HOWARD TEN B R O E C K SOCONY-VACU UM 0 1L COMPANY, INC., 400 KINOSLAND AVL.. BROOKLYN. N. Y .
Theoretically, in des i g n i n g an exchanger, an e c o n o m i c balance should be set up to determine the o p t i m u m pressure drops, since the pressure drops are dir e c t l y r e l a t e d to the over-all coefficient, and pumping costs can therefore be balanced against exchanger costs. It is possible to obtain a wide range in the values of the over-all coefficient by varying t h e p r e s s u r e drop. This is done by changing the number of Bank of Four Heat Exchang a passes, diameter of shell, Gasoline Refinery Designed and Built by The Lummus Company diameter and length of tubes, and by baffling. HE purpose of this paper is to provide a method for the I n practice this balance is seldom made since the economic economic selection of the sizes of individual units in a batpressure drop is high. Fouling and plugging would cause the uld possibly cause a prematery of exchangers. The type of batt ,operators sacrifice the last ounce in which a stream, called the “series stream” ndability. Therefore the allowunits in series and engages in heat transfer specified by the operator and is streams. Included in the battery are the cool each stream to the desired final temperature. usually specified so that the debe either tubular or coil-in-box equipment. ses to approach the desired will be considered to be water. ere are standard shell diameters and The economic selection of the size of an exchan integral number of passes, it is obvious entering temperatures have been fixed is com oefficient and pressure drop will vary with This simplicity, however, does not extend to b the entering temperature of the series stream to e etween limits, and will show f the manner in which it varies, we a variable governed by the heat transferred in the can predict the over-all coefficient The heat transferred, therefore, is dependent up the previous exchangers, all other variables having been fixed by and that it is substantially independent of area. the conditions of the problem. The multipass effect will affect the mean temperature diEerence and consequently will influence the optimum size. The I n addition to being affected by the sizes of exchangers preceding it, each exchanger affects the size of the cooler required to greatest effect will be that of one shell pass and two or more tube passes (1-2 type exchanger). All other types are intermediate cool the side stream to the desired temperature. The amount of water required for cooling is likewiseaffected. between that and countercurrent type exchangers. It is conceivable that an exchanger could be economical even if Equations are developed here for 1-2 and 2-4 type multipass exchangers and for countercurrent exchangers. The type of exthe heat recovered had no value-for example, if the series stream were subsequently heated using waste products as fuel and if a changer to select depends largely upon the ratio of the quantities heater of sufficient capacity were available. This, coupled with a of the two streams but is not entirely independent of area. The because of piping considerations or behigh water cost, would make the heat exchanger economica!. differenceswhich necessitate a floating Chave (a) equated cost of exchanger t o tained a minimum value of U At, considerin paper, the type must be predown in the range in which Nelson (8) differentiated the heat transfer e spect to the approach (t. - TI), but did not t the effect of the exchanger on reducing the si5 and exchangers and the reduction in water kind it is not necessary to Buddine (1) differentiated the heat transfer e know total costs. The cost data necessary are increment costs. respect to area and included a factor for the effect on the cooler, For example, if a 1000 square foot exchanger costs 3000 dollars but did not consider its effect on other exchangers.
-
T
64
lanuary, 1944
INDUSTRIAL AND ENGINEERING CHEMISTRY
65
The savings of each unit is a function of its area and the areas of the exchangers preceding it, but not following it. The above equation can be differentiated and equated to zero in order to determine the maximum savings:
The saving for each unit may be written
Figure 1.
