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My background is more Chemistry and Biology, not Physics, or specifically engineering with fluid dynamics, so bear with me! Also, this is a little long, so I will do my best to make it easy to follow.

The Problem:

I am trying to derive a formula to calculate air flow rates (in CFM) based on the static pressure (inches W.C.) of the HEPA filter I am blowing through for a laminar flow hood that I am building. Link I am trying to figure this out because many HVAC application fans have ratings of their output at certain static pressures, many, many more do not, and it would be really helpful to beable to derive this information on my own.

Information I have available to use:

  • Standard fan information, from the manufacturer (Brake Horsepower, Fan wheel dimensions, RPM &/or CFM).

  • Volume of the Plenum Chamber, Link, in this case >= 4 ft$^3$. (12" x 24" x 24" as it is generally accepted that the size of an adequate plenum must be >= the volume of the filter).

  • The Static Pressure of the HEPA filter, in my case 1" w.g., and although I don’t know how it relates to anything or what it means, I also have the resistance range of, 0.69.

  • The desired volume/rate of air on the intake of the filter, calculated out to 400 CFM @ 1.00" w.g. SP. ( Based upon this formula (Desired FPM Output) x (Volume of HEPA Filter) = Required CFM Input. Link. --The desired output of the entire unit, 100-200 FPM, ideally.

Questions developed from research & attempts to solve:

  • Some fans are rated at 0.00” SP all the way up through +1.5” SP. What are the standard conditions for these specs? I.E. in chemistry you have SC = 1.0 ATM @ 0$^{\circ}$C. If a fan is rated at 500 CFM without any information given about Static Pressure, what would be the static pressure? Is it 0.00” w.c.? above, or below?

  • I have stared at the Affinity, or Fan Laws, Link, for countless hours now. Specifically Law #2. SP$_2$ = SP$_1$( CFM$_2$ / CFM$_1$)$^2$. I was hoping that this would solve my problems, but I have been trying to solve for CFM$_1$, since I know I want SP$_2$ to equal 1.0” and I know I want CFM$_2$ to equal 400 @ SP$_2$. I have been operating under the assumption that a fan, at it’s highest RPM/CFM is operating at 0.0” SP. But then you can’t calculate for the obvious reasons, so this has to be wrong. So back to standard conditions for fans.

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As I understand it, you would like to develop an equation for the flow rate on the exit side of the HEPA filter based on the static pressure the fan produces.

The formula for the pressure versus flow rate fan curve will be unique to every fan. Influencing factors will be whether the fan is axial or centrifugal and for centrifugal fans, whether the fan blades are forward curving, backward curving or radial.

Other factors that will influence the exit flow rate from the HEPA filter will be associated with the design of your system:

  • shock losses – basically caused by the abruptness of changes in air direction & the lack of streamlining
  • friction from materials in the air pathway due to material roughness

  • back pressure, if it occurs

  • the resistance to airflow caused by the filter & anything else blocking the path of the air flow

To get a better answer you will need to include a diagram of your system, with dimensions.

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I don't know the whole answer to your question, but I can tell you why you are confused about the fan laws. Those laws are used if you want to change the motor attached to the fan, they don't tell you the shape of the fan curve itself, which is the locus of possible operating points for a given fan and motor combination. You are correct that at maximum flow rate, static pressure is zero. Similarly, at maximum static pressure, flow rate is zero

What you actually want is to know the relation between the static pressure (usually called head, $H$) and the flow rate $Q$ (what you have called CFM). I don't know the typical shape of fan curves, but if I were trying to do what you are doing, I'd probably just assume the fan curve was approximately $H = H_{max} - AQ^3$.

If you make that assumption, then given that you know the maximum head and maximum flow rate from the data sheet, you can estimate what the flow rate will be at your operating pressure. Unfortunately, the only way to know for sure is to experiment, unless someone else here knows the real fan curve equation. Even then, keep in mind that engineering calculations, especially those dealing with compressible fluids, can easily by 20% off. Give yourself a generous safety factor.

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Generally, the pressure drop across a filter will follow the second fan law where: $$\Delta P=C_v Q^2$$ Therefore, you only need one data point for the filter ($\Delta P_1, Q_1)$ to calculate the proportionality constant of that filter material. Then you can extrapolate for a continuous range of CFM. Assuming the filter is the biggest source of resistance, that should give you a pretty good approximation of the system curve ($Q$ vs. $\Delta P$).

From there you'll need to overlay your fan curves to find the operating point. That will be difficult if the fan specs don't include them. The operating point is where the fan curve intersects the system curve for a particular RPM, which you get to choose.

To one of your other questions, if the fan is rated at 500 CFM without giving a static pressure, I think you would have to assume that it's for the free-running condition. In other words, it's the maximum CFM without any imposed resistance to the flow. In more words - no pressure drop, or 0 static pressure. I'm not sure how useful that is because you already know there's going to be resistance to flow in the form of the filter.

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