Page 190 - Chemical process engineering design and economics

172 Chapter 4

Rate Equations
(4-4.3)
(At) LM = f(t,,w', t 2.w, tit, t4,p') — Equation 4.2 (4.4.4)
F = f(t ljW', t 2. w, t 3jP', t 4, P') — Equation 4.10 (4.4.5)

Economic Relations
C T = m ljW e'C w '+A(Cc' + C M') (4.4.6)
dC T/dt 2, w = 0 (4.4.7)

Thermodynamic Properties
(
h,, w = *IW) = C PW (ti >w - t B') (4.4.8)
h 2,w = f(t 2,w) = Cp W (t 2>w - t B') (4.4.9)

Transfer Properties
U 0 = f(heat-exchanger type', shell fluid', tube fluid') (4.4. 1 0)

Variables
2>w - t 2,w - A 0 - F - U 0 - (At) LM - C T
Degrees of Freedom
F = V-R=10-10 = 0

Because the degrees of freedom are zero, the problem is completely for-
mulated and we can now solve the equations listed in Table 4.4.1. Next, express
the total cost equation in terms of a single variable, which is the exit water tem-
perature, t, so that it can be differentiated. It is reasonable to assume that in the
2jW
temperature range of interest the heat capacity of the cooling water will not vary
appreciably. Thus, from Equations 4.4.1 and 4.4.2,
Q = (h 2;W - hi, w) nii, w (4.4.11)

and after substituting Equations 4.4.8 and 4.4.9, where t is a base temperature,
B
into Equation 4.4.11

(h 2;W-h ljW) = = m liW Cp W (t 2, w - ti, w) (4.4.12)
Therefore,

Q = mi, w Cp W (t 2jW - ti, w) (4.4.13)

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