Sohematic Arrangement o f Battery of Exchangers
andra 2000 square foot exchanger costs 5000 dollars, the increment cost is 2 dollars per square foot instead of 3 dollars as would be calculated if total costs were used. The increment cost should be multiplied by the sinking fund factor, capital recovery factor, or reciprocal of the depreciation period, depending upon the accounting system used. Symbol i will be used to represent the above factors. Maintenance in general does not vary with the size of the exchanger, since the cost for cleaning or repairing a large exchanger is not very different from that of doing the same work on a small one. If short tube life is expected, this cost should be divided by tube life and area t o give maintenance cost per year per square foot. Installation cost may or may not vary with the size of the exchanger, depending upon whether the location is easily accessible and whether an expensive structure must be provided. Piping cost is an example of installation cost that is definitely not a function of exchanger size. The increment of exchanger cost is the sum of the increment of first cost plus the increment of maintenance cost plus the increment of installation cost, and is represented by iE. HEAT. By heating the stream in the exchanger, the duty of a subsequent heater is reduced; if a new installation is under consideration, the size of the heater can be reduced. The increment cost of the heater can be expressed as dollars per year per million B. t. u. per hour. If a heater of sufficient capacity is available, the advantage of preheat from this standpoint is eliminated. The heat saved causes a decrease in the fuel consumption, and the value of the heat can be determined from the price of the fuel, its heating value, and the expected furnace efficiency. The increment cost of heat is the increment cost of the heater plus the increment fuel cost. WATER. The heat transferred to the series stream represents heat that does not have to be removed from the side streams by cooling with water. The cost of water per million B. t. u. removed can be calculated from the water charges and the number of degrees the water will be heated in the cooler. I n the case of tubular equipment the outlet water temperature is generally limited to 100" or 125" I?., as a result of more rapid corrosion at higher temperatures. In coil-in-box equipment the exit temperature is limited by its boiling point or by the entering temperature of the stream to be cooled. If the charge for water is G,the cost of water per million B. t. u. is 120 G / ( t , ti).
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DERIVATION O F EQUATIONS
A battery of exchangers is shown diagrammatically in Figure 1.
The savings per year for such a battery can be expressed as
s
=
s1 + sa
+ + S a + + sc 8 2
s 3
and differentiated
SI f Sa) bA1
=
YHf ~ Q 114 aAl
+
I
YHwa aQa 114 dAl
- iE.
bA aAi
- iEl
(4)
The term iE,A,(&, Q u ) represents the investment cost of a cooler transferring (Q1 Qg),the total cooling load, when A1 = 0, and therefore is a constant which drops out in the differentiation. But
Defining,
+
114iE, Ha = y Ua
Substituting in Equation 4,
The savings of the second exchanger when differentiated with respect to the area of the first exchanger result in the equation:
Substituting the above relations in Equation 8,
INDUSTRIAL AND ENGINEERING CHEMISTRY
66
Substituting Equations 5, 6, 7 , 9, 10, 11 in Equations 1, 2, 3,
the type of flow (whether countercurrent or multipass) except in establishing the relation (bP2/bQl)~z = 0. This was derived using the equation for countercurrent flow; but from the curves (4) for multipass exchangers, it is obvious that when U2A2/WCP and Rz are specified, there is a unique value of P . The only limitation is the requirement that d& must be proportional to dT, which limits the application to sensible heat transfer.
The partial derivatives are interrelated as may be established by the following steps: \
Vol. 36, No. 1
€-
P
WGRzPz(t2 - Tz)
Qz
'.17
0.35 But since
UzAz
m 1 R2R2 - In I
1
- RaPz 1 - Pz
0.40~
-
it follows that
-
which leads to the differential
From which
aQ &= -
2
RSPS( 1
- &Pz)
Equations 12,13, and 14 now take the forms:
Figure 2.
