DESIGN GUIDE TO ELECTRONIC COOLING
A Project
Presented to the faculty of the Department of Mechanical Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Mechanical Engineering
(Thermal and Fluid Design)
by
Timothy Michael Rau
SUMMER
2013
© 2013
Timothy Michael Rau
ALL RIGHTS RESERVED
ii
DESIGN GUIDE TO ELECTRONIC COOLING
A Project
by
Timothy Michael Rau
Approved by:
__________________________________, Committee Chair
Dr. Dongmei Zhou
__________________________________, Second Reader
Matt Neumann
____________________________
Date
iii
Student: Timothy Michael Rau
I certify that this student has met the requirements for format contained in the University format
manual, and that this project is suitable for shelving in the Library and credit is to be awarded for
the project.
__________________________, Department Chair
Dr. Susan Holl
Department of Mechanical Engineering
iv
___________________
Date
Abstract
of
DESIGN GUIDE TO ELECTRONIC COOLING
by
Timothy Michael Rau
The ability to identify heat sources and predict their temperatures across a variety of operating
conditions is key in the design of a reliable electronic system such as a computer, a server, or any
other system containing a printed circuit board assembly (PCA). The goal of this thesis is to
explore a process on how one should approach thermal design to improve the probability to
design a reliable electronic system. During the process, one should be able to define system
requirements, understand operational environment conditions, choose an appropriate cooling
method, and understand how the system will behave with these defined constraints. Using these
constraints, one can perform an initial study with hand calculations to help determine a high-level
feasibility of the design. Computational Fluid Dynamics (CFD) can then be used to create a more
complex study that will provide detailed results. Finally, a thermal mock-up should be built to
compare experimental results to CFD results or to check how well the empirical data correlates to
the theoretical data. Improvements to the CFD model can increase the accuracy of the correlated
data and increase the confidence of results for future optimization studies.
A natural convection study of a heat sink was used to demonstrate this thermal design process.
Multiple fin heights of a heat sink at different ambient temperatures were analyzed by hand
v
calculations, CFD simulations were performed using ANSYS Icepak 14.0, and a mock-up was
built using a copper slug with a Minco heater attached to simulate a processor on a PCA.
Results from this experiment demonstrate how by using known correlated data to future
investigations one can improve the accuracy of results. That is why it is always a benefit to assist
CFD simulations with empirical data. Empirical data helps improve the accuracy of a CFD
model and adds credibility to the theoretical results. This allows for improved optimization
studies and helps reduce design cycle time.
, Committee Chair
Dr. Dongmei Zhou
______________________
Date
vi
ACKNOWLEDGMENTS
I would like to thank Dr. Dongmei Zhou for her help and guidance in the completion of my
thesis. I would also like to thank everyone at California State University of Sacramento who
helped in the journey to completing this degree. Most of all, I would like to thank my family for
all of their help and support.
vii
TABLE OF CONTENTS
Page
Acknowledgments................................................................................................................... vii
List of Tables ............................................................................................................................ x
List of Figures .......................................................................................................................... xi
Chapter
1. MOTIVATION FOR STUDY ………………………………………………………….. 1
1.1 Introduction............................................................................................................ 1
2. SYSTEM DEFINTION ...................................................................................................... 3
2.1 System Requirements ............................................................................................ 3
2.2 Environmental ........................................................................................................ 6
2.3 Fans ...................................................................................................................... 10
2.4 Fan Speed Control Methods................................................................................. 12
2.5 CPU...................................................................................................................... 14
2.6 Heat Sinks ............................................................................................................ 17
3. ANALYSIS ....................................................................................................................... 19
3.1 Analysis Introduction ........................................................................................... 19
3.2 Initial Calculations ............................................................................................... 20
3.3 CFD Simulations.................................................................................................. 30
3.4 Mock-up/Prototype .............................................................................................. 32
3.4.1 Mock-up/Prototype Construction................................................................. 32
3.4.2 Mock-up Testing .......................................................................................... 35
4. EXAMPLE PROBLEM .................................................................................................... 39
4.1 Example Problem Statement ................................................................................ 39
viii
4.2 Initial Calculations ............................................................................................... 40
4.3 CFD...................................................................................................................... 43
4.4 Mock-up ............................................................................................................... 44
4.5 Iteration Results ................................................................................................... 47
4.5.1 Initial Calculation Iteration .......................................................................... 47
4.5.2 CFD Iteration ............................................................................................... 48
4.5.3 Mock-up Iteration ........................................................................................ 54
5. CONCLUSIONS AND FUTURE WORK ....................................................................... 58
5.1 Conclusion ........................................................................................................... 58
5.2 Future Work ......................................................................................................... 58
Appendix A Formulas ............................................................................................................. 59
Appendix B Nomenclature ..................................................................................................... 61
REFERENCES ...................................................................................................................... 62
ix
LIST OF TABLES
Tables
Page
1.
Table 2.1 Altitude Multiplier ……………… .....………….………………………...... 8
2.
Table 4.1 Dimensions of 23mm heat sink ........………….………………………….. 41
3.
Table 4.2 Initial heat sink calculations .............……….…………………………….. 43
4.
Table 4.3 Initial CFD results .............................………….………………………...... 43
5.
Table 4.4 Initial heat sink results ......................………….………………………….. 46
6.
Table 4.5 Altitude multiplier assessment ..........……….…………………………….. 47
7.
Table 4.6 New heat transfer coefficient ............………….………………………...... 48
8.
Table 4.7 Multiplier for orientation calculation ………….………………………….. 48
9.
Table 4.8 Second iteration of CFD results ........……….…………………………….. 53
10.
Table 4.9 23mm heat sink results .....................………….………………………...... 54
11.
Table 4.10 19.75mm heat sink results ..............………….………………………….. 55
12.
Table 4.11 16.45mm heat sink results ..............……….…………………………….. 56
x
LIST OF FIGURES
Figures
Page
1.
Figure 2.1 Example of a First Order Flow Network ... .………………………………. 4
2.
Figure 2.2 Simplified System .......................................... ……………………………. 6
3.
Figure 2.3 Thermal conductivity of moist air .....………….…………………………. 8
4.
Figure 2.4 ASHRAE allowable operating region ............... …………………………. 9
5.
Figure 2.5 Axial fan vs. Centrifugal blower ............. .………………………………. 10
6.
Figure 2.6 Series vs. Parallel.......................................... ……………………………. 11
7.
Figure 2.7 System Operating Point ........………….…………………………………. 12
8.
Figure 2.8 Leakage current graph ...................................... …………………………. 15
9.
Figure 2.9 Chip heated to 50% leakage loss ............. .………………………………. 16
10.
Figure 2.10 Collection of heat sinks extruded, bonded, and cast ....... ……………….18
11.
Figure 2.11 Folded fin heat sink with heat pipes .......………………………………. 18
12.
Figure 3.1 Example problem radiation .............................. …………………………. 22
13.
Figure 3.2 Example of a Thermal Resistance Network .... .…………………………. 24
14.
Figure 3.3 Example problem Resistances ...................... ……………………………. 24
15.
Figure 3.4 Example problem LMTD .....………….…………………………………. 26
16.
Figure 3.5 Example problem ε-NTU .................................. ………………………….27
17.
Figure 3.6 Example problem Bernoulli equation ...... .………………………………. 29
18.
Figure 4.1 Top right = Vertical, bottom left = Horizontal, bottom right = Flat ..... …. 40
19.
Figure 4.2 Convective heat transfer chart ..................... ……………………………. 42
20.
Figure 4.3 First CFD simulation ........................................ …………………………. 44
21.
Figure 4.4 Built Mock-up .............................................. ……………………………. 45
xi
22.
Figure 4.5 Mock-up with screen captures of Agilent VEE …………………………. 46
23.
Figure 4.6 CFD mesh study ........................................... ……………………………. 50
24.
Figure 4.7 CFD grid refinement study ........................... ……………………………. 52
xii
1
Chapter 1
MOTIVATION FOR STUDY
1.1 INTRODUCTION
The purpose of this paper is to investigate and develop a methodology in thermal design
for printed circuit board assemblies (PCA’s). The research will begin by focusing on system
definition and design. It will cover what the major components that typically need thermal design
attention are and environmental factors that affect the ability to cool the components. Fans and
other methods of cooling will be described along with some preferred design practices on how to
structure or layout your system or sub-system. After gaining an understanding of the system
definition, it is important to describe methods for analysis and testing. This section will describe
some important hand calculations in proving early feasibilities of design proposals followed by
the importance of using theoretical data with empirical results or the use of computational fluid
dynamics (CFD) compared with a tested mock-up. An example problem will then be performed
following the methodology established from the research described. The example problem will
be a heat sink investigation that will be defined, designed, tested, and analyzed. Hand
calculations will be performed, a CFD analysis will be completed, and the results will be
compared to real data from a physical mock-up. Comparing results from the three calculations
will then allow the methodology to be evaluated for accuracy. Results or assumptions that return
less than desired results can then be modified for improved results. These learning’s or gained
experience is what I am personally wanting to learn for future investigations as I believe that this
methodology will be able to be applied to larger scaled projects. It will also be able to be applied
to investigations with different cooling technologies. The formulas and details may change, but
2
the methodology should remain reliable. The ultimate goal of this paper is to establish a process
resulting in a reliable system with results representing an end user experience.
