Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 7
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Here V= is the free-stream velocity.A particularly interesting phenomenon connected withtransition in the boundary layer occurs with blunt bodies,e.g., spheres or circular cylinders. In the region of adversepressure gradient (i.e., 3P/bx > 0 in Fig. 1.9) the boundarylayer separates from the surface. At this location the shearstress goes to zero, and beyond this point there is a reversalof flow in the vicinity of the wall, as shown in Fig.
1.9. In this1.10CHAPTER ONEseparation region, the analysis of the flow is very difficult and emphasis is placed on the useof experimental methods to determine heat and mass transfer.Nonnewtonian Fluids. In previous parts of this section we have mentioned only newtonianfluids. Newtonian fluids are those that have a linear relationship between the shear stress andthe velocity gradient (or rate of strain), as in Eq.
1.20. The shear stress x is equal to zero whendu/dy equals zero. The viscosity, given by the ratio of shear stress to velocity gradient, is independent of the velocity gradient (or rate of strain), but may be a function of temperature andpressure.Gases, and liquids such as water, usually exhibit newtonian behavior. However, many fluids, such as colloidal suspensions, polymeric solutions, paint, grease, blood, ketchup, slurry,etc., do not follow the linear shear stress-velocity gradient relation; these are called nonnewtonian fluids. Chapter 10 deals with the hydrodynamics and heat transfer of nonnewtonianfluids.Combined Heat Transfer MechanismsIn practice, heat transfer frequently occurs by two mechanisms in parallel.
A typical exampleis shown in Fig. 1.10. In this case the heat conducted through the plate is removed from theplate surface by a combination of convection and radiation. An energy balance in this casegives~ksAd-~,. J = hZ(Tw- T,o) + oA~ (T 4- T4a)uYl(1.33)Wwhere Ta is the temperature of the surroundings, ks is the thermal conductivity of the solidplate, and ~ is the emissivity of the plate (i.e., in this special case ~1-2 = ~, as the area of theplate is much smaller than the area of the surroundings [3]). The plate and the surroundingsare separated by a gas that has no effect on radiation.There are many applications where radiation is combined with other modes of heat transfer, and the solution of such problems can often be simplified by using a thermal resistance Rthfor radiation.
The definition of Rth is similar to that of the thermal resistance for convectionand conduction. If the heat transfer by radiation, for the example in Fig. 1.10, is writtenTw-T~q=gth(1.34)jF,ow, u _L//j--YtA/4 qco~ hA(Tw-Too)-/ST eat conductedthrough wailFIGURE 1.10 Combination of conduction, convection, and radiation heattransfer.BASIC CONCEPTS OF HEAT TRANSFER1.11the resistance is given byRth--(1.35)T w - Tao A e ( T 4 _ T 4)Also, a heat transfer coefficient hr c a n be defined for radiation:hr-1- ° e ( T 4 - T4) : oe(Tw + Ta)(T 2 + TZa)Rthm(1.36)T w - TaHere we have linearized the radiation rate equation, making the heat rate proportional to atemperature difference rather than to the difference between two temperatures to the fourthpower.
Note that hr depends strongly on temperature, while the temperature dependence ofthe convection heat transfer coefficient h is generally weak.CONSERVATION EQUATIONSEach time we try to solve a new problem related to momentum, heat, and mass transfer in afluid, it is convenient to start with a set of equations based on basic laws of conservation forphysical systems. These equations include:1.2.3.4.TheTheTheThecontinuity equation (conservation of mass)equation of motion (conservation of momentum)energy equation (conservation of energy, or the first law of thermodynamics)conservation equation for species (conservation of species)These equations are sometimes called the equations o f change, inasmuch as they describe thechange of velocity, temperature, and concentration with respect to time and position in thesystem.The first three equations are sufficient for problems involving a pure fluid (a pure substance is a single substance characterized by an unvarying chemical structure).
The fourthequation is added for a mixture of chemical species, i.e., when mass diffusion with or withoutchemical reactions is present.• The control volume. When deriving the conservation equations it is necessary to select acontrol volume. The derivation can be performed for a volume element of any shape in agiven coordinate system, although the most convenient shape is usually assumed for simplicity (e.g., a rectangular shape in a rectangular coordinate system).
