The Reynolds numberis defined as Re = UD h /ν, where U is the average channel... NOMENCLATURE Roman Letter Symbols A constant or correlation constant, dimensionless Prandtl number–depende
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[512], (74)
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b = t + 1.328
Re= U c
The parameterα is the aspect ratio α = s/W, where W is the strip width.
• Flush-mounted heat sources (Section 6.6.1):
Nu= 0.486Pe0.53
se
s x
0.71
k s
k f
0.057
(6.174) Forthe rectangularpatch,
Nu=
0.60Pe0.48
c
2 s
2x s + s
0.63P
s
2A
0.18 k
sub
k f = 1
(6.175)
0.43Pe0.52 c
2 s
2x s + s
0.70P
s
2A
0.07 k
sub
k f = 10
(6.176) Here Nu is as defined forthe two-dimensional strip,
Pec=U0(s x + se )
α
A/P is the source surface area/perimeter ratio The foregoing correlations are
valid for
103≤ Pec≤ 105 5≤ x s + s /2
s ≤ 150 0.2 ≤ w s
s ≤ 5
In the foregoing,w sis the heat source height,P its length, andP wits width The channel width isW = 12 mm and the height H can vary from 7 to 30 mm The
heat source dimensions covered in the experiments areP = 12 mm, P h= 4, 8,
and 12 mm,H − P h = 3, 8, and 12 mm, and P s= 12 mm
• Isolated blocks (Section 6.6.3):
Nu= 0.150Re0.612(A∗) −0.455
H
P
0.727
(6.178)
where Nu= ¯hP /k, ¯h is the average heat transfer coefficient, and A sis the heat transfer area,
A s = 2P h P w + P P w + 2P h P
¯T s is the average surface temperature and T∞ is the stream temperature The
Reynolds numberis defined as Re = UD h /ν, where U is the average channel
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[513],(75)
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[513],(75)
velocity upstream of the heat source,D H is the channel hydraulic diameter at
a section unobstructed by the heat source, and ν is the fluid (air) kinematic
viscosity The fraction of the channel open to flow is
A∗= 1 − P w /W
P h /H
Equation (6.178) is valid for
1500≤ Re ≤ 104 0.33 ≤ P P h
≤ 1.00
0.12 ≤ P W w ≤ 1.00 0.583 ≤ P H
≤ 2.50
A realistic error bound is 5%
• Block array (Section 6.6.4):
NuP = 0.348Re0.6
where the characteristic length for both Nu and Re is the streamwise length of the block,P
• Pin fin heat sinks (Section 6.6.6): Two correlations are given:
Nu= 7.12 × 10−4C0.574
∆p a
L
0.223p
d
1.72
(6.185) where in
C ∆p=ρL3∆p
µ2
µ is the dynamic viscosity of the air Equation (6.185) was derived from the data
for5× 106< C ∆p < 1.5 × 108:
Nu= 3.2 × 10−6C0.520
PW a
L
−0.205 p
d
0.89
(6.186) where
C PW = ρ2LP w
µ3
covers a range of 1011to 1013
• Single round submerged jet impinging on an isothermal target surface (Section
6.7.2):
Nu
Pr0.42 = G
D
r ,
H D
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[514], (76)
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[514], (76)
where
f1(Re) = 2Re1/2 (1 + 0.005Re0.55 )1/2 (6.191a)
G = D r
1− 1.1(D/r)
1+ 0.1[(H/D) − 6](D/r) (6.191b)
The range of applicability of the foregoing is
2000≤ Re ≤ 4 × 105 2≤ H
D ≤ 12 2.5 ≤
r
D ≤ 7.5 0.004 ≤ A r ≤ 0.04
• Single submerged slot jet impinging on an isothermal target surface (Section
6.7.2):
Nu
Pr0.42 = 3.06Re m
(x/W) + (H/W) + 2.78 (6.193)
where
m = 0.695 − x
2W
+
H
2W
1.33
+ 3.06
−1
(6.194)
The range of applicability is
3000≤ Re ≤ 9 × 104 4≤ W H ≤ 20 4≤ W x ≤ 50
• Array of round submerged jets impinging on an isothermal target surface
(Sec-tion 6.7.2):
Nu
Pr0.42 = K
A r , H D
, G
A r , H D
where
f2(Re) = 0.5Re2/3 (6.196a)
K =
1 +
H/D
0.6/A1/2r
6
−0.05
(6.196b)
andG is given by eq (1.191b) The range of validity of the foregoing is
2000≤ Re ≤ 105 2≤ H D ≤ 12 2.5 ≤ D r ≤ 7.5 0.004 ≤ A r ≤ 0.04
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[515],(77)
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• Array of submerged slot jets impinging on an isothermal target surface (Section
6.7.2):
Nu
Pr0.42 =2
3A3/4r,0
2Re
A r /A r,0 + A r,0 /A r
2/3
(6.197) where
A r,0 =
60+ 4
h
2W − 2
2−1/2
(6.198)
with a range of validity of
1500≤ Re ≤ 4 × 104 2≤ W H ≤ 80 0.008 ≤ A r ≤ 2.5A r,0
• Single round free surface jet impinging on a square isothermal target surface
(Section 6.7.2):
Nu
Pr0.4 = C1· Rem
Di L h
D i A r + C2· Ren
L∗L h
L∗(1 − A r ) (6.202) where
A r = πD2i
4L2
h
L∗= 0.5(
√
2L h − D i ) + 0.5(L h − D i )
