# Continuity equation and mass flowing through two pipes

Can we apply continuity equation between the given two cases?
The only difference in the second case is that the pipe of diameter d2 is replaced by a pipe of diameter d3.
Will the mass flow rate be same for both the cases.

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## Answers and Replies

Chestermiller
Mentor
The mass flow rate will not necessarily be the same in both cases, but within each case, the mass flow rate through each of the two tubes will be the same. The latter is a consequence of the continuity (conservation of mass) equation.

Mohankpvk
The mass flow rate will not necessarily be the same in both cases, but within each case, the mass flow rate through each of the two tubes will be the same. The latter is a consequence of the continuity (conservation of mass) equation.
Thank you.
And the book you suggested earlier 'Transport phenomena 'by Bird stewart and lightfoot, is it suitable for beginners(under graduate students)?
If not, Is there any other good introductory reference books for beginners(for fluid mechanics)?

Chestermiller
Mentor
Thank you.
And the book you suggested earlier 'Transport phenomena 'by Bird stewart and lightfoot, is it suitable for beginners(under graduate students)?
If not, Is there any other good introductory reference books for beginners(for fluid mechanics)?
We used Transport Phenomena as undergrads.

Mohankpvk
We used Transport Phenomena as undergrads.
Thank you.

The mass flow rate will not necessarily be the same in both cases, but within each case, the mass flow rate through each of the two tubes will be the same. The latter is a consequence of the continuity (conservation of mass) equation.
Hi, I have a related question.

The equation I see in the snapshot in the OP is $A_1 v_1 = A_2 v_2$. That is: the "volume flow rate" across the two cross sections is the same.
I understand that this is because, for incompressible fluids, the "mass flow rate" should be the same, $\rho A_1 v_1 =\rho A_2 v_2$.

Basically, if the flow is incompressible, whatever mass goes through one cross section should go through the second cross section in the same time interval.
This is a version of the continuity equation, as I understand it.

Now I have a book that apparently maintains that this is always the case. Specifically the book says: since the mass flow rate through the two cross sections must be the same, we get $\rho_1 A_1 v_1 =\rho_2 A_2 v_2$. Then the book proceeds with the particular case of an incompressible fluid, $\rho_1 =\rho_2 = \rho$ and gets the equation for the volume flow rate.

I am not really convinced of this. I do not see why, if the fluid is compressible, the mass flow rate through any cross section should be the same.
Is this really the case?

Thanks a lot for your help.
Franz

Chestermiller
Mentor
Hi, I have a related question.

The equation I see in the snapshot in the OP is $A_1 v_1 = A_2 v_2$. That is: the "volume flow rate" across the two cross sections is the same.
I understand that this is because, for incompressible fluids, the "mass flow rate" should be the same, $\rho A_1 v_1 =\rho A_2 v_2$.

Basically, if the flow is incompressible, whatever mass goes through one cross section should go through the second cross section in the same time interval.
This is a version of the continuity equation, as I understand it.

Now I have a book that apparently maintains that this is always the case. Specifically the book says: since the mass flow rate through the two cross sections must be the same, we get $\rho_1 A_1 v_1 =\rho_2 A_2 v_2$. Then the book proceeds with the particular case of an incompressible fluid, $\rho_1 =\rho_2 = \rho$ and gets the equation for the volume flow rate.

I am not really convinced of this. I do not see why, if the fluid is compressible, the mass flow rate through any cross section should be the same.
Is this really the case?
It's the case for steady state flow in which, at any given location, nothing is changing with time. For unsteady flow, where things are changing with time, you can have accumulation or depletion within the control volume, and the amount leaving can exceed or be less than the amount entering.

It's the case for steady state flow in which, at any given location, nothing is changing with time. For unsteady flow, where things are changing with time, you can have accumulation or depletion within the control volume, and the amount leaving can exceed or be less than the amount entering.
Ok.... so you're saying that a stationary flow is a sufficient condition $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$, even for compressible fluids.
If instead the flow is not stationary, the above equation might not apply.

I have to think about it, but not because I'm doubting it... I need to "visualize" it in an intuitive way, and right now I'm not sure I am able to do that. But "stationary" is helpful. It probably has to do with the flux lines never crossing.

Thanks!

Chestermiller
Mentor
Ok.... so you're saying that a stationary flow is a sufficient condition $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$, even for compressible fluids.
If instead the flow is not stationary, the above equation might not apply.

I have to think about it, but not because I'm doubting it... I need to "visualize" it in an intuitive way, and right now I'm not sure I am able to do that. But "stationary" is helpful. It probably has to do with the flux lines never crossing.

Thanks!
If you have more mass entering a region than leaving, then mass must be accumulating within the region (with its density will be increasing).

If you have more mass entering a region than leaving, then mass must be accumulating within the region (with its density will be increasing).
Of course... But I still can't wrap my head around the fact that, for this to happen, the flow must be non-stationary.

