What is the equation for Cp in terms of R for an adiabatic charging process?

[Tomdickjerry]

Hi guys,
i just discovered this forum and this is my first post so apologies if i made any mistake with the how I am suppose to structure a thread. but nontheless any help would be greatly appreciated!
cheers

1. Homework Statement
4. A 250mm diameter insulated cylinder (see figure below) is fitted with a frictionless, leakproof piston, which is attached to an unstressed spring S (spring constant, k = 175kN/m). The cylinder is connected by a closed valve to the mains pipe, in which air flows at a high pressure and a temperture of 27°C. The spring side of the cylinder is open to the atmosphere (Patm = 101kPa).

The valve is now opened slowly permitting air to move the piston through a distance of 150mm, thus compressing the spring. At this point the valve is closed, trapping air in the cylinder.

a) Write down the 1st law of thermodynamics for an open system and simplify it for this charging process, which can be assumed to be adiabatic. b) Calculate:

i) The final pressure of the air in the cylinder.
ii) The final temperature of the air in the cylinder.
iii) The mass of air injected into the cylinder.

answers:
[636kPa, 340.8K, 0.0479kg]

The Attempt at a Solution

i've managed to get the first answer of P₂. but I am stuck on the 2nd part. am i suppose to use
P₁/T₁=P₂/T₂
any help would be appreciated!

 

[Simon Bridge]

Welcome to PF;

i've managed to get the first answer of P₂. but I am stuck on the 2nd part. am i suppose to use
P₁/T₁=P₂/T₂
any help would be appreciated!

... that formula would be appropriate if the volume did not change: does it?
Why not use your answer to (a) to help you here?

 

[Tomdickjerry]

Welcome to PF;
... that formula would be appropriate if the volume did not change: does it?
Why not use your answer to (a) to help you here?

yes the volume does change. but initial volume is 0 because the piston is flushed against the cylinder so i dun really thing that would work..
how do i use the part (a) answer to help me?

 

[Simon Bridge]

You asked about an equation that requires volume to remain the same, but you notice that it changes. Therefore: do not use that equation. The initial volume being zero makes no difference to these facts.

how do i use the part (a) answer to help me?

Read question (a) again.
1. what is the name of the process?
2. what was the simplified 1st Law equation you came up with?

 

[Tomdickjerry]

You asked about an equation that requires volume to remain the same, but you notice that it changes. Therefore: do not use that equation. The initial volume being zero makes no difference to these facts.Read question (a) again.
1. what is the name of the process?
2. what was the simplified 1st Law equation you came up with?

1) i think its an adiabatic system as mentioned in (a)
2) I am not sure if I am right but i got -w=m(h₂-h₂)

 

[Chestermiller]

You are working with the open system version of the first law. This is not a steady flow situation, so the equation you wrote down is not applicable. Does the internal energy of the system (i.e., the contents of the cylinder) change? You have mass entering the system, but no mass leaving. Please write down the form of the first law that applies to this situation.

 

[Tomdickjerry]

You are working with the open system version of the first law. This is not a steady flow situation, so the equation you wrote down is not applicable. Does the internal energy of the system (i.e., the contents of the cylinder) change? You have mass entering the system, but no mass leaving. Please write down the form of the first law that applies to this situation.

i see.. so what you mean is the equation should be -w+m_ih_i=m₂u₂-m₁u₁?
im not sure if there is w though but if the spring compresses then there should be work done right?

 

[Chestermiller]

i see.. so what you mean is the equation should be -w+m_ih_i=m₂u₂-m₁u₁?
im not sure if there is w though but if the spring compresses then there should be work done right?

Much better. So, if m is the mass that enters, then, in the final state:$$mu=-W+mh_{in}$$
You are correct that (shaft) work is done on the surroundings. Algebraically, what do you get for the shaft work done (on the piston)? How does the enthalpy and temperature of the air entering the system h_in and T_in compare with the enthalpy and temperature of the air in the mains pipe (assuming ideal gas)?

