Table 5.1 shows five duct cross sections that have the same hydraulic diameter.. The hydraulic diameter of the round tube coincides with the tube diameter.. All Hagen–Poiseuille flows are
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TABLE 5.1 Scale Drawing of Five Different Ducts That Have the Same Hydraulic Diameter
Circular
Square
Equilateral triangle
Rectangular(4:1)
Infinite parallel plates
Source: Bejan (1995).
Table 5.1 shows five duct cross sections that have the same hydraulic diameter
The hydraulic diameter of the round tube coincides with the tube diameter The hydraulic diameter of the channel formed between two parallel plates is twice the spacing between the plates For cross sections shaped as regular polygons,D his the diameter of the circle inscribed inside the polygon In the case of highly asymmetric cross sections,D hscales with the smallerof the two dimensions of the cross section
The general pressure drop relationship (5.2) is most often written in terms of hydraulic diameter,
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∆P = f4L
D h
1
2ρU2
(5.16)
To calculate∆P , the friction factor f must be known and it can be derived from the
flow solution The friction factors derived from the Hagen–Poiseuille flows described
by eqs (5.7) and (5.9) are
f =
24
ReD h D h = 2D parallel plates(D = spacing) (5.17) 16
ReD h D h = D round tube(D = diameter) (5.18) Equations (5.17) and (5.18) hold forlaminarflow (ReD h ≤ 2000) Friction factors for
other cross-sectional shapes are reported in Tables 5.2 and 5.3 Additional results can
be found in Shah and London (1978) All Hagen–Poiseuille flows are characterized by
TABLE 5.2 Effect of Cross-Sectional Shape on f and Nu in FullyDeveloped Duct Flow
Nu= hD h /k
Cross Section f/Re Dh B= πD h2/4
A Uniformq UniformT0
Source: Bejan (1995).
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TABLE 5.3 Friction Factors and Nusselt Numbers for Heat Transfer to Laminar Flow through Ducts with Regular Polygonal Cross Sections
Nu= hD h /k
f · Re Dh Uniform Heat Flux Isothermal Wall
Developed Developed Slug Developed Slug
Source: Data from Asako et al (1988).
where the constantC depends on the shape of the duct cross section It was shown in
Bejan (1995) that the duct shape is represented by the dimensionless group
B = πD A h /4
duct
(5.20)
and thatf · Re D h (orC) increases almost proportionally with B This correlation is
illustrated in Fig 5.4 for the duct shapes documented in Table 5.2
Consider the stream shown in Fig 5.5, and assume that the duct cross sectionA is
not specified According to the thermodynamics of open systems, the first law for the control volume of lengthdx is q p = ˙m dh/dx, where h is the bulk enthalpy of the
stream When the fluid is an ideal gas (dh = c p dT m) oran incompressible liquid with negligible pressure changes (dh = c dT m), the first law becomes
dT m
dx =
q p
In heat transfer, the bulk temperature T m is known as mean temperature It is
related to the bundle of ministreams of enthalpy (ρuc p T dA) that make up the bulk
enthalpy stream (h) shown in the upper left corner of Fig 5.5 From this observation
it follows that the definition ofT minvolves au-weighted average of the temperature
distribution over the cross section,
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Figure 5.4 Effect of the cross-sectional shape (the numberB) on fully developed friction
and heat transfer in a straight duct (From Bejan, 1995.)
T m= 1
UA
In internal convection, the heat transfer coefficienth = q /∆T is based on the
difference between the wall temperature (T0) and the mean temperature of the stream:
namely,h = q /(T0− T m).
5.3.2 Thermally Fully Developed Flow
By analogy with the developing velocity profile described in connection with Fig 5.1, there is a thermal entrance region of lengthX T In this region the thermal boundary
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u dA
m.
q⬙
q⬙
r
v u x
x x + dx
T m⫹dT m
T m
Figure 5.5 Nomenclature for energy conservation in a duct segment (From Bejan, 1995.)
layers grow and effect changes in the distribution of temperature over the duct cross section Estimates forX T are given in Section 5.4.1 Downstream fromx ∼ X T the thermal boundary layers have merged and the shape of the temperature profile across the duct no longer varies For a round tube of radiusr0, this definition of a fully developed temperature profile is
T0(x) − T (r, x)
T0(x) − T m (x) = φ
r
r0
(5.23)
The functionφ(r/r0) represents the r-dependent shape (profile) that does not depend
on the downstream positionx.
