Adiabatic compressed air and energy calculations

[Akaramos45]

Homework Statement

. Consider a pump that is required to compress air in a factory. The cylinder in the pump has an inner diameter of 2.00 cm and length 60.0 cm. Air is drawn into the pump at atmospheric pressure and 18 degrees celcius and the pump adiabatically compresses the air to a pressure of 17 atmospheres.

(a) Calculate the volume and temperature of the compressed air.

(b) What is the increase in internal energy of the gas during the compression?

(c) Suppose that the air in the pump is compressed once and the compressed air and the steel of the cylinder on the pump the come into thermal equilibrium. If the cylinder is 1.00 mm thick, what will be the increase in temperature of the steel after one compression?

How should i go about solving this question? I don't fully understand thermodynamics. I would love it if you just gave me a worked example. I find it easier to understand since my English is not very good.

Homework Equations

P1V1gamma =P2v2gamma

The Attempt at a Solution

I calculated the volume to be 3.16 me but that's wrong according to the answer and I'm quite stuck on how to calculate energy

 

[haruspex]

I calculated the volume to be 3.16 me but that's wrong

I'm trying to see your working so that I can find your mistake, but my crystal ball doesn't seem to have the necessary range.

 

[Akaramos45]

I am sorry?.

 

[Akaramos45]

Do you know how to solve this problem?

 

[haruspex]

I am sorry?.

You wrote that you got the wrong answer for the volume. My task, as a homework helper, is to find where you went wrong. But if you do not post your working I cannot possibly find your error.
Please post your working (as typed text, not as an image).

 

[Akaramos45]

I'm sorry. Here is my working.
I used PiVi^gamma = PiVi^gamma.
In this case gamma = 7/5 as air is a diatomic molecule therefore Cp/Cv =7/2R divided by 5/2R = 1.4
Vi = Pi*(0.01)^2* 0.6 = 1.885x10(-4) m^3
I reformulated it to be
Vf = Vi*(Pi/Pf)^1.4
Vf 3.57x10(-6) m^3

 

[haruspex]

Vi = Pi*(0.01)^2* 0.6

Risk of confusion between π and P_i. Use the buttons above the text area, X₂ for subscript, Σ for special characters.

I reformulated it to be
Vf = Vi*(Pi/Pf)^1.4

There's your mistake. Try that reformulation again.

 

[Akaramos45]

I calculated the temperature using TiVi^(gamma -1) = TfVf^(gamma -1)
Basically it's the same method as before.
Tf = Ti*(Vi/Vf)^2/5
Tf = (18 +273.15)(1.885x10(-4)/3.57x10(-6))^2/5
Tf = 291.15(4.887)
Tf = 1422.89 K
This doesn't seem right. I think it's because of turning 18 Degrees to Kelvin?
If i didn't it becomes 52.8 K which still doesn't sound right

 

[Akaramos45]

_Vi = _Pi*_Vi / ^Pgamma

 

[Akaramos45]

V_f= P_i*V_i^gamma/Pf^gamma

 

[Akaramos45]

I am sorry. They are all wrong you cannot swap the position of the power therefore
V_f = square root by gamma(P_i*V_i/P_f)
This gives me an answer 2.88x10-4

 

[haruspex]

V_f = square root by gamma(P_i*V_i/P_f)

Still no.
Starting with P_fV_f^γ=P_iV_i^γ, take it one step at a time and post all the steps.

 

[Akaramos45]

V_f^γ = P_i*V_i^γ/P_i
V_f = square root by γ (P_i*V_i^γ/P_i)

 

[haruspex]

V_f^γ = P_i*V_i^γ/P_i
V_f = square root by γ (P_i*V_i^γ/P_i)

Right, but "square root" is the wrong terminology. You could write it as V_f=V_i(P_f/P_i)^1/γ, or in LaTeX, $V_f=V_i\left(\frac{P_i}{P_f}\right)^{\frac 1{\gamma}}$.
What numerical answer do you get now? If you still get 2.88x10-4, please post your working.

 

[Akaramos45]

V_f = 2.49 x10(-4) m^3

 

[haruspex]

V_f = 2.49 x10(-4) m^3

Hmmm... I get the same except 10^-5, not 10^-4.

 

[Akaramos45]

oh yes sorry that is a typo. I get -5 also.

 

[Akaramos45]

With this i calculated the energy to be 654 K. Which i think is right.
I calculated this using T_f = T_i(1/17)^2/5

 

[haruspex]

With this i calculated the energy to be 654 K. Which i think is right.
I calculated this using T_f = T_i(1/17)^2/5

I agree with the number but not the algebra. I think you would have used T_f = T_i(17)^2/7.

 

[Akaramos45]

I'm sorry I do not understand how you derived it to that from:
T_iV_i^γ-1 = T_fV_f^γ-1

 

[Akaramos45]

It's fine I understand now.
For part b). I think you have to calculate the number of moles initially using the idea gas law
PV = nRT
where n = PV/RT
n = 1.013x10(5)*1.885x10(-4)/8.314*291.15
n = 19.1/2421
n =7.89x10(-3) moles
This is used in the equation ΔE = 2.5 nR ΔT:
ΔE = 2.5*7.89x10(-3)*8.314*(654 - 291.15)
ΔE = 59.5 Joules of Energy.
Therefore, there is an increase of 59.5 Joules of energy.
However, I do not understand the third question at all. I know I need to make some assumptions(This was a hint the teacher gave me.)

 

[haruspex]

I'm sorry I do not understand how you derived it to that from:
T_iV_i^γ-1 = T_fV_f^γ-1

I didn't derive it from that. The 17 is the ratio of pressures, not volumes. I derived it from T^γP^1-γ is constant.
How did you get

T_f = T_i(1/17)^2/5

?
That would give a final temperature below the starting temperature.

