fan outlet velocity distributions and calculations

8/14/2019 Fan Outlet Velocity Distributions and Calculations

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Vo (LFM) d (ft) A (ft 2) x (ft) Vx (LFM) % Difference Equation 1 1066 0.13 0.0135 1 1081 0% Equation 2 1066 0.13 0.0135 1 962 11% Equation 3 1066 0.13 0.0135 1 700 35%

Table 1. Variation in results of equations 1, 2 and 3.

The lack of accurate, easy to use, equations for fan air flow velocity in the literature has prompted us toderive our own equations based on experimentation.

3.2 Experimental Setup A cube test chamber of 1 m length was constructed out of wood. One side of the chamber was left withouta solid wall and was covered with a thin flexible screen to allow free flow of outgoing air, but preventingoutside air from coming in. On the opposite side a metal plate was placed with a cutout for the fan. Fans ofdifferent sizes required a new metal plate in order to exactly match their cross-sectional area. Inside the boxa hotwire anemometer probe was mounted on top of a XY linear stage. The stage would move the probeacross the area of the fan along its center and away from the fan. The probe was connected to ananemometer which in turn was connected to a data acquisition card. The card was connected to a PC withLabView software which would process the data. Post processing of the data into meaningful plots andfunction derivation was done in MatLab.

Equipment List:

1) XY Linear Stage: OWIS GmbH LTM 80, operated with two PS 10 position controls2) Hotwire Probe: Dantec Dynamics Probe Type 553) Anemometer: A.A. Lab Systems Anemometer AN 10034) Data Acquisition Card: National Instruments Analog to Digital USB-6009 Card5) Software: National Instruments LabView, MatLab, Excel

Figure 3. Test chamber setup.

Since the anemometer measures differences in voltage and not velocity a calibration procedure had to beperformed before measuring a new fan or restarting measurements on a new day in order to createcalibration curve function which would correlate measured voltage to air velocity. The calibration wasperformed using an airtight box connected to a compressed air source at 4 ATM through a valve and avolumetric flow rate meter. Since the box had a single opening with a known diameter the velocity of airleaving the box could be easily calculated. Measuring the air exiting the box at a known velocity with ananemometer allowed creating a second order polynomial calibration function.



4

Figure 4. Sample Anemometer Calibration Function

Once the anemometer was calibrated for a specific fan it was placed inside the chamber and connected tothe linear stage. The measurements were made at a step of 0.5 – 1 mm depending on fan size, and ateach point 60,000 samples were taken. The temperature sampling rate was 1 Hz for the duration of theentire velocity measurement.

The data was measured in 3 different configurations:

1) Vertical placementof the hotwire Figure 5

2) Horizontalplacement of thehotwire

Figure 6

3) Horizontalplacement of thehotwire and a 90degree turn of thefan (PhasePlacement)

Figure 7

Horizontal and Vertical probe configurations were necessary in order to capture the swirl of the air flow.

y = 0.0107178x 2 - 0.2166470x + 1.1564115R² = 0.9977082

00.5

11.5

2

2.5

33.5

44.5

5

-15 -10 -5 0 5 10 15

C a

l c u

l a t e d V e

l o c

i t y ( m / s )

Voltage (V)

Velocity vs. Voltage



5

Figure 8. Vector representation of Vswirl

(Equation 4)

The 3 rd configuration, 90 degrees rotation of the fan, or the Phase, was done in order to measure theasymmetry of the air flow. The reason that there is asymmetry in the flow is because the fans are not

manufactured perfectly symmetrical. On the face of each fan there is a wide vane which is used to pass thewires through. This vane creates an asymmetry in the air flow as it comes out of the fan.

Figure 9. Asymmetry of the fan

Four fans from Jaro Thermal were used in all measurements.

Fan outer diameter(mm) Radial step(mm) Axial step(mm) Startingdistance (mm) Endingdistance (mm)

25 0.5 15 14 170 11

40 0.5 15 14 89 6

60 1 15 14 164 9

90 1 30 14 284 9

Table 2. List of fans measured in the experiment.