3
Chapter 2
SYSTEM DEFINTION
2.1 SYSTEM REQUIREMENTS
Thermal system definition begins by identifying the major power consumers that need
thermal attention. These components typically are CPUs, power converters, logic controllers,
memory, storage, and IO cards. The importance of maintaining thermal limits of these
components is to allow for flawless performance and reliability of the components. Since these
power consumers are semi-conductors and silicon based they suffer from leakage current at
higher temperatures. Leakage current is when the charge tunnels through an insulting layer or
leaks out of the intended circuit causing more power to be consumed than originally intended. At
higher temperatures, this occurs more rapidly decreasing reliability with the potential to cause a
complete circuit failure. Maintaining these components at intended temperatures makes for a
reliable system and begins with the way the system is architected or designed.
Knowing the heat sources and their maximum temperatures is important, but the way the
components are positioned can help maintain intended temperatures. Understanding what
components are positioned in series or of each in terms of airflow will help determine if there
may be difficulties in cooling some components. An example of this can be described with the
help of Figure 2.1 that displays a flow network. Looking at air section 1 you can see inlet air
entering the enclosure and passing a bank of hard drives or storage. After the air passes the
storage, it splits across CPU’s and memory. The air from the memory leaves the enclosure, but
the air from the CPU’s continues across another logic controller and then IO cards before leaving
the enclosure. At every node in this network there is a temperature delta from its previous node
in the network. This means that there is an inlet temperature before the storage that then
4
increases the temperature of the air as it passes the storage produces a resulting temperature at the
next node. The heat being produced by the component and the amount of airflow passing it
determines this temperature change. As the temperature rises or preheats the air through the
system it can make it a challenge to cool the IO cards, which are at the end of a series of
components. So even though the system may operate with a cool inlet air temperature, the air
will pass over and be heated by storage, CPU’s, and a logic controller before it ever begins to
cool the IO cards. This preheated air has to potential to be at a temperature and airflow that will
not be able to cool the IO cards to an acceptable temperature. This is why it important to
understand the airflow path of a system and how the cooling of one component could affect
another.
Figure 2.1 Example of a First Order Flow Network
5
The layout of a system must not only consider temperatures rises through the system but
pressure drops or system resistances. System resistances affect airflow and if the system is not
arranged properly, it could cause the air to flow though unintended areas of the system. This may
prevent heat sinks from experiencing the proper amount of airflow to keep the component cool.
Factors that affect system resistance are friction, blockage, sharp turns, and expansion or
contraction of the fluid. Anytime a fluid passes through an area of expansion or contraction the
mass flow rate will have a change in velocity with respect to the change of cross sectional area.
This can occur with the positioning of components or a blockage. Grills or perforated sections of
metal will also have this effect on fluid flow, as will any sharp changes in direction of the flow.
A heat sink can cause this effect when improperly designed or configured in a system. A heat
sink designed with a high-pressure drop can decrease the amount of air that passes through it if
there are bypass regions nearby with a lower resistance to air flow. In an instance such as this,
baffling, or blockage, would most likely need to be created to allow the heat sink to be a lower
pressure region than the bypass area. This is why it is a balance in the design of a system.
Creating a very dense heat sink or one with a tight fin pitch may actually have to high of a flow
resistance that could prevent from air passing through it when installed in a system making it
useless in cooling the component. This can also potentially occur for an example when arranging
a CPU heat sink parallel to a bank of memory as seen in Figure 2.2. It then becomes important to
balance the resistances of the components in each path of the network so they will have the
necessary amount flow to remain cool.
6
Figure 2.2 Simplified System
2.2 ENVIRONMENTAL
Understanding the environmental conditions the system will be exposed to is an
important factor in defining the system for a better design. Some of the important factors to
understand are the ambient temperatures the system will function in, altitudes it will operate at,
and relative humidity or wet-bulb conditions it will be exposed to. The range of these
environmental conditions the system will need to operate in will define the design in terms of
robustness or what engineering design decisions will need to be made to achieve them.
The ambient temperature range the system is designed to help determine what
engineering decisions are made or how it is cooled. If the system is to be able to function in a
high ambient environment, the necessary cooling will need to be engineered into the system to
handle that environment. This can be challenging when there may not be a high temperature
delta between the ambient temperature and the maximum temperature of a component. Different
7
cooling technologies may need to be explored which could add cost to the design. A low ambient
temperature could also cause concern to where some components may need to be insulted to
maintain the design intent.
Altitude is important to thermal design since air density changes with respect to elevation
change. Essentially, air becomes less dense as you go up in altitude reducing its convective
capability. The electronics will therefore experience greater component temperature rises at
higher altitudes even while operating at constant power outputs. To account for this change in air
density, extra margin needs to be designed into the system or the product needs to have operating
specifications stating where and how it can operate. Since designing margin can be expensive,
many server specifications state an inlet ambient maximum that adjusts with altitude. For every
1000 ft. above sea level that the server operates at, the maximum inlet ambient temperature is
reduced by 1C. A product that is rated to operate at a maximum inlet ambient temperature of 45
C at sea level would then have a maximum inlet temperature of 35 C at 10,000 ft. This is a
simple correction method, but PCBs can be complicated systems with variable power
components, complex flow patterns, and variable flow. These complexities can make it difficult
to calculate changes from altitude, hence the simple conservative correction formula. A second
correction formula uses multipliers depending on how the component is cooled. This formula is
not as conservative as the first correction formula, but it still is a simple correction calculation
and is stated in Table 2.1.
8
Altitude
m(ft)
Multiplier
Fan-Cooled
Fan-Cooled
(General)
(High Power)
0
1.00
1.00
1,500 (5,000) 1.20
1.16
3,000 (10,000) 1.45
1.35
4,500 (15,000) 1.77
1.58
6,000 (20,000) 2.18
1.86
Table 2.1 Altitude Multiplier [8]
Naturally
Cooled
1.00
1.10
1.21
1.33
1.48
Humidity does not have a major effect on thermal performance. Looking at Figure 2.3
shows how the thermal conductivity of air changes with respect to humidity at different
temperatures. Most PCA’s operate below 50C ambient air environments. At that temperature,
the thermal conductivity of air varies by approximately 0.001 W/m*K from zero percent humidity
to one hundred percent humidity. This calculates that a 100W heat source would vary only 1C
between extreme humidity’s at 50C. Humidity is not considered an important variable unless you
need this extra amount of accuracy.
Figure 2.3 Thermal conductivity of moist air [9]
9
Humidity is important to pay attention to when there are pressure changes within the
system. As air changes direction or hits resistance/blockages, it can create condensation. The
condensation has the potential to accumulate within a system and can create electrical problems
causing a system to fail. It is important to specify an operating environment to prevent this from
occurring. ASHRAE has an operating region recommended for data centers as seen in Figure
2.4. The chart defines an operating region to help prevent condensation from occurring within a
system. The safety of preventing condensation is where humidity does need some attention.
Figure 2.4 ASHRAE allowable operating region [10]
10
2.3 FANS
Having a thermal and system flow network created, fans can be chosen. There are two
types of fans typically chosen for electronics cooling, centrifugal blowers (high pressure, low
flow) or axial fans (low pressure, high flow) with their fans shown in Figure2.5.
Figure 2.5 Axial fan vs. Centrifugal blower
Fans can also adjust their pressure flow curves by arranging them in series or parallel
configuration in a system. Arranging the fans parallel to each other maintains the maximum
static pressure but increases the total flow. A serial configuration of the fans increases the
maximum static pressure but maintains its total flow. These fan curves can be seen in Figure 2.6.
11
Figure 2.6 Series vs. Parallel
Having the mass flow rate required to cool a system and either an estimate or measured value of
pressure drop helps determine the fans for the system. Using fan curves with a system curve help
determine if the fans will be able to cool the system. As the demand of flow changes with respect
to the change in wattage of the system, you can see what the operating range the system will be at
along the fan curve as seen in Figure 2.7. This allows you to determine if the fan will be
adequate to cool the system and see if you need to specify a more powerful and expensive fan or
can find something less powerful and expensive. Checking flow requirements at different
altitudes the product will be used at should also be done to make sure the fans will be able to
supply enough flow as the density of the air changes.
12
Figure 2.7 System Operating Point
2.4 FAN SPEED CONTROL METHODS
There a couple different methods to control DC fan speed. The most common and
simplest method is to drive the fan with a set DC voltage. This type of DC fan is one of the
cheaper and simple fans manufactured. A power voltage control fan has an operating range of
voltage’s it can operate at. The high end of this range provides the most airflow the fan can
reliably produce, but it is also typically the loudest the fan operates at and could create acoustical
concerns for the system. The lower end of the range has not only a minimum operating voltage
but also a minimum start-up voltage. The minimum start-up voltage is the lowest voltage the fan
is guaranteed to start from a cold start. The minimum operating voltage is not always the same as
the minimum start-up voltage allowing the system to start the fan and then reduce the voltage
13
lower reducing the speed of the fan. Switching set voltages like this is one of the simplest forms
of fan control.
PWM (pulse width modulation) fans are another method of fan control. A PWM fan is
typically controlled in one of two ways. The first method is by pulsing the power to the fan to
change its speed. This method is not the preferred approach since it can stress components of the
fan reducing its reliability or leading to a premature failure. This is similar to the first method
describe in which it can operate on a standard DC fan.
