Diesel Heat Engine Cycle: Work Calculation Homework | P-V Diagram and Equations

In summary, the problem involves calculating the work done in a heat engine cycle for a Diesel engine using given pressure and volume values. The work done is calculated for each process using the appropriate equations for isobaric, isothermal, and polytropic processes. The compression and expansion index can be calculated using the given ratio of specific heats. It is important to double check calculations and ensure consistent units throughout.
  • #1
EngNoob
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Homework Statement



Engine data for a heat engine cycle, ie: Diesel.

State ...|...1...|...2...|...3...|...4...|
Pressure(Pa) | 1x10^5 | 11.764x10^5 | 11.764x10^5 | 2.207x10^5 |

Volume(m^3) |. 0.826 .|... 0.142 ...|... 0.25 ...|... 0.826 ...|

The gas has a compression and expansion index of 1.4,

determine the work done in (kJ) in the cycle.


Homework Equations



I am not sure. But to have an attempt i would say.

Constant Volume: W = 0

Constant Pressure: W = P(V_2 - V_1)

Constant Tempreture: W = P_2 * V_2 * ln (V_2/V_1)


The Attempt at a Solution



I have done a P-v diagram, and came up with.

1-2: Istothermal (Constant Tempreture)
2-3: Isobaric (Constant Pressure)
3-4: Isothermal (Constant Tempreture)
4-1: Ischoric (Constant Volume)

I am unsure about

1-2 and 3-4 as it could be a polypropic process (Temp/Press/Vol Change) but i think this has to do with the exp/comp index? but not sure how to figure out?

I came up with

-30431.859(kJ)

I think this is wrong, as this would indicate mins work.

Do i need to convert PA to KPA or MPA?

I would welcome any help, advice, or some hints/tips.
 
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  • #2



Hello there,

Thank you for sharing your attempt at a solution. It seems like you are on the right track with your understanding of the different types of processes (isothermal, isobaric, etc.) and the equations to calculate work done in each one.

To calculate the work done in each process, you will need to use the appropriate equation and plug in the given values for pressure and volume. For example, for the isobaric process (2-3), you can use the equation W = P(V2 - V1) where P is the given pressure (11.764x10^5 Pa) and V2 and V1 are the given volumes (0.142 m^3 and 0.25 m^3, respectively). Make sure to convert the pressure to the same units as the volume (either both in Pa or both in kPa, for example).

For the isothermal processes (1-2 and 3-4), you will need to use the equation W = P2 * V2 * ln(V2/V1). Again, make sure to convert the pressure to the same units as the volume before plugging in the values.

As for the compression and expansion index, you are correct in thinking that it has to do with the polytropic process. The compression index (n) is related to the ratio of specific heats (k) by the equation n = 1/(k-1). So for this problem, you can use the given value of k=1.4 to calculate the compression index. Similarly, the expansion index can be calculated using the same equation and the given value of k.

I hope this helps guide you in the right direction. Keep in mind that it's always a good idea to double check your calculations and make sure your units are consistent throughout. Good luck!
 
  • #3


I would suggest using the ideal gas law to calculate the work done in the cycle. The ideal gas law states that PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature. Since the gas in this cycle has a compression and expansion index of 1.4, we can assume that it follows the polytropic process. Therefore, we can use the equation W = (P_2V_2 - P_1V_1) / (1-n), where P_1 and V_1 are the initial pressure and volume, and P_2 and V_2 are the final pressure and volume. To convert from Pa to kPa or MPa, you can simply divide the values by 1000 or 1,000,000, respectively.

For the given data, the work done in the cycle would be:

W = ((11.764x10^5 kPa * 0.142 m^3) - (1x10^5 kPa * 0.826 m^3)) / (1-1.4) + ((2.207x10^5 kPa * 0.826 m^3) - (11.764x10^5 kPa * 0.25 m^3)) / (1-1.4)

W = -33976.3 kJ

This value is negative because the work done on the gas in the compression process is greater than the work done by the gas in the expansion process. This indicates that the cycle is not very efficient and some energy is lost in the process. You can also check your answer by calculating the net work done in the cycle using the area under the P-v diagram. The net work should be equal to the value calculated using the ideal gas law.

I hope this helps and good luck with your homework!
 

1. What is the Diesel Heat Engine Cycle?

The Diesel Heat Engine Cycle is a thermodynamic cycle that describes the process of how a diesel engine converts heat into work. It follows a four-stroke process of intake, compression, power, and exhaust.

2. How is work calculated in the Diesel Heat Engine Cycle?

Work is calculated by finding the area under the P-V (pressure-volume) diagram of the cycle. This can be done by using the mathematical formula for work, which is work = force x distance.

3. What is the significance of the P-V diagram in the Diesel Heat Engine Cycle?

The P-V diagram helps to visualize and analyze the thermodynamic processes that occur in the diesel engine. It shows the relationship between pressure and volume, and the area under the curve represents the work done by the engine.

4. What equations are used in the Diesel Heat Engine Cycle?

The ideal gas equation, also known as the Boyle's law, is used to calculate the pressure and volume at each stage of the cycle. The ideal gas law, which includes temperature and moles of gas, is also used to calculate the work done by the engine.

5. How does the efficiency of the Diesel Heat Engine Cycle compare to other heat engines?

The Diesel Heat Engine Cycle has a higher efficiency compared to other heat engines, such as gasoline engines, due to its higher compression ratio and lower heat loss. However, it may have lower efficiency compared to other advanced heat engines, such as gas turbines.

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