use of superposition to describe heat transfer from...

Use of Superpositionto 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 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


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


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


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


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


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


Mixed Convection

Buoyancy effects can be present in a forced convection flow


Recirculation in Plan View

device with heat sink


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


Thermal Wakes (1)

Thermal wake for a flush heat source


Thermal Wakes (2)

Three-dimensional representation of a wake, T (x, y)


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


By-pass and Array Flow (1)



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.


By-pass and Array Flow (2)



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.


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


Example: Fan-Cooled Enclosure (1)

power supplydisk drive


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


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.


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


——————-


Heat Transfer Correlations for Board-Level Analysis

m1•

h, Ta


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:


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]


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


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.


CFD Modeling (1)

m1•


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/


Experimental Data

• flush mounted heaters

• Ribs

• Arrays of blocks

• arrays of “heater” devices


Correlations

Flow over a flat plate

Nu = C ReaPr

b

Which length scales to use?

Proper application requires

• geometric similarity

• dynamic similarity

• thermal similarity


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


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


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)


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)


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


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


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.



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



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)



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.



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.


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



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.



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.



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


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



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

¯



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



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


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



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



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.



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.



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




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




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.



1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

90

100

row number

Tem

pera

ture

(C

)


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.


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