantee control since, although the mean shell-side temperature will increase, it cannot increase beyond Twd = T :“. In the limit of infinite shell-side flow rate, the exchanger behaves as one of constant wall temperature. Equation (Al) (see Appendix) can be used to calculate the lowest heat-transfer coefficient for which the outlet temperature can be kept constant:

Corresponding

to this, the maximum

possible value of

2% is:

E,,

= ;

-

1.

In practice, the maximum Biot number which can be accommodated will depend on an acceptable shell-side pressure drop or the adequate supply of heating fluid. The control of heat exchangers by bypass streams is discussed by Shinskey (1976). In the general case where 4 # 0 (i.e. some shell-side film resistance), increasing the shell-side flow rate, by decreasing that resistance, will increase the effectiveness of control action. Equation (7) will still represent the theoretical maximum % for which control would be possible. Implementation of this strategy must take account of the changes in the shell-side flow rate. For incompressible fluids any change in v, propagates instantaneously along the exchanger, altering the slopes of all the y characteristics. As a result, there exist areas of the mesh where the Q, /I and y characteristics do not intersect at the same point. The methods devised to overcome this problem for co- and counter-current systems are considered in turn.

Counter

current

Figure 3 shows the mesh resulting from a step change in shell-side velocity from v,, to vs2, at time t

Fig. 2. Typical characteristic mesh for an exchanger of uniform wall temperature. The nomenclature for specifying nodes at the intersection of particular a, /V and y characteristics is indicated adjacently.

P. J. FRYER and N.

2366

K. H.

SLATER

Tf =T; (i, j-l,k-1)

Fig. 3. Typical characteristic mesh for counter-current exchanger. Control action alters the shell-side fluid flow rate at ttit causing node (i + 1, j, k + 1) to move from point 1 to 2.

(I,l,k)

(I.pl,k+l)

Tf =T; tcrit -

(i+l, j-1.k)

Fig. 4. Typical characteristic mesh for co-current exchanger. Control action alters the shell-side fluid flow rate at rtit causing node (i + 1,j, k + 1) to move from point 1 to 2. = ttit_ At this time, all the shell-side characteristics alter the slope, and the time taken between nodes along the y characteristics changes from Ayi to Ay2. Figure 3 demonstrates that this change produces some nodes at which the three characteristics do not intersect simultaneously. It is necessary, therefore, to choose a node at which the three properties T,, T, and Bi are to be simultaneously determined and from which the calculation is continued. The choice of this node is made in such a way that the overall characteristic mesh is re-established as rapidly as possible. The procedure will be described for node (i + 1, j, k + 1) shown in the inset to Fig. 3. In the absence of change of slope at tcri,, all three characteristics would have normally converged at point 1. The change in vScauses the u and y characteristics to cross at point 2. Integration of the shell-side conditions to point 2 can be carried out in two stages, one of length At, = fait--t(i,

j+

1, k+ 1)

(9)

[where t(i, j + 1, k + 1) refers to the time at node (i,j + 1, and one of length

k + l)]

(10)

In order that the predictor-corrector algorithm can be employed, the tube-side conditions at point 2 are obtained by linear interpolation between those at points 1 and (i, j, k). Integration can then proceed using point 2 as the new node (i + 1, j, k + 1) from which the (i + 1)th cxcharacteristic continues. It can be seen from Fig. 3 that the new mesh is established for all y characteristics which enter the exchanger after t = fait, and that the new At, is smaller than the old one, leading to an increase in computer processing time. This effect is not severe. Co-current Figure 4 shows the mesh resulting from a step change in shell-side flow rate. In this case, it is most convenient to carry out the integration along a y rather than an u characteristic, since in our examples these propagate faster through the system. Integration along the slower-moving a characteristic would not be values of T, would be possible, as unknown encountered. The inset (a) to Fig. 4 illustrates the calculation of the first node along the (k + 1)th characteristic. In order to re-establish the mesh as rapidly as possible, the time at which the (k + 1)th characteristic enters the exchanger