Let
Zs = CsFs Za = CtF2 CiFi Z1
- ZsRsPa
- ZzRzPs
- ZaRsPs
bAl
I n a previous article ( 4 ) P was plotted against U A / w c for multipass exchangers of the 1-2 and 2-4 types. The significance of Equation 16 is that, when the slope (on a rectangular coordinate plot) is C H / Z ( t 2') the optimum exchanger size is obtained. If only one exchanger is considered, the slope term reduces to H / F ( t T). For countercurrent exchangers:
-
and substitute in the above equations. Three equations result, all of the form:
bQi =-
-
UiCiHi
A -U=
21
wc dP -=
The differential coefficient is easily obtained: &1
= WlClPl(t1
Nomograph t o Eva1 uate P
- TI)
1 - RP ln- 1 - P 1 - R
(1
- RP)(1 - P )
For multipass exchangers of the 1-2 type ( 4 , 5 ): WlCl
Substituting in Equation 15, we obtain three simultaneous equations of the form:
UA
1 2-[l+R-di77F]P wc = d i = - - s l n 2 - 1 1 + R + ~ ~ ] p
dP
-I=
d -U A
WlCl
which must be satisfied for each exchanger for the economical exchanger size. Up to this point no mention has been made of
1
-P
(1
RP +R -7 )
we
For multipass exchangers of the 2-4 type the equation is complex and must be differentiated graphically.
.
INDUSTRIAL AND ENGINEERING CHEMISTRY
January, 1944
TABLEI. Hi
Ha W
W
b
R
tl
Ta (assumed)
- TtuI - twd
ti
Ha/(ta
Hwo
-%
CF CH
-Multipass 1 4.6 6.5
(1-2)----Countercurrent---.2 3 1 2 3 5.7 7.6 4.6 5.7 7.6 7.6 9.1 6.5 7.6 9.1 70000 22400 70000 22400 100,000 100:000 1oo:ooo 100,000 100:000 1oo:ooo 0.625 0.675 0.620 0.675 0:4% 0.550 0.605 0.495 0.565 0.625 1.0 0.795 0.250 1.0 0,768 0,242 230 450 600 230 450 600 90 100 110 155 332 388
....
150 220 0.054 0.06 0.17 0.284 0.139 2.25
Za CH/Z(t T) P (Fig. 2) RP ZRP
-
Z CH/Z(t T) P (Fig. 2) RP ZRP
-
Z T) CH/Z(t P (Fig. 2) RP 11
-
TI
Tz (calcd.)
ta
- twa)F
Ha/(ta
CALCULATIONS~
.... ....
.... .... ....
0.058 0.288 0.415 0.415 150 142 168
360 437 0.023 0.06 0.17 0.253 0.139 3.13
500 560 0.020 0.06 0.17 0.250 0.151 4.60
.... ....
....
0.151 0.061 0.815 0.204 0.0308
0.108 0.0805 0.590 0.468 0.0506
.... .... .... .... .*.. .... ....
.... ....
.... .... .... ....
308 286 268
....
....
314 350 344
150 155 0.118 0.06 0.17 0.348 0.172 2.28
295 220 0.063 0.06 0.17 0.293 0.166 3.21
........
.. .. .. .. .. .. .. ..
.. .. .. .. .. .. .. .. ....
.... .... .... .... 0,058 0.262 0.490 0.490 150 154 156
0.132 0.083 0.785 0.603 0.0790
....
.... .... 296 332 218
268 368 0.034 0.06 0.17 0.264 0.165 4.75
The determination of R for the condenser must be obtained bv separate calculations involving the use of boiling point curves, per cent steam, pressure, etc. It should be remembered that the use of the logarithmic mean temperature difference for the mean temperature difference in condensing equipment is only an approximation, since the derivation of logarithmic mean temperature difference is based upon constant specific heats or its equivalent, a constant value of R. A reasonable temperature should be assumed and the per cent condensation calculated The rise in the temperature of the crude can then be calculated, and the value of R1 = (Tz- T 1 ) / ( t l- to) computed. I n this example preliminary calculations show that approximately R1 = 1.0. The data will be taken as:
Y = 1 iE = $2/sq. ft./year
H, = $0.17/million B. t. u. G = $O.Ol/thousand gal. tl = 230" F.,tz = 450°, t s 600"; Ti = SO", ti = SO", t, = 100" F. u 1 = 50 B. t. u./hr./sq. ft./'F., Uz= 40, u a 30, U. 35, Ub = 30, U, = 25
0.165 0.107 0.865 0.209 0.0346
.... .... .... .... .... .... .... .
