John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 50
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The thermal entry length, xet , turns out to be different from xe .We deal with it shortly.The hydrodynamic entry length for a pipe carrying fluid at speedsnear the transitional Reynolds number (2100) will extend beyond 100 diameters. Since heat transfer in pipes shorter than this is very often important, we will eventually have to deal with the entry region.The velocity profile for a fully developed laminar incompressible pipeflow can be derived from the momentum equation for an axisymmetricflow. It turns out that the b.l. assumptions all happen to be valid for afully developed pipe flow:• The pressure is constant across any section.• ∂ 2 u ∂x 2 is exactly zero.• The radial velocity is not just small, but it is zero.• The term ∂u ∂x is not just small, but it is zero.The boundary layer equation for cylindrically symmetrical flows is quitesimilar to that for a flat surface, eqn.
(6.13):u∂u1 dpν ∂∂u+v=−+∂x∂rρ dxr ∂rr∂u∂r(7.13)347Forced convection in a variety of configurations§7.2For fully developed flows, we go beyond the b.l. assumptions and setv and ∂u/∂x equal to zero as well, so eqn. (7.13) becomes1 dr drrdudr1 dpµ dx=We integrate this twice and get1 dpr 2 + C1 ln r + C2u=4µ dxThe two b.c.’s on u express the no-slip (or zero-velocity) condition at thewall and the fact that u must be symmetrical in r :du =0u(r = R) = 0 anddr r =0They give C1 = 0 and C2 = (−dp/dx)R 2 /4µ, so 2 dprR2−1−u=4µdxR(7.14)This is the familiar Hagen-Poiseuille2 parabolicvelocity profile.
We can2identify the lead constant (−dp/dx)R 4µ as the maximum centerlinevelocity, umax . In accordance with the conservation of mass (see Problem 7.1), 2uav = umax , so 2 ru=2 1−(7.15)uavRThermal behavior of a flow with a uniform heat flux at the wallThe b.l. energy equation for a fully developed laminar incompressibleflow, eqn. (6.40), takes the following simple form in a pipe flow wherethe radial velocity is equal to zero:u1 ∂∂T=α∂xr ∂rr∂T∂r(7.16)2The German scientist G.
Hagen showed experimentally how u varied with r , dp/dx,µ, and R, in 1839. J. Poiseuille (pronounced Pwa-zói or, more precisely, Pwä-z´ē) didthe same thing, almost simultaneously (1840), in France. Poiseuille was a physicianinterested in blood flow, and we find today that if medical students know nothing elseabout fluid flow, they know “Poiseuille’s law.”e348§7.2Heat transfer to and from laminar flows in pipesFor a fully developed flow with qw = constant, Tw and Tb increase linearlywith x.
In particular, by integrating eqn. (7.10), we findxTb (x) − Tbin =0qw P xqw Pdx =ṁcpṁcp(7.17)Then, from eqns. (7.11) and (7.1), we get∂TdTbqw Pqw (2π R)2qw α====2ṁcp∂xdxρcp uav (π R )uav RkUsing this result and eqn. (7.15) in eqn. (7.16), we obtain 2 r1 ddTqw4 1−=rRRkr drdrThis ordinary d.e. in r can be integrated twice to obtain4qw r 2r4T =−+ C1 ln r + C2Rk416R 2(7.18)(7.19)The first b.c. on this equation is the symmetry condition, ∂T /∂r = 0at r = 0, and it gives C1 = 0. The second b.c. is the definition of themixing-cup temperature, eqn. (7.6).
Substituting eqn. (7.19) with C1 = 0into eqn. (7.6) and carrying out the indicated integrations, we getC2 = Tb −7 qw R24 ksoqw RT − Tb =k r 2 1 r 47−−R4 R24(7.20)and at r = R, eqn. (7.20) givesTw − Tb =11 qw D11 qw R=24 k48 k(7.21)so the local NuD for fully developed flow, based on h(x) = qw [Tw (x) −Tb (x)], isNuD ≡qw D48= 4.364=(Tw − Tb )k11(7.22)349350Forced convection in a variety of configurations§7.2Equation (7.22) is surprisingly simple.
Indeed, the fact that there isonly one dimensionless group in it is predictable by dimensional analysis.In this case the dimensional functional equation is merelyh = fn (D, k)We exclude ∆T , because h should be independent of ∆T in forced convection; µ, because the flow is parallel regardless of the viscosity; and ρu2av ,because there is no influence of momentum in a laminar incompressibleflow that never changes direction. This gives three variables, effectivelyin only two dimensions, W/K and m, resulting in just one dimensionlessgroup, NuD , which must therefore be a constant.Example 7.1Water at 20◦ C flows through a small-bore tube 1 mm in diameter ata uniform speed of 0.2 m/s.
