Question

Explain how the LMTD method can be used to determine the heat transfer surface area of a multipass shelland-tube heat exchanger when all the necessary information, including the outlet temperatures, is given.

Short answer

**Question:** Explain how to use the Log Mean Temperature Difference (LMTD) method to determine the heat transfer surface area of a multi-pass shell and tube heat exchanger given all necessary information, including outlet temperatures. **Answer:** To determine the heat transfer surface area using the LMTD method, follow these steps: 1. Understand the LMTD method: The LMTD method is used for determining the heat transfer rate and required surface area of a heat exchanger, considering the temperature gradients between the hot and cold fluids. 2. Calculate the LMTD: Use the inlet and outlet temperatures of the hot and cold fluids to determine the LMTD using the formula: LMTD = \[\frac{(ΔT_1 - ΔT_2)}{ln(\frac{ΔT_1}{ΔT_2})}\], where ΔT_1 and ΔT_2 are the temperature differences at the inlet and outlet, respectively. 3. Determine the heat transfer rate (Q): Calculate Q using the formula Q = m × Cp × ∆T, where m is the mass flow rate, Cp is the specific heat capacity, and ∆T is the temperature difference. 4. Calculate the overall heat transfer coefficient (U): The overall heat transfer coefficient combines the fluid's conductivity and the material's resistance and is provided in the exercise or obtained from a reference source or experimental data. 5. Determine the required heat transfer surface area (A): Calculate the surface area using the formula A = \[\frac{Q}{U × LMTD}\], where Q is the heat transfer rate, U is the overall heat transfer coefficient, and LMTD is the Log Mean Temperature Difference.

Step-by-step solution

Understand the LMTD method

The Log Mean Temperature Difference (LMTD) method is used to determine the heat transfer rate and the required surface area of a heat exchanger. The LMTD is the average driving force for heat transfer across the heat exchanger, considering the temperature gradients between the hot and cold fluids.

Calculate the LMTD

Use the given inlet and outlet temperatures of the hot and cold fluids in the heat exchanger to determine the LMTD using the formula: LMTD = \[\frac{(ΔT_1 - ΔT_2)}{ln(\frac{ΔT_1}{ΔT_2})}\] Where: - ΔT_1 is the temperature difference between the hot and cold fluids at the heat exchanger's inlet - ΔT_2 is the temperature difference between the hot and cold fluids at the heat exchanger's outlet

Determine the heat transfer rate

To find the heat transfer rate, Q, you need to know the mass flow rate (m) and the specific heat capacity (Cp) of the hot and cold fluids. With this information, use the formula: Q = m × Cp × ∆T Where: - m is the mass flow rate of either the hot or cold fluid (whichever has known values) - Cp is the specific heat capacity of the fluid - ∆T is the temperature difference between inlet and outlet for the fluid

Calculate the overall heat transfer coefficient

The overall heat transfer coefficient (U) represents the effectiveness of heat transfer in the heat exchanger and combines the fluid's conductivity and the material's resistance. This coefficient should be provided in the exercise or obtained from a reference source or experimental data.

Determine the required heat transfer surface area

To calculate the heat transfer surface area (A) of the heat exchanger, use the formula: A = \[\frac{Q}{U × LMTD}\] Where: - Q is the heat transfer rate, calculated in Step 3 - U is the overall heat transfer coefficient, calculated in Step 4 - LMTD is the Log Mean Temperature Difference, calculated in Step 2