Question

A heat engine that pumps water out of an underground mine accepts \(700 \mathrm{kJ}\) of heat and produces \(250 \mathrm{kJ}\) of work. How much heat does it reject, in kJ?

Short answer

Answer: The heat engine rejects 450 kJ of heat.

Step-by-step solution

Write the energy conservation equation for a heat engine

The energy conservation equation for a heat engine states that the total energy input (Q_in) must equal the total energy output (W_out + Q_out). Mathematically, this is expressed as: Q_in = W_out + Q_out

Substitute the given values in the equation

We are given that the heat input (Q_in) is 700 kJ and the work output (W_out) is 250 kJ. We can substitute these values into the equation from Step 1: 700 = 250 + Q_out

Solve for the heat rejected (Q_out)

To find the heat rejected (Q_out), we can simply rearrange the equation from Step 2 and solve for Q_out: Q_out = Q_in - W_out Q_out = 700 - 250

Calculate the final answer

Calculate the heat rejected (Q_out) by subtracting the work output from the heat input: Q_out = 450 kJ So, the heat engine rejects 450 kJ of heat.

Key concepts

Energy Conservation in Thermodynamics

The principle of energy conservation is foundational in thermodynamics, asserting that energy cannot be created or destroyed, but it can change forms. This principle applies to heat engines, which are devices designed to convert heat energy into mechanical work. Let's delve into the energy dynamics of a typical heat engine process.

The engine receives a certain amount of heat energy, denoted as \( Q_{in} \), from a hot reservoir. Throughout the engine cycle, it then converts a portion of this heat into work (\( W_{out} \)). However, not all the input energy can be transformed into useful work due to inherent inefficiencies and the physical limitations dictated by the second law of thermodynamics. As a result, some of the energy is released back into a cooler reservoir as waste heat, denoted as \( Q_{out} \).

Understanding the Energy Flow

In clear terms, energy enters the system as heat, part of it does the actual intended work, like moving pistons or pumping water, and the remainder of the energy is expelled as less useful heat energy. The relationship between the incoming heat, work done, and waste heat is mathematically represented by the equation: \( Q_{in} = W_{out} + Q_{out} \). This equation is a concise statement of energy conservation, tailored for the function of heat engines.

Work Output in Heat Engines

In heat engines, work output represents the useful energy that is extracted from the heat energy provided. It's this work output that drives mechanical processes, such as pistons in an engine or, in the context of our exercise, the pumping of water from an underground mine.

The efficiency of a heat engine is a measure of its capability to convert the energy input (the absorbed heat) into the desired work output. This efficiency is crucial since it indicates how much of the heat energy is being effectively used and determines how much energy is essentially 'lost' as waste heat. Mathematically, the efficiency (\( \text{Eff} \)) can be calculated using the formula: \( \text{Eff} = \frac{W_{out}}{Q_{in}} \times 100\text{%} \).

Practical Implications

For example, if a heat engine takes in 700 kJ of heat and does 250 kJ of work, the work output is utilized to perform the engine's intended function. The rest of the heat is considered excess and is not part of the work output. Understanding the work output helps engineers and scientists design more efficient engines by reducing energy waste.

Heat Rejection Calculation

The calculation of heat rejection in a heat engine is tied up with the core concepts of energy balance and efficiency. Heat rejection is the amount of heat energy expelled by the heat engine into the cooler reservoir, often symbolized as \( Q_{out} \). It's an integral part of the heat engine cycle since it represents the non-usable energy that must be disposed of for the cycle to continue.

To calculate the heat rejection, one must know the initial heat input (\( Q_{in} \)) and the work output (\( W_{out} \)). As derived from our energy conservation equation, the heat rejected can be calculated using \( Q_{out} = Q_{in} - W_{out} \).

Real-World Application

In our textbook exercise, the engine accepts 700 kJ and produces 250 kJ of work. From this, we can easily compute the amount of heat rejected to be 450 kJ, indicating this is the energy not converted into work, either due to thermodynamic inefficiencies or limitations. This calculation is critical in assessing the performance of a heat engine and planning for the necessary cooling systems to manage the waste heat.