DOC

University of Nevada, Reno ME467L Intermediate Fluid Mechanics Lab

By Emily Lawson,2014-01-20 04:38
8 views 0
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