Calculating Entrained Flow in Injector Design: Challenges and Considerations

[holdingmyhill]

Hello,

I am trying to analyze an injector and having trouble coming up with the final equation for the calculation. The apparatus can be seen below.

Assuming we know the inlet flow rate and pressure (A) from the pump/fan and the pipe geometry.
The pressure out of the outlet of the nozzle is simple enough:
P_nozzle=P_inlet+1/2*rho*Q^2*(1/A_inlet²-1/A_nozzle²)

The suction velocity can be then be calculated based on this pressure.
V=sqrt(2/rho * (P_nozzle-P_atm))

However, what I cannot figure out is how to calculate the flow rate of the entrained flow. With this math, the flow is only based on the size of the outer tube.

I am thinking maybe it has to do with shear between the fluid flows? Anybody have any thoughts on this?

 

[Chestermiller]

For the entrained flow, wouldn't you just use the area of the annulus?

 

[holdingmyhill]

For the entrained flow, wouldn't you just use the area of the annulus?

Most problems I've seen worked out that is what they do. However, I'm trying to understand the calculation on what would be the max flowrate of the entrained flow. Obviously, as the annulus gets bigger, the jet would not be able to entrain the whole flow. For example, if I have a 1 inch jet and 1 foot diameter annulus; the 1 foot diameter duct wouldn't all be pulling equal amounts of air across the opening. So I'd like to try understand this maximum better and what the max CFM that could be pulled based on a certain setup.

 

[Chestermiller]

Most problems I've seen worked out that is what they do. However, I'm trying to understand the calculation on what would be the max flowrate of the entrained flow. Obviously, as the annulus gets bigger, the jet would not be able to entrain the whole flow. For example, if I have a 1 inch jet and 1 foot diameter annulus; the 1 foot diameter duct wouldn't all be pulling equal amounts of air across the opening. So I'd like to try understand this maximum better and what the max CFM that could be pulled based on a certain setup.

This is a more complicated situation than a typical Bernoulli problem. Of course, Bernoulli does not accurately describe all situations encountered in fluid mechanics. So, viscous stresses and deformations are likely to contribute to the situation you described. But, if you wish to obtain a Bernoulli-type of approximation to this problem which neglects viscous dissipation of energy, then you need to proceed a little differently. In this case, you have two entering streams and 1 exit stream. So you would need to weight the entering streams in proportion to their mass flow rates. So, you would write something like: $$\dot{m}_1(p_1+\frac{1}{2}\rho v_1^2)+\dot{m}_2(p_2+\frac{1}{2}\rho v_2^2)=\dot{m}_3(p_3+\frac{1}{2}\rho v_3^2)$$
where 1 and 2 are for the two streams at the inlet upstream locations (where the flows are uniform in the x direction) and 3 is for the one stream at the outlet downstream location (where the total flow is again uniform in the x direction). Also, $$\dot{m}_1+\dot{m}_2=\dot{m}_3$$

 

[JBA]

Note: This post was started before the above post that directly addresses the mathematics involved but I have decided to present anyway as a general description of the process.

In general, what you are dealing with is the reduction in static pressure pressure due to the jet velocity at the jet discharge and maintaining a differential between the inlet static pressure on the entrained fluid source and the discharge point of the downstream flow.
So the first question to be addressed is how much fluid will flow through the intake annulus based upon the annulus feed area and the static pressure at the discharge point of the the jet; and, second, based upon the flow friction losses through the discharge line, how long can the static pressure differential required to move the total flow volume be sustained between the entraining point and the discharge point of the system.
As a result, the ratio of the jet discharge area and the entrained fluid feed area at the downstream point of the jet discharge is mostly based upon what the velocity, and therefore the static pressure of the jet flow will be at the increased area at the downstream point of the jet discharge nozzle vs. the jet orifice area. and maintaining an efficient flow pattern in this transition is why most jet eductor systems include a diverging cone section just beyond the actual entraining section at the jet discharge.

Unfortunately the design of most of these types of systems is based more on testing than analysis, which is why the mathematics of these systems is essentially nonexistent and only presented as performance graphs by the manufacturers of these units.

 

[Chestermiller]

@JBA Are you saying that this type of problem can't be solved using computational fluid dynamics (CFD)?

 

[JBA]

No, I am not saying that; but my experience using this technology for valve designs for subcritical gas flow is that CFD has its limits and ultimately only actual flow testing will determine the delivery rates and efficiency of its results because of the fluid characteristic factors you presented in your post. As with FEA, CFD is very dependent upon meshing; and does not address the issue of laminar vs turbulent boundary flow conditions and transitions.

In designing a new steam/water eductor many years ago, I started by writing a program that was based upon configuring a converging steam/water mixing region to maintain, as much as possible, the steam nozzle discharge velocity as the steam condensed due to the heat transfer at the steam/water interface contact surface; but, in the end, until we ran the prototype flow test I had no real clue as to the viability of my procedure. Luckily, it exceeded all expectations and launched a new high performance product line.