How to calculate time of Pressurization in vessel?

[METCON]

TL;DR

"Urgent: Need Expert Advice on Calculating Pressure Buildup in Fixed Volume Vessel!"

Hello community,

I'm currently facing a challenging situation and would greatly appreciate some expert advice. Here's the scenario:

I have a fixed volume vessel with a total volume of 15458 ft³. The volumetric flow rate through this volume is 90 MMSCFD. The initial temperature of the gas is 80°F, with a Compressibility Factor of approximately 0.91 and a Molecular Weight of 22.9. The source pressure is 300 PSIG, and the normal operating pressure is 100 PSIG.

My concern is understanding the time it would take for the pressure to reach 250 PSIG if the exit of the vessel were to be suddenly shut. Assume no heat transfer and no additional information on piping dimensions.

I'm reaching out to this knowledgeable community for assistance in calculating this pressure buildup time. Any insights, formulas, or guidance on how to approach this problem would be immensely helpful.

Your expertise is highly valued, and I'm eager to hear your thoughts on this matter. Thank you in advance for your time and assistance!"

Feel free to make any adjustments or let me know if you have specific preferences!

Thanks!

-METCON

 

[renormalize]

The volumetric flow rate through this volume is 90 MMSCFD.

This flowrate unit, MMSCFD = One million standard cubic feet per day, is typically used in the natural gas industry. Are there significant safety concerns (rupture, flammability or detonation) impacted by this calculation?

 

[METCON]

@renormalize no there are no concerns with this. the system is protected by redundant safety systems to mitigate this risk.

 

[Chestermiller]

Are you willing to make the approximation that the gas behaves as an ideal gas?

 

[Dullard]

Not 'expert' advice:
In round numbers / Ideal gas: approx 2.5 minutes. More precision will not produce more accuracy without a lot more detail (actual supply flow vs pressure, thermal...).

 

[METCON]

Are you willing to make the approximation that the gas behaves as an ideal gas?

Ideal gas is ok. The entire basis is approximated.

 

[PeterDonis]

the system is protected by redundant safety systems to mitigate this risk.

Meaning, to mitigate the risk of the exit valve being shut?

If so, what is the point of the calculation you are asking about?

Generally speaking, engineering risk calculations are not something that should be done here at PF. They should be done by experts on the spot who are responsible to whoever owns the system.

 

[berkeman]

@renormalize no there are no concerns with this. the system is protected by redundant safety systems to mitigate this risk.

Can you please detail those redundant safety systems for us? Thank you.

 

[METCON]

Can you please detail those redundant safety systems for us? Thank you.

@berkeman @PeterDonis Lets assume the theoretical system has an MAWP of 500 PSIG, is and is equipped with a relief device.

I'm looking for a general approach to find the time it takes for the system to go from operating pressure to target pressure while being fed by a source pressure and flowrate. I have approached this from the work equation by integrating for change in pressure. w= ∫ VdP = P1*V1*ln( P2/P1 ). Finding the work to fill the volume from 100PSIG to 250PSIG, then finding the work done by compression , inlet flowrate 90MMSCFD and source pressure 300PSIG. Then dividing the work to fill the volume by the work done by compression to find the time. Flowrate provides the time unit. Is this the correct approach? Maybe more importantly, is my integration correct? It has been a while...

I tried to take an PV=nRT approach to this with the molar flow rate of the gas but I can't seem to make it work...

@Dullard can you explain how you got to 2.5min?

Thank you all.

 

[PeterDonis]

I'm looking for a general approach to find the time it takes for the system to go from operating pressure to target pressure while being fed by a source pressure and flowrate.

So basically you're filling the tank and you want to estimate how long it will take?

 

[erobz]

@berkeman @PeterDonis Lets assume the theoretical system has an MAWP of 500 PSIG, is and is equipped with a relief device.

I'm looking for a general approach to find the time it takes for the system to go from operating pressure to target pressure while being fed by a source pressure and flowrate. I have approached this from the work equation by integrating for change in pressure. w= ∫ VdP = P1*V1*ln( P2/P1 ). Finding the work to fill the volume from 100PSIG to 250PSIG, then finding the work done by compression , inlet flowrate 90MMSCFD and source pressure 300PSIG. Then dividing the work to fill the volume by the work done by compression to find the time. Flowrate provides the time unit. Is this the correct approach? Maybe more importantly, is my integration correct? It has been a while...

