Diesel Cycle Help: P,T @ Start & End Head Addition

[SF2K4]

Homework Statement

The pressure and temperature at the beginning of compression in an air-standard Diesel cycle are 95 kPa and 300K. At the end of the head addition process, the pressure is 7.2 MPa and 2150 K. Assume variable specific heats.

I'm not going to list the parts because all of the questions rely on knowing the V and T of State 2, which I cannot find.


2. Homework Equations

I know that P_2 = P_3 = 7.2 MPa. I also know that s_2 = s_1 = 1.70203 kJ/kg*K


3. The Attempt at a Solution

I have no idea. In order to do anything, find cycle ratios or thermal effiencies relating to the compression or head addition stage, I NEED to either know the Temperature of State 2 or the Volume or Volume Ratio of State 2, which I need the Temperature of to find it on the table, or both. Basically without the Temperature of State 2 I cannot finish this problem and I don't know what I'm missing.

 

[h2oski1326]

So for an ideal gas we can write an entropy relationship like the following:

$$ds = c_v\frac{dT}{T}-R\frac{dp}{p}$$

Integrating this yields:

$$s_2-s_1=\int_1^2c_v\frac{dT}{T}-Rln\frac{p2}{p1}$$

The trouble is $$\mbox{c_v}$$ is a function of temperature!

So, $$\mbox{\int_1^2c_v\frac{dT}{T}}$$ can be re-written as $$\mbox{s_o^2-s_o^1}$$. These quantities are tabulated for air so you should be able to bridge the gap between states 1 and 2 and find your temperature.

The rest of the cycle should just fall into place. Just remember you may need to utilize other independent equations, like the 1st law or the ideal gas law to get everything you need.
P.S. It is likely that a post like this should be put in HW help in the future.

 

[engware]

Hi there:

Here are two URLs on the Diesel Cycle operation.

Technical background information on Diesel Cycle
http://members.aol.com/engware/cycles.htm

Online calculator on Diesel Cycle operation
http://members.aol.com/engware/calc4.htm

The provided URLs should help you with Diesel Cycle technical background information and engineering calculations regarding the Diesel Cyucle operation and/or technical performance evaluation.

Here are two plots regarding the Diesel Cycle thermal efficiency and power output.

Thanks,

Gordan

 

[gwnch]

i want to ask a question about the cycle... What if we use Peng -Robinson Equation of state instead of ideal gas assumption for nitrogen for example? Calculations gets very complicated when i move into second to third and third to fourth steps...

 

[engware]

Hi there:

In terms of the thermal efficiency definition, nothing changes.

Give a try and see what happens.

In my opinion, there should be no major changes for low pressure values in terms of the operating conditions and final numeric values. However, as you indicated, the equations get complicated and the relationships for isentropic processes are no longer as simple and straight forward as in the case of ideal gas equation assumptions.

Therefore, my suggestion is to go with ideal gas assumptions, do the calculations and you know what it is for the ideal case conditions. One can always try to do different things theoretically and numerically. However, if you what to invest more time, experimental work needs to be done as well as experimental measurements.

In my opinion that would be the way to go -- do ideal case numerical calculations and some experimental work and show both sets of results.

Thanks,

Gordan