If the interest is on an average heat transfer coefficient, many empirical equations are available.. 6.5 HEAT TRANSFER FROM ARRAYS OF OBJECTS 6.5.1 Crossflow across Tube Banks This configur
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T+− δT+
+
0
dy+
where the second term on the left is the temperature difference across which heat is transferred by conduction With
H
ν =
1
PrT
ν =
κ
PrT
y++ δy+
0
(6.150a) then
T+= δT+
0 +PrT
κ ln
32.6y+
A local heat transfer coefficient can be defined based onδT0:
δT+
0 = ρc p v∗
1
Based on experimental data for roughness from spheres,
Stk = C · Re −0.20
withC ≈ 0.8 and PrT ≈ 0.9, the law of the wall can be rewritten as
T+= 1
Stk +PrT
κ ln
32.6y+
and using a procedure similar to that used for a smooth surface,
St= Cf /2
PrT + (C f /2)1/2 /Stk (6.154)
6.4.16 Some Empirical Transport Correlations
Cylinder in Crossflow The analytical solutions described in Sections 6.4.9 and 6.4.10 provide local convection coefficients from the front stagnation point (using similarity theory) to the separation point of a cylinder using the Smith–Spalding method At the front stagnation point, the free stream is brought to rest, with an accompanying rise in pressure The initial development of the boundary layer along the cylinder following this point is under favorable pressure gradient conditions;
that is,dp/dx < 0 However, the pressure reaches a minimum at some value of
x, depending on the Reynolds number Farther downstream, the pressure gradient
is adverse (i.e., dp/dx > 0), until the point of boundary layer separation where
the surface shear stress becomes zero This results in the formation of a wake For
ReD≤ 2 × 105, the boundary layer remains laminar until the separation point, which
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occurs at an arc of about 81° For higher values of ReD, the boundary layer undergoes transition to turbulence, which results in higher fluid momentum and the pushing back
of the separation point to about 140°
The foregoing changes result in large variations in the transport behavior with the angleθ If the interest is on an average heat transfer coefficient, many empirical
equations are available One such correlation for an isothermal cylinder that is based
on the data of many investigators for various ReD and Pr, such that ReD · Pr > 0.2
was proposed by Churchill and Bernstein (1977):
NuD≡ ¯hD
k = 0.30 +
0.62Re1/2
D · Pr1/3
[1+ (0.40/Pr)2/3]1/4
1+
ReD 282,000
5/84/5
(6.155)
The fluid properties are evaluated at the film temperature, which is the average of the surface and ambient fluid values
Flow Over an Isothermal Sphere Whitaker (1972) recommends the correlation
NuD = 2 +0.4Re1/2
D + 0.06Re2/3
D
Pr0.4
µ
µs
1/4
(6.156)
The uncertainty band around this correlation is±30% in the range: 0.71 < Pr380,
3.5 < ReD < 7.6 × 104, and 1.0 < µ/µ s < 3.2 All properties except µ s are evaluated at the ambient temperature
6.5 HEAT TRANSFER FROM ARRAYS OF OBJECTS 6.5.1 Crossflow across Tube Banks
This configuration is encountered in many practical heat transfer applications, includ-ing shell-and-tube heat exchangers Zhukauskas (1972, 1987) has provided compre-hensive heat transfer and pressure drop correlations for tube banks in aligned and staggered configurations (Fig 6.15) The heat transfer from a tube depends on its lo-cation within the bank In the range of lower Reynolds numbers, typically the tubes in the first row show similarheat transferto those in the innerrows ForhigherReynolds numbers, flow turbulence leads to higher heat transfer from inner tubes than from the first row The heat transfer becomes invariant with tube location following the third or fourth row in the mixed-flow regime, occurring above ReD,max The average Nusselt number from a tube is then of the form
NuD = C · Re m
D,max· Pr0.36
Pr
Prs
1/4
(6.157) for0.7 ≤ Pr < 5000, 1 < ReD,max < 2 × 104, andNL ≥ 20 and where m is as given
in Table 6.2 The Reynolds numberReD,maxin eq (6.157) is based on the maximum
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S D
A2
A2
( ) Aligned tube banka ( ) Staggered tube bankb
Figure 6.15 Tube bundle configurations studied by Zhukaukas (1987)
fluid velocity within the tube bank which occurs at the mimimum-flow cross section
For the aligned arrangement, it occurs atA1in Fig 6.15 and is given by
Vmax= ST
whereV is the upstream fluid velocity.
For the staggered configuration, the maximum fluid velocity occurs atA2if 2(SD−
D) < (ST − D), in which case
Vmax= S T
2(S D − D) V (6.159)
Otherwise, the aligned tube expression of eq (6.158) can be used
TABLE 6.2 Parameters ReD,max , C, an d m for Various Aligned and Staggered Tube
Arrangements
Aligned (S T /S L > 0.7) 1000–2× 105 0.27 0.63 Staggered (S T /S L < 2) 1000–2× 105 0.35(S T /S L )0.20 0.60
Staggered (S T /S L > 2) 1000–2× 105 0.40 0.60
Staggered (Pr> 0.70) 2× 105–2× 106 0.031(S T /S L )0.20 0.80
Staggered (Pr= 0.70) 2× 105–2× 106 0.027(S T /S L )0.20 0.80
Source: Zhukauskas (1987).
