analysis of parameter variations in heat sink effectiveness

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Analysis of Parameter Variations on

Heat Fin Effectiveness

Student name: Kjell Sadowski

Student number: 301 117 094

Report due date: Thursday December 15th, 2011

ENGINEERING THERMODYNMICS AND HEAT TRANSFER LABORATORY REPORT

MECHATRONICS SYSTEM ENGINEERING

SCHOOL OF ENGINEERING SCIENCE

SIMON FRASER UNIVERSITY


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TABLE OF CONTENTS1.0 Introduction ............................................................................................................................................. 12.0 Method of Analysis ................................................................................................................................. 13.0 Fin Analysis ............................................................................................................................................ 2

3.1 Variation of Fin Elements ................................................................................................................... 33.2 Variation of Thermal Contact Resistance ........................................................................................... 43.3 Variation of the Heat Transfer Coefficient ......................................................................................... 63.4 Variation of Thermal Conductivity ..................................................................................................... 73.5 Variations of Fin Length ..................................................................................................................... 8

4.0 Optimal Fin Parameters ........................................................................................................................ 104.1 Space Limited Design with Forced Convection................................................................................ 104.2 Space Unlimited Design with Natural Convection ........................................................................... 11

5.0 Conclusion ............................................................................................................................................ 126.0 Works Cited .......................................................................................................................................... 127.0 Appendix ............................................................................................................................................... 13

Section 1.............................................................................................................................................. 13Section 2.............................................................................................................................................. 13Section 3.............................................................................................................................................. 17Section 4.............................................................................................................................................. 20Section 5.............................................................................................................................................. 22Section 6.............................................................................................................................................. 25


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Table of Figures

Figure 1 - Default Parameter Temperature and Heat Transfer...................................................................... 3

Figure 2 - Variation of elements ................................................................................................................... 4

Figure 3 - Variation of Thermal contact resistance ....................................................................................... 5Figure 4 - Thermal contact resistance efficiencies ........................................................................................ 5

Figure 5 Variation of the Heat Transfer Coefficient .................................................................................. 6

Figure 6 - Heat Transfer Coefficient Reverse Efficiencies ........................................................................... 6

Figure 7 Variation of thermal conductivity................................................................................................ 7

Figure 8 - Thermal Conductivity efficiencies ............................................................................................... 8

Figure 9 - Variation of fin length .................................................................................................................. 9

Figure 10 - Fin Length efficiencies ............................................................................................................... 9


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1.0INTRODUCTIONThe dissipation of heat is an important characteristic in many systems. Heat is generated by the

operation of many electrical and mechanical devices, which if left unchecked, can build up and cause a

system to behave abnormally or possibly fail. This is why devices like heat sinks must be used. Heat sinks

accelerate the heat dissipation process by absorbing the heat and moving it into a geometry that will passthat heat into the environment and away from the device. Aside from the ambient conditions that a heat

sink is in, the two important mechanical properties are the material and its geometry. The material affects

how well heat is absorbed into the heat sink and how fast the heat conducts through it and the geometry

affects the dissipation of the heat.

The focus of this study was to analyze the parameters that affect the operation of a fin in a heat

sink and how their values affect the internal temperature, heat transfer and efficiency of the fin. The

results in each section display the trends parametrically and in a non-dimensionalized form.

2.0METHOD OF ANALYSISAll data produced for this study was done on a theoretical basis using numerical analysis.

MATLAB1 was used to facilitate all calculations and to simplify the process of generating figures (seen

throughout the document) and tables (see Appendix). The fundamental approach to the problem was

detailed inENSC 388: Final Project 2 by Doctor Bahrami. The numerical principle employed in this

study to analyze the fin was to apply an energy balance over fin elements of uniform length [which can

be used for] predicting the temperature within each fin element (Bahrami, 2011).

The analysis of a fin included three different elements: a base cell, an interior cell and a tip cell. If

Nelements are used to analyze a fin, there is always: 1 tip cell, 1 base cell andN-2 interior cells. Each cell

has a different equation representing the heat flow through it. The tip cell dissipates more heat than the

other two types because it is open on three sides rather than the interior and base cells which are open on

two. The base cell has a constant value for heat transfer between it and the base because the base

temperature always remains the same.

When constructing the energy balance over the three types of elements within a fin, three

different equations must be written and derived. Starting with the base cell the equation can be written as

= (1)where is conductive heat transfer from the base to the base cell and is convective heattransfer from the fin to the environment. This equation can be rearranged and written in terms of

temperatures and resistances by analyzing a thermal resistor network and using the relation

= (2)

1See attached documents; FinalProjectII.pdf and HeatSinkAnalysis_Function.pdf


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The final equation used to solve for base temperature is:

= 1 +1

+ 0.5 +1

+ 0.5 + +

2 (3)

A similar equation is written for the energy balance over an interior cell

= (4)which, when substituted with equations derived from a resistor network while applying equation (2),

becomes

= 2 +1

1 + + (5)

Finally, we turn to the tip cell. The tip cell is open ended so it has more convective heat transfer comingfrom it. In the equation for the heat transfer, we can see that the heat transfer in is equal to the heat

transfer in the sides and the tip. For n elements, we have

= + (6)which, when substitutions are made using equation (2), becomes

= 1 +1 +

1

+

1 +

1 (7)

Equations (3), (5) and (7) are the constitutive equations employed in the fin analysis. These equations are

solved iteratively in a function built in MATLAB until a specific precision it accomplished.

3.0FIN ANALYSISThis section outlines the effects of various parameters on the temperature and heat transfer in a

fin. The default parameters used in the system were:

Width: 0.05m Height: 0.005m

Length: 0.1m

Conductivity: 120 W/(mK)

Heat Transfer Coefficient: 10 W/(m2K)

Contact Resistance: 0.15 K/W

Base Temperature: 150C


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Ambient Temperature: 20C

Number of Fin Elements: 10

An analysis of these parameters with no variations provides the following graphs shown in Figure 1.

FIGURE 1 - DEFAULT PARAMETER TEMPERATURE AND HEAT TRANSFER

Each graph produced by the analysis has its axes listed in non-dimensional numbers. This means

that every length is compared to the total length (0 being the base and 1 being the tip), convective heat

transfer is given with respect to the conductive heat transfer at the base and temperature is given with

respect to the temperature at the base.

Studying the above graphs, we see temperature decline as we move through the length of the cell.

This is expected because heat transfer is high at the cells closest the base so lots of heat is transferred out

of the each cell. As we move further through the fin, we see lower temperatures and a spike at the end of

the Heat Transfer graph. This is the heat transfer at the tip which is expectantly higher than any other cell

because it has more exposed surface area than any other cell.

Efficiency graphs are also presented in some sections to show the effects of parameters on the

system with respect the maximum overall heat transfer. The following equation describes this

relationship:

=

3.1VARIATION OF FIN ELEMENTSVarying the number of elements in a fin is an easy way to see how many elements in a given fin

length are required to accurately describe the phenomenon which are being analyzed. If too few elements

are used, the graph will not display a sufficient amount of data for proper analysis and if too many

elements are used, excess computational power and time are being used to provide large quantities of

results which do not add any insight into the problem.


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FIGURE 2 - VARIATION OF ELEMENTS

Looking at the heat transfer graph in Figure 2, we notice a cascading of graphs downward as the

number of elements in a fin increases. This is because as the number of fins decreases, so does the amount

of heat transfer per element. The same principle can be applied to the Temperature graph. The important

quality about the graphs is to notice their trends. In the Heat Transfer graph the slanted line at the end,which represents heat transfer in the tip, decreases as the number of elements increases. This is due to the

fact that the difference in heat transfer becomes less noticeable as the number of elements increases.

When n=50, there is only a little perturbation at the end of the graph which would go unnoticed if the

other trends were not shown. In the temperature graphs, the relationship of temperature per element

becomes clearer as the number of elements increases. Every graph shows an inverse quadratic

relationship of temperature per element but the temperature in a cell compared to the base temperature

varies more with more elements. This will not be seen as much when fewer elements are analyzed

because less heat transfer is accounted for. This accounts for why there is so much heat transfer in the tip

when only 3 cells are analyzed; the fin is estimated to be at such a high temperature that a high

temperature gradient exists. From the temperature graph in Figure 1, we can see that when a graph is

presented at a certain resolution, the relationship still exists but over a smaller temperature spectrum.

No efficiency graph is presented for this section because efficiency is not affected by the amount

of elements used to analyze a fin. The number of elements may affect the accuracy of the efficiency that a

study produces but in this study, only the number of elements is varied for the same system so no analysis

is needed.

3.2VARIATION OF THERMAL CONTACT RESISTANCE

Thermal Contact Resistance (TCR) is the resistance between two mating surfaces which preventsheat from transferring between them. This is caused by irregularities and air gaps between the two

surfaces. On an ideal perfectly smooth surface there is no contact resistance because the two surfaces

mesh completely allowing no room for air or other materials to sit between them. Often times, the thermal

contact resistance can be decreased by adding a paste that will fill the irregularities and pockets in the

surface.


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FIGURE 3 - VARIATION OF THERMAL CONTACT RESISTANCE

Looking at the two graphs in Figure 3, we see predictable results. By varying the Thermal

Contact Resistance, we do not expect to see a change in non-dimensional heat transfer because the

proportionality still holds for the design. If the TCR is low, the fin will be closer to the temperature of the

base because it conducts more of the heat. If TCR is higher (as in the purple curve for Rc=5.0), less heat

will be absorbed into the fin. Regardless of the TCR, the temperature curve of the system remains thesame, it is only shifted vertically on the graph. The heat transfer graph absorbs the translations of the

temperature graphs because the heat transfer still has the same relative ratios.

FIGURE 4 - THERMAL CONTACT RESISTANCE EFFICIENCIES

In Figure 4, the efficiency of thermal contact resistance is clearly presented. As the thermal

contact resistance increases, the efficiency of heat transfer from the base decreases. This is expected

because less heat is transferred to the fin from the base and thus the heat sink will be doing a poorer job ofconducting heat away from the source. This graph also shows us that for small TCRs, there is little

deviation in efficiency. The first deviation arises between Rc = 0.1 and 1.0. It could be said that it would

be ideal to maintain a TCR of 0.1 or 1.0 if possible because as the TCR increases, the efficiency decreases

at an exponential rate. This property is purely dependent upon the materials surfaces which the TCR

exists between and how costly it is to achieve such low values.


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3.3VARIATION OF THE HEAT TRANSFER COEFFICIENT

The Heat Transfer Coefficient is an environmental parameter which is affected by the

environmental conditions and the presence of a device supplying motion to the system. Normally, a low

heat transfer coefficient is used when the convection is natural and based on the change in buoyancy force

of air as it is heated. For a heat transfer coefficient above 25 (Cengel, 2008), we can estimate that an

external source is applying motion to the system and thus forcing convection.

FIGURE 5 VARIATION OF THE HEAT TRANSFER COEFFICIENT

Based on the statement made in the previous paragraph, we notice a difference between the

curves produced for h=10 and 100. Above, it was said that any heat transfer coefficient above 25 can be

classified as forced. We can see that to be true from these graphs because the shape of the curve starts to

change. A more parabolic relationship starts to occur compared to the curves produced for natural

convection. The natural convection curves behave as they did before, heat transfer is mainly constant and

then spikes at the tip of the fin. Temperature also sees more variation when forced convection is applied.

