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Endorsed byThe Institution of Chemical EngineersThe Institution of Mechanical Engineers

Heat pipes – performance ofcapillary-driven designs

Issued September 1979With Amendment A

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ESDU 79012ESDU DATA ITEMS

Data Items provide validated information in engineering design and analysis for use by, or under the supervisionof, professionally qualified engineers. The data are founded on an evaluation of all the relevant information, bothpublished and unpublished, and are invariably supported by original work of ESDU staff engineers or consultants.The whole process is subject to independent review for which crucial support is provided by industrial companies,government research laboratories, universities and others from around the world through the participation of someof their leading experts on ESDU Technical Committees. This process ensures that the results of much valuablework (theoretical, experimental and operational), which may not be widely available or in a readily usable form, canbe communicated concisely and accurately to the engineering community.

We are constantly striving to develop new work and review data already issued. Any comments arising out of youruse of our data, or any suggestions for new topics or information that might lead to improvements, will help us toprovide a better service.

THE PREPARATION OF THIS DATA ITEM

The work on this particular Item was monitored and guided by the following Working Party:

on behalf of the Heat Transfer Steering Group which first met in 1966 and now has the following membership:

The Steering Group has benefitted from the participation of members from several engineering disciplines. Inparticular, Mr E.C. Firmin has been appointed to represent the interests of mechanical engineering as the nomineeof the Institution of Mechanical Engineers and Dr G.F. Hewitt has been appointed to represent the interests ofchemical engineering as the nominee of the Institution of Chemical Engineers.

The work on this Item was carried out in the Heat Transfer Group of ESDU under the supervision of Mr N. Thompson,Group Head. The member of staff who undertook the technical work involved in the initial assessment of theavailable information and the construction and subsequent development of the Item was

Dr D. Chisholm — National Engineering LaboratoryMr J.B. Goodacre — Marconi Research Laboratories, ChelmsfordMr G. Rattcliff — Isoterix LtdMr D.A. Reay — International Research and Development Co. LtdDr G. Rice — Reading University

ChairmanDr G.F. Hewitt — H.T.F.S. Atomic Energy Authority, Harwell

Vice-ChairmanProf. V. Walker — Bradford University

MembersDr T.R. Bott — Birmingham UniversityMr E.C. Firman — IndependentMr J.A. Hitchcock — Central Electricity Research LaboratoriesProf. R.H. Sabersky*

* Corresponding Member

— California Institute of Technology, USAMr E.A.D. Saunders — Whessoe LtdMr R.A. Smith — Imperial Chemical Industries LtdMr M.A. Taylor — B.N.O.C. (Developments) LtdMr N.G. Worley — Babcock and Wilcox (Operations) Ltd.

Dr A. Acton — Senior Engineer.

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ESDU 79012HEAT PIPES – PERFORMANCE OF CAPILLARY-DRIVEN DESIGNS

CONTENTS

1. NOTATION AND UNITS 1

2. INTRODUCTION 42.1 Scope 42.2 Steady-state Performance 4

3. OVERALL THERMAL RESISTANCE 7

4. MAXIMUM HEAT TRANSFER – CIRCULATION LIMIT 94.1 Overall Pressure Criterion 94.2 Maximum Capillary Pressure Difference, 94.3 Pressure Drop in Liquid, 104.4 Pressure Drop in Vapour, 11

4.4.1 Laminar Flow 114.4.2 Turbulent Flow 124.4.3 Overall Pressure Drop 12

4.5 Gravity-assisted Heat Pipes 13

5. MAXIMUM HEAT TRANSFER – OTHER LIMITS 145.1 Vapour Pressure limit 145.2 Sonic Limit 145.3 Entrainment Limit 155.4 Boiling Limit 15

6. EXAMPLE 17

7. REFERENCES AND DERIVATION 24

8. TABLE OF THERMAL RESISTANCES 26

9. PROCEDURE FOR PERFORMANCE CALCULATION 27

FIGURES 1 TO 5 30 to 32

APPENDIX A CALCULATION OF PRESSURE DROPS FOR LAMINAR VAPOUR FLOW 33

A1. NOTES ON METHOD 33

∆pσ∆pl∆pv

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HEAT PIPES – PERFORMANCE OF CAPILLARY-DRIVEN DESIGNS

1. NOTATION AND UNITS

Notes

(i) Excepting angles (degrees), the SI system of units is used throughout.

(ii) All physical properties of the working fluid are to be evaluated at saturation conditionscorresponding to the effective operating temperature of the heat pipe, , unless stated otherwisein the text.

cross-sectional area m2

speed of sound in vapour m/s

diameter m

equivalent diameter, 4 for a duct (equal to for a duct of circular cross section) or 4 (volume of pores)/(surface area of pores) for a porous material

m

local gravitational acceleration* m/s2

heat transfer coefficient W/(m2K)

fluid permeability of wick m2

specific latent heat of vaporisation J/kg

length m

reduced adiabatic section length,

transverse internal dimension of heat pipe container in plane defined by container axis and direction of gravity (see Sketch 2.1)

m

Mach number†,

perimeter of duct m

vapour pressure Pa

overall rate of heat transfer W

gas constant‡ J/(kgK)

axial Reynolds number† for liquid in wick,

For Footnotes refer to end of Notation Section

Teff

A

a

D

DE A /P D

g

h

K

L

l

la* la DEvRev( )⁄

li

M Vv a⁄

P

pv

Q·

R

Rel ρlVlDEl εµl( )⁄

Issue September 1979

1

With Amendment A, October 1980

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axial Reynolds number† in vapour,

radial Reynolds numbers for vapour flow in evaporator and condenser,

radius m

surface area m2

parameter characterising radial distribution of axial vapour velocity

absolute temperature K

effective operating temperature, (evaporator outside surface temperature + condenser outside surface temperature)

K

thickness m

superficial liquid velocity† in wick, m/s

average vapour velocity†, m/s

characteristic length in entrainment equation (see Section 5.3) m

thermal resistance K/W

ratio of specific heat capacity at constant pressure to that at constant volume**

pressure drop in liquid Pa

pressure drop in vapour Pa

maximum capillary pressure difference Pa

temperature difference K

porosity, (volume of pores)/(volume of porous material)

contact angle degrees

thermal conductivity W/(m K)

dynamic viscosity N s/m2

density kg/m3

surface tension N/m

angle made by axis of heat pipe to horizontal (see Sketch 2.1) degrees

For Footnotes refer to end of Notation Section

Rev ρvVvDEv εµv( )⁄

ReDe ,ReDcDEvRev 4 le( ) ,DEvRev 4 lc( )⁄

r

S

s

T

Teff ½

t

Vl Q AwLρl( )⁄

Vv Q AvLρv( )⁄

x

z

γ

∆pl

∆pv

∆pσ

∆T

ε

θ

λ

µ

ρ

σ

φ

2

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Subscripts

refers to adiabatic section

refers to condenser

dummy subscript representing or

refers to evaporator

refers to effective value of a quantity

refers to internal dimension

refers to liquid or liquid flow-channel

refers to maximum value of a quantity

refers to bubble nucleation

refers to external dimension

refers to vapour or vapour duct

refers to wick

refers to container wall

refers to wick capillary

* Standard value at sea level = 9.81 m/s2.† Evaluated in adiabatic section or at junction of evaporator and condenser if there is no adiabatic section.‡ R = (universal gas constant)/(molecular weight of fluid). Universal gas constant = 8 314 J/(Kmol K).** In the range of temperatures over which heat pipes are operated, lies between 1.67 (most monatomic vapours) and a

limiting value of unity (approached for vapours composed of complex molecules).

For Footnotes refer to end of Notation Section

a

c

d a e

e

eff

i

l

M

n

o

v

w

x

σ

γ

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2. INTRODUCTION

2.1 Scope

This Data Item, which is one of a series concerned with heat pipes, provides methods of estimating thesteady-state thermal performance of the most commonly-used type of heat pipe, i.e. a heat pipe that

(i) is of tubular construction with uniform cross section,

(ii) contains a single-component working fluid and no non-condensible gas,

(iii) operates with a small-pore wick,

(iv) uses capillary action in the wick to provide the driving force for liquid flow and,

(v) operates with approximately uniform surface heat fluxes.*

The Data Item is primarily intended to be used for rating purposes, that is to calculate the performance ofan existing heat pipe under prescribed boundary conditions. A procedure for this calculation is shown inthe flow charts in Section 9. The information in the Data Item is also useful for design purposes, althoughno method for optimising the design has been included because many optimising parameters exist and it isdifficult to make generalisations about their relative importance.