The second method to fan control using a PWM fan supplies a constant voltage with an
additional signal line for the PWM signal. This method requires a special fan to handle the PWM
signal that is supplied by a square wave to drive the fan speed or percent duty cycle that the fan
operates at. This is the preferred PWM method and typically is the method referred to when
discussing PWM controlled fans.
PWM has some important advantages over the other methods of fan control. One of its
most distinct advantages is its ability to adjust its fan speed from the PWM signal. This allows
fan speeds to be adjusted to optimize cooling needs of components while minimizing the power
draw of a system. For higher-powered systems, this feature is crucial as to not waste power for
the system in an attempt to meet efficiency rewards such as Energy Star. The difference in power
consumption between a high usage condition to an idle state can be over 100 Watts just in the
power draw from the fans to cool the components for these two states. For low-power systems,
this control can be important for power but also acoustics since these systems are more likely to
be used in an office environment or close to the end user. Instead of having a hard switch of
power or duty cycle, the change can be designed to occur over set amount of time. Slowly
increasing or decreasing fan speed can make the acoustics more pleasing to the ear rather than
having a hard start or stop. The last benefit to discuss is the flexibility of this design approach. If
14
it is learned that there needs to be a change to fan speeds late in a program or the design cycle
process, the change can be made relatively cheap and easy with a change to firmware or the code
that controls the fan speed. It would be more expensive and disruptive to the design schedule if
the change needed to be made with hardware. This design flexibility is a benefit in using PWM
fans.
2.5 CPU
The CPU is typically the hottest component on a PCA and needs the most attention to
maintain its operating requirements. It is important to understand how the CPU functions in order
to be able to design a better cooling solution for it. Many CPU’s have the ability to reduce its
power or throttle to save energy within a system during periods of low activity. A CPU can save
power by changing voltage supplied to the CPU, frequency it operates at, or a variety of other
methods. However the CPU may throttle its power, it is important to understand when and how it
will accomplish this so the cooling solution can be designed accordingly. If one did not
understand how and when a CPU would throttle, you may supply too much cooling potentially
wasting energy and designing an acoustically loud system if using fans. The CPU being used
may not have the ability to throttle and sufficient cooling will need to be provided even in periods
of low activity when one would typically reduce cooling. This could lead to a dangerous scenario
called thermal runaway, where an ASIC continues to heat up until it fails, due the susceptibility
CPU’s have to leakage current.
Leakage current, as defined earlier, is when the electric charge tunnels through an
insulting layer or leaks out of the intended circuit causing more power to be consumed than
originally intended. This occurs progressively more as temperature increases reducing reliability
and life of the CPU. Figure 2.8 and Figure 2.9 illustrates this event occurring on two different
15
ASIC’s that I was able to test. In Figure 2.8, you can see the case temperature of this chip rising
from 38 C to its maximum case temperature of 70 C while operating at full speed with a constant
airflow. The increase in power over the change in case temperature is its leakage current that is
around 15W. This means that of the total 130W, only 15W (11%) is lost to leakage current. You
can also observe in the results that the case temperature increases by 32 C that is essentially the
equivalent of the 35C increase of the ambient temperature.
Figure 2.8 Leakage current graph
Figure 2.9 shows similar results from a second ASIC. At lower case temperatures, the power
draw of the ASIC is constant. It is not until approximately 50C case temperature with a 76.5 watt
power draw that you begin to see an increase in power draw relative to the increase in case
temperature. This ASIC was tested a point at which 50% of the power draw is due to losses from
leakage current. Figure 2.8’s maximum case temperature is 70C while Figure 2.9’s is 80C.
16
Figure 2.9 Chip heated to 50% leakage loss
What is interesting about looking at these two plots is that the location of the maximum case
temperatures relative to the point of inflection of the power curve or the point in which the rise
becomes greater than the run. This point seems to be the ideal point to set the limit since it is the
point at which any increase in temperature would cause more power to be consumed than the
increase in temperature. It also aligns with a junction temperature of the silicon around 100C,
which is around its reliability limit.
The other importance to consider for this topic is how you design your system for the
leakage losses in your processors compared to the leakage losses in the fans cooling the product.
The power losses in the fans are fairly well known and documented; the losses in the processors
are not. Depending on your system, the losses in the CPU(s) could be greater than the losses
from your fan(s). A more efficient cooling solution may be needed in a situation such as this.
17
Most people believe that the losses in the fans are far greater than the losses in the processors but
this is still a scenario that should be tested and understood.
2.6 HEAT SINKS
Heat sinks are made in a variety shapes, sizes, materials, and technologies. They are
typically made with a material with a high thermal conductivity such as copper or aluminum.
The high thermal conductivity allows the heat to spread more easily through the surface area of
the heat sink. System constraints, such as physical space available and power to be cooled, drive
the shape and size. Some common manufacturing methods are extrusion, casting, bonded fin, and
folded fin heat sinks that can be seen in Figure 2.10 and Figure 2.11. These are some of the
more simple heat sinks or less expensive. As system constraints make the cooling solution more
difficult to achieve, more complicated or expensive heat sinks are used and continue to be
developed. Some examples of these technologies are heat pipes, liquid cooling, phase change
cooling, heat exchangers, and thermoelectric coolers. Most of these technologies described take
advantage of liquid cooling in one way or another. A thermoelectric cooler, however, works on
the principle of the Peltier effect that creates a temperature difference when a voltage is applied to
the free ends of two dissimilar materials. Some of these advanced cooling solutions are fun to
learn about but typically have cost challenges when trying to implement them. The heat sink that
meets system and reliability requirements for the least cost usually is the chosen solution.
18
Figure 2.10 Collection of heat sinks extruded, bonded, and cast [12]
Figure 2.11 Folded fin heat sink with heat pipes [13]
19
Chapter 3
ANALYSIS
3.1 ANALYSIS INTRODUCTION
Printed circuit board analysis is accomplished by using initial calculations, computational
fluid dynamics, and the testing of a mocked up system. These three approaches have different
benefits in terms of ease and speed to collect the data and when in the design cycle they should be
performed. The first approach is initial calculations, which is an early analysis for quick simple
results. The formulas used in this approach are oversimplified for speed and ease of early
feasibility studies. The primary use of these calculations is for one and two dimension analysis of
components and thermal and flow networks. Assumptions for these problems typically are
uniform heat sources, constant flow, and uniform heat spreading. The purpose of these
calculations is to be very simple and quick feasibility calculations even though it may limit the
accuracy of the results.
Computational fluid dynamics (CFD) is used for three-dimensional analysis of
components or systems. Since these calculations are performed by a computer, partial differential
equations of fluid mechanics and heat transfer are calculated for the conservation of momentum,
mass, and energy. Depending on the complexity of the model being analyzed, CFD simulations
may take minutes to days. The advantage of using CFD is the speed in which these calculations
are calculated. It also allows one to visualize temperature and velocity gradients within the fluid
or component that the initial calculations simplified to be constant. It is important to note that
this does not necessarily mean the accuracy of the results has improved. The results are only as
good as the model created which still contains assumptions for simplification. Many of the
assumptions made may be due to the limitations of the software, since many cannot handle
20
complex geometries. The software is getting better to handle these geometries, but a simple
model with a low grid count can save calculation time. That is why it is important to correlate the
theoretical data to empirical data.
The testing of a mock-up and prototype is the last analysis that should be performed to
help achieve a successful final system validation. A mock-up is a simplified construction of a
system. They can be constructed of salvaged predecessor components, heaters, simplified
sheetmetal, and even cardboard. The purpose of them is to get preliminary real world results
before spending the money or time to get prototypes made. They allow for quicker real world
feasibility studies and iterations saving potential costly fixes or prototype revisions.
3.2 INITIAL CALCULATIONS
The first formula to discuss is for thermal resistance. This is the primary formula used
for hand calculations of a simplified physical construction of a computer system. Knowing the
power of heat sources and thermal limits of components allows thermal resistances to be
calculated for heat sinks, thermal interface materials (TIMs), or any other resistive component.
The total thermal resistance is the sum of the resistance values for three the modes of heat transfer
(convection, conduction, and thermal radiation).
