Dinc Ibrahim. Refrigeration systems and applications 2th edition

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Heat Pipes 391

Table 7. 1 Some heat pipe ﬂuids and their temperature ranges.

Medium Melting Point Temperature (

◦

C) Boiling Point Temperature (

◦

C) Application Range (

◦

(at Atmospheric Pressure)

Helium −271 −261 −271 to −269

Nitrogen −210 −196 −203 to −160

Ammonia −78 −33 −60 to 100

Acetone −95 57 0 to 120

Methanol −98 64 10 to 130

Flutec PP2 −50 76 10 to 160

Ethanol –112 78 0 to 130

Water 0 100 30 to 200

Toluene −95 110 50 to 200

Mercury −39 361 250 to 650

Sodium 98 892 600 to 1200

Lithium 179 1340 1000 to 1800

Silver 960 2212 1800 to 2300

Source: Narayanan (2001).

boiling temperatures between 250 and 375 K both cryogenic heat pipes (operating in the 5 to 100 K

temperature range) and liquid-metal heat pipes (operating in the 750 to 5000 K temperature range)

have been developed and used. Figure 7.8 illustrates the possible working temperature ranges for

some of the various heat pipe ﬂuids. In addition to the thermophysical properties of the working

ﬂuid, consideration must be given to the ability of the working ﬂuid, to the wettability of the

working ﬂuid, and to the wick and wall materials. Further criteria for the selection of the working

ﬂuids, including a number of other factors such as liquid and vapor pressure and compatibility of

the materials, are considered.

7.6.3 Selection of Working Fluid

Because the basis for operation of a heat pipe is the vaporization and condensation of the working

ﬂuid, selection of a suitable working ﬂuid is perhaps the most important aspect of the design and

manufacture process. Factors affecting the selection of an appropriate working ﬂuid include

• operating temperature range,

• vapor pressure,

• thermal conductivity,

• compatibility with the wick and case materials, the stability, and

• toxicity.

It should be noted that the theoretical operating temperature range for a given heat pipe is

typically between the critical temperature and triple state of the working ﬂuid. Above the critical

temperature, the working ﬂuid exists in a vapor state and no increase in pressure will force it to

392 Refrigeration Systems and Applications

Increasing

heat

transport

capability

Cryogenic heat pipes Low temperature

heat pipes

High temperature

heat pipes

(CH

)CO

F-11

F-21

10 50 100 500 1000 5000

Temperature (°K)

Figure 7.8 Temperature ranges of some heat pipe working ﬂuids (Peterson, 1994).

return to a liquid state. As a result, when working ﬂuids are above their critical temperature, the

capillary pumping mechanism provided by the wicking structure ceases to function. Similarly, when

the operating temperature is below the triple state, the working ﬂuid exists in the solid and vapor

states. While some heat transfer may occur due to sublimation, operation in this temperature range

should be avoided (Peterson, 1994).

7.6.4 Wick or Capillary Structure

The concept of utilizing a wicking structure as part of a passive two-phase heat-transfer device

capable of transferring large quantities of heat with a minimal temperature drop was ﬁrst introduced

by Gaugler (1944).

The wick or capillary structure is porous and made of materials like steel, aluminum, nickel, or

copper in various ranges of pore sizes. They are fabricated using metal foams and, more particularly,

felts, the latter being more frequently used. By varying the pressure on the felt during assembly,

various pore sizes can be produced. By incorporating removable metal mandrels, an arterial structure

can also be molded in the felt.

Fibrous materials, like ceramics, have also been used widely. They generally have smaller pores.

The main disadvantage of ceramic ﬁbers is that they have little stiffness and usually require a

continuous support by a metal mesh. Thus, while the ﬁber itself may be chemically compatible

with the working ﬂuids, the supporting materials may cause problems. More recently, interest has

turned to carbon ﬁbers as a wick material. Carbon ﬁber ﬁlaments have many ﬁne longitudinal

grooves on their surface, have high capillary pressures, and are chemically stable. Many heat pipes

constructed using carbon ﬁber wicks seem to show a greater heat transport capability.

The main goal of the wick is to generate capillary pressure to transport the working ﬂuid from

the condenser to the evaporator. It must also be able to distribute the liquid around the evaporator

section to any area where heat is likely to be received by the heat pipe. Often, these two functions

Heat Pipes 393

require wicks of different forms. The selection of the wick for a heat pipe depends on many factors,

several of which are closely linked to the properties of the working ﬂuid.