Wide vane with wires



6

The reason the ending distance is different for each fan is because the fans were analyzed at a different

/ ratios (see Data Analysis section for an explanation), which were dimensionless parameters requiredfor similarity analysis used in the derivation of the final equation.

3.3 Data Analys isEach run of the hotwire resulted in 60,000 samples taken at each of the 200 – 500 points along the traversedirection. Each of these sample sets was averaged to a single value, thus creating a set of discrete voltagereadings. This set was calibrated using the calibration function discussed above, which turned it into avelocity profile. Thus, for every fan, there now were a number of velocity profiles for each configuration.

Once all the velocity profiles were made, each profile was averaged around the zero point to make itsymmetrical. Since each fan is not perfectly symmetrical, as was explained above, this created an error inthe final function of the velocity profile. This error was a small price to pay in return for the ease of use of theprofiles.

As a requirement for the similarity analysis, dimensionless parameters were needed to standardize thefunction for all fans regardless of their dimensions. Therefore all profiles, which were measured at specificaxial distances away from the fan, were linearly interpolated to find their value at different values of thedimensionless distance parameter, / , which was defined as the axial distance x (away from the fan)normalized by the outside diameter of the fan d.

As can be seen from the following curve if the velocity profiles are normalized by converting them intofunctions of unitless parameters, such as / , which is a ratio between any axial velocity and themaximum axial velocity for that profile, and / . , which is a ratio between any radial distance and theradial distance from the fan’s center to the point where maximum velocity was halved, then it can be seenthat all unitless velocity profiles curves fall on top of each other thus indicating that they can beapproximated into one curve. This implies that there should be one equation that can describe the velocityprofile of a fan regardless of its dimensions.

Figure 10. Similaritybetween all normalizedvelocity profiles for various

/ values. All curvesbasically fall onto thesame curve.

Finally each symmetrical profile had to be phase averaged in order to take account for the swirl of the air.The horizontal and phase profiles were averaged together to create a final, symmetrical, velocity profile foreach cross section.

-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/r 0.5umax

u / u

m a x

Normal Velocity Profile for a 60mm Fan at various X/D

X/D=0.5X/D=0.9X/D=1.25X/D=1.5X/D=1.8



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Figure 11. Samplemeasured velocity profile.Notice the asymmetry.

Figure 12. Sample averaged

velocity profile.

Figure 13. Sample averagedvelocity profile at a given/ . Notice that the profile isperfectly symmetric.

-150 -100 -50 0 50 100 1500

0.5

1

1.5

2

2.5

Radial Distance - r (mm)

V e

l o c

i t y ( m / s e c

)

Mean Velocity P rofile for a 60mm Fan, With a Horizontal Placement, at a Radial Distance of 44(mm) from the Fan

-150 -100 -50 0 50 100 1500

0.5

1

1.5

2

2.5


V e l o c

i t y ( m / s e c

)

Averaged Mean Velocity Profile for a 60mm Fan, With a Horizontal Plac ement, at a Radial Distance of 44(mm) from the Fan

-150 -100 -50 0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


V e

l o c

i t y ( m / s e c

)

Averaged Mean Velocity Profile for a 60mm Fan, With a Horizontal Placement, at X/D=1.2, meaning a Radial Distance of 72(mm) from the Fan



8

Figure 14. Sample

symmetrical phaseaveraged profile.

-80 -60 -40 -20 0 20 40 60 800

0.5

1

1.5

2

2.5


V e l o c

i t y ( m / s e c

)

Phase Avereged Velocity Profiles of a 25(mm) Fan, at Various X/D

X/D=0.9X/D=1.2X/D=1.5X/D=1.8X/D=2X/D=2.3X/D=2.6X/D=3X/D=4X/D=5X/D=6

Figure 15. All phaseaveraged profiles for a25 mm fan at various/ .

3.4 Finding the generic function for maximum velocity

In order to come up with the final equation a theoretical model of the flow had to be developed.Each velocity profile was divided into two parts, the part between the velocity peaks, and the partoutside of the peaks.