A computer Data Item in the series (Reference 7) gives data on the properties of small-pore wicks. A furtherData Item in the Series (Reference 8) gives introductory information on capillary-driven and other typesof heat pipe, plus data on types of wick, compatibility of working fluids with constructional materials,safety and transient performance.

2.2 Steady-state Performance

Sketch 2.1 illustrates the type of heat pipe considered. It comprises an evaporator, an optional adiabaticsection and a condenser. The wick structure lies in close contact with the inner wall of the container, andthe remaining space forms the vapour duct. The angle the heat pipe axis makes with the horizontal, , maybe positive (or zero) corresponding to “normal” operation or negative corresponding to gravity-assistedoperation.

* Throughout this Data Item, heat flux is defined as the rate of heat transfer per unit surface area.

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Sketch 2.1

In essence, the performance calculation detailed in the flow charts in Section 9 consists of two parts. Firstly,any unknown evaporator or condenser surface temperatures are calculated; the overall rate of heat transfer,

, is also calculated if it is not known. Secondly, the overall rate of heat transfer is compared with variouscalculated limits to ensure that they are not exceeded.

Three common types of boundary condition at either the evaporator or the condenser are considered.

(i) The outside surface heat flux is known, in which case the overall rate of heat transfer is equal tothe value of the heat flux multiplied by the appropriate surface area. (The overall rate of heat transfermay also be directly specified).

(ii) Heat is transferred from or to a source or sink at a known temperature with a known heat transfercoefficient.

(iii) The outside surface temperature is known. This is a special case of (ii) arising when the heat transfercoefficient is so large that it can be taken as infinite.

Under normal operating conditions, the heat pipe is almost isothermal although small temperaturedifferences must still exist within it. The data in Section 3 can then be used with the known boundaryconditions to estimate the overall rate of heat transfer or surface temperatures as required.

Heat is transferred in a heat pipe by a process of evaporation and condensation of the circulating fluid.Capillary action in the wick provides the driving force needed to overcome the pressure drops associatedwith the flows of liquid and vapour. Normally, the gravitational force is only significant in the liquid, andit either aids or hinders the fluid circulation in a non-horizontal pipe depending upon whether the condenser

Directionof gravity

Condenser

Adiabatic

EvaporatorVapour duct

Wick

Liquid flow

Vapour flow

Q in(heat source)

.

Q out(heat sink)

ta

te

tc

.90°

–90°

0°φ

Q·

5

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or the evaporator is uppermost. By their action on the circulation, these forces and pressure drops limit theoverall rate of heat transfer to a maximum value. the maximum value, called the circulation limit, can beobtained from data in Section 4. In some cases when is negative, the gravitational force can drive theliquid flow in the wick rather than the capillary force. Operation of a heat pipe under these conditions isnot within the scope of this Data Item, see Section 4.5.

The overall rate of heat transfer can also be limited by other factors. They are,

(i) vapour limitations (vapour pressure and sonic limits),

(ii) the tendency for a high-velocity vapour flow to entrain liquid droplets at the vapour-wick interfaceand return them to the condenser before they reach the evaporator (entrainment limit) and

(iii) the maximum heat flux that can occur in the evaporator without the onset of boiling (boiling limit).

Data for calculating these limits are given in Section 5. The calculation procedure in Section 9 is designedto avoid the vapour pressure and sonic limits. This is desirable because appreciable temperature gradientsoccur in a heat pipe as these limits are approached, making the heat pipe unsuitable for most applications.However, data for calculating these limits are included for completeness and because they can be importantduring start-up and fault conditions.

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3. OVERALL THERMAL RESISTANCE

When operating below its maximum overall rate of heat transfer, , the performance of a heat pipe canbe characterised by the overall thermal resistance*, . The actual overall rate of heat transfer, , and theoverall temperature difference between the heat source and the heat sink, , are then related by,

. (3.1)

The overall thermal resistance can be represented by the idealised network of thermal resistances, to, as shown in Sketch 3.1.

Sketch 3.1 Thermal resistances and their locations

Equations for calculating to are given in Section 8 for a circular cylinder and also for a plane sectionso that estimates can be made for heat pipes with non-circular cross sections. The thermal resistances ariseas follows.

and The thermal resistances between the heat source and the evaporator external surface andbetween the condenser external surface and the heat sink respectively. In some cases thetemperature of one or both of the evaporator and condenser external surfaces may be knownor the heat flux at one of these surfaces may be prescribed. In these cases one or both of

and need not be calculated.

and The thermal resistances across the thickness of the container wall in the evaporator andthe condenser respectively.

and The thermal resistances across the wick thickness in the evaporator and the condenserrespectively. High estimates of these thermal resistances are obtained by consideringconduction only, making the following assumptions. Firstly, that the wick is fully saturatedand that formation or condensation of vapour occurs only at the interface of the wick andvapour duct and, secondly, that convection and radiation within the wick structure arenegligible.

* In this Data Item, thermal resistance is as defined by Equation 3.1 and has units of K/W.

Q·

Mz Q

·

∆T

Q· ∆T /z=

z1z10

Condenser vapour-liquidinterface

Condenser wick(transverse resistance)

Condenser wall(transverse resistance)

Condenser externalsurface-sink

Evaporator liquid-vapourinterface

Evaporator wick(transverse resistance)

Evaporator wall(transverse resistance)

Souce-evaporatorexternal surface

Heat source

Vapour duct

Wall and wick(axial resistance)

Heat sink

Q.

Q

z1

z2

z3

z4

z9

z8

z7

z6

z5

z10

.

z1 z10

z1 z9

z1 z9

z2 z8

z3 z7

7

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Data for the effective thermal conductivity, of common types of liquid-saturated wickare available in Reference 7. These data are useful as a guide. However, apparently identicalwick structures often have different properties and for accurate work is best determinedexperimentally for the particular combination of wick and liquid concerned. A suitablemethod is described in Reference 4. If there is excess liquid present, this will tend to collectin the condenser and allowance should be made for its thermal resistance. Allowanceshould also be made for any thermal resistance arising between the wick and the containerwall due to poor contact between these parts.

and The thermal resistances that occur at the vapour-liquid interfaces in the evaporator and thecondenser respectively. In order to sustain a finite evaporation or condensation rate, atemperature difference must exist between the liquid and the vapour at their interface (seeDerivation 28). This temperature difference gives rise to a thermal resistance, but one thatis often small enough to be set equal to zero.

The effective thermal resistance of the vapour. Because there is an overall pressure loss inthe vapour between the evaporator and the condenser and the vapour is in saturatedequilibrium with the liquid, there is a corresponding temperature difference in the vapour.This temperature difference gives rise to a thermal resistance, but one that is often smallenough to be set equal to zero.

The axial thermal resistance of the container wall and the wick. As well as conducting heatthrough their thickness, the container wall and the wick both conduct heat axially. Thisresistance is the combination of the axial wall and the axial wick resistances in parallel.

In most practical heat pipes, axial conduction in the container and the wick is negligible compared with theheat transported by the vapour. A practical criterion for negligible (less than 5 per cent approximately) axialconduction is, setting the small thermal resistances , and to zero,

(3.2)

If Criterion (3.2) is satisfied, the overall thermal resistance is

, (3.3)

where, for an initial estimate at least, , and can be set to zero.

If Criterion (3.2) is not satisfied, the heat pipe is probably operating under unsuitable conditions and itsperformance is poor. The boundaries between the evaporator, the adiabatic section (if present) and thecondenser are ill-defined and many of the data in this section and in Section 4 are inapplicable. Anapproximate analysis for calculating under these conditions is given in Derivation 20.

λW

λW

z4 z6

zs

z10

z4 z5 z6

z10/ z2 z3 z7 z8+ + +( ) 20>

z znn 1=

9

∑=

z4 z5 z6

z

8

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4. MAXIMUM HEAT TRANSFER – CIRCULATION LIMIT

4.1 Overall Pressure Criterion

Pressure drops occur in both liquid and vapour as they flow along the paths shown in Sketch 2.1 and, inorder to maintain the fluid circulation, a compensatory pressure difference must be set up by capillary forcesin the wick. This pressure difference varies with the conditions of operation, but it cannot exceed a maximumvalue, , that depends on the properties of the working fluid and the wick material.

Circulation will be maintained and the heat pipe will operate normally provided that

, (4.1)

where the term accounts for the transverse hydrostatic effect in the liquid. Other transversepressure drops in the liquid are usually small and are neglected in this analysis. Methods of calculating

, and follow in Sections 4.2, 4.3 and 4.4. The maximum capillary pressure difference, ,is substantially independent of the overall rate of heat transfer, . The axial pressure drops in the liquidand vapour, and , increase with the circulation rate and therefore increase with . Therefore, aninitial value of is needed to see if Criterion (4.1) is satisfied. Knowledge that the circulation limit is notexceeded is often sufficient, in which case the design value of increased by an appropriate safety factoris used to calculate both and for substitution in Criterion (4.1). When it is necessary to calculatethe maximum overall rate of heat transfer, (circulation limit), Criterion (4.1), taken as an equality,should be solved. Sometimes it is possible to neglect , in which case can be obtained from anexplicit equation, see Section 4.4.3.

