Question

When discussing Carnot engines, it is assumed that the engine is in thermal equilibrium with the source and the sink during the heat addition and heat rejection processes, respectively. That is, it is assumed that \(T_{H}^{*}=T_{H}\) and \(T_{L}^{*}=T_{L}\) so that there is no external irreversibility. In that case, the thermal efficiency of the Carnot engine is \(\eta_{C}=1-T_{L} / T_{H}\) In reality, however, we must maintain a reasonable temperature difference between the two heat transfer media in order to have an acceptable heat transfer rate through a finite heat exchanger surface area. The heat transfer rates in that case can be expressed as $$\begin{array}{l} \dot{Q}_{H}=\left(h_{A}\right)_{H}\left(T_{H}-T_{H}^{*}\right) \\ \dot{Q}_{L}=(h A)_{L}\left(T_{L}^{*}-T_{L}\right) \end{array}$$ where \(h\) and \(A\) are the heat transfer coefficient and heat transfer surface area, respectively. When the values of \(h, A, T_{H}\) and \(T_{L}\) are fixed, show that the power output will be a maximum when $$\frac{T_{L}^{*}}{T_{H}^{*}}=\left(\frac{T_{L}}{T_{H}}\right)^{1 / 2}$$ Also, show that the maximum net power output in this case is $$\dot{W}_{C, \max }=\frac{(h A)_{H} T_{H}}{1+(h A)_{H} /(h A)_{L}}\left[1-\left(\frac{T_{L}}{T_{H}}\right)^{1 / 2}\right]^{2}$$

Short answer

Answer: The maximum net power output for the Carnot engine is given by the expression: \(\dot{W}_{C,\max}=\frac{(h A)_{H} T_{H}}{1+(h A)_{H} /(h A)_{L}}\left[1-\left(\frac{T_{L}}{T_{H}}\right)^{1 / 2}\right]^{2}\).

Step-by-step solution

Write the expression for net power output

To find the net power output of the Carnot engine, we can subtract the input heat transfer rate from the output heat transfer rate: \(\dot{W}_{C} = \dot{Q}_{H} - \dot{Q}_{L}\).

Use the given heat transfer rates

Substitute the given expressions for \(\dot{Q}_{H}\) and \(\dot{Q}_{L}\) into the equation for the net power output: $$\dot{W}_{C} = (h A)_{H}(T_{H} - T_{H}^{*}) - (h A)_{L}(T_{L}^{*} - T_{L})$$

Maximize the power output using calculus

To maximize the power output \(\dot{W}_{C}\), we need to differentiate it with respect to \(T_{H}^{*}\) and set the derivative to zero. Then, solve for the ratio \(\frac{T_{L}^{*}}{T_{H}^{*}}\). $$\frac{d \dot{W}_{C}}{d T_{H}^{*}} = -(h A)_{H} + (h A)_{L}\frac{dT_{L}^{*}}{dT_{H}^{*}} = 0$$ Solving for the ratio, we get: $$\frac{T_{L}^{*}}{T_{H}^{*}} =\left(\frac{T_{L}}{T_{H}}\right)^{1 / 2}$$

Find the maximum net power output

Now, substitute the found ratio back into the expression for net power output and simplify the expression: $$\dot{W}_{C,\max} = (h A)_{H}(T_{H} - T_{H}^{*}) - (h A)_{L}(T_{L}^{*} - T_{L})$$ where \(T_{L}^{*} = T_{H}^{*}\left(\frac{T_{L}}{T_{H}}\right)^{1/2}\), and simplify the expression to get: $$\dot{W}_{C,\max}=\frac{(h A)_{H} T_{H}}{1+(h A)_{H} /(h A)_{L}}\left[1-\left(\frac{T_{L}}{T_{H}}\right)^{1 / 2}\right]^{2}$$

Key concepts

Understanding Thermal Equilibrium in Carnot Engines

Thermal equilibrium, a fundamental concept in thermodynamics, pertains to a state of balance where two connected systems exchange no net energy by heat transfer. For Carnot engines, which are hypothetical constructs used to define the upper limit of engine efficiency, thermal equilibrium is assumed between the heat source (at temperature T_H) and the engine during heat addition, as well as between the engine and the heat sink (at temperature T_L) during heat rejection. This means the temperatures of the engine's working fluid (T_H^* and T_L^*) match the respective temperatures of the heat source and sink.

However, in real-world applications, maintaining exact thermal equilibrium is not feasible. To enable efficient heat transfer, a temperature difference must be maintained. The formula for the efficiency of an ideal Carnot engine, \text{\(\eta_{C}=1-T_{L} / T_{H}\)}, would normally not account for these actual conditions, hence the need to consider other factors such as heat transfer rates and external irreversibility when analyzing real engines.

External Irreversibility and Its Impact on Engine Performance

External irreversibility in a thermodynamic system refers to the energy losses due to a finite temperature difference during heat transfer between the system and its surroundings. It contrasts with internal irreversibility, which arises from friction, turbulence, and other factors within the system's boundary. For a Carnot engine, the assumption of no external irreversibility implies perfect, reversible processes. Nonetheless, in real engines, irreversibility is inevitable and degrades efficiency.

External irreversibility is introduced when there are actual temperature differences between the heat reservoirs and the working fluid, indicated as T_H - T_H^* and T_L^* - T_L. These differences drive the heat transfer necessary for the engine to operate. Optimizing these temperature disparities is crucial for maximizing power output, as illustrated by the exercise, which demonstrates that the net power output is maximized under a specific ratio relating these temperature differences, a balance between desired efficiency and necessary irreversibility.

How Heat Transfer Rate Affects Carnot Engine Performance

The rate of heat transfer is a vital parameter in thermodynamics, affecting how quickly a system, such as a steam engine or refrigeration cycle, can perform work or remove heat. It can be defined using the formula \text{\(\dot{Q}=hA(T_{source}-T_{system})\)}, where h is the heat transfer coefficient, A is the heat transfer area, and the temperatures T_{source} and T_{system} refer to the heat source/sink and the system, respectively.

In the context of Carnot engines, heat transfer rates for the hot (\text{\(\dot{Q}_{H}\)}) and cold (\text{\(\dot{Q}_{L}\)}) reservoirs are crucial. These rates must be substantial enough to allow for the necessary energy transfer across finite surface areas but balanced so as not to introduce excessive external irreversibility. The textbook exercise suggests that the engine's power output can be optimized by carefully managing the relationship between these heat transfer rates and the temperatures of the engine's components. The sought-after balance ensures the maximum net power output, reflecting the intricate interplay between efficiency and practical limitations in heat transfer processes.