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cen58933_ch08

# 813 2 for a circular tube p 2 r and m m ac m r and eq

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Unformatted text preview: ( R ), and Eq. 8–23 becomes T x Circular tube: x1 Tm constant. Combining Eqs. 8–20, 8–21, and 8–22 gives T x T (r) Ts x 1 Ts dTs dx · 2qs mCp R dTm dx constant (8-24) x2 FIGURE 8–13 The shape of the temperature profile remains unchanged in the fully developed region of a tube subjected to constant surface heat flux. where m is the mean velocity of the fluid. Constant Surface Temperature (Ts constant) From Newton’s law of cooling, the rate of heat transfer to or from a fluid flowing in a tube can be expressed as · Q hAs Tave hAs(Ts Tm)ave (W) (8-25) where h is the average convection heat transfer coefficient, As is the heat transfer surface area (it is equal to DL for a circular pipe of length L), and Tave is some appropriate average temperature difference between the fluid and the surface. Below we discuss two suitable ways of expressing Tave. constant) case, Tave can be In the constant surface temperature (Ts expressed approximately by the arithmetic mean temperature difference Tam as Tave Tam Ti Ts Te 2 Tb (Ts Ti) (Ts 2 Te) Ts Ti Te 2 (8-26) where Tb (Ti Te)/2 is the bulk mean fluid temperature, which is the arithmetic average of the mean fluid temperatures at the inlet and the exit of the tube. Note that the arithmetic mean temperature difference Tam is simply the average of the temperature differences between the surface and the fluid at the inlet and the exit of the tube. Inherent in this definition is the assumption that the mean fluid temperature varies linearly along the tube, which is hardly ever the case when Ts constant. This simple approximation often gives acceptable results, but not always. Therefore, we need a better way to evaluate Tave. Consider the heating of a fluid in a tube of constant cross section whose inner surface is maintained at a constant temperature of Ts. We know that the cen58933_ch08.qxd 9/4/2002 11:29 AM Page 429 429 CHAPTER 8 mean temperature of the fluid Tm will increase in the flow direction as a result of heat transfer. The energy balance on a differential control volume shown in Figure 8–12 gives · m Cp dTm h(Ts Tm)dAs (8-27) That is, the increase in the energy of the fluid (represented by an increase in its mean temperature by dTm) is equal to the heat transferred to the fluid from the tube surface by convection. Noting that the differential surface area is d(Ts Tm), dAs pdx, where p is the perimeter of the tube, and that dTm since Ts is constant, the relation above can be rearranged as d(Ts Ts Integrating from x Tm Te) gives Tm) Tm 0 (tube inlet where Tm Ts ln Ts Te Ti T hp · dx mCp (8-28) Ti) to x Ts = constant Ts Tm ∆Ti L (tube exit where ∆T e ∆T = Ts – Tm Ti hAs · mCp (8-29) (Tm approaches Ts asymptotically) 0 where As pL is the surface area of the tube and h is the constant average convection heat transfer coefficient. Taking the exponential of both sides and solving for Te gives the following relation which is very useful for the determination of the mean fluid temperature at the tube exit: Te Ts (Ts · Ti) exp( hAs /m Cp) ln[(Ts hAs Te)/(Ts Ti)] x Te Ti Ts = constant (8-30) This relation can also be used to determine the mean fluid temperature Tm(x) at any x by replacing As pL by px. Note that the temperature difference between the fluid and the surface decays exponentially in the flow direction, and the rate of decay depends on the · magnitude of the exponent hAx /m Cp, as shown in Figure 8–14. This dimensionless parameter is called the number of transfer units, denoted by NTU, and is a measure of the effectiveness of the heat transfer systems. For NUT 5, the exit temperature of the fluid becomes almost equal to the surface temperature, Te Ts (Fig. 8–15). Noting that the fluid temperature can approach the surface temperature but cannot cross it, an NTU of about 5 indicates that the limit is reached for heat transfer, and the heat transfer will not increase no matter how much we extend the length of the tube. A small value of NTU, on the other hand, indicates more opportunities for heat transfer, and the heat transfer will continue increasing as the tube length is increased. A large NTU and thus a large heat transfer surface area (which means a large tube) may be desirable from a heat transfer point of view, but it may be unacceptable from an economic point of view. The selection of heat transfer equipment usually reflects a compromise between heat transfer performance and cost. · Solving Eq. 8–29 for m Cp gives · m Cp L (8-31) FIGURE 8–14 The variation of the mean fluid temperature along the tube for the case of constant temperature. Ts = 100°C Ti = 20°C · m , Cp Te As, h · NTU = h As / mCp Te , °C 0.01 0.05 0.10 0.50 1.00 5.00 10.00 20.8 23.9 27.6 51.5 70.6 99.5 100.0 FIGURE 8–15 An NTU greater than 5 indicates that the fluid flowing in a tube will reach the surface temperature at the exit regardless of the inlet temperature. cen58933_ch08.qxd 9/4/2002 11:29 AM Page 430 430 HEAT TRANSFER Substituting this into Eq. 8–17, we obtain · Q hAs Tln (8-32) where Tln ln[(Ts Ti Te Te)...
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