Nikuradse 1933 measured the effect of surface roughness on the friction factor by coating the inside surface of pipes with sand of a measured grain size glued as tightly as possible to t
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0.05 0.04 0.04
0.000,01 0.000,05 0.0001 0.0002 0.0004 0.0006 0.001 0.002 0.004 0.008 0.01 0.015 0.02 0.006
0.0008
k s
0.01
0.02 0.03 0.04 0.05 0.1
0.000,005
0.000,001
ReD = UD v/
103 104 105 106 107 108
Laminar flow,
f = 16Re D
Smooth pipes (the Karman-Nikuradse relation)
Surface condition k S(mm) Riveted steel
Concrete Wood stave Cast iron Galvanized iron Asphalted cast iron Commercial steel or Wrought iron Drawn tubing
0.9–9 0.3–3 0.18–0.9 0.26 0.15 0.12 0.05 0.0015
Figure 5.13 also documents the effect of wall roughness It is found experimentally that the performance of commercial surfaces that do not feel rough to the touch departs from the performance of well-polished surfaces This effect is due to the very small thickness acquired by the laminarsublayerin many applications [e.g., because
UyVSL/ν is of order 102 (Bejan, 1995), whereyVSL is the thickness of the viscous sublayer] In water flow through a pipe, withU ≈ 10m/s and ν ≈ 0.01cm2/s, yVSL
is approximately 0.01 mm Consequently, even slight imperfections of the surface may interfere with the natural formation of the laminar shear flow contact spots If the surface irregularities are taller thanyVSL, they alone rule the friction process
Nikuradse (1933) measured the effect of surface roughness on the friction factor
by coating the inside surface of pipes with sand of a measured grain size glued as tightly as possible to the wall Ifk s is the grain size in Nikuradse’s sand roughness, the friction factor fully rough limit is the constant
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f
1.74 ln D
k s + 2.28
−2
(5.70)
The fully rough limit is that regime where the roughness size exceeds the order of magnitude of what would have been the laminarsublayerin time-averaged turbulent flow overa smooth surface,
k+
s = k s (τ0/ρ)1/2
The roughness effect described by Nikuradse is illustrated by the upper curves in Fig 5.13
There are several empirical relationships for calculating the time-averaged coefficient forheat transferbetween the duct surface and the fully developed flow,h = q /(T0−
T m ) The analytical form of these relationships is based on exploiting the analogy
between momentum and heat transfer by eddy rotation One of the earliest examples
is due to Prandtl in 1910 (Prandtl, 1969; Schlichting, 1960),
Prt + ( ¯u1 /U)(Pr − Pr t ) (5.72)
where St, Pr, and Prt are the Stanton number, Prandtl number, and turbulent Prandtl number,
ρc p U Pr=
ν
α Prt=
M
Equation (5.72) holds forPr≥ 0.5, and if Pr t is assumed to be 1, the factor ¯u1 /U is
provided by the empirical correlation (Hoffmann, 1937)
¯u1
U 1.5Re −1/8 D h · Pr−1/6 (5.74)
Better agreement with measurements is registered by Colburn’s (1933) empirical correlation,
St· Pr2/3f
Equation (5.75) is analytically the same as the one derived purely theoretically for boundary layerflow (Bejan, 1995) Equation (5.75) holds forPr≥ 0.5 and is to be
used in conjunction with Fig 5.13, which supplies the value of the friction factorf It
applies to ducts of various cross-sectional shapes, with wall surfaces having various degrees of roughness In such casesD is replaced by D h In the special case of a pipe
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with smooth internal surface, eqs (5.75) and (5.68) can be combined to derive the Nusselt numberrelationship
NuD= hD k = 0.023Re4/5
which holds in the range 2× 104 ≤ ReD ≤ 106 A popularversion of this is a correlation due to Dittus and Boelter (1930),
NuD = 0.023Re4/5
which was developed for0.7 ≤ Pr ≤ 120, 2500 ≤ Re D ≤ 1.24 × 105, and
L/D > 60 The Prandtl number exponent is n = 0.4 when the fluid is being heated (T0> T m ), and n = 0.3 when the fluid is being cooled (T0< T m) All of the physical
properties needed for the calculation of NuD , Re D, and Prare to be evaluated at the
bulk temperatureT m The maximum deviation between experimental data and values
predicted using eq (5.77) is on the order of 40%
