Question

Consider a double-pipe counterflow heat exchanger. In order to enhance heat transfer, the length of the heat exchanger is now doubled. Do you think its effectiveness will also double?

Short answer

Answer: Doubling the length of a double-pipe counterflow heat exchanger does not double its effectiveness. Although the heat transfer area increases, the relationship between effectiveness and length is not directly proportional.

Step-by-step solution

Understand the heat exchanger effectiveness definition

Effectiveness is defined as the ratio of actual heat transfer to the maximum possible heat transfer. Mathematically, it can be represented as: Effectiveness = \(\frac{Q_{actual}}{Q_{max}}\) Where \(Q_{actual}\) is the actual heat transfer and \(Q_{max}\) is the maximum possible heat transfer.

Identify the heat exchanger equation

For a double-pipe counterflow heat exchanger, the heat exchanger equation can be represented as: \(Q = U \times A \times \Delta T_{lm}\) Where: - \(Q\) is the rate of heat transfer - \(U\) is the overall heat transfer coefficient - \(A\) is the heat transfer area - \(\Delta T_{lm}\) is the log mean temperature difference

Relate the heat exchanger equation to the effectiveness

We can express the effectiveness in terms of the heat exchanger equation components: Effectiveness = \(\frac{UA}{C_{min}}(1-e^{-\frac{UA}{C_{min}}})\) Where \(C_{min}\) is the minimum heat capacity rate of the two fluids in the heat exchanger.

Analyze the effectiveness when doubling the heat exchanger length

When the length of the heat exchanger is doubled, the heat transfer area (\(A\)) also doubles. Let's analyze the effectiveness for the new heat exchanger where the area is \(2A\): Effectiveness (\(2A\)) = \(\frac{2UA}{C_{min}}(1-e^{-\frac{2UA}{C_{min}}})\)

Compare the effectiveness of the original and doubled heat exchanger

Now we will determine if the effectiveness doubles when the length of the heat exchanger is doubled: Effectiveness ratio = \(\frac{Effectiveness (2A)}{Effectiveness (A)}\) Effectiveness ratio = \(\frac{\frac{2UA}{C_{min}}(1-e^{-\frac{2UA}{C_{min}}})}{\frac{UA}{C_{min}}(1-e^{-\frac{UA}{C_{min}}})}\) After simplifying the equation, we get: Effectiveness ratio ≠ 2 Since the effectiveness ratio is not equal to 2, it can be concluded that doubling the length of the heat exchanger does not double the effectiveness.