These data have been found to be best correlated in the range 1000≤ ReDn ≤
51,000 forC1 = 0.516, C2 = 0.491, and n = 0.532, where the fluid properties
are evaluated at the mean of the surface and ambient fluid temperature
NOMENCLATURE
Roman Letter Symbols
A constant or correlation constant, dimensionless
Prandtl number–dependent constant, dimensionless source surface area, m2
A T total heat sink surface area, m2
A∗ fraction of channel cross section open to flow, m2
A1 flow area (aligned tube arrangement), m2
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[516], (78)
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A2 flow area (staggered tube arrangement), m2
fin height, m
B correlation constant, dimensionless
b parameter defined by eq (6.172), dimensionless
b(x) similarity function, dimensionless
C ratio of eddy to turbulent diffusivity, dimensionless
constant or correlation constant, dimensionless
C ∆p coefficient in eq (6.185), dimensionless
C Pw coefficient in eq (6.186), dimensionless
¯C coefficient in free stream velocity definition, dimensionless
C f friction coefficient, dimensionless
c(x) similarity function, dimensionless
c p specific heat, J/kg· K
D substantial derivative, dimensionless
cylinderorsphere diameter, m round jet diameter, m
D H channel hydraulic diameter, m
tube diameter, m
d(x) similarity function, dimensionless
Ec Eckert number, dimensionless
¯F Prandtl number, dimensionless
f friction factor, dimensionless
f (η) stream function, dimensionless
G parameter defined by eq (6.131), dimensionless
location outside the boundary layer, m channel height (plate spacing), m
heat transfer coefficient, W/m2· K
¯h mean heat transfer coefficient, W/m2· K
had adiabatic heat transfer coefficient, W/m2· K
hav average heat transfer coefficient, W/m2· K
h L heat transfercoefficient forlaminarboundary layer, W/m2· K
h T heat transfer coefficient for turbulent boundary layer, W/m2· K
i unit vectorinx-coordinate direction, dimensionless
i step counter, dimensionless
row counter, dimensionless
(i,j) row and column index, dimensionless
Colburn heat transfer factor, dimensionless
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[517],(79)
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j (x) similarity function, dimensionless
j unit vectoriny-coordinate direction, dimensionless
k thermal conductivity, W/m· K
k∗ plate plate/fluid thermal conductivity ratio, W/m· K
k f fluid thermal conductivity, W/m· K
k s mean roughness length scale, dimensionless
ksub substrate thermal conductivity, W/m· K
k unit vectorinz coordinate direction, dimensionless
length scale factor, m length of heat sink, m length in streamwise direction, m
Lcore core length, m
cylinderlength, m mixing length, m spacing between plates, m
1 leading edge to first block spacing, m
2 last block to trailing edge spacing, m
s heat source length, m
N L exponent, dimensionless
numberof tube rows, dimensionless
Nu Nusselt number, dimensionless
Nu mean oraverage Nusselt number, dimensionless
NuD Nusselt numberbased on diameter, dimensionless
NuD average Nusselt number based on diameter, dimensionless
NuL Nusselt numberbased on length, dimensionless
Nux Nusselt numberbased on diameter, dimensionless
numberof pins, dimensionless numberof plates in stack, dimensionless
heat source height, m
P heat source length, m
length of block, m
heat source width
Pe P´eclet number, dimensionless
Pec P´eclet numberused in eqs (6.175) and (6.176), dimensionless
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[518], (80)
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PrPrandtl number, dimensionless
PrT turbulent Prandtl number, dimensionless
fin pitch, m
p+ normalized pressure, N/m2
¯p mean or average pressure, N/m2
p∗ normalized pressure, N/m2
p m motion pressure, N/m2
heat sink dissipation, W heaterpowerinput, W
Q A direct heat transfer component, W
Q B conjugate heat transfer component through substrate, W
q A heat dissipation, blockA, W
q B heat dissipation, blockB, W
q
q volumetric heat generation, W/m3
q (i,j) rate of heat dissipation by block (i,j) in the array, W
Re Reynolds number, dimensionless
Re∗b Reynolds numberdefined by eq (6.171), dimensionless
