1、Chapter 7,College of Nuclear Science and Technology,Natural Convection Systems,1,Chapter 7,College of Nuclear Science and Technology,2,7-1 Introduction,Natural, or free convection is observed as a result of the motion of the fluid due to density changes arising from the heating process.,Example,A ho
2、t radiator used for heating a room is one example of a practical device which transfers heat by free convection,Chapter 7,College of Nuclear Science and Technology,3,The movement of the fluid in free convection results from the buoyancy forces imposed on the fluid when its density in the proximity o
3、f the heat-transfer surface is decreased as a result of the heating process.,Definition,Body Forces: The buoyancy forces that give rise to the free-convection currents.,Chapter 7,College of Nuclear Science and Technology,4,Natural Convection,In an infinite space,In enclosures,The development of ther
4、mal boundary layer will not be influenced or disturbed. It does not mean infinite in geometry.,The development of thermal boundary layer or the flow of the fluids will be influenced.,Chapter 7,College of Nuclear Science and Technology,5,7-2 Free-Convection Heat-transfer On A Vertical Flat Plate,Cons
5、ider the vertical flat plate shown in Figure 7-1. When the plate is heated, a free-convection boundary is formed. The velocity profile: At the wall the velocity is 0 because of the no-slip condition; it increases to some maximum value and then decreases to 0 at the edge of boundary layer since the “
6、free-stream conditions ” are at rest in the free-convection system.,6,Chapter 7,College of Nuclear Science and Technology,The temperature profile and velocity profile along vertical flat plate,Chapter 7,College of Nuclear Science and Technology,7,The initial boundary-layer development is laminar; bu
7、t at some distance from the leading edge, depending on the fluid properties and the temperature difference between wall and environment , turbulent eddies are formed, and transition to a turbulent boundary layer begins. Farther up the plate the boundary layer may become fully turbulent.,Chapter 7,Co
8、llege of Nuclear Science and Technology,8,To analyze the heat-transfer problem, we must obtain the differential equation of motion for the boundary layer. For this purpose we choose the x coordinate along the plate and y coordinate perpendicular to the plate as in the analyses of Chapter 5. The only
9、 new force we must consider in the derivation is the weight of element of fluid .,Equate the sum of the external forces in the x direction to the change in momentum flux through control volume,Chapter 7,College of Nuclear Science and Technology,9,The pressure gradient in the x direction results from
10、 the change in elevation up the plate. Thus,Substituting the two equations,Chapter 7,College of Nuclear Science and Technology,10,The density difference may be expressed in terms of volume coefficient of expansion ,So that,The equation of motion for the free-convection boundary layer,Chapter 7,Colle
11、ge of Nuclear Science and Technology,11,Notice that the solution for the velocity profile demands a knowledge of the temperature distribution. The energy equation for the free-convection system is the same as that for a forced-convection system at low velocity,The volume coefficient of expansion may
12、 be determined from tables of properties for the specific fluid.,Chapter 7,College of Nuclear Science and Technology,12,For ideal gas, it may be calculated from,Even though the fluid motion is the result of density variations, these variations are quite small, and a satisfactory solution to the prob
13、lem may be obtained by assuming incompressible flow, that is, =constant.,Chapter 7,College of Nuclear Science and Technology,13,For free-convection system, the integral momentum equation becomes,We observe that the functional form both the velocity and the temperature distribution must be known in o
14、rder to arrive at the solution.,7-6,Chapter 7,College of Nuclear Science and Technology,14,The following conditions apply for the temperature distribution:,The temperature distribution,7-7,Chapter 7,College of Nuclear Science and Technology,15,The conditions for velocity profile are,An additional co
15、ndition may be obtained from Equation(7-4) by noting that,Chapter 7,College of Nuclear Science and Technology,16,As in the integral analysis for forced-convection problems, we assume that the velocity profiles have geometrically similar shapes at various x distances along the plate. For the convecti
16、on problem, we now assume that the velocity may be represented as a polynomial function of y multiplied by some arbitrary function of x.,Chapter 7,College of Nuclear Science and Technology,17,Applying the four conditions to the velocity profile listed above, we have,The term involving the temperatur