 

[Tomdickjerry]

Much better. So, if m is the mass that enters, then, in the final state:$$mu=-W+mh_{in}$$
You are correct that (shaft) work is done on the surroundings. Algebraically, what do you get for the shaft work done (on the piston)? How does the enthalpy and temperature of the air entering the system h_in and T_in compare with the enthalpy and temperature of the air in the mains pipe (assuming ideal gas)?

the enthalpy and temperature of air entering the system should be the same as that in the pipe isn't it?

 

[Chestermiller]

the enthalpy and temperature of air entering the system should be the same as that in the pipe isn't it?

Yes.

From a force balance on the piston, what is the final pressure P in terms of x, k, and p_atm?

 

[Tomdickjerry]

Yes.

From a force balance on the piston, what is the final pressure P in terms of x, k, and p_atm?

the final pressure i calculated was 635.8kPa, there shouldn't be an x right since its air? assuming you meant x as in quality..? k would be 1.4 for air and Patm was given in the question at 101kPa.

 

[Chestermiller]

the final pressure i calculated was 635.8kPa, there shouldn't be an x right since its air? assuming you meant x as in quality..? k would be 1.4 for air and Patm was given in the question at 101kPa.

Please do it algebraically. There is a good reason I'm asking for this.

 

[Tomdickjerry]

the final pressure i calculated was 635.8kPa, there shouldn't be an x right since its air? assuming you meant x as in quality..? k would be 1.4 for air and Patm was given in the question at 101kPa.

okay correct me if I am wrong but what you're asking for is P=P_atm+k*x/Area_piston

 

[Chestermiller]

okay correct me if I am wrong but what you're asking for is P=P_atm+k*x/Area_piston

Perfect. Now, the final volume is V = Ax. So, again, in terms of these same parameters, what is the final value of PV?

 

[Tomdickjerry]

Perfect. Now, the final volume is V = Ax. So, again, in terms of these same parameters, what is the final value of PV?

that would make it P₂*Ax?

 

[Chestermiller]

Yes. That would be $$PV=\left(P_a+\frac{kx}{A}\right)(Ax)=P_aAx+kx^2=P_aV+kx^2$$
So now you have the two equations:$$U=mh_{in}-W$$and$$PV=P_aV+kx^2$$
What do you get if you add these two equations together? (You still haven't answered my question about the algebraic expression for W in terms of k, x, A, and P_a)

Chet

 

[Tomdickjerry]

Yes. That would be $$PV=\left(P_a+\frac{kx}{A}\right)(Ax)=P_aAx+kx^2=P_aV+kx^2$$
So now you have the two equations:$$U=mh_{in}-W$$and$$PV=P_aV+kx^2$$
What do you get if you add these two equations together?

Chet

erm I am not sure..

 

[Chestermiller]

erm I am not sure..

You're not sure how to algebraically add two equations together by adding their left sides together and their right sides together?

 

[Tomdickjerry]

You're not sure how to algebraically add two equations together by adding their left sides together and their right sides together?

oh.. PV+U=PaV+kx2+mhin−W is this what you mean?

 

[Chestermiller]

oh.. PV+U=PaV+kx2+mhin−W is this what you mean?

Yes. Now, on the left hand side of this equation, you have U + PV. What thermodynamic function is that, and what does it represent physically in this problem?

I need to go to bed now, so I won't be available for a few hours.

Chet

 

[Tomdickjerry]

Yes. Now, on the left hand side of this equation, you have U + PV. What thermodynamic function is that, and what does it represent physically in this problem?

I need to go to bed now, so I won't be available for a few hours.

Chet

U+PV=H right?

 

[Chestermiller]

U+PV=H right?

Right! So H is the enthalpy of the gas in the cylinder at the end of the process: H = mh. In our subsequent analysis, we are going to let m be the number of moles of gas in the cylinder, and h the molar enthalpy. If we substitute this into our previous equation, we obtain:
$$mh=mh_{in}-W+kx^2+P_aAx$$
So we see that, by adding PV to both sides of the equation, we have been able to express everything in terms of enthalpies. This equation then gives:
$$m(h-h_{in})=-W+kx^2+P_aAx$$
Now, we will not be able to proceed any further until you provide an equation for the total work W done by the gas on the spring and atmospheric air in terms of k, x, A, and P_a.