4.2 Maximum Capillary Pressure Difference,

Surface tension forces sustain a pressure difference between the vapour and the liquid at a curved interfacein a heat-pipe wick. The interfacial shape depends on the position of the interface within the wick and formany wicks the shape is complex. However, for the calculation of the maximum capillary pressuredifference, it is sufficient to define an effective capillary radius, , that is also the minimum value for anyparticular wick structure. Data for , which is independent of fluid physical properties, are available inReference 7 for various types of wick. These data are useful as a guide. However, apparently identical wickstructures often have different properties and for accurate work is best determined experimentally forthe particular wick concerned. Simple methods exist for measuring , see Reference 3 and Derivation 13.

The maximum capillary pressure difference available to drive the fluid circulation in a heat pipe arises inthe circumstances illustrated in Sketch 4.1. The vapour-liquid interface at the closed end of the evaporatorhas an effective radius of curvature equal to and the pressure in the vapour exceeds that in the liquid atthis point. A flat interface exists in the condenser (usually at the closed end but see Section 4.4.3) with zerointerfacial pressure difference.

Sketch 4.1

∆pσ

∆pσ ∆pl ∆pv gliρl φcos+ +≥

gliρl φcos

∆pσ ∆pl ∆pv ∆pσQ·

∆pl ∆pv Q·

Q·

Q·

∆pl ∆pvQ·

M∆pv Q

·M

∆pσ

rσrσ

rσrσ

rσ

Vapour

Liquid

Evaporator Adiabatic section Condenser

rσ

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The maximum capillary pressure difference is given by,

. (4.2)

Data for the surface tension, , and for the contact angle, , can be obtained from References 1, 2 and 6.These data are generally for clean surfaces and pure fluids. Dirt and impurity usually increase anddecrease dramatically, both changes causing reductions in . Therefore, it is essential that properattention is paid to the cleaning of components and to the fluid purity in order to achieve the heat transfercapability of the pipe.

In the absence of data for the contact angle, it is usual to set , although this is not a conservativeprocedure. Alternatively, can be obtained from an experiment with the intended wick and workingfluid, and the value substituted into Equation (4.2)

4.3 Pressure Drop in Liquid,

The liquid velocity in heat-pipe wicks is nearly always low; consequently the flow is laminar and the effectsof acceleration and deceleration of the liquid are negligible. Under these conditions, the axial pressure dropin the liquid is, taking account of frictional and hydrostatic effects,

. (4.3)

Data for the permeability, , of various wick types are available in Reference 7 together with maximumvalues of the Reynolds number, , up to which the data apply. These data are useful as a guide, but foraccurate work is best determined experimentally for the particular wick concerned. Suitable methodsare described in Reference 3 and Derivation 13.

The effective length, , in Equation (4.3) accounts for the axial variation of mass flow rate in the wick.The evaporator and condenser sections are very nearly isothermal in normal operation, and the heat fluxesinto and out of these sections do not usually vary much axially. Under these conditions, the effective lengthsof the sections are and . Because the mass flow rate is invariant in it, the effective length of anadiabatic section is simply its length, .

The pressure drop in the condenser, , is given by Equation (4.3) with

(4.4a)

and . (4.4b)

If the criterion*

(4.5)

is satisfied, the overall pressure drop, , is given by Equation (4.3) with

(4.6a)

and , (4.6b)

where if there is no adiabatic section.

* See also Section 4.4.3.

∆pσ 2σ θ /rσcos=

σ θθ

σ ∆pσ

θcos 1=rσ / θcos

∆pl

∆pl leffQµl / AwKLρl( ) glρl+ φsin=

KRel

K

leff

le /2 lc /2la

∆plc

leff lc /2=

l lc=

∆plc ∆pvc–>

∆pl

leff le /2 la lc /2+ +=

l le la lc+ +=

la 0=

10

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If Criterion (4.5) is not satisfied, the pressure drop in the condenser is ignored and the (effective) overallpressure drop, , is given by Equation (4.3) with

(4.7a)

and , (4.7b)

where if there is no adiabatic section.

4.4 Pressure Drop in Vapour,

A heat pipe is usually designed so that, under normal operating conditions, the vapour flow can beconsidered to be incompressible, i.e.

(4.8a)

and (4.8b)

The speed of sound in the vapour, needed to calculate in Criterion (4.8a), can be calculated with sufficientaccuracy from the equation for a perfect gas, . The vapour flow is also often laminar but,for high flow rates, a transition to turbulent conditions may occur.

Methods of calculating the axial pressure drop for both laminar and turbulent incompressible flows areoutlined in this section and the calculation routine is given in the flow chart of Section 9. The methods takeaccount of the axial variation of mass flow rate in the vapour duct. Frictional and inertial effects are included,but hydrostatic effects are excluded as they are not important in the vapour flow. The methods are derivedfor vapour ducts of circular cross section but for generality, and in the absence of more reliable data, an adhoc extension to non-circular cross sections has been made by introducing the equivalent diameter, .Consequently, the pressure drop can be predicted with more confidence for circular than for non-circularcross sections.

4.4.1 Laminar Flow

For laminar flow in the vapour duct , the pressure drops in the evaporator, adiabatic section(if present) and condenser are obtained separately using a method based on data in Derivation 29 andoutlined in Appendix A. The detailed application of the method is shown in the flow chart in Section 9Routine II, but briefly it is as follows. The radial Reynolds numbers, for the evaporator and for the condenser, are calculated; if there is an adiabatic section its reduced length, is also calculated.The separate pressure drops, , (if non-zero) and , can then be obtained using Figures 1 to 5.

The pressure drop in a length of the vapour duct depends on the radial distribution of the axial velocity inthat length. The equations in Appendix A show that the distribution (which is described by a parameter )varies with , and in the evaporator, adiabatic section and condenser respectively. Thedistribution is also a function of axial distance. Therefore, values of at the junctions of consecutivesections are needed to determine and ; these values can be obtained from Figures 2 and 4. Inorder to facilitate the calculation of for heat pipes either with or without adiabatic sections, theparameter is shown in Figure 5 with a dummy subscript, . If there is an adiabatic section equals ,whereas if there is no adiabatic section equals .

∆pl

leff le /2 la+=

l le la+=

la 0=

∆pv

M 0.2<∆pv /pv 0.1<

Ma γRTeff=

DE

Rev 2000<( )

ReDe ReDcla*

∆pve ∆pva ∆pvc

sReDe Rev ReDc

s∆pva ∆pvc

∆pvcs d sd sa

sd se

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4.4.2 Turbulent Flow

For predominantly turbulent flow , an overestimate of the pressure drop in the evaporatorplus the adiabatic section (if present) is obtained from the equation

. (4.9)

If there is no adiabatic section, in Equation (4.9).

In the condenser, an overestimate of the pressure drop is obtained from the equation

. (4.10)

Equations (4.9) and (4.10) are based on data for smooth tubes in Derivation 16. The equations are derivedassuming that the vapour flow is turbulent over the whole length of the vapour duct when although in reality regions of laminar and transitional flow will exist. However, if the assumption is made,conservative (high) pressure drops are obtained and uncertainties concerning regions of transitional floware largely avoided.

4.4.3 Overall Pressure Drop

At high vapour flow rates, the pressure in the condenser can rise in the flow direction and is thennegative. Under these circumstances it is possible within the condenser for the predicted pressure in theliquid to exceed the predicted pressure in the vapour (see Derivations 10 and 32). This requires the liquidsurface at the wick to be convex, protruding into the vapour space. However, the continual deposition ofliquid in the condenser in normal heat-pipe operation prevents the occurrence of a convex liquid surface.As a result, pressure equality between the liquid and vapour occurs at the condenser entrance unless(Criterion (4.5))

.

When Criterion (4.5) is satisfied, pressure equality between liquid and vapour occurs at the closed end ofthe condenser. The overall pressure drop for either laminar or turbulent flow is then given by

, (4.11)

and the overall pressure drop in the liquid is obtained from Equation (4.3) and (4.6a) (4.6b) in Section 4.3.