As the heat transfer coefficient increases, the temperature decreases in an inverse parabolic fashion. For

natural convection, the temperature remains mainly constant because the fluid around the fin is movingslower than if forced convection is applied so there is less cool fluid is present to absorb the heat.

FIGURE 6 - HEAT TRANSFER COEFFICIENT REVERSE EFFICIENCIES


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This graph provides a result that is slightly unexpected at first glance but provides some

interesting insight into the inner workings of the system. Before going in depth into the information

provided by the graph, the mathematical representation for efficiency will be written,

=

(8)

It should be noted that is the calculated heat transfer at the base and the denominator isthe maximum possible heat transfer if the entire fin at the base temperature, Tb. The heat transfer

coefficient, h, in each scenario changes to match the tested value.

Now when we look at Figure 6, we must consider how the system is reacting when each

value of the heat transfer coefficient is applied when the entire fin is at Tb. When h is 2500 and

the system experiences forced convection, the rate of heat transfer across the fin will be very

large so the denominator of the efficiency equation (8) will be very large. Looking back at the

heat transfer graph in Figure 5, we can see that heat transfer drops off very quickly and the

temperature is very low so the magnitude of heat transfer will be low to begin with and will drop

quickly. Since heat transfer is low in the system when h=2500, the numerator in the efficiency

equation (8) will be small. Putting the numerator and the denominator together, we receive a

very low efficiency because the numerator is small and the denominator is large; therefore, we

have actually calculated a reverse efficiency. The efficiency equation compares the heat transfer

that a normal fin is emitting compared to the heat transfer that a maximally hot fin is transferring.

The larger the discrepancy in these values shows the systems efficiency at removing heat from

itself.

3.4VARIATION OF THERMAL CONDUCTIVITYThermal conductivity is defined as the rate of heat transfer through a unit thickness of material

per unit area per unit temperature difference (Bahrami, 2011). It quantifies the materials ability to

conduct or store energy. If a material has a very high thermal conductivity, it is a good material to draw

away heat from a heat source. The converse can be said for a material with a low thermal conductivity.

Therefore, in our situation, we are interested to see what will happen when the material has a high thermal

conductivity.

FIGURE 7 VARIATION OF THERMAL CONDUCTIVITY


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As predicted Figure 7 shows that the higher the thermal conductivity, the more uniform the

temperature is because heat transfer through the fin is faster. If the thermal conductivity is low, it takes

longer for the heat to travel through the fin. Therefore, if the heat travels through the fin slower, the rate

of heat transfer is higher per element and the temperature towards the end of the fin will be lower. This is

because most of the heat will be lost at the part of the fin closer to the base (which is the hottest part).

Heat transfer decreases as we look at elements closer to the tip because temperature in those elements islower due to the slow transfer of it from element to element. When we look at curves of high thermal

conductivity, we see a more uniform temperature along the fin and a higher rate of heat transfer at the tip.

A more uniform temperature distribution along the fin will result in a more uniform curve for the heat

transfer too.

FIGURE 8 - THERMAL CONDUCTIVITY EFFICIENCIES

As thermal conductivity increases the temperature in the heat sink becomes more uniform andcloser to that of the maximum heat transfer. The maximum heat transfer is defined as the heat transfer that

the fin would experience if the entire fin was the same temperature as the base. For a material with a very

high thermal conductivity, such as diamond, this maximum heat transfer temperature could be

approached. A material with a very high thermal conductivity ensures a fast response to any temperature

fluctuations produced by the heat source. This means that if the temperature in the source rises quickly,

the temperature in the fin will rise quickly too and dissipate the heat quickly. Therefore, the higher the

thermal conductivity, the faster the response and the more efficient the system is at heat transfer.

3.5VARIATIONS OF FIN LENGTHFin length is a parameter of interest because it affects the internal resistance of the fin as well as

the capacity for heat storage. The length of the fin determines how exposed the fin is to the environment

and how much heat can be transferred out of it. A smaller fin will conduct and dissipate heat very quickly

whereas a larger fin will take longer to conduct heat out of a system. The length is usually constrained by

the environment and application where the heat fin is to be placed. Finned heat sinks in microelectronics

tend to be very small whereas power electronic devices will have heat sinks with large fins.


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FIGURE 9 - VARIATION OF FIN LENGTH

Figure 9 shows the results we would expect to see for fin lengths. Short lengths have near

uniform temperature with little variation in the heat transfer until the tip where a lot of heat is transferred.

Long fins have more of a gradual distribution because lots of heat is transferred near the base and less is

transferred at the tip. The further away from the base we are, the lower the temperature because lots of the

heat has already been transferred out of the fin closer to the base.

FIGURE 10 - FIN LENGTH EFFICIENCIES

Figure 10 produces both expected and unexpected results. This graph is similar to that in Figure 6

in the sense that it doesnt exactly measure efficiency. Moreover, it measures a fins ability to move heat

along itself. A higher efficiency, based on our method, means that the heat transfer at the base is closer to

the heat transfer in a fin uniformly at base temperature. The farther away the fin efficiency is from 1, the

more heat has been dissipated along the length of the fin. It is hard to classify a small fin or a large fin as

more efficient because their operation depends on more than just the fin length, it depends on the type ofconvection being used and the application of the fin in the heat sink. For a short fin, forced convection

will be acceptable to dissipate the heat. Forced convection is required for this because the short fin gets

very hot (close to base temperature) as seen in Figure 9 and is better for applications were space is an

issue. Either natural or forced convection can be used for longer fins because the heat is more evenly

distribute along the fin to give a lower mean temperature. Longer fins are used in areas where space is not

an issue.


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The unexpected results in this graph are for the first point, which has an efficiency greater than 1.

Looking through the results of the heat fin analysis, the numbers seem reasonable (see Appendix) so we

can only surmise that a problem exists with the method.

4.0OPTIMAL FIN PARAMETERSThis section is designed to discuss the optimal parameters for a fin given certain criteria. Two

criteria will be given:

Space

Convection

Two variations of these criteria will be used to assess two different situations: A space-limited design

which has forced convection (eg. h is greater than 25) and a non-space-limited design with natural

convection.

4.1SPACE LIMITED DESIGN WITH FORCED CONVECTION

In this design, we are looking at smaller fins of 0.1m or less where forced convection is applied

by an external mechanism.

Parameters

Thermal Contact Resistance: The thermal contact resistance should be between 0.1 K/W and 1.0 K/W

but 1.0 K/W with an efficiency of ~80% would work fine because the forced convection will cool the fin

relatively fast creating a large enough temperature gradient between the surfaces to draw heat at anappropriate rate.

Heat Transfer Coefficient: The heat transfer coefficient, which will be greater than 25, should be chosen

so that heat transfer occurs at a fast enough rate to cool the heat sink and maintain a steady temperature

gradient between the fin and the base. Looking at Figure 5, we notice that the temperature that is

maintained when h=100 is reasonable and in the range of 35-85% of the base temperature. This seems

like a good range to keep the heat sink at because the temperature at the base as low enough to pull heat

from the base at a reasonable rate but hot enough to maintain a high heat transfer rate. For this reason, we

would try to achieve a heat transfer coefficient between 80 and 250 (this is the upper limit for forced

convection of gases (Cengel, 2008)).

Thermal Conductivity: Heat conduction is relatively important here in order to achieve a consistent and

reasonable heat transfer rate which is why a range for thermal conductivity must be specified. Since the

fins are short, thermal conductivity is not as essential because the heat has a shorter distance to travel so a

lower range can be picked for thermal conductivity. Studying Figure 7 and 8, we can see that a k=125

would provide ~90% efficiency and a moderately uniform temperature curve. The non-dimensional heat

transfer is similar to that of k=220 and 2500 and is good for a short fin, uniform heat transfer along the


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length of the fin and lots at the tip. The tip is where the moving fluid is absorbing most of the heat so this

is ideally where the fin loses most of it.

Fin Length: Since the problem constraint asks for a heat fin shorter than 0.1m, we can look to Figures 9

and 10 for comparisons. If the fin is made to be 0.01m, the full potential of the convection may not be

reached because there isnt enough fin length to absorb heat from the base so 0.05m or 5cm may be abetter choice. Among fins of 0.01m, 0.05m and 0.1m, the temperature is nearly uniform in comparison to

the other designs. One advantage of the 0.05m fin over the 0.1m fin is the slope of the heat transfer at the

tip. The slope for the 0.05m design is greater than that for the 0.1m. We want a steeper slope because we

want more heat transfer at the tip which will be taken advantage of by the forced convection; therefore, it

is more preferable to go with the 0.05m design.

Summary of Results:

Thermal Contact Resistance: 1.0 K/W

Heat Transfer Coefficient: 80 - 250 W/(m2K)

Thermal Conductivity: 125 W/(mK)

Fin Length: 0.05m

4.2SPACE UNLIMITED DESIGN WITH NATURAL CONVECTION

In this design, space is not an issue so any length of fin can be used. No external device will be

used to help convection so natural convection is the only method of transporting heat from the sink.

Parameters

Thermal Contact Resistance: Designing for a natural convection system requires more emphasis on

storing heat in the fin and removing it from the base because heat transfer is slower, therefore, we need to

ensure heat transfer occurs at higher efficiencies. In the case of thermal contact resistance, we need to go

for a higher value than that required for the previous design, so we will specify that the thermal contact

resistance should lie between 0.01 and 0.1 but closer to 0.01 to gain that extra percent of heat transfer and

get the efficiency as close to ideal as possible.

Heat Transfer Coefficient: The heat transfer coefficient does not need to be specified directly for this

system because it is based on the environment that the heat sink is place in. Therefore, as long as h


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Fin Length: Fin length is very important in the design of a fin for natural convection. The longer the fin,

the more surface area the fin has to expel the heat and the more volume it has to store it. Noting Figure 9,

we can see that the fin length of 0.5m looks appropriate because the temperature has a noticeable inverse

quadratic relationship with length which means that temperature is dropping rapidly through the length of

the fin because enough convection is occuring (analysis was performed with a default of h=10 which

depicts natural convection so this is accurate). The heat transfer curve is also noticeably exponentialinstead of the linear relationship seen in the shorter fin models. This shows that heat transfer is tapering

off toward the end because temperature is low. The low temperature in the tip will maintain a temperature

differential within the fin ensuring conductive heat transfer through it. Looking at the realistic numbers

for the fin lengths, 0.5m seems very large for any application so perhaps it is best to take approximately

half the number and propose 0.2m for the fin length. This should still experience the same mathematical

relationships that the 0.5m model does but with less heat transfer.

