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Unformatted text preview: /(Ts Ti)] Te
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
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 ·
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
Ts 120°C 115°C 5°C
120°C 15°C 105°C
ln( Te / Ti) ln(5/105) Tln Te
Ti The heat transfer surface area is ·
Q hAs Tln → As ·
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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