# Otto Cycle Engine Homework: Total Work & Efficiency Calculation

• Recipi
In summary, a car with a six-cylinder Otto-cycle engine has a compression ratio of 10.6 and cylinder diameter of 82.5 mm. The piston moves 86.4 mm during the compression stroke and the initial air-fuel mixture pressure is 8.50 x 10^4 Pa with a temperature of 300K. Assuming 200 J of heat is added per cycle, the efficiency of the engine is used to calculate the total work done and heat released when the gas is cooled to outside temperature. The volume of the air-fuel mixture at point a is calculated using the ideal gas equation. The pressure, volume, and temperature of the gas at points b, c, and d are also calculated and can
Recipi

## Homework Statement

"A car has a six-cylinder Otto-cycle engine with compression ratio r = 10.6.

The diameter of each cylinder is 82.5 mm.

The distance that the piston moves during the compression stroke (see fig. 1) is 86.4 mm.

The initial pressure of the air-fuel mixture (at point a in fig. 2) is 8.50 x 10^4 Pa and the initial temperature is 300K (the same as the outside air).

Assume that 200 J of heat is added to each cylinder in each cycle by the burning petrol and that the gas has CV = 20.5 J.mol/K and γ = 1.40."

(a) By considering the efficiency of the engine, calculate
(i) the total work done in one cycle in each cylinder of the engine, and
(ii) the heat released when the gas is cooled to the temperature of the air outside.

(b) Calculate the volume of the air-fuel mixture at point a in the cycle.

(c) Calculate the pressure, volume, and the temperature of the gas at points b, c, and d in the cycle. In a pV-diagram, show numerical values of p, V and T for each of the four states.

(d) Compare the efficiency of this engine with the efficiency of a Carnot-cycle engine operating between the same maximum and minimum temperatures

## The Attempt at a Solution

(a)(i) has me completely baffled. I understand that the work is the area bounded by the two adiabats and the vertical isochors, but I don't see how this is related to the efficiency if η = 1 - r1-γ = 1 - (|QC| / QH).

That relation gives me a value for (a)(ii) of |QC| = QH*r1-γ - 200*10.6-0.4 = 77.8 J, though; is this on the right lines?

(b) I tried to use the relationship for an adiabatic process TVγ-1 = constant, so:
Ta(rV)γ-1 = TbVγ-1
However, the 'V' terms just cancel here. I then considered the ideal gas equation V=nRT/P, but there is no value for the number of moles of working substance.

(c) I think I will be able to do on my own once I am pointed in the right direction for (b); right now I feel like I am missing information I need to be able to do the question, but filling in the first few gaps should help me enough.

(d) Again, once I actually have the value the maximum temperature I can use the Carnot efficiency η = 1 - TC/TH to compare the value of η ≈ 0.611 for this Otto cycle. I suspect it should be lower given the nature of the Carnot engine?

My biggest problem with thermodynamics at the moment is a massive unfamiliarity with many of the key relations between variables; I have a nagging feeling I'm either overlooking the obvious or am unable to find the relationship that would make the questions tractable in my notes.

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I've had more of a look at (a)(i), and I've gotten a bit further, although unless I can use the efficiency to get past the problem I've encountered in my last line, I'm not sure I've done it the way the question wants (if it's right at all)?

The issue now seems to be finding Td?

Edit: Okay, I think I got the efficiency into the picture, but the value I come out with seems ridiculous; 1320J of work for 200J heat input seems all kinds of wrong?

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## 1. What is an Otto Cycle Engine?

An Otto Cycle Engine is a type of internal combustion engine that operates on the principle of a four-stroke cycle, consisting of an intake stroke, compression stroke, power stroke, and exhaust stroke. It is commonly used in automobiles and other vehicles.

## 2. How is the total work of an Otto Cycle Engine calculated?

The total work of an Otto Cycle Engine is calculated by multiplying the pressure and volume change during each stroke and summing them up for all four strokes. The formula for total work is W = P(V2-V1).

## 3. What is the efficiency of an Otto Cycle Engine?

The efficiency of an Otto Cycle Engine is the ratio of the net work output to the total heat input. It can be calculated using the formula efficiency = (Wnet/Qin) * 100%, where Wnet is the net work output and Qin is the total heat input.

## 4. What factors affect the efficiency of an Otto Cycle Engine?

The efficiency of an Otto Cycle Engine is affected by factors such as the compression ratio, air-fuel ratio, ignition timing, and engine design. Higher compression ratios, proper air-fuel mixture, and optimal ignition timing can result in higher efficiency.

## 5. How can the efficiency of an Otto Cycle Engine be improved?

The efficiency of an Otto Cycle Engine can be improved by implementing technologies such as turbocharging, variable valve timing, and direct fuel injection. Regular maintenance and tuning of the engine can also help improve its efficiency.

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