Question

(a) In a two-stage Carnot heat engine, a quantity of heat $\left|Q_1\right|$ is absorbed at a temperature $T_1$, work $\left|W_1\right|$ is done, and a quantity of heat $\left|Q_2\right|$ is expelled at a lower temperature $T_2$, by the first stage. The second stage absorbs the heat expelled by the first, does work $\left|W_2\right|$, and expels a quantity of heat $\left|Q_3\right|$ at a lower temperature $T_3$. Prove that the efficiency of the combination is $(T_1-T_3)/T_1.(b)$ A combination mercury-steam turbine takes saturated mercury vapor from a boiler at ${469}^{\circ }\mathrm{C}$ and exhausts it to heat a steam boiler at ${238}^{\circ }\mathrm{C}$. The steam turbine receives steam at this temperature and exhausts it to a condenser at ${37.8}^{\circ }\mathrm{C}$. Calculate the maximum efficiency of the combination.

Short answer

The efficiency of a two-stage Carnot heat engine is given by $(T_1-T_3)/T_1$, and the maximum efficiency of the combination mercury-steam turbine is calculated using this equation.

Step-by-step solution

Understand Carnot's theorem and efficiency

Carnot's theorem states that no heat engine can be more efficient than a Carnot engine (an ideal reversible heat engine) operating between the same temperatures. The efficiency of a heat engine is defined as the ratio of the work done to the heat absorbed - $\eta =\frac{W}{Q_{\text{absorbed}}}$. For a Carnot engine, this can also be stated in terms of the temperatures of the hot and cold reservoirs - ${\eta }_{\text{Carnot}}=1-\frac{T_{\text{cold}}}{T_{\text{hot}}}$.

Calculate the efficiency of a two-stage Carnot heat engine

Considering the first part of this problem where a two-stage Carnot engine is described. In the first stage, the engine absorbs heat $Q_1$ at temperature $T_1$, does work $W_1$ and expels heat $Q_2$ at a lower temperature $T_2$. By the definition of efficiency, the efficiency of this stage is: $\frac{W_1}{Q_1}=1-\frac{T_2}{T_1}$. For the second stage, it absorbs the heat expelled by the first stage $Q_2$ at temperature $T_2$, does work $W_2$, and expels a quantity of heat $Q_3$ at a still lower temperature $T_3$; the efficiency of this second stage can be given by: $\frac{W_2}{Q_2}=1-\frac{T_3}{T_2}$. From the first law of thermodynamics, $Q=W+Q^{\prime }$, where $Q$ is the absorbed heat, $W$ is the work done, and $Q^{\prime }$ is the expelled heat, we can express $W_2$ in terms of $Q_1$, $Q_3$, and $T_3$ as follows: $W_2=Q_2-Q_3=Q_1-Q_3$. Substituting this into the second stage efficiency gives: $\frac{Q_1-Q_3}{Q_1}=1-\frac{T_3}{T_2}$, which simplifies to $\frac{Q_3}{Q_1}=\frac{T_3}{T_2}$. Since $T_2=T_1$, we can simplify this further to obtain $Q_3=Q_1\times \frac{T_3}{T_1}$. Thus, the total work done $W$ is $Q_1-Q_3=Q_1-Q_1\times \frac{T_3}{T_1}=Q_1\times (1-\frac{T_3}{T_1})$, and therefore the efficiency of the combination, $\eta =\frac{W}{Q_{\text{absorbed}}}=\frac{Q_1(1-\frac{T_3}{T_1})}{Q_1}=1-\frac{T_3}{T_1}=\frac{T_1-T_3}{T_1}$. Hence we've proved the given equation for the efficiency of a two-stage Carnot heat engine.

Calculate the efficiency of a combination mercury-steam turbine

Now, calculate the maximum efficiency of a combination mercury-steam turbine using the efficiency formula obtained in Step 2. We convert all the temperatures to the Kelvin scale (as thermal efficiencies must be calculated in absolute temperature) where $T_1={469}^{\circ }\mathrm{C}=469+273.15=742.15\mathrm{K}$, $T_2={238}^{\circ }\mathrm{C}=238+273.15=511.15\mathrm{K}$, $T_3={37.8}^{\circ }\mathrm{C}=37.8+273.15=310.95\mathrm{K}$. Now, by substituting these values into the efficiency formula, we get $\eta =\frac{T_1-T_3}{T_1}=\frac{742.15-310.95}{742.15}$. Calculating this number will give the maximum efficiency of the combination mercury-steam turbine.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that studies the relationships between heat, work, temperature, and energy. It provides the fundamental principles that govern all heat engines, including the Carnot engine. A key concept in thermodynamics is the idea of the thermodynamic cycle. This cycle is used in heat engines to transform thermal energy into mechanical work. The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed. It can only be changed from one form to another. This law is essential for understanding how heat engines operate.
In the context of the Carnot heat engine, thermodynamics helps us to understand how energy is transferred between the system's different components, such as the hot reservoir, cold reservoir, and working substance.

• Heat: Energy in transit due to temperature differences.
• Work: Energy transferred by a process that results in motion.
• Temperature: A measure of the average kinetic energy of the particles in a system.

Efficiency

Efficiency in thermodynamics, specifically for heat engines, refers to the engine's ability to convert absorbed heat into useful work. For a Carnot engine, efficiency is determined by the temperatures of the hot and cold reservoirs. This is expressed by the formula:
$$\eta =1-\frac{T_{\text{cold}}}{T_{\text{hot}}}$$
This formula highlights that the efficiency depends entirely on the absolute temperatures (measured in Kelvin). The larger the difference between the hot and cold reservoir temperatures, the higher the efficiency.

• A Carnot engine is an idealized engine that operates on a reversible cycle and achieves maximum efficiency.
• The efficiency is always less than 100% due to practical limits, such as energy losses.
• Engineers use this efficiency concept to design more effective and energy-saving engines.

Understanding efficiency includes acknowledging the inherent limits imposed by thermodynamics, reminding us that energy transformations are not perfect.

Temperature Conversion

Temperature conversion is critical when calculating the efficiency of heat engines, as thermodynamic equations require temperatures to be in absolute units, i.e., Kelvin. The Kelvin scale is an absolute temperature scale where 0 Kelvin (K) is the point of absolute zero, the theoretical point where molecular motion ceases.
To convert temperatures from Celsius to Kelvin, you simply add 273.15 to the degrees Celsius.

• For example, to convert 469°C to Kelvin:  $$T_{\text{K}}=469+273.15=742.15\text{K}$$
• The same calculation applies to other temperatures, ensuring all thermodynamic calculations are consistent with thermodynamic principles.

In the context of the given exercise, converting all temperatures to Kelvin ensures that calculations of engine efficiency are accurate.

Turbine

A turbine is a device that extracts energy from a fluid flow and converts it into useful work. Turbines are commonly used in power generation and propulsion systems, making them crucial components in many heat engines.
The efficiency and operation of a turbine depend on how well it can convert the thermal energy of steam or gas into mechanical energy. Turbines work based on thermodynamic principles, using the high-pressure steam or gas to rotate blades around a shaft, thus performing mechanical work.

• In a steam turbine, for instance, steam enters the turbine at high temperature and pressure, expands through the turbine, and leaves at a lower temperature and pressure.
• The expansion causes the turbine to spin, generating power that can be used for electrical generation or mechanical work.
• Turbines are key to maximizing the efficiency of thermodynamic cycles when integrated into systems like steam or gas power plants.

A deeper understanding of turbines' functioning and properties enables the designing of more efficient and powerful engines in both industrial and domestic applications.