Relationship between the pressure gradient and mass flow of a gas

[Thomot512]

Hi everyone,
I would like to know how to calculate the diameter of a pipe when we know the desired mass flow, the gas type, and the pressure at both end.

I have these requirements :
Gas : O₂
Molecular weight : 0.032 [kg / mol]
Desired mass flow : 0.32 [kg / s]
Pressure in the gas tank : p1 = 53.6 * 10⁵ [Pa]
Pressure at the end of the pipe (constant) : p2 = 38.6 * 10⁵ [Pa]
Temperature of the gas in the tank : T = 293.15 [°K]
There is no height difference (the pipe is horizontal)

I'm a bit stuck with this one.
Could someone tell me how to proceed?
Thanks a lot.
Greetings.

 

[Chestermiller]

Hi everyone,
I would like to know how to calculate the diameter of a pipe when we know the desired mass flow, the gas type, and the pressure at both end.

I have these requirements :
Gas : O₂
Molecular weight : 0.032 [kg / mol]
Desired mass flow : 0.32 [kg / s]
Pressure in the gas tank : p1 = 53.6 * 10⁵ [Pa]
Pressure at the end of the pipe (constant) : p2 = 38.6 * 10⁵ [Pa]
Temperature of the gas in the tank : T = 293.15 [°K]
There is no height difference (the pipe is horizontal)

I'm a bit stuck with this one.
Could someone tell me how to proceed?
Thanks a lot.
Greetings.

Is this a homework problem?

 

[Thomot512]

No it's a problem I have for something I would like to build as a School project.

 

[NFuller]

Assuming the pipe is much longer than it is wide, the flow will be laminar and the Hagen–Poiseuille equation will hold. See here https://en.wikipedia.org/wiki/Hagen–Poiseuille_equation.

 

[Chestermiller]

No it's a problem I have for something I would like to build as a School project.

I hope you are aware to the safety issues involved in using oxygen. Is there going to be a safety audit performed by a qualified professional prior to use of this equipment?

You also need to specify the length of the pipe.

What is your background in fluid mechanics? Have you had a course? If you knew the length, diameter, and exit pressure of the pipe, would you know how to calculate the required inlet pressure? Do you know the definition of the Reynolds number?

 

[Chestermiller]

Assuming the pipe is much longer than it is wide, the flow will be laminar and the Hagen–Poiseuille equation will hold. See here https://en.wikipedia.org/wiki/Hagen–Poiseuille_equation.

This is not a proper criterion for whether the flow is laminar (and when Hagen-Poiseuille will be applicable).

 

[NFuller]

This is not a proper criterion for whether the flow is laminar (and when Hagen-Poiseuille will be applicable).

For the low pressures described it should be sufficient since for a long thin pipe the flow velocity will be small, allowing for viscous forces to dominate the flow throughout most of the pipe.

 

[Chestermiller]

For the low pressures described it should be sufficient since for a long thin pipe the flow velocity will be small, allowing for viscous forces to dominate the flow throughout most of the pipe.

That remains to be seen. Suppose I told you that, in this system, for a 1" pipe, the Reynolds number would be 500000. Would you still argue that the flow would be laminar?

You regard 56 atm as a low pressure?

 

[NFuller]

You regard 56 atm as a low pressure?

I actually read it as 5.6 atm, whoops.

That remains to be seen. Suppose I told you that, in this system, for a 1" pipe, the Reynolds number would be 500000. Would you still argue that the flow would be laminar?

The OP should definitely check that the Reynolds number isn't huge for their final set up. If it is, then the dimensions of the pipe should be changed to make it smaller. It would be a very involved task to calculate the flow rate at high Reynolds number and the OP should make every effort to avoid that regime.

 

[Chestermiller]

I actually read it as 5.6 atm, whoops.

The OP should definitely check that the Reynolds number isn't huge for their final set up. If it is, then the dimensions of the pipe should be changed to make it smaller. It would be a very involved task to calculate the flow rate at high Reynolds number and the OP should make every effort to avoid that regime.

I totally disagree. It is a very straightforward calculation. The inlet and outlet pressures are already specified, so there is only one diameter that will give this pressure drop. If the flow happens to be turbulent, so what.

