Use of Superposition
to Describe Heat Transfer
from Multiple Electronic Components
Gerald Recktenwald
Portland State University
Department of Mechanical Engineering
[email protected]
Convection from PCBs
These slides are a supplement to the lectures in ME 449/549 Thermal Management Measurements
c 2006, Gerald W. Recktenwald, all rights reserved. The material is provided to enhance
and are the learning of students in the course, and should only be used for educational purposes. The
material in these slides is subject to change without notice.
The PDF version of these slides may be downloaded or stored or printed only for noncommercial,
educational use. The repackaging or sale of these slides in any form, without written consent of
the author, is prohibited.
The latest version of this PDF file, along with other supplemental material for the class, can be
found at www.me.pdx.edu/~gerry/class/ME449. Note that the location (URL) for this web
site may change.
Version 0.81
Convection from PCBs
May 30, 2006
page 1
Overview
• Overview of the Physics
• Experimental Data
• Superposition and the adiabatic
heat transfer coefficient
• Sample Calculation
Convection from PCBs
page 2
Heat Transfer Modes
Vin, Tin
radiation
convection
conduction in the board
•
•
•
•
•
conduction within devices and attached heat sinks
conduction in the multilayer, composite PCB
forced and natural convection from devices and heat sinks
radiation between devices and adjacent boards
radiation between the fins of a heat sink
Convection from PCBs
page 3
Geometrical Complexity
H
B2
B1
L1
S12
L2
B3
S23
L3
• multiple length scales: large boxes and small components
• irregularly shaped flow passages with blockages
• three-dimensional flow patterns around heat sinks and in the wake of discrete
components
• internal board configurations may change in the field
Convection from PCBs
page 4
Fully-Developed Flow
Hydrodynamically fully-developed flow:
• velocity field is independent of the flow direction
dp
•
= constant
dx
Thermally fully-developed flow:
• flow is hydrodynamically fully-developed
• heat transfer coefficient is independent of the flow direction
Flow over arrays of blocks in a channel exhibits fully-developed behavior after the third or
fourth row of blocks
Convection from PCBs
page 5
Laminar, Transitional, and Turbulent Flow
Industrial equipment tends to be turbulent flow
• little or no noise constraint
⇒ high flow velocities
• high power consumption equipment
Office equipment tends to have transitional flow
• equipment must be relatively quiet
⇒ lower flow velocities
Convection from PCBs
page 6
Natural Convection Applications
Some equipment uses natural convection only
• low power devices
⇒ battery power makes fan use “expensive”
• portable test equipment
• optimize internal heat conduction paths
conduct heat to external case
use of heat pipes in lap-top computers
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page 7
Mixed Convection
Buoyancy effects can be present in a forced convection flow
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page 8
Recirculation in Plan View
device with heat sink
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page 9
Recirculation in Elevation View
Experiments by Sparrow, Niethammer and Chaboki [3]
H
b
Nu =
Nufd
1.00
1.46
Re = 3700
Convection from PCBs
1.49
1.30
1.21
1.15
t
b–t = 1
H
5
page 10
Thermal Wakes (1)
Thermal wake for a flush heat source
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page 11
Thermal Wakes (2)
Three-dimensional representation of a wake, T (x, y)
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page 12
Unmixed Temperature Profile
Flow tends to organize into
• By-pass flow above the devices
• Array flow around the devices
bypass flow, above blocks
array flow, between blocks
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page 13
By-pass and Array Flow (1)
bypass flow, above blocks
array flow, between blocks
By-pass flow
• Higher velocity than array flow
• Streamlines are topologically simple
• Relatively higher turbulent fluctuation at interface between by-pass flow and top of
blocks. Flow may still be considered unsteady laminar for many applications.
• Gross flow features may be predicted with CFD.
Convection from PCBs
page 14
By-pass and Array Flow (2)
bypass flow, above blocks
array flow, between blocks
Array flow
• Lower velocities than by-pass flow
• Streamlines are topologically complex: many recirculation zones
• Very hard to accurately predict the details because of small scale flow features.