Simulation of heat exchanger control is selected carefully. Since the nodes (i, 1, k) and (i - 1, 2, k) are known, and At, is constant, the position of the second node of the (k + 1)th characteristic [node (i, 2, k + I)] may be readily located. Node (i+ 1, 1, k + 1) is then located by using the local value of Ay. is known, integration along the y*+ 1 Once t(i+l,I,k+l) characteristic can begin. The calculation of node (i+ 1, j, k+ 1) is shown in inset (b) to Fig. 4. The two-stage procedure described for the counter-current case must again be adopted. The tube-side conditions at point 2 are again found by interpolation between point 1 and node (i, j, k). The mesh is then re-established using point 2 as the new node (i + l,j, k + 1) from which ai + f emerges. Figure 4 demonstrates that the new mesh is soon established. The resulting At, is larger than the old one, leading to a decrease in CPU time at the expense of slight numerical accuracy. As the flow rate increases, At, tends to At, and the y characteristic flattens out. CONTROL

ACTION

0.2 0.0

--=

d T,,

For co-current

1 -exp(

-

250 I 4

I 2

0

100

I 6

tength

(ml

I

I

killlinles in s , I a

10

=,,I

EVALUATION

The previous subsection has described the techniques necessary to allow the effects of process control to be simulated. In this section, the procedures employed to evaluate the necessary extent of control action at the end of each a characteristic will be described. Some estimate of the sensitivity of the fluid output temperature to changes in shell-side (or wall) temperature resulting from a buildup of solids can be obtained by differentiating the temperature distributions given in the Appendix. For uniform Twd, dTpUT

2367

-z).

340

(all tlmesin

V

I

330[0

2

4

b

Length

5.)

I

8

I

10

(ml

(11)

flow,

%=&[I-exp(y)z]. For counter-current

(12) flow, Time

(5)

(13) where B=*

WC

(14)

WfC ff

2.5,(d’

T

I

I

1150000

and >

OndL

1 .

(15)

Should T BUT fall below the required value, then the appropriate temperature correction can be approximated as 6T iN (or Twd) = ( T P*(required)

s

- T p”) / Length

s

(or Twd).

We emphasize that this procedure is approximate as it neglects any change in Bi over the small time

Fig. 5. Variation

(e-01

of (a) local Bi, (b) tube-side process fluid

temperature Tf, (c)E and wall temperature T,, and (d) local heat fluxes for model fouling by milk in the constant wall temperature exchanger ( T$ = 373 K) specified in Table 1.

2368

P. J. FRYER

and N. K. H. SLATER

increments between u characteristics. In the simulations conducted here, this correction factor was found to yield stable control of the heat exchanger. At the end of each a characteristic, eq. (16) was applied, using the appropriate value for dTPUT/dTfN calculated using the value of 0 at that point. The correction required was then applied to the next y characteristic to enter the exchanger. The sensitivity of T PUT to changes in w, may not be as readily approximated since T,, K and u also change. A much simpler empirical approximation was employed in this situation: SW = c ( T PUT (required)

- T PUT)

250

1

3

2 Length (m)

(17) 380 8,

where c is a constant. A value of c = 0.01 was found to give stable control for the simulations. RESULTSAND

I

O.OiO

DISCUSSION

A similar set of performance calculations to that described in our previous paper (Fryer and Slater, 1986) has been. considered. The effects of process control on the heat exchangers of Table 1 have thus been simulated. Although the overall quantity of heat transferred to the process fluid remains constant, it will be seen that the required control action is particularly exchanger-dependent. Uniform wall temperature exchangers Figures S(a) and 6(a) show the distribution of Bi at various times in the two uniform wall temperature exchangers, for which the initial wall temperatures were 373 and 390 K, respectively. As with uncontrolled operation, fouling proceeds to an asymptotic level although for both exchangers the final extent of fouling is higher than that for the uncontrolled cases. The values of Bi for the shorter, hotter exchanger are at al1 times much higher than those for the longer exchanger and at long times they increase up to 3.4 [Fig. 6(a)] at the exit as compared with 0.85 [Fig. 5(a)]. These levels of fouling are greater than those for uncontrolled operation (0.85 and 0.63, respectively). Figures 5(b) and 6(b) show that the tube-side outlet temperatures have been maintained constant, but that the temperature profiles became less steep as fouling proceeded. The changes in wall temperature required to maintain the output temperature constant are shown in Figs 5(c) and 6(c), together with the variation of Bi the mean value of Bi for the entire exchanger. The results are also summarized in Table 2. For practical purposes it is significant that the hotter exchanger required an overall wall temperature increase of nearly 90 K, compared to less than 6 K for the cooler exchanger. Such an increase may well be beyond the scope of sensible plant operation. Some explanation of this disparity can be seen in the values of q/q0 given in Figs 5(d) and 6(d). Since the total amount of heat transferred to the process fluid must remain constant, any decrease in flux in one section of the exchanger must be matched by a corresponding increase elsewhere. Such an increase in flux requires an enhancement of the local temperature driving force AT. If the

