In developing our model for a heat exchanger, we assumed that the overall heat transfer coefficient, \(\mathcal{U}\), was a constant. In most instances, especially if the temperature difference between hot and cold fluids changes greatly through the exchanger, \(U\) will vary. Consider the case where \(\mathcal{U}\) varies linearly with the temperature difference, \(T_{h}-T_{c}\). Develop a new design equation for the heat exchanger with this information based on the analysis of Section 11.5.