Objectives
1. To gain a working knowledge of the hot film anemometer.

2. To make fundamental measurements of both free and forced convection heat transfer along with measurements of flow velocity and turbulence intensity.

3. To develop working correlations for Nusselt number vs. Grashof number or Reynolds number for free or forced convection heat transfer.

Background

Even though the cylinder is a simple geometric shape, the viscous flow of a fluid past a cylinder is complicated by the developing boundary layer flow around the cylinder or along its axis. An experimental approach is frequently used because it yields a direct measurement of the flow and heat transfer parameters. From properly designed experiments one can determine correlations of nondimensional numbers which can be used to predict heat transfer results for other fluids and other flows.

The hot wire or hot film anemometer has been a very important research tool in the study of fluid mechanics. The actual hot film sensor that "measures" the flow velocity is a miniature cylindrical element mounted between two stainless steel fingers. The sensor becomes one of the resistive elements in a four- corner Wheatstone bridge which heats the sensor to a constant temperature by adjusting the power dissipated in the sensor such that its resistance (and hence temperature) remains constant. That's why it is called a constant temperature hot film anemometer

The sensor is fabricated by mounting a quartz fiber ( .005 in.) between two metallic "fingers" and then a layer of platinum is vapor-deposited over the fiber forming the cylindrical element. A very thin quartz layer is sputtered over this platinum film to electrically insulate the film from being grounded if the sensor is used in a liquid which conducts electricity. Figure 1 shows a sketch of the hot film probe. 

Figure 1. The hot film anemometer.

While the front of the box that houses the hot film electronics looks complicated with all of its dials and switches, it is nothing more than a four-corner Wheatstone bridge shown below in Figure 2.

Figure 2. Basic circuit of the hot film anemometer.

The two upper resistances are equal (40 ohms each), R₁ is a variable resistor R_s is the sensor itself and the 17.4 resistances account for internal and cable resistances. A brief description of how the bridge works is now given. When the bridge circuit is activated, a voltage called the bridge voltage is imposed on the top node. This causes an electric current to flow down both legs of the bridge. If R₁ is higher than the sensor resistance, the voltage input into the D-C differential amplifier will be non zero. The amplifier communicates this to the power supply and causes the bridge voltage to increase or decrease until the current passing through the sensor heats it to the exact temperature (and hence resistance) so that the bridge is balanced and the amplifier has zero voltage as its input. This adjustment happens very fast, usually on time scale of microseconds. When a fluid having a fluid temperature lower than the hot operating temperature of the sensor flows over the sensor it cools the sensor causing the bridge voltage to increase until the sensor temperature is again constant. A plot of the fundamental measurements of bridge voltage and velocity yields a calibration curve which demonstrates a monotonically increasing voltage with flow. Figure 3, below, shows a typical calibration curve where the bridge voltage is plotted against mass flux (lb/ft² -s).

Theory
Let the ambient temperature be given by T_a and the sensor temperature be given by T_s. The electrical resistance of the sensor, R_s, can be approximated by a linear relationship:

R_s/R_a = 1 + (T_s - T_a )

where R_a is the resistance under ambient conditions and  is the temperature coefficient of resistance which is determined for each sensor at the factory and is specified on the shipping box of the probe. The resistance ratio R_s/R_a is called the overheat ratio. The temperature difference, T_s - T_a, is called the overheat and has units of temperature.

Figure 3. Examples of hot film calibration data taken on several different days.

The power dissipated in the heated hot film sensor is given by

P = I²R_s

and this must equal the instantaneous rate of heat transfer between the hot film and the surrounding fluid (provided the sensor mass is negligible and the conduction loss to the supports is neglected).

Thus, P = Q = hA_s (T_s - T_a)

where h is the average convective heat transfer coefficient and A_s in the surface area of the hot film

(A_s = dL, d = diameter, L = length).

The Nusselt number is based on the hot film diameter for forced convection and for free convection with the axis of the probe horizontal.

When the axis of the probe is vertical the length of the probe becomes the characteristic length and the Nusselt number is given by

Similarly, the Reynolds number is based on the free-stream velocity and diameter,

In the case of free convection the free stream velocity is zero and the appropriate free convection parameter is the Grashof number,

or

where the characteristic length is d or L, depending on probe orientation, "g" is the acceleration due to gravity and "ß" is the volumetric coefficient of expansion for air,

where T_f is the "average film temperature" in absolute units.

