Heat Transfer Enhancement (Raw Data)
Tables 5.10 and 5.11 summarize the data collected within the rib roughened channel at an approximate Reynolds number of 30,000. As the tables indicate, data was collected on two walls within the square channel: a smooth wall and a wall roughened with Vshaped ribs.
It is convenient to compare the wall temperatures measured in the actual test to those obtained during the heat loss calibration. Figure 5.5 compares the wall temperature distributions on the ribbed and smooth walls with those obtained from the high and low sets of heat loss data. As the figure indicates the temperature distributions measured during the heat loss calibration are more uniform through the channel than distributions obtained during the actual heat transfer enhancement test. These actual distributions can vary depending on the type of heater used to heat the test section walls. Figure 5.5 also shows how the “high” and “low” sets of heat loss data bracket the temperatures measured on both the smooth and ribbed walls.
TABLE 5.10
Temperature Distributions Acquired at Re = 32,050
x (cm) 
Smooth7„ (°C) 
Roughened7„ (°C) 
5.16 
58.3 
57.5 
15.5 
64.1 
59.8 
25.5 
61.7 
59.5 
35.8 
61.8 
61.7 
45.8 
65.8 
65.4 
56.1 
69.8 
66.9 
66.3 
70.2 
69.5 
76.4 
74.5 
72.7 
86.6 
75.5 
76.8 
96.6 
75.4 
76.5 
Inlet Bulk 
28.1 

Outlet Bulk 
41.4 

Room 
22.5 
TABLE 5.11
Voltage and Resistance Measurements at Re = 32,050
Voltage (V) 
Resistance (Й) 

Smooth wall 
28.9 
6.0 
Roughened wall 
33.1 
6.0 
FIGURE 5.5 Comparison of wall temperature distributions in actual test and heat loss calibration.
Heat Transfer Enhancement Data Reduction
Recalling that the goal is to determine the heat transfer enhancement in the rib roughened channel, it is necessary to recognize how the heat transfer enhancement will be evaluated. In the study shown with this example, the heat transfer enhancement is presented in terms of the Nusselt number ratio (Nu/Nu_{0}). This is a convenient method for comparing the increased heat transfer in the rib roughened channel (Nu) to that of a smooth channel (Nu_{0}). In order to determine the Nusselt number, the heat transfer coefficient must first be calculated, and the heat transfer coefficient can be determined the convective heat transfer equation.
The local wall temperatures have been measured directly with the thermocouples mounted in the copper plates. The local bulk temperature, T_{h} v, must be determined from linear interpolation, and the local, convective heat flux is the difference of the power input to the heaters and the heat loss determined from the calibration.
Using the measured inlet and outlet temperatures, the local bulk temperature can be determined at each location in the channel. Table 5.12 presents the bulk temperature distribution through the channel. As this is the average coolant temperature through the channel, this bulk temperature distribution is valid for both the ribbed wall and the smooth wall of the square channel.
The measured wall temperatures are compared to the interpolated bulk fluid temperature in Figure 5.6. As shown in the figure, a moderate temperature difference (between the wall and fluid) is maintained through the length of the channel. One would expect the wall temperature distribution and the bulk fluid temperature
TABLE 5.12
Bulk Temperature Distribution through Rib Roughened Channel
x (cm) 

