1. Difficulties with LMTD Method

From the design equation: \[Q = UA\Delta T_{\text{lm}}\] From energy balance: \[Q = \dot{m}_cC_{Pc}(T_{co}-T_{ci}) = \dot{m}_hC_{Ph}(T_{hi}-T_{ho})\] The LMTD method of heat exchanger design is difficult to use if we want to predict the performance of a heat exchanger.

Here we would know: \(\dot{m}_c, \dot{m}_h, T_{ci}, T_{hi}, U\), and \(A\)

However, we would not know: \(T_{co} \quad \text{or} \quad T_{ho}\)

Hence, we cannot find: \(Q \quad \text{or} \quad \Delta T_{\text{lm}}\)

To solve the above problem with the usual LMTD method:

  • We could guess a value for \(T_{ho}\) or \(T_{co}\), find \(Q\) from a heat balance, and then find \(Q\) from \(UA\Delta T_{\text{lm}}\).

  • Using this value of \(Q\), find new values of \(T_{ho}\) and \(T_{co}\).

  • We would need to progressively alter our guess until the first and second step values of \(T\) were equal.

This iterative method can readily be done by computer, but a direct method can also be used. This direct method is known as the Effectiveness-NTU method (or \(\varepsilon\)-NTU method)