The flow is fully developed at a pointbeyond which a constant heat flux of 6000 W/m2 is imposed. Howmuch farther down the tube will the water reach 74◦ C at its hottestpoint?Solution. As a fairly rough approximation, we evaluate propertiesat (74 + 20)/2 = 47◦ C: k = 0.6367 W/m·K, α = 1.541 × 10−7 , andν = 0.556×10−6 m2 /s. Therefore, ReD = (0.001 m)(0.2 m/s)/0.556×10−6 m2 /s = 360, and the flow is laminar.
Then, noting that T isgreatest at the wall and setting x = L at the point where Twall = 74◦ C,eqn. (7.17) gives:Tb (x = L) = 20 +qw P4qw αLL = 20 +ṁcpuav DkAnd eqn. (7.21) gives74 = Tb (x = L) +soor4qw α11 qw D11 qw D= 20 +L+48 kuav Dk48 k11 qw D uav kL= 54 −D48 k4qw α11 6000(0.001)0.2(0.6367)L= 54 −= 1785D480.63674(6000)1.541(10)−7Heat transfer to and from laminar flows in pipes§7.2so the wall temperature reaches the limiting temperature of 74◦ C atL = 1785(0.001 m) = 1.785 mWhile we did not evaluate the thermal entry length here, it may beshown to be much, much less than 1785 diameters.In the preceding example, the heat transfer coefficient is actuallyrather largeh = NuD0.6367k= 4.364= 2, 778 W/m2 KD0.001The high h is a direct result of the small tube diameter, which limits thethermal boundary layer to a small thickness and keeps the thermal resistance low. This trend leads directly to the notion of a microchannel heatexchanger.
Using small scale fabrication technologies, such as have beendeveloped in the semiconductor industry, it is possible to create channels whose characteristic diameter is in the range of 100 µm, resulting inheat transfer coefficients in the range of 104 W/m2 K for water [7.2]. If,instead, liquid sodium (k ≈ 80 W/m·K) is used as the working fluid, thelaminar flow heat transfer coefficient is on the order of 106 W/m2 K — arange that is usually associated with boiling processes!Thermal behavior of the flow in an isothermal pipeThe dimensional analysis that showed NuD = constant for flow with auniform heat flux at the wall is unchanged when the pipe wall is isothermal.
Thus, NuD should still be constant. But this time (see, e.g., [7.3,Chap. 8]) the constant changes toNuD = 3.657,Tw = constant(7.23)for fully developed flow. The behavior of the bulk temperature is discussed in Sect. 7.4.The thermal entrance regionThe thermal entrance region is of great importance in laminar flow because the thermally undeveloped region becomes extremely long for higherPr fluids. The entry-length equation (7.12) takes the following form for351352Forced convection in a variety of configurations§7.2the thermal entry region3 , where the velocity profile is assumed to befully developed before heat transfer starts at x = 0:xet 0.034 ReD PrD(7.24)Thus, the thermal entry length for the flow of cold water (Pr 10) can beover 600 diameters in length near the transitional Reynolds number, andoil flows (Pr on the order of 104 ) practically never achieve fully developedtemperature profiles.A complete analysis of the heat transfer rate in the thermal entry region becomes quite complicated.
The reader interested in details shouldlook at [7.3, Chap. 8]. Dimensional analysis of the entry problem showsthat the local value of h dependson uav , µ, ρ, D, cp , k, and x—eightvariables in m, s, kg, and J K. This means that we should anticipate fourpi-groups:NuD = fn (ReD , Pr, x/D)(7.25)In other words, to the already familiar NuD , ReD , and Pr, we add a newlength parameter, x/D. The solution of the constant wall temperatureproblem, originally formulated by Graetz in 1885 [7.6] and solved in convenient form by Sellars, Tribus, and Klein in 1956 [7.7], includes an arrangement of these dimensionless groups, called the Graetz number:Graetz number, Gz ≡ReD Pr Dx(7.26)Figure 7.4 shows values of NuD ≡ hD/k for both the uniform walltemperature and uniform wall heat flux cases.
The independent variablein the figure is a dimensionless length equal to 2/Gz. The figure alsopresents an average Nusselt number, NuD for the isothermal wall case:DhDNuD ≡=kk31LLh dx01=LL0NuD dx(7.27)The Nusselt number will be within 5% of the fully developed value if xet 0.034 ReD PrD for Tw = constant. The error decreases to 1.4% if the coefficient is raisedfrom 0.034 to 0.05 [Compare this with eqn. (7.12) and its context.].
For other situations,the coefficient changes. With qw = constant, it is 0.043 at a 5% error level; when the velocity and temperature profiles develop simultaneously, the coefficient ranges betweenabout 0.028 and 0.053 depending upon the Prandtl number and the wall boundary condition [7.4, 7.5].§7.2Heat transfer to and from laminar flows in pipesFigure 7.4 Local and average Nusselt numbers for the thermal entry region in a hydrodynamically developed laminar pipeflow.where, since h = q(x) [Tw −Tb (x)], it is not possible to average just q or∆T . We show how to find the change in Tb using h for an isothermal wallin Sect. 7.4.
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