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TABLE 6.3 ParameterC2for Various Tube Rows and Configurations
Aligned 0.70 0.80 0.86 0.90 0.92 0.95 0.97 0.98 0.99
(Re D,max > 1000)
Staggered 0.83 0.87 0.91 0.94 0.95 0.97 0.98 0.98 0.99
(Re D,max100–1000) Staggered 0.64 0.76 0.84 0.89 0.92 0.95 0.97 0.98 0.99
(Re D,max > 1000) Source: Zhukauskas (1987).
It is noted that all properties except Prs are evaluated at the arithmetic mean of the fluid inlet and outlet temperatures The constantsC and m are listed in Table 6.2,
where it can be observed that forS T /S L < 0.7, heat transferis poorand the aligned
tube configuration should not be employed
IfNL < 20, a corrected expression can be employed:
NuL
NL<20 = C2· NuL
NL≥20
whereC2 = 1.0 foraligned tubes in the range Re D,max and is provided for other conditions in Table 6.3
6.5.2 Flat Plates
Stack of Parallel Plates Consider a stack of parallel plates placed in a free stream (Fig 6.16) The envelope for the plate stack has a fixed volume,W (width)
×L (length) ×H (height) Morega et al (1995) conducted numerical analyses on a
two-dimensional model whereW L, t (the plate thickness), equal to L/20 and q
(the heat flux) uniform over the plate surfaces, excluding the edges The question addressed by Morega et al is the number of plates needed to maximize the heat transfer performance from the entire plate stack Fewer plates than the optimum
Figure 6.16 Plate stack (From Morega et al., 1995.)
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reduce the amount of heat transfer area in a given volume A greater population
of plates than the optimum increases the resistance to flow through the stack and hence diverts the flow to the free stream zone outside the stack The heat transfer performance is represented by the nondimensional hot-spot temperature,
θhot=k(Tmax− T0 )
whereTmaxis the maximum surface temperature on the plate surface,T0is the free stream temperature, andk is the fluid thermal conductivity.
Figure 6.17 showsθhotversus the number of plates in the stack,n The parameter
is the Reynolds numberReL = U0 L/ν, where U0 is the free stream velocity and
ν is the kinematic viscosity of the fluid Figure 6.17 also shows the curves derived
from Nakayama et al (1988), where the experimental data were obtained with the finned heat sinks having nearly isothermal surfaces Despite the difference in thermal boundary conditions, the two groups of curves imply a smooth transition of optimum
n from low to high Reynolds number regimes Morega et al (1995) proposed a
relationship for optimumn:
nopt 0.26(H/L)Pr1/4· Re
1/2 L
1+ 0.26(t/L)Pr1/4· Re1/2
L
(6.161)
where Pr is the Prandtl number Equation (6.161) is applicable to cases where Pr≥
0.7 and n 1.
Figure 6.17 Nondimensional hot spot temperature (θhot) versus the number of plates (n).
(From Morega et al., 1995.)
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Offset Strips Offset arrangements of plates or strips (Fig 6.1d) reproduced with
the addition of the symbols in Fig 6.18 offerthe advantage of heat transferenhance-ment The boundary layer development is interrupted at the end of the strip and then resumes from the leading edge of the strip in the downstream row This arrangement allows the boundary layer to be held thin everywhere in the strip array A similar effect can be obtained with the in-line arrangement of strips, but offsetting the strips from row to row introduces additional favorable effects on heat transfer The fluid in the offset strips has a longer distance to travel after leaving the trailing edge of a strip
to the leading edge of the next strip than in the in-line counterpart This elongated distance results in a longerelapsed time forthe diffusion (ordispersion) of momen-tum and heat When the fluid velocity is high enough, vortex shedding or turbulence attains a high level in the intervening space between the strips Thus, the strips after the third row come to be exposed to highly dynamic flow; thermal wakes shed from the upstream strips tend to be diluted by increased level of turbulence so that the indi-vidual strip is washed by a cooler stream than in the in-line strip arrangement Because
of these advantages, the offset strip array has been studied by many researchers and used widely in compact heat exchangers
Figure 6.18 includes a table showing the relative strip thicknesses (t/) and the
relative strip spacings (s/) studied by DeJong et al (1998) Those rectangles painted
black in the sketch of the strip array are the strips covered with naphthalene to measure the mass transfer rate The mass transfer data were converted to the heat transfer coefficient using the analogy between mass and heat transfer The experimental data reveal row-by-row variations of heat transfer coefficient that depend on the Reynolds number Figure 6.19 shows the friction factorf and Colburn j-factor j plotted as a
function of the Reynolds numberRe They are defined as
1 2 3 4 5 6 7 8 Row
Flow
l
s t
geometrical parameter experimentalgeometry numericalgeometry
Geometrical parameters for experimental and numerical arrays
Figure 6.18 Offset strips Black strips were naphthalene coated to measure the mass transfer coefficient The table shows the dimensions (From DeJong et al., 1998.)
