Solve Carnot Engine: Efficiency (T1-T3)/T1

In summary, the efficiency of a two stage Carnot engine is the amount of work done divided by the amount of heat flow into the system.
  • #1
LandOfStandar
61
0
[SOLVED] Carnot engine - please help asp

I just typed a whole question and as I typed it I solved it, lol!

Homework Statement



In the 1st stage of a 2-stage Carnot engine, energy is absorbed as heat Q1 at temp T1, work W1 is done, and energy is expelled as heat Q2 at lower temp T2. The 2nd stage absorbs that energy as heat Q2, does work W2 and expels energy as heat Q3 at a still lower temp T3. Prove that the efficiency of the engine is (T1 - T3)/T1

Stage 1 Qin=Q1
Qex=Q2
T2 less then T1
W1

Stage 2 Qin=Q2
Qex=Q3
T3 less then T2
W2

Homework Equations



e =Qex/W = (Qin-Qex)/Qin = (Tin-Tex)/Tin

The Attempt at a Solution



e(stage1) = (T1-T2)/T1
e(stage1) = (T2-T3)/T2

How do you put them together?

or

is this all wrong? how do I approach this?
 
Physics news on Phys.org
  • #2
LandOfStandar said:

The Attempt at a Solution



e(stage1) = (T1-T2)/T1
e(stage1) = (T2-T3)/T2

How do you put them together?

or

is this all wrong? how do I approach this?

Start with the definition of efficiency for each of the cycles and the combined cycle:

E1 = W1/Qh1
E2 = W2/Qh2

What is Ec in terms of W1, W2, Qh1 and Qh2?

What are Qh1 and Qh2 in terms of T1, T2, T3?

AM
 
  • #3
E1 = W1/Qh1 = (Qh1-Ql1)/Qh1 = (T1 - T2)/T1

what do you mean?

E2 = W2/Qh2 = (Qh2-Ql2)/Qh2 = (T2 - T3)/T2
 
  • #4
LandOfStandar said:
E1 = W1/Qh1 = (Qh1-Ql1)/Qh1 = (T1 - T2)/T1

what do you mean?

E2 = W2/Qh2 = (Qh2-Ql2)/Qh2 = (T2 - T3)/T2

So what is the overall efficiency in terms of the total work done (W1+W2) and the total heat flow into the system (Qh1+Qh2)?

AM
 
  • #5
(T1 -T2 + T2 - T3) / (T1 + T2) = (T1 - T3) / (T1 + T2)

the problem is the T2 on the bottom
 
  • #6
LandOfStandar said:
(T1 -T2 + T2 - T3) / (T1 + T2) = (T1 - T3) / (T1 + T2)

the problem is the T2 on the bottom

Since the heat flow into the system is just Q1 (the heat flow into the second engine is the output heat of the first):

[tex]\eta_{total} = (W_1 + W_2) / Q_1[/tex]

But W1 = Q1-Q2 and W2 = Q2-Q3, so

[tex]\eta_{total} = (Q_1-Q_3) / Q_1 = (T_1-T_3)/T_1[/tex]

AM
 
  • #7
thank you that makes since

I now understand the question
 

1. How does a Carnot engine work?

A Carnot engine works by using a reversible thermodynamic cycle to convert heat energy into mechanical work. It operates between two heat reservoirs, with one at a higher temperature (T1) and the other at a lower temperature (T2). The engine absorbs heat from the higher temperature reservoir, converts some of it into work, and then rejects the remaining heat to the lower temperature reservoir. This process is repeated in a continuous cycle to produce work.

2. What is the formula for calculating the efficiency of a Carnot engine?

The efficiency of a Carnot engine can be calculated using the formula (T1-T2)/T1, where T1 is the temperature of the higher reservoir and T2 is the temperature of the lower reservoir. This formula is based on the Carnot cycle, which is a theoretical maximum for the efficiency of a heat engine.

3. Why is the Carnot engine considered to be the most efficient heat engine?

The Carnot engine is considered to be the most efficient heat engine because it operates on a reversible thermodynamic cycle, which is the most efficient way to convert heat energy into work. This means that it has the highest possible efficiency for a given temperature difference between the two reservoirs. No other heat engine can match the efficiency of a Carnot engine.

4. How does the temperature difference affect the efficiency of a Carnot engine?

The efficiency of a Carnot engine is directly proportional to the temperature difference between the two reservoirs. This means that a larger temperature difference (T1-T2) will result in a higher efficiency. However, as the temperature difference approaches zero, the efficiency also approaches zero. This is because a very small temperature difference would require an infinite amount of time to complete a cycle, making it impossible to produce any work.

5. Are there any real-world applications of the Carnot engine?

While the Carnot engine itself is a theoretical concept, its principles and efficiency limits have been applied to real-world systems such as gas turbines, refrigerators, and heat pumps. These systems are not as efficient as a Carnot engine, but they are based on the same principles and can achieve high levels of efficiency when designed and operated correctly.

Similar threads

The maximum efficiency of two continuous processes
    • Introductory Physics Homework Help
Replies
1
Views
652
Open Gas Turbine - calculate T2, T3, T4, efficiency
    • Introductory Physics Homework Help
Replies
1
Views
968
How to show that Carnot engine is more efficient?
    • Introductory Physics Homework Help
Replies
1
Views
1K
Calculating work done by a Carnot engine
    • Introductory Physics Homework Help
Replies
14
Views
8K
Carnot engine efficiency problem
    • Introductory Physics Homework Help
Replies
3
Views
3K
Net heat in a thermodynamic cycle
    • Introductory Physics Homework Help
Replies
2
Views
698
Question about Carnot theorem proof
    • Introductory Physics Homework Help
Replies
5
Views
1K
Thermodynamics problem involving engine efficiency
    • Introductory Physics Homework Help
Replies
20
Views
2K
Work done by a Carnot Engine. Melting ice
    • Introductory Physics Homework Help
Replies
2
Views
2K
How Do Carnot's Efficiency Formulas for Heat Engines Align?
    • Thermodynamics
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top