E • CONVECTIVE HEAT TRANSFER AND FRICTION 



property variations, it is evident that the property ratios in Eq. 2-5, 2-6, 

 and 3-2' will be functions of wall temperature T^ and pressure, as well as 

 of T/Tv,. A separate calculation must therefore be made for each wall 

 temperature (or wall Prandtl number) and for each pressure and each 

 value of j8. Calculations with all the properties variable were carried out 

 for supercritical water at a pressure of 5000 lb/in. ^ (critical pressure « 

 3200 lb/in.2) [14]. 



The results of the calculations for heating the water can be sum- 

 marized as follows [14, discussion by Eckert]: The properties in the 

 Nusselt and Reynolds numbers in Fig. E,4c are evaluated at the temper- 

 ature at which the specific heat Cp assumes its maximum value T^, as long 

 as that temperature remains between the wall temperature and the fluid 

 bulk temperature. If T^, is higher than Ty, the properties are evaluated at 

 Tw) if it is lower than T^, they are evaluated at approximately T^,. The 

 properties in the Prandtl number are evaluated at the wall temperature 

 in all cases. Although this reference temperature rule correlates the results 

 for supercritical water, it does not work as well for the results for carbon 

 dioxide to be discussed later, possibly because the carbon dioxide was 

 closer to the critical point. 



An alternative method of analysis of heat transfer to supercritical 

 water [61] assumed that the relation between u* and y* for /3 = can be 

 used for flow with heat transfer if w* is redefined as j^du/y/r^/p and y* 

 is written as jldyy/r^/p/ (m/p) , where local values of the properties are 

 used. Although the justification of this assumption is not clear, the results 

 from that analysis are in reasonable agreement with those in the analysis 

 given here. 



Bringer and Smith recently measured heat transfer coefficients for 

 carbon dioxide in the critical region [63]. The measurements were for 

 turbulent flow in a tube at a pressure of 1200 lb/in. ^ abs (critical pres- 

 sure = 1070) and fluid temperatures between 70 and 120°F (critical 

 temperature = 88°F). They also calculated heat transfer coefficients by 

 the analytical method given here, that is, by using Eq. 2-5, 2-6, 3-1', 

 and 3-2'. 



Fig. E,5f, E,5g, E,5h, and E,5i give a comparison between analytical 

 and experimental results. The Nusselt number is plotted against the 

 Reynolds number (fluid properties evaluated at the wall) for various 

 values of ^, the heat flux parameter. The effects of /3 are large, and in 

 some cases the trends reverse as jS increases. The agreement between 

 theory and experiment is very good. Inasmuch as these tests represent 

 extreme variations of fluid properties, they lend considerable support to 

 the theory given here. When it was attempted to correlate the data by 

 conventional methods, deviations on the order of 50 to 120 per cent were 

 obtained. 



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