𝑅1−2 =
𝑇1 − 𝑇2
, °𝐶/𝑊
𝑄
𝑘𝐴
(𝑇𝐿
𝐿
Conduction (1-d)
𝑄𝑥 = −
Convection
𝑄 = ℎ𝐴(𝑇𝑤 − 𝑇𝑓 )
Radiation
𝑄𝑛𝑒𝑡 =
− 𝑇𝑂 )
𝐴𝜀𝜎(𝑇𝑤4
−
(𝑇𝐿 −𝑇𝑂 )
𝐿
, 𝑅𝑐𝑜𝑛𝑑 = 𝑘𝐴
𝑅𝑐𝑜𝑛𝑑
(𝑇𝑤 −𝑇𝑓 )
1
, 𝑅𝑐𝑜𝑛𝑣 = ℎ𝐴
𝑅
or
𝑄𝑥 = −
or
𝑄=
4
𝑇𝑠𝑢𝑟
) or
𝑐𝑜𝑛𝑣
(𝑇𝑤 −𝑇𝑠𝑢𝑟 )
,
𝑅𝑟𝑎𝑑
1
2 +𝑇 2
𝐴𝜀𝜎(𝑇𝑤 +𝑇𝑎𝑚𝑏 )(𝑇𝑤
𝑎𝑚𝑏 )
𝑄𝑟𝑎𝑑 = −
𝑅𝑟𝑎𝑑 =
21
Electric Circuit Analogy
𝑉
𝐼=𝑅
𝑄=
∆𝑇
𝑅
𝑅𝑡𝑜𝑡 = 𝑅1 + 𝑅2
𝑅𝑡𝑜𝑡 = 𝑅𝑐𝑜𝑛𝑑 + (𝑅
1
𝑐𝑜𝑛𝑣
+𝑅
1
𝑟𝑎𝑑
)−1
Thermal resistance values can follow a basic electric circuit. Typical assumptions used in
this analogy between an electric circuit and thermal resistance are steady state and 1-d heat
transfer that can be used to describe 3-d effects. Ohm’s Law and thermal resistance translate very
well with current and power being interchangeable while voltage compares with change in
temperature, and resistance with thermal resistance. Thermal resistance can be found for a
component with the summation of resistance values for the three modes of heat transfer. A
thermal network can now be created to represent an electronics system with simple hand
calculations. In some instances a transient analysis needs to be added to the network. It can be
represented in the network by thermal capacitance. Situations in which this is important is the
time needed for a system to find a thermal equilibrium as it either heats up or cools down. It can
also help with calculations for fatigue due to thermal cycling or time allowed to hot swap
components before access panels need to be closed to maintain the thermal feasibility. Creating a
thermal network is good for quick feasibility calculations and first approximations of conductive
problems with simple boundary conditions.
The first mode of heat transfer covered is radiation. “Radiation is important to consider
when there is little air movement around the heat source but may be insignificant when forced
circulation is present.” [1] When this quote is applied to a typical electronics system, radiation is
not a major factor to consider. Most electronics systems have forced conduction and make the
radiation an insignificant contributor to the thermal solution. The only time radiation exchange
becomes important is if you were designing a solution without forced air circulation or a solution
22
that have components close together, very hot, and large surface areas. A radiation calculation
seen in Figure 3.1 demonstrates the relatively small amount of wattage.
Figure 3.1 Example problem radiation
Convection is the next mode of heat transfer covered. There are two primary categories
of convection: forced convection and buoyancy induced convection. Forced convection can be
further divided into external flow and internal flow. Some general guidelines for determining the
type of flow is through the use of the Reynolds number. Laminar flow is defined as having a
Reynolds number is less than 2000 for internal flow and 2 x 10^5 for external flow. The
Reynolds number for turbulent flow is greater than 4000 for internal flow and 5 x 10^5 for
external flow. The Grashof number is used for natural convection problems by replacing the
inertia force values in the Reynolds number with buoyancy force. Other dimensionless numbers
used in calculating thermal boundary layers are Nusselt and Prandtl number. Typically these
23
values are not used very often for quick calculations because the geometry of the problem tends
to be to complex. One problem where it is possible to use these values is in heat sink analyses
that are typically simple open channels. Nusselt values help determine the heat transfer that
occurs in an open channel allowing you to then calculate an overall resistance value of a heat sink
with its total number of channels. This also assumes that there is no flow bypass and uniform
temperature through the heat sink. Buoyancy induced convection is not used for forced
convection calculations since the forced airflow inertia forces overcomes the buoyancy forces. It
is important for natural convection calculations where the buoyancy force of the heated air creates
the air movement. The direction of gravity is important for buoyancy induced convection
problems. The reason for this is that an upward facing heat source (or against gravity) provides
twice the heat transfer of a downward facing plate when properly ventilated.
gβ𝐿3 (𝑇𝑤 −𝑇∞ )
𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒
=
𝑣3
𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒
𝜌v𝐿
v𝐿
𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒
= 𝑣 = 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒
𝜇
Reynolds Number 𝑅𝑒 =
Grashof Number 𝐺𝑟 =
Nusselt Number 𝑁𝑢 = 𝑘
Prandtl Number 𝑃𝑟 =
ℎ𝐿
ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑏𝑦 𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛
𝑓𝑙𝑢𝑖𝑑
𝑐𝑝 𝜇
𝑘𝑓𝑙𝑢𝑖𝑑
= ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑏𝑦 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
=
𝑣
𝛼
=
𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 ℎ𝑒𝑎𝑡
The primary equation for conductive heat transfer is the Fourier equation. This equation
leads to the definition of thermal resistance. Thermal resistance values allow for a complex heat
flow path to be simplified down to a series and parallel network of 1-d heat paths similar to
Figure 3.2. An excellent first order estimate can be easily achieved with a second assumption of
uniform heat spreading. Later as empirical data is collected, a correction value can be added for
spreading and contact resistances for better correlation to the theoretical and future calculations.
An example resistance calculation can be seen in Figure 3.3.
24
Figure 3.2 Example of a Thermal Resistance Network
Figure 3.3 Example problem resistances
25
Once the methods to calculate overall resistance are known, the initial calculations are
simple and good first order estimates. You can add more analysis to get a better approximation
though. Some of the extra steps you can do are to calculate fin efficiency of the heat sinks and
adjust for conductive resistance. This will give a more accurate approximation for a heat sink.
Another analysis that can be done is the use of Log Mean Temperature Difference (LMTD)
which an example can be seen in Figure 3.4. This method accounts for changes in temperature
along the length of the heat exchanger whether it is a heat sink to air or a liquid-air heat
exchanger for analysis on a liquid loop. The last method to discussed is the use of ε-NTU. This
method is primarily used for liquid-air heat exchange which an example is worked out in
Figure3.5. This heat exchange effectiveness is actually the ratio of actual heat transfer to the
maximum possible rate of heat transfer. Determining this value can be elaborate, but may
become necessary in liquid loops.
𝑚𝑐𝑝 (𝑇𝑐,𝑜𝑢𝑡 − 𝑇𝑐,𝑖𝑛 ) = ℎ𝐴∆𝑇𝑙𝑚𝑡𝑑
∆𝑇𝑙𝑚𝑡𝑑 =
(𝑇ℎ,𝑜𝑢𝑡 − 𝑇𝑐,𝑜𝑢𝑡 ) − (𝑇ℎ,𝑖𝑛 − 𝑇𝑐,𝑖𝑛 )
𝑙𝑛
(𝑇ℎ,𝑜𝑢𝑡 − 𝑇𝑐,𝑜𝑢𝑡 )
(𝑇ℎ,𝑖𝑛 − 𝑇𝑐,𝑖𝑛 )
∆𝑇𝑙𝑚𝑡𝑑 = (∆𝑇𝑜𝑢𝑡 − ∆𝑇𝑖𝑛 )⁄(𝑙𝑛(∆𝑇𝑜𝑢𝑡 − ∆𝑇𝑖𝑛 ))
𝑄𝑎𝑐𝑡𝑢𝑎𝑙 = 𝜀𝑄𝑚𝑎𝑥
26
Figure 3.4 Example problem LMTD
The best way to use these formulas is to create an excel spreadsheet. Creating an excel
spreadsheet setup for variable geometry of a heat sink allows one to quickly optimize the design
of the heat sink. Variables such as fin count, fin height which changes the surface area, air flow,
and material constant values should be included in the spreadsheet for evaluating design changes.
Inputting the values of these variables can quickly allow one to analyze a design by all three
approaches (Sum of Resistances, LMTD, and ε-NTU) discussed so far. In simple cases, all three
approaches will return similar if not the same result as seen in Figure 3.3, Figure 3.4, and Figure
3.5. It is still important to understand the basics of the LMTD and ε-NTU method to better
understand heat exchanger analysis though.
27
Figure 3.5 Example problem ε-NTU
Energy and the Bernoulli Equation is the start point for discussions on fluid flow, head
loss, and flow resistance. Most energy balance problems can be solved with the Bernoulli
equation whether it is frictional or frictionless. The Bernoulli equation is the sum of mechanical
energy at a point in a system that can be used to calculate change in energy or pressure at 2 points
in the system finding a head loss. Head loss is an irreversible loss of mechanical energy to
friction (major) losses and flow separation (minor losses). The Kinetic energy coefficient (α) can
also be used to find head loss with a value of two for fully developed laminar flow and
approximately one for fully developed turbulent flow. For calculations across developing flow,
α1 would not equal α2 because the kinetic energy is changing between the two points. This is
28
usually insignificant compared to friction and minor losses and can be ignored. An example
problem is demonstrated in Figure 3.6.
