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# University of Nevada, Reno ME467L Intermediate Fluid Mechanics Lab

By Emily Lawson,2014-01-20 04:38
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University of Nevada, Reno ME467L Intermediate Fluid Mechanics Lab

University of Nevada, Reno

ME467L Intermediate Fluid Mechanics Lab

Experiment # 3: Hot-film Anemometer

Calibration

Student’s Names: Henry J. McCubbin

Date Lab Submitted: April 7, 2006

Assumptions

There were two major assumptions made in order to calibrate the anemometer for this lab

experiment. The first relates the energy equation, and the second relates to the

uncertainty in measurement of the hot-film voltage.

First of all, in order to calibrate this anemometer a pressure differential was measured

across the flow to calculate the velocity. Both flow conservation (Equation 1) and energy

conservation (Equation 2) were used to establish the velocity.

VA?VA (1) 1122

1122 (2) P??V??gh?P??V??gh11122222

Combining these equations, the result is Equation 3.

?2P? (3) V22??A2?1???2??A1??

Equation 3 indicates that as long as Ais much larger than A then the ratio of the areas 1 2squared can be neglected. It was assumed that A was at least 20 times the size of A, 12

therefore the ration was neglected for the purpose of not having to make exact

measurements of the wind-tunnel geometry. The validation of this is shown in Table 1 as

the error associated with the inclusion of an area ratio of 1/20 is only 0.13%.

Table 1

Vc [m/s] Percent P1 [Pa] Vc [m/s] (A1/A2=1/20) Error

1726.78 58.77 58.84 0.13%

1225.94 49.52 49.58 0.13%

994.20 44.59 44.65 0.13%

737.56 38.41 38.46 0.13%

505.82 31.81 31.85 0.13%

239.21 21.87 21.90 0.13%

0.00 0.00 0.00 0.00%

249.17 22.32 22.35 0.13%

498.35 31.57 31.61 0.13%

750.01 38.73 38.78 0.13%

996.70 44.65 44.70 0.13%

1250.85 50.02 50.08 0.13%

1507.50 54.91 54.98 0.13%

2

1744.22 59.06 59.14 0.13%

Secondly, when reading the voltage from the hot-film anemometer, there was fluctuation

in the readout. This fluctuation was significantly steady at 0.005 V. This fluctuation was

neglected since it only amounted to 0.1% error or less. When making readings it will be

sufficient to estimate the voltage to the nearest 0.005 V and the fluctuation will be

inconsequential.

Data

The data collected during the laboratory experiment is presented in Table 2.

Table 2

Density Patm Patm Tatm [C] of Air [atm] [in] [kg/m^3]

1 29.72 25 1

P1 [in] P1 [m] P1 [Pa] Vc [m/s] (Vc)^.5 Vhw [V] +/- V

6.93 0.18 1726.78 58.77 7.67 6.960 0.005

4.92 0.12 1225.94 49.52 7.04 6.670 0.005

3.99 0.10 994.20 44.59 6.68 6.500 0.005

2.96 0.08 737.56 38.41 6.20 6.270 0.005

2.03 0.05 505.82 31.81 5.64 6.010 0.005

0.96 0.02 239.21 21.87 4.68 5.590 0.005

0.00 0.00 0.00 0.00 0.00 3.000 0.005

1.00 0.03 249.17 22.32 4.72 5.600 0.005

2.00 0.05 498.35 31.57 5.62 5.980 0.005

3.01 0.08 750.01 38.73 6.22 6.270 0.005

4.00 0.10 996.70 44.65 6.68 6.485 0.005

5.02 0.13 1250.85 50.02 7.07 6.665 0.005

6.05 0.15 1507.50 54.91 7.41 6.820 0.005

7.00 0.18 1744.22 59.06 7.69 6.945 0.005

Figure 1 displays the correlation between the Velocity and the output voltage directly,

and Figure 2 displays the King’s Equation correlation given by Equation 4.

2V?A?BV (4) HFC

3

Velocity as a Function of Output Voltage

8.000

7.000

6.000

5.000

4.000

Velocity (m/s)

3.000

2.000

1.000

0.0000.0010.0020.0030.0040.0050.0060.0070.00

Voltage (V)

Figure 1

King's Correlation60.000

50.000

40.000

30.000Voltage^2

20.000

10.000

0.0000.001.002.003.004.005.006.007.008.009.00

SQRT(Velocity)

Figure 2

4

Calculations

The uncertainty in King’s equation was calculated statistically (and the results are the

reason for neglecting the error associated with the voltage fluctuation). Because King’s

equation is much more useful that the direct correlation due to linearity and the

unnecessary difficulty in calculating uncertainty in nonlinear curve-fits, the exact

uncertainty associated with the direct correlation was not calculated.

The basic linear regression equation was the starting point:

y?(??ts)?(??ts)x (5) 0n?2,?/2?1n?2,?/2?01

The coefficients for that equation were calculated using the following equations and the

Students-t distribution tables.

??xx?i?y (6) ????i12xx()?i?????

2??x1 (7) ?y?????i02nxx()????i??

22ˆ(y?y)?(y?y)??i2 (8) r?2(y?y)?i

22(1?r)(y?y)?i (9) s?n?2

s (10) s??12(x?x)?i

2x1s?s? (11) ?20n(x?x)?i

5

Equation 5 can be used for the hot-film data and is rearranged as Equation (12).

2 (12) V?(A?ts)?(B?ts)VHFn?2,?/2An?2,?/2BC

The following set of equations give the equations and their confidence intervals. The

interpretation of these equations is: given some value of V we can be XX% confident Cthat the average corresponding value of V is given. HF

99%

2 V?(7.838?2.054)?(5.147?0.329)VHFC

2 V?(7.838?26.21%)?(5.147?6.39%)VHFC

95%

2 V?(7.838?1.465)?(5.147?0.235)VHFC

2 V?(7.838?18.70%)?(5.147?4.56%)VHFC

80%

2 V?(7.838?0.912)?(5.147?0.146)VHFC

2 V?(7.838?11.64%)?(5.147?2.84%)VHFC

It is obvious from the uncertainty above that the error from fluctuation is negligible in

comparison as error values are only approximately 0.1% compared to uncertainty values

of order of magnitude difference. Error brackets depicting the small fluctuation errors, as

a result, would be somewhat uninformative. The polynomial regression of Figure 1 is

presented in Figure 3 and the linear regression of Figure 2 is presented in Figure 4.

6

Velocity as a Function of Output Voltage

8.000

7.000

6.000y = -7E-07x432 + 0.0001x - 0.0069x + 0.2215x + 3.00062R = 0.99985.000

4.000

Velocity (m/s)

3.000

2.000