817 we obtain q has tln 8 32 where tln lnts ti te

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Unformatted text preview: /(Ts Ti)] Te Ti ln( Te / Ti) (8-33) is the logarithmic mean temperature difference. Note that Ti Ts Ti and Te Ts Te are the temperature differences between the surface and the fluid at the inlet and the exit of the tube, respectively. This Tln relation appears to be prone to misuse, but it is practically fail-safe, since using Ti in place of Te and vice versa in the numerator and/or the denominator will, at most, affect the sign, not the magnitude. Also, it can be used for both heating (Ts Ti and Te) and cooling (Ts Ti and Te) of a fluid in a tube. The logarithmic mean temperature difference Tln is obtained by tracing the actual temperature profile of the fluid along the tube, and is an exact representation of the average temperature difference between the fluid and the surface. It truly reflects the exponential decay of the local temperature difference. When Te differs from Ti by no more than 40 percent, the error in using the arithmetic mean temperature difference is less than 1 percent. But the error increases to undesirable levels when Te differs from Ti by greater amounts. Therefore, we should always use the logarithmic mean temperature difference when determining the convection heat transfer in a tube whose surface is maintained at a constant temperature Ts. EXAMPLE 8–1 Water enters a 2.5-cm-internal-diameter thin copper tube of a heat exchanger at 15°C at a rate of 0.3 kg/s, and is heated by steam condensing outside at 120°C. If the average heat transfer coefficient is 800 W/m2 C, determine the length of the tube required in order to heat the water to 115°C (Fig. 8–16). Steam Ts = 120°C Water 15°C 0.3 kg/s 115°C D = 2.5 cm FIGURE 8–16 Schematic for Example 8–1. Heating of Water in a Tube by Steam SOLUTION Water is heated by steam in a circular tube. The tube length required to heat the water to a specified temperature is to be determined. Assumptions 1 Steady operating conditions exist. 2 Fluid properties are constant. 3 The convection heat transfer coefficient is constant. 4 The conduction resistance of copper tube is negligible so that the inner surface temperature of the tube is equal to the condensation temperature of steam. Properties The specific heat of water at the bulk mean temperature of (15 115)/2 65°C is 4187 J/kg °C. The heat of condensation of steam at 120°C is 2203 kJ/kg (Table A-9). Analysis Knowing the inlet and exit temperatures of water, the rate of heat transfer is determined to be · Q · m Cp(Te Ti) (0.3 kg/s)(4.187 kJ/kg °C)(115°C The logarithmic mean temperature difference is 15°C) 125.6 kW cen58933_ch08.qxd 9/4/2002 11:29 AM Page 431 431 CHAPTER 8 Te Ti Ts Ts 120°C 115°C 5°C 120°C 15°C 105°C Te Ti 5 105 32.85°C ln( Te / Ti) ln(5/105) Tln Te Ti The heat transfer surface area is · Q hAs Tln → As · Q h Tln 125.6 kW (0.8 kW/m2 · °C)(32.85°C) 4.78 m2 Then the required length of tube becomes As DL → L As D 4.78 m2 (0.025 m) 61 m Discussion The bulk mean temperature of water during this heating process is 65°C, and thus the arithmetic mean temperature difference is Tam 120 65 55°C. Using Tam instead of Tln would give L 36 m, which is grossly in error. This shows the importance of using the logarithmic mean temperature in calculations. 8–5 I LAMINAR FLOW IN TUBES We mentioned earlier that flow in tubes is laminar for Re 2300, and that the flow is fully developed if the tube is sufficiently long (relative to the entry length) so that the entrance effects are negligible. In this section we consider the steady laminar flow of an incompressible fluid with constant properties in the fully developed region of a straight circular tube. We obtain the momentum and energy equations by applying momentum and energy balances to a differential volume element, and obtain the velocity and temperature profiles by solving them. Then we will use them to obtain relations for the friction factor and the Nusselt number. An important aspect of the analysis below is that it is one of the few available for viscous flow and forced convection. In fully developed laminar flow, each fluid particle moves at a constant axial velocity along a streamline and the velocity profile (r) remains unchanged in the flow direction. There is no motion in the radial direction, and thus the velocity component v in the direction normal to flow is everywhere zero. There is no acceleration since the flow is steady. Now consider a ring-shaped differential volume element of radius r, thickness dr, and length dx oriented coaxially with the tube, as shown in Figure 8–17. The pressure force acting on a submerged plane surface is the product of the pressure at the centroid of the surface and the surface area. The volume element involves only pressure and viscous effects, and thus the pressure and shear forces must balance each other. A force balance on the volume element in the flow direction gives (2 rdrP)x (2 rdrP)x dx (2 rdx )r (2 rdx )r dr 0 (8-34) τr dr Px Px dx τr (r) dr R r x dx max FIGURE 8–17 Free body diagram of a cylindrical fluid element of radius r, thickness dr, and length dx oriented coaxially with a horizontal tube in fully developed steady flow. cen58933_ch08.qxd 9/4/2002 11:29 AM Page 432 4...
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