I'm trying to picture the situation in my head. The flow becomes slower as the fluid proceeds. As the velocity decreases (with distance) the density increases. Of course, if the decrease in velocity is proportional to the increase in density, the same mass of fluid goes through any section.
Is it possible for the density to increase faster than the velocity decrease? Like the fluid becomes very compressed?

Maybe I get it.

The book I have defines a stationary flow as a flow whose velocity field does not change in time. But maybe this is not enough, it is not just a matter of velocity. Should everything be time independent? Is this what you mean by steady state flow?

Thanks again

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Chestermiller
Mentor
No. By steady state, what I mean is that, at a given location in the flow channel, the velocity and density don't change with time. However, they do change with spatial position within the flow channel.

Yes, I'm clear on that now.
So in a steady flow both the velocity and the density should be time-independent at a given point in space. I guess that this applies to all the properties of the fluid, including e.g. its pressure and temperature.

What I'm not clear about is whether a stationary velocity field is a sufficient condition for a steady state flow.

If the velocity only depends on the spatial position, but does not depend on time, can we conclude that all the properties of the flowing fluid (pressure, density...) are time independent?
If this is the case, is there a simple argument showing that?
Or in principle the velocity of a fluid can be time-independent while its density for instance changes in time?

I'm asking this because the book I have just says that "in a stationary flow velocity does not change with time" and seems to give mass flow conservation for granted, without (explicitly) assuming that the density should be time-independent too .
It seems to me that one of the two thing should be assumed. Either the density is also time-independent, and hence mass flow conservations seems a natural consequence, or mass flow conservation is assumed, and hence the time independence of the density is a natural consequence.

Am I making sense here?

Chestermiller
Mentor
Yes, I'm clear on that now.
So in a steady flow both the velocity and the density should be time-independent at a given point in space. I guess that this applies to all the properties of the fluid, including e.g. its pressure and temperature.

What I'm not clear about is whether a stationary velocity field is a sufficient condition for a steady state flow.

If the velocity only depends on the spatial position, but does not depend on time, can we conclude that all the properties of the flowing fluid (pressure, density...) are time independent?
If this is the case, is there a simple argument showing that?
Or in principle the velocity of a fluid can be time-independent while its density for instance changes in time?

I'm asking this because the book I have just says that "in a stationary flow velocity does not change with time" and seems to give mass flow conservation for granted, without (explicitly) assuming that the density should be time-independent too .
It seems to me that one of the two thing should be assumed. Either the density is also time-independent, and hence mass flow conservations seems a natural consequence, or mass flow conservation is assumed, and hence the time independence of the density is a natural consequence.

Am I making sense here?
Steady state is defined as a flow and heat transfer situation in which, at each position in space, nothing is changing with time. So, at any given position, velocity, temperature, density, pressure etc. are all constant. In most flow and heat transfer systems, all you need to do is have a constant inlet flow rate and then wait long enough for steady state to be achieved.

As far as stationary is concerned, even though I have had years and years of experience with fluid dynamics, I am not familiar with the use of that term. What does it mean to you?

Hi Chestermiller,
apologies for the delayed reply, and thanks a lot again for your help.

I have to mention that the book I am referring to is a translation (into Italian) of an American text by Cutnell and Johnson. Perhaps my translation back into English is wrong, and I should say "steady state" when I read "stazionario". Apologies, I'm not an expert in fluid dynamics and I assumed I could use the word
"stationary", because it means "not changing in time" in English too.
But I get what you say: this term is not used in the context of fluid dynamics.

What I do not like/understand very much in the book is that it refers to velocity only when it defines a steady state for a fluid flowing. It does not say that not only velocity, but everything else should be constant in time.
Of course velocity does not change in time for a steady state: it is a necessary condition. However I am not sure that it's a sufficient condition.

Chestermiller
Mentor
Hi Chestermiller,
apologies for the delayed reply, and thanks a lot again for your help.

I have to mention that the book I am referring to is a translation (into Italian) of an American text by Cutnell and Johnson. Perhaps my translation back into English is wrong, and I should say "steady state" when I read "stazionario". Apologies, I'm not an expert in fluid dynamics and I assumed I could use the word
"stationary", because it means "not changing in time" in English too.
But I get what you say: this term is not used in the context of fluid dynamics.

What I do not like/understand very much in the book is that it refers to velocity only when it defines a steady state for a fluid flowing. It does not say that not only velocity, but everything else should be constant in time.
Of course velocity does not change in time for a steady state: it is a necessary condition. However I am not sure that it's a sufficient condition.
Well, in a strictly fluid dynamics development, the flow is typically considered isothermal, and the only parameter varying is velocity (and density for an incompressible fluid). Later, when heat transfer is introduced, the definition of steady state is (naturally) expanded to also include temperature.