For applications in which influence of temperature on properties is significant, the Siederand Tate (1936) modification of eq (5.76) is recommended:
NuD = 0.027Re4/5
D · Pr1/3
µ µ0
0.14
(5.78)
This correlation is valid for 0.7 ≤ Pr ≤ 16,700 and Re D > 104 The effect of temperature-dependent properties is taken into account by evaluating all the prop-erties (except µ0) at the mean temperature of the stream, T m The viscosityµ0 is evaluated at the wall temperatureµ0 = µ(T0 ) Equations (5.76)–(5.78) can be used
for ducts with constant temperature and constant heat flux
More accurate correlations of this type were developed by Petukhov and Kirilov (1958) and Petukhov and Popov (1963); respectively:
1.07 + 900/Re D − 0.63/(1 + 10Pr) + 12.7(f/2)1/2 (Pr2/3 − 1) (5.79a)
and
1.07 + 12.7(f/2)1/2
forwhichf is supplied by Fig 5.13 Additional information is provided by Petukhov
(1970) Equation (5.79a) is accurate within 5% in the range 4000≤ ReD ≤ 5 × 106
and 0.5 ≤ Pr ≤ 106 Equation (5.79b) is an abbreviated version of eq (5.79a) and was modified by Gnielinski (1976):
NuD = (f/2)(Re D− 103)Pr
1+ 12.7(f/2)1/2
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which is accurate within±10% in the range 0.5 ≤ Pr ≤ 106 and 2300 ≤ ReD ≤
5× 106 The Gnielinski correlation of eq (5.80) can be used in both constant-q and constant-T0applications Two simpler alternatives to eq (5.80) are (Gnielinski, 1976)
NuD = 0.0214Re0D .8− 100Pr0.4 (5.81a) valid in the range
0.5 ≤ Pr ≤ 1.5 104≤ ReD ≤ 5 × 106
and
NuD = 0.012Re0D .87− 280Pr0.4 (5.81b) valid in the range
1.5 ≤ Pr ≤ 500 3× 103≤ ReD ≤ 106
The preceding results refer to gases and liquids, that is, to the range Pr≥ 0.5 For
liquid metals, the most accurate correlations are those of Notter and Sleicher (1972):
NuD =
6.3 + 0.0167Re0.85
D · Pr0.93 q0 = constant (5.82)
4.8 + 0.0156Re0.85
D · Pr0.93 T0= constant (5.83) These are valid for 0.004 ≤ Pr ≤ 0.1 and 104 ≤ ReD ≤ 106 and are based
on both computational and experimental data All the properties used in eqs (5.82) and (5.83) are evaluated at the mean temperatureT m The mean temperature varies with the position along the duct This variation is linear in the case of constantq
and exponential when the duct wall is isothermal (see Section 5.7) To simplify the recommended evaluation of the physical properties at theT m temperature, it is
convenient to choose as representative mean temperature the average value
T m= 1
In this definition,TinandToutare the bulk temperatures of the stream at the duct inlet and outlet, respectively (Fig 5.14)
5.7 TOTAL HEAT TRANSFER RATE
The summarizing conclusion is that in both laminar and turbulent fully developed duct flow the heat transfer coefficienth is independent of longitudinal position This
feature makes it easy to express analytically the total heat transfer rateq (watts)
between a stream and duct of lengthL,
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⌬Tin
⌬Tin
Stream, ( )T x m Stream, ( )T x
m
Wall ( )
T xWall0( )
T x0
Wall
Tout
Tout
T0
with uniform heat flux
In this expressionA w is the total duct surface,A w = pL The effective
tempera-ture difference between the wall and the stream is the log-mean temperatempera-ture differ-enceT lm.
When the wall is isothermal (Fig 5.14) the log-mean temperature difference is
∆T lm= ∆Tin − ∆Tout
Equations (5.85) and (5.86) express the relationship among the total heat transfer rate
q, the total duct surface conductance hA w, and the outlet temperature of the stream.
Alternatively, the same equations can be combined to express the total heat transfer rate in terms of the inlet temperatures, mass flow rate, and duct surface conductance,
In cases where the heat transfer coefficient varies longitudinally,h(x), the h factoron
the right side of eq (5.87) represents theL-averaged heat transfer coefficient: namely,
¯h = L1 L
0
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5.7.2 Wall Heated Uniformly
In the analysis of heat exchangers (Bejan, 1993), it is found that the applicability of eqs (5.85) and (5.86) is considerably more general than what is suggested by Fig
5.14 Forexample, when the heat transferrateq is distributed uniformly along the
duct, the temperature difference∆T does not vary with the longitudinal position.
