Question

Consider a heat exchanger that has an NTU of \(0.1\). Someone proposes to triple the size of the heat exchanger and thus triple the NTU to \(0.3\) in order to increase the effectiveness of the heat exchanger and thus save energy. Would you support this proposal?

Short answer

Answer: Tripling the size of the heat exchanger to triple the NTU value to 0.3 does result in increased effectiveness of the heat exchanger. However, the energy savings observed may not be significant due to diminishing returns and the practical implications such as increased costs and space requirements.

Step-by-step solution

Understand the NTU Method

The NTU (Number of Transfer Units) is a dimensionless parameter that indicates the size and performance of a heat exchanger. It is the ratio of the actual heat transfer rate to the maximum possible heat transfer rate. The effectiveness of a heat exchanger can be calculated using the following equation: Effectiveness, \(\epsilon = \frac{Q_{actual}}{Q_{max}}\) Where: \(\epsilon\) = Effectiveness \(Q_{actual}\) = Actual heat transfer rate (W) \(Q_{max}\) = Maximum heat transfer rate (W) Using the NTU method, the effectiveness of the heat exchanger is related to the NTU value and the heat capacity rate ratio \(C_{min}/C_{max}\). Generally, an increase in the heat exchanger's size (and consequently the NTU value) results in an increase in effectiveness.

Evaluating the proposal - Tripling the NTU value

To evaluate the proposal of tripling the NTU, consider the initial NTU value of \(0.1\). With the proposal, the NTU value will be tripled to \(0.3\). The effectiveness would depend on the type of heat exchanger and the heat capacity ratio. For the sake of simplicity, let's assume a simple counterflow heat exchanger. The effectiveness formula for such a heat exchanger is given by: \(\epsilon = \frac{1 - e^{-(NTU)(1+C_r)(1-e^{-NTU(1-C_r)})}}{(1 + C_r)(1-e^{-NTU(1-C_r)})}\) Where: \(C_r\) = heat capacity rate ratio = \(C_{min}/C_{max}\) \(NTU\) = Number of Transfer Units Now, we do not know the \(C_r\) for our current scenario. To truly evaluate the effectiveness of the proposed idea, we must consider the relationship between NTU and effectiveness while keeping \(C_r\) constant.

Illustrate the relationship between NTU and effectiveness

One way to illustrate this relationship is to create a graph of effectiveness versus NTU while keeping the \(C_r\) constant. When you create such a graph, you would observe that the effectiveness increases with increasing NTU value. However, it is also important to note that the effectiveness has a diminishing return as the NTU value gets larger, and the curve will asymptotically approach \(100\%\) (complete energy transfer) without reaching it.

Conclusion - Evaluating the proposal

Although increasing the size of the heat exchanger can result in higher effectiveness, it is essential to consider the practical implications of tripling the size of the heat exchanger. A larger heat exchanger may result in higher capital costs, increased maintenance requirements, and potentially a larger footprint. In conclusion, while increasing NTU does increase the effectiveness of the heat exchanger, the effectiveness gain per NTU decreases when NTU is increased. Therefore, tripling the size of the heat exchanger to triple the NTU to \(0.3\) may save some energy but may not be worth the investment considering the diminishing returns and the practical implications such as costs and space requirements.

Key concepts

Heat Exchanger Effectiveness

Heat exchanger effectiveness is an integral concept in understanding how well a heat exchanger is performing its task of transferring heat from one fluid to another. It is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate within the heat exchanger.

In simple terms, it is a measure of how close the heat exchanger comes to reaching its maximum potential. The effectiveness, represented by the symbol \( \epsilon \), can be described by the equation:\
\[ \epsilon = \frac{Q_{actual}}{Q_{max}} \]
where \( Q_{actual} \) is the actual rate of heat transferred and \( Q_{max} \) is the theoretical maximum rate of heat transfer if the exiting fluid temperature were to reach the entering temperature of the hot/cold stream.

In practice, this parameter helps in deciding whether a heat exchanger is efficient enough for a particular application or whether modifications, such as increasing the size to raise the NTU, are warranted.

Heat Transfer Rate

The heat transfer rate in a heat exchanger ties directly into its effectiveness. It represents the quantity of heat transferred between the two fluids inside the heat exchanger per unit time, typically measured in watts (W). The actual heat transfer rate, \( Q_{actual} \), can be affected by several factors including the size of the heat exchanger, the materials used, the flow arrangement, and the temperature gradient.

Engineers aim to maximize this rate to improve the heat exchanger's performance while considering the constraints of the system. For example, by enlarging the surface area through which the heat is transferred, they can increase the heat transfer rate up to a point dictated by the laws of diminishing returns relative to size and cost.

Heat Capacity Rate Ratio

Understanding the heat capacity rate ratio, often symbolized as \( C_r \) and defined as the ratio \( C_{min}/C_{max} \), where \( C_{min} \) and \( C_{max} \) are the minimum and maximum heat capacity rates of the fluids involved, is essential when analyzing or designing heat exchangers.

This ratio impacts how the heat exchanger's effectiveness will respond to changes in the NTU. It's an indication of the relative capacity of one fluid to hold heat compared to the other fluid. When this ratio is balanced, the heat exchanger can work more effectively, and adjustments to the NTU can be more predictable in their impact on effectiveness.

Diminishing Returns in Heat Exchanger Performance

The concept of diminishing returns in the context of heat exchanger performance refers to the observation that after a certain point, increasing the size of the heat exchanger and hence the NTU yields progressively smaller improvements in effectiveness.

As illustrated in the step-by-step solution, making the heat exchanger three times larger to triple the NTU from \(0.1\) to \(0.3\) will increase effectiveness, but not linearly. The effectiveness curve will asymptotically approach 100% as NTU is increased, but never actually reach it, indicating that there's a practical limit to the benefits gained from just enlarging the heat exchanger. Beyond a specific NTU value, the cost, space, and added complexity may outweigh the marginal gains in effectiveness, which is crucial to consider when proposing such upgrades.