Question

Explain how the maximum possible heat transfer rate \(\dot{Q}_{\max }\) in a heat exchanger can be determined when the mass flow rates, specific heats, and inlet temperatures of the two fluids are specified. Does the value of \(\dot{Q}_{\max }\) depend on the type of the heat exchanger?

Short answer

Based on the given information, determine the maximum possible heat transfer rate (\(\dot{Q}_{\max}\)) in a heat exchanger. 1. \(\dot{m}_{1}\) = mass flow rate of fluid 1 (kg/s) 2. \(\dot{m}_{2}\) = mass flow rate of fluid 2 (kg/s) 3. \(C_{p1}\) = specific heat of fluid 1 (J/kg·K) 4. \(C_{p2}\) = specific heat of fluid 2 (J/kg·K) 5. \(T_{1,in}\) = inlet temperature of fluid 1 (°C or K) 6. \(T_{2,in}\) = inlet temperature of fluid 2 (°C or K) Answer: To determine the maximum possible heat transfer rate (\(\dot{Q}_{\max}\)), follow the steps below: 1. Identify the minimum and maximum mass flow rates, denoted as \(\dot{m}_{min}\) and \(\dot{m}_{max}\). 2. Calculate the heat capacity rates for both fluids using the specific heats and mass flow rates: \(C_{1}=\dot{m}_{1} \cdot C_{p1}\) and \(C_{2}=\dot{m}_{2} \cdot C_{p2}\). 3. Determine the minimum and maximum heat capacity rates, denoted as \(C_{min}\) and \(C_{max}\). 4. Calculate the maximum possible temperature difference: \(\Delta T_{max}=\left|T_{1,in}-T_{2,in}\right|\) 5. Calculate the maximum possible heat transfer rate: \(\dot{Q}_{max}=C_{min} \cdot \Delta T_{max}\) The result will be the maximum possible heat transfer rate (\(\dot{Q}_{\max}\)) in the heat exchanger, independent of the type of heat exchanger used.

Step-by-step solution

Identify the minimum and maximum mass flow rates

Based on the mass flow rates of both fluids (\(\dot{m}_{1}\) and \(\dot{m}_{2}\)), we need to identify the fluids with the minimum and maximum mass flow rates. Let's denote them as \(\dot{m}_{min}\) and \(\dot{m}_{max}\).

Calculate the heat capacity rates

Using the specific heats (\(C_{p1}\) and \(C_{p2}\)) and mass flow rates for both fluids, we can calculate the heat capacity rates for each fluid: $$C_{1}=\dot{m}_{1} \cdot C_{p1}$$ $$C_{2}=\dot{m}_{2} \cdot C_{p2}$$

Determine the minimum and maximum heat capacity rates

Based on the calculated heat capacity rates, we need to determine the fluids with the minimum and maximum heat capacity rates. Let's denote them as \(C_{min}\) and \(C_{max}\).

Calculate the maximum possible temperature difference

The maximum possible temperature difference between the two fluids can be determined by comparing their inlet temperatures. If \(T_{1,in}\) and \(T_{2,in}\) denote the inlet temperatures of fluids 1 and 2, respectively, the maximum possible temperature difference (\(\Delta T_{max}\)) can be calculated as: $$\Delta T_{\max }=\left|T_{1, \mathrm{in}}-T_{2, \mathrm{in}}\right|$$

Calculate the maximum possible heat transfer rate

Now that we have all the necessary information, we can calculate the maximum possible heat transfer rate (\(\dot{Q}_{\max}\)) using the following formula: $$\dot{Q}_{\max }=C_{\min } \cdot \Delta T_{\max }$$ This calculation is independent of the type of heat exchanger used, as it only takes into consideration the maximum energy transfer between the two fluids based on their properties and inlet temperatures.