1 1

330’0

3.5y 3.0

I

I 3

2

Length (m)

1

I

I

I - 480

-

460

iz

- 440 420

I 4000

4

0.0;

1000

2000

2

3000

Ttme (5)

I

I

1

-I 200000

150000

2 z

100000

“d

50000

‘9

0

0.5

1.0

I.5

2.0

2.5

3.0

0

Length Cm)

Fig. 6. Variation of (a) local Bi, (b) tube-side process fluid temperature T,, (c) E and wall temperature T,, and (d) local heat fluxes for model fouling by milk in the constant wall temperature exchanger ( T g = 390 K) specified in Table 1. initial AT is small at any point, then a unit change in T, produces a greater proportional effect upon heat flux than if AT were much larger. Since the AT values are much smaller for the cooler of the two exchangers (3 K

2369

Simulation of heat exchanger control

Table 2. Effect of fouling on the exchangers of Table 1 if TFUT is kept constant

to)

(1) TiN or T,,,,, adjustment Change in T iN

Exchanger

or L(K)

T;=

373-378.6 390-472.3 3-399.1 3-33.3

373 K 390K Co-current Countercurrent Tg=

Bi, 0.622 2.390 0.821 1.502

(2) Change in ws Exchanger Co-current Counter-current

Change in wS O.Sl.47

Length

(m)

Bi, 0.762

not possible 440

as opposed to 20 K), less control action is necessary to maintain product output conditions; the long exchanger is thus more sensitive to incremental changes in T,. It was previously shown that local heat fluxes increase at the exit even in uncontrolled operation of the TWd= 373 K exchanger. In this case, the increase arose from enhanced values of AT produced by the lowering of Tras fouling builds up. Corresponding to the substantially greater values of T, for the short

exchanger, the ultimate value of E is much larger than that for the hotter exchanger (Table 2) which may well cause cleaning problems for the removal of a thicker deposit. Co- and counter-current exchangers Change in T iN.The predicted distributions of Bi, Tt, T,, 0 and the change in T iNare given in Figs 7 and 8 for the counter- and co-current exchangers. In both cases, Bi again increases along the tube, though most markedly for the counter-current exchanger. The change in shell-side inlet temperature required to maintain the product outlet temperature constant is much greater for the counter-current exchanger-43 K, as opposed to 9 K for the co-current exchanger. The increases in q/go, also shown in Figs 7 and 8, are, however, much more marked in the case of the co-current exchanger, reflecting the initially smaller values of AT in the exit region. As with the cooler uniform wall temperature exchanger discussed above, the co-current exchanger is more easily controlled as a result of the initially small values of AT. Again the suggestion is that compact exchangers pose more severe control problems. Change in shell-sidepow rate. Examination of eq. (8) shows that in the case of exchangers designed to perform the same thermal duty, the value of g,,.,,, beyond which control is not possible depends solely on the ratio P/D, which varies with exchanger configuration. For the two exchangers considered here, Bi, = 0.341 for the counter-current and 1.07 for the co-current system. Since in the absence of control the final values of % are in the region of 0.75 (Table 2), then this strategy cannot be used to control the counter-current exchanger successfully.

fb)

4ooo

420 7

I

I

1000 _

0

I

Lengfh

(m)

3

4

(c)

2.5 I,40 2.0

-430 -420

g

-410

E P

- 400 I 3000 Tcme

2.0

400k?90

(s)

(d)

100000 00000

1.5 6I y 1.0 O-

60000

CT 5 b

40000

0.5 A,, timesin 5. 0.0 0

I 1

I

2 Length

3

I

4

0

Cm)

Fig. 7. Variation of (a) local Bzxb) shell- and tube-side fluid temperatures T, and T,, (c) Bi and shell-side fluid inlet temperature TfN, and (d) local heat fluxes for model fouling by milk in the counter-current heat exchanger specified in Table I.