17、e difference, , and maybe incorporated into the function so that,7-8,Chapter 7,College of Nuclear Science and Technology,18,A plot of Equation7-8 is given in Figure 7-2.,Chapter 7,College of Nuclear Science and Technology,19,Substituting Equations 7-7 and 7-8 into Equation 7-6 and carrying out the i
18、ntegrations and differentiations yields,The integral form of the energy equation for the free-convection system is,7-9,Chapter 7,College of Nuclear Science and Technology,20,And when the assumed velocity and temperature distributions are inserted into this equation and the operations are performed,
19、these results,We assume the following exponential functional variations for Ux and ,7-11,Chapter 7,College of Nuclear Science and Technology,21,Introducing relations into Equations 7-9 and 7-11 gives,And,Chapter 7,College of Nuclear Science and Technology,22,The two equations may be solved for the c
20、onstants C1 and C2 to give,Chapter 7,College of Nuclear Science and Technology,23,The resultant expression for the boundary-layer thickness is,Definition,Grashof Number Gr,buoyancy forces,viscous forces,Chapter 7,College of Nuclear Science and Technology,24,The heat-transfer coefficient may be evalu
21、ated from,Using the temperature distribution of Equation 7-7, one obtains,Chapter 7,College of Nuclear Science and Technology,25,So that the dimensionless equation for the heat-transfer coefficient becomes,7-22,Equation (7-22) gives the variation of the local heat-transfer coefficient along the vert
22、ical plate. The average heat-transfer coefficient may then be obtained by performing the integration,Chapter 7,College of Nuclear Science and Technology,26,For the variation given by Equation 7-22, the average coefficient is,Chapter 7,College of Nuclear Science and Technology,27,7-3 Empirical Relati
23、ons For Free Convection,The following functional form for a variety of circumstances can represent the average free-convection heat-transfer coefficients.,The subscript f indicates that the properties in the dimensionless groups are evaluated at the film temperature,Chapter 7,College of Nuclear Scie
24、nce and Technology,28,Definition,Rayleigh Number: The product of Grashof and Prandtl number is called Rayleigh number,Chapter 7,College of Nuclear Science and Technology,29,Characteristic Dimensions,The characteristic dimension to be used in the Nusselt and Grashof numbers depends on the geometry of
25、 the problem. Experimental data for free-convection problems appear in a number of references, with some conflicting results.,The purpose of the sections that follow is to give the results in summary form that may be easily used for calculation purposes.,Chapter 7,College of Nuclear Science and Tech
26、nology,30,The functional form of Equation (7-25) is used for many of these presentations, with the values of the constants C and m specified for each case. Table 7-1 provides a summary of the values of these correlations constants .,Chapter 7,College of Nuclear Science and Technology,31,Chapter 7,Co
27、llege of Nuclear Science and Technology,32,7-4 Free Convection From Vertical Planes And Cylinders,For vertical surfaces, the Nusselt and Grashof numbers are formed with L, the height of the surface as the characteristic dimension.,The general criterion is that a vertical cylinder may be treated as a
28、 vertical flat plate when,Chapter 7,College of Nuclear Science and Technology,33,For isothermal surfaces, the values of the constants C and m are given in Table 7-1 with the appropriate references noted for further consultation.,More complicated relations which are applicable over wider ranges of Ra
29、yleigh number:,Chapter 7,College of Nuclear Science and Technology,34,Constant-Heat-Flux Surfaces,Extensive experiments have been reported for free convection from vertical and inclined surfaces to water under constant-heat-flux conditions. In such experiments, the results are presented in terms of
30、a modified Grashof number,Where qw is the wall heat flux in watts per square meter.,Chapter 7,College of Nuclear Science and Technology,35,The local heat-transfer coefficients were correlated by the following relation for the laminar range.,It is to be noted that the criterion for laminar flow expre
31、ssed in terms of Gr* is not the same as that expressed in terms Grx.,7-31,Chapter 7,College of Nuclear Science and Technology,36,For the turbulent region, the local heat-transfer coefficients are correlated with,7-32,All properties in Equation (7-31) and (7-32) are evaluated at the local film temper
32、ature.,Although these experiments were conducted for water, the resulting correlations are shown to work for air as well.,Chapter 7,College of Nuclear Science and Technology,37,The average heat-transfer coefficient for the constant-flux case must be obtained through a separate application of Equatio
33、n 7-23. Thus, for laminar region, using equation 7-31 to evaluate hx,Chapter 7,College of Nuclear Science and Technology,38,At this point we may note the relationship between the correlations in the form of Equation7-25 and those just presented in terms of Gr*x=GrxNux. Writing Equation 7-25 as a loc