When Criterion (4.5) is not satisfied, the pressure drops in the vapour and liquid in the condenser are ignored.The (effective) overall pressure drop in the vapour for either laminar or turbulent flow is then given by

, (4.12)

and the (effective) overall pressure drop in the liquid is obtained from Equations (4.3) and (4.7a) (4.7b) inSection 4.3.

Rev 2000≥( )

∆pve ∆pva+

ρvVv2

-------------------------------- 0.158

DEvRev¼

-------------------------le

2.75---------- la+

0.93+=

la 0=

∆pvc

ρvVv2

------------ 0.158

DEvRev¼

-------------------------lc

2.75----------

0.93–=

Rev 2000≥

∆pvc

∆plc ∆pvc–>

∆pv ∆pve ∆pva ∆pvc++=

∆pv ∆pve ∆pva+=

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Calculated values of are used in conjunction with Criterion (4.1) and calculated values of and

as discussed in Section 4.1 to ensure that the circulation limit is not exceeded. In some cases, wherethe pressure drop in the liquid flowing through the wick is relatively large, it is possible to simplify thecalculation of the circulation limit by neglecting the pressure drop in the vapour, , altogether. However

should always be calculated at least once to see if this is a reasonable approximation. When isnegligible, the circulation limit is given explicitly as

, (4.13)

,

with (4.6a) ,

and (4.6b) ,

where if there is no adiabatic section.

4.5 Gravity-assisted Heat Pipes

The methods in Sections 4.1 to 4.4 can be used to estimate the circulation limit of a gravity-assisted heatpipe provided that during operation the criterion

(4.14)

is satisfied*, and provided there is no excess liquid present to form a pool in the bottom of the evaporator.A gravity-assisted heat pipe can be successfully operated under other conditions, but the methods in thisData Item are not appropriate to this type of operation. Further information can be obtained fromDerivation 31.

* The requirement is that the pressure in the liquid falls in the flow direction throughout the heat pipe, but it follows from Equation 4.3 thatthis is always the case when Criterion (4.14) is satisfied.

∆pv ∆pσ∆pl

∆pv∆pv ∆pv

Q·

M

AWKLρl

leff µl

----------------------2σ θcos

rσ-------------------- glρl φ gliρl φcos–sin–

=

Q·

M 0≥

leff le /2 la lc /2+ +=

l le la lc+ +=

la 0=

φ 0°<( )

∆plc 0>

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5. MAXIMUM HEAT TRANSFER – OTHER LIMITS

5.1 Vapour Pressure limit

Vapour pressures are low but necessarily exceed zero at temperatures close to the bottom of the operationalrange of a heat pipe. The minimum vapour pressure, which occurs at the closed end of the condenser, canbe very small. The pressure drop in the vapour duct, , is then constrained by this effectively zeropressure and by the low vapour pressure existing at the closed end of the evaporator. Because increaseswith the overall rate of heat transfer, , the constraint on requires (if not otherwise limited) to beapproximately limited to a value, called the vapour pressure limit in this text.*

The vapour pressure limit can be a severe restriction to heat transfer at the lowest operating temperatures,and appreciable axial temperature gradients can occur as the limit is approached. Therefore, heat pipes arenearly always designed to operate well away from this limit. Provided that Criterion (4.8b) is satisfied, thevapour pressure limit will not be encountered. However, the limit may be important during start up.

The maximum overall rate of heat transfer can be calculated from the following theoretical equation,substantiated by rather sparse experimental data in Derivations 22 and 24.

, (5.1)

where .

In Equation (5.1), and are evaluated at the temperature of the closed end of the evaporator; and are evaluated at this temperature and the pressure .

5.2 Sonic Limit

At low temperatures in the operational range, the vapour velocity in a heat pipe can become comparablewith the speed of sound in the vapour. If the vapour velocity equals the local speed of sound at a point inthe flow path, the flow is then choked and the overall rate of heat transfer is limited to a value called thesonic limit. Appreciable axial and radial temperature gradients occur in the vapour under choked flowconditions and, therefore, heat pipes are seldom designed to operate at the sonic limit. Provided thatCriterion (4.8a) is satisfied, the sonic limit will not be encountered. However, it is useful to know the valueof the sonic limit in the consideration of start-up and fault conditions.

The maximum overall rate of heat transfer can be estimated from Equation (5.2) which is a theoreticalequation from Derivations 12 and 14.

, (5.2)

where and are evaluated at the temperature of the closed end of the evaporator† and is the saturatedvapour density at this temperature.

* The vapour pressure limit is alternatively called the viscous limit in the literature, see Derivation 22 for example.† Ideally in Equation (5.2) should be evaluated at the temperature of the evaporator exit, but this temperature will not often be known.

However, can usually be evaluated at the temperature of the closed end of the evaporator. This is acceptable because latent heats ofvaporisation are usually only weakly temperature dependent and the absolute temperature falls by 25 per cent or less along the length ofthe evaporator for choked flow.

∆pv∆pv

Q· ∆pv Q

·

Q·

M AvDEv2

Lpvρv / 64µvleff( )=

leff le /2 la lc /2+ +=

pv L pvµv pv

Q·

M

AvaLρv

2 1 γ+( )--------------------------=

a L ρv

LL

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The speed of sound in the vapour can be estimated from the equation for a perfect gas*, . Ifthe specific heat ratio, , is not known, it can be taken equal to unity† with only small (and conservative)error in as given by Equation (5.2).

Equation (5.2) shows that the heat flux is a function of the fluid physical properties only. Therefore,in this approximation, sonic heat flux limits apply to all heat pipes. Useful tabulations of as afunction of the temperature of the closed end of the evaporator are given in Reference 5 for water andvarious metallic vapours.

Equation (5.2) does not take account of the viscosity of the vapour and predicts that sonic conditions occurat the evaporator exit. Data in Derivations 23, 26 and 33 show that the effects of vapour viscosity aretwofold. Firstly, at low temperatures viscous effects in the evaporator cause to be appreciably lowerthan predicted by Equation (5.2). Secondly, if there is an adiabatic section, sonic conditions occur at theexit of this section rather than at the evaporator exit, and is again lower than predicted by Equation (5.2).However, the scant available data suggest that these effects are fairly small for most heat pipes, and thateven at the lowest temperatures actual sonic limits will usually be greater than two-thirds of the valuepredicted by Equation (5.2). Larger errors can occur if either or is large (> 40) and a specialanalysis may then be required. Suitable methods of performing this analysis are outlined in Derivations 23and 26.

5.3 Entrainment Limit

At high vapour velocities, shear forces between the vapour and the liquid at their interface can be sufficientto tear off droplets of liquid from the wick surface. These droplets become entrained in the vapour flowand are carried towards the closed end of the condenser, thus bypassing the normal circulation route andmaking no appreciable contribution to the axial rate of heat transfer. If the entrainment becomes severe, apoint can be reached when insufficient liquid is returned to the evaporator and a dry-out occurs. The overallrate of heat transfer that produces this condition is the entrainment limit.

At present, the experimental evidence for the entrainment limit in capillary-driven heat pipes is sparse andinconclusive (see Derivations 11, 30 and 31). Although it is therefore not possible to predict the entrainmentlimit for this type of heat pipe, the overall rate of heat transfer for the onset of entrainment can be deducedon theoretical grounds and can be taken as a working maximum value. An equation from Derivation 11gives this rate of heat transfer as

, (5.3)

where is a characteristic dimension of the wick surface. For those experimental data from Derivation 11in which an entrainment limit was suspected, Equation (5.3) predicts conservative (low) values of themaximum overall rate of heat transfer when is taken equal to twice the minimum capillary radius of thescreen wicks used in the experiments. In the absence of data for other types of wick it is therefore suggestedthat should be used to calculate from Equation (5.3) for all small-pore wick structures.

5.4 Boiling Limit

The temperature drop across the liquid-saturated wick in a heat pipe increases with the transverse heat fluxin the evaporator. The liquid in the evaporator wick is superheated, and at a high enough value of thetransverse heat flux, the degree of superheat is sufficient to initiate boiling within the wick. No comparable

* It is common to use an index of isentropic expansion to calculate the choking velocity of a saturated vapour flow. However, this indexwill not be known for many heat-pipe working fluids and, furthermore, data in Derivation 23 indicate that appreciable error in the calculatedsonic limit of a heat pipe is unlikely to be introduced by assuming that the vapour behaves as a perfect gas.