Note that the maximum capillary head generated by a wick increases with decrease in pore size.

The wick permeability increases with increasing pore size. Another feature of the wick which must

be optimized is its thickness. The heat transport capability of the heat pipe is raised by increasing

the wick thickness. The overall thermal resistance at the evaporator also depends on the conductivity

of the working ﬂuid in the wick. There are some other necessary properties of the wick, namely,

compatibility with the working ﬂuid and wettability.

The two most important properties of a wick are the pore radius and the permeability. The pore

radius determines the pumping pressure that the wick can develop. The permeability determines

the frictional losses of the ﬂuid as it ﬂows through the wick. The most common types of wicks

that are used are as follows (Narayanan, 2001):

• Sintered powder metal. This process will provide high power handling, low temperature gradi-

ents, and high capillary forces for antigravity applications. A complex sintered wick with several

vapor channels and small arteries is used to increase the liquid ﬂow rate. Very tight bends in the

heat pipe can be achieved with this type of structure.

• Grooved tube. The small capillary driving force generated by the axial grooves is adequate for

low-power heat pipes when operated horizontally or with gravity assistance. The tube can be

readily bent. When used in conjunction with screen mesh, the performance can be considerably

enhanced.

• Screen mesh or cable or ﬁber. This type of wick is used in the majority of the products and

provides readily variable characteristics in terms of power transport and orientation sensitivity,

according to the number of layers and mesh counts used.

Figure 7.9 shows several heat pipe wick structures. It is important to select the proper wick

structure for your application. The above list is in order of decreasing permeability and decreasing

pore radius. Grooved wicks have a large pore radius and a high permeability; as a result, the pressure

losses are low and the pumping head is also low. Grooved wicks can transfer high heat loads in

a horizontal or gravity-aided position, but cannot transfer large loads against gravity. The powder

metal wicks at the opposite end of the list have small pore radii and relatively low permeability. They

are limited by pressure drops in the horizontal position but can transfer large loads against gravity.

The wicking structure has two functions in heat pipe operation:

• providing the mechanism by which the working ﬂuid is returned from the condenser to the

evaporator and

• ensuring that the working ﬂuid is evenly distributed over the evaporator surface.

Figure 7.9 Wick structures (Garner, 1996) (Reproduced by permission of Flomerics, Inc.).

394 Refrigeration Systems and Applications

Wrapped screen Sintered metal Axial groove

Simple homogeneous

(a)

Current composite

(b)

Slab Pedestal artery Spiral artery Tunnel artery

Monogroove

Advanced designs

(c)

Axial groove

(non-constant

groove width)

Double wall

artery

Channel wick

Figure 7.10 Some common heat pipe wicking conﬁgurations and their structures. (a) Simple homogeneous,

(b) current composite, and (c) advanced designs (Peterson, 1994).

Figure 7.10 illustrates several common wicking structures presently in use, along with several

more advanced concepts under development. In order to provide a ﬂow path with low ﬂow resistance

through which the liquid can be returned from the condenser to the evaporator, an open porous

structure with a high permeability is desirable. However, to increase the capillary pumping pressure,

a small pore size is necessary. Solutions to this apparent dichotomy can be achieved through the

use of a nonhomogeneous wick made of several different materials or through composite wicking

structures similar to those shown in Figure 7.10b.

The wicking structure has two functions in heat pipe operation: it is both the vehicle and the

mechanism through which the working ﬂuid returns from the condenser to the evaporator and it

ensures that the working ﬂuid is evenly distributed circumferentially over the entire evaporator

surface. Figure 7.10 illustrates several common wicking structures presently in use, along with

several composite and high capacity concepts under development. As shown in Figure 7.10, the

various wicking structures can be divided into three broad categories as follows:

• Homogeneous structures. Homogeneous wicks typically comprise of a single material and

are distributed uniformly along the axial length of the heat pipe. The most common types of

homogeneous wicks include wrapped screen, sintered metal, axially grooved and crescent as

shown in Figure 7.10a.

Heat Pipes 395

• Current composite. In order to provide a ﬂow path with low ﬂow resistance through which

the liquid can be returned from the condenser to the evaporator, an open porous structure with

a high permeability is desirable. However, to increase the capillary pumping pressure, a small

pore size is necessary. Solutions to this apparent dichotomy can be achieved through the use

of a nonhomogeneous wick made of several different materials or through a composite wicking

structure. Composite wicks are typically comprised of a combination of several types or porosities

of materials and/or conﬁgurations. Examples of these types of wick structures are illustrated in

Figure 7.10b.