The part of the data between the peaks was assumed to act as the wake behind a row of cylindricalbars. The solution to this problem is defined as follows:

Figure 16. Flow pattern behind a rowof bars.

2

31 cos(2 )

8r

l xuu

(Equation 5)

– Flooding velocity (i.e. the wind velocity that blows on the bars)

-150 -100 -50 0 50 100 1500

0.5

1

1.5

2

2.5


V e

l o c i t y ( m / s e c

)

Phase Averaged Mean Velocity Profile for a 60mm Fan at X/D=1.2, meaning a Radial Distance of 72(mm) from the Fan



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u – Air velocity in the wake

– Velocity defined as

λ – Distance between two velocity peaks

l – Mixing length (a flow dynamics value that represents how efficient the energy transfer is in a fluid)

x – Axial distance

r – Radial distance

Source: Hermann Schlichting, Boundary-Layer Theory , 7th

edition, McGraw-Hill, 1979, p. 744, eq. 24.41.

The part of the data outside of the peaks was assumed to act as a circular turbulent jet the solutionto which is defined as follows:

Figure 17. Streamlines ina circular turbulent jet.

20 2

3 18 1

14

K u

x

0

1 34

K y x

(Equation 6)

u – Velocity in the jet

η – Dimensionless distance coordinate

K – Constant kinematic momentum ⁄ ε o – The virtual kinematic viscosity of the fluid ⁄

x – Axial distance

y – Since velocity u is maximum when λ 2⁄ , Schlichting’s variable y is defined as λ 2⁄ , which isthe center of the jet

r – Radial distance

Source: Hermann Schlichting, Boundary-Layer Theory , 7 th edition, McGraw-Hill, 1979, p. 748, eq. 24.46.

3.5 Finding the generic func tion for the data insi de the peaks

Equation 5 can be rearranged in the following manner, since velocity u umax when r λ/2:

2 2

max3 3

11

8 8u

u u u ul x l x

(Equation 7)

For simplicity, an arbitrary parameter A can be defined ,when r λ/2:

2

3

18

Al x

(Equation 8)

Which then can be substituted into equation 7:

2

max 3

11 1

8u u A u

l x

(Equation 9)



10

Finally, equation 9 can be rearranged into the following form:

2

3

max

max

11 cos(2 )

8

1cos(2 )

1 1

1 cos(2 )1 1

r u u

l x

A r u

A A

A A r u

A A

(Equation 10)

To further simplify the problem an arbitrary parameter Bcan be defined:

1 A

B A

(Equation 11)


max

1 cos(2 )u r

B Bu

(Equation 12)

Equation 12 is the unitless velocity parameter for the wake part of the velocity profile.

3.6 Finding the generic func tion for the data outs ide the peaks

It can be seen that at maximum velocity η 0 in equation 6. The definition for η as defined inequation 6 can be substituted back into that equation’s definition for u which will result in thefollowing equation:

max22

2 20

3 11 64 2

uu

K r x

(Equation 13)

To further simplify the problem an arbitrary parameter Ccan be defined:

20

364

K C

(Equation 14)


22max

2

1

1

2

uu C

r

x

(Equation 15)

Equation 15 is the unitless velocity parameter for the jet flow part of the velocity profile.

3.7 Derivation of the equation for the maximum air velocity

For any fan its maximum volumetric flow rate Q 3 ⁄ is provided by the manufacturer in thefan’s datasheet, usually in the form of a fan performance curve. But it can also be calculated, which



11

is what was done in this study. (See Appendix B, Step 2, for details.) Using that as a knownparameter maximum air velocity can be derived as follows:

Q uA (Equation 16)

u – Air velocity

A – Cross-sectional area of the flow

The momentum of an air flowing system is defined as follows:

2 J u A (Equation 17)

ρ – Air density

Which further can be defined as kinematic momentum ⁄ as follows:

2 J K u A

(Equation 18)

Equation 18 can be further simplified as follows, thus defining the kinematic momentum in terms ofvolumetric flow rate, Q:

2 2Q QK A

A A

(Equation 19)

2 2

2 2

2 2 4out in

out in

D D A D D

(Equation 20)

2

2 2

4

out in

QK

D D (Equation 21)

out D d - The outer diameter of the fan at the edge of the blades

in D - The inner diameter of the fan, where the blades begin (i.e. the diameter of the fan’s hub)

Besides the definition in equation 21 the Kinematic Momentum of a fluid is also defined as follows:

22 2 2

0 02

2 2wake jet K u rdr u rdr u rdr

(Equation 22)

Source: Hermann Schlichting, Boundary-Layer Theory , 7 th edition, McGraw-Hill, 1979, p. 748, footnote.

