Heat exchangers are commonly used in industry, and the proper design of a heat exchanger depends on many variables. In analyzing heat exchangers, it is often convenient to work with an overall heat transfer coefficient, known as a U-factor. The U-factor is defined by an expression analogous to Newton’s law of cooling. Moreover, engineers also use the logarithmic mean temperature difference (LMTD) to determine the temperature driving force for heat transfer in heat exchangers.
Special Reference: John R. Thome, Engineering Data Book III. Wolverine Tube Inc. 2004.
These terms and methods widely used to select an appropriate heat exchanger are described below.
Heat Exchanger Analysis – Performance Calculation
Overall Heat Transfer Coefficient
A heat exchanger typically involves two flowing fluids separated by a solid wall. Many of the heat transfer processes encountered in the industry involve composite systems and even involve a combination of both conduction and convection. Heat is first transferred from the hot fluid to the wall by convection, through the wall by conduction, and then to the cold fluid again by convection.
It is often convenient to work with an overall heat transfer coefficient, known as a U-factor with these composite systems. The U-factor is defined by an expression analogous to Newton’s law of cooling:
The overall heat transfer coefficient, U, is related to the total thermal resistance and depends on the geometry of the problem. For example, heat transfer in a steam generator involves convection from the bulk of the reactor coolant to the steam generator inner tube surface, conduction through the tube wall, and convection (boiling) from the outer tube surface to the secondary side fluid.
In cases of combined heat transfer for a heat exchanger, there are two values for h. The convective heat transfer coefficient (h) for the fluid film inside the tubes and a convective heat transfer coefficient for the fluid film outside the tubes. The tube wall’s thermal conductivity (k) and thickness (Δx) must also be accounted for.
Logarithmic Mean Temperature Difference – LMTD
To solve certain heat exchanger problems, engineers often use a log mean temperature difference (LMTD), which is used to determine the temperature driving force for heat transfer in heat exchangers. LMTD is introduced because the temperature change that takes place across the heat exchanger from the entrance to the exit is not linear.
The heat transfer through the wall of the heat exchanger at a given location is given by the following equation:
Here, the overall heat transfer coefficient value can be assumed as a constant. On the other hand, the temperature difference continuously varies with location (especially in counter-flow arrangement). To determine the total heat flow, either the heat flow should be summed up using elemental areas and the temperature difference at the location. More conveniently, engineers can average the value of temperature difference. The heat exchanger equation can be solved much easier by defining a “Mean Temperature Difference” (MTD). It can be seen from the figure that the temperature difference varies along with the flow, and the arithmetic average may not be the real average; therefore, engineers use the logarithmic mean temperature difference. The “Logarithmic Mean Temperature Difference” (LMTD) is a logarithmic average of the temperature difference between the hot and cold feeds at each end of the heat exchanger. The larger the LMTD, the more heat is transferred. It can be seen from the figure that the temperature difference varies along with the flow, and the arithmetic average may not be the real average.
For a heat exchanger that has two ends (which we call “A” and “B”) at which the hot and cold streams enter or exit on either side, the LMTD is defined as:
The heat transfer is then given by:
This holds both for parallel-flow arrangement, where the streams enter from the same end, and for counter-flow arrangement, they enter from different ends.
In a cross-flow, where one system, usually the heat sink, has the same nominal temperature at all points on the heat transfer surface, a similar relation between exchanged heat and LMTD holds, but with a correction factor. A correction factor is also required for other more complex geometries, such as a shell and tube exchanger with baffles.
LMTD – Condensers and Boilers
Steam generators and condensers are also examples of components found in nuclear facilities where the concept of LMTD is needed to address certain problems. When the subcooled water enters the steam generator, it must be heated up to its boiling point, and then it must be evaporated. Because evaporation occurs at a constant temperature, it cannot be used as a single LMTD. In this case, the heat exchanger must be treated as a combination of two or three (when superheat occurs) heat exchangers.
Pinch Point – Heat Exchanger
Another interesting aspect of the design is that the temperature difference is, known as ‘pinch,’ can limit the performance of heat exchangers if the areas and flow rates are not properly designed. The pinch point is the location in the heat exchanger where the temperature difference between hot and cold fluid is minimum at that location.
NTU Effectiveness Method
The log means temperature difference (LMTD) method discussed in the previous section is easy to use in heat exchanger analysis when the inlet and the outlet temperatures of the hot and cold fluids are known or can be determined from an energy balance. Therefore, the LMTD method is very suitable for determining the size and performance of a heat exchanger.
When direct knowledge of the LMTD is not available, the NTU method (Number of Transfer Units method) can be used. This method is based on a dimensionless parameter called the heat transfer effectiveness, defined as:
As can be seen, the effectiveness is the ratio between the actual heat transfer rate and the maximum possible heat transfer rate. To define the effectiveness of a heat exchanger, we must first determine the maximum possible heat transfer rate, qmax, for the heat exchanger.
Further reading:
- Fundamentals of Heat and Mass Transfer, 7th Edition. Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera. John Wiley & Sons, Incorporated, 2011. ISBN: 9781118137253.
- Heat and Mass Transfer. Yunus A. Cengel. McGraw-Hill Education, 2011. ISBN: 9780071077866.
Example: Calculation of Heat Exchanger
Consider a parallel-flow heat exchanger used to cool oil from 70°C to 40°C using water available at 30°C. The outlet temperature of the water is 36°C, and the rate of flow of oil is 1 kg/s. The specific heat of the oil is 2.2 kJ/kg K. The overall heat transfer coefficient U = 200 W/m2 K.
Calculate the logarithmic mean temperature difference. Determine the area of this heat exchanger required for this performance.
- LMTD
The logarithmic mean temperature difference can be calculated simply using its definition:
- Area of Heat Exchanger
To calculate the area of this heat exchanger, we have to calculate the heat flow rate using the mass flow rate of oil and LMTD.
The required area of this heat exchanger can be then directly calculated using the general heat transfer equation:
