Understanding Heat Exchanger Effectiveness

The effectiveness of a heat exchanger, often denoted by \(\epsilon\), is a measure of its performance. It is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate.

Mathematically, the effectiveness is given by:

\(\epsilon = \frac{Q_{actual}}{Q_{max}}\)

Where \(Q_{actual}\) is the actual heat transferred, and \(Q_{max}\) is the maximum possible heat transfer rate if one fluid undergoes a temperature change equal to the maximum temperature difference available, which is the difference between the inlet temperatures of the hot and cold fluids.

Effectiveness and NTU Relationship

The effectiveness of a heat exchanger is also related to the Number of Transfer Units (NTU) and the capacity rate ratio (\(C_r\)). The NTU is a dimensionless parameter that indicates the size of the heat exchanger and is defined as:

\(\rm NTU = \frac{UA}{C_{Min}}\)

Where U is the overall heat transfer coefficient, A is the heat transfer area, and \(C_{Min}\) is the minimum heat capacity rate of the two fluids (\(C = \dot{m}c_p\)).

The capacity rate ratio is defined as \(C_r = \frac{C_{Min}}{C_{Max}}\), where \(C_{Max}\) is the maximum heat capacity rate.

Effectiveness for Double Pipe Counter-Flow Heat Exchanger with \(C_r = 1\)

For a double pipe counter-flow heat exchanger, the general formula for effectiveness (\(\epsilon\)) is:

\(\epsilon = \frac{1 - e^{-NTU(1-C_r)}}{1 - C_r e^{-NTU(1-C_r)}}\)

We are given the specific condition that \(C_r = \frac{C_{Min}}{C_{Max}} = 1\). When \(C_r = 1\), the denominator of the general formula becomes \(1 - 1 \cdot e^{-NTU(1-1)} = 1 - e^0 = 1 - 1 = 0\), and the numerator becomes \(1 - e^{-NTU(1-1)} = 1 - e^0 = 1 - 1 = 0\). This results in an indeterminate form (0/0).

For the special case of a counter-flow heat exchanger with \(C_r = 1\), the effectiveness formula needs to be derived differently (often using L'Hopital's rule on the general formula or directly from energy balance equations). The correct formula for this specific case is:

\(\epsilon = \frac{NTU}{1 + NTU}\)

Let's check this formula:

• As \(NTU \to 0\), \(\epsilon \to 0/1 = 0\), which makes sense for a very small heat exchanger.
• As \(NTU \to \infty\), \(\epsilon = \lim_{NTU \to \infty} \frac{NTU}{1 + NTU} = \lim_{NTU \to \infty} \frac{1}{1/NTU + 1} = \frac{1}{0+1} = 1\). For \(C_r=1\), perfect heat exchange (\(\epsilon=1\)) is theoretically possible with infinite heat transfer area in a counter-flow arrangement.

Comparing this derived formula with the given options:

• Option 1: \(\rm 1+\frac{1}{NTU}\)
• Option 2: \(\rm \frac{NTU}{NTU-1}\)
• Option 3: \(\rm \frac{NTU}{NTU+1}\)
• Option 4: \(\rm \frac{1+NTU}{1-NTU}\)

The derived formula \(\epsilon = \frac{NTU}{1 + NTU}\) matches Option 3.

Conclusion

For a double pipe counter flow heat exchanger with the capacity rate ratio \(\frac{C_{Min}}{C_{Max}}=1\), the effectiveness of the heat exchanger is given by \(\frac{NTU}{1 + NTU}\).