How Do You Calculate the Initial Pressure in a Polyprotic Expansion Process?

[ttb90]

Polyprotic Expansion HELP!

Homework Statement

Nitrogen at 100°C and specific volume of 0.1846 m^3/kg undergoes a polyprotic expansion process such that pv^(1.2)=const and the final pressure is 100 kPa. A constant pressure compression is then performed to bring the volume back to 0.1846 m^3/kg. Finally a constant volume pressurisation occurs to bring the Nitrogen back to its initial pressure. Nitrogen is a perfect gas with R= 0.297 kJ/kg K and Cv=0.745 kJ/kg K.

i) Determine the initial pressure

ii) Determine T and v at the end of the polyprotic expansion

iii) Sketch 3 processes on a P-V diagram

iv) Calculate the net work done at the conclusion of all three processes

v) Calculate the net heat transfer at the conclusion of all three processes.

Homework Equations

p1v1^λ=p2v2^γ
w= Rx(T1-T2)/(n-1)

The Attempt at a Solution

So for the first section I am struggling to keep up with all the processes that are going on. My logic was since the first process is expanding as the volume increases the pressure must decrease therefore our expected initial pressure must be higher than 100 kPa. However I am not entirely sure on how to calculate it. I cannot use the ideal gas law as its not ideal to begin with and the pv^(1.2)=const has limited use as I need the final v for the expansion to be able to use it..


Can someone help me through this question.. I am really struggling with it. Thanks in advance.

 

[Simon Bridge]

What is a polyprotic expansion? What does it mean? (Sounds like what I'd call an adiabatic expansion.)

You are missing a couple of relations:
P(V/m)=RT for each corner on the graph.
Work = area under the PV diagram.
... W = (V₂-V₁)P ... (isobaric)
... W = 0 ... (isochoric)
... W = $K\frac{V_2^{-0.2}-V_1^{-0.2}}{-0.2}$ ... (adiabatic)
... ... K=PV^1.2

state1:
V₁/m=0.1846 m^3/kg
P₁=?
T₁=373K

... get the idea?

 

[ttb90]

Sorry you are right. The question states the term polytropic expansion which would mean adiabatic in this case. I think polytropic is a general case where n=γ is adiabatic , n=1 is isothermal and n=0 is isobaric.

When you say P(V/m)=RT for each corner on the graph, do you mean that because they are just points on the corners. So we can use that relationship and assume it is ideal. I initially used another form of the same relationship but with specific volume and did the following:

T=100+273.15=373.15 K
v(specific volume) = 0.1846, R (nitrogen)= 0.297

Pv=RT
p(initial) =RT/v = (0.297 x 10^3 x 373.15)/0.1846 = 600 kPa

When it asks for the T at the end of the expansion, I used this relationship below

(P1/P2)^((γ-1)/γ)=T2/T1

(600/100)^((1.2-1)/1.2)= T2/373.15

T2= 503.04 K

Would this be correct ?

Not sure how i would continue with the v calculation but this is what i have done so far.

 

[Simon Bridge]

When you say P(V/m)=RT for each corner on the graph, do you mean that because they are just points on the corners. So we can use that relationship and assume it is ideal. I initially used another form of the same relationship but with specific volume and did the following:

Well V/m is the specific volume - that's why I parenthesized it off - Nitrogen is not an ideal gas so it is a good idea to adjust R for this, correct. Point is you can get an equation of state. What level is this aimed at?

T=100+273.15=373.15 K
v(specific volume) = 0.1846, R (nitrogen)= 0.297

Pv=RT
p(initial) =RT/v = (0.297 x 10^3 x 373.15)/0.1846 = 600 kPa

looks like what I'd do in your place all right. Only I would have recorded the units for R - just for the record. (J/kg.K = kPa.m^2/kg.K)

When it asks for the T at the end of the expansion, I used this relationship below

(P1/P2)^((γ-1)/γ)=T2/T1

(600/100)^((1.2-1)/1.2)= T2/373.15

T2= 503.04 K

Would this be correct ?

If you looked up that relation - try deriving it.
The way to check processes and results is to see if they make sense in terms of the problem. Is the result consistent with what you know about adiabatic processes?
(Sooner or later you will have to deal with situation nobody knows the answer to - nobody to turn to to see if you are right - so you have to learn how to tell if you've done it right all by yourself while the problems are still known.)

Not sure how i would continue with the v calculation but this is what i have done so far.

Your graph usually tells you what to do next.
Usual process, list what you know for each corner.
Compare with the state equation.

(hint: you are given P, you have just calculated T, and you know R for Nitrogen.)

 

[DeeNos]

I would approach this problem by first understanding the concept of polyprotic expansion. Polyprotic expansion is a process in which a gas undergoes multiple expansions and compressions, with the final pressure and volume being equal to the initial pressure and volume. In this case, the gas is nitrogen and the process involves three steps: polyprotic expansion, constant pressure compression, and constant volume pressurization.

To answer the first question, we can use the equation p1v1^λ=p2v2^γ, where λ and γ are exponents that can be calculated using the specific heat ratio, γ = Cp/Cv. Since the final pressure is 100 kPa and the specific volume is 0.1846 m^3/kg, we can plug these values into the equation and solve for the initial pressure, p1. This will give us the initial pressure before the polyprotic expansion process begins.

For the second question, we need to use the ideal gas law, pV = nRT, to calculate the final temperature and volume at the end of the polyprotic expansion. Since we know the final pressure and the specific volume at the end of the expansion, we can solve for the final temperature and volume using the ideal gas law. This will give us the values for T and v at the end of the polyprotic expansion.

To sketch the three processes on a P-V diagram, we can plot the initial and final points for each process and connect them with a line. The polyprotic expansion will be a curved line since the pressure and volume are changing, while the constant pressure compression and constant volume pressurization will be straight lines.

To calculate the net work done at the conclusion of all three processes, we can use the equation w= Rx(T1-T2)/(n-1), where R is the gas constant and T1 and T2 are the initial and final temperatures, respectively. We can calculate the work done for each process and then add them together to get the net work done.

Finally, to calculate the net heat transfer at the conclusion of all three processes, we can use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. We can use the values of work done and the change in internal energy to calculate the net heat transfer.

In conclusion, understanding the concept of polyprot