Summary of Results:

Thermal Contact Resistance: 0.1 K/W

Heat Transfer Coefficient:


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7.0APPENDIX

SECTION 1

V a r i a t i o n 1 : D e f a u l t

N o n - D i m e n s i o n a l

L e n g t h

H e a t T r a n s f e r ( W ) T e m p e r a t u r e ( C )

N o r m a l

N o n

D i m e n s i o n a l N o r m a l N o n D i m e n s i o n a l

0 . 1 1 . 3 8 5 5 2 7 0 6 3 0 . 1 0 8 5 2 1 1 1 2 1 4 5 . 9 5 7 0 0 5 7 0 . 9 7 3 0 4 6 7 0 5

0 . 2 1 . 3 4 3 7 9 0 2 4 0 . 1 0 5 2 5 2 0 8 4 1 4 2 . 1 6 2 7 4 9 1 0 . 9 4 7 7 5 1 6 6

0 . 3 1 . 3 0 6 9 7 2 3 1 5 0 . 1 0 2 3 6 8 3 2 8 1 3 8 . 8 1 5 6 6 5 0 . 9 2 5 4 3 7 7 6 7

0 . 4 1 . 2 7 4 9 3 6 4 7 2 0 . 0 9 9 8 5 9 1 2 7 1 3 5 . 9 0 3 3 1 5 7 0 . 9 0 6 0 2 2 1 0 4

0 . 5 1 . 2 4 7 5 5 9 3 6 0 . 0 9 7 7 1 4 8 2 1 1 3 3 . 4 1 4 4 8 7 2 0 . 8 8 9 4 2 9 9 1 5

0 . 6 1 . 2 2 4 7 4 0 7 3 7 0 . 0 9 5 9 2 7 5 5 7 1 3 1 . 3 4 0 0 6 7 0 . 8 7 5 6 0 0 4 4 6

0 . 7 1 . 2 0 6 3 9 0 7 0 7 0 . 0 9 4 4 9 0 2 9 5 1 2 9 . 6 7 1 8 8 2 4 0 . 8 6 4 4 7 9 2 1 6

0 . 8 1 . 1 9 2 4 4 4 0 7 5 0 . 0 9 3 3 9 7 9 2 8 1 2 8 . 4 0 4 0 0 6 8 0 . 8 5 6 0 2 6 7 1 2

0 . 9 1 . 1 8 2 8 4 3 7 6 1 0 . 0 9 2 6 4 5 9 8 5 1 2 7 . 5 3 1 2 5 1 0 . 8 5 0 2 0 8 3 4

1 1 . 4 4 5 1 8 5 2 9 5 0 . 1 1 3 1 9 3 8 3 1 2 7 . 0 5 0 7 6 2 6 0 . 8 4 7 0 0 5 0 8 4

SECTION 2

V a r i a t i o n 2 : n = 3


L e n g t h


N o r m a l

N o n


0 . 3 3 3 1 . 4 1 4 2 5 6 1 1 2 0 . 3 1 2 9 0 3 9 9 5 1 4 8 . 5 6 8 7 3 7 5 0 . 9 9 0 4 5 8 2 5

0 . 6 6 7 1 . 4 0 2 8 5 4 7 2 4 0 . 3 1 0 3 8 1 4 3 9 1 4 7 . 5 3 2 2 4 7 6 0 . 9 8 3 5 4 8 3 1 7

1 1 . 7 1 3 9 9 0 1 5 8 0 . 3 7 9 2 2 0 1 1 6 1 4 6 . 9 6 2 2 3 3 9 0 . 9 7 9 7 4 8 2 2 6


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V a r i a t i o n 2 : n = 2 0


L e n g t h


N o r m a l

N o n


0 . 0 5 1 . 3 6 2 5 3 7 1 5 6 0 . 0 7 0 3 5 2 3 7 2 1 4 3 . 8 6 7 0 1 4 2 0 . 9 5 9 1 1 3 4 2 8

0 . 1 1 . 2 9 6 5 1 8 2 2 4 0 . 0 6 6 9 4 3 5 9 3 1 3 7 . 8 6 5 2 9 3 1 0 . 9 1 9 1 0 1 9 5 4

0 . 1 5 1 . 2 3 5 2 5 0 1 3 2 0 . 0 6 3 7 8 0 1 1 5 1 3 2 . 2 9 5 4 6 6 5 0 . 8 8 1 9 6 9 7 7 7

0 . 2 1 . 1 7 8 5 0 7 1 4 4 0 . 0 6 0 8 5 0 2 8 4 1 2 7 . 1 3 7 0 1 3 1 0 . 8 4 7 5 8 0 0 8 7

0 . 2 5 1 . 1 2 6 0 7 9 2 3 0 . 0 5 8 1 4 3 2 5 5 1 2 2 . 3 7 0 8 3 9 1 0 . 8 1 5 8 0 5 5 9 4

0 . 3 1 . 0 7 7 7 7 3 4 3 3 0 . 0 5 5 6 4 9 0 6 4 1 1 7 . 9 7 9 4 0 3 0 . 7 8 6 5 2 9 3 5 4

0 . 3 5 1 . 0 3 3 4 1 0 4 5 4 0 . 0 5 3 3 5 8 4 5 5 1 1 3 . 9 4 6 4 0 4 9 0 . 7 5 9 6 4 2 6 9 9

0 . 4 0 . 9 9 2 8 2 7 2 8 9 0 . 0 5 1 2 6 3 0 0 9 1 1 0 . 2 5 7 0 2 6 2 0 . 7 3 5 0 4 6 8 4 2

0 . 4 5 0 . 9 5 5 8 7 2 8 0 3 0 . 0 4 9 3 5 4 9 2 5 1 0 6 . 8 9 7 5 2 7 6 0 . 7 1 2 6 5 0 1 8 4

0 . 5 0 . 9 2 2 4 1 1 5 5 3 0 . 0 4 7 6 2 7 2 0 8 1 0 3 . 8 5 5 5 9 5 7 0 . 6 9 2 3 7 0 6 3 8

0 . 5 5 0 . 8 9 2 3 1 8 4 2 3 0 . 0 4 6 0 7 3 3 9 9 1 0 1 . 1 1 9 8 5 6 7 0 . 6 7 4 1 3 2 3 7 8

0 . 6 0 . 8 6 5 4 8 3 5 2 1 0 . 0 4 4 6 8 7 8 2 3 9 8 . 6 8 0 3 2 0 1 2 0 . 6 5 7 8 6 8 8 0 1

0 . 6 5 0 . 8 4 1 8 0 6 0 . 0 4 3 4 6 5 2 7 3 9 6 . 5 2 7 8 1 8 2 1 0 . 6 4 3 5 1 8 7 8 8

0 . 7 0 . 8 2 1 1 9 9 8 7 0 . 0 4 2 4 0 1 3 0 9 9 4 . 6 5 4 5 3 3 6 3 0 . 6 3 1 0 3 0 2 2 4

0 . 7 5 0 . 8 0 3 5 8 7 1 5 7 0 . 0 4 1 4 9 1 9 0 5 9 3 . 0 5 3 3 7 7 9 2 0 . 6 2 0 3 5 5 8 5 3

0 . 8 0 . 7 8 8 9 0 4 4 6 8 0 . 0 4 0 7 3 3 7 8 9 9 1 . 7 1 8 5 8 8 0 . 6 1 1 4 5 7 2 5 3

0 . 8 5 0 . 7 7 7 0 9 5 6 4 5 0 . 0 4 0 1 2 4 0 6 9 0 . 6 4 5 0 5 8 6 5 0 . 6 0 4 3 0 0 3 9 1

0 . 9 0 . 7 6 8 1 1 8 9 0 4 0 . 0 3 9 6 6 0 5 6 8 9 . 8 2 8 9 9 1 2 4 0 . 5 9 8 8 5 9 9 4 2

0 . 9 5 0 . 7 6 1 9 3 9 1 6 5 0 . 0 3 9 3 4 1 4 8 8 9 . 2 6 7 1 9 6 7 7 0 . 5 9 5 1 1 4 6 4 5

1 0 . 9 3 0 9 3 0 0 2 0 . 0 4 8 0 6 7 0 4 5 8 8 . 9 5 7 7 7 9 2 6 0 . 5 9 3 0 5 1 8 6 2

V a r i a t i o n 2 : n = 5 0


L e n g t h


N o r m a l

N o n


0 . 0 2 1 . 3 5 0 3 5 7 4 6 8 0 . 0 5 9 0 6 0 7 1 9 1 4 2 . 7 5 9 7 6 9 9 0 . 9 5 1 7 3 1 7 9 9

0 . 0 4 1 . 2 7 1 4 7 4 3 0 7 0 . 0 5 5 6 1 0 5 9 8 1 3 5 . 5 8 8 5 7 3 4 0 . 9 0 3 9 2 3 8 2 2

0 . 0 6 1 . 1 9 7 2 5 2 7 8 0 . 0 5 2 3 6 4 3 6 4 1 2 8 . 8 4 1 1 6 1 8 0 . 8 5 8 9 4 1 0 7 9

0 . 0 8 1 . 1 2 7 4 2 0 4 9 0 . 0 4 9 3 1 0 1 0 2 1 2 2 . 4 9 2 7 7 1 8 0 . 8 1 6 6 1 8 4 7 9


19/30

16

0 . 1 1 . 0 6 1 7 2 1 1 8 9 0 . 0 4 6 4 3 6 6 0 5 1 1 6 . 5 2 0 1 0 8 1 0 . 7 7 6 8 0 0 7 2 1

0 . 1 2 0 . 9 9 9 9 1 3 7 1 2 0 . 0 4 3 7 3 3 3 2 6 1 1 0 . 9 0 1 2 4 6 5 0 . 7 3 9 3 4 1 6 4 3

0 . 1 4 0 . 9 4 1 7 7 1 2 5 6 0 . 0 4 1 1 9 0 3 4 3 1 0 5 . 6 1 5 5 6 8 8 0 . 7 0 4 1 0 3 7 9 2

0 . 1 6 0 . 8 8 7 0 8 0 3 4 7 0 . 0 3 8 7 9 8 3 2 1 1 0 0 . 6 4 3 6 6 7 9 0 . 6 7 0 9 5 7 7 8 6

0 . 1 8 0 . 8 3 5 6 4 0 3 0 . 0 3 6 5 4 8 4 8 3 9 5 . 9 6 7 3 0 0 0 4 0 . 6 3 9 7 8 2

0 . 2 0 . 7 8 7 2 6 2 2 0 . 0 3 4 4 3 2 5 6 5 9 1 . 5 6 9 2 9 0 9 5 0 . 6 1 0 4 6 1 9 4

0 . 2 2 0 . 7 4 1 7 6 8 5 3 4 0 . 0 3 2 4 4 2 8 0 4 8 7 . 4 3 3 5 0 3 0 6 0 . 5 8 2 8 9 0 0 2

0 . 2 4 0 . 6 9 8 9 9 2 1 7 3 0 . 0 3 0 5 7 1 8 9 8 3 . 5 4 4 7 4 3 0 3 0 . 5 5 6 9 6 4 9 5 4

0 . 2 6 0 . 6 5 8 7 7 6 1 6 9 0 . 0 2 8 8 1 2 9 5 9 7 9 . 8 8 8 7 4 2 6 8 0 . 5 3 2 5 9 1 6 1 8

0 . 2 8 0 . 6 2 0 9 7 2 7 3 4 0 . 0 2 7 1 5 9 5 4 6 7 6 . 4 5 2 0 6 6 7 1 0 . 5 0 9 6 8 0 4 4 5

0 . 3 0 . 5 8 5 4 4 3 1 7 5 0 . 0 2 5 6 0 5 5 8 6 7 3 . 2 2 2 1 0 6 8 1 0 . 4 8 8 1 4 7 3 7 9

0 . 3 2 0 . 5 5 2 0 5 6 8 7 7 0 . 0 2 4 1 4 5 3 6 7 7 0 . 1 8 6 9 8 8 7 9 0 . 4 6 7 9 1 3 2 5 9

0 . 3 4 0 . 5 2 0 6 9 1 3 6 8 0 . 0 2 2 7 7 3 5 3 6 7 . 3 3 5 5 7 8 9 3 0 . 4 4 8 9 0 3 8 6

0 . 3 6 0 . 4 9 1 2 3 1 2 9 2 0 . 0 2 1 4 8 5 0 3 2 6 4 . 6 5 7 3 9 0 1 9 0 . 4 3 1 0 4 9 2 6 8

