Forclose block spacing, s/P = 0.25, cavity-type flow is formed in the interblock space, indicating that the forward and back surfaces of the block do not contribute much to the heat tran
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where Nu is as defined for the two-dimensional strip, Pec = U o (2x s + s )/2α and A/P
is the source surface area/perimeter ratio The correlations of eqs (6.175) and (6.176) are valid for 103 ≤ Pec ≤ 105, 5 ≤ (2x s + s ≤)2 s ≤ 150, and 0.2 ≤ w s / s ≤ 5, wherew sis the heat source width Additional detailed treatments of the subject have been compiled by Ortega (1996)
6.6.2 Two-Dimensional Block Array
Figure 6.22 shows the pertinent dimensions, all nondimensionalized by the plate spac-ing for the two-dimensional block array Arvizu and Moffat (1982) performed exper-iments for heat transfer from aluminum blocks in forced airflow in such channels
The parameter ranges covered areP h = 1/2, 1/4.6, and 1/7, s = 0.5, 1, 2, 3, 6, and
8, and 2200< Re P h = UP
h /ν < 12,000 P
h = dimensional block height For s ≥ 2,
the heat transfercoefficients fora fixedP halmost collapse into a single curve The
effect ofP hon heat transferis large fortightly placed blocks; fors = 0.5, when P h
is increased from 1/7 to 1/2, the Nusselt number almost doubles However, Lehman and Wirtz (1985) found a near collapse of heat transfer data of variousP h ranging from 1/2 to 1/6 withs = P handP /P h= 4 The data of Lehman and Wirtz (1985) were obtained for 1000< Re P < 12,000 and2
3 ≤ P ≤ 2
A visualization study was also conducted by Lehman and Wirtz and it revealed modes of convection that depend on the block spacing Forclose block spacing,
s/P = 0.25, cavity-type flow is formed in the interblock space, indicating that the
forward and back surfaces of the block do not contribute much to the heat transfer
When the spacing is widened tos/P = 1, significant cavity-channel flow interac-tions were observed
Davalath and Bayazitoglu (1987) performed numerical analysis on a three-row block array placed in a parallel-plate channel Heat transfer correlations were derived for the cases where the following dimensions are fixed at,P = s = 0.5, P h =
0.25, 1 = 3.0, 2 = 9.5, and t = 0.1 The Reynolds numberis defined as Re =
U0H/ν, where U0is the average velocity in the channel (with an unobstructed cross section),H is the channel height, and ν is the fluid kinematic viscosity The average
Nusselt numberis Nu = ¯h/k, where ¯h is the average heat transfer coefficient
overthe block surface, is the dimensional block length, and k f is the fluid thermal conductivity The correlation between Nu, Re, and Pris given in the form
P l
P h
t
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for the Case of Blocks onanInsulated Plate
Block
for the Case of Blocks on a Conducting Plate
Block
from the Bottom Surface of a Plate to the Total Heat Dissipationby Block
Percent of Total Heat
k∗
The correlation constants,A, B, and C are given in Table 6.4 for the blocks on
an insulated plate and in Table 6.5 forthe blocks on a conducting plate having the same thermal conductivity as the block Table 6.6 shows the ratios, expressed as a percentage, of the heat transfer rate from the bottom surface of the plate to the total heat dissipation by the block Herek∗
plate is the ratio of thermal conductivity of the plate to the fluid thermal conductivity
6.6.3 Isolated Blocks
Roeller et al (1991) and Roeller and Webb (1992) performed experiments with the protruded rectangular heat sources mounted on a nonconducting substrate The pertinent dimensions are shown in Fig 6.23, where it is observed thatH and W are
the height and the width, respectively, of the channel where the heat source/substrate composite is placed,P h is the height,P the length, and P w the width of the heat
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source The channel width is fixed atW = 12 mm, and the height H was allowed to
vary from 7 to 30 mm The heat source dimensions covered by the experiments were
P = 12 mm; P h = 4, 8, and 12 mm, H −P h = 3, 8, and 12 mm, and P w = W = 12
mm The heat transfer correlation is given by
Nu= 0.150Re0.632 (A∗) −0.455H
P
−0.727
(6.178)
where Nu= ¯hP /k and where in
A s ( ¯T s − T∞)
¯h is the average heat transfer coefficient, q the heat transfer rate, A sthe heat transfer
area,
A s = 2P h P w + P P w + 2P h P
¯T s the average surface temperature, andT∞the free stream temperature The Reyn-olds numberis defined as Re = UD H /ν, where U is the average channel velocity
upstream of the heat source,D H the channel hydraulic diameter at a section unob-structed by the heat source, andν the fluid (air) kinematic viscosity A∗is the fraction
of the channel cross section open to flow:
A∗= 1 − P W w P H h
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Airflow
Q
Q B
Q A
Block
and Park, 1996.)
Equation (6.178) is valid for1500 ≤ Re ≤ 10,000, 0.33 ≤ P h /P ≤ 1.0, 0.12 ≤
P w /W ≤ 1.0, and 0.583 ≤ H/P ≤ 2.5 Here, a realistic error bound is 5%.
