Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 81
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It is therefore given by Fourier’s law asBOOKCOMP, Inc. — John Wiley & Sons / Page 527 / 2nd Proofs / Heat Transfer Handbook / Bejan19.097pt PgVar———Normal Page* PgEnds: Eject[527], (3)Figure 7.1 Natural convection flow over a vertical surface, together with the coordinatesystem.q = h̄A(Tw − T∞ )*528123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONq = −kA∂T∂y(7.2)0Here the temperature gradient is evaluated at the surface, y = 0, in the fluid and k isthe thermal conductivity of the fluid.
From this equation it is obvious that the naturalconvection flow largely affects the temperature gradient at the surface, since theremaining parameters remain essentially unaltered. The analysis is therefore directedat determining this gradient, which in turn depends on the nature and characteristicsof the flow, temperature field, and fluid properties.The heat transfer coefficient h̄ represents an integrated value for the heat transferrate over the entire surface, since, in general, the local value hx would vary with thevertical distance from the leading edge, x = 0, of the vertical surface.
The local heattransfer coefficient hx is defined by the equationq = hx (Tw − T∞ )(7.3)where q is the rate of heat transfer per unit area per unit time at a location x, where thesurface temperature difference is Tw − T∞ , which may itself be a function of x. Theaverage heat transfer coefficient h̄ is obtained from eq. (7.3) through integration overthe entire surface area.
Both h̄ and hx are generally given in terms of a nondimensionalparameter called the Nusselt number Nu. Again, an overall (or average) value Nu, anda local value Nux , may be defined asNu =h̄LkNux =hx xk(7.4)where L is the height of the vertical surface and thus represents a characteristicdimension.The fluid far from the vertical surface is stationary, since an extensive medium isconsidered. The fluid next to the surface is also stationary, due to the no-slip condition.Therefore, flow exists in a layer adjacent to the surface, with zero vertical velocity oneither side, as shown in Fig. 7.2. A small normal velocity component does exist at theedge of this layer, due to entrainment into the flow. The temperature varies from Twto T∞ .
Therefore, the maximum vertical velocity occurs at some distance away fromthe surface. Its exact location and magnitude have to be determined through analysisor experimentation.The flow near the bottom or leading edge of the surface is laminar, as indicatedby a well-ordered and well-layered flow, with no significant disturbance. However,as the flow proceeds vertically upward or downstream, the flow gets more and moredisorderly and disturbed, because of flow instability, eventually becoming chaoticand random, a condition termed turbulent flow. The region between the laminarand turbulent flow regimes is termed the transition region. Its location and extentdepend on several variables, such as the temperature of the surface, the fluid, and thenature and magnitude of external disturbances in the vicinity of the flow.
Most ofthe processes encountered in nature are generally turbulent. However, flows in manyBOOKCOMP, Inc. — John Wiley & Sons / Page 528 / 2nd Proofs / Heat Transfer Handbook / Bejan[528], (4)Lines: 105 to 137———1.76207pt PgVar———Normal PagePgEnds: TEX[528], (4)BASIC MECHANISMS AND GOVERNING EQUATIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445529[529], (5)Figure 7.2 Velocity and temperature distributions in natural convection flow over a verticalsurface.industrial applications, such as those in electronic systems, are often in the laminaror transition regime.
A determination of the regime of the flow and its effect on theflow parameters and heat transfer rates is therefore important.Natural convection flow may also arise in enclosed regions. This flow, which isgenerally termed internal natural convection, is different in many ways from the external natural convection considered in the preceding discussion on a vertical heatedsurface immersed in an extensive, quiescent, isothermal medium.
Buoyancy-inducedflows in rooms, transport in complete or partial enclosures containing electronicequipment, flows in enclosed water bodies, and flows in the liquid melts of solidifying materials are examples of internal natural convection. In this chapter we discussboth external and internal natural convection for a variety of flow configurations andcircumstances.7.27.2.1BASIC MECHANISMS AND GOVERNING EQUATIONSGoverning EquationsThe governing equations for a convective heat transfer process are obtained by considerations of mass and energy conservation and of the balance between the rate ofmomentum change and applied forces.
