# Efficiency of a Heat Engine

## Homework Statement

A hypothetical engine, with an ideal gas as the working substance, operates on the cycle shown in Figure 1. Show that the efficiency of this engine is

$$e = 1 - \frac{1}{\gamma}\left(\frac{1 - \frac{p_3}{p_1}}{1 - \frac{v_3}{v_1}}\right)$$

Where $$\gamma = \frac{c_p}{c_v}$$

## Homework Equations

Efficiency = Benefit/Cost

Benefit = Work out

Cost = Heat in

PV = nRt

$$\Delta U = Q_{hot} - Q_{cold} - Work$$

Work = pdv

## The Attempt at a Solution

The Graph is attached, but basically it has a Adiabatic curve, which at the bottom goes up vertically, then left horizontally, back to make a cycle.

So far I have:

E = Work/Cost

Work = pdv

Cost = Q

$$w = pdv$$

Since p is not constant, use ideal law,

$$w = \frac{nRt}{v}dv$$

thus

$$w = \frac{nRt}{v}dv$$

$$w = nRt [ln(v)]^{v_1}_{v_2}$$

I also have:

$$e = 1 - \frac{Q_{cold}}{Q_{hot}}$$,

from

$$\Delta U = Q_{hot} - Q_{cold} - Work$$

Which is cylci and this delta u = 0.

Many thanks,

TFM

#### Attachments

• Heat Engine Graph.jpg
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Redbelly98
Staff Emeritus
Homework Helper
Efficiency = Benefit/Cost

Benefit = Work out

Cost = Heat in

PV = nRt

ΔU = Qhot - Qcold - Work
That's a good start. By calculating "Work out" and "Heat in", you'll be able to figure out the efficiency.

There's a problem with
$$w = nRt [ln(v)]^{v_1}_{v_2}$$

Note that T is not constant along path 31, so it must be treated as a function of V in doing the integral. Better yet forget T, keep things in terms of P. How are P and V related for an adiabatic process? (Hint: it involves γ)

Well, that would be:

$$PV^{\gamma} = Constant$$

Would you rearrange for p, and insert into:

work = pdv

Redbelly98
Staff Emeritus
Homework Helper
Yes. And another hint:
"the constant" = P1 V1γ or P3 V3γ (Choose either one)

Well, wouldn't it be:

$$p_1v_1 = p_3v_3$$

?

Redbelly98
Staff Emeritus
Homework Helper
No. That would be true for an isothermal process, since
PV = nRT = constant only if T=constant​

But this is an adiabatic process. It's not isothermal, so T changes.

I see, so I should rearrange like so,

$$pv^{\gamma} = C$$

$$p = \frac{C}{v^{\gamma}}$$

insert c

$$p = \frac{C}{v^{\gamma}}$$

insert into work:

$$Work = \frac{C}{v^{\gamma}}dv$$

Integrate:

$$Work = C\int v^{-\gamma}}dv$$

$$Work = C\left[ \frac{\gamma + 1}{v^{\gamma + 1}} \right]^{limits}$$

So does this look right?

Redbelly98
Staff Emeritus
Homework Helper
Integrate:

$$Work = C\int v^{-\gamma}}dv$$
Looks good up to here.

$$Work = C\left[ \frac{\gamma + 1}{v^{\gamma + 1}} \right]^{limits}$$

So does this look right?
Mmm, no, try that integral again.

Okay, the integral of x^a is (x^a+1) / (a + 1)

so if we say a = -gamma, x= v

$$\left[ \frac{v^{-\gamma} + 1}}{-\gamma + 1} \right]$$

$$C\left[ \frac{1}{(-\gamma + 1)v^{-\gamma + 1}} \right]$$

$$C = p_1V_1^{\gamma}$$

$$p_1V_1\left[ \frac{1}{(-\gamma + 1)v^{-\gamma + 1}} \right]$$

Does this look okay now?

Redbelly98
Staff Emeritus
Homework Helper
Uh, the algebra looks rather messy here. Can you double-check it and clean things up?

Hmm, can't see why they thing split in two. So:

{I'll use f for gamma, because gamma doesn't seem clear in latex}

$$C\int{v^{-f}}dv$$

$$C\left[\frac{v^{-f + 1}}{-f + 1}\right]$$

$$C\left[\frac{1}{v^{f + 1}(-f + 1)}\right]$$

$$C = pv^f$$

$$\left[\frac{pv^f}{v^{f + 1}(-f + 1)}\right]^{limit}_{limit}$$

Does this look okay?

Redbelly98
Staff Emeritus
Homework Helper
Two things:

1. in the denominator, the exponent should be f-1, not f+1

2. in the numerator, let's call those p1 and v1^f, since they are constants.

Otherwise it looks good.

okay so:

$$\left[\frac{p_1v_1^f}{v^{f - 1}(f - 1)}\right]^{limit}_{limit}$$

Better?

The contents inside the parentheses were correct before. You just had to change the exponent.

