Question

A test is conducted to determine the overall heat transfer coefficient in a shell-and-tube oil-to-water heat exchanger that has 24 tubes of internal diameter \(1.2 \mathrm{~cm}\) and length \(2 \mathrm{~m}\) in a single shell. Cold water \(\left(c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters the tubes at \(20^{\circ} \mathrm{C}\) at a rate of $3 \mathrm{~kg} / \mathrm{s}\( and leaves at \)55^{\circ} \mathrm{C}\(. Oil \)\left(c_{p}=2150 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$ ) flows through the shell and is cooled from \(120^{\circ} \mathrm{C}\) to \(45^{\circ} \mathrm{C}\). Determine the overall heat transfer coefficient \(U_{i}\) of this heat exchanger based on the inner surface area of the tubes. Answer: $8.31 \mathrm{~kW} / \mathrm{m}^{2} \mathrm{~K}$

Short answer

Answer: The key steps to determine the overall heat transfer coefficient in a shell-and-tube heat exchanger are: 1. Calculate the mass flow rate of oil using the energy balance equation. 2. Find the total inner surface area of tubes using the number of tubes, their diameter, and length. 3. Determine the overall heat transfer coefficient using the heat transfer rate, total inner surface area, and log mean temperature difference.

Step-by-step solution

Calculate the mass flow rate of oil

To determine the mass flow rate of oil, we can apply the energy balance equation, which states that the energy transferred from the hot fluid (oil) is equal to the energy absorbed by the cold fluid (water). \(Q = m_w c_{p_w} (T_{w_2} - T_{w_1}) = m_o c_{p_o} (T_{o_1} - T_{o_2})\) Where: \(Q\) is the heat transfer rate (W), \(m_w\) and \(m_o\) represent the mass flow rates of water and oil (kg/s), respectively, \(c_{p_w}\) and \(c_{p_o}\) denote the specific heat capacities of water and oil (J/kg·K), \(T_{w_1}\) and \(T_{w_2}\) are the initial and final temperatures of water (°C), and \(T_{o_1}\) and \(T_{o_2}\) stand for the initial and final temperatures of oil (°C). Given \(m_w = 3 \mathrm{~kg/s}\), \(T_{w_1} = 20^\circ\mathrm{C}\), \(T_{w_2} = 55^\circ\mathrm{C}\), \(T_{o_1} = 120^\circ\mathrm{C}\), and \(T_{o_2} = 45^\circ\mathrm{C}\), we can now calculate the mass flow rate of oil.

Find the inner surface area of tubes

The total inner surface area of tubes can be found using the following equation: \(A_i = N \pi D_i L\) Where: \(A_i\) is the total inner surface area of tubes (m²), \(N\) is the number of tubes, \(D_i\) is the internal diameter of the tubes (m), and \(L\) is the length of the tubes (m). Plugging in the values given in the problem, \(N = 24\), \(D_i = 0.012\;\text{m} = 1.2\;\text{cm}\), and \(L = 2\;\text{m}\):

Determine the overall heat transfer coefficient

Using the heat transfer rate found in Step 1 and the total inner surface area determined in Step 2, we can now calculate the overall heat transfer coefficient \(U_i\) by applying the following equation: \(Q = U_i A_i \Delta T_{lm}\) Where: \(\Delta T_{lm}\) is the log mean temperature difference (K). To find \(\Delta T_{lm}\), we can use the following formula: \(\Delta T_{lm} = \dfrac{(T_{o1} - T_{w2}) - (T_{o2} - T_{w1})}{\ln((T_{o1} - T_{w2}) / (T_{o2} - T_{w1}))}\) Plugging in the temperature values and solving for \(\Delta T_{lm}\), we can then substitute the values to find \(U_i\).