Equation 22 can be solved using standard methods of integration, such as integration by parts.(See Appendix A for the detailed derivation.) Then the result can be inserted into equation 21 whichresults in the following equation:

122 2 2 2

2 2 2max

2 2

4 3 1615 32

16 8 32 384out in

Qu B B x C x

C D D

(Equation 23)

B and C are empirical constants. However, B is a constant that is always the same regardless of fansizes and axial distance, whereas C changes with fan size and axial distance away from the fan.

It was necessary to find the numerical values of B and C.

By applying the similarity principal described above in section 3.3 a single set of curves wasderived for the four fans measured and then averaged into a single curve.



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Figure 18. Average velocity profiles for 4 fansmeasured.

Figure 19. Single fitted average velocityprofile.

The final fitted curve in figure 19 is described by the following function:

max max

max

max

0.5 max 0.5

22max0.5 max

0.5

0.57705 : 1 0.22982 cos 5.44423 0.22982

10.57705:

1 2.3169 0.57705

u u

u

u

r u r r u r

u r uu

r u r r

(Equation 24)

The analysis was further normalized by creating dimensionless parameter λ ⁄ , which was in turnnormalized verses dimensionless parameter / . The results of this normalization can beseen on the following plot:

Figure 20. λ ⁄ normalized verses / for different fans.

From each curve in figure 20, linear regressions were extracted, as shown on figures 21 and 22.

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/r 0.5umax

u / u

m a x

Average velocity profiles for different fan diameters

Fan diameter -25(mm)Fan diameter -40(mm)Fan diameter -60(mm)Fan diameter -90(mm)

-1.5 -1 -0.5 0 0.5 1 1.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/r 0.5umax

u / u

m a x

The Curve Fitted to the Average Velocity Profile Curve

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

X/D

/ D

/D vs. X/D for the different fans

Fan Diameter - 25

Least squares linear regression, /D=0.29086(X/D)+0.68973, R 2=0.91089Fan Diameter - 40



Least squares linear regression, /D=0.73289(X/D)+0.28375, R 2=0.93774



13

Figure 21. Slope extraction. Figure 22. Offset extraction.

0.0075935 0.10077s d (Equation 25)s – Slope of the curve as a function of d

0.0074483 0.83851o d (Equation 26)

o – Offset of the curve as a function of d

The linear regressions for slope and offset can be used to calculate λ as follows:

20.0075935 0.10077 0.0074483 0.83851

xs od d

sx od d x x d d

(Equation 27)

Even though the fit of each individual slope and offset function seems to be rather off, theoverall regression of λ ⁄ remains accurate as can be seen on figure 23. The error growsas x ⁄ grows, but for x 3⁄ , the fit is rather good as can be seen on figure 24.

Figure 23. Comparison between actual linear regression of λ ⁄ vs. calculated one.(The linear regression is the measured trend line, while the linear fit is the calculated trend line.)