P. J. FRYER

2370

0

I

2

3

4

5

6

and N.K.

H. SLATER

“‘0

7

I

I

I

2

1

Length(m)

3

I

4

Length

I

5

6

7

I

(ml

(b) 400,

I Shell

380

380 -

z -

370 P

c

370

4000,

250

\IxyI L

I-==-

0

360

350

3.

340

r

I

3300

1

I

1

I

2

3

4

Length

5

6

7j

-‘0

r

I

I

1

2

(m)

3

4

Length

l_O,(c’

I

5

I

b

7

Cm)

,400

Time

(5)

Time

d) I

I

’

timcsan s

3.0

200000

2.5

$ 2

50000

T

00000

2.0 8 Yo-

(5)

(d)

200000 AUlimc5

in

5.

150000

1.5

100000

8 s 8 O-

1.0 50000

0

1

2

3

4

Length

5

6

0

7

50000

0.5 “‘0

(ml

2

3 Length

4

5

6

7

0

Cm)

Fig. 8. Variation

of (a) local Bid(b) shell- and tube-side fluid

Fig. 9. Variation

temperatures

and

temperatures T, and T,, (c) Bi and shell-side fluid flow rate, and (d) local heat fluxes for mode1 fouling by milk in the cocurrent exchanger specified in Table 1.

T,

Tf,

(c) Bi

and

she&side

fluid

inlet

temperature T iN, and (d) local heat fluxes for model fouling by milk in the co-current heat exchanger specified in Table 1.

of (a) local_Bi, (b) shell- and tube-side fluid

2371

Simulation of heat exchanger control Figure 9 shows the change in temperature

profiles,

E, ws and q/q,, occurring during simulation of the cocurrent exchanger. In this case, control is possible although the shell-side flow rate increases three-fold, perhaps leading to an unacceptably high pressure drop. The effect of increasing the flow rate is to raise the values of T, along the exchanger. As the flow rate increases, the incremental change in T, at any point for unit increase in ws decreases; the system thus becomes less sensitive to control action as fouling increases. Overall we note that control of ws leads to a lower value of Bi,

than does control

of T kN.

Thermodynamic eficiencies The second law eficiency of heat exchangers can be used to evaluate the effects of control action upon exchanger efficiency. If heat Sq is rejected from a source at temperature T, (or T,) to a sink at T,-, the entropy change is given by SS=Sq

(

;+

(18)

*>

and thus the total entropy change for an exchanger of length L can be found from As=nd/;

($-$)&(.-T&Ix.

(19)

Since at any stage in the simulation T,, T, and Bi are known for all x, the entropy generation rate can be found. Table 3 lists the initial entropy generation rates for each of the exchangers specified in Table 1 and those after 4000 s of fouling by milk. As might be expected, exchangers in which the heat source temperature is varied most dramatically display the severest increase in entropy generation rate; that is, the short uniform wall temperature and the countercurrent exchangers. Equation (19) underestimates the total entropy generation rate as it neglects that contribution due to the pressure drop down the exchanger tubes. In probability the pressure gradient would increase for the more severe fouling which would further detract from the efficiency of the hotter exchangers.

been amended to permit a consideration of controlled exchanger performance. The resulting procedure has been demonstrated for four different configurations of exchanger and a variety of control strategies. Fouling was assumed to proceed according to a local model for solids accumulation from milk although other locai fouling rate models could well be accommodated. The simulations indicate the typical extent ofcontrol action required for the various exchanger designs and the rate at which Bi varies with time. Generally, the greater the heat transfer area which is used to perform a given thermal duty, the less the amount of control action required to maintain the process stream target temperature. Further, such exchangers with small values of AT are thermodynamically more efficient. In practice the size of an exchanger adopted for a particular task would be determined by an economic balance between equipment capital cost and benefits which accrue from enhanced thermal efficiency and reduced cleaning costs. Finally, it has been demonstrated that adjustment of the shell-side flow rate provides only limited capacity for control and is therefore less effective than variation of the shell-side inlet fluid temperature. Acknowledgetftenrs-One of us (P.J.F.) is indebted to the Science and Engineering Research Council and the APV Co. Ltd. for the provision of a CASE award during the tenure of which this work was conducted.