† See also Section 1.

a γRTv=γ

Q·

M

Q·

M /AvQ·

M /Av

Q·

M

Q·

M

le /DEv la /DEv

Q·

M AvL ρvσ x⁄=

x

x

x 2rσ= Q·

M

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effect occurs in the condenser because the liquid is subcooled (below its saturation temperature) there.Boiling in the wick is undesirable because it interferes with the circulation of liquid and makes it necessaryto use empirical methods to predict the performance of a heat pipe rather than the methods given in thisData Item. Therefore, the onset of boiling is taken to define the boiling limit, although it may sometimesbe possible to operate the heat pipe at a higher evaporator transverse heat flux. Further data are availablein Derivations 16, 18 and 28 and the references contained therein for those cases in which it is essential tooperate a heat pipe with boiling in the evaporator wick.

The boiling limit is a limit to the transverse heat flux, and this leads to an equivalent limit to the overallrate of heat transfer. The maximum overall rate of heat transfer can be calculated using the followingtheoretical equation from data in Derivation 27.

. (5.4)

From a comparison of experimental data in Derivations 13, 17, 19 and 25 it is recommended that thenucleation radius, , in Equation (5.4) should be taken equal to m. Smaller values of canoccur, but larger values are unlikely provided that the heat pipes are carefully processed to eliminatenon-condensible gas. The thermal resistance, , in Equation (5.4) can be obtained from equations inSection 8. The terms is the maximum capillary pressure difference, obtainable from data inSection 4.2. Strictly, the actual capillary pressure difference, equal to the right-hand side of Criterion (4.1),should be used in place of . However, it is rare for to be comparable with so the extracomplexity of calculation involved in using the actual capillary pressure difference is not usually justified.

Q·

M

Teff

Lz3ρv

---------------2σrn

------- ∆pσ–

=

rn 2 106–× rn

z3∆pσ

∆pσ ∆pσ 2σ /rn

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6. EXAMPLE

A vertical heat pipe operating against terrestrial gravity is required to convey 100 W from an electricalheater to water in a closed container at about 65° C. The arrangement shown in Sketch 6.1 is proposed. Thecopper heat pipe container is circular in cross section with an outside diameter of 15 mm and a wall thicknessof 1 mm. The annular wick (porosity 0.38) is made of sintered copper particles with diameter (beforesintering) in the range from 125 to 180 , and the vapour duct diameter is 4 mm. The working fluidis water. Taking the thermal conductivity of copper as 380 W/(m/ K), check that the heat pipe can performthe proposed duty

.

Sketch 6.1

µm µm

110 mm

80 mm

50 mm

15 mm

Electricalheater

Water at 65°C

1 mm4 mm

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The calculation procedure outlined in the flow charts of Section 9 will be followed.

.

Therefore the Prandtl number .

.

(1) The overall rate of heat transfer, , and the temperature of the heat sink are known. Thereforethe outside surface temperatures of the evaporator and condenser have to be calculated usingthe data in Section 3. Heat is transferred from the heat-pipe condenser to the water in thecontainer by free convection. The heat transfer coefficient is obtained from ESDU 77031.Because is known, the temperature of the outside surface of the condenser can becalculated before to inclusive are evaluated.

(1a) A first estimate of the temperature of the outside surface of the condenser is 85°C, giving asurface to bulk fluid temperature difference, , of 85 – 65 = 20 K, and film temperatureof 65 + ½ (20) = 75° C.

(1b) Properties of water at the film temperature are (where is the specific heat capacity atconstant pressure and is the volumetric expansion coefficient):

(1c) The Grashof number based on the characteristic height, (equal to ), is

(1d) Therefore . Because exceeds

109, the free-convective flow is turbulent and the Nusselt number, , is given byEquation (A2.1) of ESDU 77031.

(1e)

,

giving .

(1f) Therefore, W/(m2K).

Q·

Q·

z2 z8

∆T×

cpβ

cp 4.19 103× J/ kg K( ) ,=

β 0.58 103– × 1/K ,=

λ 0.666 W/(m K) ,=

µ 0.374 103–× N s/m

2=

ρ 0.975 103× kg/m

3=

Pr cpµ /λ 4.19 103

0.374 103–/0.666××× 2.35= = =

H lc

Gr H( ) βglc3ρ2∆T /µ2

.=

0.58 103–

9.81 0.11( )3 0.975 103×( )

220/ 0.374 10

3–×( )2

×××××=

1.03 109×=

Gr H( )Pr 1.03 109

2.35×× 2.42 109×= = Gr H( )Pr

Nu H( )

Nu H( )[ ]½ 0.825 0.387 GrHPr/ 1 0.492/Pr( )9 16⁄+[ ]

16 9⁄

1 6⁄

+=

0.825 0.387 2.42 109/ 1 0.492/2.35( )9 16⁄

+[ ]16 9⁄

×

+=

1 6⁄

Nu H( ) hclc /λ 185= =

hc 185 0.666/0.11× 1120 = =

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The temperature of the outside surface of the condenser is 65.0 + 17.8 = 82.8°C (356.0 K)

(1g) From Section 8, K/W.

(1h) Therefore, a second estimate of the surface to bulk fluid temperature difference is

K. A second estimate of the film temperture is

.

(1i) The calculation is repeated with these second estimates of and the film temperature toobtain further estimates until sufficient accuracy is obtained. The results of the calculationsare tabulated below. Note that the calculated values of film temperature are close enough tothe first estimate (75°C) for physical properties at 75°C to be used throughout, and that

always exceeds 109 indicating turbulent flow.

20.0 17.2 18.0 17.7 17.8

filmtemperature, °C 75.0 73.6 74.0 73.9 –

2.42 2.08 2.18 2.14 –

z 9, K/W 0.172 0.180 0.177 0.178 –

(1j) The thermal resistances , , and are calculated from the equations in Section 8. Theeffective thermal conductivity of the wick is needed for and and, because a value of thisthermal conductivity is not available, it is estimated from the equation for sintered particle wicksin Reference 7. The thermal conductivity of the water working fluid, , is taken as 0.672W/(m K), the value at the condenser outside surface temperature of 83°C. The thermalconductivity of the solid material of the wick, , is taken as 380 W/(m K).

.

z9 1/ hcSc( ) 1/ 1120 π 0.015 0.11×××( ) 0.172= = =

∆T Q·

z9 100 0.172× 17.2= = =

65.0 ½ 17.2( )×+ 73.6°C=

∆T

Gr H( )Pr

∆T ,K

Gr H( )Pr /109

z2 z3 z7 z8z3 z7

λl

λs

λw

λs2λs λl 2ελs λl––+

2λs λl ελs λl–+ +------------------------------------------------------------≈

380 2 380 0.672 2 0.38 380 0.672–( )××–+×[ ]×2 380 0.672 0.38 380 0.672–( )×+ +×

---------------------------------------------------------------------------------------------------------------------------------------= 198 W/(m K)=

z2

loge ro /ri( )

2π leλx

----------------------------- loge 0.0075/0.0065( )2 π 0.05 380×××

---------------------------------------------------- 1.20 103– K/W .×= = =

z3

loge ri /rv( )

2π leλw

---------------------------- loge 0.0065/0.002( )2 π 0.05 198×××------------------------------------------------- 18.9 10

3– K/W .×= = =

z7

loge ri /rv( )

2π lcλw

---------------------------- loge 0.0065/0.002( )2 π 0.11 198×××------------------------------------------------- 8.61 10

3– K/W .×= = =

z8

loge ro /ri( )

2π lcλx

----------------------------- loge 0.0075/0.0065( )2 π 0.11 380×××

---------------------------------------------------- 0.545 103– K/W .×= = =

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Therefore, the temperature difference between the outside surfaces of the evaporator andcondenser is, neglecting the small thermal resistances , and ,

The temperature of the outside surface of the evaporator is 82.8 + 2.9 = 85.7°C (358.9 K).

(2) The effective operating temperature of the heat pipe, 357 K(84º) C.

Physical properties of the water working fluid at this temperature that are required later are:

,

,

,

N s/m2,

N s/m2,

kg/m3,

kg/m3,

N/m.

The molecular weight of water is 18.0; therefore J/(kg K).

(3) From Section 8, ; therefore

Therefore, , and Criterion (3.2) is easily

satisfied.

(4) The maximum capillary pressure difference is calculated. A value of the minimum capillaryradius for the sintered wick is not available, so it is estimated from the equation in Reference 7. Inthis equation, is the average diameter of the particles used to manufacture the wick.

m.