Convection from PCBs
page 15
Hierarchy of Analysis Strategies
In order of increasing effort:
•
•
•
•
hand calculation of energy balance
use of heat transfer correlations for board-level analysis
resitive network of entire enclosure
Conduction modeling in the board: fluid flow is treated only as a convective boundary
coefficient.
• PCBCAT layer-based models
• Full 3-D CFD models of conjugate heat transfer
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page 16
Example: Fan-Cooled Enclosure (1)
disk
drive
Convection from PCBs
power supply
page 17
Fan-Cooled Enclosure (2)
ΣQ1
ΣQ2
ΣQ4
X
m1
ΣQ3
m2
m3
Q1 = ṁ1cp (To,1 − Ti,1)
P
Q1
ṁ1cp
P
Q2
= Ti,2 +
ṁ2cp
P
Q3
= Ti,3 +
ṁ3cp
⇒ To,1 = Ti,1 +
To,2
To,3
Convection from PCBs
page 18
Fan-Cooled Enclosure (3)
What contributes to
P
Qi?
• Power dissipation of devices
• Heat loss directly through the cabinet to the ambient
• Heat gain/loss through the PCB to an adjacent channel containing other board
Perhaps individual control volumes should be connected into a thermal network.
Convection from PCBs
page 19
Board-Level Energy Balance
Tm(x)
Tout
Tin
x
•
min
Q1
A
B
Q2
C
Q3
D
E
• 3D effects
fan wake
non-uniform inlet
blockage by obstacles including heat sinks
• Channel by-pass and unmixed temperature profile
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page 20
——————-
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page 21
Heat Transfer Correlations for Board-Level Analysis
m• 1
h, Ta
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page 22
Heat Transfer Correlations for Board-Level Analysis
• Energy balance only gives the air
temperature.
• We need values for thermal resistances to
estimate junction temperatures.
• Thermal resistance of heat sink comes
from heat sink manufacturer.
(But
does that test data apply to your
configuration?)
• Other convective resistances are estimated
from heat transfer coefficients.
• General correlations for heat transfer
coefficients from arbitrary devices on a
PCB do not exist.
Convection from PCBs
R values for a high
performance CPU:
Rsa ∼ 0.4 W/C
Heat
Sink
Case (cover)
Rim ∼ 0.1 W/C
Rjc ∼ 0.3 W/C
Die
Substrate
Q
Rjb
Rba
page 23
Resistive Network Models
SINDA:
http://www.webcom.com/~crtech/sinda.html
http://www.indirect.com/user/sinda/
See also Thermal Computations for Electronic Equipment, by Gordon Ellison [2]
Convection from PCBs
page 24
Conduction Modeling (1)
Internal resistance can be obtained from finite-element analysis of conduction heat
transfer inside the device. This data is usually supplied by the device manufacturer,
because only they know the details of the internal construction.
m• 1
h, Ta
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page 25
Conduction Modeling (2)
•
•
•
•
Need heat transfer coefficient at all fluid-solid interfaces.
Analysis is a standard procedure with most FEM packages.
Practical limit to the geometric detail
Analysis time is short compared to model building time.
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page 26
CFD Modeling (1)
m• 1
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page 27
CFD Modeling (2)
• Significant investment in model development
=⇒ CFD model run time is often short compared to model building time.
• Detailed solution still requires significant computing requirements
• Momentum equations are nonlinear
• Turbulence models
• Inlet vents and fans need to be modeled.
• Practical limit to the geometric detail
• CFD packages for electronic cooling
FlothermTM http://www.flomerics.com/
IcePackTM http://www.fluent.com/
Convection from PCBs
page 28
Experimental Data
•
•
•
•
flush mounted heaters
Ribs
Arrays of blocks
arrays of “heater” devices
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page 29
Correlations
Flow over a flat plate
a
Nu = C Re Pr
b
Which length scales to use?
Proper application requires
• geometric similarity
• dynamic similarity
• thermal similarity
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page 30
Heat Transfer Coefficients
Vin, Tin
s
Experimental Procedure
1. Adjust flow rate
2. Set power level of each block
3. Wait for thermal equilibrium
4. Measure temperature of each block
5. Compute heat transfer coefficient
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page 31
Which heat transfer coefficient?
