9.78.

In essentially all cases of reactor cooling where forced convection is used, the coolant is under turbulent flow conditions. Satisfactory pre­dictions of heat-transfer coefficients in long, straight channels of uniform cross section can be made on the assumption that the only variables in­volved are the mean velocity of the fluid coolant, the diameter (or equiv­alent diameter) of the coolant channel, and the density, heat capacity, viscosity, and thermal conductivity of the coolant. From the fundamental differential equations or by the use of the methods of dimensional analysis, f it can be shown that heat-transfer coefficients for turbulent flow conditions can be expressed in terms of three dimensionless moduli; one of these is the Reynolds number, already defined, and the others are the Nusselt number (Nu) and the Prandtl number (Pr), defined by

9.79. As a result of numerous experimental studies of heat transfer, various expressions relating the three moduli have been proposed; one of these, for an ordinary (nonmetal) fluid in a long, straight channel, is the Dittus-Boelter correlation,

(9.35)

fSee standard texts on heat transfer, e. g., General References for this chapter.

Nu = 0.023Re° 8Pr° 4,

with all the physical properties evaluated at the bulk temperature of the fluid. In a modified form,

Nu = 0.023Re08Pr0 33,

the physical properties, except the specific heat, are the values at the film temperature, i. e., in the laminar (film) layer of fluid adjacent to the surface. This is taken as the arithmetic mean of the wall (or surface) temperature and the bulk fluid temperature.

Example 9.6. Calculate the heat-transfer coefficient for the water cool­

ant in Example 9.5.

From the data in the Appendix, к at 311°C is estimated to be 0.518 W/m • К and Pr about 1.06. Hence, from equation (9.36),

9.80. It is seen from equation (9.35) that if the viscosity, thermal con­ductivity, density, and specific heat of the coolant are known, the heat — transfer coefficient can be estimated for turbulent flow of given velocity in a pipe or channel of specified diameter (or equivalent diameter). The results appear to be satisfactory for values of Re in excess of about 10,000, and for Pr values of from 0.7 to 120. This range of Prandtl numbers includes gases and essentially all liquids, except liquid metals; the latter have very low Prandtl numbers, primarily because of their high thermal conductivity but also often because of their low viscosity and heat capacity. The cor­relations for liquid metals will, therefore, be considered separately (§9.84 et seq.).

9.81.

Where large differences exist between the temperatures of the solid and of the fluid, the associated variations in the physical properties of the coolant can influence the heat transfer. As indicated, such variation is most significant for the viscosity, and in order to make allowance for it the relationship

has been proposed, where |x is the viscosity at the bulk fluid temperature, and is that at the wall temperature. This equation, used in conjunction with the equivalent diameter concept, has been found to be satisfactory for predicting local and average heat-transfer coefficients for ordinary fluids flowing through thin, rectangular channels [6].