Question

Under what conditions is the heat transfer relation $$ \dot{Q}=\dot{m}_{c} c_{p c}\left(T_{c, \text { out }}-T_{c, \text { in }}\right)=\dot{m}_{h} c_{p h}\left(T_{h, \text { in }}-T_{h, \text { out }}\right) $$ valid for a heat exchanger?

Short answer

Answer: The heat transfer relation is valid for a heat exchanger when the mass flow rates and specific heat capacities remain constant for both fluids, the fluids are single-phase, homogeneous mixtures, without phase changes or significant temperature-dependent specific heat capacities.

Step-by-step solution

Identify the terms in the heat transfer relation equation.

The given heat transfer equation is: $$ \dot{Q} = \dot{m}_c c_{p c} (T_{c, out} - T_{c, in}) = \dot{m}_h c_{p h} (T_{h, in} - T_{h, out}) $$ Where, \(\dot{Q}\) is the rate of heat transfer, \(\dot{m}_c\) and \(\dot{m}_h\) are the mass flow rates of the cold and hot fluids, respectively, \(c_{p c}\) and \(c_{p h}\) are the specific heat capacities of the cold and hot fluids, respectively, at constant pressure, \(T_{c, in}\) and \(T_{c, out}\) are the temperature of the cold fluid at the inlet and outlet, respectively, \(T_{h, in}\) and \(T_{h, out}\) are the temperature of the hot fluid at the inlet and outlet, respectively.

Identify the conditions when mass flow rates are constant.

The mass flow rates of both cold and hot fluids (\(\dot{m}_c\) and \(\dot{m}_h\)) must be constant for the given heat transfer relation to hold. This means that there should be no external forces, leaks, blockages or any other factors causing the mass flow rates to vary throughout the heat exchanger.

Identify the conditions when specific heat capacities are constant.

The specific heat capacities of both cold and hot fluids (\(c_{p c}\) and \(c_{p h}\)) must also be constant. This implies that both fluids should: 1. Be single-phase and homogeneous mixtures, and no phase changes (e.g., evaporation or condensation) should occur during the heat exchange process. 2. Have specific heat capacities that are not significantly dependent on temperature.

Identify the conditions for the validity of the heat transfer relation.

Based on the analysis in Steps 2 and 3, the heat transfer relation will be valid under the following conditions: 1. The mass flow rates of both cold and hot fluids must be constant, with no external forces, leaks, blockages, or any other factors causing variation. 2. The specific heat capacities of both cold and hot fluids must remain constant throughout the heat exchange process. 3. Both fluids should be single-phase and homogeneous mixtures, with no phase changes occurring during the heat exchange process. 4. The specific heat capacities of both fluids should not be significantly dependent on temperature. In conclusion, the heat transfer relation is valid for a heat exchanger when the mass flow rates and specific heat capacities remain constant for both fluids, and the fluids are single-phase, homogeneous mixtures, without phase changes or significant temperature-dependent specific heat capacities.