0 . 3 8 0 . 4 6 3 5 6 8 5 9 8 0 . 0 2 0 2 7 5 1 4 6 6 2 . 1 4 2 5 9 9 8 3 0 . 4 1 4 2 8 3 9 9 9

0 . 4 0 . 4 3 7 6 0 1 4 9 8 0 . 0 1 9 1 3 9 4 2 5 9 . 7 8 1 9 5 4 3 3 0 . 3 9 8 5 4 6 3 6 2

0 . 4 2 0 . 4 1 3 2 3 4 7 7 3 0 . 0 1 8 0 7 3 6 9 5 7 . 5 6 6 7 9 7 5 4 0 . 3 8 3 7 7 8 6 5

0 . 4 4 0 . 3 9 0 3 7 8 7 1 4 0 . 0 1 7 0 7 4 0 3 3 5 5 . 4 8 8 9 7 3 9 8 0 . 3 6 9 9 2 6 4 9 3

0 . 4 6 0 . 3 6 8 9 4 9 5 3 5 0 . 0 1 6 1 3 6 7 8 3 5 3 . 5 4 0 8 6 6 7 7 0 . 3 5 6 9 3 9 1 1 2

0 . 4 8 0 . 3 4 8 8 6 8 2 9 1 0 . 0 1 5 2 5 8 4 8 7 5 1 . 7 1 5 2 9 9 1 7 0 . 3 4 4 7 6 8 6 6 1

0 . 5 0 . 3 3 0 0 6 1 3 9 6 0 . 0 1 4 4 3 5 9 2 8 5 0 . 0 0 5 5 8 1 4 8 0 . 3 3 3 3 7 0 5 4 3

0 . 5 2 0 . 3 1 2 4 5 9 5 1 8 0 . 0 1 3 6 6 6 0 7 3 4 8 . 4 0 5 4 1 0 7 3 0 . 3 2 2 7 0 2 7 3 8

0 . 5 4 0 . 2 9 5 9 9 8 1 8 5 0 . 0 1 2 9 4 6 1 0 2 4 6 . 9 0 8 9 2 5 9 3 0 . 3 1 2 7 2 6 1 7 3

0 . 5 6 0 . 2 8 0 6 1 6 6 6 4 0 . 0 1 2 2 7 3 3 5 9 4 5 . 5 1 0 6 0 5 7 9 0 . 3 0 3 4 0 4 0 3 9

0 . 5 8 0 . 2 6 6 2 5 8 6 4 8 0 . 0 1 1 6 4 5 3 8 1 4 4 . 2 0 5 3 3 1 6 3 0 . 2 9 4 7 0 2 2 1 1

0 . 6 0 . 2 5 2 8 7 1 1 1 6 0 . 0 1 1 0 5 9 8 4 9 4 2 . 9 8 8 2 8 3 2 6 0 . 2 8 6 5 8 8 5 5 5

0 . 6 2 0 . 2 4 0 4 0 5 0 9 7 0 . 0 1 0 5 1 4 6 2 2 4 1 . 8 5 5 0 0 8 8 5 0 . 2 7 9 0 3 3 3 9 2

0 . 6 4 0 . 2 2 8 8 1 4 5 0 9 0 . 0 1 0 0 0 7 6 8 3 4 0 . 8 0 1 3 1 9 0 . 2 7 2 0 0 8 7 9 3

0 . 6 6 0 . 2 1 8 0 5 6 9 9 3 0 . 0 0 9 5 3 7 1 8 3 9 . 8 2 3 3 6 2 9 7 0 . 2 6 5 4 8 9 0 8 6

0 . 6 8 0 . 2 0 8 0 9 2 7 3 3 0 . 0 0 9 1 0 1 3 7 3 3 8 . 9 1 7 5 2 1 1 6 0 . 2 5 9 4 5 0 1 4 1

0 . 7 0 . 1 9 8 8 8 5 3 5 7 0 . 0 0 8 6 9 8 6 6 9 3 8 . 0 8 0 4 8 6 9 9 0 . 2 5 3 8 6 9 9 1 3

0 . 7 2 0 . 1 9 0 4 0 0 7 3 8 0 . 0 0 8 3 2 7 5 7 6 3 7 . 3 0 9 1 5 8 0 1 0 . 2 4 8 7 2 7 7 2

0 . 7 4 0 . 1 8 2 6 0 7 9 5 2 0 . 0 0 7 9 8 6 7 4 2 3 6 . 6 0 0 7 2 2 8 7 0 . 2 4 4 0 0 4 8 1 9


20/30

17

0 . 7 6 0 . 1 7 5 4 7 8 0 6 3 0 . 0 0 7 6 7 4 9 0 2 3 5 . 9 5 2 5 5 1 2 0 . 2 3 9 6 8 3 6 7 5

0 . 7 8 0 . 1 6 8 9 8 5 1 3 7 0 . 0 0 7 3 9 0 9 2 3 5 . 3 6 2 2 8 5 1 7 0 . 2 3 5 7 4 8 5 6 8

0 . 8 0 . 1 6 3 1 0 5 0 1 3 0 . 0 0 7 1 3 3 7 4 3 4 . 8 2 7 7 2 8 4 2 0 . 2 3 2 1 8 4 8 5 6

0 . 8 2 0 . 1 5 7 8 1 6 3 5 7 0 . 0 0 6 9 0 2 4 3 3 4 . 3 4 6 9 4 1 5 5 0 . 2 2 8 9 7 9 6 1

0 . 8 4 0 . 1 5 3 0 9 9 4 3 5 0 . 0 0 6 6 9 6 1 2 5 3 3 . 9 1 8 1 3 0 4 8 0 . 2 2 6 1 2 0 8 7

0 . 8 6 0 . 1 4 8 9 3 7 1 9 8 0 . 0 0 6 5 1 4 0 8 1 3 3 . 5 3 9 7 4 5 2 9 0 . 2 2 3 5 9 8 3 0 2

0 . 8 8 0 . 1 4 5 3 1 4 0 5 2 0 . 0 0 6 3 5 5 6 1 5 3 3 . 2 1 0 3 6 8 3 8 0 . 2 2 1 4 0 2 4 5 6

0 . 9 0 . 1 4 2 2 1 6 9 7 7 0 . 0 0 6 2 2 0 1 5 8 3 2 . 9 2 8 8 1 6 0 8 0 . 2 1 9 5 2 5 4 4 1

0 . 9 2 0 . 1 3 9 6 3 4 2 9 7 0 . 0 0 6 1 0 7 1 9 9 3 2 . 6 9 4 0 2 6 9 6 0 . 2 1 7 9 6 0 1 8

0 . 9 4 0 . 1 3 7 5 5 6 8 2 3 0 . 0 0 6 0 1 6 3 3 6 3 2 . 5 0 5 1 6 5 7 5 0 . 2 1 6 7 0 1 1 0 5

0 . 9 6 0 . 1 3 5 9 7 6 6 3 3 0 . 0 0 5 9 4 7 2 2 4 3 2 . 3 6 1 5 1 2 1 2 0 . 2 1 5 7 4 3 4 1 4

0 . 9 8 0 . 1 3 4 8 8 8 2 3 0 . 0 0 5 8 9 9 6 2 3 2 . 2 6 2 5 6 6 4 0 . 2 1 5 0 8 3 7 7 6

1 0 . 1 6 4 8 0 7 1 8 1 0 . 0 0 7 2 0 8 1 8 8 3 2 . 2 0 7 9 3 9 3 5 0 . 2 1 4 7 1 9 5 9 6

SECTION 3

V a r i a t i o n 3 : R c = 0 K / W


L e n g t h


N o r m a l

N o n

D i m e n s i o n a l N o r m a l

N o n

D i m e n s i o n a l

0 . 1 1 . 4 0 6 2 3 9 8 4 5 0 . 1 0 8 5 0 5 4 5 3 1 4 7 . 8 3 9 9 8 5 9 0 . 9 8 5 5 9 9 9 0 6

0 . 2 1 . 3 6 3 8 7 2 9 4 2 0 . 1 0 5 2 3 6 4 2 3 1 4 3 . 9 8 8 4 4 9 3 0 . 9 5 9 9 2 2 9 9 5

0 . 3 1 . 3 2 6 5 0 0 0 5 7 0 . 1 0 2 3 5 2 7 3 1 1 4 0 . 5 9 0 9 1 4 2 0 . 9 3 7 2 7 2 7 6 2

0 . 4 1 . 2 9 3 9 8 1 9 6 5 0 . 0 9 9 8 4 3 6 3 6 1 3 7 . 6 3 4 7 2 4 1 0 . 9 1 7 5 6 4 8 2 7

0 . 5 1 . 2 6 6 1 9 3 2 4 8 0 . 0 9 7 6 9 9 4 5 9 1 3 5 . 1 0 8 4 7 7 1 0 . 9 0 0 7 2 3 1 8 1

0 . 6 1 . 2 4 3 0 3 2 0 3 1 0 . 0 9 5 9 1 2 3 3 9 1 3 3 . 0 0 2 9 1 1 9 0 . 8 8 6 6 8 6 0 7 9

0 . 7 1 . 2 2 4 4 0 6 6 7 8 0 . 0 9 4 4 7 5 2 0 7 1 3 1 . 3 0 9 6 9 8 0 . 8 7 5 3 9 7 9 8 7

0 . 8 1 . 2 1 0 2 5 1 1 1 8 0 . 0 9 3 3 8 2 9 6 4 1 3 0 . 0 2 2 8 2 8 9 0 . 8 6 6 8 1 8 8 5 9

0 . 9 1 . 2 0 0 5 0 6 9 5 3 0 . 0 9 2 6 3 1 1 0 5 1 2 9 . 1 3 6 9 9 5 7 0 . 8 6 0 9 1 3 3 0 5

1 1 . 4 6 6 7 6 5 9 6 0 . 1 1 3 1 7 5 6 4 7 1 2 8 . 6 4 9 3 3 0 4 0 . 8 5 7 6 6 2 2 0 2


21/30

18

V a r i a t i o n 3 : R c = 0 . 0 1 K / W


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 4 0 4 8 3 9 5 6 8 0 . 1 0 8 5 0 6 5 4 4 1 4 7 . 7 1 2 6 8 8 0 . 9 8 4 7 5 1 2 5 3

0 . 2 1 . 3 6 2 5 1 5 2 8 0 . 1 0 5 2 3 7 5 1 4 1 4 3 . 8 6 5 0 2 5 5 0 . 9 5 9 1 0 0 1 7

0 . 3 1 . 3 2 5 1 7 9 9 2 5 0 . 1 0 2 3 5 3 8 1 8 1 4 0 . 4 7 0 9 0 2 3 0 . 9 3 6 4 7 2 6 8 2

0 . 4 1 . 2 9 2 6 9 4 4 4 5 0 . 0 9 9 8 4 4 7 1 5 1 3 7 . 5 1 7 6 7 6 8 0 . 9 1 6 7 8 4 5 1 2

0 . 5 1 . 2 6 4 9 3 3 5 6 0 . 0 9 7 7 0 0 5 2 9 1 3 4 . 9 9 3 9 6 0 . 8 9 9 9 5 9 7 3 3

0 . 6 1 . 2 4 1 7 9 5 5 0 7 0 . 0 9 5 9 1 3 4 1 3 2 . 8 9 0 5 0 0 6 0 . 8 8 5 9 3 6 6 7 1