NOTATION

a B Bi

heat transfer area per unit volume, m-’ ratio of enthalpic flow rates [eq. (A2)] Biot number, dimensionless local overall fouling factor (h@JA,)

Bi

dimensionless average overall fouling factor (for whole exchangers) specific heat capacity, J kg-’ K-’ diameter of tube, m activation energy for solids deposition from milk, kJ mol - ’ local heat-transfer coefficient, W m-’ K-’ solids deposition rate constant, s- ’ solids removal rate constant, s-l variable defined by eq. (A5) exchanger length, m local heat flux, W mm2 gas constant, J mol-’ fouling resistance 2 K ,“-I-2 w-l time, s time at which control action is implemented, s temperature, K heat-transfer coefiicient, overall local Wm-‘K-’ average overall heat-transfer coeficient, Wm-‘K-l fluid velocity, ms -’ fluid mass flow, kg s-l axial distance along heat exchanger, m

CP d E

CONCLUSIONS

The simulation techniques which we previously developed to model overali heat exchanger performance on the basis of a local fouling rate equation have Table 3. Entropy generation rates in the exchangers specified in Table 1 Initial generation Exchanger

Uniform

To = 373 K Uniform Ti= 390 K Co-current ( T fN controlled) Counter-current ( T iN

controlled)

rate (W K-l) 6.31 10.43 8.54

8.18

4000s

generation rate (W K-l) 7.73 25.43 10.71

17.51

T u 0 V W Z

P. J. FRYER

2372 Greek

and N. K. H. SLATER

Sundaram,K. M. and Froment, G. F., 1979. Chem. Engng Sci.

letters

s

Ata(i3, Y) AT 1 P

thickness, m time increment between nodes on the a( /3, y) characteristic, s temperature difference, K thermal conductivity, W m-’ K-l bulk density, kg rnp3 surface shear stress, N m-2 variable employed in eq. (1) and defined in the text

34, 635. Taborek, J., Aoki, T., Ritter, R. B., Palen, J. W. and Knudsen, J. G., 1972, C.E.P. 68(2), 59; 68(7), 69. APPENDIX

IN HEAT EXCHANGERS This appendix summarizes the equations which describe the steady-state temperature profiles in the types __ ofexchanger considered when clean. Constant wall temperature T&? = Twd - (Twd -TiN)exp(

Subscripts

d f i LM 0 S

t W

co

: STEADY-STATE TEMPERATURE PROFILES

of the deposit of the tube-side fluid of the fluid deposit interface log mean time = 0 of the shell-side fluid of the tube side of the tube wall at infinite fouling time

-sz).

Co-current B=S w CP, WfCPf

Tf(4

=

T’N+I s

&

B

T;N_(

T~“-T;N)

[

exp - (?)=I] T,(r) = TfN -;(Tf(s,-T[N).

Superscripts

IN 0

OUT

at the exchanger inlet of a clean tube at the exchanger outlet

Counter-current K = exp TOUT_

s

REFERENCES

Acrivos, A., 1956, Ind. Engng Chem. 48, 703. Fryer, P. J. and Slater,N. K. H., 1986, Chem. Engng J. 31.97.

Kern, D. Q. and Seaton, R. E., 1959, Br. them. Engng 4,258. Shinskey, F. G., 1976, Process Control Systems, p. 228.

McGraw-Hill, New York.

Tf@==&

-

K

1 ~-~ WBCPS

WfCPf

T:N(K-l)+T;NK(B-l)

z-T$‘“T+(TF”T-T;N) [ x

1

exp -(y)-$&.z] T,(z) = TFUT

BK-1

)

OndL

1