Page 191 - Chemical process engineering design and economics

Process Heat Transfer 173

Use Equations 4.4.13 and 4.4.3 to eliminate m and A from the total cost
li W 0
equation, Equation 4.4.6. Thus,

Q 6 C w Q (C c + CM)
C T=———————— + ——————— (4.4.14)
CPW (t 2,w - ti. w) U 0 F (At) LM

With the exception of F, all the parameters in both terms in Equation 4.4.14
are constants. In order to obtain a first approximation for the exit-water tempera-
ture and to simplify the derivation, assume that F is constant. To obtain an eco-
nomically-viable heat exchanger, let F = 0.85.
After substituting the logarithmic-mean temperature difference, Equation
4.4.4, into Equation 4.4.14 we find that

FQ9C W Q(C C + C M) (t4,p-t llW)
= ————————— + —————————————————— In —————— (4.4.15)
C T
Cpw (t 2,W - ti,w) U 0 F [ (t4 jP - ti\n) - (taj
Equation 4.4.15 can now be differentiated with respect to t,w After setting
2
the derivative equal to zero, and rearranging the equation, it is found that
U 0 6 C w (A^ - At 2) 2 (At!-At 2)
(4.4.16)
Cp W(C F + C M) (t 2,w-ti,w) 2 At 2 At 2

where At t = t^f - t\\N and At 2 = ts ;p - t 2jw Equation 4.4.16 is dimensionless. The
t
optimum cooling-water temperature, , is obtained from Equation 4.4.16 by
2;W
iteration.

APPROACH TEMPERATURE DIFFERENCES

Frequently, an approximate value of the optimum exit-water temperature is all that
is required, and a rule-of-thumb will be satisfactory. Table 4.4 lists the approach
temperature difference, which is the difference between the two terminal tempera-
tures of two passing streams, for several heat exchangers. Several approach tem-
perature differences were taken from Ulrich [8]. For refrigerants, Ulrich's range of
10 to 50°C is on the high side. Frank [7] recommends a range of 3 to 5°C whereas
Walas [3] recommends a value of 5.6°C or less.

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