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CHAPTER 5 Forced Convection: Internal Flows
ADRIAN BEJAN
Department of Mechanical Engineering and Materials Science Duke University
Durham,North Carolina
5.1 Introduction 5.2 Laminar flow and pressure drop 5.2.1 Flow entrance region 5.2.2 Fully developed flow region 5.2.3 Hydraulic diameter and pressure drop 5.3 Heat transfer in fully developed flow 5.3.1 Mean temperature
5.3.2 Thermally fully developed flow 5.4 Heat transfer in developing flow 5.4.1 Thermal entrance region 5.4.2 Thermally developing Hagen–Poiseuille flow 5.4.3 Thermally and hydraulically developing flow 5.5 Optimal channel sizes forlaminarflow
5.6 Turbulent duct flow 5.6.1 Time-averaged equations 5.6.2 Fully developed flow 5.6.3 Heat transfer in fully developed flow 5.7 Total heat transfer rate
5.7.1 Isothermal wall 5.7.2 Wall heated uniformly 5.8 Optimal channel sizes forturbulent flow 5.9 Summary of forced convection relationships Nomenclature
References
An internal flow is a flow configuration where the flowing material is surrounded by
solid walls Streams that flow through ducts are primary examples of internal flows
Heat exchangers are conglomerates of internal flows This class of fluid flow and
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convection phenomena distinguishes itself from the class of an external flow, which
is treated in Chapters 6 (forced-convection external) and 7 (natural convection) In an external flow configuration, a solid object is surrounded by the flow
There are two basic questions for the engineer who contemplates using an internal flow configuration One is the heat transfer rate, or the thermal resistance between the stream and the confining walls The other is the friction between the stream and the walls The fluid friction part of the problem is the same as calculation of the pressure drop experienced by the stream over a finite length in the flow direction The fluid friction question is the more basic, because friction is present as soon as there is flow, that is, even in the absence of heat transfer This is why we begin this chapter with the calculation of velocity and pressure drop in duct flow The heat transfer question is supplementary, as the duct flow will convect energy if a temperature difference exits between its inlet and the wall
To calculate the heat transfer rate and the temperature distribution through the flow, one must know the flow, or the velocity distribution When the variation of temperature over the flow field is sufficiently weak so that the fluid density and viscosity are adequately represented by two constants, calculation of the velocity field and pressure drop is independent of that of the temperature field This is the case in all the configurations and results reviewed in this chapter When this approximation
is valid, the velocity field is “not coupled” to the temperature field, although, as already noted, correct derivation of the temperature field requires the velocity field
as a preliminary result, that is, as an input
The following presentation is based on the method developed in Bejan (1995)
Alternative reviews of internal flow convection are available in Shah and London (1978) and Shah and Bhatti (1978) and are recommended
5.2.1 Flow Entrance Region
Considerthe laminarflow through a two-dimensional duct formed between two parallel plates, as shown in Fig 5.1 The spacing between the plates isD The flow
velocity in the inlet cross section (x = 0) is uniform (U) Mass conservation means thatU is also the mean velocity at any position x downstream,
U = A1
whereu is the longitudinal velocity component and A is the duct cross-sectional area
in general Boundary layers grow along the walls until they meet at the distance
x ≈ X downstream from the entrance The length X is called entrance length or
flow (hydrodynamic) entrance length, to be distinguished from the thermal entrance
length discussed in Section 5.4.1 In the entrance length region the boundary layers coexist with a core in which the velocity is uniform (U c) Mass conservation and the
fact that the fluid slows down in the boundary layers requires thatU c > U.
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Figure 5.1 Developing flow in the entrance region of the duct formed between two parallel plates (From Bejan, 1995.)
The lengthX divides the duct flow into an entrance region (0 < x ≤ X) and a fully
developed flow region (x ≥ X) The flow friction and heat transfer characteristics
of the entrance region are similar to those of boundary layer flows The features
of the fully developed region require special analysis, as shown in Section 5.2.2
The entrance lengthX is indicated approximately in Fig 5.1 This is not a precise
dimension, for the same reasons that the thickness of a boundary layer (δ) is known only as an order-of-magnitude length The scale ofX can be determined from the
scale ofδ, which according to the Blasius solution is
δ ∼ 5x
Ux ν
−1/2
The transition from entrance flow to fully developed flow occurs atx ∼ X and
δ ∼ D/2, and therefore it can be concluded that
X/D
where ReD = UD/ν The heat transfer literature also recommends the more precise
value 0.04 in place of the 10−2in eq (5.2) (Schlichting, 1960), although as shown in Fig 5.2 and 5.3, the transition from entrance flow to fully developed flow is smooth
The friction between fluid and walls is measured as the local shear stress at the wall surface,
τx (x) = µ ∂u
∂y
y=0
orthe dimensionless local skin friction coefficient
C f,x= 1τw
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10
102
12
x D/
ReD
C fx
Figure 5.2 Local skin-friction coefficient in the entrance region of a parallel-plate duct
(From Bejan, 1995.)