The alternative to the definition in eq (5.23) is the scale analysis of the same regime (Bejan, 1995), which shows that the heat transfer coefficient must be con-stant (x-independent) and of order k/D The dimensionless version of this second
definition is the statement that the Nusselt numberis a constant of order1:
Nu =hD
k = D
∂T /∂r
r=r0
The second part of the definition refers to a tube of radiusr0 The Nu values compiled
in Tables 5.2 and 5.3 confirm the constancy and order of magnitude associated with thermally fully developed flow The Nu values also exhibit the approximate propor-tionality with theB number that characterizes the shape of the cross section (Fig.
5.4) In Table 5.3, slug flow means that the velocity is distributed uniformly over the
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cross section,u = U Noteworthy are the Nu values for a round tube with uniform
wall heat flux
Nu = 48
11 = 4.364 uniform wall heat flux and a round tube with isothermal wall
Nu = 3.66 isothermal wall
5.4.1 Thermal Entrance Region
The heat transfer results listed in Tables 5.2 and 5.3 apply to laminar flow regions where the velocity and temperature profiles are fully developed They are valid in the downstream sectionx, where x > max(X, X T ) The flow development length X is
given by eq (5.2) The thermal lengthX T is determined from a similar scale analysis
by estimating the distance where the thermal boundary layers merge, as shown in Fig 5.6 When Pr 1, the thermal boundary layers are thicker than the velocity
boundary layers, and consequently,X T X The Prandtl numberPris the ratio of
the molecularmomentum and thermal diffusivities,ν/α When Pr 1, the thermal
boundary layers are thinner, andX Tis considerably greater thanX The scale analysis
of this problem shows that for both Pr 1 and Pr 1, the relationship between X T
andX is (Bejan, 1995)
X T
Pr1
Pr1 0
0
X
X
X T⬃ PrX
Figure 5.6 Prandtl number effect on the flow entrance lengthX and the thermal entrance
lengthX T (From Bejan, 1995.)
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Equations (5.25) and (5.2) yield the scale
X T ≈ 10−2Pr· D h· ReD h (5.26)
which is valid overthe entire Prrange
5.4.2 Thermally Developing Hagen–Poiseuille Flow
When Pr 1, there is a significant length of the duct (X < x < X T) overwhich the velocity profile is fully developed, whereas the temperature profile is not If the
x-independent velocity profile of Hagen–Poiseuille flow is assumed, it is possible
to solve the energy conservation equation and determine, as an infinite series, the temperature field (Graetz, 1883) The Pr= ∞ curve in Fig 5.7 shows the main
features of the Graetz solution for heat transfer in the entrance region of a round tube with an isothermal wall (T0) The Reynolds numberReD = UD/ν is based on
the tube diameterD and the mean velocity U The bulk dimensionless temperature
of the stream (θ∗
m), the local Nusselt number(Nux), and the averaged Nusselt number (Nu0−x) are defined by
100
10
3
Nux
Pr = ⬁ 5 2 0.7
x D/
Re PrD
3.66 0.1
1
*m
Nu0⫺x(Pr = )⬁
*m(Pr = )⬁
Figure 5.7 Heat transfer in the entrance region of a round tube with isothermal wall (From Bejan, 1995; drawn based on data from Shah and London, 1978, and Hornbeck, 1965.)