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1
0.1
0.01
0.001
numerical simulation Joshi and Webb (1987) correlation
Re
Figure 6.19 Colburnj-factors and friction factors versus Reynolds number The curves were
based on the Joshi and Webb correlations [eqs (6.166)–(6.169)] (From DeJong et al., 1998.)
Re=U cνd h (6.162) whereU c is the flow velocity at the minimum free-flow area, d h is the hydraulic diameter,
d h= 2(s − t)
l + t
ν is the fluid kinematic viscosity, and
f = 2∆pcore ρU2
c
d h
4Lcore
(6.163)
where∆pcoreis the pressure drop across the entire eight-row test section,Lcoreis the total length of the strip array,ρ is the fluid density, and
j = Nu
RePr
where Nu is the Nusselt number, andm = 0.40 [DeJong et al (1998)] or0.30 [Joshi
and Webb (1987)] The originalj-factorin DeJong et al (1998) is defined using the
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Sherwood and Schmidt numbers For the sake of consistency throughout this section, they are replaced by heat transfer parameters in
Nu=dhhav
wherehavis the average heat transfer coefficient andk is the fluid thermal
conduc-tivity
The curves in Fig 6.19 show the correlations proposed by Joshi and Webb (1987)
With the transition Reynolds number denoted as Re*, in the laminar flow range where
Re≤ Re∗,
f = 8.12Re −0.74
d h
−0.41
j = 0.53Re −0.50
d h
−0.15
whereα is the aspect ratio; α = s/W, with W taken as the strip width measured
normal to the page in Fig 6.18 In the turbulent flow range where Re≥ Re∗+ 1000,
f = 1.12Re −0.36
d h
−0.65 t
d h
0.17
(6.168)
j = 0.21Re −0.40
dh
−0.24
t dh
0.02
(6.169) The transition Reynolds number Re∗is given by
Re∗= Re∗
b dh
with
Re∗b= 257
s
1.23t
0.58
(6.171)
b = t + 1.328
and
Re=Uνc (6.173) The solid symbols in Fig 6.19 are the results of numerical simulation on the same strip Figure 6.19 shows the state-of-the-art accuracy in the predictions off and j.
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6.6 HEAT TRANSFER FROM OBJECTS ON A SUBSTRATE
Figure 6.20 shows a classification of the situations and models encountered in
elec-tronics cooling applications: (a) a heated strip that is flush bonded to the substrate surface, (b) a rectangular heat source which is also flush to the substrate surface, (c)
an isolated two-dimensional block, (d) a two-dimensional block array, (e) a rectan-gularblock and ( f ) an array of rectangular blocks Ortega et al (1994) suggest that
in many practical situations the substrate is a heat conductor, so that a heat path from the heat source through the substrate to the fluid flow cannot be ignored in what is
called conductive/convective conjugate heat transfer The analytical solution of the
Figure 6.20 Configurations of heat sources on substrate (From Ortega et al., 1994.)
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conjugate heat transfer problem is difficult, and numerical solutions or experiments have been used in the studies reported in the literature
6.6.1 Flush-Mounted Heat Sources
Heat transfer considerations in flush-mounted heat sources involve less complications than in those forfluid flow The flow is described by the solution forboundary layer flow or duct flow, particularly where the flow is laminar and conjugate heat transfer
is important in laminar flow cases In turbulent flow, the heat transfer coefficient on the heat source surface is high enough to reduce the heat flow to the substrate to an insignificant level Gorski and Plumb (1990, 1992) performed numerical analyses on
the two-dimensional (Fig 6.20a) and rectangular (Fig 6.20b) patch problems The
cross section of the heat source and the substrate is shown in Fig 6.21 The fluid flow is described by the analytical solution of Blasius forthe laminarboundary layer, while the substrate is assumed to be infinitely thick and the numerical solutions are correlated for the two-dimensional strip by
Nu= 0.486Pe0.53
s xs
0.71k
sub
kf
0.057
(6.174)
where Nu= ¯h s /kf , ¯h is the average heat transfer coefficient based on the heat source
area (per unit width normal to the page in Fig 6.21),sis the heat source length,xsis the distance between the leading edge of the substrate and that of the heat source,ksub
is the substrate thermal conductivity, andkfis the fluid thermal conductivity Pe is the P´eclet number, Pe= U0 x s /α, where U0is the free stream velocity andα is the fluid
thermal diffusivity Equation (6.174) correlates the numerical solutions within 5% in the parameter ranges 103≤ Pe ≤ 105, 0.10 ≤ ksub/k f ≤ 10, and 5 ≤ x s / s≤ 100
Forthe rectangularpatch:
Nu=
0.60Pe0.48 c
2s
2xs + s
0.63
P s
2A
0.18
ksub
kf = 1
(6.175)
0.43Pe0.52 c
2 s
2xs + s
0.70P
s
2A
0.07 k
sub
kf = 10
(6.176)
Figure 6.21 Two-dimensional flush-mounted heat source