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Unformatted text preview: 20 x/D • The Nusselt number reaches a constant value at a distance of less than 10 diameters, and thus the flow can be assumed to be fully developed for x 10D. • The Nusselt numbers for the uniform surface temperature and uniform surface heat flux conditions are identical in the fully developed regions, and nearly identical in the entrance regions. Therefore, Nusselt number is insensitive to the type of thermal boundary condition, and the turbulent flow correlations can be used for either type of boundary condition. Precise correlations for the friction and heat transfer coefficients for the entrance regions are available in the literature. However, the tubes used in practice in forced convection are usually several times the length of either entrance region, and thus the flow through the tubes is often assumed to be fully developed for the entire length of the tube. This simplistic approach gives reasonable results for long tubes and conservative results for short ones. 8–4 . GENERAL THERMAL ANALYSIS You will recall that in the absence of any work interactions (such as electric resistance heating), the conservation of energy equation for the steady flow of a fluid in a tube can be expressed as (Fig. 8–10) Q Ti ·CT m pi I Te · m Cp Te Energy balance: · · Q = m Cp(Te – Ti ) FIGURE 8–10 The heat transfer to a fluid flowing in a tube is equal to the increase in the energy of the fluid. · Q · m Cp(Te Ti) (W) (8-15) where Ti and Te are the mean fluid temperatures at the inlet and exit of the · tube, respectively, and Q is the rate of heat transfer to or from the fluid. Note that the temperature of a fluid flowing in a tube remains constant in the absence of any energy interactions through the wall of the tube. The thermal conditions at the surface can usually be approximated with constant) or reasonable accuracy to be constant surface temperature (Ts · constant surface heat flux (qs constant). For example, the constant surface cen58933_ch08.qxd 9/4/2002 11:29 AM Page 427 427 CHAPTER 8 temperature condition is realized when a phase change process such as boiling or condensation occurs at the outer surface of a tube. The constant surface heat flux condition is realized when the tube is subjected to radiation or electric resistance heating uniformly from all directions. Surface heat flux is expressed as · qs hx (Ts (W/m2) Tm) (8-16) where hx is the local heat transfer coefficient and Ts and Tm are the surface and the mean fluid temperatures at that location. Note that the mean fluid temperature Tm of a fluid flowing in a tube must change during heating or cooling. Therefore, when hx h constant, the surface temperature Ts must change · · when qs constant, and the surface heat flux qs must change when Ts con· stant. Thus we may have either Ts constant or qs constant at the surface of a tube, but not both. Next we consider convection heat transfer for these two common cases. · Constant Surface Heat Flux (qs · In the case of qs constant) T constant, the rate of heat transfer can also be expressed as · Q q·s As · m Cp(Te Ti) (W) Entrance region Fully developed region Ts (8-17) Te Then the mean fluid temperature at the tube exit becomes Te Ti · qs As · m Cp Ti Note that the mean fluid temperature increases linearly in the flow direction in the case of constant surface heat flux, since the surface area increases linearly in the flow direction (As is equal to the perimeter, which is constant, times the tube length). · The surface temperature in the case of constant surface heat flux qs can be determined from · qs h(Ts Tm) → Ts Tm · qs h dTm · q s(pdx) → dx · qs p · m Cp dTs dx · qs = constant constant L Ti x Te FIGURE 8–11 Variation of the tube surface and the mean fluid temperatures along the tube for the case of constant surface heat flux. · δ Q = h (Ts – Tm ) dA Tm (8-20) where p is the perimeter of the tube. · Noting that both qs and h are constants, the differentiation of Eq. 8–19 with respect to x gives dTm dx 0 (8-19) In the fully developed region, the surface temperature Ts will also increase linearly in the flow direction since h is constant and thus Ts Tm constant (Fig. 8–11). Of course this is true when the fluid properties remain constant during flow. The slope of the mean fluid temperature Tm on a T-x diagram can be determined by applying the steady-flow energy balance to a tube slice of thickness dx shown in Figure 8–12. It gives · m Cp dTm Tm · qs ∆T = Ts – Tm = –– h (8-18) (8-21) Tm + dTm . · m Cp(Tm + d Tm) m CpTm Ts dx FIGURE 8–12 Energy interactions for a differential control volume in a tube. cen58933_ch08.qxd 9/4/2002 11:29 AM Page 428 428 HEAT TRANSFER Also, the requirement that the dimensionless temperature profile remains unchanged in the fully developed region gives Ts x Ts since Ts T Tm Tm 0 → T (r) Ts 2 Ts1 x · qs T x 0 dTs dx T x → (8-22) dTs dx · qs p · Cp m dTm dx constant (8-23) Then we conclude that in fully developed flow in a tube subjected to constant surface heat flux, the temperature gradient is independent of x and thus the shape of the temperature profile does not change along the tube (Fig. 8–13). · 2 For a circular tube, p 2 R and m m Ac m...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.

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