### Flow Calculations

In order meet our target flow of 170 liters/minute (6 CFM), we have to calculate the requirements of our fan. Through our prototyping processes, we have discovered that consumer computer fans are not capable of delivering the airflow that we needed, so we took some time to get some estimates on the minimum requirements for fans which would deliver the necessary airflow. The 3M Versaflo has a high mode which delivers a high setting of 250 litres/minute (9 CFM) as well, so we want to see what would be necessary for that.

#### Air Hose

The air hose can be a major contributor to the total pressure drop across the system. The pressure drop of compressed air through a pipe is as follows:

$$d_p=7.57*10^4\dfrac{q^{1.85}L}{d^5p}$$

Compressed Air - Pressure Loss in Pipe Lines

• $d_p$: pressure drop ($kg/cm^2$)
• $q$: volumetric air flow at atmospheric conditions ($m^3/min$)
• $L$: length of the pipe ($m$)
• $d$: inner diameter of the pipe ($mm$)
• $p$: initial pressure ($kg/cm^2$)

From the article Pressure Loss in Hoses, the estimate the relative friction factor of various materials in comparison to a steel pipe. A rough corrugated hose will have a similar geometry as the metal corrugated hose listed in the article. Medical flow hoses are usually wire reinforced plastic with the metal spiral on the outside of the hose allowing the inner lining to be smooth. Based on those figures, we have calculated some approximate pressure drops on different diameters of hoses. Here the important part to note is that the pressure drop exponentionally decreases with an increase in diameter.

Diameter $in$ Length $ft$ Friction Multiplier Pressure drop $inH_2O$
1 3 1.5 - 2.5 0.145 - 0.252
1.25 3 1.5 - 2.5 0.047 - 0.080
1.5 3 1.5 - 2.5 0.019 - 0.031

#### Filter Efficiency

The pressure drop generated by the filter is dependent on the linear flow rate through the filter, thus the filter linear flow rate is inversely correlated with the total area of the filter. As most filters are pleated, this area is larger than the area of the individual filter.

$$A=w\sqrt{h^2+(pd)^2}, f_l=f_v/A$$

• $A$ = total filter media area
• $w$: width of filter package
• $h$: height of filter package
• $d$: depth of filter package
• $p$: the total number of heightwise pleats over the entire package
• $f_l$: linear flow for the filter media
• $f_v$: target volumetric flow

Recent advances in antimicrobial air filter (Khoiruddin et al., 2018)

Filter media Width $in$ Height $in$ Depth $in$ Pleats Area $in^2$ Required Flow $m/s$ Pressure drop $inH_2O$
HEPA Kenmore EF-1 4 6 0.7 46 195 0.023 ~0.2
HEPA Honeywell HPA100 10.2 6.5 1.6

#### Fan Power & Requirements

The work exerted by the fan is proprotional to the air moved as well as the pressure increase generated. If we were to look at an ideal fan, the power would be given as:

$$p_i={\Delta}p*q$$

Fans - Efficiency and Power Consumption

• $p_i$: ideal power consumption ($w$)
• ${\Delta}p$: total pressure increase from the fan ($Pa$, $N/m^3$)
• $q$: volumetric air flow of the fan ($m^3/s$)

Physical fans, however, are not able to deliver this ideal efficiency. They have a efficiency factor defined as:

$$\mu_f=\frac{p_f}{p_i}$$

For smaller fans, this efficiency factor tends to be quite poor overall, which gives a lot of room for individual fans to have different performance characteristics due to fan design and the parts used. Understanding Fan Efficiency Grades (FEG) shows that the same efficiency grade fan will have significantly lower efficiency when the impeller diameter is under ten inches. Extrapolating from the graph in the paper, a mid grade 40mm fan will have an efficiency around 20%.

Knowing the total pressure drop incurred by the selected hose and filter, we can take a look a the different fans available. The two primary categories of fans are blower and axial fans. We are looking for a fan which can exert the required static pressure to overcome all the pressure drops of the system at the target volumetric flow. Fan specifications define the peak static pressure when there no flow and the peak flow when there is no static pressure generated. In real conditions, there will both some amount of static pressure generated and some flow necessary. To see if a specified fan meets our requirements we have to look at its datasheet and the static pressure versus airflow graph. If at the desired airflow, the fan can achieve the necessary pressure, the fan will be a good candidate.

Given the estimates above with a margin of error incorporated for unaccounted friction, we may see a total pressure drop of 0.5$inH_2O$ with a 1" diameter hose. If we were to utilize a wider hose, 1.25" the pressure drop could be as low as 0.25$inH_2O$.

Make Model Static pressure $in H_2O$ Flow $CFM$ Current $A$ Power $W$
Sanyo Denki 9GA0412P3H01 1.39 19.1 0.25 3.36
Sanyo Denki 9GA0412P3G01 1.81 21.5 0.39 4.68
Sanyo Denki 9GA0412P3J01 2.15 23.7 0.49 5.88
Sanyo Denki 9GA0412P3K01 3.21 28.6 0.92 11.02