34、al heat-transfer form gives,Inserting Grx=Gr*x/Nux gives,Or,Chapter 7,College of Nuclear Science and Technology,39,Thus, when the “characteristic ” values of m for laminar and turbulent flow are compared to the exponents on Gr*x, we obtain,Laminar, m=1/3,Laminar, m=1/4,Chapter 7,College of Nuclear S
35、cience and Technology,40,While the Gr* formulation is easier to employ for the constant-flux case, we see that the characteristic exponents fit nicely into the scheme that is presented for the isothermal surface correlations.,It is also interesting to note the variation of hx with x in the two chara
36、cteristic regimes.,For laminar,For turbulent,Chapter 7,College of Nuclear Science and Technology,41,7-5 Free Convection From Horizontal Cylinders,The values of the constants C and n are given in Table 7-1. The predictions of Morgan are the most reliable for Gr Pr of approximately,A more complicated
37、expression for use over a wider range of Gr Pr :,Chapter 7,College of Nuclear Science and Technology,42,A simpler equation is available from Reference 70 but is restricted to laminar range of,Heat transfer from horizontal cylinders to liquid metals may be calculated from Reference 46,Chapter 7,Colle
38、ge of Nuclear Science and Technology,43,7-6 Free Convection From Horizontal Plates,Isothermal Surfaces,It indicates that better agreement with experimental data can be achieved by calculating the characteristic dimension with,L=A/P,A area P the perimeter of the surface,Chapter 7,College of Nuclear S
39、cience and Technology,44,Constant Heat Flux,The experiments have produced the following correlations for constant heat flux on a horizontal plate.,For heated surface facing upward,And,Chapter 7,College of Nuclear Science and Technology,45,For heated surface facing downward,In these equations all pro
40、perties except are evaluated at a temperature Te defined by,To the heat flux by,Chapter 7,College of Nuclear Science and Technology,46,The Nusselt number is formed as before,Irregular Solids,There is no general correlation that can be applied to irregular solids. The results of Reference 77 indicate
41、 that Equation (7-25) may be used with C=0.775 and m=0.208 for a vertical cylinder with height equal to diameter.,Chapter 7,College of Nuclear Science and Technology,47,7-7 Free Convection From Inclined Surface,The angle that the plate makes with the vertical is designated , with positive angles ind
42、icating that the heater surface faces downward, as shown in Figure 7-7.,Chapter 7,College of Nuclear Science and Technology,48,For the inclined plate facing downward with approximately constant heat flux the following correlation was obtained for the average Nusselt number,7-43,In Equation 7-43 all
43、properties except are evaluated at a reference temperature Te defined by,Chapter 7,College of Nuclear Science and Technology,49, is evaluated at a temperature of,For almost-horizontal plates facing downward, that is 88 90 , an additional relation was obtained as,For an inclined plate with heated sur
44、face facing upward the empirical correlations become more complicated. For angles between -15 and -75 a suitable correlation is,Chapter 7,College of Nuclear Science and Technology,50,The quantity of Grc is a critical Grashof relation indicating when the Nusselt number starts to separate from the lam
45、inar relation of Equation7-43 and is given in the following tabulation:,Chapter 7,College of Nuclear Science and Technology,51,For GreGrc, the first term of Equation 7-46 is dropped out. There is some evidence to indicate that the above relations may also be applied to constant-temperature surfaces.
46、,In turbulent AIR region, the following empirical correlation was obtained:,Chapter 7,College of Nuclear Science and Technology,52,Gr*x is the same for the vertical plate when the heated surface faces upward. It is replaced by Gr*cos when the heated suraface face downward.,For inclined cylinders , i
47、t indicates that laminar heat-transfer under constant-heat-flux conditions may be calculated with the following relation,Chapter 7,College of Nuclear Science and Technology,53,The is the angle the cylinders makes with the vertical; that is 0corresponds to a vertical cylinder. Properties are evaluate
48、d at the film temperature except , which is evaluated at ambient conditions.,Chapter 7,College of Nuclear Science and Technology,54,7-8 Nonnewtonian Fluids,When the shear-stress viscosity relation of the fluid does not obey the simple newtonian expression of Equation5-1, the above equations for free-convection heat transfer do not apply. Extremely viscous polymer and lubricants are examples of fluids with Nonnewtonian behavior. Successful analytical and experimental studies have been carried out with such fluids, but the