So:

$$\left[\frac{p_1v_1^f}{v^{f - 1}(f + 1)}\right]^{limit}_{limit}$$

Is this better?

Redbelly98
Staff Emeritus
Homework Helper
Compare that with post #11, and my comments in post #12. Does it look like you made only the changes I said, or did you change something else too? (Hint: I wouldn't ask that question if what you wrote were correct.)

As a side note, this problem involves a lot of algebraic manipulation. And there is more of that to come. You will have to be very careful with the algebra if you are going to get this one.

I'm willing to help with reasoning things out, etc., but when symbols mysteriously change, or +/- signs keep appearing/disappearing for no reason, it gets very tedious for anybody assisting you.

Please, please try to be careful with the algebra. Double check things yourself, please don't keep relying on us to catch these numerous little errors.

Okay from P11:

$$\left[\frac{pv^f}{v^{f + 1}(-f + 1)}\right]^{limit}_{limit}$$

So,

1. in the denominator, the exponent should be f-1, not f+1

$$\left[\frac{pv^f}{v^{f - 1}(-f + 1)}\right]^{limit}_{limit}$$

2. in the numerator, let's call those p1 and v1^f, since they are constants.

$$\left[\frac{p_1v_1^f}{v^{f - 1}(-f + 1)}\right]^{limit}_{limit}$$

Better?

Redbelly98
Staff Emeritus
Homework Helper
Yes, good.

Okay so:

$$Work = \left[\frac{p_1v_1^f}{v^{f - 1}(-f + 1)}\right]^{limit}_{limit}$$

So I need to put this back into:

$$\Delta U = Q_{hot} - Q_{cold} - Work$$

$$\Delta U = Q_{hot} - Q_{cold} - \left[\frac{p_1v_1^f}{v^{f - 1}(-f + 1)}\right]^{limit}_{limit}$$

Now for the cycle $$\Delta U = 0$$

$$0 = Q_{hot} - Q_{cold} - \left[\frac{p_1v_1^f}{v^{f - 1}(-f + 1)}\right]^{limit}_{limit}$$

Thus

$$\left[\frac{p_1v_1^f}{v^{f - 1}(-f + 1)}\right]^{limit}_{limit} = Q_{hot} - Q_{cold}$$

Is this okay?

Redbelly98
Staff Emeritus
Homework Helper
Let's come back to that expression later. In the interest of organizing our thoughts, I suggest making a table showing W and Q for each of the three paths:

Work:
W31 = ?
W12 = ?
W23 = ?

Heat:
Q31 = ?
Q12 = ?
Q23 = ?

Once you fill in that table it should be easier to figure out the total Work, and the Heat Input, and from there figure out the efficiency.

We'll come back to the W31 expression we've been working on for the last 7-or-so posts, but first:
On which paths, if any, is W or Q equal to 0? Fill in those parts of the chart first, and we'll go from there.

Okay so
The work will be equal to 0 when dv = 0. this is the vertical straight line, which is path 23

For adiabatic, the heat that enters or leaves the system is 0, so path 13 is 0

For path 12, the work is the area under the curve so is thus (V3 - V1)*(P1)

Work:
W31 = ?
W12 = (V3 - V1)*(P1)
W23 = 0

Heat:
Q31 = 0
Q12 = ?
Q23 = ?

Redbelly98
Staff Emeritus
Homework Helper
Good, we've got all the easy parts and now we have to figure out the rest.

Let's get back to calculating W31:
$$\left[\frac{p_1v_1^f}{v^{f - 1}(-f + 1)}\right]^{limit}_{limit}$$
That's right so far. Next put in the limits on that integral and work out what W31 is.

Okay so would the limits be v3 and v 1, thus:

$$\left[\frac{p_1v_1^f}{v^{f - 1}(-f + 1)}\right]^{v_3}_{v_1}$$

$$\left[\frac{p_1v_1^f}{v_3^{f - 1}(-f + 1)}\right] - \left[\frac{p_1v_1^f}{v_1^{f - 1}(-f + 1)}\right]$$

Redbelly98
Staff Emeritus
Homework Helper
Those are the right limits, but in the wrong order. To see why, note that path 31 starts at "3" and ends at "1". So the limits are

$$\left[ \text{...} \right] ^{v1}_{v3}$$

EDIT: including figure from 1st post: Okay, so:

$$\left[\frac{p_1v_1^f}{v_1^{f - 1}(-f + 1)}\right] - \left[\frac{p_1v_1^f}{v_3^{f - 1}(-f + 1)}\right]$$

Okay changed it. So now

Work:
W31 = $$\left[\frac{p_1v_1^f}{v_1^{f - 1}(-f + 1)}\right] - \left[\frac{p_1v_1^f}{v_3^{f - 1}(-f + 1)}\right]$$
W12 = (V3 - V1)*(P1)
W23 = 0

Heat:
Q31 = 0
Q12 = ?
Q23 = ?