Bernoulli Equation
𝑃+
1
2
𝜌v 2 + 𝜌𝑔ℎ = 𝐸
Potential energy + Kinetic energy + Gravitational energy = total energy
Static pressure + Dynamic pressure + Gravitational pressure = total pressure
Electric Circuit Analogy for pressure
𝑉
𝐼=𝑅
v𝐴 = 𝑉 2̇ =
𝑅𝑡𝑜𝑡 = 𝑅1 + 𝑅2
𝑙 𝜌
𝜁𝜌
𝑅 = 𝑓 𝑑 2𝐴2 = 2𝐴2
∆𝑃
𝑅
√𝑅𝑡𝑜𝑡 = (
1
√𝑅1
+
1
√𝑅2
)−1
29
Figure 3.6 Example problem Bernoulli equation
30
The importance of the Bernoulli equation is allows loss coefficients to be represented by
resistive elements. This goes back to the electric circuit analogy and thermal resistance which
now can also represent head loss. Volumetric flow resembles current and change in pressure
represents voltage. Resistance values for head loss resemble resistance and acts the same in a
series layout. For a parallel layout, the square roots of all resistance values are performed. Now
is behaves the same as a standard parallel circuit. One can now calculate flow resistance values
across heat sinks, through perforated plates, a sudden expansion or contraction, sharp bends, or
anything else that may create a head loss. A circuit of all flow resistances can be created to
represent the airflow through a system to know where air movement would need to be improved
or flow bypasses adjusted.
3.3 CFD SIMULATIONS
Computational Fluid Dynamics (CFD) is an important tool to quickly perform thermal
investigations. It can quickly calculate theoretical results for temperature, linear and volumetric
airflow, flow visualization, and pressure drop across a defined area in an air path. Configurations
simulated should match the final product. Knowing the power levels for components, acceptable
maximum temperatures of these components, supported ambient conditions, and corresponding
fan setting should be used to simulate the corner cases of the product. It is important to know
these configurations since CFD should not be the sole means for system concept evaluation.
Results from these simulations can be used to correlate to empirical tests that match the
configurations from the theoretical tests. Adjustments to the CFD models can be made to better
correlate to real world results before performing an optimization study. Many times adjustments
to the meshing or gridding can help. You may learn that a previous assumption made in the
construction of your CFD model is not good and needs to be adjusted for better CFD results.
31
CFD results should include the power level and fan curve assumptions, temperature
monitoring point locations, fan curve operating point(s), system inlet ambient temperature(s),
particle flow paths, and ducting design(s) used. Simulated results should report the configuration
tested such as a fan failure, no failures, or optional hardware load configuration. Graphic results
should report using a consistent temperature scale and units (Celsius) with a range and resolution
that is fine enough to indicate components that are meeting or exceeding their specifications. If a
mock-up has been made and tested the corresponding results should be included for comparison.
Any other assumptions or hidden items in the visualization should be listed as such.
A list of important components to analyze and their corresponding temperature limits
should be made. Two corresponding temperature limits should be included in the list, the
maximum temperature limit and the reliability limit. The maximum temperature is the
temperature at which the component would begin to have data corruption or permanent damage.
The reliability limit temperature is the temperature at which the component will maintain the
intended reliability for that component. It is important to list these two values since under any
fault or failure condition the maximum temperature should not be exceeded. Under normal
operating conditions, the goal would be that the components would not exceed their reliability
temperature. If only one temperature is listed for a component, it is typically considered the
reliability limit and its temperature should never exceed it no matter if under normal operating
conditions or a fault condition.
It is important to improve the accuracy of the CFD results by assisting with empirical
data. Whenever possible, one can improve the accuracy of the model by constructing the model
with test data. Examples of data to input into the model when possible are an airflow impedance
curve through a dense sub-assembly, P-Q curve for a fan assembly with grills, fan assemblies in
series, a fan within a power supply, or a free-wheeling and locked rotor fan assembly. Some of
32
this data can be inputted by using component data sheets or performing wind tunnel tests to input
the resulting data. This information will not only speed a simulation’s creation, but also improve
the accuracy of the results.
People enjoy visual results and CFD models can deliver them. Unfortunately, just
because there is a pretty picture that allows people to visualize temperature and flow results it
does not mean they are accurate. This is why it is important to build a physical mock-up to
correlate results especially with new designs. As a product evolves with generational
improvements, a mock-up may become less important with historical data to compare to.
CFD results should not be interpreted as absolute results but as trends the product will
resemble. Visualizing flow for complicated systems are difficult to understand from testing alone
and CFD can help provide these trends in layout concepts. The more complicated the CFD model
the more important it is to view results as trends. Simplified and well-defined models, such as a
component like a heat sink investigation, provide results that are more accurate but should still
not be interpreted as an absolute result. The more one relies on CFD results as an absolute, the
greater the risk one accepts in the process and being disappointed if the final results do not align.
3.4 MOCK-UP/PROTYPE
3.4.1 MOCK-UP/PROTYPE CONSTRUCTION
Mock-ups are made for multiple reasons, but the primary purpose is to correlate
theoretical data to empirical data. Information that can be learned from a mock-up is pressure
drop curves, airflow tests, thermal test points, and preliminary acoustic measurements. A
preliminary physical system can be built to evaluate and correlate designs to CFD models.
Pressure drop points can be measured and compared to CFD results. This is true for airflow rates
and temperature test points. Improvements can then be made to the CFD models until data
33
correlates across corner case test points for the system. Once results correlate between a CFD
model and mock-up, confidence can be gained in the theoretical models to iterate on designs in an
effort to optimize. The mock-up can be quickly modified to test and validate the optimized
design. The advantage to this approach is to prove the system design before builds when changes
could mean schedule impacts or costly fixes.
Inlet ambient temperature sensing is an important measuring point for analysis of test
data and for the system. Some systems are built with temperature sensing and use ambient
monitor points to determine behavior within the system. These systems may have a true inlet
ambient point to measure system ambient temperature and it is important that this point does not
experience preheated air. It should not be affected by airflow changes or any other changes
within the system as well. Air temperature monitor points are also used to correlate data and help
determine thermal resistances of components. A thermocouple, made of two dissimilar
conductors that produce a voltage with changes in temperature, can be used to measure these
temperatures. The position of these thermocouples is important when evaluating a component or
an airflow path. A proper position of the thermocouple should be at the center, both vertically
and horizontally, of the inlet space to be monitored. This could apply to an inlet of a heat sink, an
airflow path, a vent, or grill. It can be useful to also place a thermocouple on the outlet of heat
sink or airflow path to use the temperature rise for a resistance calculation for the component
being measured. It may be necessary to add some thermal mass or a piece of metal to the
thermocouple. If the thermocouple is located in a turbulent airflow with mixing temperatures,
you can get variability of the temperature readings. The addition of some thermal mass can help
in smoothing the variability of the readings.
Heat sources are important in the construction of a mock-up. Mock-ups are typically
built before hardware prototypes are available to test. This means heaters are needed to simulate
34
heat sources for the system. Any heat source in the system that needs monitoring or is believed to
make a thermal impact to the system should be included in the mock-up. Components that need a
heater are typically processors, memory, hard drives (HDD’s), voltage regulators, PCI cards, etc.
Some vendors may make a thermal test vehicle (TTV) that emulates the heat profile that the real
component will produce. These TTV’s are driven by a power supply and only function is to be a
heat source that emulates the real component. If a vendor does not produce a TTV, you can build
your own. There are a variety of resistors that can be purchased to emulate a component. It is
important that you choose a solution that will meet the power density and form factor of the
component. One style of heater I like to use is a flexible Kapton heater. They are very thin and
can be made for a variety of wattages and surface areas. A copper slug can be machined with a
Kapton heater applied to emulate a chip that has a copper lid. A thermocouple can be used with
the slug to measure for maximum case temperatures and only a power supply is needed to create
the maximum wattage. This is a relative simple approach to construct heat sources for a mock-up
before functional boards are available.
Sense lines are important to improve accuracy when testing a mock-up. When using a
power supply to power fans or TTV’s, the voltage set may not be the actual voltage the load
experiences. The wires carrying the current can experience a voltage drop due to the resistance of
the cable and length delivering a voltage less than desired. Using larger gauge wire can help
prevent this voltage drop, but it is typically not practical or sufficient in correcting the problem.
This is why it is important to use sense lines to solve this potential problem by improving
accuracy of your test setup and increasing confidence in your results. A sense line is a light
gauge wire that is wired on either side of the load to measure the voltage drop across the
component of interest. The measured voltage drop can then be used to correctly dial in the
intended voltage. A sense line can be used with most power supplies and wired into the sense
35
terminals of the power supply to display the actual voltage across the powered load. The display
is then setup to see and adjust any voltage drift that may occur of the duration of the test as
components heat up and equalize.
Systems designed to have optional components that may or may not be loaded should be
simulated in mock-up testing. Hard drives, PCI card, memory, redundant power supplies, or even
optional CPU’s are some examples of these components. The airflow and impedance of the
system can change the thermal design of the system when a component is loaded or not loaded. It
is sometimes therefore necessary to test this condition and determine if it is necessary for a blank
to be installed when a component is not. A blank mimics the impedance or air blockage the
component would have created if it were installed. These are usually a designed plastic or
sheetmetal component that is a low cost and a drop-in replacement. If a blank will be used in the
final product, it is important to test these configurations and validate they work as intended.
3.4.2 MOCK-UP/PROTYPE TESTING
Early in the development cycle, many tests are performed on a bench top. It is important
to know how to project the results of this testing to different test environments since temperature
chamber resources may be limited. The important variables that need attention in this testing are
the ambient temperature, airflow of the fans, and power of the components. Other variables may
be important to your test depending on what system results you are looking for such as elevation.