 

[Tomdickjerry]

Right! So H is the enthalpy of the gas in the cylinder at the end of the process: H = mh. In our subsequent analysis, we are going to let m be the number of moles of gas in the cylinder, and h the molar enthalpy. If we substitute this into our previous equation, we obtain:
$$mh=mh_{in}-W+kx^2+P_aAx$$
So we see that, by adding PV to both sides of the equation, we have been able to express everything in terms of enthalpies. This equation then gives:
$$m(h-h_{in})=-W+kx^2+P_aAx$$
Now, we will not be able to proceed any further until you provide an equation for the total work W done by the gas on the spring and atmospheric air in terms of k, x, A, and P_a.

would that just mean moving the W to one side and everything else to the other?

 

[Chestermiller]

would that just mean moving the W to one side and everything else to the other?

No. The work is the force integrated over the displacement. In terms of k and x, how much elastic energy is stored in the spring? How much work is required to push the atmosphere back a distance of x if the area of the piston is A and the pressure of the atmosphere is P_a? What is the sum of these?

 

[Chestermiller]

Another way of figuring out the work W is to recognize that, at any time t during the process, the force F exerted by the gas on the piston is:
$$F=k\xi +P_aA$$where $\xi$ is the piston displacement at time t. So, the amount of work that the gas does on the piston between time t and time t + dt is given by: $$dW=(k\xi +P_aA)d\xi$$ To get the total work W, you integrate this between $\xi = 0$ and $\xi = x$, where x is the final displacement.

 

[Tomdickjerry]

Another way of figuring out the work W is to recognize that, at any time t during the process, the force F exerted by the gas on the piston is:
$$F=k\xi +P_aA$$where $\xi$ is the piston displacement at time t. So, the amount of work that the gas does on the piston between time t and time t + dt is given by: $$dW=(k\xi +P_aA)d\xi$$ To get the total work W, you integrate this between $\xi = 0$ and $\xi = x$, where x is the final displacement.

that would make W=[kξ²/2+PaAξ]₀^x?

 

[Chestermiller]

that would make W=[kξ²/2+PaAξ]₀^x?

Yes, so substitute the integration limits please.

 

[Tomdickjerry]

Yes, so substitute the integration limits please.

x would be 150mm..?

 

[Chestermiller]

x would be 150mm..?

Yes, but I want the result algebraically in terms of k, x, P_a, and A. I want to hold off at substituting numbers in until later.

 

[Chestermiller]

As you responded in a private conversation: W=[kx2/2+PaAx]. So now what do we get if we substitute this into our equation:
$$m(h-h_{in})=-W+kx^2+P_aAx$$
Chet

 

[Tomdickjerry]

As you responded in a private conversation: W=[kx2/2+PaAx]. So now what do we get if we substitute this into our equation:
$$m(h-h_{in})=-W+kx^2+P_aAx$$
Chet

m(h−hin)=−[kx²/2+PaAx]+kx²+PaAx ?

 

[Chestermiller]

m(h−hin)=−[kx²/2+PaAx]+kx²+PaAx ?

Collect terms?

 

[Tomdickjerry]

Collect terms?

m(h−hin)=1/2kx²?

 

[Chestermiller]

Correct. Pretty simple, huh?

This is all that needs to be done on the right hand side of the equation. I'll be back in a little while to get you started working on the left hand side of the equation. Right now, I'm on my iPhone.

 

[Tomdickjerry]

Correct. Pretty simple, huh?