20 30 40 50 60 70 80 900.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fan diameter - D[mm]

S l o p e o f

t h e

L i n e a r

R e g r e s s

i o n

f o r

t h e

V a r i o u s

F a n s

Slope of the Linear Regression

Linear Fit to the Slopes: Slope=0.0075935D+0.10077; R 2=0.84471

20 30 40 50 60 70 80 900.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fan diameter - D[mm]

O f f s e t o

f t h e

L i n e a r

R e g r e s s

i o n

f o r

t h e

V a r i o u s

F a n s

Offset of the Linear Regression

Linear Fit to the Offsets: Offset=-0.0074483D+0.83851; R 2=0.58181

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

X/D

/ D

The Linear Regressions for /D for Various Fans vs. The Linear Fit to t hese Regressions

Linear Regression - 25mm Fan

Linear Fit - 25mm Fan, R 2=0.82108Linear Regression - 40mm Fan



Linear Fit - 90mm Fan, R 2=0.93121



14

Figure 24. Linear Regression of . ⁄ as a function of x ⁄ for different fan diameters

From figure 24 the formula for . can be extracted, which in turn was used to calculate thevalues of B and C. Note that units for . , x, and din the following equations are mm, since thedata on the graphs was measured and correlated as such.

max0.5 0.36853 0.47291ur xd d

(Equation 28)

By using equations 12 and 24, when r 0 the value for B can be calculated:

max

1 cos(0) 1 2 0.54036

1 0.540360.22982

2

u B B B

u

B

(Equation 29)

max

14

uu

is picked arbitrarily for simplicity, and then by using equation 24 it was calculated that

max0.5

1.234

u

r

r . Then equation 28 can be used to get a simplified equation for Cin terms of x

and dsince . and λ are functions of x.Both x and d must be in the units of mm in thisequation .

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

X/D

r 0 . 5

U m a x / D

r 0.5Umax

/D vs. X/D for the different fans

Fan Diameter - 25(mm)Fan Diameter - 40(mm)Fan Diameter - 60(mm)Fan Diameter - 90(mm)

Least squares linear regression, r 0.5Umax

/D=0.36953(X/D)+0.47291, R 2=0.89491



15

max

max

max

max

max

max

2 22 2

2 0.520.5

2

0.520.5

2

2

0.50.5

2

3

1 1 14

1 12 2

2 12

2

0.404385 0.164315 3.72415 3.7967 10

uu

uu

uu

C C r r r x x r

C r r

x r

xC

r r

r

xC

x d d x d

2

(Equation 30)

4 Results

A single equation that describes the maximum air velocity, umax,from a fan was derived:

122 2 2 2

2 2 2max

2 2

4 3 1615 32

16 8 32 384out in

Qu B B x C x

C D D

(Equation 23)

B 0.22982

2

230.404385 0.164315 3.72415 3.7967 10

xC

x d d x d

d = The outer diameter of the fan at the edge of the blades – units: mm

x = Axial distance away from the fan where velocity of air is being obtained – units: mm

To calculate use x and d in mm. The resulting value for C is a unitl ess number.

20.0075935 0.10077 0.0074483 0.83851d x x d d d = The outer diameter of the fan at the edge of the blades – units: mm

x = Axial distance away from the fan where velocity of air is being obtained – units: mm

To calculate λ use x and d in mm. But when the calculated value for λ is plugged back into equation foru max (eq. 23) convert it to meters, since the units of u max are meters/sec.

Dout - The outer diameter of the fan at the edge of the blades – units: meters

Din – The inner diameter of the fan, where the blades begin (i.e. the diameter of the fan’s hub) –units: meters

Q- Volumetric flow rate (know from the fan datasheet) – units: meters 3/sec

In order to recreate the complete velocity profile plot the value for umax needs to be substituted backinto the equations for wake (equation 12) and jet flow (equation 15). λ is known from equation 27

and r and x are given by the definition of the problem, since they are locations at which the velocityprofile needs to be calculated. Note that λ, r,and xmust be in meters and umax must be inmeters/sec when plugged back into equations 28 and 29.

max 1 cos(2 )r

u u B B

(Equation 28)



16

max22

212

uu

C r

x

(Equation 29)

Plotting equations 28 and 29 on the same graph results in the complete velocity profile as shownon figure 25 in red.

Figure 25. Comparison between measured and calculated velocity profiles for a 25 mm fan.

The maximum error of the calculated velocity profile when compared to the measured one is roughly 16%.This is a much more accurate correlation to actual measurements than the existing solutions summarizedin section 3.1, which could deviate from reality by over 100% and did not represent the actual type of airflow from a fan. It is possible to further reduce this error by performing more measurements of various fans,which would result in tighter phase average profiles thus improving the overall equations and constantvalues for B and C.