Assuming constant airflow and power levels of components, test results can be extrapolated to
different ambient temperatures due to the inherent linearity of most thermal testing. An example
of this is if testing was completed at an ambient temperature of 22C but want to know results if
was done at 40C, you can add the difference in these temperature to your test point results. In
this example, it would be adding 18C to test results to predict temperature results at the airflow
36
and power settings tested. Similarly, testing could then be done with different fan speeds with
their temperatures projected to as if the tests were performed at a different ambient temperature to
quickly validate the setting would be good in that condition. It is important to note that there is a
potential for poor results with this approach to testing and it is important to note your assumptions
to your testing and results. One example of this is leakage current of any of your powered
components. The approach described assumed constant power levels across the projected
ambient temperatures. This is not always a valid assumption and a different bench top test
approach may be needed for results that are more accurate.
The use of a wind tunnel or air chamber testing can be used to determine pressure drops
or volumetric airflow curves for systems or components. A fan is one component that this testing
can be valuable in correlating CFD models. A single fan, fans in parallel, or fans in series can be
tested to generate airflow curves that can then be imported into a CFD model for analysis. Fan
fault conditions such as locked rotor or freewheeling curves could be created. These curves can
then be used to evaluate system configuration changes, fault conditions, or airflow speed changes
of the fan. They can also be used in the evaluation of grills, venting in a system, system
impedance, or volumetric airflow measurements. This testing is useful for data correlation and
optimization studies.
Understanding the air movement with the system is important to understand when testing
a mock-up. It becomes important when testing configurations of a system. There are two
important aspects to consider which are understanding pre-heat of components and how it affects
downstream components and in optional load configurations how non-loading a component and
change the impedance of the channel affecting airflow of parallel channels.
Understanding what components pre-heat another when testing a mock-up is important.
Testing a worst-case condition for a component requires that it experience the maximum inlet
37
temperature to the component. Therefore, when testing a downstream component within a
system it is important to have it experience its maximum inlet temperature. This can be affected
by adjustable fan speeds or the non-loading of components that adjust airflow.
The potential for air bypass to affect the cooling efficiency of various parts of the system
can vary significantly, especially when there are optional load configurations to the system. In
this situation, it is important to have blanks that can be loaded that match the resistance as if the
actual component is loaded. This can help balance your system by not changing the airflow path
that could potentially cause another component to overheat when this component is a non-load. If
possible, it is a benefit to design your blank to automatically deploy itself when the component is
not installed. An end user has a tendency to dispose of blanks when a component is loaded and if
they choose to load the component later on they no longer have the blank to load and potentially
compromise the system cooling. Inlet and exhaust vents should be checked that they are working
as intended. This can be checked with the use of an anemometer or smoke kit to measure and
visualize the airflow.
Simulating fan faults is an important scenario to be tested. Two fan failures to be tested
are a locked fan rotor and a freewheeling fan rotor. Both should be tested to determine which has
the worst impact to the system thermally. A locked fan rotor is relatively simple to simulate by
using a cable to through the fan to lock the position of the rotor. A freewheeling fan is a little
more difficult to simulate but can be achieved by disconnecting power to the fan to simulate. If
the presence of other fans is unable to cause the fan rotor to rotate it can then be considered a lock
rotor.
Fans flow direction needs to be validated since they do not always operate as expected.
If not sized appropriately for the pressure drop and other fans in the system, they can actually
impede airflow, or even allow air to flow in the direction opposite to that desired. One scenario
38
that this could happen is with a fan in a power supply. It is easy to overlook smaller fans within a
subsystem that can be configured in parallel with a system fan. The system fan can be a much
more powerful fan that could potentially over power the smaller fan and stall it or even cause it to
spin backwards.
39
Chapter 4
EXAMPLE PROBLEM
4.1 EXAMPLE PROBLEM STATEMENT
A demonstration of this design methodology will be implemented in the testing of a heat
sink. The process will analyze a heat sink with initial calculations, CFD, and a mock-up will be
built and tested to see how well the process works. The results of this first pass study will then be
used to predict the results of similarly built heat sinks with changes in fin height at a different
ambient condition to test how well the process correlates to these changes.
I have chosen to test an extruded aluminum crosscut heat sink with a fin height of 23mm
made by Delta Electronics. The initial heat sink will be tested at an ambient temperature of 10C
for three different power levels of 5, 10, and 15 watts. Three potential use orientations of flat,
vertical, and horizontal will be tested for the effects of gravity as seen in Figure 4.1. The heat
sink will be tested at a simulated elevation, in a Russells Technical Products RHD-64 altitude
chamber, of sea level, 5,000, 10,000, and 15,000ft. All of these tests will be accomplished with
no airflow to check the natural convection performance of the heat sink. Humidity will not be
monitored in this testing.
40
Figure 4.1 Top right = Vertical, bottom left = Horizontal, bottom right = Flat
Results of the initial testing will be used to correlate the processes. Initial assumptions or
predictions will be evaluated and adjusted for a second iteration and improved results. These new
assumptions will be used to predict two heat sinks with changes in fin height from 23mm fins to
19.75mm and 16.45mm. The heat sinks will also be tested at a higher ambient temperature of
30C. All of these tests will occur at sea level and no other altitudes.
4.2 INITIAL CALCULATIONS
Initial calculations used for the heat sink are resistance and power formulas for
convection. These formulas will be used throughout the initial calculations and subsequent
iterations to predict resulting heat sink temperatures. There is second unknown for h, heat
41
transfer coefficient, that will be used to estimate power for the first iteration and then correlated
with results to improve subsequent calculations and predictions.
Convection
row
6
𝑄 = ℎ𝐴(𝑇𝑤 − 𝑇𝑓 ) or 𝑄 =
Heat Sink dimensions in mm
column
short wall
8
1.5
long wall
3.75
(𝑇𝑤 −𝑇𝑓 )
𝑅𝑐𝑜𝑛𝑣
1
1
, 𝑅𝑐𝑜𝑛𝑣 = ℎ𝐴 𝑅 = ℎ𝐴
fin height
23
𝑠
Area
11592
meters^2
0.01159
Table 4.1 Dimensions of 23mm heat sink
I have never done any natural convection investigations and have no idea what value the heat
transfer coefficient should be. I used the Figure 4.2 graph to make an initial prediction. Using
the heat sink dimensions and a first test condition of 5 watts, I calculate an approximate heat
transfer of 431 watts/m^2. Depending on the temperature difference between the ambient air
temperature and heat sink temperature determines the convective heat transfer coefficient. I am
going to estimate a 40C delta, which provides a value of 10. With an estimate value for
convective heat transfer coefficient, I am able to calculate an initial estimate for a resulting heat
sink temperature.
42
Figure 4.2 Convective heat transfer chart [11]
Using the convection power formula and the estimated convective heat transfer coefficient I
calculate the results seen in Table 4.2. This is a very first iteration for initial calculations. I will
compare this data with a first pass CFD and then mock-up results. The results of this first
iteration will then be used to improve assumptions for a second iteration and improved accuracy.
𝑄 = ℎ𝐴(𝑇𝑤 − 𝑇𝑓 )
5 = 10(0.1159)(𝑇𝑤 − 10)
𝑇𝑤 = 53.13
43
10C Ambient Temperature
Flat Heat Sink Orientation
Power Watts
Heat Sink Temp C
5
53.13
10
76.27
15
119.40
Table 4.2 Initial heat sink calculations
4.3 CFD
Using the system test definitions and the heat sink dimensions, a preliminary CFD model
was created using ANSYS ICEPAK 14.0. All simulations were performed using this software.
Since I had no real world test data to correlate results with, I chose variables for air and materials
from the ANSYS library of materials included in the software. I began investigation with a
minimum element in a gap of three and iterations of 50. I will later perform a CFD study for
accuracy of the model with respect to these variables when I am able to correlate to mock-up
data. In the meantime, I arbitrarily chose these values to get first iteration results. These can be
seen in Table 4.3 and a screen capture of one of the simulations in Figure 4.3. The first iteration
shows some differences in results and mock-up results will determine how much work needs to
be completed to improve correlation.
10C Ambient Temperature
Flat Heat Sink Orientation
Power Watts
Heat Sink Temp C
5
53.13
10
76.27
15
119.4
Table 4.3 Initial CDF results
CFD
54.5
84.6
109
44
Figure 4.3 First CFD simulation
4.4 MOCK-UP
A simplified mock-up was built to test the heat sinks in the three orientations. A copper
slug was machined with a 13.7 mm by 13.7 mm raised pedestal to simulate the contact area of a
small ASIC that had Shin-Etsu 7783 as a thermal interface material. A Kapton Minco heater was
attached to the base of the copper slug that is able to provide the adjustable wattage up to 15
watts. A piece of sheetmetal was bent with four standoffs press-fitted to allow the FR4 board,
which is similar to PCA material, to attach by screws. The FR4 had heat sink anchors glued to it
that allows the heat sink to attach and holds the simulated ASIS heater to stay in place as seen in
Figure 4.4.
45
Figure 4.4 Built Mock-up
Two different chambers were used for environmental control in testing the heat sinks, a Russells
Technical Products RHD-64 altitude chamber and a RTP RD-64 temperature/humidity chamber.