This is all that needs to be done on the right hand side of the equation. I'll be back in a little while to get you started working on the left hand side of the equation. Right now, I'm on my iPhone.

alright thanks! for the left side do i just use cv(t₂-t₁) i figured since the it isn't constant pressure process

 

[Chestermiller]

alright thanks! for the left side do i just use cv(t₂-t₁) i figured since the it isn't constant pressure process

Actually, no, but you're on the right track. The molar enthalpy of a pure single phase substance can be expressed uniquely as a function of temperature T and pressure P: h = h(T,P). The molar heat capacity C_p is, by definition, the partial derivative of h with respect to T at constant P:
$$C_p\equiv \left(\frac{\partial h}{\partial T}\right)_P$$For an ideal gas, molar enthalpy is a function only of temperature T. So the partial derivative becomes a total derivative: $$\frac{dh}{dT }=C_p$$For an ideal gas, this equation applies even if the pressure is not constant. The quantity $C_p$ is called the heat capacity at constant pressure only because, in measuring $C_p$ experimentally, not only is the change in enthalpy dh at constant pressure equal to $C_pdT$, but so also is the amount of heat added dQ. So, under constant pressure conditions, $C_p$ can be measured experimentally by directly determining the amount of heat added. If the pressure is not constant, then dQ is not equal to $C_pdT$ but dh (for an ideal gas) is still equal to C_pdT.

So, since for an ideal gas, dh is always equal to $C_pdT$, irrespective of whether the pressure is constant, we have:
$$(h-h_{in})=C_p(T-T_{in})$$where h is the final molar enthalpy of the air in the cylinder, $h_{in}$ is the molar enthalpy of the air in the tank, T is the final temperature of the air in the cylinder, and $T_{in}$ is the temperature of the air in the tank (27 C = 300 K).

Now m is the final number of moles of air in the cylinder. From the ideal gas law, please (algebraically) express m as a function of PV, R, and T, where R is the universal gas constant.

Air is a diatomic gas. In the ideal gas region, how is the molar heat capacity $C_p$ of a diatomic ideal gas related to the ideal gas constant R?

Chet

 

[Tomdickjerry]

Actually, no, but you're on the right track. The molar enthalpy of a pure single phase substance can be expressed uniquely as a function of temperature T and pressure P: h = h(T,P). The molar heat capacity C_p is, by definition, the partial derivative of h with respect to T at constant P:
$$C_p\equiv \left(\frac{\partial h}{\partial T}\right)_P$$For an ideal gas, molar enthalpy is a function only of temperature T. So the partial derivative becomes a total derivative: $$\frac{dh}{dT }=C_p$$For an ideal gas, this equation applies even if the pressure is not constant. The quantity $C_p$ is called the heat capacity at constant pressure only because, in measuring $C_p$ experimentally, not only is the change in enthalpy dh at constant pressure equal to $C_pdT$, but so also is the amount of heat added dQ. So, under constant pressure conditions, $C_p$ can be measured experimentally by directly determining the amount of heat added. If the pressure is not constant, then dQ is not equal to $C_pdT$ but dh (for an ideal gas) is still equal to C_pdT.

So, since for an ideal gas, dh is always equal to $C_pdT$, irrespective of whether the pressure is constant, we have:
$$(h-h_{in})=C_p(T-T_{in})$$where h is the final molar enthalpy of the air in the cylinder, $h_{in}$ is the molar enthalpy of the air in the tank, T is the final temperature of the air in the cylinder, and $T_{in}$ is the temperature of the air in the tank (27 C = 300 K).

Now m is the final number of moles of air in the cylinder. From the ideal gas law, please (algebraically) express m as a function of PV, R, and T, where R is the universal gas constant.

Air is a diatomic gas. In the ideal gas region, how is the molar heat capacity $C_p$ of a diatomic ideal gas related to the ideal gas constant R?

Chet

m=pv/rt.
i don't know what a diatomic gas is it wasn't covered in my syllabus but I am assuming you're referring to the k=1.4?

 

[Chestermiller]

m=pv/rt.
i don't know what a diatomic gas is it wasn't covered in my syllabus but I am assuming you're referring to the k=1.4?

Yes. So, if $\frac{C_p}{C_v}=1.4$ and $C_p-C_v=R$ (you are familiar with this equation, correct?), what is the equation for C_p expressed exclusively in terms of R?