• Advanced designs. Most of these are relatively new (Figure 7.10c) and consist of variations

on the composite wicking structures. Again, the two functions of the wicking structure (i.e.,

circumferential distribution and axial ﬂuid transport) are achieved by different segments of the

capillary structure. The basic design of this advanced capacity conﬁguration consists of two large

axial channels, one for vapor ﬂow and the other for liquid ﬂow. In this type of heat pipe, several

improvements result from the separation of the liquid and vapor channels. First, because the axial

liquid transport can be handled independently from the circumferential distribution, a high heat

transport capacity can be achieved. Second, by separating the two channels, the viscous pressure

drop normally associated with heat pipes in which the liquid and vapor ﬂows occur within the

same channel can be greatly reduced. Third, with the majority of the ﬂuid located in an external

artery, heat transfer in the evaporator and condenser takes place across a relatively thin ﬁlm of

liquid in the circumferential wall grooves, thereby increasing the heat-transfer coefﬁcient. While

somewhat different in shape, the basic principle of operation of the other advanced designs is the

same: Separate the circumferential distribution and axial liquid ﬂow to maximize the capillary

pumping and reduce the liquid pressure drop.

7.7 Operational Principles of Heat Pipes

Inside the container is a liquid under its own pressure that enters the pores of the capillary material,

wetting all internal surfaces. Applying heat at any point along the surface of the heat pipe causes the

liquid at that point to boil and enter a vapor state. When that happens, the liquid picks up the latent

heat of vaporization. The gas, which then has a higher pressure, moves inside the sealed container

to a colder location where it condenses. Thus, the gas gives up the latent heat of vaporization and

moves heat from the input to the output end of the heat pipe. Heat pipes have an effective thermal

conductivity many thousands of times that of copper. The heat transfer or transport capacity of a

heat pipe is speciﬁed by its “Axial Power Rating (APR).” It is the energy moving axially along the

pipe. The larger the heat pipe diameter, the greater is the APR. Similarly, the longer the heat pipe

the lesser is the APR. Heat pipes can be built in almost any size and shape.

Heat pipes transfer heat by the evaporation and condensation of a working ﬂuid. As stated earlier,

a heat pipe is a vacuum-tight vessel which is evacuated and partially backﬁlled with a working

ﬂuid. As heat is input at the evaporator, ﬂuid is vaporized, creating a pressure gradient in the

pipe. This pressure gradient forces the vapor to ﬂow along the pipe to the cooler section where

it condenses, giving up its latent heat of vaporization. The working ﬂuid is then returned to the

evaporator by capillary forces developed in the porous wick structure or by gravity.

A heat pipe is said to be operating against gravity when the evaporator is located above the con-

denser. In this orientation, the working ﬂuid must be pumped against gravity back to the evaporator.

All heat pipes have wick structures that pump the working ﬂuid back to the evaporator using the

capillary pressure developed in the porous wick. The ﬁner the pore radius of a wick structure,

the higher the heat pipe can operate against gravity. A thermosiphon is similar to a heat pipe, but

has no wick structure and will only operate gravity aided.

A heat pipe (Figure 7.7) consists of a vacuum-tight envelope, a wick structure, and a working

ﬂuid. The heat pipe is evacuated and then backﬁlled with a small quantity of working ﬂuid, just

396 Refrigeration Systems and Applications

enough to saturate the wick. The atmosphere inside the heat pipe is set by an equilibrium of liquid

and vapor. As heat enters at the evaporator this equilibrium is upset, generating vapor at a slightly

higher pressure. This higher pressure vapor travels to the condenser end where the slightly lower

temperatures cause the vapor to condense, giving up its latent heat of vaporization. The condensed

ﬂuid is then pumped back to the evaporator by the capillary forces developed in the wick structure.

This continuous cycle transfers large quantities of heat with very low thermal gradients. A heat

pipe’s operation is passive, being driven only by the heat that is transferred. This passive operation

results in high reliability and long life.

