Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 96
Текст из файла (страница 96)
The experimentation involves the measurement ofsolid and fluid temperatures and of heat flux in a multitude of ways.DEFINITION OF TERMSThe local convective heat flux from a point on a body is often expressed through Newton'slaw of cooling, generalized asq'~ =peUeSt(iw - ie,eff)(6.1)Enthalpy is used as the measure of the thermal driving potential to broaden the application ofEq. 6.1 to thermally perfect gases with temperature-dependent specific heats wherei= f cp(T) dT(6.2)With bodies having constant surface temperatures, at low speedsie,eff = ie = cpTe(6.3)At high speeds where frictional heating takes place,ie,eff = ie + r(O)(u2/2) = iaw(6.4)where r(0) is the recovery factor, having approximate values for air of 0.85 and 0.9 for laminar and turbulent flow, respectively.For bodies with nonuniform surface temperature distributions, ie,eff depends not only onthe conditions at the boundary layer edge but also on the distribution of the surface temperature upstream of the location being considered.For constant fluid properties, Eq.
6.1 correlates both theoretical and experimental resultsfor a wide range of flow and temperature conditions through the single parameter Stantonnumber St. As will be seen in subsequent sections, this correlation is even useful when St isdependent on ie or ie,eff and the equation is nonlinear.The Reynolds analogy, defined as the ratio of the Stanton number to the local skin frictioncoefficient St/(cr/2 ) is a function of the Prandtl number and is extremely useful for estimatingheat transfer. Pressure drop can be used to predict heat transfer in pipes, and the skin frictioncan be used to predict Stanton number for external flows.When mass transfer of a foreign gas occurs at a surface, an equation similar to Eq. 6.1 isemployed to define the local mass flux of the species i of a binary mixture asj i w - peUeCmi(giw- gie)The Reynolds analogy can be extended to express(6.5)Cmi/(Q/2) as a function of the Lewis number.TWO-DIMENSIONAL LAMINAR BOUNDARY LAYERUniform Free-Stream ConditionsThe most studied configuration for forced convection has been the "flat plate," a surface atconstant pressure with a sharp leading edge.
The simplicity of this configuration so facilitatesFORCED CONVECTION, EXTERNAL FLOWS6.3the solution of the boundary layer equations, even for a variety of surface boundary conditions, that the bulk of heuristic theoretical boundary layer research is identified with the flatplate. These results are useful because much that is learned can be extended to more realisticbody shapes using computer codes and applied directly to platelike surfaces (e.g., supersonicaircraft wings or fins having wedge cross sections and attached shock waves).Uniform Surface TemperatureGoverning DifferentialEquations.
For a fluid as general as a gas in chemical equilibrium,the boundary layer equations for laminar flow over a flat plate are:(pu) + -q7 (pv)= 0aypu -~x + pv ~OlOl(6.6)= 0y l.t0 [ B OI((6.7)1 )0(u2/2)]PU-~x+PVffffy=~y -~rT-~y +B 1 - - ~ r r0y(6.8)with the boundary conditionsx=0x>0y>0;y---)oo;y=0;U -.--~ U eI = lel---> Ieu=0v=0U -- U e(6.9)I = iw = constantor0iay -0forI = i,wThe leading edge of the plate is located at x = 0.
The surface boundary conditions at y = 0reflect the assumed conditions of zero mass transfer, a prescribed uniform temperature including the case of zero heat flux and an implied condition of a smooth surface.A stream function ~ defined asOVpU = pe --~-ypV =--pe OV0X(6.10)immediately satisfies Eq. 6.6.F l u i d W i t h C o n s t a n t Properties.When the density and viscosity in Eqs. 6.6 and 6.7 areconstant, the velocity field is independent of the temperature field.
Blasius [1] collapsed thepartial differential equations (Eqs. 6.6 and 6.7) to a single ordinary differential equation bytransforming the coordinate system from x and y to ~ and 11, defined as"- X"I] = # y U e / ' l ) e X(6.11)The stream function defined by Eq. (6.10) is expressed asV = m(;)f(q)wherem(~)--VUel)e~and the velocity components are expressed in terms of the similarity variable 1"1uu---~=f'(rl)v1 [rlf'(rl) - f(rl)]ue - 2W~ex e(6.12)6.4C H A P T E R SIXwhere f(rl) is the solution of the ordinary differential equation(6.13)f'" + a//eff'= 0with the boundary conditions11=0f=0,f'=011 ~ oof' ~ 1(6.14)The u velocity profile is shown in Fig.
6.1. The velocity ratio reaches a value of 0.99 at aboundary layer thickness of8X5- ~~VRe-/~--x(6.15)/Ji|ftm11jJiv/ "0~20.40.6a81.0u/usFIGURE 6.1 Similarvelocity profile in the laminar boundarylayer on a fiat plate---constant fluid properties.The local skin friction coefficient isci_2f'(0)_ 0.332~//peUeX/l.te(6.16)W/-~exeThe average skin friction coefficient for the length of the plate up to x is defined as-dy l fxo cf2 -x _ ~dx-0.664V/Rexe(6.17)Equation 6.17 indicates that on a fiat plate the average skin friction coefficient is equal totwice the local skin friction coefficient at the trailing edge.Experimental verification of the Blasius theory has been hindered by the difficulty inreproducing the ideal fiat plate boundary conditions in the laboratory.
