A double-pipe heat exchanger is constructed of a copper \((k=380 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) inner tube of internal diameter \(D_{i}=\) \(1.2 \mathrm{~cm}\) and external diameter \(D_{o}=1.6 \mathrm{~cm}\) and an outer tube of diameter \(3.0 \mathrm{~cm}\). The convection heat transfer coefficient is reported to be \(h_{i}=700 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the inner surface of the tube and \(h_{o}=1400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on its outer surface. For a fouling factor \(R_{f, i}=0.0005 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\) on the tube side and \(R_{f, o}=\) \(0.0002 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\) on the shell side, determine \((a)\) the thermal resistance of the heat exchanger per unit length and \((b)\) the overall heat transfer coefficients \(U_{i}\) and \(U_{o}\) based on the inner and outer surface areas of the tube, respectively.

Question: Calculate the thermal resistance of the heat exchanger per unit length and the overall heat transfer coefficients based on the inner and outer surface areas, given the convection coefficients, fouling factors, and tube dimensions. Answer: To calculate the thermal resistance of the heat exchanger per unit length (R_tot) and the overall heat transfer coefficients (U_i and U_o), follow these steps: 1. Calculate the tube wall's thermal resistance per unit length (R_t) using the formula: \[ R_{t} = \frac{\ln\frac{D_{o}}{D_{i}}}{2 \pi k} \] 2. Calculate inner and outer convective resistance, considering fouling factors (R_i and R_o) using the formulas: \[ R_{i} = \frac{1}{h_{i} \pi D_{i}} + R_{f, i} \] and \[ R_{o} = \frac{1}{h_{o} \pi D_{o}} + R_{f, o} \] 3. Determine the total thermal resistance per unit length (R_tot) by adding up all the resistances: \[ R_{tot} = R_{i} + R_{t} + R_{o} \] 4. Calculate the overall heat transfer coefficients based on the inner and outer surface areas (U_i and U_o) using the formulas: \[ U_{i} = \frac{1}{R_{tot}} \pi D_{i} \] and \[ U_{o} = \frac{1}{R_{tot}} \pi D_{o} \ ] Plug in the given values and perform the calculations to obtain the thermal resistance of the heat exchanger per unit length (R_tot), and the overall heat transfer coefficients (U_i and U_o).