Based on inlet temperature:
hin,i =
Qconv,i/Ai
Tb,i − Tin
Based on local, mean fluid temperature:
hm,i =
Qi/Ai
Tb,i − Tm,i
Pi
Tm,i = Tin +
j=1
Qj
ṁcp
Based on adiabatic wall temperature:
had,i =
Convection from PCBs
Qconv,i/Ai
Tb,i − Tad,i
page 32
Superposition Principle (1)
Consider flow in a tube with an arbitrary axial variation in heat input.
ξ
x
∆x
u(r)
R
r
hydrodynamically
fully-developed flow
qw''(x)
Energy Equation
∂T
k ∂
ρcpu(r)
=
∂x
r ∂r
„
«
∂T
r
∂r
Boundary conditions
˛
∂T ˛˛
= 0 (symmetry)
˛
∂r r=0
Convection from PCBs
˛
∂T ˛˛
00
k
=
q
w (x)
˛
∂r r=R
page 33
Superposition Principle (2)
General solution is
+
Tw,ad(x ) − Tin
R
=
k
Z
x+
+
00
g(x − ξ) qw (ξ) dx
0
where g(x+) is the superposition kernel function
+
g(x ) = 4 +
`
´
2
X exp −γm
x+
m
2 A
γm
m
For a single heated patch this reduces to
+
Tw,ad(x ) − Tin =
Convection from PCBs
Q
+
g(x − ξ)
4ṁcp
page 34
Interpretation of Kernel Function (1)
∆x
r
.
m
y
Q
Tw,ad(x)
Tm
Tin
∆Tm
∆Tm
∆Tm
y
Tin
Convection from PCBs
T(y)
Tin Tm
Tw,ad
Tin
Tw,ad
page 35
Interpretation of Kernel Function (2)
Energy balance gives increase in mean fluid temperature
∆Tm =
Q
ṁcp
Solve equation defining Tw,ad for g(x+)
Tw,ad(x+) − Tin
g(x − ξ) =
Q/(4ṁcp)
+
Tw,ad(x+) − Tin
= 4
∆Tm
Convection from PCBs
page 36
Application to PCB Heat Transfer (1)
3
2
m=1
n=1
2
3
4
The adiabatic temperature of a block is
the temperature it attains when it is has
zero internal heat generation.
Note that if no blocks are heated, then Tad,i = Tin. Remember that “adiabatic” in this
context means unheated, not insulated.
Convection from PCBs
page 37
Application to PCB Heat Transfer (2)
The temperature difference between block i and the inlet air can be decomposed as
Tb,i − Tin = (Tb,i − Tad,i) + (Tad,i − Tin)
Convection from PCBs
Tb,i
=
average surface temperature
of heated block i.
Tb,i − Tad,i
=
temperature rise due to selfheating
Tad,i − Tin
=
temperature rise due to heat
inputs from other heated
elements
(1)
page 38
Application to PCB Heat Transfer (3)
The adiabatic temperature rise of block i due to heat input from all blocks is
Tb,i
n
X
Qj ∗
gi,j
= Tin +
ṁcp
j=1
(2)
Interpret as sum of two major contributions
n
X
Tb,i − Tin =
j=1,
|
Qconv,j ∗
gi,j
ṁc
p
j6=i
{z
}
+
upstream contribution
Qconv,i ∗
g
ṁcp i,i
| {z }
(3)
self-heating
Temperature rise due to self-heating is rise due to self-heating alone is
Tb,i − Tad,i =
Convection from PCBs
Qconv,i
had,iAi
(4)
page 39
Application to PCB Heat Transfer (4)
Equating the right hand side of Equation (4) with the second term on the right hand side
of Equation (3) gives
Qconv,i
Qconv,i ∗
=
g
(5)
had,iAi
ṁcp i,i
Thus,
ṁcp
∗
gi,i =
(6)
had,iAi
∗
Equation (6) shows that gi,i
and had,i are intrinsically related. This is no accident since
∗
both gi,i and had,i are derived from measurements in which only block i is heated.
