7. Forced convection in a variety of configurations
John Richard Thome
20 avril 2008
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 1 / 64 7.2 Heat transfer to and from laminar flows in pipes
Figure 7.1 The development of a laminar velocity profile in a pipe.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 2 / 64 4.1 Heat transfer to and from laminar flows in pipes
Development of a laminar flow m˙ : The mass flow rate [kg/s] uav : The average velocity Ac : The cross-sectional area of the pipe m˙ = ρuav Ac (7.1) Fully developed flow ∂u = 0 or v = 0 (7.2) ∂x The notion of the mixing-cup, or bulk are important. We define enthalpy and ˆ temperature, hb and Tb. The bulk enthalpy for the fluid flowing through a cross section of the pipe : Z ˆ ˆ m˙ hb ≡ ρuhdAc (7.3) Ac If we assume that fluid pressure variation in the pipe are too small to affect the thermodynamic state much and if we assume a constant value of ˆ h = cp(T − Tref )dAc . R ρucpTdA Ac Tb = (7.5) mC˙ p
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 3 / 64 7.2 Heat transfer to and from laminar flows in pipes
In other word. rate of flow of enthalpy through a cross section T = b rate of flow of heat capacity through a cross section
A fully developed flow, from the thermal standpoint, is one for which the relative shape of the temperature profile does not change with x. We state this mathematically as. ∂ Tw − T = 0 (7.7) ∂x Tw − Tb Where T generally depends on x and r.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 4 / 64 7.2 Heat transfer to and from laminar flows in pipes
Figure 7.2 The thermal development of flows in tubes with a uniform wall heat flux and with a uniform wall temperature (the entrance region) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 5 / 64 7.2 Heat transfer to and from laminar flows in pipes
The entrance region If we consider a small length of pipe, dx long with perimeter P, then its surface area is Pdx and an energy balance on it is. ˆ dQ = qw Pdx =md ˙ hb =mc ˙ pdTb (7.8) − (7.9)
so that dTb qw P = (7.10) dx mc˙ p
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 6 / 64 7.2 Heat transfer to and from laminar flows in pipes
Figure 7.3 The thermal behavior of flows in tubes with a uniform wall heat flux and with a uniform temperature (the thermally developed region) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 7 / 64 7.2 Heat transfer to and from laminar flows in pipes
The thermally developed region The condition to obtain a developed profile. The convection coefficient, h, is constant in the fully developed flow. When qw is constant, then Tw − Tb will be constant in fully developed flow.
∂T dTb qw P = = = cst (7.11) ∂x dx mc˙ p
In the uniform wall temperature case, the temperature profile keeps the same shape, but its amplitude decreases with x, as does qw .
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 8 / 64 7.2 Heat transfer to and from laminar flows in pipes
The velocity profile in laminar tube flows u D Re ≡ av D ν
For a fully developed velocity profile, the hydrodynamic entry length xe.
xe ' 0.03ReD (7.12) D The velocity profile for a fully developed laminar incompressible pipe flow can be derived from the momentum equation for an axisymetric flow. It turns out that the boundary layer assumption all happen te be valid for a fully developed pipe flow : The pressure is constant across any section. ∂2/∂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. ∂u ∂u 1 dp ν ∂ ∂u u + v = − + r (7.13) ∂x ∂r ρ dx r ∂r ∂r
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 9 / 64 7.2 Heat transfer to and from laminar flows in pipes
The solution for the velocity profile in laminar tube flows is
R2 dp r 2 u(r) = − 1 − (7.14) 4µ dx R
Average velocity R2 dp uav = − 8µ dx
In accordance with the conservation of mass, 2uav = umax , so u r 2 = 2 1 − (7.15) uav R
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 10 / 64 7.2 Heat transfer to and from laminar flows in pipes
Thermal behavior of a flow with a uniform heat flux at the wall Fully developed laminar region. In cylindrical coordinates, the energy equation is (the radial velocity is equal to zero) : ∂T 1 ∂ ∂T u = α r (7.16) ∂x r ∂r ∂r
∂T dTb qw P = = (7.17b) ∂x dx mc˙ p For the constant surface heat flux boundary condition, i.e. (∂2T /∂x 2) = 0. 2 1 d dT qw r r = 4 1 − (7.18) r dr dr Rk R 2 4 4qw r r T = − + C1lnr + C2 (7.19) Rk 4 16R2 Since the temperature remains finite at r=0, then C1 = 0. From the requirement that T (R) = Tb where Tb varies with x, it follows that C2 is :
7 qw R C2 = Tb − 24 k
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 11 / 64 7.2 Heat transfer to and from laminar flows in pipes
The temperature profile for fully developed laminar flow for constant surface heat flux is thus :
2 4 qw R r 1 r 7 T = Tb + − − (7.20) k R 4 R 24
And at r = R, the last equation gives.
11 qw R 11 qw D Tw − Tb = = (7.21) 24 k 45 k
So the local NuD for fully developed flow, based on h(x) = qw /[Tw (x) − Tb(x)], is
qw D 11 NuD ≡ = = 4.364 (7.22) (Tw − Tb)k 48
00 for qs constant.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 12 / 64 7.2 Heat transfer to and from laminar flows in pipes
Thermal behavior of a flow in a isothermal pipe The dimensional anlysis that showed NuD = cst for flow with a uniform heat flux at the wall is unchanged when the pipe walll is isothermal. For fully developed flow and for Tw = constant, the convective heat transfer coefficient is :
NuD = 3.66 (7.23)
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 13 / 64 7.2 Heat transfer to and from laminar flows in pipes
The thermal entrance region The entry-length equation takes the following form for the thermal entry region, where the velocity profile is assumed to be fully developed before heat transfer starts at x=0. xet ' 0.034ReDPr (7.24) D
ReD PrD Graetz number : Gz ≡ x For an isothermal wall, the following curve ffits are available for the Nusselt number in thermally developing flow.
