An engineer redesigns a heat engine and improves its thermal...

An engineer redesigns a heat engine and improves its thermal efficiency from $20 \%$ to $25 \% .$ (a) Does the ratio of the heat input to heat output (1) increase, (2) remain the same, or (3) decrease? Explain. (b) What is the engine's change in $Q_{\mathrm{h}} / Q_{\mathrm{c}}$ ?

Key Concepts

Thermal Efficiency

Thermal efficiency is a measure of a heat engine’s ability to convert the heat input into useful work. It is defined as the ratio of the work output (W) to the heat input (Q?), often expressed as ? = W/Q?. Because some of the input heat is lost as waste heat (Qc), thermal efficiency can also be written as ? = 1 - Qc/Q?. An improvement in efficiency indicates that less heat is wasted and more of the input energy is being converted into work.

Relationship Between Heat Input and Waste Heat

The relationship between the heat input (Q?) and the waste heat (Qc) is critical in understanding engine performance. Since efficiency can be expressed as ? = 1 - Qc/Q?, it follows that the ratio Q?/Qc = 1/(1 - ?). An increase in efficiency leads to a decrease in Qc relative to Q?, and consequently, an increase in the ratio Q?/Qc. This relationship quantitatively expresses how improvements in engine efficiency yield a higher proportion of the input heat being utilized for work.

Transcript

00:01 Here we examine what improving the efficiency of a heat engine does.

00:07 So a reminder that a heat engine makes use of the natural flow of energy from a hotter reservoir to a colder reservoir.

00:16 So some of the flow of heat can be used for useful work.

00:23 The general expression for the efficiency of any heat engine is the amount of work, that you get out of it divided by the amount of heat absorbed from the high -temperature reservoir.

00:41 And what the qc represents is heat expelled, so that is considered to be waste heat.

00:51 And what we would like to see happen with efficiency improving is that your amount of waste gets lowered.

01:01 And let's see if that's true.

01:05 Now, this expression for efficiency isn't alone very helpful because we don't know whether the efficiency getting bigger is due to less heat being absorbed or more work coming out of the system.

01:21 So we will need to use the first law of thermodynamics to kind of tease out how this all works.

01:29 First law.

01:32 Not first of, first law of thermo, which is basically energy conservation.

01:43 There's no internal energy because we're representing a cycle and there's no change in that in an entire cycle.

01:54 But the difference between the heat absorbed and the heat expelled with an absolute value, is equal to the work done.

02:07 So that's energy conservation.

02:10 And we can eliminate the work using our efficiency equation.

02:20 So that is just equal to qh times efficiency...

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