Correlations for the Test Plate

To evaluate the results of the experimental trials, the h values that were found were compared to theoretical correlations.

The following flat-plate correlations were used to calculate boundary layer thicknesses and verify experimental heat transfer data. The correlations used were reported by Incropera et al. [21].

Boundary layer thickness is defined as the value for which . Two different correlations exist, one each for laminar (found using a similarity method on the continuity and momentum equation), and turbulent boundary layers. To determine

turbulence, a critical Reynolds number was assumed to fall between and . Also, air properties were evaluated at a film temperature of 300 K for convenience, with a kinematic viscosity _{. The equation to calculate the Reynolds}

number based on x is

where x is the stream-wise distance from the start of the boundary layer. The correlation for laminar boundary layer thickness on a flat surface is:

.

The correlation for turbulent boundary layer thickness on a flat surface is:

.

The boundary layer thickness equations were not needed for any calculations involving the flat plate, but were applied to calculations involving the wind tunnel test section in Section 4.5.2: Tunnel Side Wall Effects.

In a similar manner, there exists a Nusselt correlation for a flat plate, found using similarity methods on the energy equation. The Nusselt number for a flat plate is a

function of the Reynolds and Prandtl numbers. Table 4.6 contains a summary of Nusselt number correlations for both laminar and turbulent flow, as well as two boundary conditions: constant surface temperature and constant heat flux.

Table 4.6 Nusselt Correlations: Correlations corresponding to different conditions.

Conditions Nusselt Correlation

Laminar, (4.11) Turbulent, (4.12) Laminar, (4.13) Turbulent, (4.14)

For these experiments, a Prandtl number of 0.707 was used based on a 300 K air film temperature. In this manner, h can be found from generally accepted correlations and compared to h values returned by the inverse method to check for accuracy.

4.5.1.2 Velocity Measurement

Since calculation of the Reynolds number was needed, an accurate measurement of velocity was required. For this, a pitot tube and Dwyer Series 475 Mark III digital manometer was used. The manometer had a reported accuracy of W.C. and a range of 0-1.000” W.C. The pitot tube was inserted into the top of the wind tunnel slightly ahead of the leading edge of the flat plate, halfway between the plate and top of the test section. The fan was activated with heater off, and pressure readings were taken at 1% fan intervals from 10% to 20%. The pressure readings were then translated into velocity readings using Equation 4.15, where is the pressure differential reading from the manometer, and is the density of air, evaluated at film temperature, 1.193 .

(4.15)

Table 4.7 shows the range of velocities corresponding to the pressure differences measured with the Dwyer manometer.

Table 4.7 Velocity: A range of wind tunnel velocities measured.

% Blower (Pa) V 10 0.016 3.287 2.35 11 0.023 4.724 2.81 12 0.028 5.751 3.11 13 0.035 7.189 3.47 14 0.042 8.627 3.80 15 0.049 10.065 4.11 16 0.060 12.324 4.55 17 0.067 13.762 4.80 18 0.078 16.022 5.18 19 0.090 18.487 5.57 20 0.102 20.951 5.93

4.5.1.3 Boundary Layer Start Position

An accurate value of x, the distance from the start of the boundary layer, was needed to calculate both Reynolds and Nusselt numbers. In an ideal flat plate, this value would be the distance from the leading edge. However, the reality is that a blunt edge creates a stagnation point at the leading edge of the flat plate [51]. This causes the flow to separate from the leading section of the surface of the plate, and reattach at some distance downstream, as seen in Figure 4.19. The flat plate was positioned in the center of the wind tunnel test section such that there was 10 cm between the top surface and the wall, and 10 cm between the bottom surface and the wall.

Figure 4.19 Separation Bubble: A blunted edge on a flat plate creates a separation bubble at the leading edge when it is immersed in a flow.

A rounded edge was added to the leading edge of the plate to minimize this effect. The distance from the leading edge of the plate to the reattachment point, , needed to be found to determine experimental x values.

To find , the image series from experimental trials was examined. From these, the start of the boundary layer could be identified as the position where the TLCs first started to change color. It

should be recognized this position will change as velocity changes. Since red is the first color the TLCs experience, the red signal was examined.

Figure 4.20 shows an image closely corresponding to the start of the red signal to illustrate the location of the boundary layer start. The leading edge of the plate

corresponds to the left edge of the image (the leading edge of the plate being the edge of the top of the half-circle leading edge). The green light at the lower edge of the image is an LED to denote activation of the tunnel air heater.

FLOW Separation bubble

Boundary layer

Figure 4.20 Boundary Layer Start: This image corresponds to the point where the TLCs just start to change color. The left edge of the image corresponds to the plate leading edge. It can be seen the

boundary layer starts at a distance downstream of the leading edge.

The pixel location of the start of the red signal near the middle of the plate was correlated to position. For this trial (blower at 16.4%), was found to be 7/8” (2.22 cm) from the leading edge of the plate (considered to be the top of the rounded edge). The positions of the three surface-mount thermocouples (designated Position1, Position 2, and Position 3) were 1”, 4”, and 7” from the leading edge, respectively. Accounting for the position of , the distance from the start of the boundary layer of the three

thermocouples (x) becomes 1/8”, 3 1/8”, and 6 1/8”. 4.5.1.4 Reynolds Numbers

With V and x found, the Reynolds numbers could now be calculated. A kinematic viscosity of air of 1.59E-05 was used. Table 4.8 contains a summary of these

Reynolds numbers.

Pos.1

Pos.2

Table 4.8 Reynolds Numbers: A summary of the Reynolds numbers at different positions on the test plate.

% V Reynolds Number

Pos. 1 Pos. 2 Pos. 3 10 2.35 469.01 11725.24 26737.24 11 2.81 562.32 14058.07 32056.82 12 3.11 620.44 15511.03 35370.04 13 3.47 693.67 17341.86 39544.91 14 3.80 759.88 18997.06 43319.28 15 4.11 820.77 20519.17 46790.17 16 4.55 908.23 22705.83 51776.44 17 4.80 959.75 23993.80 54713.43 18 5.18 1035.55 25888.63 59034.22 19 5.57 1112.35 27808.85 63412.93 20 5.93 1184.19 29604.78 67508.22

To determine whether the flow is laminar or turbulent, these values are to be compared to a critical Reynolds number, a number that denotes transition to turbulence. It has been found experimentally that the critical Reynolds number varies between and [21]. Referring to Table 4.8, this implies that all flows were laminar. However, roughness of the plate, flow inconsistencies, etc., could induce turbulence. Therefore, h values for both laminar and turbulent conditions were compared to

experimental h values. These Reynolds numbers were used to calculate Nusselt number correlations to compare with the h values calculated with inverse methods.