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Use of Superpositionto Describe Heat Transfer
from Multiple Electronic Components
Gerald Recktenwald
Portland State University
Department of Mechanical Engineering
Convection from PCBs
These slides are a supplement to the lectures in ME 449/549 Thermal Management Measurementsand are c© 2006, Gerald W. Recktenwald, all rights reserved. The material is provided to enhancethe learning of students in the course, and should only be used for educational purposes. Thematerial 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 ofthe author, is prohibited.
The latest version of this PDF file, along with other supplemental material for the class, can befound at www.me.pdx.edu/~gerry/class/ME449. Note that the location (URL) for this website may change.
Version 0.81 May 30, 2006
Convection from PCBs 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
conduction in the board
radiationconvection
Vin, Tin
• 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
L1 L2 L3
B3B2B1
S12 S23
H
• 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
dx= constant
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
Convection from PCBs page 7
Mixed Convection
Buoyancy effects can be present in a forced convection flow
Convection from PCBs page 8
Recirculation in Plan View
device with heat sink
Convection from PCBs page 9
Recirculation in Elevation View
Experiments by Sparrow, Niethammer and Chaboki [3]
NuNufd
= 1.00 1.46 1.49 1.30 1.21 1.15
H
t
b
b – tH
=Re = 3700 15
Convection from PCBs page 10
Thermal Wakes (1)
Thermal wake for a flush heat source
Convection from PCBs page 11
Thermal Wakes (2)
Three-dimensional representation of a wake, T (x, y)
Convection from PCBs 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
Convection from PCBs 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
Convection from PCBs page 16
Example: Fan-Cooled Enclosure (1)
power supplydisk drive
Convection from PCBs page 17
Fan-Cooled Enclosure (2)
m3
m2
m1
ΣQ4 ΣQ3
ΣQ1
ΣQ2
XQ1 = m1cp (To,1 − Ti,1)
⇒ To,1 = Ti,1 +
PQ1
m1cp
To,2 = Ti,2 +
PQ2
m2cp
To,3 = Ti,3 +
PQ3
m3cp
Convection from PCBs page 18
Fan-Cooled Enclosure (3)
What contributes toP
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
Q1 Q2 Q3min•
Tin
Tout
A B C D E
Tm(x)
x
• 3D effects
� fan wake
� non-uniform inlet
� blockage by obstacles including heat sinks
• Channel by-pass and unmixed temperature profile
Convection from PCBs page 20
——————-
Convection from PCBs page 21
Heat Transfer Correlations for Board-Level Analysis
m1•
h, Ta
Convection from PCBs 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.
Heat Sink
Case (cover)
SubstrateDie
Q
Rjb
Rba
Rjc ∼ 0.3 W/C
Rim ∼ 0.1 W/C
Rsa ∼ 0.4 W/C
R values for a high performance CPU:
Convection from PCBs page 23
Resistive Network Models
SINDA:
http://www.webcom.com/~crtech/sinda.htmlhttp://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.
m1•
h, Ta
Convection from PCBs 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.
Convection from PCBs page 26
CFD Modeling (1)
m1•
Convection from PCBs 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
Convection from PCBs page 29
Correlations
Flow over a flat plate
Nu = C ReaPr
b
Which length scales to use?
Proper application requires
• geometric similarity
• dynamic similarity
• thermal similarity
Convection from PCBs page 30
Heat Transfer Coefficients
Vin, Tins
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
Convection from PCBs 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
Tm,i = Tin +
Pij=1 Qj
mcp
Based on adiabatic wall temperature:
had,i =Qconv,i/Ai
Tb,i − Tad,i
Convection from PCBs page 32
Superposition Principle (1)
Consider flow in a tube with an arbitrary axial variation in heat input.
xu(r)
hydrodynamically fully-developed flow
ξ
r
∆x
qw(x)''
R
Energy Equation
ρcpu(r)∂T
∂x=
k
r
∂
∂r
„r
∂T
∂r
«Boundary conditions
∂T
∂r
˛r=0
= 0 (symmetry) k∂T
∂r
˛r=R
= q′′w(x)
Convection from PCBs page 33
Superposition Principle (2)
General solution is
Tw,ad(x+)− Tin =
R
k
Z x+
0
g(x+ − ξ) q
′′w(ξ) dx
where g(x+) is the superposition kernel function
g(x+) = 4 +
Xm
exp`−γ2
m x+´
γ2m Am
For a single heated patch this reduces to
Tw,ad(x+)− Tin =
Q
4mcp
g(x+ − ξ)
Convection from PCBs page 34
Interpretation of Kernel Function (1)
Tin
∆Tm
Q
m.
y
r
y
Tw,adTinTin
Tw,ad(x)
Tw,adTinTm
Tm
T(y)
∆x
∆Tm∆Tm
Convection from PCBs page 35
Interpretation of Kernel Function (2)
Energy balance gives increase in mean fluid temperature
∆Tm =Q
mcp
Solve equation defining Tw,ad for g(x+)
g(x+ − ξ) =
Tw,ad(x+)− Tin
Q/(4mcp)
= 4Tw,ad(x
+)− Tin
∆Tm
Convection from PCBs page 36
Application to PCB Heat Transfer (1)
n = 1 2 3 4
m = 12
3
The adiabatic temperature of a block isthe temperature it attains when it is haszero 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) (1)
Tb,i = average surface temperature
of heated block i.
