Polytropic exponent, thermodynamics question

[Studious_stud]

Homework Statement

Air is compressed to 6% of its initial volume in the cylinder of a diesel engine. In the process, its pressure increased from 1 bar to 41 bar.

Calculate the polytropic exponent for this process and comment on the associated heat transfer.

Homework Equations

PV^λ = constant

The Attempt at a Solution

Pinital (Vinital)^λ = Pfinal(Vfinal)^λ

Pinitial = 1 bar
P final = 41 bar
V initial = 100%
V final = 94%

Well I'm pretty much not sure if I'm going about solving this (probably simple) problem in the right way. Anyone care to help?

Also do you have to change the units from bar to pascals?

Thanks dudes and dudettes

 

[rude man]

Whew! "Polytropic exponent!" Ya live & you learn!

OK, so you know p1 and V1, giving you one equation with unknowns C and n:
p1V1^n = C
Then, what are p2(p1) and V2(V1)? Write the equation involving pp2 and V2 in therms of p1 and V1. That's a second equation with two unknowns C and n.

Solve pre h.s. algebra.

Now for part 2: if you assume an ideal gas, you can compute T1 and T2 (assume w.l.o.g. 1 mole of air), use the 1st law to express ΔU = (const.)ΔT, then work = ∫pdV and finally ΔQ = ΔU + W.

 

[Studious_stud]

Whew! "Polytropic exponent!" Ya live & you learn!

OK, so you know p1 and V1, giving you one equation with unknowns C and n:
p1V1^n = C
Then, what is p2(p1

I don't quite understand, do I know V1? :S

No volume is given in the question but can you just use percentage values instead?

So I'll have

P1V1^n = C

and

P2V2^n = C

?

Thanks by the way rude man!

 

[Q_Goest]

Hi Studios,

The Attempt at a Solution

Pinital (Vinital)^λ = Pfinal(Vfinal)^λ

Pinitial = 1 bar
P final = 41 bar
V initial = 100%
V final = 94%

You have the right equation to solve for the polytropic exponent, but you need to better define V_i and V_f. If "Air is compressed to 6% of its initial volume..." then can you write a very short equation that relates V_i and V_f? Something like V_i = V_f C where C is a constant? Once you do that, you can find initial and final conditions so you can solve for the polytropic exponent.

Do you know how the polytropic exponent varies depending on heat flux? For an adiabatic (no heat transfer) process, the process is isentropic, so what do you think the polytropic exponent would be equal to in that case? For the case where temperature is constant, PV = mRT = constant, right? In that case, what would be the polytropic exponent and how would you describe the heat transfer to the air during the process?

 

[rude man]

Sorry, my post was totally incomplete.

Q Goest is giving you good leads.

One thing that bothers me is I think we're all assuming an adiabatic process, which was not given but maybe was so defined in the relevant chapter of the textbook. I believe that assumption is generally made, so maybe I'm just being picky. For that matter we weren't given authority to assumean ideal gas either - but what can one accomplish without that?

I suppose if n computes to cp/cv for air then we're entitled to assume isentropy.