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θ∗
m= T m − T0
Tin− T0
(5.27)
Nux = q x D k(T0− T m ) (5.28)
Nu0−x = q0−x D
In these expressions,T m (x) is the local mean temperature, Tinthe stream inlet tem-perature,q x the local wall heat flux, andq0−x the heat flux averaged fromx = 0 to
x, and ∆T lmthe logarithmic mean temperature difference,
∆T lm= [T0− Tin(x)] − (T0− T m )
ln [T0− T m (x)/(T0− Tin)] (5.30)
The dimensionless longitudinal position plotted on the abscissa is also known asx∗:
x∗ = x/D
This group, and the fact that the knees of the Nu curves occur atx1/2
∗ ≈ 10−1, support
theX T estimate anticipated by eq (5.26)
The following analytical expressions are recommended by a simplified alternative
to Graetz’s series solution (L´evˆeque, 1928; Drew, 1931) The relationships for the
Pr= ∞ curves shown in Fig 5.7 are (Shah and London, 1978)
Nux =
1.077x∗−1/3 − 0.70 x∗≤ 0.01
3.657 + 6.874(103x∗) −0.488 e −57.2x∗ x∗ > 0.01 (5.32)
Nu0−x =
1.615x∗−1/3 − 0.70 x∗≤ 0.005
1.615x∗−1/3 − 0.20 0.005 < x∗< 0.03
3.657 + 0.0499/x∗ x∗ > 0.03
(5.33)
The thermally developing Hagen–Poiseuille flow in a round tube with uniform heat fluxq can be analyzed by applying Graetz’s method (the Pr= ∞ curves in Fig
5.8) The results for the local and overall Nusselt numbers are represented within 3%
by the equations (see also Shah and London, 1978)
Nux∗=
3.302x∗−1/3 − 1.00 x∗≤ 0.00005
1.302x∗−1/3 − 0.50 0.00005 < x∗ ≤ 0.0015
4.364 + 8.68(103x∗) −0.506 e −41x∗ x∗ > 0.001
(5.34)
Nu0−x =
1.953x∗−1/3 x∗ ≤ 0.03
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100
10
3
Nux
Pr = ⬁ 5 2 0.7
x D/
Re PrD
4.36
Nu0⫺x(Pr = )⬁
Figure 5.8 Heat transfer in the entrance region of a round tube with uniform heat flux (From Bejan, 1995; drawn based on data from Shah and London, 1978, and Hornbeck, 1965.)
where
Nux∗ = q k [T D
0(x) − T m (x)]
Nu0−x = q D
k ∆Tavg
with
∆Tavg= 1
x
x 0
dx
T0(x) − T m (x)
−1
(5.36)
Analogous results are available for the heat transfer to thermally developing Hagen–Poiseuille flow in ducts with othercross-sectional shapes The Nusselt num-bers for a parallel-plate channel are shown in Fig 5.9 The curves for a channel with isothermal surfaces are approximated by (Shah and London, 1978)
Nu0−x =
1.233x∗−1/3 + 0.40 x∗ ≤ 0.001
7.541 + 6.874(103x∗) −0.488 e −245x∗ x∗> 0.001 (5.37)
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Nu0−x =
1.849x∗−1/3 x∗ ≤ 0.0005
1.849x∗−1/3 + 0.60 0.0005 < x∗≤ 0.006
7.541 + 0.0235/x∗ x∗> 0.006
(5.38)
If the plate-to-plate spacing isD, the Nusselt numbers are defined as
Nux = q (x)D h
k [T0− T m (x)]
Nu0−x =q0−x D h
k ∆T lm
whereD h = 2D and ∆T lmis given by eq (5.30)
The thermal entrance region of the parallel-plate channel with uniform heat flux and Hagen–Poiseuille flow is characterized by (Shah and London, 1978)
Nux =
1.490x∗−1/3 − 0.40 0.0002 < x∗≤ 0.001
8.235 + 8.68(103x∗)0.506 e −164x∗ x∗ > 0.001
(5.39)
100
10
3
Nux
Nux
Nu0⫺x
Nu0⫺x
x D/
Re PrD h h
8.23 7.54 Uniform
wall temperature
Uniform wall heat flux
Figure 5.9 Heat transfer in the thermal entrance region of a parallel-plate channel with Hagen–Poiseuille flow (From Bejan, 1995; drawn based on data from Shah and London, 1978.)
...Nu0⫺x(Pr = )⬁
Figure 5.8 Heat transfer in the entrance region of a round tube with uniform heat flux (From Bejan, 1995; drawn based on data from Shah and London,... 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
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8.23 7.54 Uniform
wall temperature
Uniform wall heat flux
Figure 5.9 Heat transfer in the thermal entrance region of a parallel-plate channel with Hagen–Poiseuille