dA = dQ/U DT (11-1)

where dA is the element of surface area required to transfer an amount of heat dQ at a point in the exchanger where the overall heat transfer coefficient is U and where the overall bulk temperature difference between the two streams is DT. The overall heat-transfer coefficient is related to the individual film heat-transfer coefficients and fouling and wall resistances by Eq. (11-2). Basing Uo on the outside surface area Ao results in

Uo = 1/ (1/ho + Rdo + xAo/KwAwm + (1/hi + Rdi)Ao/A ) (11-2)

Equation (11-1) can be formally integrated to give the outside area required to transfer the total heat load QT:

To integrate Eq. (11-3), Uo and DT must be known as functions of Q. For some problems, Uo varies strongly and nonlinearly throughout the exchanger. In these cases, it is necessary to evaluate Uo and DT at several intermediate values and numerically or graphically integrate. For many practical cases, it is possible to calculate a constant mean overall coefficient Uom from Eq. (11-2) and define a corresponding mean value of DTm, such that

Ao = QT /Uμm dTm (11-4)

Care must be taken that Uo does not vary too strongly, that the proper equations and conditions are chosen for calculating the individual coefficients, and that the mean temperature difference is the correct one for the specified exchanger configuration.

Mean Temperature Difference The temperature difference between the two fluids in the heat exchanger will, in general, vary from point to point. The mean temperature difference (DTm or MTD) can be calculated from the terminal temperatures of the two streams if the following assumptions are valid:

- All elements of a given fluid stream have the same thermal history in passing through the exchanger.*
- The exchanger operates at steady state.
- The specific heat is constant for each stream (or if either stream undergoes an isothermal phase transition).
- The overall heat-transfer coefficient is constant.
- Heat losses are negligible.