I tried to take an PV=nRT approach to this with the molar flow rate of the gas but I can't seem to make it work...

@Dullard can you explain how you got to 2.5min?

Thank you all.

Are you measuring the source pressure and the flowrate, or just the source pressure?

 

[METCON]

Are you measuring the source pressure and the flowrate, or just the source pressure?

Source pressure and flowrate is known. 300 PSIG, 90MMSCFD.

 

[Dullard]

"@Dullard can you explain how you got to 2.5min?"

I can:
Think in 'Bar-Ft3'
Your initial and final conditions are specified in your post. The 'tank content' change is the tank volume (fixed) multiplied by the pressure change (10 Bar). The supply flow rate is in Ft3/min. content change / flow rate = time.

This doesn't account for non-idealities or thermal effects, but it's a decent approximation.

 

[erobz]

You can also go this route(I think), which would be a bit more detailed, but also probably not nearly detailed enough...

let $m$ be the mass of the gas in the tank.

$$ \frac{dm}{dt}= \rho_s Q \tag{1} $$

where the RHS is a constant by your approximation

Using the Ideal Gas Law to find $m$ ( not likely valid due to the dynamics)

$$ P = \frac{R_{air}}{V}mT \tag{2}$$

$P$ is the instantaneous tank pressure
$V$ tank volume
$R_{air}$ specific ideal gas constant
$T$ instantaneous tank gas temperature

sub (2) Into (1) for $m$

$$ \frac{V}{R_{air}}\frac{d}{dt}\left( \frac{P}{T} \right)= \rho_s Q \tag{3} $$

Then with an adiabatic expansion we have:

$$ T = T_s \left( \frac{P}{P_s} \right)^{(k-1)/k} \tag{4} $$

$T_s$ is the temp of the source gas
$P_s$ is the source pressure
$k$ is the ratio of specific heats

sub (4) into (3) for $T$

$$ \frac{V}{R_{air}}\frac{d}{dt}\left( \frac{P}{T_s \left( \frac{P}{P_s} \right)^{(k-1)/k}} \right)= \rho_s Q \tag{5} $$

Then you can simplify and solve (5) for $P(t)$ by separation of variables with initial condition $P(0) = P_o$

Take care to use absolute temp, pressures, etc... and your volumetric flowrate is a standardized cfm, I believe you should be careful with how that factors into the RHS value for mass flowrate.

 

[erobz]

When I solve the equation above, I get about $1.95 ~\rm{min}$ for air. That agrees very well with @Dullard 's approximation. However, it is an unrealistic representation as the tank pressure will not approach line pressure asymptotically with this constant mass flowrate assumption - it shoots right past and keeps going. I think the best we can say with this simplistic model is this is a minimum possible time to reach $254.7 ~\rm{psia}$.

 

[Chestermiller]

Assume that the gas is ideal, and let the subscript 0 represent the initial steady state parameters for the tank and the subscript 1 represent the parameters when the pressure in the tank reaches $P_1$. Let $m_0$ represent the steady sate moles of gas in the tank and $\delta m=m_1-m_0$. Then the ideal gas law gives: $$P_0V=m_0RT_0$$ and $$P_1V=(m_0+\delta m)RT_1$$
Assume that the gas being fed to the tank is at the same temperature as the steady state temperature $T_0$. Taking as our closed adiabatic system the initial steady state moles in the tank $m_0$ plus the number of moles $\delta m$ injected until the tank pressure is $P_1$, we have $$(m_0+\delta m)C_v(T-T_0)=P_{ext}\delta v=P_{ext}\frac{\delta m RT_0}{P_{ext}}=\delta m RT_0$$where $\delta v$ is the volume injected at 300 psi. Solving this for the final temperature gives: $$\frac{T_1}{T_0}=1+(\gamma-1)\frac{\delta m}{m_0+\delta m}$$. The temperature ratio can be eliminated from the above 3 equations to give $\frac{\delta m}{m_0}$ as a function of $P_1/P_0$.