Both heat pipes and thermosiphons operate on a closed two-phase cycle and utilize the latent

heat of vaporization to transfer heat with very small temperature gradients. However, the operation

of these two devices is signiﬁcantly different. In a heat pipe, as illustrated earlier, heat added to the

bottom portion of a thermosiphon vaporizes the working ﬂuid. During this phase change process,

the ﬂuid picks up the heat associated with its latent heat of vaporization. Because the vapor in the

evaporator region is at a higher temperature and hence at a higher pressure than the vapor in the

condenser, the vapor rises and ﬂows to the cooler condenser where it gives up the latent heat of

vaporization (buoyancy forces assist this process). Gravitational forces then cause the condensate

ﬁlm to ﬂow back down the inside of the heat pipe wall where it can again be vaporized. Although

the inside surface of a thermosiphon may occasionally be lined with grooves or a porous structure

to promote return of the condensate to the evaporator or increase the heat-transfer coefﬁcient,

thermosiphons principally rely upon the local gravitational acceleration for the return of the liquid

from the evaporator to the condenser. By deﬁnition, then, for proper operation the evaporator of a

thermosiphon must be located below the condenser or dryout of the evaporator will occur.

Alternatively, heat pipes utilize some sort of capillary wicking structure to promote the ﬂow of

liquid from the condenser to the evaporator and as a result can be used in a horizontal orientation,

microgravity environments, or even applications where the capillary structure must “pump” the

liquid against gravity from the evaporator to the condenser. It is this single characteristic – the

dependence of the local gravitational ﬁeld to promote the ﬂow of the liquid from the condenser to

the evaporator – that differentiates thermosiphons from heat pipes.

7.7.1 Heat Pipe Operating Predictions

Historically, the use of metallic heat sinks has been sufﬁcient to provide the required thermal

management for most electronic cooling applications. However, with the new breed of compact

devices dissipating larger heat loads, the use of metallic heat sinks is sometimes limited because

of the weight and physical size required. Accordingly the use of heat pipes is becoming a solution

of choice.

The performance of natural convection heat sinks is directly dependent on the effective surface

area; more effective surface area results in better performance. A heat pipe embedded into the base

material of a standard aluminum extrusion can reduce the overall temperature difference along the

base material, tending to isothermalize the base material. In essence, the localized heat source is

spread equally along the length of the heat pipe, increasing the overall efﬁciency of the heat sink.

Although an embedded heat pipe heat sink is slightly more expensive because of the added cost of

the heat pipe, it is an easy method of improving the performance of a marginal extrusion.

A more elegant approach is to design a heat sink that fully utilizes the characteristics of a heat

pipe. Typical extruded heat sinks have limited aspect ratios and thick ﬁns, which result in lower

surface area per length. The material thickness adds unnecessary weight, and, more importantly,

obstructs the cooling air ﬂow. To alleviate the extrusion limits, bonded ﬁn heat sinks have been

developed which allow the use of a tall, thin ﬁn, which optimizes cooling ﬂow. But bonded ﬁn heat

sinks can also be limited by the conduction losses in the base plate for concentrated heat sources.

Heat Pipes 397

A heat pipe used in conjunction with parallel plate ﬁns provides more efﬁcient surface area with

minimum volume demands. This design application is useful when there is not enough physical

volume or airﬂow above the device to use an extrusion, and allows the designer much latitude

in component arrangement. The heat pipe can transport the heat to a “remote” parallel plate ﬁn

stack that has enough volume to dissipate the heat. Heat pipes can be designed into most electronic

devices for various power levels, and may even allow the use of a natural convection heat sink

(DeHoff and Grubb, 2000).

Predicting or developing an optimum heat pipe thermal solution requires use of theoretical and

empirical relationships, wisdom and design experience, and knowledge of the application design

parameters and system ﬂexibilities. For design concepts and preliminary designs, it can be useful

to have a guideline for heat pipe performance. These are general performance guidelines based

on a “standard” powder metal wick structure. Alternative powders and production techniques are

available that may increase the performance by upward of 500%. The above operating limitations

can be summarized to predict heat pipe performances based on three orientation categories and the

performance proﬁles for such cases are shown in Figure 7.11.

(a)

(b)

(c)

250

200

150

100

Heat pipe capacity

(Watts)

Heat pipe capacity

(Watts-Inches)

350

300

250

200

150

100

0 50 100 150 200 250 300

Operating temperature (°C)

0 50 100 150 200 250 300

Operating temperature (°C)

0 50 100 150 200 250 300

Operating temperature (°C)

0.375"

0.25"

4 mm

3 mm

0.375"

0.25"

4 mm

3 mm

Performance factor

1.0

0.8

0.6

0.4

0.2

0.0

1 in.