Tb,i − Tad,i = temperature rise due to self-
heating
Tad,i − Tin = temperature rise due to heat
inputs from other heated
elements
Convection from PCBs 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 = Tin +
nXj=1
Qj
mcp
g∗i,j (2)
Interpret as sum of two major contributions
Tb,i − Tin =
nXj=1, j 6=i
Qconv,j
mcp
g∗i,j| {z }
upstream contribution
+Qconv,i
mcp
g∗i,i| {z }
self-heating
(3)
Temperature rise due to self-heating is rise due to self-heating alone is
Tb,i − Tad,i =Qconv,i
had,iAi
(4)
Convection from PCBs 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) givesQconv,i
had,iAi
=Qconv,i
mcp
g∗i,i (5)
Thus,
g∗i,i =
mcp
had,iAi
(6)
Equation (6) shows that g∗i,i and had,i are intrinsically related. This is no accident since
both g∗i,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 =
nXj=1, j 6=i
Qconv,j
mcp
g∗i,j +
Qconv,i
had,iAi
(7)
With measured values of g∗i,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 g∗i,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
mcp
g∗2,1 +
Q2
mcp
g∗2,2 +
Q3
mcp
g∗2,3 (8)
Since Q2 = Q3 = 0 in this experiment, the preceding equation reduces to
Tb,2 = Tin +Q1
mcp
g∗2,1 (9)
Solving for g∗2,1 gives
g∗2,1 =
Tb,2 − Tin
Q1/(mcp)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
g∗2,1 =
Twake,2,1 − Tin
Q1/(mcp)(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 g∗3,1.
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 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
Convection from PCBs page 45
Anderson and Moffat Correlation (1)
Sz
Sx
z
x
flow direction
Lx
Lz
top view side view
H
B
rownumber
12345678
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
g∗1 = max˘0.8 g∗, 1
¯g∗2 = 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
u′max = 0.82
„Um
−∆Prow
ρ
(H − B)
Lx
«(1/3)
where Um is the velocity in the bypass region
Um =V H
H − B
Convection from PCBs page 48
Anderson and Moffat Correlation (4)
For fully-developed region
g∗ = 1 + β1 exp (−α1N) + β2 exp (−α2N)
For first two rows
g∗1 = max˘0.8 g∗, 1
¯g∗2 = max
˘0.95 g∗, 1
¯α1 = 0.31 u
′max + 1.91
α2 = 0.098 u′max + 0.19
β1 =1
1.13
„mcp/A
32.2 u′max + 14.4− 1
«β2 = 0.13 β1
Convection from PCBs page 49
Example Calculation (1)
parameter value
H 0.0214 m
B 0.0095 m
Lx 0.0375 m
Sx 0.0502 m
Lz 0.0465 m
Sz 0.0592 m
Table 1: Geometrical parameters for the example calculations.
ρ = 1.185 kg/m3
cp = 1005 J/(kg K)
V = 7.1 m/s −∆Prow = 7.78 N/m2
Convection from PCBs page 50
Example Calculation (2)
A = 0.00334 m2
Um = 12.8m/s
m = 1.06× 10−2
kg/s per row
u′max = 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 Q (W )
8 12
7 18
6 14
5 7
4 2
3 13
2 11
1 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“T e,n − Tin
”1=
Q1
m cp
g∗1(n− 1)
n g∗1(n− 1)“
T e,n − Tin
”1
(C)
8 1.000 1.40
7 1.033 1.45
6 1.158 1.62
5 1.351 1.89
4 1.654 2.32
3 2.214 3.10
2 4.438 6.22
1 27.503 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“T e,n − Tin
”2=
Q2
m cp
g∗2(n− 2)
n g∗2(n− 2)“
T e,n − Tin
”2
(C)
8 1.227 1.26
7 1.375 1.41
6 1.604 1.65
5 1.964 2.02
4 2.629 2.70
3 5.270 5.42
2 32.660 33.58
1 0 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“T e,n − Tin
”3=
Q3
m cp
g∗(n− 3)
n g∗(n− 3)“
T e,n − Tin
”3
(C)
8 1.448 1.76
7 1.689 2.05
6 2.068 2.51
5 2.768 3.36
4 5.547 6.74
3 34.378 41.77
2 0 0
1 0 0
Table 5: Temperature rise due to heat dissipated in row 3.
Convection from PCBs page 55
Example Calculation (7)
n T e,n − Tin (C)
8 57.6
7 72.2
6 54.9
5 30.8
4 18.2
3 50.3
2 39.8
1 38.6
Table 6: Total temperature rise for modules.
Convection from PCBs page 56
Example Calculation (8)
1 2 3 4 5 6 7 80
10
20
30
40
50
60
70
80
90
100
row number
Tem
pera
ture
(C
)
Convection from PCBs page 57
References
[1] A. M. Anderson and R. J. Moffat. The adiabatic heat transfer coefficient and the superposition kernel function: Part 1–datafor 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 rectangularmodules encountered in electronic equipment. International Journal of Heat and Mass Transfer, 25(7):961–973, 1982.
Convection from PCBs page 58