FE Heat Transfer: Heat Exchangers Review

Heat exchangers are a cornerstone of thermal systems engineering, appearing in power plants, HVAC systems, and industrial processes. For the FE exam, you must move beyond simple definitions to confidently select and apply the correct analysis method to solve sizing or performance problems. This review focuses on the two primary methodologies—the Log Mean Temperature Difference (LMTD) method and the Effectiveness-NTU method—and the critical concepts that connect them, ensuring you can tackle any related problem efficiently.

The Overall Heat Transfer Coefficient

The analysis of any heat exchanger begins with the overall heat transfer coefficient, denoted as $U$. This parameter encapsulates the total thermal resistance to heat flow between the two fluid streams. It is defined by the rate equation: $$q=UA\Delta T_m$$ where $q$ is the total heat transfer rate, $A$ is the heat transfer surface area, and $\Delta T_m$ is a suitably averaged temperature difference between the hot and cold fluids.

The overall coefficient $U$ is calculated by summing the individual resistances: the convective resistance on the hot side, the conductive resistance of the wall material, and the convective resistance on the cold side, along with any fouling factors. For a plane wall, this is expressed as: $$\frac{1}{U}=\frac{1}{h_h}+\frac{R_{f,h}}{A}+\frac{L}{kA}+\frac{R_{f,c}}{A}+\frac{1}{h_c}$$ where $h$ are convective coefficients, $R_f$ are fouling resistances, $L$ is wall thickness, and $k$ is thermal conductivity. For cylindrical tubes, the expression uses areas to account for the changing surface area from inner to outer radius. On the FE exam, you may need to calculate $U$ from given resistances or use it directly in the core heat exchanger equations.

The Log Mean Temperature Difference (LMTD) Method

The LMTD method is directly derived from the rate equation $q=UA\Delta T_m$. It is most straightforwardly applied when all four terminal temperatures (hot in, hot out, cold in, cold out) are known or can be easily determined from an energy balance. The key is calculating the correct Log Mean Temperature Difference.

For pure parallel-flow and counterflow arrangements, the LMTD is calculated using the same formula but with different definitions for the temperature differences at each end of the exchanger. $$\Delta T_{lm}=\frac{\Delta T_1-\Delta T_2}{\ln (\Delta T_1/\Delta T_2)}$$

For a counterflow exchanger, $\Delta T_1=T_{h,in}-T_{c,out}$ and $\Delta T_2=T_{h,out}-T_{c,in}$. For a parallel-flow exchanger, $\Delta T_1=T_{h,in}-T_{c,in}$ and $\Delta T_2=T_{h,out}-T_{c,out}$. Counterflow almost always yields a higher LMTD for the same terminal temperatures, meaning it requires less surface area for the same heat duty—a critical design insight.

The basic application procedure is: 1) Perform energy balances ($q={\dot{m}}_h{c}_{p,h}(T_{h,in}-T_{h,out})={\dot{m}}_c{c}_{p,c}(T_{c,out}-T_{c,in})$) to find any unknown temperatures or $q$. 2) Determine $\Delta T_1$ and $\Delta T_2$ based on flow arrangement. 3) Calculate $\Delta T_{lm}$. 4) Solve $q=UA\Delta T_{lm}$ for the unknown (typically $A$ or $U$).

Correction Factor for Multi-Pass and Crossflow Exchangers

Real-world heat exchangers are rarely pure parallel or counterflow; they are often shell-and-tube with multiple passes or crossflow arrangements. These complex flows result in a less effective temperature driving force. To use the LMTD method for these designs, a correction factor, $F$, is introduced: $$q=UAF\Delta T_{lm,cf}$$ Here, $\Delta T_{lm,cf}$ is the LMTD calculated as if the exchanger were in pure counterflow with the same inlet and outlet temperatures. The factor $F$ (where $0<F\le 1$) then corrects this value downward to account for the less efficient flow arrangement.

On the FE exam, you will likely be given a chart or an equation to determine $F$ based on two dimensionless parameters: $$P=\frac{t_2-t_1}{T_1-t_1}\text{and}R=\frac{T_1-T_2}{t_2-t_1}$$ where $T$ refers to shell-side fluid temperatures and $t$ refers to tube-side fluid temperatures. It is crucial to correctly identify the shell-side and tube-side streams from the problem statement. The LMTD method with correction factor is the preferred approach for rating problems (finding outlet temperatures) when the flow arrangement and $F$ chart are available.

The Effectiveness-NTU Method

When the outlet temperatures are unknown, solving with the LMTD method becomes an iterative, tedious process. This is where the Effectiveness-NTU method shines. It is specifically designed for performance calculation problems where the inlet conditions, flow rates, $U$, and $A$ are known, and the outlet temperatures must be found.