0 . 7 1 . 2 2 3 1 8 8 7 6 9 0 . 0 9 4 4 7 6 2 5 8 1 3 1 . 1 9 8 9 7 9 0 . 8 7 4 6 5 9 8 6

0 . 8 1 . 2 0 9 0 4 7 3 3 3 0 . 0 9 3 3 8 4 0 0 7 1 2 9 . 9 1 3 3 9 3 9 0 . 8 6 6 0 8 9 2 9 3

0 . 9 1 . 1 9 9 3 1 2 8 9 3 0 . 0 9 2 6 3 2 1 4 1 1 2 9 . 0 2 8 4 4 4 9 0 . 8 6 0 1 8 9 6 3 2

1 1 . 4 6 5 3 0 7 0 7 2 0 . 1 1 3 1 7 6 9 1 3 1 2 8 . 5 4 1 2 6 4 6 0 . 8 5 6 9 4 1 7 6 4

V a r i a t i o n 3 : R c = 0 . 1 K / W


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 3 9 2 3 6 2 5 9 5 0 . 1 0 8 5 1 6 0 8 1 4 6 . 5 7 8 4 1 7 8 0 . 9 7 7 1 8 9 4 5 2

0 . 2 1 . 3 5 0 4 1 7 9 1 1 0 . 1 0 5 2 4 7 0 5 2 1 4 2 . 7 6 5 2 6 4 6 0 . 9 5 1 7 6 8 4 3 1

0 . 3 1 . 3 1 3 4 1 6 8 9 5 0 . 1 0 2 3 6 3 3 1 7 1 3 9 . 4 0 1 5 3 5 9 0 . 9 2 9 3 4 3 5 7 3

0 . 4 1 . 2 8 1 2 2 1 9 4 3 0 . 0 9 9 8 5 4 1 5 1 3 6 . 4 7 4 7 2 2 1 0 . 9 0 9 8 3 1 4 8 1

0 . 5 1 . 2 5 3 7 0 9 0 2 2 0 . 0 9 7 7 0 9 8 8 5 1 3 3 . 9 7 3 5 4 7 5 0 . 8 9 3 1 5 6 9 8 3

0 . 6 1 . 2 3 0 7 7 7 3 5 4 0 . 0 9 5 9 2 2 6 6 7 1 3 1 . 8 8 8 8 5 0 4 0 . 8 7 9 2 5 9 0 0 2

0 . 7 1 . 2 1 2 3 3 6 4 7 4 0 . 0 9 4 4 8 5 4 4 7 1 3 0 . 2 1 2 4 0 6 8 0 . 8 6 8 0 8 2 7 1 2

0 . 8 1 . 1 9 8 3 2 0 8 9 6 0 . 0 9 3 3 9 3 1 2 1 2 8 . 9 3 8 2 6 3 3 0 . 8 5 9 5 8 8 4 2 2

0 . 9 1 . 1 8 8 6 7 3 1 1 3 0 . 0 9 2 6 4 1 2 0 4 1 2 8 . 0 6 1 1 9 2 1 0 . 8 5 3 7 4 1 2 8 1

1 1 . 4 5 2 3 0 7 5 2 2 0 . 1 1 3 1 8 7 9 8 7 1 2 7 . 5 7 8 3 3 5 0 . 8 5 0 5 2 2 2 3 3


22/30

19

V a r i a t i o n 3 : R c = 1 . 0 K / W


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 2 7 8 8 6 1 8 8 5 0 . 1 0 8 5 8 9 8 2 1 1 3 6 . 2 6 0 1 7 1 4 0 . 9 0 8 4 0 1 1 4 3

0 . 2 1 . 2 4 0 3 6 2 7 2 7 0 . 1 0 5 3 2 0 8 0 8 1 3 2 . 7 6 0 2 4 7 9 0 . 8 8 5 0 6 8 3 2

0 . 3 1 . 2 0 6 3 9 7 9 3 8 0 . 1 0 2 4 3 6 8 1 4 1 2 9 . 6 7 2 5 3 9 9 0 . 8 6 4 4 8 3 5 9 9

0 . 4 1 . 1 7 6 8 4 2 3 3 3 0 . 0 9 9 9 2 7 2 0 9 1 2 6 . 9 8 5 6 6 6 6 0 . 8 4 6 5 7 1 1 1 1

0 . 5 1 . 1 5 1 5 8 3 0 1 9 0 . 0 9 7 7 8 2 4 0 8 1 2 4 . 6 8 9 3 6 5 4 0 . 8 3 1 2 6 2 4 3 6

0 . 6 1 . 1 3 0 5 2 7 9 0 1 0 . 0 9 5 9 9 4 5 9 1 2 2 . 7 7 5 2 6 3 7 0 . 8 1 8 5 0 1 7 5 8

0 . 7 1 . 1 1 3 5 9 5 3 3 9 0 . 0 9 4 5 5 6 8 2 4 1 2 1 . 2 3 5 9 3 9 9 0 . 8 0 8 2 3 9 5 9 9

0 . 8 1 . 1 0 0 7 2 4 9 0 3 0 . 0 9 3 4 6 3 9 7 9 1 2 0 . 0 6 5 9 0 0 2 0 . 8 0 0 4 3 9 3 3 5

0 . 9 1 . 0 9 1 8 6 5 3 4 4 0 . 0 9 2 7 1 1 7 0 2 1 1 9 . 2 6 0 4 8 5 8 0 . 7 9 5 0 6 9 9 0 5

1 1 . 3 3 4 0 2 9 3 2 5 0 . 1 1 3 2 7 4 1 6 1 1 8 . 8 1 6 9 8 7 0 . 7 9 2 1 1 3 2 4 7

V a r i a t i o n 3 : R c = 5 . 0 K / W


L e n g t h


N o r m a l

N o n


N o n


0 . 1 0 . 9 3 9 0 3 6 9 0 7 0 . 1 0 8 7 0 1 8 5 2 1 0 5 . 3 6 6 9 9 1 5 0 . 7 0 2 4 4 6 6 1

0 . 2 0 . 9 1 0 7 9 7 2 2 6 0 . 1 0 5 4 3 2 8 5 9 1 0 2 . 7 9 9 7 4 7 8 0 . 6 8 5 3 3 1 6 5 2

0 . 3 0 . 8 8 5 8 8 0 4 2 7 0 . 1 0 2 5 4 8 5 1 8 1 0 0 . 5 3 4 5 8 4 3 0 . 6 7 0 2 3 0 5 6 2

0 . 4 0 . 8 6 4 1 9 5 5 3 4 0 . 1 0 0 0 3 8 2 9 9 9 8 . 5 6 3 2 3 0 4 0 . 6 5 7 0 8 8 2 0 3

0 . 5 0 . 8 4 5 6 6 1 0 0 4 0 . 0 9 7 8 9 2 7 6 3 9 6 . 8 7 8 2 7 3 0 7 0 . 6 4 5 8 5 5 1 5 4

0 . 6 0 . 8 3 0 2 0 9 5 9 6 0 . 0 9 6 1 0 4 1 2 5 9 5 . 4 7 3 5 9 9 6 1 0 . 6 3 6 4 9 0 6 6 4

0 . 7 0 . 8 1 7 7 8 2 7 5 9 0 . 0 9 4 6 6 5 6 0 9 9 4 . 3 4 3 8 8 7 2 2 0 . 6 2 8 9 5 9 2 4 8

0 . 8 0 . 8 0 8 3 3 5 9 6 6 0 . 0 9 3 5 7 2 0 5 9 9 3 . 4 8 5 0 8 7 7 8 0 . 6 2 3 2 3 3 9 1 9

0 . 9 0 . 8 0 1 8 3 2 8 6 7 0 . 0 9 2 8 1 9 2 6 7 9 2 . 8 9 3 8 9 7 0 3 0 . 6 1 9 2 9 2 6 4 7

1 0 . 9 7 9 6 7 1 6 1 3 0 . 1 1 3 4 0 5 6 8 9 2 . 5 6 8 2 6 7 6 0 . 6 1 7 1 2 1 7 8 4


23/30

20

SECTION 4

V a r i a t i o n 4 : h = 0 . 1 W / ( m 2 . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 0 . 0 1 4 2 9 6 6 7 7 0 . 1 4 9 8 5 0 8 8 2 1 4 9 . 9 6 9 7 8 8 1 0 . 9 9 9 7 9 8 5 8 7

0 . 2 0 . 0 1 4 2 9 3 6 6 2 0 . 1 4 9 8 1 9 2 8 3 1 4 9 . 9 4 2 3 8 1 6 0 . 9 9 9 6 1 5 8 7 7

0 . 3 0 . 0 1 4 2 9 1 0 7 4 0 . 1 4 9 7 9 2 1 5 2 1 4 9 . 9 1 8 8 5 0 . 9 9 9 4 5 9

0 . 4 0 . 0 1 4 2 8 8 8 9 0 . 1 4 9 7 6 9 2 6 4 1 4 9 . 8 9 8 9 9 9 2 0 . 9 9 9 3 2 6 6 6 1

0 . 5 0 . 0 1 4 2 8 7 0 4 2 0 . 1 4 9 7 4 9 8 9 2 1 4 9 . 8 8 2 1 9 7 4 0 . 9 9 9 2 1 4 6 4 9

0 . 6 0 . 0 1 4 2 8 5 5 3 1 0 . 1 4 9 7 3 4 0 5 4 1 4 9 . 8 6 8 4 5 9 8 0 . 9 9 9 1 2 3 0 6 6

0 . 7 0 . 0 1 4 2 8 4 2 8 3 0 . 1 4 9 7 2 0 9 7 4 1 4 9 . 8 5 7 1 1 5 4 0 . 9 9 9 0 4 7 4 3 6

0 . 8 0 . 0 1 4 2 8 3 3 2 3 0 . 1 4 9 7 1 0 9 1 1 1 4 9 . 8 4 8 3 8 7 3 0 . 9 9 8 9 8 9 2 4 9

0 . 9 0 . 0 1 4 2 8 2 5 7 9 0 . 1 4 9 7 0 3 1 2 1 1 4 9 . 8 4 1 6 3 0 9 0 . 9 9 8 9 4 4 2 0 6

1 0 . 0 1 7 5 2 8 0 2 9 0 . 1 8 3 7 2 0 3 6 6 1 4 9 . 8 3 7 2 5 5 4 0 . 9 9 8 9 1 5 0 3 6

V a r i a t i o n 4 : h = 1 W / ( m 2 . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 0 . 1 4 2 5 1 5 2 0 1 0 . 1 0 2 3 9 8 8 2 1 4 9 . 5 5 9 2 7 4 1 0 . 9 9 7 0 6 1 8 2 7

0 . 2 0 . 1 4 2 0 5 6 7 3 6 0 . 1 0 2 0 6 9 4 0 7 1 4 9 . 1 4 2 4 8 7 5 0 . 9 9 4 2 8 3 2 5

0 . 3 0 . 1 4 1 6 4 9 3 8 1 0 . 1 0 1 7 7 6 7 1 8 1 4 8 . 7 7 2 1 6 4 5 0 . 9 9 1 8 1 4 4 3

0 . 4 0 . 1 4 1 2 9 2 7 7 4 0 . 1 0 1 5 2 0 4 9 1 1 4 8 . 4 4 7 9 7 6 1 0 . 9 8 9 6 5 3 1 7 4

0 . 5 0 . 1 4 0 9 8 6 0 9 1 0 . 1 0 1 3 0 0 1 3 6 1 4 8 . 1 6 9 1 7 3 3 0 . 9 8 7 7 9 4 4 8 9

