Question

In the heat transfer relation \(\dot{Q}=U A_{s} \Delta T_{\operatorname{lm}}\) for a heat exchanger, what is \(\Delta T_{\operatorname{lm}}\) called? How is it calculated for a parallel-flow and counter-flow heat exchanger?

Short answer

To summarize, the term \(\Delta T_{\operatorname{lm}}\) represents the log mean temperature difference, which is essential for heat transfer calculations in heat exchangers. It helps account for the varying temperature differences between the hot and cold fluids within the exchanger. Two types of heat exchangers, parallel-flow and counter-flow, have different formulas to calculate \(\Delta T_{\operatorname{lm}}\). For both types, the formula is: $$ \Delta T_{\operatorname{lm}} = \frac{\Delta T_1 - \Delta T_2}{\ln{\frac{\Delta T_1}{\Delta T_2}}}, $$ with \(\Delta T_1\) and \(\Delta T_2\) representing temperature differences at different points in the heat exchanger. Finally, the log mean temperature difference (\(\Delta T_{\operatorname{lm}}\)) is used in the heat transfer relation to calculate the heat transfer rate.

Step-by-step solution

Definition of \(\Delta T_{\operatorname{lm}}\)

The term \(\Delta T_{\operatorname{lm}}\) stands for "log mean temperature difference." It is an essential parameter in the heat transfer calculations of a heat exchanger and represents the average temperature difference between the hot and cold fluids across the heat exchanger. It takes into account the varying temperature differences across the exchanger and averages them for more accurate calculations.

Parallel-flow heat exchanger

In a parallel-flow heat exchanger, the hot and cold fluids flow in the same direction. To calculate the log mean temperature difference for a parallel-flow heat exchanger, use the following formula: $$ \Delta T_{\operatorname{lm}, \parallel} = \frac{\Delta T_1 - \Delta T_2}{\ln{\frac{\Delta T_1}{\Delta T_2}}}, $$ where \(\Delta T_1\) is the temperature difference between the hot and cold fluids at the inlet (entrance) of the heat exchanger, and \(\Delta T_2\) is the temperature difference between the hot and cold fluids at the outlet (exit) of the heat exchanger.

Counter-flow heat exchanger

In a counter-flow heat exchanger, the hot and cold fluids flow in opposite directions. To calculate the log mean temperature difference for a counter-flow heat exchanger, use the following formula: $$ \Delta T_{\operatorname{lm}, \text{counter}} = \frac{\Delta T_1 - \Delta T_2}{\ln{\frac{\Delta T_1}{\Delta T_2}}}, $$ where \(\Delta T_1\) is the temperature difference between the hot and cold fluids at one end of the heat exchanger, and \(\Delta T_2\) is the temperature difference between the hot and cold fluids at the other end of the heat exchanger. In both cases, the log mean temperature difference (\(\Delta T_{\operatorname{lm}}\)) is used in the heat transfer relation, given by: $$ \dot{Q} = U A_s \Delta T_{\operatorname{lm}}, $$ where \(\dot{Q}\) is the heat transfer rate, \(U\) is the overall heat transfer coefficient, and \(A_s\) is the heat transfer surface area. By understanding the differences between parallel-flow and counter-flow heat exchangers and their respective formulas to calculate \(\Delta T_{\operatorname{lm}}\), the student can effectively perform accurate heat transfer calculations for different heat exchanger configurations.

Key concepts

Heat Exchanger Efficiency

Heat exchanger efficiency is a critical measure of how well a heat exchanger performs its tasks of transferring heat between two or more fluids. This efficiency can be defined as the ratio of the actual heat transfer to the maximum possible heat transfer, given the same inlet and outlet temperatures for the hot and cold fluids. An efficient heat exchanger is one that transfers the maximum amount of heat with the least amount of energy loss.

Key factors impacting the efficiency of a heat exchanger include the design of the unit (parallel-flow or counter-flow), the materials used for construction, the surface area available for heat transfer, and the overall heat transfer coefficient. By optimizing and maintaining these aspects, heat exchangers can operate with greater efficiency, saving energy and reducing costs.

Parallel-Flow Heat Exchanger

In a parallel-flow heat exchanger, the fluids flow in the same direction, entering and exiting the exchanger side by side. This setup means the temperature gradient between the fluids is greatest at the entrance and decreases along the flow path. As a consequence, the efficiency of heat exchange may be lower compared to the counter-flow configuration since the temperature difference, which drives heat transfer, diminishes along the exchanger.

When calculating the efficiency or performance of a parallel-flow heat exchanger, it is essential to consider not only the log mean temperature difference \(\Delta T_{\operatorname{lm}, \parallel}\) but also recognize that this type of exchanger may require more surface area or a higher overall heat transfer coefficient to achieve similar performance to a counter-flow design under the same conditions.

Counter-Flow Heat Exchanger

The counter-flow heat exchanger stands apart due to its fluids moving in opposite directions. This arrangement provides a more consistent temperature gradient between the hot and cold fluids across the length of the exchanger, which typically allows for better heat transfer efficiency than in parallel-flow design. The counter-flow arrangement maintains a higher driving force for heat transfer throughout the device, making it effective for systems where space is limited or where the greatest energy recovery is desired.

A counter-flow heat exchanger can often achieve a closer approach temperature — the minimum temperature difference between the outlet temperatures of the fluids — which is an indicator of a highly efficient heat exchange process. Understanding the calculation of log mean temperature difference \(\Delta T_{\operatorname{lm}, \text{counter}}\) for a counter-flow heat exchanger is fundamental in designing and optimizing these systems.

Overall Heat Transfer Coefficient

The overall heat transfer coefficient, denoted as \(U\), is a composite measure that describes how well a heat exchanger can move heat between two fluids. It incorporates the characteristics of the exchanger's materials, the surface area through which the heat is being transferred, and the nature of the fluids involved (including flow configuration and properties).

The coefficient \(U\) plays a vital role in the equation \(\dot{Q} = U A_s \Delta T_{\operatorname{lm}}\) because it directly affects the heat transfer rate \(\dot{Q}\). A higher value of \(U\) indicates better efficiency and effectiveness in transferring heat. To maximize the performance of a heat exchanger, engineers work to optimize the overall heat transfer coefficient by selecting appropriate materials, maximizing surface area, and designing for the most efficient flow configuration.