Assuming that pure water and clean materials have been used in the heat pipe construction( unaffected by impurity), Equation (4.2) gives:

z4 z5 z6

∆T Q z2 z3 z7 z8+ + +( ) 100 1.20 18.9 8.61 0.545+ + +( ) 103–××= =

100 0.029× 2.9 K .= =

Teff ½ 356.0 358.9+( ) × ==

L 2.29 106× J/kg=

pv 55.7 103 Pa×=

γ 1.3=

µl 0.339 103–×=

µv 11.4 106–×=

ρl 0.969 103×=

ρv 0.345=

σ 61.8 103–×=

R 8314= /18.0 462=

z10

le la lc+ +

Axλx Awλw+-----------------------------------=

z100.05 0.08 0.11+ +

π 0.0075( )2 0.0065( )2–[ ] 380 π 0.0065( )2 0.002( )2

–[ ]×+×× 198×----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

0.24/ 44.0 106–

380 120 106–

198××+××( )= 5.93 K/W .=

z10/ z2 z3 z7 z8+ + +( ) 5.93/ 0.029( ) 204= =

d

rσ 0.21d 0.21 125 180+( ) 106–

2⁄×× 0.21 153 106–×× 32.1 10

6–×= = = =

θ 0° ,σ=

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(5) The rate of heat transfer is multiplied by a safety factor (see Section 4.1). A factor of 1.5 is

chosen, so becomes W.

(6) The Reynolds number for the flow of liquid in the adiabatic section is calculated to ensure thatthe data in Section 4.3 apply.

m/s.

Using the equation for sintered particle wicks from Reference 7,

Hence

For a sintered particle wick, can be up to 10 (see Reference 7). Because the Reynolds

number for the liquid flow is a maximum in the adiabatic section, the data in Section 4.3 applythroughout the heat pipe.

(7) The permeability of the wick is not known so a value is estimated from the equation for sinteredparticle wicks in Reference 7.

m2.

From Equation (4.3) and (4.4a)(4.4b), since ,

Criterion (4.14) is satisfied . Note that this is always the case for heat pipes operating

against gravity.

(8) The pressure drop in the vapour is calculated using Routine II of the flow chart.

∆pσ2σ θcos

rσ-------------------- 2 61.8 10

3–1×××

32.1 106–×

--------------------------------------------------- 3850 = = =

Q· 1.5 100× 150=

Vl Q· / AwLρl( ) 150/ 120 10

6–2.29 10

60.969 10

3×××××( )= =

0.563 103–× )=

DEl2dε

3 1 ε–( )--------------------- 2 153 10

6–0.38×××

3 1 0.38–( )×--------------------------------------------------------- 62.5 10

6–× m.= = =

Rel ρlVlDEl / εµl( )=

0.969 103

0.563 103–

62.5 106–× / 0.38 0.339 10

3–××( )××××=

0.265=

Rel

Kd

2ε3

150 1 ε–( )2------------------------------

153 106–×( )

20.38( )×

3

150 1 0.38–( )× 2-------------------------------------------------------------- 22.3 10

12–×= = =

φ 90°=

∆plc lc /2( )Q· µl / AWKLρl( ) glcρl+ φsin=

0.11/2( )= 150 0.339 103–/ ×××

120 106–

22.3 1012–

2.29 106

0.969 103×××××××( )

9.81 0.11 0.969 103

1××××+

471 1050 += 1520 Pa. =

∆plc 0>( )

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(8a)

m/s.

.

Pa.

(8b) Criterion (4.8a) is satisfied , and the sonic limit will not be encountered. The flow is

laminar since , and so the data discussed in Section 4.4.1 are used to calculate the

pressure drops in each section of the vapour duct.

(8c)

(8d) From Figure 1, , and therefore Pa.

From Figure 2, .

(8e) There is an adiabatic section; therefore is calculated.

.

(8f) From Figure 3, , and therefore

Pa.

From Figure 4, , and in this case, .

(8g) .

(8h) From Figure 5, , and therefore, pa.

(8i) Criterion (4.5) is satisfied so that

Pa.

(8j) . Criterion (4.8b) is satisfied ,and the vapour pressure limit will not be encountered. Since Criterion (4.8a) was earlier shown tobe satisfied, the methods in Routine II are appropriate for the calculation of .

(9) Returning to Routine I, there is no need to calculate and recalculate because was givenas input data. Because Criterion (4.5) has been shown to be satisfied, the pressure loss in theliquid is given by Equations (4.3) and (4.6a)(4.6b).

Vv Q·

/ AvLρv( ) 150/ π 0.002( )2 2.29 106

0.345××××[ ] 15.1= = =

M Vv /a Vv / γRTeff 15.1/ 1.3 462 357×× 0.033= = = =

Rev ρvVvDEv /µv 0.345 15.1 0.004/11.4 106–××× 1830 = = =

ρvVv2

0.345 15.1( )× 278.7= =

M 0.2<( )Rev 2000<

ReDe DEvRev /4 le 0.004 1830/ 4 0.05×( ) × 36.6= = =

∆pve / ρvVv2( ) 1.38= ∆pve 1.38 78.7× 109= =

se 3.09=

la*

la* la / DEvRev( ) 0.08/ 0.004 1830 ×( ) 1.09 10

2–×= = =

∆pva / la*ρvVv

2( ) 42.0=

∆pva 42.0 1.09 102–×× 78.7× 36.0= =

sa 2.35= sd sa=

ReDc DEvRev / 4 lc( ) 0.004 1830/× 4 0.11×( ) 16.6= = =

∆pvc / ρvVv2( ) 1.09–= ∆pvc 1.09– 78.7× 85.8–= =

∆ plc ∆pvc–>( )

∆pv ∆pve= ∆pva ∆pvc+ + 109 36.0 85.8–+ 59.2= =

∆pv /pv 59.2 55.7 103×( )⁄ 1.06 10

3–×= = ∆pv /pv 0.1<( )

∆pv

z5 Q·

Q·

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Pa.

(10) A value of is not required in this example and it is only necessary to check that Criterion

(4.1) is satisfied. The right-hand side of Criterion (4.1) is

Pa.

The left-hand side of Criterion (4.1) has already been calculated in step (4); it isPa. Therefore the criterion is satisfied and the heat pipe will not be

circulation-limited under the proposed operating conditions.

(11) The overall rate of heat transfer for the onset of entrainment is given by Equation (5.3)

Therefore, since , entrainment is unimportant in this example.

(12) The overall rate of heat transfer for the onset of boiling in the evaporator is given byEquation (5.4).

Therefore, boiling is unimportant in this example. It should be noted, however, that the boilinglimit will not be as high as calculated here unless care has been taken during manufacture toeliminate all traces of non-condensible gas.

It is concluded that the heat pipe is suitable for the proposed duty, and will not fail due to having reached circulation, entrainment or boiling limits. The vapour pressure and sonic limitswill also not be encountered in steady-state operation.

∆pl le /2 la lc /2+ +( )Q· µl / AWKLρl( ) g le la lc+ +( )+= ρl φsin

0.05/2 0.08 0.11/2+ +( ) 150 0.339 103–/120 10

6–22.3×××××= 10

12–×

2.29 106

0.969×× 103×× ) 9.81 0.05 0.08 0.11+ +( )× 0.969 10

3× 1××+

1370 = 2280 + 3650 =

Q·

M

∆pl ∆pv gliρl φcos+ + 3650 59.2 9.81 0.013 0.969 103–× 0×××+ +=

3710 =

∆pσ 3 850 =

Q·

M AvL ρvσ x⁄ AvL ρvσ 2rσ( )⁄= =

π 0.002( )22.29 10

60.345 61.8 10

3–2 32.1× 10

6–×( )⁄×××××× 524 W.= =

Q·

Q·

M<

Q·

M

Teff

Lz3ρv

--------------- 2σrn

------- ∆pσ–

=

357

2.29 106

18.9 103–

0.345××××--------------------------------------------------------------------------------------=

2 61.8 103–××

2 106–×

---------------------------------------- 3850 –

1390 W.=

Q

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7. REFERENCES AND DERIVATION

References

The References given are recommended sources of information supplementary to that in this Data Item.

1. – International critical tables of numerical data, Physics, Chemistry andTechnology, Vol. 4, pp. 435-475, McGraw-Hill, 1933.

2. – Landolt-Börnstein, numerical data and functional relationships inPhysics, Chemistry, Astronomy, Geophysics, Technology, Vol. II, Part 3,pp. 404-486, Springer-Verlag, 1956 (in German).

3. LANGSTON, L.S.KUNZ, H.R.

Liquid transport properties of some heat pipe wicking materials. Am.Soc. mech. Engrs, Paper 69-HT-17, 1969.

4. SOLIMAN, M.M.GRAUMANN, D.W.BERENSON, P.J.

Effective thermal conductivity of dry and liquid-saturated sintered fibermetal wicks. Am. Soc. mech. Engrs, Paper 70-HT/SpT-40, 1970.

5. De MICHELE, D.W.DAVIS, M.V.

Vapour transport limits of liquid metal heat pipes. Nuclear Technology,Vol. 15, pp. 366-383, 1972.