0 . 6 0 . 1 4 0 7 2 9 2 3 7 0 . 1 0 1 1 1 5 5 8 3 1 4 7 . 9 3 5 6 6 9 6 0 . 9 8 6 2 3 7 7 9 7

0 . 7 0 . 1 4 0 5 2 1 3 8 1 0 . 1 0 0 9 6 6 2 3 6 1 4 7 . 7 4 6 7 0 9 7 0 . 9 8 4 9 7 8 0 6 4

0 . 8 0 . 1 4 0 3 6 2 6 9 2 0 . 1 0 0 8 5 2 2 1 7 1 4 7 . 6 0 2 4 4 7 6 0 . 9 8 4 0 1 6 3 1 7

0 . 9 0 . 1 4 0 2 5 2 4 0 6 0 . 1 0 0 7 7 2 9 7 5 1 4 7 . 5 0 2 1 8 7 6 0 . 9 8 3 3 4 7 9 1 7

1 0 . 1 7 2 0 5 2 5 0 7 0 . 1 2 3 6 2 1 7 1 5 1 4 7 . 4 4 6 3 0 1 7 0 . 9 8 2 9 7 5 3 4 4


24/30

21

V a r i a t i o n 4 : h = 1 0 W / ( m 2 . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 3 8 5 5 2 7 0 6 3 0 . 1 0 8 5 2 1 1 1 2 1 4 5 . 9 5 7 0 0 5 7 0 . 9 7 3 0 4 6 7 0 5

0 . 2 1 . 3 4 3 7 9 0 2 4 0 . 1 0 5 2 5 2 0 8 4 1 4 2 . 1 6 2 7 4 9 1 0 . 9 4 7 7 5 1 6 6

0 . 3 1 . 3 0 6 9 7 2 3 1 5 0 . 1 0 2 3 6 8 3 2 8 1 3 8 . 8 1 5 6 6 5 0 . 9 2 5 4 3 7 7 6 7

0 . 4 1 . 2 7 4 9 3 6 4 7 2 0 . 0 9 9 8 5 9 1 2 7 1 3 5 . 9 0 3 3 1 5 7 0 . 9 0 6 0 2 2 1 0 4

0 . 5 1 . 2 4 7 5 5 9 3 6 0 . 0 9 7 7 1 4 8 2 1 1 3 3 . 4 1 4 4 8 7 2 0 . 8 8 9 4 2 9 9 1 5

0 . 6 1 . 2 2 4 7 4 0 7 3 7 0 . 0 9 5 9 2 7 5 5 7 1 3 1 . 3 4 0 0 6 7 0 . 8 7 5 6 0 0 4 4 6

0 . 7 1 . 2 0 6 3 9 0 7 0 7 0 . 0 9 4 4 9 0 2 9 5 1 2 9 . 6 7 1 8 8 2 4 0 . 8 6 4 4 7 9 2 1 6

0 . 8 1 . 1 9 2 4 4 4 0 7 5 0 . 0 9 3 3 9 7 9 2 8 1 2 8 . 4 0 4 0 0 6 8 0 . 8 5 6 0 2 6 7 1 2

0 . 9 1 . 1 8 2 8 4 3 7 6 1 0 . 0 9 2 6 4 5 9 8 5 1 2 7 . 5 3 1 2 5 1 0 . 8 5 0 2 0 8 3 4

1 1 . 4 4 5 1 8 5 2 9 5 0 . 1 1 3 1 9 3 8 3 1 2 7 . 0 5 0 7 6 2 6 0 . 8 4 7 0 0 5 0 8 4

V a r i a t i o n 4 : h = 1 0 0 W / ( m 2 . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 2 . 0 0 1 7 6 8 5 3 0 . 1 8 1 9 0 5 7 8 7 1 2 9 . 1 0 6 9 8 6 6 0 . 8 6 0 7 1 3 2 4 4

0 . 2 1 0 . 0 2 2 6 2 6 3 5 0 . 1 5 1 9 0 8 7 5 6 1 1 1 . 1 1 4 7 8 5 0 . 7 4 0 7 6 5 2 3 4

0 . 3 8 . 4 1 0 9 4 2 5 9 9 0 . 1 2 7 4 8 1 1 4 9 6 . 4 6 3 1 1 4 5 4 0 . 6 4 3 0 8 7 4 3

0 . 4 7 . 1 0 7 6 1 3 5 6 7 0 . 1 0 7 7 2 7 1 2 7 8 4 . 6 1 4 6 6 8 7 9 0 . 5 6 4 0 9 7 7 9 2

0 . 5 6 . 0 6 4 8 2 4 7 9 9 0 . 0 9 1 9 2 2 0 1 3 7 5 . 1 3 4 7 7 0 9 0 . 5 0 0 8 9 8 4 7 3

0 . 6 5 . 2 4 4 3 4 1 0 2 2 0 . 0 7 9 4 8 6 2 8 4 6 7 . 6 7 5 8 2 7 4 7 0 . 4 5 1 1 7 2 1 8 3

0 . 7 4 . 6 1 6 0 5 1 7 0 5 0 . 0 6 9 9 6 3 5 6 5 6 1 . 9 6 4 1 0 6 4 1 0 . 4 1 3 0 9 4 0 4 3

0 . 8 4 . 1 5 6 9 2 8 7 9 4 0 . 0 6 3 0 0 4 8 3 1 5 7 . 7 9 0 2 6 1 7 6 0 . 3 8 5 2 6 8 4 1 2

0 . 9 3 . 8 5 0 1 1 4 3 4 4 0 . 0 5 8 3 5 4 5 7 3 5 5 . 0 0 1 0 3 9 4 9 0 . 3 6 6 6 7 3 5 9 7

1 4 . 5 2 1 7 3 3 6 0 2 0 . 0 6 8 5 3 4 0 2 5 5 3 . 4 9 4 3 2 2 9 8 0 . 3 5 6 6 2 8 8 2


25/30

22

V a r i a t i o n 4 : h = 2 5 0 0 W / ( m 2 . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 4 6 . 2 8 7 5 0 8 4 0 . 6 0 3 1 4 6 3 2 7 7 3 . 1 9 5 4 5 7 6 0 . 4 8 7 9 6 9 7 1 7

0 . 2 5 8 . 0 5 5 2 5 8 6 4 0 . 2 3 9 3 6 2 9 9 4 4 1 . 1 1 1 0 0 3 1 4 0 . 2 7 4 0 7 3 3 5 4

0 . 3 2 3 . 0 3 9 9 3 1 8 5 0 . 0 9 4 9 9 4 1 2 8 . 3 7 8 1 5 7 0 4 0 . 1 8 9 1 8 7 7 1 4

0 . 4 9 . 1 4 4 0 1 0 2 0 . 0 3 7 7 0 0 9 3 7 2 3 . 3 2 5 0 9 4 6 2 0 . 1 5 5 5 0 0 6 3 1

0 . 5 3 . 6 2 9 4 0 1 7 3 7 0 . 0 1 4 9 6 4 0 9 6 2 1 . 3 1 9 7 8 2 4 5 0 . 1 4 2 1 3 1 8 8 3

0 . 6 1 . 4 4 1 0 1 4 9 8 0 . 0 0 5 9 4 1 3 3 4 2 0 . 5 2 4 0 0 5 4 5 0 . 1 3 6 8 2 6 7 0 3

0 . 7 0 . 5 7 2 7 4 9 7 2 4 0 . 0 0 2 3 6 1 4 5 9 2 0 . 2 0 8 2 7 2 6 3 0 . 1 3 4 7 2 1 8 1 8

0 . 8 0 . 2 2 8 7 7 4 1 8 8 0 . 0 0 0 9 4 3 2 4 1 2 0 . 0 8 3 1 9 0 6 1 0 . 1 3 3 8 8 7 9 3 7

0 . 9 0 . 0 9 3 8 0 3 8 0 1 0 . 0 0 0 3 8 6 7 5 5 2 0 . 0 3 4 1 1 0 4 7 0 . 1 3 3 5 6 0 7 3 6

1 0 . 0 5 4 3 4 8 0 7 1 0 . 0 0 0 2 2 4 0 7 8 2 0 . 0 1 6 1 0 3 1 3 0 . 1 3 3 4 4 0 6 8 8

SECTION 5

V a r i a t i o n 5 : k = 2 W / ( m . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 0 9 9 7 7 7 7 4 0 . 3 7 1 8 4 1 0 2 8 1 1 9 . 9 7 9 7 9 4 5 0 . 7 9 9 8 6 5 2 9 7

0 . 2 0 . 6 9 1 0 4 3 8 6 8 0 . 2 3 3 6 4 5 8 1 1 8 2 . 8 2 2 1 6 9 8 4 0 . 5 5 2 1 4 7 7 9 9

0 . 3 0 . 4 3 4 3 3 8 0 1 2 0 . 1 4 6 8 5 2 1 2 5 9 . 4 8 5 2 7 3 8 4 0 . 3 9 6 5 6 8 4 9 2

0 . 4 0 . 2 7 3 1 8 4 2 8 9 0 . 0 9 2 3 6 5 1 4 2 4 4 . 8 3 4 9 3 5 3 3 0 . 2 9 8 8 9 9 5 6 9

0 . 5 0 . 1 7 2 1 2 7 6 6 3 0 . 0 5 8 1 9 7 3 2 9 3 5 . 6 4 7 9 6 9 3 4 0 . 2 3 7 6 5 3 1 2 9

0 . 6 0 . 1 0 8 9 3 5 6 0 1 0 . 0 3 6 8 3 1 7 3 8 2 9 . 9 0 3 2 3 6 4 4 0 . 1 9 9 3 5 4 9 1

0 . 7 0 . 0 6 9 7 0 4 7 9 4 0 . 0 2 3 5 6 7 5 8 2 2 6 . 3 3 6 7 9 9 4 4 0 . 1 7 5 5 7 8 6 6 3

0 . 8 0 . 0 4 5 8 0 4 9 2 3 0 . 0 1 5 4 8 6 9 0 1 2 4 . 1 6 4 0 8 3 8 7 0 . 1 6 1 0 9 3 8 9 2

0 . 9 0 . 0 3 1 9 7 7 3 2 7 0 . 0 1 0 8 1 1 7 1 4 2 2 . 9 0 7 0 2 9 7 7 0 . 1 5 2 7 1 3 5 3 2

1 0 . 0 3 0 9 0 3 7 5 8 0 . 0 1 0 4 4 8 7 3 4 2 2 . 2 8 9 1 6 7 2 6 0 . 1 4 8 5 9 4 4 4 8


26/30

23

V a r i a t i o n 5 : k = 2 0 W / ( m . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 3 2 0 6 9 8 0 1 0 . 1 5 2 8 5 0 1 8 9 1 4 0 . 0 6 3 4 5 5 5 0 . 9 3 3 7 5 6 3 7

0 . 2 1 . 1 5 9 6 6 1 1 7 9 0 . 1 3 4 2 1 2 6 8 9 1 2 5 . 4 2 3 7 4 3 5 0 . 8 3 6 1 5 8 2 9

0 . 3 1 . 0 2 4 1 3 2 6 1 4 0 . 1 1 8 5 2 7 3 7 2 1 1 3 . 1 0 2 9 6 5 0 . 7 5 4 0 1 9 7 6 6