Agilent VEE 6.0 was used was in automating the testing and data collection to drive the power
supply and collect thermocouple readings. An Agilent Data Acquisition Unit 34970A was used
to collect the thermocouple readings. One of these setups can be seen with a screen capture of the
program in Figure 4.5. The results from the first iteration of the mock-up testing are shown in
Table 4.4. As you can see, there are improvements that need to be made for prediction of future
studies and correlation of the results.
46
Figure 4.5 Mock-up with screen captures of Agilent VEE
10C Ambient Temperature
Flat Heat Sink Orientation
Power
Heat Sink Temp CFD MockWatts
C
up
5
53.13
54.5
46.8
10
76.27
84.6
72.6
15
119.4
109
93.5
Table 4.4 Initial heat sink results
47
Using the data collect at different altitudes allows the altitude multiplier to be validated with
mock-up results. Table 4.5 shows that the multiplier appears to be conservative especially as
altitude increases. Another potential explanation for this is error in the testing from the altitude
chamber used. The error is the internal fan inside the chamber that helps maintain a constant air
temperature and pressure. I positioned the mock-up underneath a large volume cardboard box in
an attempt to prevent any air movement occurring around the heat sink. Thermal couples that
were placed around the heat sink showed that the volume of the box was sufficient to not create
an increase in ambient temperature, but I did see some erratic readings in the thermal couple that
was located above the heat sink its heat flume. Theoretically, this should have been very
consistent readings with laminar flow occurring. The relatively quick temperatures changes
indicated that was some turbulence and that is why I believe there may have been some airflow
occurring in the experiment.
Multiplier
1
1.1
1.21
1.33
10C
Altitude
0
5
10
15
Q=5W
T calc
45.80
50.38
55.42
60.91
T actual
45.8
46.8
49.9
52.6
Q=10W
T calc
69.50
76.45
84.10
92.44
T actual
69.5
74.5
78.8
82.1
Q=15W
T calc
92.10
101.31
111.44
122.49
T actual
92.1
96.7
102
109
Table 4.5 Altitude multiplier assessment
4.5 ITERATION RESULTS
4.5.1 INITIAL CALCULATION ITERATION
The initial calculations were off the mock-up values from as much as 3.7 C to 25.9 C.
Most of these differences can be attributed to the estimate on the convective heat transfer
48
coefficient. Now that there is tested data to correlate to, I will calculate the convective heat
transfer coefficient and use this new convective heat transfer coefficient value for the second
iteration of calculations. The new heat transfer coefficient seen in Table 4.6 was then used for
calculation of the new orientations. This same process was also followed to create a multiplier
between orientations of the heat sink. The results of this correlation are in Table 4.7.
10C
Power
T
flat A =0.011592
h
5
46.8
11.72
10
69.3
14.55
15
89.8
16.22
Table 4.6 New heat transfer coefficient
Power
5
10
15
Flat
45.8
69.5
92.1
Multipler
1.022
1.045
1.015
1.027
Vert
46.8
72.6
93.5
Multiplier
1.083
1.055
1.053
1.064
Hor
49.6
73.3
97
Table 4.7 Multiplier for orientation calculation
4.5.2 CFD ITERATION
The first iteration of CFD results had temperature differences from 1C to 10C. To
improve the correlation of the data I did a few studies and some constant changes to improve
results. The first thing I did was attempt to get the data to correlate better across the different
altitudes. To do this I calculated specific air densities for the different altitudes at the
temperatures tested. I imputed this value instead of the standard air values in the simulation and
saw a significant improvement. With mock-up in hand, I was also able to perform some CFD
studies to check that I was running enough iterations in the simulation and had sufficient grid
49
refinement to improve accuracy. I started by checking my model at 40, 60, and 80 iterations. I
was able to notice a resulting temperature change 0.7C between 40 and 60 iterations and 0.1C
between 60 and 80 iterations. I chose to run the rest of the simulations at 60 iterations. The
results of this mesh study can be seen in Figure 4.6.
50
Figure 4.6 CFD mesh study
51
The second CFD study performed was to check the gridding. I continued to use the Mesher-HD
in ICEPAK and adjusted the minimum elements in a gap at 1, 3, and 5. I ended up with mesh
counts of 130k, 317k, and 588k cells. Resulting temperature varied from 1.3 between 130k and
317k and 0.2C between 317k and 528k. The rest of the simulations were performed with 317k
grid cells. The importance of these studies are not only to demonstrate accuracy in the model but
save time in how long the models take to perform the simulation calculations and converge.
Figure 4.7 displays the results of the grid refinement study.
52
Figure 4.7 CFD grid refinement study
53
With CFD studies completed and a variable change done to improve correlation and accuracy, the
rest of the simulations were completed with the results seen in Figure 4.8.
10C
Flat Test
Alt Tc
0
45.8
5
46.8
10 49.9
15 52.6
5W
CFD
Tc
46.8
49.3
52.6
56.4
10C
Flat Test
Tc
0
69.5
5
74.5
10 78.8
15 82.1
10W
CFD
Tc
69.3
74.4
79.9
85.4
10C 15W
Flat Test CFD
Tc
Tc
0
92.1 89.8
5
96.7 96.1
10 102 103.3
15 109 111.1
delta
-1
-2.5
-2.7
-3.8
10C
Vert Test
Alt
Tc
0
46.8
5
48.6
10 50.8
15 53.7
5W
CFD
Tc
47.2
51.3
56.2
61.9
0.2
0.1
-1.1
-3.3
10C
Vert Test
Tc
0
72.6
5
76.3
10 81.3
15 85.7
10W
CFD
Tc
70.9
77.8
85.6
94.8
2.3
0.6
-1.3
-2.1
10C 15W
Vert Test CFD
Tc
Tc
0
93.5 91.5
5
100 100.4
10 107 110.8
15 111 123.2
-0.4
-2.7
-5.4
-8.2
Hor
Alt
0
5
10
15
Hor
1.7
-1.5
-4.3
-9.1
0
5
10
15
Hor
2
-0.4
-3.8
-12
0
5
10
15
10C
Test
Tc
49.6
50.3
52.1
55.4
5W
CFD
Tc
47.5
50.2
53.1
56.1
2.1
0.1
-1
-0.7
10C
Test
Tc
73.3
78.6
84.4
87
10W
CFD
Tc
72.3
76.8
81.6
86.6
1
1.8
2.8
0.4
10C
Test
Tc
97
103
110
111
15W
CFD
Tc
93.5
99.7
106.2
112.7
3.5
3.3
3.8
-1.7
Table 4.8 Second iteration of CFD results
As seen in the results, you can see a much improved correlation and accuracy. There are still
outliers at higher altitudes but I believe that still may be due to some airflow in the test chamber.
Overall the results have improved and now the learning’s from these studies can be applied to
predict not only this heat sink, but two new heat sinks and at a different ambient temperature.
54
4.5.3 MOCK-UP ITERATION
The final test configurations are for all three heat sinks at an ambient temperature of 30C.
This will be the third iteration for the 23mm heat sink but first at 30C ambient temperature, and a
first pass for the 19.75mm and 16.45mm heat sinks. Results can be viewed in Table 4.9, Table
4.10, and Table 4.11.
H=23mm
Flat
30C
5W
Hand Calculations
CFD
Test
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
66.9
7.6
65.8
6.5
59.3
10
89.1
6.1
89.3
6.3
83
15
109.9
6.9
109.5
6.5
103
H=23mm
30C
Horizontal
5W
Hand Calculations
CFD
Test
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
69.2
4.6
67.6
3
64.6
10
93.2
-1.5
92.1
-2.6
94.7
15
114.9
4.9
113.4
3.4
110
H=23mm
30C
Vertical
5W
Hand Calculations
CFD
Test
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
67.9
6.6
67
5.7
61.3
10
91.1
5.6
90.9
5.4
85.5
15
112
7
111.3
6.3
105
Table 4.9 23mm heat sink results
55
H=19.75mm
Flat
30C
5W
Hand Calculations
CFD
Test
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
72.9
9.9
70.5
7.5
63
10
98.8
7.7
96.6
5.5
91.1
15
123
12
119.7
8.7
111
H=19.75mm
30C
Horizontal
5W
Hand Calculations
CFD
Test
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
75.6
8.1
70.1
2.6
67.5
10
103.7
5.2
96.5
-2
98.5
15
128.9
12.9
119.3
3.3
116
H=19.75mm
30C
Vertical
5W
Hand Calculations
CFD
Test
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
74.1
9.2
69.9
5
64.9
10
101.2
8.9
95.8
3.5
92.3
15
125.5
14.5
118.4
7.4
111
Table 4.10 19.75mm heat sink results
56
H=16.45mm
30C
5W
Flat
Hand Calculations
CFD
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
81.5
14.6
74.7
7.8
66.9
10
112.6
15.9
104
7.3
96.7
15
141.7
27.7
129.6
15.6
114
H=16.45mm
30C
5W
Horizontal
Hand Calculations
CFD
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
84.8
14.3
73
2.5
70.5
10
118.4
16.4
101.8
-0.2
102
15
148.7
26.7
126.8
4.8
122
H=16.45mm
30C
5W
Vertical
Hand Calculations
CFD
Power
Tcase
Delta to Test
Tcase
Delta to Test
Tcase
5
82.9
13.4
73
3.5
69.5
10
115.4
17.8
102.1
4.5
97.6
15
144.7
28.7
127.5
11.5
116
Test
Test
Test
Table 4.11 16.45mm heat sink results
Results for the 23mm heat sink are within 7.6C for all orientations and power levels. I
like to see that results for power levels within each orientation are within 3C of each other except
for one reading that is 5C. This demonstrates consistent results that are correlating.