2 in.

5 in.

10 in.

Figure 7.11 Performance curves: (a) for gravity-aided operation, (b) for horizontal operation, and (c) for

various heights against gravity (Courtesy of Thermacore International, Inc.).

398 Refrigeration Systems and Applications

7.7.1.1 Gravity-Aided Orientation

The evaporator is at a lower elevation than the condenser. The gravity-aided orientation is the most

efﬁcient, since the heat pipe acts as a thermosiphon and gravity will return the condensed ﬂuid to

the evaporator. A sintered powder wick structure may still be needed to handle the heat ﬂux in the

evaporator. Heat pipe operation is typically limited by the ﬂooding limit or the boiling limit (at

elevated temperatures above 175

◦

C). These two limitations are greatly affected by the diameter of

the heat pipe: a larger diameter heat pipe will carry more power. Figure 7.11a can be used as a

guideline for the selection of a “standard” copper–water powder wick heat pipe in the gravity-aided

orientation. The area below each curve is the allowable operating region. For the miniature heat

pipes (3 and 4 mm) use the greater of the “gravity-aided” and “horizontal” curves.

7.7.1.2 Horizontal Orientation

The horizontal orientation relies on the wick structure to provide the capillary pressure to return

the condensed ﬂuid to the evaporator. The heat pipe operation is typically limited by the capillary

limit. This limitation is greatly affected by the diameter of the heat pipe (a larger diameter heat

pipe will carry more power) and the length of the heat pipe (a longer heat pipe will carry less

power). A useful parameter is the effective length (DeHoff and Grubb, 2000):

eff

= L

adi

+ 0.5(L

eva

+ L

con

) (7.1)

where L

eff

is the effective length of the heat pipe, m; L

adi

is the length of the adiabatic section of

the heat pipe, m; L

eva

is the length of the evaporator section of the heat pipe, m; and L

con

is the

length of the adiabatic section of the heat pipe, m.

In conjunction with this, Figure 7.11b can be used as a guideline for the selection of a “standard”

copper–water powder wick heat pipe in the horizontal orientation. The capacity of a heat pipe can

be determined by taking the appropriate value from the ﬁgure in “watt-inches” and dividing by the

effective length. For example, a 0.25 in. OD heat pipe with a total length of 8 in., an evaporator

length of 1 in., and a condenser length of 5 in. operated at 25

◦

C gives an effective length of 5 in.

Therefore the heat pipe can carry 20 W [100 W-in. (from Figure 7.11b)/5 in.].

7.7.1.3 Against Gravity Orientation

The evaporator is at a higher elevation than the condenser. Heat pipe operation against gravity

orientation relies solely on the wick structure to return the condensed ﬂuid up to the higher evapo-

rator. Again the heat pipe operation is limited by the capillary limit. This orientation is very similar

to the horizontal orientation, except that the effects of gravity must be accounted for. A larger

elevation difference between the evaporator and the condenser results in a lower power capacity.

Figure 7.11c shows the performance factor that can be used as a guideline for the selection of

a “standard” copper–water powder wick heat pipe in the horizontal orientation. The performance

factor must be applied to the capacity obtained from Figure 7.11b. For example, the heat pipe from

the previous example operated 5 in. against gravity would carry 14 W [0.7 (from Figure 7.11c) ×

20 W from above] (DeHoff and Grubb, 2000).

7.7.2 Heat Pipe Arrangement

As mentioned above, the orientation and layout of a heat pipe design are critical. When the design

allows, the heat source should be located below or at the same elevation as the cooling section for

best performance. This orientation allows gravity to aid the capillary action, and results in a greater

Heat Pipes 399

heat-carrying capability. If this orientation is unacceptable, then a sintered powder wick structure

will be necessary. Additionally, heat pipes have the ability to adhere to the physical constraints

of the system, and can be bent around obstructions. For a cylindrical heat pipe, the typical bend

radius is three times the heat pipe diameter. Tighter bend radii are possible, but may reduce the

heat-transfer capability. Since a bend in the heat pipe will have a small impact on performance,

the number of bends should be limited. A good rule of thumb is to assume a 1

◦

C temperature

loss for each 90

◦

bend, if the heat pipe is not operated near one of the limitations. The heat pipe

is also capable of being ﬂattened (a 3-mm diameter heat pipe can be ﬂattened to 2 mm) (DeHoff

and Grubb, 2000). Again, ﬂattening has a minimal effect on performance if the vapor space is not

collapsed or the heat pipe is not operating near a limitation. The heat pipe can conform to the

system space restrictions and can transport the heat from the source to the ﬁn stack or heat sink,

where the heat is effectively dissipated (DeHoff and Grubb, 2000).