These equations may be written, for constantviscosity µ and zero bulk viscosity, as (Gebhart et al., 1988)Dρ∂ρ=+ V · ∇ρ = −ρ∇ · VDt∂tBOOKCOMP, Inc. — John Wiley & Sons / Page 529 / 2nd Proofs / Heat Transfer Handbook / Bejan(7.5)Lines: 137 to 155———0.69102pt PgVar———Normal PagePgEnds: TEX[529], (5)530123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTION∂VDVµ=ρ+ V · ∇V = F − ∇p + µ∇ 2 V + ∇(∇ · V)Dt∂t3DT∂TDpρcp= ρcp+ V · ∇T = ∇ · (k∇T ) + q + βT+ µΦvDt∂tDtρ(7.6)(7.7)where V is the velocity vector, T the local temperature, t the time, F the body forceper unit volume, cp the specific heat at constant pressure, p the static pressure, ρthe fluid density, β the coefficient of thermal expansion of the fluid, Φv the viscousdissipation (which is the irreversible part of the energy transfer due to viscous forces),and q the energy generation per unit volume.
The coefficient of thermal expansionβ = −(1/ρ)(∂ρ/∂T )p , where the subscript p denotes constant pressure. For a perfectgas, β = 1/T , where T is the absolute temperature. The total, or particle, derivativeD/Dt may be expressed in terms of local derivative as ∂/∂t + V · ∇.[530], (6)As mentioned earlier, in natural convection flows, the basic driving force arisesfrom the temperature (or concentration) field. The temperature variation causes adifference in density, which then results in a buoyancy force due to the presenceLines: 155 to 188of the body force field.
For a gravitational field, the body force F = ρg, where g———is the gravitational acceleration. Therefore, it is the variation of ρ with temperature5.08006pt PgVarthat gives rise to the flow. The temperature field is linked with the flow, and all———the preceding conservation equations are coupled through variation in the density ρ.Long PageTherefore, these equations have to be solved simultaneously to determine the velocity, * PgEnds: Ejectpressure, and temperature distributions in space and in time.
Due to this complexityin the analysis of the flow, several simplifying assumptions and approximations aregenerally made to solve natural convection flows.[530], (6)In the momentum equation, the local static pressure p may be broken down intotwo terms: one, pa , due to the hydrostatic pressure, and other other, pd , the dynamicpressure due to the motion of the fluid (i.e., p = pa + pd ). The former pressurecomponent, coupled with the body force acting on the fluid, constitutes the buoyancyforce that is driving mechanism for the flow.
If ρ∞ is the density of the fluid in theambient medium, we may write the buoyancy term asF − ∇p = (ρg − ∇pa ) − ∇pd = (ρg − ρ∞ g) − ∇pd = (ρ − ρ∞ )g − ∇pd(7.8)If g is downward and the x direction is upward (i.e., g = −ig, where i is the unitvector in the x direction and g is the magnitude of the gravitational acceleration, asis generally the case for vertical buoyant flows), thenF − ∇p = (ρ∞ − ρ)gi − ∇pd(7.9)and the buoyancy term appears only in the x-direction momentum equation. Therefore, the resulting governing equations for natural convection are the continuity equation, eq. (7.5), the energy equation, eq. (7.7), and the momentum equation, whichbecomesρDVµ= (ρ − ρ∞ )g − ∇pd + µ∇ 2 V + ∇(∇ · V)Dt3BOOKCOMP, Inc.
— John Wiley & Sons / Page 530 / 2nd Proofs / Heat Transfer Handbook / Bejan(7.10)BASIC MECHANISMS AND GOVERNING EQUATIONS1234567891011121314151617181920212223242526272829303132333435363738394041424344457.2.2531Common ApproximationsThe governing equations for natural convection flow are coupled, elliptic, partialdifferential equations and are therefore of considerable complexity. Another problemin obtaining a solution to these equations lies in the inevitable variation of the densityρ with temperature or concentration.
Several approximations are generally made tosimplify these equations. Two of the most important among these are the Boussinesqand the boundary layer approximations.The Boussinesq approximations involve two aspects. First, the density variationin the continuity equation is neglected. Thus, the continuity equation, eq. (7.5), becomes ∇ · V = 0. Second, the density difference, which causes the flow, is approximated as a pure temperature or concentration effect (i.e., the effect of pressureon the density is neglected). In fact, the density difference is estimated for thermalbuoyancy asρ∞ − ρ = ρβ(T − T∞ )(7.11)These approximations are employed very extensively for natural convection.
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