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I'm trying to make a generally simple calculation but I'm not sure if I'm approaching this correctly. So as I'm making assumptions solving this problem, please comment or correct me wherever I'm wrong - fluid dynamics is a dark past for me.

I have a chamber in which I input gas with known volumetric/mass flow which exits the chamber with the same volumetric/mass flow. The equipment is supposed to produce incompressible and steady flow through small holes throughout the width. So let's say the gas is flowing in the $x$ direction and exits after known length $L$.

Assumption 1: I assumed Euler's equation with no force applied: $$ \frac{D \vec u}{Dt} = - \frac{\vec \nabla p }{\rho} $$

There is a pressure difference between the two ends, lets say $p_1$ in the input and $p_2$ in the output.

Assumption 2: Since the equipment is well designed, I would assume a linear pressure function across the chamber after equilibrium like ($x=0$ is the input $x=L$ is the output): $$ p(x) = p_1 + \frac{p_2 - p_1}{L}x$$

By also calculating the material derivative, I find the result:

$$\frac{\partial u_x}{\partial t} + \vec u\vec\nabla u_x = - \frac{\Delta p}{\rho L} $$

which leads to: \begin{equation} \frac{\partial u_x}{\partial t} + u_x\frac{\partial u_x}{\partial x}= - \frac{\Delta p}{\rho L} \; \; \; (1) \end{equation}

since my velocity isn't a function of $y$ or $z$.

Question 1: Is the gas accelerating in the chamber? Because if it is, I surely shouldn't rule out the first term. Because if I do, solving the differential equation I get:

$$ u_2^2 - u_1^2 = - \frac{\Delta p}{\rho} $$

which doesn't help much. I could calculate the average of the velocities to assume that's the gas velocity in the chamber but that feels like an oversimplification.

Question 2: Solving equation (1) would give me a velocity that is a function of $x$ and $t$. Is this possible for a steady and incompressible flow? Because the idea of the velocity being a function of $t$ in the chamber worries me. I cannot calculate with the equipment in my hands how much time it takes for a "portion" of the gas to exit the chamber.

Sorry if that's a long post and thanks for any help in advance.

Edit for future readers: Simulation or compressible flow analysis is necessary, as gas flow should always be compressible

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What you have derived in your Q1 is essentially (save for a missing factor 1/2) Bernoulli’s equation which states:

$$ P + \rho v^2 / 2 = const $$

In the absence of external force and assuming a flow which is steady state, inviscid, incompressible and constant temperature.

So from here you could solve for the velocity at one end of the pipe given a boundary condition of the other velocity and a known pressure and density.

I am not 100% sure what you are actually trying to solve for since the question is slightly unclear, but hopefully that helps to clarify what you have derived.

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  • $\begingroup$ Oh yes, I missed a 2. Thanks! That makes sense now, so if I suppose the gas is not accelerating (I don't think it is but I'm not sure), if I calculate the entry velocity I can calculate the output velocity. What I'm originally trying to calculate is the velocity of the gas, to calculate the average velocity of a gas molecule moving with the gas. $\endgroup$ – Stamatis Tzanos Aug 12 at 21:51
  • $\begingroup$ The starting assumptions are invalid. Gases are compressible, so as pressure drop occurs under steady mass-flow conditions, the volumetric flow rate is increasing and the gas is experiencing an acceleration. Bernoulli's equation doesn't apply for this case. Also, pressure is not linear with length. If you simulate gas flows (which I recently did), you will find that there is a quadratic acceleration involved, whereby downstream pressure drop rapidly increases with velocity. $\endgroup$ – David White Aug 12 at 22:27
  • $\begingroup$ @DavidWhite I don't want to fully simulate the gas flow, I know this is more complex. I'm trying to roughly compute the velocity of the gas / molecules after equilibrium in the chamber in a steady input/output flow. So how do you believe I should approach this? $\endgroup$ – Stamatis Tzanos Aug 13 at 8:08
  • $\begingroup$ @StamatisTzanos, you probably want to find a way to simulate your problem, and use simulation results to develop an equation (polynomial?) that closely matches your results. $\endgroup$ – David White Aug 13 at 13:26
  • $\begingroup$ I don't have some simulation program in my hands at the moment, nor I need to simulate this entirely. I'm trying to make a rough estimation on paper about the velocity of the gas or the molecules. You suggest such a rough estimation is not possible? $\endgroup$ – Stamatis Tzanos Aug 13 at 15:07

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