In a thermally fully developed region the derivative

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Unformatted text preview: ion, the derivative of (Ts T)/(Ts Tm) with respect to x is zero by definition, and thus (Ts T)/(Ts Tm) is independent of x. Then the derivative of (Ts T)/(Ts Tm) with respect r must also be independent of x. That is, Ts r Ts T Tm ( T/ r) r Ts Tm rR R f(x) (8-9) Surface heat flux can be expressed as q·s hx(Ts Tm) k T r → hx rR k( T/ r) r Ts Tm R (8-10) which, from Eq. 8–9, is independent of x. Thus we conclude that in the thermally fully developed region of a tube, the local convection coefficient is constant (does not vary with x). Therefore, both the friction and convection coefficients remain constant in the fully developed region of a tube. Note that the temperature profile in the thermally fully developed region may vary with x in the flow direction. That is, unlike the velocity profile, the temperature profile can be different at different cross sections of the tube in the developed region, and it usually is. However, the dimensionless temperature profile defined above remains unchanged in the thermally developed region when the temperature or heat flux at the tube surface remains constant. During laminar flow in a tube, the magnitude of the dimensionless Prandtl number Pr is a measure of the relative growth of the velocity and thermal boundary layers. For fluids with Pr 1, such as gases, the two boundary layers essentially coincide with each other. For fluids with Pr 1, such as oils, the velocity boundary layer outgrows the thermal boundary layer. As a result, cen58933_ch08.qxd 9/4/2002 11:29 AM Page 425 425 CHAPTER 8 the hydrodynamic entry length is smaller than the thermal entry length. The opposite is true for fluids with Pr 1 such as liquid metals. Consider a fluid that is being heated (or cooled) in a tube as it flows through it. The friction factor and the heat transfer coefficient are highest at the tube inlet where the thickness of the boundary layers is zero, and decrease gradually to the fully developed values, as shown in Figure 8–8. Therefore, the pressure drop and heat flux are higher in the entrance regions of a tube, and the effect of the entrance region is always to enhance the average friction and heat transfer coefficients for the entire tube. This enhancement can be significant for short tubes but negligible for long ones. h or f hx fx Entrance Fully region developed region x Entry Lengths Lh The hydrodynamic entry length is usually taken to be the distance from the tube entrance where the friction coefficient reaches within about 2 percent of the fully developed value. In laminar flow, the hydrodynamic and thermal entry lengths are given approximately as [see Kays and Crawford (1993), Ref. 13, and Shah and Bhatti (1987), Ref. 25] Lt Fully developed flow Thermal boundary layer Velocity boundary layer Lh, laminar Lt, laminar 0.05 Re D 0.05 Re Pr D (8-11) Pr Lh, laminar (8-12) For Re 20, the hydrodynamic entry length is about the size of the diameter, but increases linearly with the velocity. In the limiting case of Re 2300, the hydrodynamic entry length is 115D. In turbulent flow, the intense mixing during random fluctuations usually overshadows the effects of momentum and heat diffusion, and therefore the hydrodynamic and thermal entry lengths are of about the same size and independent of the Prandtl number. Also, the friction factor and the heat transfer coefficient remain constant in fully developed laminar or turbulent flow since the velocity and normalized temperature profiles do not vary in the flow direction. The hydrodynamic entry length for turbulent flow can be determined from [see Bhatti and Shah (1987), Ref. 1, and Zhi-qing (1982), Ref. 31] Lh, turbulent 1.359 Re1/4 (8-13) The hydrodynamic entry length is much shorter in turbulent flow, as expected, and its dependence on the Reynolds number is weaker. It is 11D at Re 10,000, and increases to 43D at Re 105. In practice, it is generally agreed that the entrance effects are confined within a tube length of 10 diameters, and the hydrodynamic and thermal entry lengths are approximately taken to be Lh, turbulent Lt, turbulent 10D (8-14) The variation of local Nusselt number along a tube in turbulent flow for both uniform surface temperature and uniform surface heat flux is given in Figure 8–9 for the range of Reynolds numbers encountered in heat transfer equipment. We make these important observations from this figure: • The Nusselt numbers and thus the convection heat transfer coefficients are much higher in the entrance region. FIGURE 8–8 Variation of the friction factor and the convection heat transfer coefficient in the flow direction for flow in a tube (Pr 1). cen58933_ch08.qxd 9/4/2002 11:29 AM Page 426 426 HEAT TRANSFER 800 700 Nux, T (Ts = constant) · Nux, H (q s = constant) Nux, T Nux, H 600 500 D 400 105 200 FIGURE 8–9 Variation of local Nusselt number along a tube in turbulent flow for both uniform surface temperature and uniform surface heat flux [Deissler (1953), Ref. 4]. 105 Re = 2 300 6 0 2 4 6 8 10 104 3 100 104 104 12 14 16 18...
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