6. REALE, F.CANNAVIELLO, M.

Wetting and surface properties of refrigerants to be used in heat pipes.Proceedings of the 2nd International Heat Pipe Conference, Bologna,Italy, pp. 771-792, 1976.

7. ESDU Heat pipes – properties of common small-pore wicks. ESDU 79013,ESDU International plc, London, 1979.

8. ESDU Heat pipes – general information on their use, operation and design.ESDU 80013, ESDU International plc, London, 1980.

Derivation

The Derivation lists selected sources that have assisted in the preparation of this Data Item.

9. BUSSE, C.A. Pressure drop in the vapor phase of long heatpipes. Record of IEEEThermionic Conversion Specialists Conference, Palo Alto, California,USA, pp. 391-398, 1967.

10. ERNST, D.M. Evaluation of theoretical heat pipe performance. Record of IEEEThermionic Conversion Specialists Conference, Palo Alto, California,USA, pp. 349-354, 1967.

11. KEMME, J.E. High performance heat pipes. Record of IEEE Thermionic ConversionSpecialists Conference, Palo Alto, California, USA, pp. 355-358, 1967.

12. LEVY, E.K. Theoretical investigation of heat pipes operating at low vapor pressures.Trans. Am. Soc. mech. Engrs, Series B, J. Engng Indust., Vol. 90 pp.547-552, 1968.

13. PHILLIPS, E.C.HINDERMAN, J.D.

Determination of properties of capillary media useful in heat pipedesign. Am. Soc. mech. Engrs, paper 69-HT-18, 1969.

14. DEVERALL, J.E.KEMME, J.E.FLORSCHUTZ, L.W.

Sonic limitations and start-up problems of heat pipes. Los AlamosScientific Laboratory Report LA-4518-MS, University of California,Los Alamos, New Mexico, USA, 1970.

15. BANKSTON, C.A.SMITH, J.H.

Incompressible laminar vapor flow in cylindrical heat pipes. Am. Soc.mech. Engrs, Paper 71-WA/HT-15, 1971.

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16. CHISHOLM, D. The heat pipe. Mills and Boon, London, 1971.

17. SILVERSTEIN, C.C. Surface heat flux for incipient boiling in liquid metal heat pipes. NuclearTechnology, Vol. 12, pp. 56-62, 1971.

18. GOLUBA, R.W.VANSANT Jr, J.H.

Evaporation from a capillary-wetted surface. Am. Soc. mech. Engrs,paper 72-WA/HT-12, 1972.

19. MARCUS, B.D. Theory and design of variable conductance heat pipes. NASA CR-2018,National Aeronautics and Space Administration, USA, 1972.

20. SUN, K.H.TIEN, C.L.

Simple conduction model for theoretical steady-state heat pipeperformance. AIAA J1, Vol. 10, pp. 1051-1057, 1972.

21. BANKSTON, C.A.SMITH, J.H.

Vapor flow in cylindrical heat pipes. Am. Soc. mech. Engrs, Paper73-HT-P, 1973.

22. BUSSE, C.A. Theory of the ultimate heat transfer limit of cylindrical heat pipes. Int. J.Heat Mass Trans., Vol. 16, pp. 169-186, 1973.

23. LEVY, E.K.CHOU, S.F.

The sonic limit in sodium heat pipes. Trans. am. Soc. mech. Engrs,Series C, J. Heat Transfer, Vol. 95, pp. 218-223, 1973.

24. VINZ, P.BUSSE, C.A.

Axial heat transfer limits of cylindrical sodium heat pipes between 25W/cm2 and 15.5 kW/cm2. Proceedings of the 1st International Heat PipeConference, Stuttgart, West Germany, Paper 2-1, 1973.

25. COLE, R. Boiling nucleation. Advances in Heat Transfer, Vol. 10 pp. 85-166,Academic Press, London, 1974.

26. BROVALSKY, Y.A.BYSTROV, P.I.MELNIKOV, M.V.

The method of calculation and investigation of high-temperature heatpipe characteristics taking into account the vapour flow compressibility,friction and velocity profile. Proceedings of the 2nd International HeatPipe Conference, Bologna, Italy, pp. 113-122, 1976.

27. CHI, S.W. Heat pipe theory and practice. Hemisphere Publishing Corporation,Washington/London, 1976.

28. DUNN, P.D.REAY, D.A.

Heat pipes. Pergamon Press, Oxford, 1976.

29. KADANER, Ya.S.RASSADKIN, Yu.P.

Laminar vapor flow in a heat pipe. J. engng Phys., Vol. 28, pp. 140-146,1976.

30. KEMME, J.E. Vapor flow considerations in conventional and gravity-assist heat pipes.Proceedings of the 2nd International Heat Pipe Conference, Bologna,Italy, pp. 11-22, 1976.

31. BUSSE, C.A.KEMME, J.E.

The dry-out limits of gravity-assist heat pipes with capillary flow. Acollection of technical papers – 3rd International Heat Pipe Conference,Palo Alto, California, USA, pp. 41-48, 1978.

32. EASTMAN, G.Y.ERNST, D.M.

High performance, high temperature heat pipes. A collection of technicalpapers – 3rd International Heat Pipe Conference, Palo Alto, California,USA, pp. 268-273, 1978.

33. KEMME, J.E.KEDDY, E.S.PHILLIPS, J.R.

Performance investigations of liquid-metal heat pipes for space andterrestrial applications. A collection of technical papers – 3rdInternational Heat Pipe conference, Palo Alto, California, USA, pp.260-267, 1978.

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8. TABLE OF THERMAL RESISTANCES

Location Equation for Circular Cylinder Equation for Plane Section

Source – EvaporatorExternal Surface

Evaporator Wall(transverse resistance)

Evaporator Wick(transverse resistance)

EvaporatorLiquid – Vapour interface

Vapour Duct

CondenserVapour – Liquid Interface

Condenser Wick(transverse resistance)

Condenser Wall(transverse resistance)

Condenser ExternalSurface – Sink

Wall and Wick(axial resistance)

S

S

Plane sectionCircular cylinder

l

l

Ax

Ax

txtw

rv

ro

ri

Aw

Aw

z11

heSe

-----------=

z2

loge ro /ri( )

2π leλx

-----------------------------= z2

tx

Seλx

-----------=

z3

loge ri /rv( )

2π leλw

----------------------------= z3

tw

Seλw

------------=

z4RT

3eff / 2π( )

L2leriρv

-----------------------------------= z4

2πRTeff3

L2Seρv

-------------------------=

zs

Teff∆pv

LQρv

-------------------=

z6RT

3eff / 2π( )

L2lcriρv

-----------------------------------= z6

2πRTeff3

L2Scρv

-------------------------=

z7

loge ri /rv( )

2π lcλw

----------------------------= z7

tw

Scλw

------------=

z8

loge ro /ri( )

2π lcλx

-----------------------------= z8

tx

Scλx

-----------=

z91

hcSc

-----------=

z10

le la lc+ +

Axλx Awλw+-----------------------------------=

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9. PROCEDURE FOR PERFORMANCE CALCULATION

START

IsQ known

?

Input dataQ plus one temperature*

ortwo temperatures* Are

both evaporatorand condenser outsidesurface temperatures

known?

Set Teff = ½ (evaporator outside surface temperature + condenser outside surface temperatures)

Check that Criterion (3.2) is satisfied

Calculate ∆pa(Section 4.2)

Multiply Q byrequired safety factor

Calculate Vl, ∆Eland Rel

Calculate z4 and z6 (Section 3)

.

.

.Calculate unknown outsidesurface temperature(s) fromdata in Section 3 ignoringz1, z9 as appropriate andsetting z4, z5 and z6equal to zero

Calculate Q and unknownutside surface temperatu-re(s) from data in Section 3ignoring z1, z9 as appropr-iate and setting z4, z5 and z6equal to zero

.

.

Calculate Q from data inSection 3 ignoring z1 and z9and setting z5 equal to zero;also set z4 and z6 equal tozero if they are smallcompared with the other z,s

.Calculate Q and unknownutside surface temperatu-re(s) from data in Section 3ignoring z1, z9 as appropr-iate and setting z4, z5 and z6equal to zero

No No

Yes

Yes

Continued onnext page

NOTE

* A known temperature may be:(i) the temperature of the outside surface of either the evaporator or the condenser.(ii) the temperature of either the heat source of the heat sink.

In case (ii), the heat transfer coefficient in the surrounding medium is required as well as the source or sink temperature. The appropriateoutside surface temperature may have to be estimated and an iterative method used to determine this temperature accurately. Time willbe saved if to inclusive are set equal to zero in the initial stages of the calculation.z2 z3

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Output∆plc for

Routine II

IsRel less than

maximum value forwick (see Section

4.3)?

