Calculating the mass of air in a pressurised air tank

[Gixer1127]

Homework Statement

Looking for some pointers in where I am going wrong with this if possible please

Relevant Equations

pV=nRT pV = (m/Mr)RT)

Hey folks, any help will be greatly appreciated. I have, what I thought, was a fairly simple equation to follow to determine the mass of air in a pressurised air tank. See below question and my attempt at solving using the pV = (m/Mr)RT) equation.
A pressurised air tank supplies compressed air to a new hybrid air engine. When the hybrid engine is not running the absolute pressure in the tank is 320 bar and the temperature of the air is 50°C.
Calculate the temperature of the air in the high pressure air tank when the air engine starts and the absolute pressure in the tank drops to 280 bar. Volume remains constant.

Using the General Gas Law: P1xV1/T1 = P2xV2/T2

Simplify the equation to P1/T1 = P2/T2 as there is no change to the volume.

Step 1) Convert the known temperature to Kelvin = T1 = 50⁰C + 273 = 323K

Insert values into the equation:
320/323 = 280/T2

T2 = (280/320) x 323 =282.62K

The temperature of the air in the high pressure air tank at 280 bar is 282.62K
Calculate the mass of air in the pressurised air tank using the temperature and pressure from part (a) and the information given here below.

Using the Characteristic Gas Law equation pV=nRT which can also be written as pV = (m/Mr)RT)

P is the pressure in Pa

V is the volume in m³

N is the amount of gas (in moles)

T is the temperature in K

R is the universal gas constant and is always 8.314J·K‾1

M is the mass of gas in g

Mr is the molecular mass of the gas (Air = 28.96g/mol)

Tank volume = 450 litres ( 0.45m³)

pV = (m/Mr)RT Change equation to find the mass m = (pV Mr)/RT

((280x10⁵)x0.45x28.96) / (8.314 x 282.62) = 15529g = 155.3kg

I'm pretty sure the mass of air in a 450litre air tank is not 155.3kg's but this is the only answer I keep getting so what have I missed?

 

[BvU]

Hi,

Pretty weird scenario. What causes the pressure to drop by 40 Bar ?

Simplify the equation to P1/T1 = P2/T2 as there is no change to the volume.

So what, exactly do you think DOES change ?

And: what mass do you get for the 320 Bar case ?

$\$

 

[Chestermiller]

Homework Statement: Looking for some pointers in where I am going wrong with this if possible please
Relevant Equations: pV=nRT pV = (m/Mr)RT)

Hey folks, any help will be greatly appreciated. I have, what I thought, was a fairly simple equation to follow to determine the mass of air in a pressurised air tank. See below question and my attempt at solving using the pV = (m/Mr)RT) equation.
A pressurised air tank supplies compressed air to a new hybrid air engine. When the hybrid engine is not running the absolute pressure in the tank is 320 bar and the temperature of the air is 50°C.
Calculate the temperature of the air in the high pressure air tank when the air engine starts and the absolute pressure in the tank drops to 280 bar. Volume remains constant.

Using the General Gas Law: P1xV1/T1 = P2xV2/T2

Simplify the equation to P1/T1 = P2/T2 as there is no change to the volume.

Step 1) Convert the known temperature to Kelvin = T1 = 50⁰C + 273 = 323K

Insert values into the equation:
320/323 = 280/T2

T2 = (280/320) x 323 =282.62K

The temperature of the air in the high pressure air tank at 280 bar is 282.62K
Calculate the mass of air in the pressurised air tank using the temperature and pressure from part (a) and the information given here below.

Using the Characteristic Gas Law equation pV=nRT which can also be written as pV = (m/Mr)RT)

P is the pressure in Pa

V is the volume in m³

N is the amount of gas (in moles)

T is the temperature in K

R is the universal gas constant and is always 8.314J·K‾1

M is the mass of gas in g

Mr is the molecular mass of the gas (Air = 28.96g/mol)

Tank volume = 450 litres ( 0.45m³)

pV = (m/Mr)RT Change equation to find the mass m = (pV Mr)/RT

((280x10⁵)x0.45x28.96) / (8.314 x 282.62) = 15529g = 155.3kg

I'm pretty sure the mass of air in a 450litre air tank is not 155.3kg's but this is the only answer I keep getting so what have I missed?

I. confirm your result.

 

[Gixer1127]

Hi,

Pretty weird scenario. What causes the pressure to drop by 40 Bar ?So what, exactly do you think DOES change ?

And: what mass do you get for the 320 Bar case ?