The method uses three key dimensionless parameters:

1. Heat capacity rate: $C=\dot{m}{c}_p$. You will have $C_h$ and $C_c$.
2. Heat capacity rate ratio: $C_r=C_{min}/C_{max}$.
3. Number of Transfer Units: $NTU=UA/C_{min}$. This is a measure of the size of the heat exchanger.
4. Effectiveness: $ϵ=q/q_{max}$. The effectiveness is the ratio of the actual heat transfer to the maximum theoretically possible heat transfer, which would occur in an infinitely large counterflow exchanger: $q_{max}=C_{min}(T_{h,in}-T_{c,in})$.

The core of the method is the $ϵ$-NTU relationship, which is a function of the flow arrangement and $C_r$. For example, for a counterflow exchanger: $$ϵ=\frac{1-\exp [-NTU(1-C_r)]}{1-C_r\exp [-NTU(1-C_r)]}\text{for }{C}_r<1$$ $$ϵ=\frac{NTU}{1+NTU}\text{for }{C}_r=1$$ Similar, distinct formulas exist for parallel-flow, shell-and-tube, and crossflow arrangements. The standard procedure is: 1) Calculate $C_h$, $C_c$, and identify $C_{min}$ and $C_{max}$. 2) Calculate $C_r$ and $NTU$. 3) Use the correct $ϵ$-NTU correlation for the given flow arrangement to find $ϵ$. 4) Calculate actual heat transfer: $q=ϵC_{min}(T_{h,in}-T_{c,in})$. 5) Use energy balances to find both outlet temperatures.

Standard NTU Correlations and Method Selection

You are expected to know the standard NTU correlations for common flow arrangements. The key ones for the FE exam are:

• Parallel-flow: $ϵ=\frac{1-\exp [-NTU(1+C_r)]}{1+C_r}$
• Counterflow: As shown in the previous section.
• Shell-and-tube (1 shell pass, 2, 4, ... tube passes): ${ϵ}_1=2{\left\{1+C_r+\sqrt{1+C_r^2}\frac{1+\exp [-NTU\sqrt{1+C_r^2}]}{1-\exp [-NTU\sqrt{1+C_r^2}]}\right\}}^{-1}$
• Crossflow (both fluids unmixed): A more complex correlation often given in a chart or table.

The most critical skill for the FE exam is selecting the appropriate analysis method. Follow this decision tree:

1. Are all four terminal temperatures known or easily found? → Use the LMTD method.
2. Is the exchanger not simple counter/parallel flow? → Use the LMTD method with correction factor $F$.
3. Are the outlet temperatures unknown? → Use the Effectiveness-NTU method.

For quick comparisons, remember: The Effectiveness-NTU method is best for performance problems (find $ϵ$, find $T_{out}$). The LMTD method is best for sizing problems (find $A$) or when $F$ factors are involved.

Common Pitfalls

1. Using the wrong LMTD formula for flow arrangement: The most common error is misidentifying $\Delta T_1$ and $\Delta T_2$. Always sketch the temperature distribution. For counterflow, the hot inlet and cold outlet are at the same end; for parallel flow, the inlets are together.
2. Misapplying the correction factor $F$: Students often use the LMTD for the actual flow arrangement instead of the counterflow LMTD in the formula $q=UAF\Delta T_{lm,cf}$. Remember, $F$ is a multiplier for the counterflow LMTD. Also, incorrectly identifying parameters $P$ and $R$ for the $F$ chart will lead to an incorrect, and often physically impossible (F > 1), answer.
3. Miscounting $C_{min}$ in the NTU method: The entire Effectiveness-NTU method hinges on correctly identifying the fluid with the minimum heat capacity rate ($C_{min}$). All formulas use $NTU=UA/C_{min}$ and $q_{max}=C_{min}(T_{h,in}-T_{c,in})$. Switching $C_{min}$ and $C_{max}$ invalidates the calculation.
4. Using the incorrect $ϵ$-NTU correlation: Each flow arrangement has a unique formula. Using the parallel-flow correlation for a counterflow problem, for example, will yield a different (and lower) effectiveness, leading to a wrong answer. If a formula is not provided, you are expected to know the standard ones.

Summary

• The overall heat transfer coefficient $U$ sums all thermal resistances and is central to the basic heat exchanger equation $q=UA\Delta T_m$.
• The LMTD method is ideal for sizing problems where terminal temperatures are known. Remember to calculate $\Delta T_{lm}$ correctly for parallel vs. counterflow and apply a correction factor $F$ for complex geometries using the counterflow LMTD.
• The Effectiveness-NTU method is the preferred tool for performance problems where outlet temperatures are unknown. It relies on the dimensionless parameters effectiveness ($ϵ$), NTU, and capacity ratio ($C_r$).
• Success on the FE exam depends on selecting the correct method: Use LMTD for sizing/knowable temperatures; use $ϵ$-NTU for finding unknown outlet temperatures.
• Avoid fatal errors by carefully identifying $C_{min}$, using the proper LMTD end-point temperatures, and selecting the $ϵ$-NTU correlation that matches the given flow arrangement.