10
100
16
x DRe D
C f x, ReD
( )C f0⫺xReD
Figure 5.3 Local and average skin friction coefficients in the entrance region of a round tube
(From Bejan, 1995.)
Figure 5.2 shows a replotting (Bejan, 1995) of the integral solution (Sparrow, 1955) forC f,x in the entrance region of a parallel-plate duct The dashed-line asymptote
indicates theC f,x estimate based on the Blasius solution forthe laminarboundary
layer between a flat wall and a uniform free stream (U) If numerical factors of order
1 are neglected, the boundary layer asymptote is
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C f,x≈
Ux
ν
−1/2
or
C f,x· ReD ≈
D · Re D
−1/2
The solid-line asymptote,C f,x· ReD = 12, represents the skin friction solution for the fully developed flow region (Section 5.2.2)
The local skin friction coefficient in the entrance region of a round tube is indicated
by the lower curve in Fig 5.3 This is a replotting (Bejan, 1995) of the solution reported by Langhaar (1942) The upper curve is for the averaged skin friction co-efficient,
C f
0−x = 1
x
x 0
or
C f
0−x = 1 ¯τ
where
¯τ = x1
x 0 τw(ξ) dξ The horizontal asymptote serves both curves,
C f,x = 16 = C f
0−x and represents the solution for fully developed skin friction in a round tube as shown subsequently in eq (5.18)
5.2.2 Fully Developed Flow Region
The key feature of the flow in the region downstream ofx ∼ X is that the transverse
velocity component (v = 0 in Fig 5.1) is negligible In view of the equation formass conservation,
∂u
∂x+
∂v
∂y = 0
the vanishing ofv is equivalent to ∂u/∂x = 0, that is, a velocity distribution that does
not change ordoes not develop furtherfrom onex to the next This is why this flow
region is called fully developed It is considered as defined by
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v = 0 or ∂u
This feature is a consequence of the geometric constraint that downstream ofx ∼ X,
the boundary layer thicknessδ cannot continue to grow In this region, the length scale for changes in the transverse direction is the constantD, not the freely growing
δ, and the mass conservation equation requires that U/L ≈ v/D, where L is the flow
dimension in the downstream direction Thev scale is then v ≈ UD/L and this scale
vanishes asL increases, that is, as the flow reaches sufficiently far into the duct.
Another consequence of the full development of the flow is that the pressure is essentially uniform in each constant-x cross section (∂P /∂y = 0) This feature is derived by substitutingv = 0 into the momentum (Navier–Stokes) equation for the
y direction With reference to Fig 5.1, the pressure distribution is P (x), and the
momentum equation forthe flow directionx becomes
dP
dx = µ
d2u
Both sides of this equation must equal the same constant, because at most, the left side is a function ofx and the right side a function of y That constant is the pressure
drop per unit length,
∆P
L = −
dP dx
The pressure drop and the flow distributionu(y) are obtained by solving eq (5.6)
subject tou = 0 at the walls (y ± D/2), where y = 0 represents the center plane of
the parallel plate duct:
u(y) =3
2U
1−
y
D/2
2
(5.7)
with
U = 12µD2
−dP dx
(5.8)
In general, for a duct of arbitrary cross section, eq (5.6) is replaced by
dP
dx = µ ∇2u = constant
where the Laplacian operator∇2accounts only for curvatures in the cross section,
∇2= ∂y ∂22 +∂z ∂22
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that is,∂2/∂x2= 0 The boundary conditions are u = 0 on the perimeter of the cross
section Forexample, the solution forfully developed laminarflow in a round tube
of radiusr0is
u = 2U
1−
r
r0
2
(5.9) with
U = r02
8µ
−dP
dx
(5.10)
This solution was first reported by Hagen (1839) and Poiseuille (1840), which is
why the fully developed laminarflow regime is also called Hagen–Poiseuille flow or
Poiseuille flow.
5.2.3 Hydraulic Diameter and Pressure Drop
Equations (5.8) and (5.10) show that in fully developed laminarflow the mean veloc-ityU (orthe mass flow rate ˙m = ρAU) is proportional to the longitudinal pressure
gradientP /L In general, and especially in turbulent flow, the relationship between
˙m and ∆P is nonlinear Fluid friction results for fully developed flow in ducts are
reported as friction factors:
f = 1τw
whereτwis the shearstress at the wall Equation (5.11) is the same as eq (5.3), with the observation that in fully developed flow,τwandf are x-independent.
The shearstressτwis proportional to∆P /L This proportionality follows from the
longitudinal force balance on a flow control volume of cross sectionA and length L,
wherep is the perimeter of the cross section Equation (5.12) is general and is
independent of the flow regime Combined with eq (5.11), it yields the pressure drop relationship
∆P = f pL
A
1
2ρU2
(5.13) whereA/p represents the transversal length scale of the duct:
r h=A
D h=4Ap hydraulic diameter (5.15)