Question

It is well established that the thermal efficiency of a heat engine increases as the temperature \(T_{L}\) at which heat is rejected from the heat engine decreases. In an effort to increase the efficiency of a power plant, somebody suggests refrigerating the cooling water before it enters the condenser, where heat rejection takes place. Would you be in favor of this idea? Why?

Short answer

Answer: Theoretically, refrigerating the cooling water before it enters the condenser would decrease the temperature at which heat is rejected (T_L). According to the Carnot efficiency formula, this would result in an increase in the power plant's thermal efficiency. However, practical aspects such as the additional energy required for refrigeration and potential operational complexities should be considered, as they could offset the efficiency gain or make the solution inefficient or unfeasible.

Step-by-step solution

Understand the concept of thermal efficiency

Thermal efficiency is a measure of how well a heat engine converts heat energy into useful work. It is defined as the ratio of useful work performed by the engine to the heat energy input. Mathematically, thermal efficiency (\(\eta\)) can be expressed as: \(\eta = \frac{W}{Q_H}\) where \(W\) is the work done by the engine and \(Q_H\) is the heat energy input.

Examine the relationship between thermal efficiency and heat rejection temperature

It is mentioned that the thermal efficiency of a heat engine increases as the temperature (\(T_{L}\)) at which heat is rejected decreases. We can use the Carnot efficiency as an example to understand the relationship between efficiency and temperatures. The Carnot efficiency is given by: \(\eta_{Carnot} = 1 - \frac{T_L}{T_H}\) where \(T_H\) is the temperature of the heat source. From this equation, it is clear that as \(T_{L}\) decreases, the efficiency \(\eta_{Carnot}\) increases.

Assess the proposed idea of refrigerating the cooling water

The suggested idea is to refrigerate the cooling water before it enters the condenser, where heat rejection takes place. By doing this, the temperature \(T_{L}\) at which the heat is rejected will decrease. According to the Carnot efficiency formula, this would result in an increase in thermal efficiency.

Consider practical aspects and make a conclusion

Although the theoretical analysis suggests that decreasing \(T_{L}\) by refrigerating the cooling water could potentially increase the efficiency of the power plant, there are several practical aspects to consider. Refrigerating the cooling water would require additional energy input, which could offset the gain in efficiency. Moreover, maintaining lower temperatures and dealing with the potential issues related to refrigeration and temperature control could create complexities in the power plant operation. Considering these practical aspects, although the theoretical basis for the suggestion might be valid, it may not be the most effective or feasible solution to increase the efficiency of the power plant in practice.