7.8 Heat Pipe Performance

The above sections provided an overview of heat pipes and their performance. More important,

though, is the proper use of the heat pipe in a heat sink and the increased heat sink capabilities

that are provided by the utilization of heat pipes. Most applications use a remote ﬁn stack design,

which consists of an aluminum evaporator block (heat input section), the heat pipe (heat transport

section), and aluminum ﬁns (heat dissipation section).

Over the last 10 years, a host of computationally inclined heat pipe investigators in the United

States have been busy modeling heat pipe transient operation. The difﬁculty of transient heat pipe

modeling can be immense, especially if a simulation of the frozen start-up problem is attempted.

Important mechanisms related to transient heat pipe operation include the transition from free

molecule to continuum ﬂow in the vapor space, the migration of the melt front in capillary structures,

mass transfer between the liquid and vapor regions, compressibility effects and shock formation

in the vapor ﬂow, and the possibility of externally imposed body forces on the working ﬂuid in

its liquid phase. Performance-limiting mechanisms during power transitions in recently proposed

heat pipe systems include evaporator entrainment, freeze-out of the working ﬂuid inventory in the

condenser, evaporator capillary limits, and nucleate boiling departure in the evaporator.

A thermal resistance network, analogous to electrical circuits, is the quickest way to predict

the overall performance of a parallel plate/heat pipe heat sink. The thermal resistance network is

considered a good approach to determine design feasibility (DeHoff and Grubb, 2000). The heat

pipe heat sink can be represented by a resistance network, as shown in Figure 7.12. Although this

network neglects the interface between the device and heat sink, it can easily be added.

Each of the above resistances can be solved to calculate an associated temperature drop using

the Fourier’s law of conduction equation as follows:

Q = kA

T

(7.2)

and with the thermal resistance it can be rewritten as

Q =

T

(7.3)

Therefore, the temperature drop in the evaporator section (block) can be calculated by the

conduction heat transfer for the evaporator section (R



eva

T

eva

(7.4)

400 Refrigeration Systems and Applications

eva

air

fin

int

Figure 7.12 Thermal resistance network for a heat pipe (Courtesy of Thermacore International, Inc.).

The loss associated with the interface between the evaporator block and the heat pipe (R

int

)

can be calculated using the thermal resistance of the interface material, which is typically solder



int

≈ 0.5

◦

C/W · cm

) or thermal epoxy (R



int

≈ 1.0

◦

C/W · cm

), and the interface area.

T

int



int

πD

eva

(7.5)

The detailed analysis of heat pipe is rather complex. The total thermal resistance of a heat pipe

is the sum of the resistances due to conduction through the evaporator section wall and wick,

evaporation or boiling, axial vapor ﬂow, condensation, and conduction losses back through the

condenser section wick and wall. A rough guide for a copper/water heat pipe with a powder metal

wick structure is to use 0.2

◦

C/W · cm

for the thermal resistance at the evaporator and condenser

(applied over the heat input/output areas) and 0.02

◦

C/W · cm

for axial resistance (applied over the

cross-sectional area of the vapor space) in the following equation:

T



eva

πD

eva



ax−hp

πD





con

πD

con

(7.6)

The resistance in transferring the heat from the ﬁn to the air (R

) is calculated using the

convection coefﬁcient as follows:

T

ﬁn

(7.7)

The conductive losses that are associated with the ﬁn (R

ﬁn

) are governed by the ﬁn efﬁciency

which is deﬁned as

ﬁn

tanh(m

ﬁn

eff

)

ﬁn

eff

(7.8)

where

ﬁn



ﬁn

(7.9)

So, the temperature drop in the ﬁn results in

T

ﬁn

= T

(1 − η

ﬁn

) (7.10)

The temperature drop in the air by cooling can be written as

T

air



˙mc



(7.11)

The overall performance of the sink is the sum of the individual temperature drops as follows:

T

total

= T

eva

+ T

int

+ T

ﬁn

+ T

air

(7.12)