In actual situations of cooling electronic packages the heat flow generally follows two paths, one directly from the package surface to the coolant flow, and the other from the package through the lead pins or solder balls to the printed wiring board (PWB), then through the PWB, and finally, from the PWB surfaces to the coolant flow Figure 6.24 depicts heat flows through such paths;Q is the total heat generation,
Q Athe direct heat transfer component, andQ Bthe conjugate heat transfer component through the substrate, hence
Q = Q A + Q B
which is due to Nakayama and Park (1996) Equation (6.178) can be used to esti-mateQ A.
The ratioQ B /Q Adepends on the thermal resistance between the heat source block
and the substrate (the block support in Fig 6.24 simulating the lead lines or the solder balls), the thermal conductivity and the thickness of the substrate, and the surface heat transfer coefficient The estimation ofQ B is a complex process, particularly where the lowerside of the substrate is not exposed to the coolant flow, which makesQ B
find its way through only the upper surface This is the case often encountered in electronic equipment Convective heat transfer from the upper surface is affected by flow development around the heat source block, which is three-dimensional, involv-ing a horseshoe vortex and the thermal wake shed from the block, leadinvolv-ing to a rise
in the local fluid temperature above the free stream temperature Nakayama and Park (1996) studied such cases using a heat source block typical to electronic package, 31
mm× 31 mm × 7 mm A good thermal bond between the block and the substrate,
of the order ofR = 0.01 K/W, and a high thermal conductance of the substrate, such
as that of a 1-mm-thick copperplate, maximizes the contribution of conjugate heat transferQ Bto the total heat dissipationQ, raising the ratio Q B /Q to a value greater
than 0.50
6.6.4 Block Arrays
Block arrays are common features of electronic equipment, particularly, large systems where a number of packages of the same size are mounted on a large printed wiring
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board (PWB) and cooled by air in forced convection Numerical analysis of three-dimensional airflow over a block array is possible only when a fully developed situation is assumed For fully developed flow, a zone around a block is carved out, and a repeating boundary condition is assumed on the upstream and downstream faces
of the zone In general, the analysis of flow and heat transfer over an entire block array depends too much upon computational resources, and experiments are frequently the sole means of investigation However, experiments can be costly, especially when
it is desired to cover a wide range of cases where the heat dissipation varies from package to package To reduce the demand for experimental (and computational as well) resources, a systematic methodology has been developed
Consider the block array displayed in Fig 6.25 BlockA dissipates q Aand block
B, q B Assume forthe moment that the otherblocks are inactive; that is, they do not dissipate The temperature of air over blockB can be written as
Tair ,B = T0+ θB/A q A (6.179) whereT0is the free stream temperature andθB/A in the second term represents the effect of heat dissipation from blockA Equation (6.179) is based on the superposi-tion of solusuperposi-tions that is permissible because of the linearity of the energy equasuperposi-tion.
However, the factor θB/Aresults from nonlinear phenomena of dispersion of warm
airfrom blockA and is a function of the relative location of B to A and the flow
velocity To findθB/Aby experiment, blockA may be energized while block B
re-mains inactive (q B = 0) A measurement of the surface temperature of block B is
then divided byq A Becauseq B = 0, its measured surface temperature is called the
adiabatic temperature.
The next step is to activate blockB and inactivate block A(q A = 0) The heat transfer coefficient measured in this situation is called the adiabatic heat transfer
coefficient and is denoted as had ,B When both blockA and block B are energized,
the heat transfer from blockB is driven by the temperature difference T s,B − Tair,B,
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whereT s,B is the surface temperature of blockB and Tair ,B is given by eq (6.179)
Again using the superposition principle, the heat flux at blockB is
q B = had,B
T s,B − Tair,B
(6.180)
OnceθB/A is determined, it is straightforward to estimateq B (or T s,B whenq B is specified) forany value ofq Afrom eqs (6.179) and (6.180)
Extension of this concept to a general case includes taking account of the contri-butions of all blocks upstream of block B in the equation forTair ,B:
Tair ,B = T0+
i,j
where(i,j) is the row and column index and the summation is performed for all the
packages upstream of blockB.
Although the foregoing concept looks convenient at first sight, it is a tedious and time-consuming task to determineθB/(i,j) Except fora limited numberof cases, there have been few correlations that relateθB/(i,j)to the geometrical and flow parameters
Moffat and Ortega (1988) summarized the work on this subject, and Anderson (1994) extended the concept to the case of conjugate heat transfer
The heat transfer data corresponding to the adiabatic heat transfer coefficient in downstream rows where the flow is fully developed were reported by Wirtz and Dykshoorn (1984) The data were correlated by the equation
NuP = 0.348Re0.6
where the characteristic length for Nu and Re is the streamwise length of the block,
P .