Calculations

The Nusselt number

The definition of the Nusselt number is

where

h = average heat transfer coefficient

d = characteristic length (can be "d" or "L")

k_f = air thermal conductivity evaluated at T_f

The heat transfer is given by
Q = hA_s(T_s - T_a)

where
A_s = probe surface area

T_s = probe or sensor temperature

T_a = ambient temperature

Now, Q = power dissipated = I²R_s where I is the electric current going through the sensor. The current going through the sensor is determined by measuring the bridge voltage, e_b. From Figure 2 the bridge voltage and current through the sensor are related by

e_b = I (R_s + 40 + 17.4)

By successive substitution one can show that

In the above equation the "40" is from the bridge upper resistance and the "17.4" is the probe cable and internal resistance. R_s is equal to R₁ the resistance set on the decade box on the front of the electronics panel, e_b is measured and T_s is found by knowing the overheat ratio, R_s/R_a. R_a is determined experimentally by using the bridge circuit.

The Grashof number

From its definition

where properties are evaluated at the film temperature T_f .

The Reynolds number

where properties are evaluated at T_f and the velocity is measured with a manometer.

Turbulence Intensity

Turbulence Intensity is defined by:

where the numerator is the RMS (Root Mean Square) value of the velocity fluctuation u'. The denominator is the mean velocity as measured with anemometer.

In order to determine the turbulence intensity we will depend on the observation that TI is usually a small number on the order of a few percent. This means that if we consider the calibration curve in the form of e_b vs. velocity (similar to Figure 3) we would find that small fluctuations in velocity will bring about small changes in bridge voltage at any instant of time. That being the case, the change of bridge voltage with velocity (i.e., the slope of the calibration curve) doesn't change very much at any operating point where the mean or time average velocity is constant. The instantaneous bridge voltage is given by a local linearization of the calibration curve

but since

u =  + u'

and

e_b = _b + e' _b

then

_b + e'_b = A + B + Bu'

A time average of the previous equation yields , which, when subtracted from the previous equation, yields a relationship between the fluctuating bridge voltage and the fluctuating velocity

Square, time average and take the square root of the above to get

Substituting this relationship into the definition of the turbulence intensity one finds

where the numerator is the RMS value of the bridge voltage (measured experimentally with a true RMS meter) divided by the slope of the calibration curve at the operating point (i.e., steady mean velocity) and the denominator is the measured steady mean velocity. The problem now becomes one of determining the slope of the calibration curve at the operating point. This could be done with a polynomial fit of the data but this method is subject to large errors when the number of data points is small which is the case here. A better way would be to observe the fact that King's law gives a good linear fit to your calibration data (See procedure section). You will have the coefficients C₁ and C₂ when you do the linear fit of the data and test it with the "r" parameter. Hence,

Taking the derivative of the above and solving

The constant C₂ comes from your King's law fit and the bridge voltage and velocity are measured during the experiment.

Procedure

1. Measure the bridge voltage from the hot film in still air with the probe mounted in the horizontal position for a range of overheat ratios from 1.05 to 1.45 in increments of 0.05. Turn the probe vertically and measure the bridge voltage at the same overheat ratios.

2. Measure the bridge voltage and velocity for three overheats of 35 ºC, 65 ºC and 95 ºC

3. Use the hot film to determine the turbulence intensity in the jet flow for a range of speeds from 1000 to 11,000 fpm. Take these data at the 95° C overheat while you are taking the other data for 2, above.

Report

1. Use the data obtained in 1, above, to calculate Nu and Gr at each operating point for both orientations of the probe. Plot Nu vs. Gr for both orientations and discuss the reasons for any observed differences in the heat transfer rates for the different orientations. Compare your results for the horizontal orientation to appropriate empirical correlations in the literature. Discuss the results of this comparison. What do the Nu and Gr represent and what does their functional relationship tell you?

2. Plot your bridge voltage vs. velocity data. How are the bridge voltage and the velocity related?

3. Use the data obtained in 2, above, to test the validity of King's law which states that the square of the bridge voltage should be a linear function of the square root of the flow velocity. Plot your data as (_b)² vs. ()^.5 for the three overheat ratios and test for a linear fit of these data in this form.

4. Calculate the Nusselt number and Reynolds number for the data in 2 above. Plot your Nu vs. Re data on a single plot. What physical parameter is this plot independent of? Why?

5. Use the data determined in 3, above, to calculate the turbulence intensity and plot TI vs. velocity. What trends for the flow regime are observed in this plot.