 

[NFuller]

From what I understand, turbulent flow through a pipe is related to this https://en.wikipedia.org/wiki/Moody_chart which contains variables which may be difficult to calculate.

It is a very straightforward calculation. The inlet and outlet pressures are already specified, so there is only one diameter that will give this pressure drop. If the flow happens to be turbulent, so what.

I'm not aware of this calculation.

 

[Chestermiller]

From what I understand, turbulent flow through a pipe is related to this https://en.wikipedia.org/wiki/Moody_chart which contains variables which may be difficult to calculate.

I'm not aware of this calculation.

Did you really think that we engineers don't know how to calculate in advance the pressure drop for turbulent flow in a pipe? How do you think we design piping systems and select pumps for industrial chemical plants and power plants? Please tell me that you don't think that the flow is laminar in most of these systems.

 

[NFuller]

Did you really think that we engineers don't know how to calculate in advance the pressure drop for turbulent flow in a pipe? How do you think we design piping systems and select pumps for industrial chemical plants and power plants?

I never made any such accusation of engineers. However, I imagine that a simple calculation not involving the length of the pipe must assume the fluid to be inviscid. Again, the Reynolds number should be calculated to see whether or not this is a valid assumption, i.e. for a very long narrow pipe it is not.

Please tell me that you don't think that the flow is laminar in most of these systems.

When fluid enters a pipe it is turbulent but after some distance $L$ the flow will become laminar. This distance is defined by the shape of the boundary layer. For a circular pipe of diameter $D$ the distance after the entrance point where the flow becomes almost completely laminar is approximately
$$L_{lam}\approx 0.05D\;Re$$
The region where the flow is strongly turbulent is much shorter and can be approximated as
$$L_{turb}\approx 1.36D\;Re^{1/4}$$

 

[Chestermiller]

I never made any such accusation of engineers. However, I imagine that a simple calculation not involving the length of the pipe must assume the fluid to be inviscid. Again, the Reynolds number should be calculated to see whether or not this is a valid assumption, i.e. for a very long narrow pipe it is not.

When fluid enters a pipe it is turbulent but after some distance $L$ the flow will become laminar. This distance is defined by the shape of the boundary layer. For a circular pipe of diameter $D$ the distance after the entrance point where the flow becomes almost completely laminar is approximately
$$L_{lam}\approx 0.05D\;Re$$
The region where the flow is strongly turbulent is much shorter and can be approximated as
$$L_{turb}\approx 1.36D\;Re^{1/4}$$

Most of this is incorrect. I am hereby asking you to cease from further posting to this thread. In my judgment as a mentor (as well as someone with substantial practical experience with fluid mechanics), you are not qualified to give advice to the OP in solving his problem, and your advice will only add to his confusion. If you post to this thread again, I'm going to issue infraction points for misinformation. This can lead to a ban from Physics Forums if compounded.

 

[Thomot512]

I hope you are aware to the safety issues involved in using oxygen. Is there going to be a safety audit performed by a qualified professional prior to use of this equipment?

You also need to specify the length of the pipe.

What is your background in fluid mechanics? Have you had a course? If you knew the length, diameter, and exit pressure of the pipe, would you know how to calculate the required inlet pressure? Do you know the definition of the Reynolds number?

To answer your question about my background, I'm in last years of my Bachelor in science of aviation, I had quite a few class in aerodynamics. I had some fluid dynamics in first years where we had to deal with pipe, but only with incompressible fluids. Yes I know what the Reynolds number is. And yes I'm aware of the safety issue due to the very high pressure, due to oxygen, and the rest.

I'm actually designing a rocket propulsion system for a small probe rocket meant to go to an altitude of 3000 m.
After I finished the Nozzle preliminary design I ended up with the number I gave you.
I don't know right now the length of the pipe since I don't know where the O₂ tank will be in the rocket. Might be near the nose to avoid stability issue or near the combustion chamber to simplify the system. So between 0.1-3 m.

And yes a few teacher are supervising the construction and safety issue, and the test are going to be made in an actual blast chamber.

 

[Chestermiller]

To answer your question about my background, I'm in last years of my Bachelor in science of aviation, I had quite a few class in aerodynamics. I had some fluid dynamics in first years where we had to deal with pipe, but only with incompressible fluids. Yes I know what the Reynolds number is. And yes I'm aware of the safety issue due to the very high pressure, due to oxygen, and the rest.