Whenever uniformpressure was attained and the effects of a real leading edge were accounted for, however, itwas found that the preceding calculated results were always verified to within the accuracy ofthe experiment.Pohlhausen [2] utilized the Blasius coordinate system and velocity distribution to evaluatethe convective heating processes within the constant-property boundary layer on a flat plate.He solved two problems:FORCED CONVECTION, EXTERNAL FLOWS6.51.
The convective heat transfer rate to a plate with uniform surface temperature for fluidspeeds sufficiently low to make viscous dissipation negligible2. The temperature attained by an insulated plate (zero surface heat transfer) when exposedto a high-speed stream where viscous dissipation is importantThe latter is the plate thermometer or adiabatic wall problem. Eckert and Drewitz [3] showedthat the general problem of heat transfer to a uniform-surface-temperature plate in constantproperty high-speed flow is merely the superposition of the two Pohlhausen solutions.For a uniform-surface-temperature plate in a low-speed flow (U 2 << 2cpT), Eq. 6.8 simplifies toc)T~9T ~t ~2Tpu -~x + 9v 3y - Pr ~9y2(6.18)with the boundary conditionsx=0x>0y>0;y=0;y~oo;T=TeT=TwT ~ TeWhen Eq.
6.18 is transformed to the independent variables (Eq. 6.11) and the new normalized dependent variableT-TeY0(rl) = Tw- Te(6.19)is introduced, the ordinary homogeneous differential equation that results isYg' + ½Pr fYg = 0(6.20)where fis the Blasius stream function. The transformed boundary conditions are1"1=0 g o = l }rl+~Yo-+O(6.21)Solutions for Pr = 0.5 and 1.0 are shown in Fig. 6.2 as solid curves. The abscissa of this figure is the thermal boundary layer thickness parameter rlH, consisting of the Blasius boundarylayer similarity parameter multiplied by Pr 1/3.The close agreement of the two solid curves suggests for Pr near unity that the thermal boundary layer thickness where Y0 = 0.01 is inverselyproportional to approximately Pr 1/3or~ir5Pr -1/38V'RexexmxPr -1/3(6.22)Thus, fluids with Pr less than unity have thermal boundary layers that are thick relative totheir flow boundary layers.
Conversely, fluids with Pr greater than unity have relatively thinthermal boundary layers.This latter condition suggests a particularly simple solution of Eq. 6.20 for very large Pr [4]because the temperature variations occur where the velocity distribution is still linear in 1"1(see Fig. 6.1 for 11< 2.0). The linear velocity condition in Eq. 6.20 permits expressing Y0 explicitly in terms of 11//. The solution for this case of large Pr is shown as a dashed line in Fig.
6.2and agrees quite well with the calculations based on the more exact velocity distributions forPr near unity. This agreement indicates that Eq. 6.22 is applicable over a large range of Prfrom values characteristic of gases to those for heavy oils.The local Stanton number found by Pohlhausen is represented very well bySt=a form consistent with the parameter 1"1..0.332Pr -2/3~Xe(6.23)6.6CHAPTER SIX1.00.8I0.6\vPri!).90.4--------0.50.223456r}H-- prl/3yVue/(VeX )FIGURE 6.2 Temperature distributions in the laminar boundary layer on a flat plate at uniform temperature----constant property, low-speed flow.The modified Reynolds analogy from Eqs. 6.16 and 6.23 isSt = ~ Pr -2/3(6.24)The excellent agreement of this formula with the precise numerical results of Pohlhausen [2]over a large range of Pr is shown graphically in Fig. 6.4 (solid curves are the numericalresults).
The dashed line, labeled 1.02 P r -2/3, results from the analysis employing a linear velocity distribution throughout the boundary layer [5]. Equation 6.24 has been shown to be consistent with experimental results through a successive series of data correlations dating backto Colburn [6].The average Stanton number up to station x isS~ = _1xSt dx = 2St(6.25)For an insulated plate in high-speed flow with constant properties, Eq.
6.8 combined withEq. 6.7 reduces topu -~x + pv ~ -Pr /)y------Y+ - Ce kaY](6.26)with boundary conditionsx=0x>0y>0;T - Tey --->oo; T--->TeaTy = 0;~y - 0(6.27)FORCED CONVECTION, EXTERNAL FLOWS6.7When the independent variables are transformed to the Blasius variables and a new dependent variableT-Te(6.28)r(]]) - Re212Ceis introduced into Eq. 6.26, the ordinary inhomogeneous equation that results isr" + ½Pr f r' = -2Pr f,,2(6.29)rl=0r'=~}1-1---->oo r ---->(6.30)with boundary conditionsThe solutions of this problem are indicated in Fig. 6.3. The temperature distributionsshown are based on calculations employing exact velocity distributions for Pr near unity anda linear velocity distribution for very large Pr. These temperatures have been normalized bythe temperature rise at the surface, and the abscissa is the rlH utilized in Fig.
6.2.m.0,---x.,-...~08N\XxkN\0.6it,.....,....."00.40.5~-~",%0.20' xI234nil:w ,., __., . _ . . . . .56789Pr'/3Y'~/ue/(VeX)FIGURE 6.3 Temperature profiles in the laminar boundary layer on an insulatedplate---constant-property, high-speed flow.Figure 6.3 shows less correlation for different Pr than was exhibited for the uniformsurface-temperature case in Fig. 6.2. The implication here is that the thermal boundary layerproduced by viscous dissipation grows at a rate different from Pr -1/3 for all but the very largevalues of Pr.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