0 
0.0 
28.1 (inlet) 
5.16 
0.0508 
28.88 
15.5 
0.152 
30.43 
25.5 
0.251 
31.94 
35.8 
0.352 
33.49 
45.8 
0.451 
35.0 
56.1 
0.552 
36.55 
66.3 
0.652 
38.08 
76.4 
0.752 
39.61 
86.6 
0.852 
41.14 
96.6 
0.951 
42.65 
101.6 
1.0 
43.4 (outlet) 
FIGURE 5.6 Comparison of wall temperature and bulk fluid temperature distributions through the square channel.
distribution to be parallel curves in the fully developed region of the channel. While both the ribbed and smooth walls maintain approximately the same slope from x = 36.886.6 cm, they are not changing at the exact rate as the fluid temperature. While each wall in the square channel is exposed to the same heat flux, circumferentially the heat flux is not uniform. Therefore, the temperature distributions slightly vary from those predicted from internal heat transfer theory.
With the regional wall and air (bulk) temperatures known, it is now necessary to determine the regional rate of convective heat transfer ((X,,,). The net rate of heat transfer is the difference between the heat transfer supplied to the channel and the miscellaneous heat losses, as shown by Equation (5.24). Therefore, both the regional heat applied to the channel and the regional heat losses must be determined.
Table 5.11 shows the measured resistance for the two foil heaters and the voltage supplied (via variable transformers) to these heaters. With these measurements, the total power (heat) supplied to the channel can be determined with Equation (5.25).
While the total power supplied to the channel is useful, power supplied to each copper plate is needed for the calculation of the regional heat transfer coefficient. For the present case, the total power supplied to the channel is evenly distributed to the 10 copper plates on each wall. The total power supplied to each channel as well as the fraction supplied to each copper plate on the surface are summarized in Table 5.13.
For the determination of the heat losses, it is necessary to determine the amount of power supplied to each copper plate during the “low” and “high” sets of heat loss
TABLE 5.13
Power Input to Each Copper Plate within the Channel
Total Q,„ (watts) 
Regional Q,„_, (watts) 