$\$

No idea what causes the pressure to drop, this is all the info I was given in the question. However, I have done what you have suggested with checking the 320bar and original temp and got a final mass of 155291g. In my original post I had missed a 4 (155294). So the only things that have changed are the temperature and the mass?

Sorry Chester, I had missed a 4 from the final figure It should have been 155294. Is this anything like you got?

 

[Chestermiller]

No idea what causes the pressure to drop, this is all the info I was given in the question. However, I have done what you have suggested with checking the 320bar and original temp and got a final mass of 155291g. In my original post I had missed a 4 (155294). So the only things that have changed are the temperature and the mass?

Yes. Dividing by 1000 gives 155

 

[BvU]

final mass of 155291g

No way you can quote six digit precision for mass when you are given two (or at best three) digit precision to start with.

So you get 155 kg mass at 320 Bar (absolute pressure).
When you apply

Simplify the equation to P1/T1 = P2/T2 as there is no change to the volume

you don't just assume the volume is the same, but you also assume the amount of air is the same. No wonder you get back the same 155 kg !

Pretty weird scenario. What causes the pressure to drop by 40 Bar ?

Well, it seems the scenario is that the cylinder is left to cool down from 50 $^\circ$C to 10 degrees; perhaps overnight? What that has to do with

when the air engine starts and the absolute pressure in the tank drops to 280 bar

is a mystery to me.

 

[Juanda]

you don't just assume the volume is the same, but you also assume the amount of air is the same. No wonder you get back the same 155 kg !

I think that's the issue. The problem states that the volume of the tank remains constant (the tank is rigid) but it doesn't say the mass of air inside such tank remains constant. In fact, that tank SUPPLIES air to the engine. That mass of air leaving the tank would be the cause behind the pressure drop.

A pressurised air tank supplies compressed air to a new hybrid air engine.

I will try to solve the problem without considering the amount of air in the tank to be constant. If that's right there should be enough information in the problem statement to solve it that way.

 

[BvU]

If that's right there should be enough information in the problem statement to solve it that way

Seems to me there isn't.
What is the complete problem statement, verbatim ?

$\$

 

[Juanda]

I will try to solve the problem without considering the amount of air in the tank to be constant. If that's right there should be enough information in the problem statement to solve it that way.

Are we getting the original problem statement with all the information? I feel like there must be something we are missing. For example, I thought the problem statement to just be:

A pressurised air tank supplies compressed air to a new hybrid air engine. When the hybrid engine is not running the absolute pressure in the tank is 320 bar and the temperature of the air is 50°C.
Calculate the temperature of the air in the high pressure air tank when the air engine starts and the absolute pressure in the tank drops to 280 bar. Volume remains constant.

I was struggling to define State 1 and later a little bit of additional key information was dropped.

Tank volume = 450 litres ( 0.45m³)

Is there any additional information? For example, the amount of mass that left the pressurized tank when it started supplying air to the engine. Assuming air is not constant in the tank but the tank volume is, this is as far as I can get.

Known information is:
$P_1 = 320 bar = 32 \times 10^6 Pa ; \ T_1 = 50^{\circ}C=323K; \ V_1=V_2=0.45m^3 ; \ P_2 = 280bar=28 \times 10^6Pa$

State 1:
$P_1 V_1 = m_1 R_{air} T_1 \rightarrow m_1 = \frac{P_1T_1}{R_{air} T_1}=155.33kg$

State 2:
$P_2 V_2 = m_2 R_{air} T_2$

I got 1 equation but two unknowns ($m_2; \ T_2$). I feel like the problem statement is incomplete because the mass being conserved in the tank doesn't make much sense. Regarding the fact $m_1=155kg$ all I can say is that the math seems right. I don't know how realistic such an engine would be but that's not really the point of the exercise I believe.

 

[Chestermiller]

Seems to me there isn't.
What is the complete problem statement, verbatim ?

$\$

There is enough information given if you assume that the tank is insulated and the gas expansion within the tank is adiabatic (even with gas leaving the tank). I'm just not sure that the OP is far enough advanced to address this.

 

[Juanda]

There is enough information given if you assume that the tank is insulated and the gas expansion within the tank is adiabatic (even with gas leaving the tank). I'm just not sure that the OP is far enough advanced to address this.

How much additional information did you need to add? A thermally insulated tank and adiabatic expansion are what I read in your post.
Anything else? Maybe adiabatic expansion up to atmospheric pressure? Amount of energy leaving the tank?

I would love to see the process to solve it. Problems with limited available information are not very common in books since they need to have close solutions but they certainly are common in the real world where sometimes it is necessary to complete the problem with "guesses" about the environment where the phenomenon is happening.