Convection from PCBs
page 40
Application to PCB Heat Transfer (5)
Substituting Equation (5) into Equation (3) gives
Tb,i − Tin =
n
X
j=1,
Qconv,j ∗
Qconv,i
gi,j +
ṁc
had,iAi
p
j6=i
(7)
∗
With measured values of gi,j
and had,i, Equation (7) uses superposition to compute the
effect of any power distribution on the temperature of each block in the domain. All that
∗
remains is a procedure for determining gi,j
from the experimental data.
Convection from PCBs
page 41
Measuring had for a 3 Block Experiment (1)
Measure had,i for i = 1 and Tad,i for i = 2, 3:
1. adjust flow rate
2. turn heat on for block 1
3. turn off heat for block 2 and block 3
4. wait for thermal equilibrium
5. measure temperatures of all three blocks
Convection from PCBs
page 42
Measuring had for a 3 Block Experiment (2)
Write out Equation (2) for i = 2, j = 1:
Tb,2 = Tin +
Q1 ∗
Q2 ∗
Q3 ∗
g2,1 +
g2,2 +
g
ṁcp
ṁcp
ṁcp 2,3
(8)
Since Q2 = Q3 = 0 in this experiment, the preceding equation reduces to
Tb,2
Q1 ∗
= Tin +
g
ṁcp 2,1
(9)
∗
Solving for g2,1
gives
∗
g2,1 =
Tb,2 − Tin
Q1/(ṁcp)
only block 1 is heated
(10)
Because only block 1 is heated, Tb,2 − Tin is the temperature rise of block 2 due to heat
input at block 1.
Convection from PCBs
page 43
Measuring had for a 3 Block Experiment (3)
Define
Twake,i,j = temperature of block i when only block j is heated.
The term “wake” is suggestive of the mechanism of heating: Twake,i,j > Tin because
block i is downstream of block j .
Thus, when only block 1 is heated, the value of Tb,2 is Twake,1,2, and Equation (10) is
∗
g2,1 =
Twake,2,1 − Tin
Q1/(ṁcp)
(11)
Remember that the simplification that leads from Equation (8) to Equation (11) is valid
because only block 1 is heated.
∗
Similar calculation (from same experiment) gives g3,1
.
Convection from PCBs
page 44
Measuring had for a 3 Block Experiment (4)
Repeat measurements to obtain data for following table
Measured Temperatures
Heat Inputs
Convection from PCBs
Block 1
Block 2
Block 3
Q1
0
0
Tself,1
Twake,2,1
Twake,3,1
0
Q2
0
Twake,1,2
Tself,2
Twake,3,2
0
0
Q3
Twake,1,3
Twake,2,3
Tself,3
page 45
Anderson and Moffat Correlation (1)
top view
Sz
side view
Lz
Sx
Lx
x
row
number
8
7
6
5
4
3
2
1
H
B
z
flow direction
Anderson and Moffat [1] found
• g ∗(x) was related to correlation for had
• no interaction between columns
• fully-developed flow after third row
Convection from PCBs
page 46
Anderson and Moffat Correlation (2)
For fully-developed region
g ∗ = 1 + β1 exp (−α1N ) + β2 exp (−α2N )
For first two rows
g1∗
˘
¯
∗
= max 0.8 g , 1
˘
¯
∗
∗
g2 = max 0.95 g , 1
Convection from PCBs
page 47
Anderson and Moffat Correlation (3)
Dimension analysis gives a relationship for maximum possible turbulence fluctuations in
the channel
„
«(1/3)
−∆P
(H
−
B)
row
0
ũmax = 0.82 Um
ρ
Lx
where Um is the velocity in the bypass region
Um =
Convection from PCBs
VH
H−B
page 48
Anderson and Moffat Correlation (4)
For fully-developed region
g ∗ = 1 + β1 exp (−α1N ) + β2 exp (−α2N )
For first two rows
g1∗
˘
¯
∗
= max 0.8 g , 1
˘
¯
∗
∗
g2 = max 0.95 g , 1
0
α1 = 0.31 ũmax + 1.91
0
α2 = 0.098 ũmax + 0.19
„
«
1
ṁcp/A
β1 =
−1
1.13 32.2 ũ0max + 14.4
β2 = 0.13 β1
Convection from PCBs
page 49
Example Calculation (1)
parameter
H
B
Lx
Sx
Lz
Sz
value
0.0214 m
0.0095 m
0.0375 m
0.0502 m
0.0465 m
0.0592 m
Table 1: Geometrical parameters for the example calculations.