Ribbed wall 
182.6 
18.26 
Smooth wall 
139.2 
13.92 
TABLE 5.14
Power Input to Each Copper Plate during the Heat Loss Calibration
"Low" Total Q, 
"Low" Regional Q_{u} 
"High" Total Q_{H} 
"High" Regional Q_{H>} 
(watts) 
(watts) 
(watts) 
(watts) 
0.770 
0.0770 
2.67 
0.267 
data. Using the power supplied (through voltage and resistance measurements), the total power and regional power distributions can be determined. The power supplied to the heaters during the heat loss calibration is shown in Table 5.14.
Linear interpolation can be used to determine the actual heat loss during the heat transfer enhancement test. Using the measured wall temperatures and power inputs during the calibration and the measured wall temperature during the actual test, the heat losses can be approximated. As the driving temperature difference for the heat losses is the temperature difference between the heated wall and the ambient room, the ambient room temperature must be considered in the calculation of the heat losses. As shown in Equation (5.26), the room temperature must be recorded during for each set of calibration data, as well as during the Reynolds number test.
Rearranging Equation (5.26) to solve for the actual, regional heat loss, Q_{OSS t}■ yields Equation (5.27). It should be noted that the expression for the regional heat loss shown in Equation (5.27) could take several forms depending on how linear interpolation was used with the known data.
Combining the heat supplied to each copper plate with the heat losses, the net heat transfer can be determined as shown in Equation (5.24). The heat flux distributions for each copper plate are summarized in Tables 5.15 and 5.16.
TABLE 5.15
Heat Flux Distributions for the Ribbed Wall
x (cm) 
Qin,x (W) 
Qlos.,* (W) 
Q„«., (W) 
5.16 
18.26 
0.136 
18.12 
15.5 
18.26 
0.148 
18.11 
25.5 
18.26 
0.144 
18.12 
35.8 
18.26 
0.164 
18.10 
45.8 
18.26 
0.180 
18.08 
56.1 
18.26 
0.184 
18.08 
66.3 
18.26 
0.210 
18.05 
76.4 
18.26 
0.227 
18.03 
86.6 
18.26 
0.256 
18.00 
96.6 
18.26 
0.255 
18.01 
TABLE 5.16
Heat Flux Distributions for the Smooth Wall
x (cm) 
О™.* (W) 
Qi„,„ (W) 
Owl* (W) 
5.16 
13.92 
0.142 
13.78 
15.5 
13.92 
0.178 
13.74 
25.5 
13.92 
0.159 
13.76 
35.8 
13.92 
0.165 
13.76 
45.8 
13.92 
0.183 
13.74 
56.1 
13.92 
0.202 
13.72 
66.3 
13.92 
0.215 
13.71 
76.4 
13.92 
0.239 
13.68 
86.6 
13.92 
0.247 
13.67 
96.6 
13.92 
0.248 
13.67 
With the net rate of heat transfer (Q_{net t}) and the difference between the regional wall and fluid temperatures (T_{w x}  T_{hx}), it is possible to determine the regional heat transfer coefficients using Equation (5.23). The surface area (A,) used in this calculation is the area of each individual copper plate. In the study considered in this example, the surface area of each copper plate is:
The calculated heat transfer coefficients are shown in Table 5.17. These values were arrived at using Equation (5.23), and the data presented in Tables 5.10, 5.12, 5.15, and 5.16 (wall temperatures, bulk temperatures, ribbed wall net rate of heat transfer, smooth wall net rate of heat transfer).
TABLE 5.17
Regional Heat Transfer Coefficients on the Ribbed and Smooth Walls
x (cm) 
h_{x} (W/m^{2}K) Ribbed Wall 
h_{x} (W/rrHK) Smooth Wall 
5.16 
122.7 
90.7 
15.5 
119.5 
79.1 
25.5 
127.6 
89.6 
35.8 
124.3 
94.1 
45.8 
115.2 
86.4 
56.1 
115.4 
79.9 
66.3 
111.3 
82.7 
76.4 
105.6 
76.0 
86.6 
97.8 
77.1 
96.6 
103.1 
80.9 
The heat transfer coefficients can be nondimensionalized, and presented in terms of the Nusselt number. The nondimensional Nusselt number is defined in Equation (5.29).
The hydraulic diameter was previously defined in Equation (5.20); for the square channel considered in this example, the hydraulic diameter is equal to the length of side of the channel (£),, = 5.08 cm). Taking the thermal conductivity of air at the inlet bulk temperature, the regional Nusselt numbers can be calculated, and are presented in Table 5.18. Also, the streamwise dimension (x) has been nondimensionalized with the channel hydraulic diameter.
Finally, when investigating roughness schemes which will enhance heat transfer, it is desirable to quantify the amount of heat transfer enhancement. While several options are available to express the level of heat transfer enhancement, the most widely used method is through the Nusselt number ratio. This ratio compares the measured Nusselt number to that expected in a simple smooth tube with fully developed, turbulent flow. The DittusBoelter/McAdams correlation is commonly used to approximate the Nusselt number for fully developed, turbulent flow in a smooth tube, and this correlation is shown in Equation (5.30).
Using the Reynolds number presented in the paper (Re = 32,050), and the Prandtl number evaluated at the inlet bulk temperature, the Nusselt number expected in a smooth tube is 80.25. Using this value, and the data presented in Table 5.18, the
TABLE 5.18
Regional Nusselt Numbers on the Ribbed and Smooth Walls
x/D_{h} 
Nu, Ribbed Wall 
Nu„ Smooth Wall 
1.01 
240.6 
178.0 
3.05 
234.4 
155.1 
5.02 
250.2 
175.7 
7.05 
243.8 
184.7 
9.02 
226.0 
169.5 
11.05 
226.4 
156.8 
13.05 
218.3 
162.2 
15.05 
207.1 
149.0 
17.05 
191.9 
151.2 
19.01 
202.2 
158.7 
TABLE 5.19
Regional Nusselt Number Ratios on the Ribbed and Smooth Walls
x/D_{h} 
; Nu i IN^j, Ribbed Wall 
InuJ, Smooth Wall 
1.01 
3.00 
2.22 
3.05 
2.92 
1.93 
5.02 
3.12 
2.19 
7.05 
3.04 
2.30 
9.02 
2.82 
2.11 
11.05 
2.82 
1.95 
13.05 
2.72 
2.02 
15.05 
2.58 
1.86 
17.05 
2.39 
1.88 
19.01 
2.52 
1.98 
regional Nusselt number ratios can be calculated using Equation (5.31). These ratios expressing the level of heat transfer enhancement over a smooth tube are shown in Table 5.19.
FIGURE 5.7 Regional Nusselt number ratio distributions in the square channel.
The final results shown in Table 5.19 can be plotted to show the heat transfer enhancement through the channel with two walls lined with Vshaped ribs. Figure 5.7 can be compared to Figure 7 of the referenced paper at Re = 32,050 [3].