6.6.5 Plate Fin Heat Sinks
While the plate stack discussed in Section 4.2.1 allows bypass flow in two-dimen-sional domain, bypass flow around an actual heat sink is three-dimentwo-dimen-sional Numer-ical analysis of such flow is possible but very resource demanding Ashiwake et al
(1983) developed a method that allows approximate but rapid estimation of the heat sink performance In their formulation the bypass flow rate is estimated using the balance between the dynamic pressure in front of the fin array and the flow resistance
in the interfin passages Although the validity of the modeling was well corroborated
by the experimental data, the method requires computations of the several pressure balance and heat transfer correlations, and the task of developing a more concise formulation of heat sink performance estimation remains
Presently, the performance estimation is largely in the realm of empirical art, although Ledezma et al (1996) and Bejan and Sciubba (1992) clearly show optimum spacings which agree with the theory The heat sink is placed in a wind tunnel and the thermal resistance is measured The thermal resistance is defined as
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1993.)
R = T Q
b − T0
whereQ is the powerinput to the heaterbonded to the bottom of the heat sink, T bthe
temperature at the bottom surface of the heat sink, andT0the airflow temperature in front of the heat sink Figure 6.26 shows examples of thermal resistance data, where
U is the free stream velocity All the data were obtained with aluminum heat sinks
having a 22 mm× 22 mm base area Of course, there is trade-off between the heat transfer performance and the cost of heat sink Conventional extruded heat sinks (see Fig 6.26) are at the lowest in the cost ranking but also in the performance ranking
The heat sink having 19 thin fins (0.15 mm thick) on the 22-mm span provides low thermal resistance, particularly at high air velocities, but the manufacture of such a heat sink requires a costly process of bonding thin fins to the base
6.6.6 Pin Fin Heat Sinks
As electronic systems become compact, the path for cooling airflow is constrained
This means increased uncertainty in the direction of airflow in front of the heat sink
The performance of plate fin heat sinks degrades rapidly as the direction of air flow deviates from the orientation of the fins The pin fin heat sink has a distinct advantage
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Airflow
Fan
L
a
over the plate fin heat sink in that its performance is relatively insensitive to the direction of the airflow
Figure 6.27 shows a scheme that exploits the advantage of pin fin heat sink to the fullest extent A small axial fan is mounted above the fin heat sink, and airis blown from above to the heat sink The airflow is longitudinal to those pins in the central area, and the pins in the perimeter are exposed to cross flow Recently, the scheme has become popularforcooling CPU chips in a constrained space
The work of Wirtz et al (1997) provides a guide for the estimation of the perfor-mance of a pin fin heat sink/fan assembly The dimensions of the pin fin heat sinks tested by Wirtz et al (1997) are as follows:
Dimensionless pin diameter: d/L = 0.05
Dimensionless pin height: a/L = 0.157 − 0.629
Fin pitch-to-diameterratio: p/d = 2.71 − 1.46
Numberof pins on a row ora column: n = 8, 10, and 14
The fan used in the experiment has overall dimensions of 52 mm× 52 mm ×
10 mm, a 27-mm-diameter hub, and a 50-mm blade shroud diameter The overall heat transfer coefficientU is defined as
whereQ is the heat transfer rate, A T the total surface area of the heat sink, and∆T
the temperature difference between the fin base and the incoming air The Nusselt numberis defined as Nu= UL/k, where k is the fluid thermal conductivity Two
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types of correlations are proposed, one for a given pressure rise maintained by the fan,∆p,
Nu= 7.12 × 10−4C0.574
∆p
a
L
0.223 p
d
1.72
(6.185) where in
C ∆p= ρL2∆p
µ2
5× 106< C ∆p < 1.5 × 108
µ is the dynamic viscosity of the air
The other correlation is for a given fan powerP W:
Nu= 3.2 × 10−6C0.520
P W
a
L
−0.205 p
d
0.89
(6.186) where
C P W = ρLP W
µ3
1011< C P W < 1013
Wirtz et al (1997) also reported on experimental results obtained with square and diamond-shaped pins
6.7 TURBULENT JETS
Jets are employed in a wide variety of engineering devices In cases where the jets are located far from solid walls, they are classified as free shear flows In most cases, how-ever, solid walls are present and affect the flow and heat transfer significantly These flows can take many configurations Some of the practically important cases for heat transfer include wall jets and jets impinging on solid surfaces as indicated in Fig 6.28
Wall jets are frequently employed in turbomachinery applications and are not discussed here Jet impingement on surfaces is of interest in materials packaging and electronics cooling
6.7.1 Thermal Transport in Jet Impingement
Due to the thin thermal and hydrodynamic boundary layers formed on the impinge-ment surface, the heat transfer coefficients associated with jet impingeimpinge-ment are large, making these flows suitable forlarge heat flux cooling applications The relationship for impingement of a jet issuing from a nozzle at a uniform velocity and at ambient temperatureT e, with a surface at temperatureT s, can be written as
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Jet ( ) Film cooling with tangential injectiona
( ) Transpiration coolingb
Porous surface
Fluid injection Nozzle
Nozzle
Figure 6.28 Several configurations of jet cooling arising in applications