5 Summary A clear methodology was shown for deriving equations for maximum velocity of air blown from a fan, aswell as equations to plot the velocity profile. Engineers can easily port these equations into a spreadsheetwhich will quickly calculate for them the maximum velocity of air at any given distance from the fan as wellas tell them what kind of air velocity a specific electronic component will see when positioned in front of thefan. The error margin of these calculations will be roughly 16% and can be further improved by moremeasurements of fans.

Bibliography/References

[1] ACGIH Industrial Ventilation A Manual of Recommended Practice, 25th Edition, ACGIH, 2004.[2] Howard Goodfellow, Esko Tahti, Industrial Ventilation Design Guidebook , Academic Press, 2001.[3] Benoit Cushman-Roisin, Environmental Fluid Mechanics , Future Wiley Publication, March 2010,

http://engineering.dartmouth.edu/~cushman/books/EFM.html [4] Hermann Schlichting, Boundary-Layer Theory , 7 th edition, McGraw-Hill, 1979.[5] J. Hennissen, W Temmerman, J Berghmans, K Allaert, Modeling of Axial Fans for Electronic

Equipment , Eurotherm Seminar No. 45: Thermal Management of Electronic Systems, September 20-22, 1995, IMEC, Leuven, Belgium.

-40 -30 -20 -10 0 10 20 30 400

0.5

1

1.5

2

2.5

3


V e

l o c

i t y ( m / s e c

)

Comparison between Measured and Calculated Data for a 25mm Fan, at X=14mm

Measured Data

Calculated Data, R 2=0.77919



17

Appendix A

Kinematic Momentum Integral (eq. 22) Solution

Step 1:2

2 2 2

0 0

2

2 2wake jet K u rdr u rdr u rdr

Step 2: max 1 cos 2wake

r u u B B

Step 3:

2

2max

2 2 2 2max

1 cos 2

1 2 2 1 cos 2 cos 2

wake

r u u B B

r r u B B B B B

Step 4: 2 2 2 2 2max ( 1 2 2 ( 1) cos 2 cos 2 )wake

r r u rdr u B B r B B r B dr

Step 5:

2 2 2 2 2 2max max max

2 2 2 2 2max 1 max 2 max 3

1 2 2 1 cos 2 cos 2

1 2 2 1

r r u B B rdr u B B r dr u B r dr

u B B I u B B I u B I

Step 6: 2

1 12r I rdr C

Step 7: 2 cos 2 r

I r dy

Integration by parts is used to solve this integral

Step 8:

, ' cos 2

sin 2

' 1, sin 22 2

r u r v

r

r u v

Step 9:

2

2 2' sin 2 sin 2 sin 2 cos 22 2 2 2

r r r r I uv u vdy r dr r C



19

Step 16:

Step 17:

22 max

42

212

jet

uu

C r

x

Step 18:

22 max

42

212

jet

uu rdr rdr

C r

x

This integral is too complex to solve analytically, so a mathematical software package, such asMathcad, Maple, or Mathematica needs to be used to arrive at the following solution

Step 19:

5 33 2

2 2 6 4 2max

322

2

2max

15 40 33 162 2 2

961

2

5arctan

32 2

jet

C C C r r r

x u x x xC C u rdr r

x

u x C r C

xC

Step 20:

The integration boundaries are between2 and . The first term is zero for and is

2 2max

6 x u

C for

2

, and so:

Step 21:

2 2 22 1 1 2 2max max

max

2

2 2 2max

2

5 1tan tan 0 15 32

6 32 192

2 15 3296

jet

jet

x u u xu rdr x C x u

C C C

u rdr x C x uC

max22

212

jet

uu

C r

x



20

Step 22:

22 2

02

2 22 2 2 2

max

2

4 3 1615 32

4 2 8 96

wake jet K u rdr u rdr

B B x C x uC

Step 23: 2

2 2

4

out in

QK

D D

Step 24: 2 2 2

2 2 2 2max 2 2

4 3 16 415 32

4 2 8 96 out in

Q B B x C x u

C D D

Step 25:

12 2 2

2 2 2 2

max 2 2

12 2 2 2 2

2 2 2

2 2

4 4 3 1615 324 2 8 96

4 3 1615 32

16 8 32 384

out in

out in

Qu B B x C xC D D

Q B B x C x

C D D

Answer: 1

2 2 2 2 22 2 2

max 2 2

4 3 1615 32

16 8 32 384out in

Qu B B x C x

C D D



21

Appendix BSample solution of Maximum Air Velocity using the equations presented

in this paper

This example illustrates how to calculate the maximum air velocity and the velocity profile for a 25mm fan at an axial distance of 14mm.

Step 1: Fan dimensions are measured:

Dout = d = 25 mm = 0.025 m

Din = 15 mm = 0.015 m

Step 2: Q is calculated in one of two ways:

Option 1: Fan flow rate, Q, can be taken directly from the manufacturer’s datasheet.

Typical Fan Performance Curve from the fan manufacturer’s datasheet. Maximum flowrate Q is shown for different fan types on the x axis of the plot.



22

Option 2: Q can be calculated using the velocity profile measured for x = 14 mm using the followingchart:

From this chart values for radial distance r and velocity u can be picked off at every point. For thesake of this example we assumed that there are approximately 400 points on the chart. Thesevalues can be used in solving the integral in equation 22 by recalling equations 16 - 19:

Momentum of a fluid, J, is defined as:

2 J Au

ρ – Density of the fluid

A – Cross sectional area through which the fluid is flowing

u – Velocity of the fluid

Kinematic momentum of a fluid, K, is defined as:

2 J K Au

Volumetric flow rate, Q, is defined as:

Q uA

And therefore by substitution

2QK

A

By using equation 22 and the data from the velocity vs. radial distance chart Kcan be solved for.Since there are 400 data points on the chart and for each data point there is a value of u and r, theintegral becomes the sum of all of those data points. dr is defined as Δr the measurement stepbetween each point which was equal to 0.5 mm. This leads us to the following equation:

-40 -30 -20 -10 0 10 20 30 400

0.5

1

1.5

2

2.5

3


V e

l o c

i t y ( m / s e c

)

Mean Velocity vs. Radial Distance for a 25mm Fan, at an axial distance X=14mm



23

400 4002 2 2

0 00

2 2 2 (0.5)i i i ii i

K u rdr u rdr u r

It is not practical to solve this sum by hand, but it can be easily done in MatLab.

A is the cross sectional area of the fan’s blades (without the hub), which is known as well:

2 2out in A D D

Dout – The outer diameter of the fan at the edge of the blades Din – The inner diameter of the fan, where the blades begin (i.e. the diameter of the fan’shub)

The equation can now be rearranged to solve for Q:

400

2 2 2 2 2 2 4 3

0

2 (0.5) 9.9202 10 secout in out in i ii

Q KA D D K D D u r m

Step 3: Calculate maximum velocity using equation 23:

2

23

2

23

0.404385 0.164315 3.72415 3.7967 1014

0.404385 14 0.164315 25 3.72415 25 3.7967 14 10 25

1.68725

xC

x d d x d

2

2

0.0075935 0.10077 0.0074483 0.83851

0.0075935 25 14 0.10077 14 0.0074483 25 0.83851 25

20.4 0.0204

d x x d d

mm m

122 2 2 2

2 2 2max 2 2

2 2 22 2

2 2 22

4 3 1615 3216 8 32 384

4 3 160.22982 0.22982 0.0204

0.00099202 16 8 32

0.025 0.01515 0.0204 0.014 1.68725 32 0.014

384 1.68725

o i

Qu B B x C xC D D

12

2.1 secm

Step 4: Recreate the velocity profile:

Insert u max into equations 28 and 29:

max 1 cos(2 ) 2.1 1 0.22982cos 2 0.229820.0204

1.617378 0.482622cos 308

r r u u B B

r

fan outlet velocity distributions and calculations

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