0.0018Gz1/3 NuD = 3.657 + (7.28) (0.04 + Gz−2/3)2
0.0668Gz1/3 NuD = 3.657 + (7.29) 0.04 + Gz−2/3
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 14 / 64 7.2 Heat transfer to and from laminar flows in pipes
For fixed qw , a more complicated formular reproduces the exact result for local Nusselt number (7.30). 1/3 4 NuD = 1.302Gz − 1 for 2x10 ≤ Gz 1/3 4 NuD = 1.302Gz − 0.5 for 667 ≤ Gz ≤ 2x10 0.506 −41/Gz NuD = 4.364 + 0.263Gz e for 0 ≤ Gz ≤ 667
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 15 / 64 7.2 Heat transfer to and from laminar flows in pipes
Figure 7.4 Local and average Nusselt numbers for the thermal entry region in a hydrodynamically developed laminar pipe flow.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 16 / 64 7.3 Turbulent pipe flow
Turbulent entry length The entry length xe and xet are generally shorter in turbulent flow than in laminar flow.
xet ≈ 10 D
Figure 7.1 : Thermal entry lengths, xet /D, which NuD will be no more than 5 per cent above its fully developed value in turbulent flow.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 17 / 64 7.3 Turbulent pipe flow
When heat transfer begins at the inlet to a pipe, the velocity and temperature profiles develop simultaneously. The entry length is then very strongly affected by the shape of the inlet. For example, an inlet that induces vortices in the pipe, such as a sharp bend or contraction, can create a much longer entry length than occurs for a thermally developing flow. These vortices may require 20 to 40 diameters to die out.
For various types of inlets, Bhatti and Shah provide the following correlation (value in Figure 7.2):
NUD C = 1 + n for Pr = 0.7 (7.31) NUoo (L/D)
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 18 / 64 7.3 Turbulent pipe flow
Figure 7.2 : Constants for the gas-flow simultaneous entry length correlation, eqn. (7.31), for various inlet configurations.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 19 / 64 7.3 Turbulent pipe flow
Illustrative experiment
Figure 7.5 : Heat transfer to air flowing in a 1 in. I.D., 60 in. long pipe
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 20 / 64 7.3 Turbulent pipe flow
Illustrative experiment
Figure 7.5 shows average heat transfer data given by Kreith for air flowing in a 1 in. I.D. isothermal pipe 60 in. in length. Let us see how these data compare with what we know about pipe flows thus far.
For laminar flow, NUD ≈ 3.66 at ReD = 750. This is the correct value for an isothermal pipe. However, the pipe is too short for flow to be fully developed over much, if any, of its length. Therefore NUD is not constant in the laminar range. The rate of rise of NUD with ReD becomes very great in the transitional range, which lies between ReD = 2100 and about 5000 in this case. Above ReD ≈ 5000, 0.8 the flow is turbulent and it turns out that NUD ≈ ReD .
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 21 / 64 7.3 Turbulent pipe flow
The Reynolds analogy and heat transfer
h Cf /2 St = = (7.32) 0.68 p ρcpuav 1 + 12.8 (Pr − 1) Cf /2
This should not be used at very low Pr’s, but it can be used in either uniform qw or uniform Tw situations.
The frictional resistance (coefficient de perte de charge ≤ coefficient de frottement) to flow in a pipe is normally expressed in terms of the Darcy-Weisbach friction factor. head loss 4p f ≡ = ( . ) 2 2 7 33 pipe lenght uav L ρuav D 2 D 2 where ∆p is the pressure drop in a pipe of length L.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 22 / 64 7.3 Turbulent pipe flow
So 8τw f = 2 = 4Cf (7.34) ρuav
For a fully developed laminar flow, we obtain.
(f /8)ReDPr NUD = p (7.35) 1 + 12.8(Pr 0.68 − 1) f /8
For turbulent flows, the frictional resistance depends of the roughness. Equation (7.35) can be used directly along with Fig. 7.6 to calculate the Nusselt number for smooth-walled pipes.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 23 / 64 7.3 Turbulent pipe flow
Figure 7.6 : Pipe friction factors. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 24 / 64 7.3 Turbulent pipe flow
Historical formulation The local Nusselt number for fully developed turbulent flow in a smooth circular tube is obtained from the Chilton-Colburn analogy.
Cf f NuD = = StPr 2/3 = Pr 2/3 (7.36/37) 2 8 ReDPr
For, 20000 < ReD < 300000, substituting the friction factor, we can find. f 0.046 = Cf = 0.2 (7.38) 4 ReD So equation becomes 4/5 1/3 NuD = 0.023ReD Pr
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 25 / 64 7.3 Turbulent pipe flow
A more widely used correlation however is the Dittus-Boelter equation.
0.8 0.4 NuD = 0.0243ReD Pr (7.39)
These equations are intended for reasonably low temperature differences under which properties can be evaluated at a mean temperature Tb + Tw /2. The Colburn equation can be modified in the following way for liquids. 0.14 0.8 1/3 µb NuD = 0.023ReD Pr (7.40) µw
The above correlations are applicable to constant temperature and constant heat flux wall condition with accuracies of about 25 percent.
John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 26 / 64 7.3 Turbulent pipe flow
Entrance region