This case is illustrated in Fig 5.14, where it was again assumed thatA, p, h, and c p
are independent ofx Geometrically, it is evident that the effective value ∆T lmis the same as the constant∆T recorded all along the duct,
Equation (5.89) is a special case of eq (5.86): namely, the limit∆Tin /∆Tout → 1
The optimization of packing of channels into a fixed volume, which in Section 5.5 was outlined forlaminarduct flow, can also be pursued when the flow is turbulent (Bejan and Morega, 1994) With reference to the notation defined in Fig 5.10, where the dimension perpendicular to the figure isW, the analysis consists of intersecting
the two asymptotes of the design: a few wide spaces with turbulent boundary layers and many narrow spaces with fully developed turbulent flow The plate thickness (t)
is not negligible with respect to the spacingD The optimal spacing and maximal
global conductance of theHWL package are
Dopt/L (1 + t/Dopt)4/11 = 0.071Pr −5/11· Be−1/11 (5.90)
and
qmax
HLW ≤ 0.57
k
L2(Tmax− T0 )Pr4/99
Dopt
−67/99
· Be47/99 (5.91)
where Be= ∆PL2/µα These results are valid in the range 104 ≤ ReD h ≤ 106and
106 ≤ ReL ≤ 108, which can be shown to correspond to the pressure drop number range 1011≤ Be ≤ 1016
5.9SUMMARY OF FORCED CONVECTION RELATIONSHIPS
• Laminar flow entrance length:
X/D
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• Skin friction coefficient definition:
C f,x= 1τw
• Laminar fully developed (Hagen–Poiseuille) flow between parallel plates with
u(y) =3
2U
1−
y
D/2
2
(5.7)
with
12µ
−dP
dx
(5.8)
• Laminar fully developed (Hagen–Poiseuille) flow in a tube with diameter D:
u = 2U
1−
r
r0
2
(5.9)
with
U = r02
8µ
−dP dx
(5.10)
• Hydraulic radius and diameter:
r h= A
D h= 4A
• Friction factor:
f =
τw 1
24
16
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• Pressure drop:
∆P = f4L
D h
1
2ρU2
(5.16)
• Nusselt number (see Tables 5.1 through 5.3):
∂T /∂r
r=r0
• Laminar thermal entrance length:
• Thermally developing Hagen–Poiseuille flow (Pr = ∞):
• Round tube, isothermal wall:
Nux =
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)
• Round tube, uniform heat flux:
Nux =
4.364 + 8.68(103x∗) −0.506 e −41x∗ x∗> 0.001
(5.34)
Nu0−x =
• Parallel plates, isothermal surfaces:
Nu0−x =
1.233x −1/3
7.541 + 6.874(103x∗) −0.488 e −245x∗ x∗> 0.001 (5.37)
Nu0−x =
1.849x∗−1/3 + 0.60 0.0005 < x∗ ≤ 0.006
7.541 + 0.0235/x∗ x∗> 0.006
(5.38)
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• Parallel plates, uniform heat flux:
Nux =
8.235 + 8.68(103x∗) −0.506 e −164x∗ x∗ > 0.001
(5.39)
Nu0−x =
2.236x −1/3∗ + 0.90 0.001 < x∗≤ 0.01
8.235 + 0.0364/x∗ x∗≥ 0.01
(5.40)
• Thermally and hydraulically developing flow:
• Round tube, isothermal wall:
Nux = 7.55 + 0.024x∗−1.14
0.0179Pr0.17 x −0.64
∗ − 0.14
1+ 0.0358Pr0.17 x −0.64
Nu0−x = 7.55 + 0.024x∗−1.14
1+ 0.0358Pr0.17 x −0.64
∗
(5.42)
∆P
1
2ρU2 = 13.74(x+)1/2+1.25 + 64x+− 13.74(x+)1/2
1+ 0.00021(x+)−2 (5.43)
x+=x/D
• Round tube, uniform heat flux:
Nux
4.3641+ (Gz/29.6)21/6
=
1+
Gz/19.04
1+ (Pr/0.0207)2/31/2
1+ (Gz/29.6)21/3
3/2
1/3
(5.46)
• Optimal channel sizes:
• Laminarflow, parallel plates:
Dopt
L 2.7Be −1/4 Be=
∆PL2
qmax
HLW 0.60
k
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• Staggered plates:
Dopt
L 5.4Pr −1/4
ReL L b
−1/2
(5.50) forthe range
Pr= 0.72 102≤ ReL≤ 104 0.5 ≤ Nb
L ≤ 1.3
• Bundle of cylinders in cross flow:
Sopt
D 1.59
(H/D)0.52
˜
P0.13· Pr0.24 P =˜ ∆PD2
forthe range
D ≤ 200
Sopt
D 1.70
(H/D)0.52
T D − T∞
4.5
with
ReD = U∞D
ν 140≤ ReD ≤ 14,000
• Array of pin fins with impinging flow:
Sopt
L 0.81Pr −0.25· Re−0.32 L (5.54)
forthe range
0.06 ≤ D L ≤ 0.14 0.28 ≤ H L ≤ 0.56 0.72 ≤ Pr ≤ 7
10≤ ReD ≤ 700 90≤ ReL≤ 6000
• Turbulent duct flow:
Dopt/L
1+ t/Dopt4/11 = 0.071Pr −5/11· Be−1/11 (5.90)
... (Fig 5.14)5.7 TOTAL HEAT TRANSFER RATE
The summarizing conclusion is that in both laminar and turbulent fully developed duct flow the heat transfer coefficienth is independent... combined to express the total heat transfer rate in terms of the inlet temperatures, mass flow rate, and duct surface conductance,
In cases where the heat transfer coefficient varies longitudinally,h(x),... − ∆Tout
Equations (5.85) and (5.86) express the relationship among the total heat transfer rate
q, the total duct surface conductance hA w, and