cen58933_ch08

# But the values of the properties at a cross section

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Unformatted text preview: over the cross section. But the values of the properties at a cross section may change with time unless the flow is steady. 8–2 I MEAN VELOCITY AND MEAN TEMPERATURE In external flow, the free-stream velocity served as a convenient reference velocity for use in the evaluation of the Reynolds number and the friction cen58933_ch08.qxd 9/4/2002 11:29 AM Page 421 421 CHAPTER 8 coefficient. In internal flow, there is no free stream and thus we need an alternative. The fluid velocity in a tube changes from zero at the surface because of the no-slip condition, to a maximum at the tube center. Therefore, it is convenient to work with an average or mean velocity m, which remains constant for incompressible flow when the cross sectional area of the tube is constant. The mean velocity in actual heating and cooling applications may change somewhat because of the changes in density with temperature. But, in practice, we evaluate the fluid properties at some average temperature and treat them as constants. The convenience in working with constant properties usually more than justifies the slight loss in accuracy. The value of the mean velocity m in a tube is determined from the requirement that the conservation of mass principle be satisfied (Fig. 8–2). That is, · m m Ac (r, x)dAc (8-1) =0 max max (a) Actual m (b) Idealized FIGURE 8–2 Actual and idealized velocity profiles for flow in a tube (the mass flow rate of the fluid is the same for both cases). Ac · where m is the mass flow rate, is the density, Ac is the cross sectional area, and (r, x) is the velocity profile. Then the mean velocity for incompressible flow in a circular tube of radius R can be expressed as Ac R (r, x)dAc m (r, x)2 rdr 0 Ac 2 R 2 R2 R (r, x)rdr (8-2) 0 Therefore, when we know the mass flow rate or the velocity profile, the mean velocity can be determined easily. When a fluid is heated or cooled as it flows through a tube, the temperature of the fluid at any cross section changes from Ts at the surface of the wall to some maximum (or minimum in the case of heating) at the tube center. In fluid flow it is convenient to work with an average or mean temperature Tm that remains uniform at a cross section. Unlike the mean velocity, the mean temperature Tm will change in the flow direction whenever the fluid is heated or cooled. The value of the mean temperature Tm is determined from the requirement that the conservation of energy principle be satisfied. That is, the energy transported by the fluid through a cross section in actual flow must be equal to the energy that would be transported through the same cross section if the fluid were at a constant temperature Tm. This can be expressed mathematically as (Fig. 8–3) · E fluid · m CpTm · m · Cp T m Ac CpT dAc Tm · m · m Cp R 0 CpT( m( 2 rdr) 2 R )Cp 2 2 mR R T(r, x) 0 Tmin (a) Actual (8-3) · where Cp is the specific heat of the fluid. Note that the product mCpTm at any cross section along the tube represents the energy flow with the fluid at that cross section. Then the mean temperature of a fluid with constant density and specific heat flowing in a circular pipe of radius R can be expressed as · CpT m Ts (r, x) rdr (8-4) Tm (b) Idealized FIGURE 8–3 Actual and idealized temperature profiles for flow in a tube (the rate at which energy is transported with the fluid is the same for both cases). cen58933_ch08.qxd 9/4/2002 11:29 AM Page 422 422 HEAT TRANSFER Note that the mean temperature Tm of a fluid changes during heating or cooling. Also, the fluid properties in internal flow are usually evaluated at the bulk mean fluid temperature, which is the arithmetic average of the mean temperatures at the inlet and the exit. That is, Tb (Tm, i Tm, e)/2. Laminar and Turbulent Flow In Tubes Flow in a tube can be laminar or turbulent, depending on the flow conditions. Fluid flow is streamlined and thus laminar at low velocities, but turns turbulent as the velocity is increased beyond a critical value. Transition from laminar to turbulent flow does not occur suddenly; rather, it occurs over some range of velocity where the flow fluctuates between laminar and turbulent flows before it becomes fully turbulent. Most pipe flows encountered in practice are turbulent. Laminar flow is encountered when highly viscous fluids such as oils flow in small diameter tubes or narrow passages. For flow in a circular tube, the Reynolds number is defined as mD Re Circular tube: Dh = 4(πD2/4) =D πD a Rectangular duct: Dh = b 4ab 2ab = 2(a + b) a+b Circular tubes: FIGURE 8–4 The hydraulic diameter Dh 4Ac /p is defined such that it reduces to ordinary diameter for circular tubes. Turbulent Dh Pipe wall FIGURE 8–5 In the transitional flow region of 2300 Re 4000, the flow switches between laminar and turbulent randomly. 4Ac p 4 D2/4 D D It certainly is desirable to have precise values of Reynolds numbers for laminar, transitional, and turbulent flows, but this is not the case in practice. This is because the transition from laminar to turbulent flow also depends on the degree of disturbance of the flow by surface roughness, pipe vibrations, and the fluctuations in the flow. Under most...
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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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