E • CONVECTIVE HEAT TRANSFER AND FRICTION 

 where 



The property ratios are therefore functions of 1 — jST*, and Eq. 2-5 and 

 2-6 can be solved simultaneously to obtain velocity and temperature dis- 

 tributions for various values of jS. As in the case of constant properties, 

 the variations of heat transfer and shear stress across the passage have 

 but a slight effect on the velocity and temperature distributions for 

 Reynolds numbers above 10,000 [i^, Fig- H] so that in most cases t/tw 

 and g/gw are set equal to one in Eq. 2-5 and 2-6. The quantity |3 is a heat 

 flux parameter and is a measure of the effect of variable fluid properties. 

 For constant properties, /? = 0. 



Most of the experimental and analytical work on variable properties 

 has been carried out for air as a fluid. Some of the work for air will there- 

 fore be considered at this point, even though the present section is mainly 

 concerned with liquids. The analyses for various fluids differ significantly 

 only in the assumptions made for the variation of their properties with 

 temperature. 



Analysis for air. In this case the viscosity, thermal conductivity, 

 and density are considered variable with temperature. The Prandtl num- 

 ber and specific heat are considered constant, inasmuch as their variations 

 with temperature are of a lower order of magnitude than those of the 

 other properties. The viscosity of air and many common gases varies 

 approximately as the 0.68 power of the absolute temperature for tem- 

 peratures between and 2000°F. From the assumptions of constant 

 Prandtl number and specific heat, the thermal conductivity must vary 

 with temperature in the same way as the viscosity, or 



7 / m\ 0.68 



i-K = [vJ =(1-^2-*)"'' (5-2) 



From the perfect gas law and the assumption of constant static pressure 

 across the passage, 



P Tw 1 



T 1 - jsr* 



(5-3) 



By substituting Eq. 5-2 and 5-3 into Eq. 2-5, 2-6, 3-1', and 3-2', and 

 letting t/tw = q/qw = 1, Eq. 2-5 and 2-6 can be integrated numerically or 

 analytically for the regions close to and away from the wall [14,13]. The 

 molecular shear stress and heat transfer terms are neglected in the region 

 away from the wall. The values of the constants n and k are 0.124 and 

 0.36 as determined from the data in Fig. E,4a. The constants k and n 

 and the function F in Eq. 3-1 and 3-2 have the same values for variable 

 properties as for constant properties if the assumptions for e„ in Art. 3 



<304) 