0 . 4 0 . 9 1 1 1 2 9 3 9 1 0 . 1 0 5 4 4 9 0 1 2 1 0 2 . 8 2 9 9 4 4 7 0 . 6 8 5 5 3 2 9 6 4

0 . 5 0 . 8 1 8 1 6 1 8 2 2 0 . 0 9 4 6 8 9 4 6 6 9 4 . 3 7 8 3 4 7 4 3 0 . 6 2 9 1 8 8 9 8 3

0 . 6 0 . 7 4 3 1 8 4 6 2 0 . 0 8 6 0 1 2 0 2 5 8 7 . 5 6 2 2 3 8 1 7 0 . 5 8 3 7 4 8 2 5 4

0 . 7 0 . 6 8 4 5 4 4 4 6 6 0 . 0 7 9 2 2 5 3 4 2 8 2 . 2 3 1 3 1 5 0 8 0 . 5 4 8 2 0 8 7 6 7

0 . 8 0 . 6 4 0 9 5 2 5 7 6 0 . 0 7 4 1 8 0 2 6 1 7 8 . 2 6 8 4 1 5 9 7 0 . 5 2 1 7 8 9 4 4

0 . 9 0 . 6 1 1 4 4 6 3 5 6 0 . 0 7 0 7 6 5 3 7 6 7 5 . 5 8 6 0 3 2 3 8 0 . 5 0 3 9 0 6 8 8 3

1 0 . 7 3 0 6 9 2 5 4 9 0 . 0 8 4 5 6 6 2 6 2 7 4 . 1 2 5 3 7 4 0 2 0 . 4 9 4 1 6 9 1 6

V a r i a t i o n 5 : k = 8 0 W / ( m . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 3 7 6 5 8 8 0 5 4 0 . 1 1 3 4 0 1 3 6 2 1 4 5 . 1 4 4 3 6 8 5 0 . 9 6 7 6 2 9 1 2 4

0 . 2 1 . 3 1 7 3 9 1 2 6 8 0 . 1 0 8 5 2 4 8 1 5 1 3 9 . 7 6 2 8 4 2 6 0 . 9 3 1 7 5 2 2 8 4

0 . 3 1 . 2 6 5 4 3 2 6 8 9 0 . 1 0 4 2 4 4 5 4 2 1 3 5 . 0 3 9 3 3 5 4 0 . 9 0 0 2 6 2 2 3 6

0 . 4 1 . 2 2 0 4 2 4 7 1 0 . 1 0 0 5 3 6 8 4 8 1 3 0 . 9 4 7 7 0 0 9 0 . 8 7 2 9 8 4 6 7 3

0 . 5 1 . 1 8 2 1 1 4 1 3 1 0 . 0 9 7 3 8 0 8 7 8 1 2 7 . 4 6 4 9 2 1 0 . 8 4 9 7 6 6 1 4

0 . 6 1 . 1 5 0 2 9 0 3 4 0 . 0 9 4 7 5 9 2 7 9 1 2 4 . 5 7 1 8 4 9 1 0 . 8 3 0 4 7 8 9 9 4

0 . 7 1 . 1 2 4 7 7 2 2 8 9 0 . 0 9 2 6 5 7 1 3 8 1 2 2 . 2 5 2 0 2 6 2 0 . 8 1 5 0 1 3 5 0 8

0 . 8 1 . 1 0 5 4 2 1 6 4 1 0 . 0 9 1 0 6 3 0 6 1 2 0 . 4 9 2 8 7 6 5 0 . 8 0 3 2 8 5 8 4 3

0 . 9 1 . 0 9 2 1 2 6 2 2 5 0 . 0 8 9 9 6 7 8 0 2 1 1 9 . 2 8 4 2 0 2 2 0 . 7 9 5 2 2 8 0 1 5

1 1 . 3 3 1 3 6 5 8 6 8 0 . 1 0 9 6 7 6 0 2 3 1 1 8 . 6 1 9 6 9 3 9 0 . 7 9 0 7 9 7 9 6


27/30

24

V a r i a t i o n 5 : k = 1 2 5 W / ( m . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 3 8 6 2 8 2 7 3 6 0 . 1 0 8 1 3 1 7 4 8 1 4 6 . 0 2 5 7 0 3 3 0 . 9 7 3 5 0 4 6 8 9

0 . 2 1 . 3 4 6 0 3 1 4 2 2 0 . 1 0 4 9 9 2 0 9 6 1 4 2 . 3 6 6 4 9 2 9 0 . 9 4 9 1 0 9 9 5 3

0 . 3 1 . 3 1 0 5 0 9 6 6 7 0 . 1 0 2 2 2 1 3 5 6 1 3 9 . 1 3 7 2 4 2 5 0 . 9 2 7 5 8 1 6 1 6

0 . 4 1 . 2 7 9 5 9 0 6 1 2 0 . 0 9 9 8 0 9 6 3 2 1 3 6 . 3 2 6 4 1 9 2 0 . 9 0 8 8 4 2 7 9 5

0 . 5 1 . 2 5 3 1 5 9 4 8 8 0 . 0 9 7 7 4 7 9 7 2 1 3 3 . 9 2 3 5 8 9 8 0 . 8 9 2 8 2 3 9 3 2

0 . 6 1 . 2 3 1 1 2 3 4 0 7 0 . 0 9 6 0 2 9 1 3 1 3 1 . 9 2 0 3 0 9 7 0 . 8 7 9 4 6 8 7 3 1

0 . 7 1 . 2 1 3 3 9 8 5 2 8 0 . 0 9 4 6 4 6 5 6 8 1 3 0 . 3 0 8 9 5 7 1 0 . 8 6 8 7 2 6 3 8 1

0 . 8 1 . 1 9 9 9 2 4 5 6 4 0 . 0 9 3 5 9 5 5 8 2 1 2 9 . 0 8 4 0 5 1 3 0 . 8 6 0 5 6 0 3 4 2

0 . 9 1 . 1 9 0 6 4 8 1 0 4 0 . 0 9 2 8 7 2 0 0 7 1 2 8 . 2 4 0 7 3 6 7 0 . 8 5 4 9 3 8 2 4 5

1 1 . 4 5 4 9 8 1 3 2 8 0 . 1 1 3 4 9 0 3 2 2 1 2 7 . 7 7 6 3 9 4 7 0 . 8 5 1 8 4 2 6 3 1

V a r i a t i o n 5 : k = 2 2 0 W / ( m . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 3 9 4 5 2 3 7 1 3 0 . 1 0 4 1 6 7 8 2 9 1 4 6 . 7 7 4 8 8 3 0 . 9 7 8 4 9 9 2 2

0 . 2 1 . 3 7 0 5 3 3 9 1 9 0 . 1 0 2 3 7 5 8 4 5 1 4 4 . 5 9 3 9 9 2 6 0 . 9 6 3 9 5 9 9 5 1

0 . 3 1 . 3 4 9 2 7 5 0 1 9 0 . 1 0 0 7 8 7 8 5 2 1 4 2 . 6 6 1 3 6 5 4 0 . 9 5 1 0 7 5 7 6 9

0 . 4 1 . 3 3 0 7 0 2 8 1 2 0 . 0 9 9 4 0 0 5 5 1 4 0 . 9 7 2 9 8 2 9 0 . 9 3 9 8 1 9 8 8 6

0 . 5 1 . 3 1 4 7 7 4 0 3 3 0 . 0 9 8 2 1 0 7 0 5 1 3 9 . 5 2 4 9 1 2 1 0 . 9 3 0 1 6 6 0 8 1

0 . 6 1 . 3 0 1 4 5 7 0 7 3 0 . 0 9 7 2 1 5 9 5 8 1 3 8 . 3 1 4 2 7 9 3 0 . 9 2 2 0 9 5 1 9 6

0 . 7 1 . 2 9 0 7 1 8 9 1 8 0 . 0 9 6 4 1 3 8 4 1 1 3 7 . 3 3 8 0 8 3 4 0 . 9 1 5 5 8 7 2 2 3

0 . 8 1 . 2 8 2 5 4 0 2 5 2 0 . 0 9 5 8 0 2 9 1 3 1 3 6 . 5 9 4 5 6 8 3 0 . 9 1 0 6 3 0 4 5 6

0 . 9 1 . 2 7 6 8 9 8 6 7 1 0 . 0 9 5 3 8 1 4 9 9 1 3 6 . 0 8 1 6 9 7 3 0 . 9 0 7 2 1 1 3 1 6

1 1 . 5 6 3 2 8 3 7 5 4 0 . 1 1 6 7 7 3 8 3 1 3 5 . 7 9 8 7 9 6 6 0 . 9 0 5 3 2 5 3 1


28/30

25

V a r i a t i o n 5 : k = 1 0 0 0 W / ( m . K )


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 4 0 4 4 1 4 6 6 2 0 . 1 0 2 6 4 6 8 9 1 4 7 . 6 7 4 0 6 0 2 0 . 9 8 4 4 9 3 7 3 4

0 . 2 1 . 3 9 9 0 0 4 5 1 9 0 . 1 0 2 2 5 1 4 7 1 4 7 . 1 8 2 2 2 9 0 . 9 8 1 2 1 4 8 6

0 . 3 1 . 3 9 4 1 9 1 8 9 5 0 . 1 0 1 8 9 9 7 2 1 1 4 6 . 7 4 4 7 1 7 7 0 . 9 7 8 2 9 8 1 1 8

0 . 4 1 . 3 8 9 9 7 4 1 2 5 0 . 1 0 1 5 9 1 4 5 1 4 6 . 3 6 1 2 8 4 1 0 . 9 7 5 7 4 1 8 9 4

0 . 5 1 . 3 8 6 3 4 4 0 8 4 0 . 1 0 1 3 2 6 1 3 5 1 4 6 . 0 3 1 2 8 0 4 0 . 9 7 3 5 4 1 8 6 9

0 . 6 1 . 3 8 3 3 0 0 9 0 7 0 . 1 0 1 1 0 3 7 1 3 1 4 5 . 7 5 4 6 2 7 9 0 . 9 7 1 6 9 7 5 1 9

0 . 7 1 . 3 8 0 8 3 8 1 1 4 0 . 1 0 0 9 2 3 7 1 1 4 5 . 5 3 0 7 3 7 6 0 . 9 7 0 2 0 4 9 1 7

0 . 8 1 . 3 7 8 9 5 6 5 8 5 0 . 1 0 0 7 8 6 1 9 2 1 4 5 . 3 5 9 6 8 9 6 0 . 9 6 9 0 6 4 5 9 7

0 . 9 1 . 3 7 7 6 5 0 7 8 3 0 . 1 0 0 6 9 0 7 5 2 1 4 5 . 2 4 0 9 8 0 3 0 . 9 6 8 2 7 3 2 0 2

1 1 . 6 8 9 8 6 0 3 1 2 0 . 1 2 3 5 0 9 7 5 2 1 4 5 . 1 7 4 8 3 7 9 0 . 9 6 7 8 3 2 2 5 3

SECTION 6

V a r i a t i o n 6 : L = 0 . 0 1 m


L e n g t h


N o r m a l

N o n


N o n


0 . 1 0 . 1 4 2 4 1 5 2 9 3 0 . 0 8 4 8 4 2 4 3 8 1 4 9 . 4 6 8 4 4 7 9 0 . 9 9 6 4 5 6 3 1 9

0 . 2 0 . 1 4 1 8 5 1 7 6 5 0 . 0 8 4 5 0 6 7 2 2 1 4 8 . 9 5 6 1 4 9 9 0 . 9 9 3 0 4 1

