In a parallel flow heat exchanger, if the heat capacity ratio is 1, then what is the maximum value of effectiveness?

Explanation:

Effectiveness of a Parallel Flow Heat Exchanger

Definition: In heat exchanger analysis, the effectiveness (ε) is a measure of how well a heat exchanger transfers heat relative to its maximum possible heat transfer. It is defined as the ratio of the actual heat transfer to the maximum possible heat transfer that could occur if the outlet temperature of one fluid reached the inlet temperature of the other fluid.

Mathematically, the effectiveness (ε) is expressed as:

ε = Q_actual / Q_max

Where:

• Q_actual: The actual heat transfer rate in the heat exchanger.
• Q_max: The maximum possible heat transfer rate, which occurs when one of the fluids undergoes the maximum possible temperature change.

For a parallel flow heat exchanger, the two fluids enter the heat exchanger at their respective inlet temperatures and flow in the same direction. The effectiveness of such a heat exchanger depends on the heat capacity ratio (C_r) and the number of transfer units (NTU).

Heat Capacity Ratio (C_r):

The heat capacity ratio is defined as:

C_r = C_min / C_max

Where:

• C_min: The smaller heat capacity rate of the two fluids (C = ṁ × c_p).
• C_max: The larger heat capacity rate of the two fluids.

When the heat capacity ratio (C_r) is equal to 1, it implies that both fluids have the same heat capacity rate.

Effectiveness for Parallel Flow Heat Exchanger:

The effectiveness of a parallel flow heat exchanger is given by the following equation:

ε = [1 - exp(-NTU × (1 + C_r))] / [1 + C_r]

Where:

• NTU: Number of Transfer Units, a dimensionless parameter that represents the size and effectiveness of the heat exchanger.
• C_r: Heat capacity ratio.

Analysis for C_r = 1:

When C_r = 1, the equation for effectiveness simplifies to:

ε = [1 - exp(-2 × NTU)] / 2

The maximum value of effectiveness occurs when NTU approaches infinity (i.e., a very large heat exchanger). In this case:

exp(-2 × NTU) → 0

Therefore, the effectiveness becomes:

ε = [1 - 0] / 2 = 0.5

Conclusion: For a parallel flow heat exchanger with a heat capacity ratio (C_r) of 1, the maximum value of effectiveness is 0.5.