 

[Akaramos45]

yes sorry i typed it wrong what i meant was:
TiViγ-1 = TfVfγ-1
T_f = T_f(Vi/Vf)
to find Tf

 

[Akaramos45]

I do not understand how to do part c). Could you please help me with that?

 

[haruspex]

Tf = Tf(Vi/Vf)

It is hard to follow your work if you keep making typos. Please check it before posting.
Did you mean T_f=T_i(V_f/V_i)^1-γ?

 

[Akaramos45]

Yes.
It seems our formulas are different. I mean the structure of them. But either ways they give the same answer.

 

[Akaramos45]

Can we please do part C?. I know this is very bad and irritating for you. I am sorry for the trouble.

 

[haruspex]

Can we please do part C?. I know this is very bad and irritating for you. I am sorry for the trouble.

No, it's fine, I'm just trying to get you to be clearer and more careful in writing out your work, as much for your benefit as mine.

For c, you need to know the specific heat of the steel.
If the final temperature is θ, what is the gain in internal energy of the cylinder and what is the internal energy loss of the gas?

 

[Akaramos45]

The specific heat of steel is 448 Kg-1 K-1. The density of steel is 7900 Kg m-3. I'm sorry I do not understand what you are asking

 

[Akaramos45]

Doesn't the gas gain energy during the compression due to the work done?

 

[Akaramos45]

The teacher also said I could assume that the final temperature is close to the initial temperature, so that all the of the energy in the hot gas is transferred to the steel

 

[haruspex]

Doesn't the gas gain energy during the compression due to the work done?

Yes, but we've done that part. We are now interested in what happens as heat from the compressed gas is transferred to the steel.

 

[Akaramos45]

The steel and the compressed gas come into thermal equilibrium. Does that mean due to a difference in temperature heat transfer occurs?. If so, How do I calculate the increase in the the temperature of the steel if i do not know the initial temperature of the steel

 

[haruspex]

The teacher also said I could assume that the final temperature is close to the initial temperature, so that all the of the energy in the hot gas is transferred to the steel

Ok. I would have said it a bit differently... The specific heat of the gas is very small compared to that of the cylinder, so nearly all the gained internal energy will finish up in the steel.

The steel and the compressed gas come into thermal equilibrium. Does that mean due to a difference in temperature heat transfer occurs?

Yes. The steel will heat up and the gas cool down until they reach the same temperature. It does not say so, but you need to ignore any heat loss from the cylinder to the surroundings.

How do I calculate the increase in the the temperature of the steel if i do not know the initial temperature of the steel

You can assume it started at the same temperature as the gas did, 18C.

 

[Akaramos45]

I see. How do I calculate the increase in temperature of the steel. What formulas would I use?.
I am sorry this is my first doing this sort of problem. Could you show me an example?

 

[haruspex]

I see. How do I calculate the increase in temperature of the steel. What formulas would I use?.
I am sorry this is my first doing this sort of problem. Could you show me an example?

You know the heat energy going into it, its volume, density and specific heat. If you don't know an equation connecting those with temperature change, look up specific heat.

 

[Akaramos45]

the volume of the steel?

 

[Akaramos45]

I am looking up specific heat

 

[haruspex]

the volume of the steel?

Yes. You are given the internal diameter, the length and the thickness.

 

[Akaramos45]

The energy lost by the gas + the energy gained by the steel cylinder = 0 Since the energy is conserved.
The mass of the steel cylinder = Volume x density = (π(0.01)^2 - π(0.009)^2)*0.6 = 3.58 x 10 (-5)* 7900 = 0.283 Kg
M_sC_s(T_f - 18) + M_gC_g(T_f- 18) = 0

 

[Akaramos45]

I do not know the mass of air, or it's specific heat.
So would it be correct if i said the energy.
Also I do not know what to do from here.

 

[haruspex]

(π(0.01)^2 - π(0.009)^2)*

The 2cm is the internal diameter, not the external diameter.

MgCg(Tf- 18)

No, the gas is cooling from the high temperature found previously to T_f. But anyway, we established that you can just write that the heat transferred to the steel is the whole of the internal energy gained by the gas during compression, and you already calculated that.

 

[Akaramos45]

Do you mean to use the internal energy that was calculated in part b?

 

[Akaramos45]

Density of steel = π(0.011)^2 - π(0.01)^2 * 0.6 = 3.96 x10(-5) * 7900 = 0.313 Kg

 

[Akaramos45]

(0.313)(448)(T_f - 18) = 59.5 J
Tf - 291.15 = 59.5/(0.313*448)
Tf = 291.57 K
Therefore there is an increase of 0.42 degrees

 

[Akaramos45]

I think this is way it should be done. I see that you are offline. Thank you very much for you help. This might be wrong, but it's too late now the homework is due tomorrow. If there is any mistakes with please just list them so i can take care of it late

 

[haruspex]

But anyway, we established that you can just write that the heat transferred to the steel is the whole of the internal energy gained by the gas during compression, and you already calculated that.

whoops! That's wrong.
Even though the gas cools, it is still at a higher pressure than it started. This means it retains some of the work done.
You need to calculate the pressure when it returns to about 18C.

 

[Akaramos45]

I am so confused. Could you please give show me how it meant to be done?

 

[Akaramos45]

Excuse me, Mr.Haruspex it's getting quite late for me could you please show me what you mean?.

 

[haruspex]

Excuse me, Mr.Haruspex it's getting quite late for me could you please show me what you mean?.

Sorry, but I'm not that expert in this area myself, and I don't want to lead you astray. I need to check some things before I give a firm answer. I understand that this will be too late for you tonight.