I
.
.
268 388 368
5
The temperatures of the series stream are assumed, and Table I is filled in down to and including CH. Then starting with exchanger 3, the values of C H / Z ( t - T) are calculated and the values of P are found from Figure 2. The temperatures are calculated, the values of F revised, and the procedure is repeated. It should be noted that the area can be calculated for countercurrent exchangers when P is known by the equation,
1
CF
CH/Z(t
z - T)
U A - ln_
wc
P
RP ZRP t - T
.... 150
#a
148 162
T2
Ha/(ta
CH/Z(t
- two) F CF Z
- T)
P RP ZRP t - T
Tz
67
0.105 0.335 0.164 0.076 0.197 0.455 0.455
.... 150
148
0.0560 0,0316 302 311 289 350 272 356 0.044 0.274 0.151 0.120 0.086 0.585 0.466 0.0559 302 288
0.036 0.266 0.161 0.161 0.092 0.785 0.196 0.0316 312 349
The slope is a function of P and R. Figure 2 is a nomograph to evaluate P . The R scale goes up to 1.0 which covers the majority of cases. If €2 is greater than 1.0, place 1/R on the R scale and read RP on the P scale. Throughout the derivation the over-all coefficients and specific heats were considered independent of temperature. This is obviously incorrect, but it is believed the error will not be serious. The solution involves assuming series-stream temperatures. The equations then provide values of P for each exchanger. The temperatures are calculated, and these values will be found to be very close to the final solution. However it is desirable to repeat the operation to eliminate the errors of the first assumption. To show how quickly the successive approximations approach the true value, the assumed temperature rise in the battery of multipass exchangers (Table I) was taken as only 30° F., whereas the economic rise is 269' F. The first results were slightly in error, but the second results were sufficiently close t o the true values.
- RP
1 - P
1-€2
and for multipass exchangers by means of curves (4). I n practice it will be necessary to revise the values of specific heat for each new temperature. This was omitted in the example for simplification. NOMENCLATURE
A C c E
= = = =
ares of exchanger, sq. ft. specific heat of series stream specific heat of side stream increment exchanger cost. dollars/sa. f t .
G H H
=
cost of water, dollars/thousand gal.
= 114iE/YU
d,
= = =
value of increment heat, dollars/million B. t. u. cost of water, dollars/million B. t. u. removed rate of depreciation or amortization
i p1 = !Lz&
- T, - Ti wici tl - t. WC1 = saving, dollars/year t.
Q R1 S
T t
ti
tw
= Geat &nsferred, B. t. u./hr. '2'2
=-i-
= = = =
U
=
Y
=
2
=
W = w =
temnerature of series stream. O P. temperature of side streF. inlet water temperature, P. outlet water temperature, O F . over-all coefficient of heat transfer rate of series stream Ib./hr. rate of side stream, ib./hr. fraction of ear in operation defined in zerivation
Subscripts 1, 2, 3 = exchangers 1, 2, 3, respectively a, b, c = coolers following exchangers 1, 2, 3, respectively
ILLUSTRATIVE CALCULATION
An example will show the method of using these equations. The system represents a preheat battery for a crude oil topping unit. The first exchanger is a condenser for the overhead product and reflux, the second is an intermediate reflux heat exchanger and the third is a side-stream exchanger.
LlTERATURE C I T E D
(1) Buddine, N. T., Oil Gus J., 34, No. 33, 34-6 (1936). (2) Chave, C.T., Refiner Natural Gasoline Mfr., 17,4 (1938). (3) Nelson, W.L.,Ibid., 15, 293 (1936). (4) Ten Broeck, Howard, IND. ENQ.CHEM.,30, 1041 (1938). (5) Underwood, A. J. V., J. Inst. Petroleum Tech.. 20,145 (1934).