The 19.75mm heat sink was a first iteration at predicting results. The CFD results
are consistent with results within 4C of each other with one outlier that is 6C. The initial
calculation could some improvement. Some calculations are as much as 15C off from
57
mock-up results. The calculations vary within an orientation, which signifies that there
could be some improvement to the multiplier used for better correlation.
The 16.45mm heat sink results did not fare as well as the 19.75mm heat sink. The
CFD results were good with some outliers at high power. The initial calculations were
not good. They were from 13.4C to as much as 28.7C off the mock-up values. This is
not good and needs improvement. This was a first iteration and did not produce results as
close to mock-up values as I would have liked to see. This demonstrates the importance
of iterating to improve accuracy and correlate theoretical and empirical data.
58
Chapter 5
CONCLUSION AND FUTURE WORK
5.1 CONCLUSION
The methodology presented is a reliable approach to thermal PCA design. It demonstrated
quick feasibility studies to a built mock-up with progressively improved accuracy in results by
iterating and correlating data. Initial hand calculations were as much as 26C off tested results but
improved to 3C after one iteration and were 7C off when correlated to a new ambient
temperature. Similar results were seen using CFD that began 15.5C off tested results but
improved to 2C in one iteration and 6.5C correlated to a new ambient temperature. Empirical
data helped improve the accuracy in the process and increased credibility to the theoretical
results. This allowed for improved optimization studies and helped reduce design cycle time. I
believe this is a reliable design methodology that can be used for any thermal investigation.
Formulas will change to meet the design requirements, but the process can remain the same.
5.2 FUTURE WORK
Next steps would be to continue to iterate on the process for improved results. A
compact model of the heat sink would be developed to reduce grid count but maintain resulting
data that correlates well. This compact model would then be able to be used within a full system
investigation with increased confidence in its accuracy. Results from this investigation can be
leveraged to perform an optimization study of similar heat sink for improved heat sink
performance. Using a CFD program to change geometry variables of the heat sink and then
simulate results can help develop a heat sink faster than prototyping and testing.
59
APPENDIX A
FORMULAS
𝑅1−2 =
𝑇1 − 𝑇2
, °𝐶/𝑊
𝑄
𝑄𝑥 = −
Conduction (1-d)
𝑄𝑥 = −
Conduction (1-d)
(𝑇𝐿 −𝑇𝑂 )
,
𝑅𝑐𝑜𝑛𝑑
− 𝑇𝑂 )
𝑅𝑐𝑜𝑛𝑑 =
𝐿
𝑘𝐴
𝑄 = ℎ𝐴(𝑇𝑤 − 𝑇𝑓 )
Convection
𝑄=
Convection
(𝑇𝑤 −𝑇𝑓 )
1
, 𝑅𝑐𝑜𝑛𝑣 = ℎ𝐴
𝑅𝑐𝑜𝑛𝑣
4
𝑄𝑛𝑒𝑡 = 𝐴𝜀𝜎(𝑇𝑤4 − 𝑇𝑠𝑢𝑟
)
Radiation
Radiation
𝑘𝐴
(𝑇𝐿
𝐿
𝑄𝑟𝑎𝑑 = −
(𝑇𝑤 −𝑇𝑠𝑢𝑟 )
,
𝑅𝑟𝑎𝑑
2
2
𝑤 +𝑇𝑎𝑚𝑏 )(𝑇𝑤 +𝑇𝑎𝑚𝑏 )
𝑉
𝐼=𝑅
Electric Circuit Analogy
1
1
1
1
𝑠
𝐶 = 𝜌𝑉𝑐𝑝
Thermal Capacitance
𝑄 = 𝑉̇ 𝜌𝑐𝑝 ∆𝑇 = 𝑚̇𝑐𝑝 ∆𝑇
𝑚̇ = 𝜌𝑖𝑛 𝑣𝑖𝑛 𝐴𝑐 = 𝜌𝑜𝑢𝑡 𝑣𝑜𝑢𝑡 𝐴𝑐
Reynolds Number 𝑅𝑒 =
Grashof Number 𝐺𝑟 =
Nusselt Number 𝑁𝑢 =
gβ𝐿3 (𝑇𝑤 −𝑇∞ )
𝑣3
𝜌v𝐿
𝜇
ℎ𝐿
𝑘𝑓𝑙𝑢𝑖𝑑
𝑣
𝑓𝑙𝑢𝑖𝑑
2
𝑅 = ℎ𝐴
Convective Resistance
𝑐𝑝 𝜇
∆𝑇
𝑅
𝑄=
𝑅𝑡𝑜𝑡 = (𝑅 + 𝑅 )−1
𝑅𝑡𝑜𝑡 = 𝑅1 + 𝑅2
Prandtl Number 𝑃𝑟 = 𝑘
1
𝑅𝑟𝑎𝑑 = 𝐴𝜀𝜎(𝑇
=𝛼=
𝜂𝑓 =
=
=
v𝐿
𝑣
=
𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒
= 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒
𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒
𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒
ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑏𝑦 𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛
ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑏𝑦 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛
𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 ℎ𝑒𝑎𝑡
tanh 𝑚𝐿
𝑚𝐿
60
𝑚𝐿 = √
ℎ𝑃
𝐿
𝑘𝐴
𝑚𝑐𝑝 (𝑇𝑐,𝑜𝑢𝑡 − 𝑇𝑐,𝑖𝑛 ) = ℎ𝐴∆𝑇𝑙𝑚𝑡𝑑
∆𝑇𝑙𝑚𝑡𝑑 =
(𝑇ℎ,𝑜𝑢𝑡 − 𝑇𝑐,𝑜𝑢𝑡 ) − (𝑇ℎ,𝑖𝑛 − 𝑇𝑐,𝑖𝑛 )
𝑙𝑛
(𝑇ℎ,𝑜𝑢𝑡 − 𝑇𝑐,𝑜𝑢𝑡 )
(𝑇ℎ,𝑖𝑛 − 𝑇𝑐,𝑖𝑛 )
∆𝑇𝑙𝑚𝑡𝑑 = (∆𝑇𝑜𝑢𝑡 − ∆𝑇𝑖𝑛 )⁄(𝑙𝑛(∆𝑇𝑜𝑢𝑡 − ∆𝑇𝑖𝑛 ))
𝑄𝑎𝑐𝑡𝑢𝑎𝑙 = 𝜀𝑄𝑚𝑎𝑥
𝑃+
Bernoulli Equation
1
𝜌v 2
2
+ 𝜌𝑔ℎ = 𝐸
Potential energy + Kinetic energy + Gravitational energy = total energy
Static pressure + Dynamic pressure + Gravitational pressure = total pressure
(
𝑃1
v12
𝑃2
v22
+ 𝛼1 + 𝑔𝑧1 ) − ( + 𝛼2 + 𝑔𝑧2 ) = ℎ𝑙𝑇
𝜌
2
𝜌
2
Head Loss ℎ𝑙𝑇 = ℎ𝑙 + ℎ𝑙𝑚
𝑙𝑣 2
Major Losses (friction) ℎ𝑙 = 𝑓 𝑑2
Minor Losses (flow separation) ℎ𝑙𝑚 = 𝑘
Electric Circuit Analogy for pressure
𝑅𝑡𝑜𝑡 = 𝑅1 + 𝑅2
𝑅=𝑓
𝑉
𝐼=𝑅
√𝑅𝑡𝑜𝑡 = (
1
√𝑅1
𝑙 𝜌
𝜁𝜌
= 2
2
𝑑 2𝐴
2𝐴
1
Power = 2 ∙ 𝐶 ∙ 𝑉 2 ∙ 𝑓
+
v2
2
v𝐴 = 𝑉 2̇ =
1
√𝑅2
)−1
∆𝑃
𝑅
61
APPENDIX B
NOMENCLATURE
R
T
Q
k
A
L
h
ε
σ
I
V
C
ρ
𝑐𝑝
t
𝜇
𝑣
v
m
𝜂
E
g
ℎ𝑙𝑇
𝑓
𝑉̇
𝜁
𝑐𝑓𝑚
d
P
Resistance, Flow resistance
Temperature
Heat transfer rate / Power
Thermal conductivity
Area
Length
Heat transfer (film, convection) coefficient
Surface emissivity
Stefan-Boltzmann constant
Electric current
Volume / Voltage
Capacitance
Density
Specific Heat at constant pressure
Time
Dynamic (absolute) viscosity
Kinematic viscosity
Velocity
mass
Fin efficiency
Energy
Acceleration due to gravity
Head loss
Friction factor
Volumetric flow rate
Overall loss coefficient
Cubic feet per minute
diameter
Pressure
C/W, N/m2 / (m3 /s)2
C or K
W
W/(m K)
m2
m
W/(m2 K)
W/(m2 K 4 )
A
m3 , V
kJ/K
kg/m3
kJ/(kg K)
s
kg/(m s)
m2 /s
m/s
kg
kJ
m/s 2
m2 /s2 = J/kg
m3 /s
ft 3 /60s
m
N/m2
62
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63
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