ReD Reynolds numberbased on diameter, dimensionless
ReD,max Reynolds numberat maximum flow, dimensionless
Rek roughness Reynolds number, dimensionless
ReL Reynolds numberbased on length, dimensionless
Re Reynolds numberdefined by eq (6.173), dimensionless
RePh Reynolds numberbased onP hdefined in Section 6.6.2,
dimensionless
ReP Reynolds numberbased onP defined in Section 6.6.2,
dimensionless
ReT critical Reynolds number, dimensionless
Rex Reynolds numberbased onx, dimensionless
Re∗ transition Reynolds number, dimensionless
r boundary layer ratio, dimensionless
r c recovery factor, dimensionless
r0 distance from axis to surface, m
S D diagonal tube spacing, m
S L longitudinal tube spacing, m
S T transverse tube spacing, m
St Stanton number, dimensionless
Stk roughness Stanton number, dimensionless
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[519],(81)
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clearspace between blocks, m
s x x-coordinate distance, m
s z distance to bounding surface, m
¯T average or mean temperature, K
¯T s average surface temperature, K
Tair,B temperature of air at blockB, K
T b temperature in buffer region, K
temperature at bottom surface of heat sink, K
Tmax maximum surface temperature, K
Tref reference temperature, K
T s surface temperature, K
T s,B surface temperature of blockB, K
T∗ normalized temperature in eq (6.8), dimensionless
T+
b normalized buffer temperature, K
T0 free stream temperature, K
air temperature at front of heat sink, K
T∞ ambient temperature, K
plate thickness, m nondimensional substrate thickness, dimensionless
U overall heat transfer coefficient, W/m2· K
velocity scale factor, m/s free stream velocity, m/s
U∞ velocity in undisturbed flow, m/s
U0 average velocity in unobstructed channel, m/s
u x-coordinate velocity, m/s
u+ normalizedx-coordinate velocity, m/s
u+
∞ normalized free streamx-coordinate velocity, m/s
u o free streamx-coordinate velocity, m/s
¯u mean oraveragex-coordinate velocity, m/s
u∗ normalizedx-coordinate velocity, m/s
Vmax maximum velocity, m/s
v y-coordinate velocity, m/s
v0 free streamy-coordinate velocity, m/s
v∗ friction velocity in turbulent flow, m/s
v+ normalizedy-coordinate velocity, m/s
¯v mean oraveragey-coordinate velocity, m/s
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[520], (82)
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channel width, m
z-coordinate velocity, m/s
slot jet width, m
w∗ normalized,z-coordinate velocity, m/s
x0 x coordinate, m
x unheated starting length, m
x∗ normalizedx-coordinate velocity, m/s
x s leading edge to heat source distance, m
y∗ normalizedy-coordinate velocity, m/s
z∗ normalizedz-coordinate velocity, m/s
∆p pressure difference, N/m2
∆pcore core pressure difference, N/m2
∆T temperature difference, K
Greek Letter Symbols
α thermal diffusivity, m2/s
aspect ratio, dimensionless
β volumetric expansion coefficient, K−1
wedge angle, rad constant, pressure difference, N/m2
δc conduction thickness, m
δT thermal boundary layer, m
H eddy diffusivity, m2/s
η similarity variable, dimensionless
similarity function, dimensionless
ηB Blasius similarity variable, dimensionless
ηδ thickness of boundary layer, dimensionless
θ normalized temperature, dimensionless
angle, rad
θB/A effect of heat dissipation from blockB on block A, K/W
θB/(i,j) contribution of all blocks upstream of blockB, K/W
θhot hot spot temperature, dimensionless
κ von K´arm´an constant, dimensionless
µ dynamic viscosity, kg/m· s
µs dynamic viscosity at surface or wall temperature, kg/m· s
ν kinematic viscosity, m2/s
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τb mean shearstress, N/m2
τo free stream shear stress, N/m2
τT turbulent shear stress, N/m2
Φ viscous dissipation, s−2
φ(x,y) stream function, dimensionless
φ(η) similarity function, dimensionless
ψ(x,y) stream function, dimensionless
Roman Letter Subscripts
direct heat transfer component
ad,B adiabatic heat transfer coefficient on blockB
air,b air temperature at blockB
conjugate heat transfer component block designator
B/A effect on blockB by dissipation from block A b/(i,j) contribution of upstream block
recovery factor
friction
(i,j) row and column index
P particular part of solution