I'm actually designing a rocket propulsion system for a small probe rocket meant to go to an altitude of 3000 m.
After I finished the Nozzle preliminary design I ended up with the number I gave you.
I don't know right now the length of the pipe since I don't know where the O₂ tank will be in the rocket. Might be near the nose to avoid stability issue or near the combustion chamber to simplify the system. So between 0.1-3 m.

And yes a few teacher are supervising the construction and safety issue, and the test are going to be made in an actual blast chamber.

I'm really relieved to know that you have has some significant experience with fluid mechanics, and even know how to solve this if the fluid is treated as incompressible.

I've looked up the critical properties of oxygen and calculated the reduced temperature and pressure at the inlet and exit conditions, assuming that the temperature doesn't change. This shows that the compressibility factor Z at the inlet and outlet conditions is 0.96 and 0.97, respectively. As a result, we can use either value in quantifying the P-V-T behavior of the oxygen. That is,$$\rho=\frac{PM}{0.97RT}$$where $\rho$ is the density, M is the molecular weight, and R is the universal gas constant. I also looked up the viscosity of oxygen under these conditions, and it is about $\mu=2.0\times 10^{-5}\ \frac{kg}{m-s}$ at the pipe conditions. From this, you can calculate the Reynolds number for the flow from the equation $$Re=\frac{\rho vD}{\mu}=\frac{4\dot{m}}{\pi D\mu}$$
where $\dot{m}$ is the mass flow rate. What do you get from this equation when you calculate the Reynolds number for the case of a 2.54 cm ID pipe?

From a force balance on the flow, we know that the pressure gradient along the tube is given by: $$\frac{dP}{dz}=-\frac{4}{D}\tau_W$$ where $\tau_w$ is the shear stress at the wall.

I think I'll stop here for now. I will continue once you tell me whether you are more comfortable working with the fanning friction factor or the Darcy-Weisbach friction factor.

 

[Thomot512]

First I want to thank you for the help you are providing.

In school we used the Darcy-Weisbach friction factor.

Using this equation $Re=\frac{4\dot{m}}{\pi D\mu}$, I get $Re=802.04\times 10^{3}$

I also would like to know if you could give me some source for the compressibility factor and for the viscosity.

Since I'm going to write a report it would be nice to be able to have some references.

 

[Chestermiller]

First I want to thank you for the help you are providing.

In school we used the Darcy-Weisbach friction factor.

Using this equation $Re=\frac{4\dot{m}}{\pi D\mu}$, I get $Re=802.04\times 10^{3}$

I also would like to know if you could give me some source for the compressibility factor and for the viscosity.

Since I'm going to write a report it would be nice to be able to have some references.

I confirm your calculation of the Reynolds number.

I looked up the critical properties of oxygen on line and calculated the reduced temperature and reduced pressure. The plot of compressibility factor Z as a function of reduced temperature and reduced pressure is in Fundamentals of Engineering Thermodynamics by Moran et al. I looked up the viscosity of oxygen at 20 C and low pressure on line, and then used a plot of reduced viscosity as a function of reduced pressure and reduced temperature in Transport Phenomena by Bird et al to ascertain the (negligible) effect of non-ideality on the viscosity.

For Darcy-Weisbach, the wall shear stress is given by: $$\tau_w=\frac{1}{2}\rho v^2\frac{f}{4}$$where f is the Darcy-Weisbach friction factor (a function of the Reynolds number). So, the pressure gradient becomes: $$\frac{dP}{dz}=-\frac{1}{2}\frac{\rho v^2}{D} f$$We also know that$$\dot{m}=\rho v\left(\frac{\pi D^2}{4}\right)$$So, eliminating v, we have:$$\frac{dP}{dz}=-\frac{8}{\rho}\frac{\dot{m}^2}{\pi^2D^5} f$$
The next step is to substitute the equation for the density in terms of the temperature and pressure into this relationship. What do you get when you do this?

 

[Thomot512]

For a 2.54 cm pipe, I get: $$-\frac{8}{pM}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}f = -150.58\times10^{3}f$$
I'm sorry but I could not figure out how to calculate the Darcy-Weisbach friction factor for a turbulent flow, which is probably the case with a Reynolds number this high.