0 . 3 0 . 1 4 1 3 3 9 2 7 6 0 . 0 8 4 2 0 1 4 1 2 1 4 8 . 4 9 0 2 5 0 7 0 . 9 8 9 9 3 5 0 0 5

0 . 4 0 . 1 4 0 8 7 7 8 3 5 0 . 0 8 3 9 2 6 5 1 3 1 4 8 . 0 7 0 7 5 8 8 0 . 9 8 7 1 3 8 3 9 2

0 . 5 0 . 1 4 0 4 6 6 1 7 3 0 . 0 8 3 6 8 1 2 7 1 4 7 . 6 9 6 5 2 1 2 0 . 9 8 4 6 4 3 4 7 5

0 . 6 0 . 1 4 0 1 0 4 8 0 . 0 8 3 4 6 5 9 8 6 1 4 7 . 3 6 7 9 9 9 5 0 . 9 8 2 4 5 3 3 3

0 . 7 0 . 1 3 9 7 9 2 2 0 7 0 . 0 8 3 2 7 9 7 6 2 1 4 7 . 0 8 3 8 2 4 7 0 . 9 8 0 5 5 8 8 3 1

0 . 8 0 . 1 3 9 5 2 9 3 3 6 0 . 0 8 3 1 2 3 1 5 9 1 4 6 . 8 4 4 8 5 1 0 . 9 7 8 9 6 5 6 7 3

0 . 9 0 . 1 3 9 3 1 4 5 7 9 0 . 0 8 2 9 9 5 2 2 1 4 6 . 6 4 9 6 1 7 1 0 . 9 7 7 6 6 4 1 1 4

1 0 . 4 5 5 3 9 7 3 8 6 0 . 2 7 1 2 9 8 2 8 4 1 4 6 . 4 9 9 2 7 4 0 . 9 7 6 6 6 1 8 2 7


29/30

26

V a r i a t i o n 6 : L = 0 . 0 5 m


L e n g t h


N o r m a l

N o n


N o n


0 . 1 0 . 7 0 2 9 6 0 5 1 4 0 . 1 0 1 6 9 2 2 9 3 1 4 7 . 8 1 1 0 0 2 6 0 . 9 8 5 4 0 6 6 8 4

0 . 2 0 . 6 9 1 5 7 4 3 5 6 0 . 1 0 0 0 4 5 1 3 8 1 4 5 . 7 4 0 7 9 1 9 0 . 9 7 1 6 0 5 2 8

0 . 3 0 . 6 8 1 4 5 1 5 6 5 0 . 0 9 8 5 8 0 7 4 6 1 4 3 . 9 0 0 2 8 4 5 0 . 9 5 9 3 3 5 2 3

0 . 4 0 . 6 7 2 5 7 2 9 0 4 0 . 0 9 7 2 9 6 3 3 4 1 4 2 . 2 8 5 9 8 2 5 0 . 9 4 8 5 7 3 2 1 6

0 . 5 0 . 6 6 4 9 1 8 5 9 3 0 . 0 9 6 1 8 9 0 3 9 1 4 0 . 8 9 4 2 8 9 7 0 . 9 3 9 2 9 5 2 6 4

0 . 6 0 . 6 5 8 4 7 5 1 5 6 0 . 0 9 5 2 5 6 9 1 3 1 3 9 . 7 2 2 7 5 5 6 0 . 9 3 1 4 8 5 0 3 7

0 . 7 0 . 6 5 3 2 2 6 9 1 4 0 . 0 9 4 4 9 7 6 8 7 1 3 8 . 7 6 8 5 2 9 8 0 . 9 2 5 1 2 3 5 3 2

0 . 8 0 . 6 4 9 1 6 5 9 8 1 0 . 0 9 3 9 1 0 2 2 1 1 3 8 . 0 3 0 1 7 8 3 0 . 9 2 0 2 0 1 1 8 8

0 . 9 0 . 6 4 6 2 8 1 0 7 0 . 0 9 3 4 9 2 8 8 1 1 3 7 . 5 0 5 6 4 9 1 0 . 9 1 6 7 0 4 3 2 7

1 0 . 9 3 7 5 5 5 8 3 1 0 . 1 3 5 6 2 9 5 2 7 1 3 7 . 1 9 4 4 7 8 8 0 . 9 1 4 6 2 9 8 5 9

V a r i a t i o n 6 : L = 0 . 1 m


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 . 3 8 5 5 2 7 0 6 3 0 . 1 0 8 5 2 1 1 1 2 1 4 5 . 9 5 7 0 0 5 7 0 . 9 7 3 0 4 6 7 0 5

0 . 2 1 . 3 4 3 7 9 0 2 4 0 . 1 0 5 2 5 2 0 8 4 1 4 2 . 1 6 2 7 4 9 1 0 . 9 4 7 7 5 1 6 6

0 . 3 1 . 3 0 6 9 7 2 3 1 5 0 . 1 0 2 3 6 8 3 2 8 1 3 8 . 8 1 5 6 6 5 0 . 9 2 5 4 3 7 7 6 7

0 . 4 1 . 2 7 4 9 3 6 4 7 2 0 . 0 9 9 8 5 9 1 2 7 1 3 5 . 9 0 3 3 1 5 7 0 . 9 0 6 0 2 2 1 0 4

0 . 5 1 . 2 4 7 5 5 9 3 6 0 . 0 9 7 7 1 4 8 2 1 1 3 3 . 4 1 4 4 8 7 2 0 . 8 8 9 4 2 9 9 1 5

0 . 6 1 . 2 2 4 7 4 0 7 3 7 0 . 0 9 5 9 2 7 5 5 7 1 3 1 . 3 4 0 0 6 7 0 . 8 7 5 6 0 0 4 4 6

0 . 7 1 . 2 0 6 3 9 0 7 0 7 0 . 0 9 4 4 9 0 2 9 5 1 2 9 . 6 7 1 8 8 2 4 0 . 8 6 4 4 7 9 2 1 6

0 . 8 1 . 1 9 2 4 4 4 0 7 5 0 . 0 9 3 3 9 7 9 2 8 1 2 8 . 4 0 4 0 0 6 8 0 . 8 5 6 0 2 6 7 1 2

0 . 9 1 . 1 8 2 8 4 3 7 6 1 0 . 0 9 2 6 4 5 9 8 5 1 2 7 . 5 3 1 2 5 1 0 . 8 5 0 2 0 8 3 4

1 1 . 4 4 5 1 8 5 2 9 5 0 . 1 1 3 1 9 3 8 3 1 2 7 . 0 5 0 7 6 2 6 0 . 8 4 7 0 0 5 0 8 4


30/30

V a r i a t i o n 6 : L = 0 . 5 m


L e n g t h


N o r m a l

N o n


N o n


0 . 1 6 . 3 8 6 9 3 5 9 4 1 0 . 1 4 5 7 7 9 5 4 9 1 3 6 . 1 2 6 1 0 8 0 . 9 0 7 5 0 7 3 8 7

0 . 2 5 . 7 0 0 7 9 2 3 9 5 0 . 1 3 0 1 1 8 5 6 6 1 2 3 . 6 5 0 7 7 0 8 0 . 8 2 4 3 3 8 4 7 2

0 . 3 5 . 1 1 9 1 3 5 9 1 2 0 . 1 1 6 8 4 2 4 6 3 1 1 3 . 0 7 5 1 9 8 4 0 . 7 5 3 8 3 4 6 5 6

0 . 4 4 . 6 3 1 2 9 5 0 9 1 0 . 1 0 5 7 0 7 6 6 9 1 0 4 . 2 0 5 3 6 5 3 0 . 6 9 4 7 0 2 4 3 5

0 . 5 4 . 2 2 8 3 0 8 6 4 5 0 . 0 9 6 5 0 9 6 4 6 9 6 . 8 7 8 3 3 8 9 9 0 . 6 4 5 8 5 5 5 9 3

0 . 6 3 . 9 0 2 7 8 6 3 4 8 0 . 0 8 9 0 7 9 7 1 5 9 0 . 9 5 9 7 5 1 7 9 0 . 6 0 6 3 9 8 3 4 5

0 . 7 3 . 6 4 8 7 4 2 8 0 3 0 . 0 8 3 2 8 1 2 6 1 8 6 . 3 4 0 7 7 8 2 3 0 . 5 7 5 6 0 5 1 8 8

0 . 8 3 . 4 6 1 5 2 4 1 9 3 0 . 0 7 9 0 0 8 0 6 3 8 2 . 9 3 6 8 0 3 5 1 0 . 5 5 2 9 1 2 0 2 3

0 . 9 3 . 3 3 7 6 8 2 5 9 9 0 . 0 7 6 1 8 1 4 2 2 8 0 . 6 8 5 1 3 8 1 6 0 . 5 3 7 9 0 0 9 2 1

1 3 . 4 2 3 8 1 8 3 0 4 0 . 0 7 8 1 4 7 4 3 9 7 9 . 5 4 4 6 6 6 1 6 0 . 5 3 0 2 9 7 7 7 4

V a r i a t i o n 6 : L = 1 . 0 m


L e n g t h


N o r m a l

N o n


N o n


0 . 1 1 2 . 0 0 8 7 6 4 0 8 0 . 1 8 2 5 6 7 5 3 1 2 9 . 1 7 0 5 8 2 5 0 . 8 6 1 1 3 7 2 1 7

0 . 2 1 0 . 0 3 7 2 4 1 5 8 0 . 1 5 2 5 9 4 7 5 4 1 1 1 . 2 4 7 6 5 0 7 0 . 7 4 1 6 5 1 0 0 5

0 . 3 8 . 4 3 3 7 1 3 8 8 9 0 . 1 2 8 2 1 6 5 5 1 9 6 . 6 7 0 1 2 6 2 7 0 . 6 4 4 4 6 7 5 0 8

0 . 4 7 . 1 3 9 3 7 4 1 3 4 0 . 1 0 8 5 3 8 8 8 8 8 4 . 9 0 3 4 0 1 2 2 0 . 5 6 6 0 2 2 6 7 5

0 . 5 6 . 1 0 6 7 3 9 9 5 8 0 . 0 9 2 8 3 9 8 9 8 7 5 . 5 1 5 8 1 7 8 0 . 5 0 3 4 3 8 7 8 5

0 . 6 5 . 2 9 7 9 4 4 6 9 6 0 . 0 8 0 5 4 3 8 9 9 6 8 . 1 6 3 1 3 3 6 0 . 4 5 4 4 2 0 8 9 1

0 . 7 4 . 6 8 3 3 0 9 9 1 6 0 . 0 7 1 1 9 9 6 9 4 6 2 . 5 7 5 5 4 4 6 9 0 . 4 1 7 1 7 0 2 9 8

0 . 8 4 . 2 4 0 3 0 3 4 3 4 0 . 0 6 4 4 6 4 7 2 9 5 8 . 5 4 8 2 1 3 0 4 0 . 3 9 0 3 2 1 4 2

0 . 9 3 . 9 5 2 6 6 2 2 5 6 0 . 0 6 0 0 9 1 7 6 1 5 5 . 9 3 3 2 9 3 2 4 0 . 3 7 2 8 8 8 6 2 2

1 3 . 8 9 6 4 3 8 8 3 0 . 0 5 9 2 3 7 0 0 5 5 4 . 6 3 5 0 1 1 8 3 0 . 3 6 4 2 3 3 4 1 2

analysis of parameter variations in heat sink effectiveness

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