3
ρ = 1.185 kg/m
cp = 1005 J/(kg K)
2
V = 7.1 m/s
− ∆Prow = 7.78 N/m
Convection from PCBs
page 50
Example Calculation (2)
2
A = 0.00334 m
Um = 12.8m/s
ṁ = 1.06 × 10
−2
kg/s
per row
0
ũmax = 2.44 m/s
α1 = 2.6685
α2 = 0.4298
β1 = 29.5387
β2 = 3.8400
Convection from PCBs
page 51
Example Calculation (3)
row
8
7
6
5
4
3
2
1
Q (W )
12
18
14
7
2
13
11
15
Table 2: Power dissipated by modules in the example caluclation.
Convection from PCBs
page 52
Example Calculation (4)
The temperature rise in row n due to heat dissipated by the module in row 1 is
”
“
Q1 ∗
T e,n − Tin =
g1 (n − 1)
1
ṁ cp
n
g1∗ (n − 1)
8
7
6
5
4
3
2
1
1.000
1.033
1.158
1.351
1.654
2.214
4.438
27.503
“
”
T e ,n − Tin
1
(C)
1.40
1.45
1.62
1.89
2.32
3.10
6.22
38.56
Table 3: Temperature rise due to heat dissipated in row 1.
Convection from PCBs
page 53
Example Calculation (5)
The temperature rise in row n due to heat dissipated by the module in row 2 is
“
”
Q2 ∗
T e,n − Tin =
g (n − 2)
2
ṁ cp 2
n
g2∗ (n − 2)
8
7
6
5
4
3
2
1
1.227
1.375
1.604
1.964
2.629
5.270
32.660
0
“
”
T e ,n − Tin
2
(C)
1.26
1.41
1.65
2.02
2.70
5.42
33.58
0
Table 4: Temperature rise due to heat dissipated in row 2.
Convection from PCBs
page 54
Example Calculation (6)
The temperature rise in row n due to the heat dissipated by row three is
“
”
Q3 ∗
T e,n − Tin =
g (n − 3)
3
ṁ cp
n
g ∗ (n − 3)
8
7
6
5
4
3
2
1
1.448
1.689
2.068
2.768
5.547
34.378
0
0
“
”
T e ,n − Tin
3
(C)
1.76
2.05
2.51
3.36
6.74
41.77
0
0
Table 5: Temperature rise due to heat dissipated in row 3.
Convection from PCBs
page 55
Example Calculation (7)
n
8
7
6
5
4
3
2
1
T e ,n − Tin (C)
57.6
72.2
54.9
30.8
18.2
50.3
39.8
38.6
Table 6: Total temperature rise for modules.
Convection from PCBs
page 56
Example Calculation (8)
100
90
80
Temperature (C)
70
60
50
40
30
20
10
0
1
Convection from PCBs
2
3
4
5
row number
6
7
8
page 57
References
[1] A. M. Anderson and R. J. Moffat. The adiabatic heat transfer coefficient and the superposition kernel function: Part 1–data
for arrays of flatpacks for different flow conditions. Journal of Electronic Packaging, 114(1):14–21, 1992.
[2] Gordon N. Ellison. Thermal Computations for Electronic Equipment. Robert Krieger Publishing Co., Malabar, FL, 1989.
[3] E. M. Sparrow, J. E. Niethammer, and A. Chaboki. Heat transfer and pressure drop characteristics of arrays of rectangular
modules encountered in electronic equipment. International Journal of Heat and Mass Transfer, 25(7):961–973, 1982.
Convection from PCBs
page 58