 

[Chestermiller]

For a 2.54 cm pipe, I get: $$-\frac{8}{pM}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}f = -150.58\times10^{3}f$$

Actually, it isn't quite right to treat the pressure as constant here. But your algebra is correct. So, we have:
$$\frac{dP}{dz}=-\frac{8}{PM}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}f $$If we multiply both sides of this equation by P, we obtain:$$P\frac{dP}{dz}=-\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}f $$
For a constant diameter, the entire right hand side of this equation is constant. What do you get (algebraically) if you integrate this from z = 0 to z = L? Notice that, for a compressible gas, the key parameter is the square of the pressure, rather than the pressure.

I'm sorry but I could not figure out how to calculate the Darcy-Weisbach friction factor for a turbulent flow, which is probably the case with a Reynolds number this high.

No problem. You need to get it off a graph. Google Darcy-Weisbach plot, Moody diagram, or Darcy-Weisbach diagram. What do you get for a Re of 800000?

 

[Thomot512]

Wow differential equation, not my specialty but I'll try.

So we've: $$P\frac{dP}{dz}=-\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}f$$
which lead us to: $$P\times dP=-\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}fdz$$
and then: $$\int P dP=\int_0^L -\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}fdz$$
we end up with : $$ P+C=-\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}f(L-0)$$

For a stainless steel pipe (make sense to me because of the $O_2$, but you may have a better(lighter) solution) I found an $\epsilon$ of 0.015 mm and D = 25.4mm which leads to a relative roughness of: $$\frac {\epsilon}{D} = 590.55\times10^{-6}$$
And with the moody Diagramm of Wikipedia it gives me approximately: $f=0.0125$ for a Re of 800000

 

[Chestermiller]

Wow differential equation, not my specialty but I'll try.

So we've: $$P\frac{dP}{dz}=-\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}f$$
which lead us to: $$P\times dP=-\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}fdz$$
and then: $$\int P dP=\int_0^L -\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}fdz$$
we end up with : $$ P+C=-\frac{8}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}f(L-0)$$

This is not quite correct. It should read: $$ P_i^2-P_f^2=\frac{16}{M}\frac{0.97RT\dot{m}^{2}}{\pi^{2}D^{5}}fL$$

For a stainless steel pipe (make sense to me because of the $O_2$, but you may have a better(lighter) solution) I found an $\epsilon$ of 0.015 mm and D = 25.4mm which leads to a relative roughness of: $$\frac {\epsilon}{D} = 590.55\times10^{-6}$$
And with the moody Diagramm of Wikipedia it gives me approximately: $f=0.0125$ for a Re of 800000

Excellent. So, now, for a 2.54 cm pipe ID, what is the required pipe length L to be consistent with the prescribed inlet and outlet pressures? (Make sure you are careful about units when you substitute into our equation)

 

[Thomot512]

So if I understand correctly $P_i^{2}-P_f^{2}$ comes from the fact that when do the Integral we have to make $\frac{P^{2}}{2}$ and the difference comes from the fact that we make the integral over the length of the pipe? But where goes the - on the right side?

So we have $$(P_i^{2}-P_f^{2})\frac{M}{16}\frac{\pi^{2}D^{5}}{0.97fRT\dot{m}^{2}}=L=794.76[m]$$
That's quite long, I should change the pressure difference in my design.

 

[Chestermiller]

So if I understand correctly $P_i^{2}-P_f^{2}$ comes from the fact that when do the Integral we have to make $\frac{P^{2}}{2}$ and the difference comes from the fact that we make the integral over the length of the pipe? But where goes the - on the right side?

$L-0 = L$

So we have $$(P_i^{2}-P_f^{2})\frac{M}{16}\frac{\pi^{2}D^{5}}{0.97fRT\dot{m}^{2}}=L=794.76[m]$$
That's quite long, I should change the pressure difference in my design.

Before you do that, you owe it to yourself to recheck the calculation to make sure the arithmetic is right and all the units match.

Another thing you can do in the design is use a larger pipe diameter. Increasing the diameter by a factor of 2 reduces the length by a factor of > 32.

 

[Thomot512]

Thanks a lot for your help!