For illustration purposes, different coordinate systems are shown in Fig. 1.11. In selecting a control volume we(x, y, z)(r,9IIIIzIzIII,~i.x...Y(a)FIGURE 1.11(c) spherical.. I x.=y.~.(r,O, ¢,)z)1I i =Y-°'~ ~///'Ill1I~ ._~_r,Jf/x,~-"~¢"''"~'x(b)"y(c)Coordinate systems: (a) rectangular, (b) cylindrical,1.12CHAPTERONEhave the option of using a volume fixed in space, in which case the fluid flows through theboundaries, or a volume containing a fixed mass of fluid and moving with the fluid.
The former is known as the eulerian viewpoint and the latter is the lagrangian viewpoint. Bothapproaches yield equivalent results.• The partial time derivative OB/Ot. T h e partial time derivative of B(x, y, z, t), where B is anycontinuum property (e.g., density, velocity, temperature, concentration, etc.), represents thechange of B with time at a fixed position in space. In other words, 3B/Ot is the change of Bwith t as seen by a stationary observer.• Total time derivative dB/dt. T h e total time derivative is related to the partial time derivativeas follows:dBOBtit-dx OBdy OB+ ¥ -ffx + ¥dz OB(1.37)+ d--;where dx/dt, dy/dt, and dz/dt are the components of the velocity of a moving observer.Therefore, dB/dt is the change of B with time as seen by the moving observer.• Substantial time derivative DB/Dt. This derivative is a special kind of total time derivativewhere the velocity of the observer is just the same as the velocity of the stream, i.e., theobserver drifts along with the current:DBDt-i)B~)Bi)B~)B~-7 + u -~x + v oy-X--+ w ~ z(1.38)where u, v, and w are the components of the local fluid velocity V.
The substantial timederivative is also called the derivative following the motion. T h e sum of the last three termson the right side of Eq. 1.38 is called the convective contribution because it represents thechange in B due to translation.The use of the operator D / D t is always made when rearranging various conservation equations related to the volume element fixed in space to an element following the fluid motion.The operator D / D t may also be expressed in vector form:DDt- - - + ( V . V)3t(1.39)Mathematical operations involving V are given in many textbooks.
Applications of V invarious operations involving the conservation equations are given in Refs. 6 and 10. Table 1.1gives the expressions for D / D t in different coordinate systems.TABLE 1.1Substantial Derivative in Different Coordinate SystemsRectangular coordinates (x, y, z):D/)/)/)/)Dt - i)t + u--~x + V-~y + w ~)zCylindrical coordinates (r, 0, z):D/)/)v0 /)Dt - i)t + v, ~r + --r - ~ + Vz OzSpherical coordinates (r, 0, ~)"D/)/)v0 /)v,/)Dt - ~)t + v, -~r + ~r - ~ + rsin0 /){~BASIC CONCEPTS OF HEAT TRANSFER1.13The Equation of ContinuiWFor a volume element fixed in space,3p _ - ( V - p V )3tnet rate ofmass effiux perunit volume(1.40)The continuity equation in this form describes the rate of change of density at a fixed point inthe fluid. By performing the indicated differentiation on the right side of Eq.
1.40 and collecting all derivatives of p on the left side, we obtain an equivalent form of the equation ofcontinuity:DpDt --p(V.V)(1.41)The continuity equation in this form describes the rate of change of density as seen by anobserver "floating along" with the fluid.For a fluid of constant density (incompressible fluid), the equation of continuity becomes:V- V = 0(1.42)Table 1.2 gives the equation of continuity in different coordinate systems.TABLE 1.2 Equation of Continuity in Different Coordinate SystemsRectangular coordinates (x, y, z):c3pC3C3C3a-S-+ ~ (p") + -b-;y(pv)+ ~ (pw): oCylindrical coordinates (r, 0, z):c3p 1 C31,9C3c3t + --r-~-r (prvr) +--r - ~ (pv0) + -~z (pVz) = 0Spherical coordinates (r, 0, ¢):1C3c3p 1 C31C3(pv,) =0C3t + ~ -~r (pr2vr) + ~ sinr 0 --C30(pve sin 0) + ~r sin 0 mc3(1)Incompressible flowRectangular coordinates (x, y, z):c3u c3vc3wa x + ~ y + ~ =°Cylindrical coordinates (r, O, z):1 C31 c3ver c3r (rVr) + - - r - - ~c3Vz=0Spherical coordinates (r, 0, ~):1 C31C31 c3v,r 2 c3r (r2vr) + r sin 0 C30 (v0 sin 0) + r sin 0 C3~ - 01.14CHAPTERONEThe Equation of Motion (Momentum Equation)The momentum equation for a stationary volume element (i.e., a balance over a volume element fixed in space) with gravity as the only body force is given by3pVOt= - ( V .
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