Evaluation of Liquid-Liquid THEODORE w. EVANS
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S ESTIMATIXG the performance of any
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piece of equipment, it is generally useful to have some ideal unit to which the apparatus may be compared. Thus, in distillation the performance of a fractionating column is usually expressed by stating the column to be equivalent to so many perfect plates. In the case of heat exchangers it has been propoqed that the efficiency be defined as the ratio of the heat actually transferred t o the maximum that could be exchanged by an infinitely large exchanger ( 2 ) . While this means of defining the efficiency of a heat exchanger is logical, taken by itself it is somewhat inconvenient. I n the first place, the efficiency defined in this fashion bears no simple relationship to the overall heat transfer coefficient or to the total surface; the functional relationship involved is cumbersome and exponential. I n the second place, the heat exchanger is compared to an ideal unit of infinite surface. This makes it difficult to visualize the ideal unit and especially to visualize just what change in surface or heat transfer coefficient is necessary in going, for example, from 50 per cent efficiency to 7 5 per cent. In the following paragraphs two situations have been treated. The first involves the calculation of the maximum temperature change which can occur when a fixed quantity of one material is heated or cooled by successive small amounts of a second material-for example, the cooling of the contents of a n autoclave by circulating water through a coil inside the kettle. This case does not seem to have been treated heretofore. The second involves the countercurrent flow of two liquids in a heat exchanger. For this case it is shown that the performance may be expressed in terms of so many ideal heat exchangers, and that this number is directly pro-
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portional to both the over-all heat transfer coefficient and the total surface of the exchanger. This provides a ready means of visualizing the performance of the exchanger in question, and a t the same time it enables the effects of changes in the over-all heat transfer coefficient and heating surface to be easily predicted. It is assumed throughout that the specific heat of each liquid involved is constant over the temperature range considered.
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OR the first case, suppose a mass ml of a material of specific heat c1 is originally a t a temperature TI. A second material of total mass m, specific heat cz, and temperature Tlris divided into n equal portions, mz/n = Am,. The first material is then brought to thermal equilibrium with one portion of the second material, for example, by placing the materials in a container separated into two compartment? by a thin metal mall, the container being perfectly insulated externally. Under these circumstances both materials will reach the same temperature, T ( l ) and , by a heat balance,
Son- let the second material be removed and replaced by a fresh portion of mass Ana2 and temperature TZ,and thermal equilibrium again reached. On repeating this process n time$, of the first material is given by the final temperature T(") T ( n ) = kinTI
+ (1 - kin)TS
For the maximum heat transfer to occur, it is evident that n qhould approach infinity, na2 remaining constant, while Am? approaches zero. Calling the final temperature of the first material T under these circumstances, we have:
SOT,\T/(TI - T 2 )represents the fraction of the total initial temperature difference, TI - T?, through which the temperature of the first material has changed. This equation consequently shows the maximum temperature change that can be secured from a fixed ratio of ml to m2 by this method. Hence, if it is desired to change the temperature of m, pounds of the first material from TI t o T, this equation can be solved for m?,and the value so found represents the minimum amount of the second material required to effect the desired temperature change. If in practice it is found that m pounds are actually required, then nz,/m is a measure of the efficiency of the process. The situation presented above may be regarded x u the limiting case for a batch of liquid in a tank which is being cooled by water passing through a coil immersed in the liquid, the water entering the coil a t a constant temperature. T,. As the volume of the coil is made smaller and smaller, and the rate of flow regulated so that the exit water from the coil is the same as that of the liquid in the tank, the temperature change will approach that given in the preceding 1212
Heat Exchangers In the consideration of countercurrent heat exchangers it is possible to introduce the concept of a perfect or ideal heat exchanger. By this means the performance of a given exchanger may be readily evaluated and visualized as the equivalent of so many ideal units; this number is directly proportional to both the total heat transmitting surface and heat transfer coefficient. A simple expression is given for the surface equivalent to each ideal unit.
equation. In Figure 1 the function 1001T) ( T I - T,i is plotted as the urdinate arid m2-2/niIclas the abbcissa.
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HE second case is that in which the two liquids move
countercurrent to each other so that a continuous process is realized. This is exactly the case in the flow of tn-o liquids in opposite directions through two concentric pipes, and approximately the case in the ordinary multipaes tubular exchanger. To investigate thiq care it is useful to employ the concept of an ideal or unit heat exchanger. This ideal h e a t exchanger m a y be visualized a s a fi6.2 closed box divided internally into two compartments by a metal s h e e t , t h e entire box b e i n g perfectly insuI I lated externally. If one liquid a t a temperature To is put in one cornpartment, a second a t T 2 in the other compartment, and the apparatus is allowed to s t a n d , h e a t will be t r a n s f e r r e d through the metal wall from the h o t t e r to the colder liquid until t h e r ina 1 I equilibrium is reached, a t w h i c h p o i n t both liquids will be a t the same temperature, TI. S o w consider a series T,*; of n of t h e s e i d e a l exchangers working countercurrently as diagrammed in Figure 2 . Here mi pounds of the first material (specific heat cl, temperature To)are fed into the upper compartment of the first exchanger. To the lower compartment of the nth exchsnger are fed m2 pounds of the second material (specific heat c2, temperature T , J . Let the temperatures in the exchangers, once the steady state and equilibrium are reached, I)e TI, T,, , . . . , , T,. To picture its operation, consider, for example, exchanger 2. This contains ml pounds of t'he first material in the upper Compartment and m2 pounds of the second in the lower compartment, both a t a temperature
i
T
,
.,'Z
These materials are nuw r e n m ed separately, and in the upper coiiipartment are placed ml pounds of the first material taken from the upper compartment of exchanger 1, the temperature being TI,and in the lo\$er compartment rn? pounds of material taken from the lower compartment of exchanger 3 a t 7'1. At the same time the materials removed froin the upper and lower compartments of exchanger 2 are placed in the upper and lower compartments of exchangers 3 and 1, respectively; a t no time do the materials taken from the exchangers mix. When thermal equilibrium is again establiqhed, both compartments of exchanger 2 are found a t T ? , etc., and the process is repeated. While this process has been described in this batchwise fashion for clarity, it may just a4 well be pictured as continuous. For example., m , pounds of the first material flow in per hour, passing successively through the upper Compartments from 1 to n , while nz2 pounds of the second material flow per hour through the lox-er compartments from n to 1. The ideal exchangers operate so that both liquids l e a w any exchanger a t the same temperature: I . e., while the liquids enter exchanger 3 at temperature. T7 and T,, respectively, both leave a t TS. To calculate the temperature change which theye n exchangers can produce, it i. necessary to write a heat balance from unit to unit. Doing this and setting mlcl = k,, m?c- = k ? ,.2: = k,, k,, we have:
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1213
ki (To - Ti)= kz (TI - T ? ) kl (TI - Tz) = k* (Tz - T3) ,
kl
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(Tn-l- T,)
Solving these equation
