Distribution of temperatures

Fig. 5 shows the averaged temperatures for the channel walls and the bulk fluid:


Z (m)

Fig. 5. Average temperature for different walls and bilk fluid, a = 2.43 and Re = 250.Example caption for


A linear increase in bulk mean temperature along the tube length can be appreciated. This is a natural result of energy balance under uniform heat flux. The temperature difference between the channel walls and the fluid attains its minimum at the channel entrance region, and gradually reaches a constant value. This is in agreement with the variation of Nusselt number as will be discussed in a following section. In figure 5, the variation of the average T of each wall and the mean bulk T, obtained with the CFD simulations is shown. It can be noticed that bulk T has a linear variation and Twalls an exponential as it was expected.

3.3. Nusselt number

The average Nu number for each wall is obtained through the local Nu as:


Nuz, ave = — j NUzdl (Eq. 5)

Lc 0

where Lc is the width of each wall.

image092 Подпись: (WNUave,bottom + WNUave,top + 2HNUave,side ) 2(W + H) Подпись: (Eq. 6)

The overall average Nu for each aspect ratio is obtained as the proportional addition of the average Nu of each wall [1].

In the figure 6, the variation of the Nu with the Re, for each aspect ratio is shown

Подпись: 0 250 500 750 1000 1250 1500 1750 2000 ReDh Fig.6. Average Nusselt numbers. 14

13.5 13

12.5 12

11.5 11

10.5 10

9.5 9

8.5 8

7.5 7

6.5 6

5.5 5

image096 image097

The thermal behaviour of the system can be understood as a function of the aspect ratio (a) and the Reynolds number. However, in order to obtain replicable results for tubes of any length, diameter, etc. A dimensionless parameter is defined as the equivalent distance of the fully developed flow from the entrance of the tube (L+, dimensionless length). This parameter has a similar definition to X+ [6], but is dependent of the total length of the tube. This parameter is known as the Graetz variable, and is defined as:

In the figure 7 the variation of Nu with the L+ is shown. Making a minimum quadratic residual analysis a correlation can be obtained. Its mathematical expression is:

Подпись:NuDh = 5.811( Г)-°’237

3.4. Thermal resistance

image099 Подпись: (Eq. 9)

The operation of the heat sink is usually evaluated by the value of the thermal resistance, defined as:

Подпись: Fig. 8. Comparision of thermal resistance: numerical vs experimental (Aspect ratio = 2.43).

Where Tw out is the fluid outlet temperature and Tin the fluid inlet temperature.

In literature we can find some correlations or values of the thermal resistance for linear concentrating systems:

Table 3. Inverse heat transfer coefficients for a=2.43.


Mass flow (kg/m2s)

1/hc (m2K/W)






















Подпись: Table 2. Thermal resistances Authors 1/hc (m2K/W) Chenlo y Cid 8.7x10-4 (Re = 5000) Coventry. 1.3x10-3 (Mass flow = 0.348 kg/m2s)

To compare with these values we should calculate the thermal resistance per unit area, obtaining the table 3.

The values of thermal resistance to the proposed sink are lower compared to those presented by Chenlo and Cid and Coventry. It should be mentioned that the values that can be compared to a greater degree

[5] , were acquired with the prototype working under real conditions. The values shown of the sink

design are made at the laboratory, so when comparing it is necessary to be critical quantifying differences.

5. Conclusion

The heat exchange properties of the aluminium improve increasing the aspect ratio of its cross section, in addition the pressure drop or in consequence the pumping power is higher when the hydraulic diameter (which is directly related with the cross section) is lower.

Nevertheless, a big aspect ratio implies a much more difficult mechanical procedures, suck as hydraulic connections, isolation. Moreover, is necessary to mention that the main aluminium factories don’t manufacture pipes of one centimetre width with aspect ratios higher than 2.43.

Attending to this explanations, the pipe selected to be include in the PVT systems under concentration is with a cross section of 20x10cm2 (a = 2.43).


This work was supported by the MCYT (Spain) (ENE2007-65410).


[1] P. Lee, S. V. Garimella, D. Liu. Investigation of heat transfer in rectangular microchannels. International Journal of Heat and Mass Transfer 48 (2005) 1688-1704.

[2] J. I. Rosell, X. Vallverdu, M. A. Lechon, M. Ibanez. Design and simulation of a low concentrating photovoltaic/thermal system. Energy Conversion and Management 46 (2005), 3034-3046.

[3] F. Chenlo, M. Cid, A linear concentrator photovoltaic module: analysis of non-uniform illumination and temperature effects on efficiency, Sol. Cells 20 (1987) 27-39.

[4] L. W. Florschuetz, C. R. Truman, D. E. Metzger, Streamwise flow and heat transfer distributions for jet array impingement with crossflow, J. Heat Transfer 103 (1981) 337-342.

[5] J. S. Coventry, Performance of the CHAPS collectors, Conference record, Destination Renewables — ANZSES 2003, Melbourne, Australia, 2003, pp. 144-153.

[6] F. P. Incropera, D. P. DeWitt, Fundamentals of Heat andMass Transfer, fourth ed, Wiley, New York, 1996.

[7] R. F. Russell, Uniform temperature heat pipe and method of using the same, Patent US4320246, 1982, USA.

[8] M. W. Edenburn, Active and passive cooling for concentrating photovoltaic arrays, Conference record, 14th IEEE PVSC, 1980, pp. 776-776.

[9] M. J. O’Leary, L. D. Clements, Thermal-electric performance analysis for actively cooled, concentrating photovoltaic systems, Sol. Energy 25 (1980) 401-406.

Добавить комментарий

Ваш e-mail не будет опубликован. Обязательные поля помечены *