 

[Chestermiller]

How much additional information did you need to add? A thermally insulated tank and adiabatic expansion are what I read in your post.
Anything else? Maybe adiabatic expansion up to atmospheric pressure? Amount of energy leaving the tank?

I would love to see the process to solve it. Problems with limited available information are not very common in books since they need to have close solutions but they certainly are common in the real world where sometimes it is necessary to complete the problem with "guesses" about the environment where the phenomenon is happening.

Are you familiar with the equation $Pv^{\gamma}=Constant$ for adiabatic expansion of an ideal gas, where v is the molar volume and $\gamma=C_p/C_v$?

 

[Juanda]

Are you familiar with the equation $Pv^{\gamma}=Constant$ for adiabatic expansion of an ideal gas, where v is the molar volume and $\gamma=C_p/C_v$?

I didn't consider that could be used in open systems. I always saw it used in piston-like systems. When moving to open systems enthalpy is usually king. I need to recheck my books about this.

If it can indeed be used then I can establish a relation between the initial mass in the tank and the final mass of air in the tank after some of it goes into the engine.

Using the information from #9
$v_1 = \frac{V_1}{m_1}=0,002896906 \ m^3/kg$

Using the adiabatic expansion as suggested by @Chestermiller in #12. ($\gamma = 1.4$)
$P_1v_1^\gamma =P_2v_2^\gamma \rightarrow v_2=e^\frac{\ln\left ( \frac{P_1v_1^\gamma}{P_2} \right ) }{\gamma} =0,003186818 \ m^3/kg$

The mass that remains in the tank will then be
$m_2 = \frac{V_2}{v_2}=141,20 \ kg$

It is now possible to find the temperature of the air remaining in the tank with
$P_2 V_2 = m_2 R_{air} T_2 \rightarrow T_2=\frac{P_2 V_2}{R_{air} m_2 }=310,9 \ K$

So the math at least seems consistent. We assumed air leaves the tank which causes a drop in pressure. The results agree with this. The mass decreases and temperature decreases as well.

I assume it must also be possible to know the energy that left the system to go into the engine with
$\Delta U_{tank} = c_{v_{air}}\left ( m_2T_2-m_1T_1 \right )$
(It'll be negative because the tank decreases its potential energy to transfer work to the engine)

Is that correct?

 

[Chestermiller]

I didn't consider that could be used in open systems. I always saw it used in piston-like systems. When moving to open systems enthalpy is usually king. I need to recheck my books about this.

If it can indeed be used then I can establish a relation between the initial mass in the tank and the final mass of air in the tank after some of it goes into the engine.

Using the information from #9
$v_1 = \frac{V_1}{m_1}=0,002896906 \ m^3/kg$

Using the adiabatic expansion as suggested by @Chestermiller in #12. ($\gamma = 1.4$)
$P_1v_1^\gamma =P_2v_2^\gamma \rightarrow v_2=e^\frac{\ln\left ( \frac{P_1v_1^\gamma}{P_2} \right ) }{\gamma} =0,003186818 \ m^3/kg$

The mass that remains in the tank will then be
$m_2 = \frac{V_2}{v_2}=141,20 \ kg$

It is now possible to find the temperature of the air remaining in the tank with
$P_2 V_2 = m_2 R_{air} T_2 \rightarrow T_2=\frac{P_2 V_2}{R_{air} m_2 }=310,9 \ K$

So the math at least seems consistent. We assumed air leaves the tank which causes a drop in pressure. The results agree with this. The mass decreases and temperature decreases as well.

I assume it must also be possible to know the energy that left the system to go into the engine with
$\Delta U = c_{v_{air}}\left ( m_1T_1-m_2T_2 \right )$

Is that last statement correct?

You can do this analysis using either the open system or the closed system version of the 1st law of thermodyamics. The open system version would read $$dU=d(un)=hdn$$which leads to $$nC_vdT=RTdn$$.

Using the closed system version of the 1st law of thermodynamics involves recognizing that the final mass of air remaining in the tank has undergone an adiabatic reversible expansion, pushing the air ahead of it (i.e., doing work on it) into the engine, and also recognizing that, during the process, it exchanges no heat with the air ahead of it (since the temperature is uniform in the tank).

Both these methods give exactly the same answer(s).

 

[Juanda]

Using the closed system version of the 1st law of thermodynamics involves recognizing that the final mass of air remaining in the tank has undergone an adiabatic reversible expansion, pushing the air ahead of it (i.e., doing work on it) into the engine, and also recognizing that, during the process, it exchanges no heat with the air ahead of it (since the temperature is uniform in the tank).

Yep, that's how I was visualizing it. Having your confirmation makes me sure of my understanding.
Thanks!