Question

A heat engine receives heat from a heat source at \(1200^{\circ} \mathrm{C}\) and rejects heat to a heat \(\operatorname{sink}\) at \(50^{\circ} \mathrm{C}\). The heat engine does maximum work equal to \(500 \mathrm{kJ}\). Determine the heat supplied to the heat engine by the heat source and the heat rejected to the heat sink.

Short answer

Answer: The heat supplied by the heat source to the heat engine is 640.13 kJ, and the heat rejected to the heat sink is 140.13 kJ.

Step-by-step solution

Calculate the efficiency of the heat engine

To calculate the efficiency, we should first convert the given temperatures from Celsius to Kelvin. Then, using the Carnot efficiency formula, we can find the efficiency of the heat engine. The formula to convert Celsius to Kelvin is: \(K = ^{\circ}C + 273.15\) So, for the heat source temperature: \(T_H = 1200 + 273.15 = 1473.15 \mathrm{K}\) And for the heat sink temperature: \(T_C = 50 + 273.15 = 323.15 \mathrm{K}\) Now, we calculate the efficiency of the heat engine using Carnot's efficiency: \(\eta = 1 - \frac{T_C}{T_H} = 1 - \frac{323.15}{1473.15}\)

Compute the efficiency

Using the temperatures we found in Step 1, let's compute the efficiency: \(\eta = 1 - \frac{323.15}{1473.15} = 1 - 0.2191 = 0.7809\)

Find the heat supplied from the heat source to the heat engine

We are given the maximum work done by the heat engine, which is \(500 \mathrm{kJ}\). We can use the efficiency and the work to find the heat supplied (\(Q_H\)) from the heat source to the heat engine using the following formula: \(W = \eta \cdot Q_H\) Let's rearrange the formula to find \(Q_H\): \(Q_H = \frac{W}{\eta} = \frac{500}{0.7809} = 640.13 \mathrm{kJ}\)

Calculate the heat rejected to the heat sink

We can now find the heat rejected to the heat sink (\(Q_C\)) by using the heat supplied and the work done: \(Q_C = Q_H - W = 640.13 - 500 = 140.13 \mathrm{kJ}\)

Finalize and present the findings

Now, we can provide the answers for the heat supplied to the heat engine by the heat source and the heat rejected to the heat sink: 1. The heat supplied by the heat source to the heat engine is \(640.13 \mathrm{kJ}\). 2. The heat rejected to the heat sink is \(140.13 \mathrm{kJ}\).

Key concepts

Heat Engine

A heat engine is a fascinating piece of technology that turns thermal energy into mechanical work. In simpler terms, it's like a machine that can harness the power from heat and turn it into motion that we can use to do things like driving cars or generating electricity. How does it do this magic? Well, a heat engine operates on the principle of heat flowing from a hotter place (the heat source) to a cooler place (the heat sink), and in the process, some of that heat can be converted into work.

Think of it as a clever intermediary: the heat doesn't just lazily drift from hot to cold, the engine makes it do a little dance that pumps out work we can use! This concept is crucial to industries and technologies, from the internal combustion engines in vehicles to power stations that supply electricity to our homes and workplaces.

Temperature Conversion

Temperature conversion is a bit like a language translation for science – it's all about understanding 'how hot' in multiple ways. Just as you might speak in Fahrenheit, Celsius, or Kelvin, different scientific equations and principles require temperature to be stated in the most useful unit. To chat with the laws of thermodynamics, we usually use Kelvin, as it sets absolute zero, the chilliest possible temperature, as zero Kelvin (0 K).

Converting from Celsius to Kelvin is a walk in the park: simply add 273.15 to the Celsius value. So, if your tea is steeping at a nice 100 degrees Celsius, it's actually 373.15 Kelvin. This step is super important when calculating things like Carnot efficiency because the formulas work seamlessly with Kelvin.

Heat Transfer

Heat transfer is a bit like socializing energy style! It's the process of heat moving around, spreading out from where it's warm to where it's cooler. There are a few key players in the world of heat transfer: conduction, where heat passes through solid objects that are touching; convection, where heat swirls around with fluids like air or water; and radiation, where heat beams through empty spaces like sunshine. When we talk about heat engines, we're usually interested in how much heat is transferred into the engine to do work and then how much is let go – like passing a ball in a game. Efficient engines grab a lot of heat, turn most of it into work, and then don't let too much escape unused. Understanding heat transfer is not only essential to improve heat engines but also to everyday life, like when you're cooking or keeping your house warm.