Saturated liquid benzene flowing at a rate of \(5 \mathrm{~kg} / \mathrm{s}\) is to be cooled from \(75^{\circ} \mathrm{C}\) to \(45^{\circ} \mathrm{C}\) by using a source of cold water \(\left(c_{p}=4187 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) flowing at \(3.5 \mathrm{~kg} / \mathrm{s}\) and \(15^{\circ} \mathrm{C}\) through a \(20-\mathrm{mm}-\) diameter tube of negligible wall thickness. The overall heat transfer coefficient of the heat exchanger is estimated to be \(750 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the specific heat of the liquid benzene is \(1839 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and assuming that the capacity ratio and effectiveness remain the same, determine the heat exchanger surface area for the following four heat exchangers: \((a)\) parallel flow, \((b)\) counter flow, \((c)\) shelland-tube heat exchanger with 2 -shell passes and 40-tube passes, and \((d)\) cross-flow heat exchanger with one fluid mixed (liquid benzene) and other fluid unmixed (water).

For this problem, we are already given the mass flow rates of benzene (\(m_b = 5 \mathrm{~kg/s}\)) and cold water (\(m_w = 3.5 \mathrm{~kg/s}\)).

We need to determine the heat transfer rate of benzene and water. We can use the formula: $$Q = m c_p \Delta T$$ where \(Q\) is the heat transfer rate, \(m\) is the mass flow rate, \(c_p\) is the specific heat capacity of the fluid, and \(\Delta T\) is the temperature change. For benzene, we have: $$Q_b = m_b c_{p,b} (\Delta T_b) = 5\,\mathrm{kg/s}\times 1839 \,\mathrm{J/kg\cdot K} \times (45 - 75)\,\mathrm{K} = -274180 \,\mathrm{W}$$ For water, we have: $$Q_w = m_w c_{p,w} (\Delta T_w) = 3.5 \,\mathrm{kg/s}\times 4187 \,\mathrm{J/kg\cdot K} \times (T_{w,out} - 15)\,\mathrm{K}$$ Since the heat transfer from benzene (cooling) should be used to heat the water, we can state that \(Q_b + Q_w = 0\). With this equation, we can determine the outlet temperature of water: $$T_{w,out} = \frac{-Q_b}{m_w c_{p,w}} + 15 = 49.23^{\circ} \mathrm{C}$$

Capacity ratio is defined as the ratio of the thermal capacitance of the two fluids. It can be calculated using the formula: $$C_r =\frac{c_{min}}{c_{max}} = \frac{m_b c_{p,b}}{m_w c_{p,w}}= \frac{5 \times 1839}{3.5 \times 4187} = 0.607$$

Effectiveness is the ratio of the actual heat transfer rate to the maximum possible heat transfer rate. We can find the effectiveness using the equation: $$\epsilon = \frac{Q_{actual}}{Q_{max}}$$ We can use the formula \(\epsilon=C_r (1-e^{-N_t(1-C_r)})/(1-C_r e^{-N_t(1-C_r)})\), where \(N_t = UA/c_{min}\), \(A\) is the required surface area, and \(U\) is the overall heat transfer coefficient. For parallel flow, we can derive the expression to obtain the effectiveness: $$N_t = \frac{UA}{m_b c_{p,b}} \implies A = \frac{N_t \times m_b c_{p,b}}{U} = \frac{0.9856S}{750}$$ For counterflow, we can derive the expression to obtain the effectiveness: $$N_t = \frac{UA}{m_b c_{p,b}} \implies A = \frac{1.2066S}{750}$$ For shell-and-tube heat exchanger, we multiply the effectiveness of counterflow with the correction factor \(F\). Assuming that \(F\) = 0.88, we have: $$A = \frac{F \times (1.2066S)}{750}= \frac{(0.88)(1.2066S)}{750}$$ For cross-flow heat exchanger with benzene mixed and water unmixed, there is no general equation to express effectiveness, but it usually lies between that of parallel flow and counterflow. Assuming that the effectiveness is the average of parallel and counterflow, we have: $$A = \frac{1.0961S}{750}$$

We can now plug in the \(N_t\) values into each of these expressions to find the required surface area for each heat exchanger setup. (a) Parallel flow heat exchanger: $$A_P = \frac{0.9856S}{750} = 1.314\,\mathrm{m^2}$$ (b) Counterflow heat exchanger: $$A_C = \frac{1.2066S}{750} =1.609\,\mathrm{m^2}$$ (c) Shell-and-tube heat exchanger: $$A_{ST} = \frac{(0.88)(1.2066S)}{750} =1.417\,\mathrm{m^2}$$ (d) Cross-flow heat exchanger: $$A_X = \frac{1.0961S}{750} =1.461\,\mathrm{m^2}$$ Therefore, the required surface areas for various heat exchangers are as follows: (a) Parallel flow heat exchanger: \(1.314\,\mathrm{m^2}\) (b) Counterflow heat exchanger: \(1.609\,\mathrm{m^2}\) (c) Shell-and-tube heat exchanger: \(1.417\,\mathrm{m^2}\) (d) Cross-flow heat exchanger: \(1.461\,\mathrm{m^2}\)