Page 189 - Chemical process engineering design and economics

Process Heat Transfer 171

tion reduces to Equations 4.4.2, which states that the heat transferred to the water
is equal to the change in enthalpy of the water.
Because the cost of a heat exchanger depends on its size, and because its
size will depend on the heat-transfer rate, a rate equation must be introduced. The
rate equation is given by Equation 4.4.3. The logarithmic-mean temperature dif-
ference in Equation 4.4.3 is given by Equation 4.4.4. Because perfect countercur-
rent flow can never be achieved in an actual heat exchanger, the logarithmic-mean
temperature difference correction factor, F, is needed. For simplicity, Equation
4.10, discussed earlier, is expressed as Equation 4.4.5, which states that F depends
only on the terminal temperatures, once a particular heat exchanger is selected.
Several cooler sizes will cool a process fluid to a specified temperature, but
there is only one that is the most economical. If the cooling-water exit tempera-
ture increases, less water is needed and its cost will be less. To achieve a high exit-
water temperature, however, requires more heat-exchanger surface area and con-
sequently a more costly heat exchanger. Equation 4.4.6, the total annual cost, ex-
presses the trade-off between the cost of cooling water and the cost of a heat ex-
changer. The total cost consists of the sum of the first term, which is the cooling-
water cost, and the second term, which is the installed cost of the heat exchanger
and the maintenance cost. In Equation 4.4.6, C equals the cost of water per
w
pound, C c capital cost of the heat exchanger per square foot, and C M the mainte-
nance cost per square foot. Because we want to obtain the optimum cooling-water
exit temperature and therefore, an optimally-sized heat exchanger, the total cost
should be the minimum. Therefore, the derivative of the total cost with respect to
the exit water temperature dC T/ dt 2>w is set equal to zero.
Finally, to complete the formulation of the problem, we need system prop-
erty data. For this particular problem, enthalpy, a thermodynamic property, is
required for the energy balance and the overall heat-transfer coefficient, a transfer
property, is needed for the rate equation. These system property relationships are
given by Equations 4.4.8 to 4.4.10. The economic balance also requires cost data.

Table 4.4.1 Summary of Equations for Calculating the Optimum
Cooling-Water Exit Temperature _______________________

First subscript: Process stream = 1,2,3 or 4
Second subscript: Component = p or w

Mass Balances
(4.4.1)

Energy Balances
Q = h 2, w m 2, w - hi,w nii.w (4-4.2)

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