Calculate ∆plc(Section 4.3)

Calculate ∆pl(Section 4.3)

Calculate entrainmentlimit (Section 5.3)

Solve Criterion (4.1)treated as an equality

Yes

Yes

Calculate QMfrom Eqn.(4.13)

Routine IICalculate ∆pv

Is ∆plc > 0?Criterion

(4.14)

Didinput datainclude Q

?

Isa circulationlimit required

?

Is∆pv negligiblecompared with

∆pl?

IsQ < QM

?

Calculate boilinglimit (Section 5.4)

IsQ < QM

?

IsQ < QM

?

.

.

.

..

. .

..

FINISH

Q too large, data inSection 4.3 are

inapplicable

.

Q too large, heat pipecirculation limited

.

Q too large, heat pipeentrainment limited

.

Q too large, heat pipeboiling limited

.

Is Criterion (4.1)

satisfied?

Heat pipe is notcapillary driven, data

inapplicable

Calculate z5 (Section 3) andcheck that it is negligiblecompared with other z's.

If not, recalculate Q

No

No

Yes

No

Yes

Yes

Yes

No

No

No

Yes

No

Yes

No

Yes

No

Continued fromprevious page

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RETURN

ENTER

CalculateVv = Q/(AvLρv),M, Rev and ρvV 2

v

Is M > 0.2?Criterion

(4.8a)

Isthere anadiabaticsection

?

IsRev < 2000

?

.

CalculateReDe = (DEvRev)/4le

Calculate ∆pve + ∆pva fromEqn. (4.9). If there is noadiabatic section, le = 0

Calculate ∆pvcfrom Eqn. (4.10)

CalculateReDe = (DEvRev)/4lc

Calculatel*a = la/(DEvRev)

sd = sa

Obtain ∆pve/(ρvV 2v) and

hence ∆pve from Fig. 1and se from Fig. 2

Obtain ∆pvc/(ρvV 2v) and

hence ∆pvc from Fig. 5

∆pv = ∆pve + ∆pva + ∆pvcEqn. (4.11)

∆pv = ∆pve + ∆pvaEqn. (4.12)

∆pva = 0and sd = seObtain ∆pva/(l*aρvV 2

v) andhence ∆pva from Fig. 3

and sa from Fig. 4

No

NoIs

∆plc > – ∆pvc? Criterion

(4.5)

Is∆pv/pv < 0.1?

Criterion(4.8b)

Input∆plc fromRoutine

I

Q too large for value ofAv. Sonic limit may

be encountered

.No

No

Q too large for values ofAv, le, la, lc. Vapour pressure

limit may be encountered

.No

Yes

Yes

Yes

Yes

Yes

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FIGURE 1 EVAPORATOR SECTION – PRESSURE DROP FOR LAMINAR VAPOUR FLOW

FIGURE 2 EVAPORATOR SECTION – EXIT VELOCITY DISTRIBUTION PARAMETER FOR LAMINAR VAPOUR FLOW

ReDe

2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 8100 101 102 103 104

∆pve

ρvVv2

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

1 10

ReDe

0.2 0.5 0.2 0.51.0

∆pve

ρvVv2

2

4

6

8

10

12

ReDe

2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 810-1 100 101 102 103

se

1.5

2.0

2.5

3.0

3.5

1

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FIGURE 3 ADIABATIC SECTION – PRESSURE DROP FOR LAMINAR VAPOUR FLOW

FIGURE 4 ADIABATIC SECTION – EXIT VELOCITY DISTRIBUTION PARAMETER FOR LAMINAR VAPOUR FLOW

la*

2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 810-4 10-3 10-2 10-1 100

∆pva

la*ρvVv2

30

32

34

36

38

40

42

44

46

48

50

52

54

56se

3.45

3.2

3.0

2.8

2.6

2.4

2.2

2.1

2.0

1

la*

2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 810-4 10-3 10-2 10-1

sa

1.5

2.0

2.5

3.0

3.5 se3.45

3.2

3.0

2.8

2.6

2.4

2.22.12.0

31

ESDU Copyright material. For current status contact ESDU.

ES

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79012APOUR FLOW

0.4

ReDc

0.5 2 51.0

10

3.452.0

sd

32

FIGURE 5 CONDENSER SECTION–PRESSURE “DROP” FOR LAMINAR V

ReDc

2 3 4 5 6 7 8 2 3 4 5 6 7 8100 101 102

∆pvc

ρvVv2

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

3.453.02.62.22.0

sd

For ReDc > 30 and

all sd, take = 0∆pvc

ρvVv2

1

0.2

∆pvc

ρvVv2

0

1

2

3

4

5

6

7

8

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APPENDIX A CALCULATION OF PRESSURE DROPS FOR LAMINAR VAPOUR FLOW

A1. NOTES ON METHOD

The method of calculating the pressure drops for laminar, incompressible vapour flow given in Section 4.4.1is based on a theoretical analysis for ducts of circular cross section in Derivation 29. The followingassumptions are made in the analysis.

(i) The heat fluxes over the surfaces of the evaporator and condenser are uniform but not necessarilyequal.

(ii) The effects of shear forces between opposed vapour and liquid flows are negligible.

(iii) The boundary-layer approximations apply: radial pressure gradients are negligible and the secondaxial derivative of the axial velocity can be neglected in comparison with the second radialderivative of the axial velocity.

The details of the analysis, in which the radial distribution of the axial velocity is approximately describedin terms of a parameter, s, that in general depends on the axial co-ordinate and on the radial Reynoldsnumber, are explained in Derivation 29 and the references contained therein. However, a brief summary,including the relevant equations, is given here.

In the evaporator section, is independent of the axial co-ordinate at distances greater than about one ductdiameter from the upstream end of the evaporator, and consequently the normalised* velocity distributionis invariant with axial distance, see Sketch A1.1.

Sketch A1.1 Typical nomralised velocity distribution in evaporator

* An axial velocity is normalised by dividing it by the cross-sectional average axial velocity at the same axial location.

s

Direction ofaxial velocity

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The parameter lies between 2.0 and 3.45, satisfying the Equation*

. (A1.1)

The value of at the downstream end (exit) of the evaporator, , is therefore obtained fromEquation (A1.1). Figure 2 shows as a function of , obtained from Equation (A1.1).

The static pressure drop for the whole evaporator can be calculated from the equation

. (A1.2)

Figure 1 shows as a function of , obtained from Equations (A1.1) and (A1.2).

In the adiabatic section (if present), the parameter decreases in the direction of the axial velocity fromits value at the section entrance towards a limiting value of 2 (corresponding to Poiseuille flow with aparabolic velocity distribution). Typical velocity distributions are illustrated in Sketch A1.2.

Sketch A1.2 Typical normalised velocity distributions in adiabatic section

At the entrance to the adiabatic section, equals (obtained from Equation (A1.1)) because the evaporatorand adiabatic sections are joined to one another. At the exit of the adiabatic section, equals , obtainablefrom the equation

. (A1.3)

Figure 4 shows as a function of the reduced length , obtained from Equation , for the complete rangeof values of .

* The equation in Derivation 29 that corresponds to Equation A1.1 is incorrectly given (in the notation of this Data Item) as

.

and Figure 2 in that Derivation is consequently incorrect. Note that in this Data Item, radial Reynolds number is defined using the(equivalent) diameter as the characteristic dimension, and both and are taken to be positive. Other definitions exist, forexample the radial Reynolds number may be defined using the radius of the vapour duct (Derivation 21), and it is often taken to be negativein the evaporator and positive in the condenser (Derivations 21 and 29).

s

ReDe4 4 s

2–( ) s 1+( )

s s2

2s– 5–( )---------------------------------------=

2

ReDe ReDc

ReDe2 4 s

2–( ) s 1+( )

2

s s2

2s– 5–( )----------------------------------------------=

s sese ReDe

∆pve

ρvVv2

------------se 2+

ReDe

---------------=se 2+

se 1+---------------+

∆pve / ρvVv2( ) ReDe

s

Direction ofaxial velocity

s ses sa

la

DvRev

---------------- la* s

25+

144 s 1+( )2 s 2+( )------------------------------------------------- 13

216--------- loge s 1+( )– +–=

364------ loge s 2+( ) 23

1728 ---------------- loge s 2–( )

sa

se

sa la*

se

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The static pressure drop for the whole adiabatic section can be calculated from the equation*

. (A1.4)

Figure 3 shows as a function of , obtained from Equations and (A1.4), for the completerange of values of .

In the condenser section, decreases in the flow direction from its value at the section entrance , to thevalue at the downstream end, , that depends on the radial Reynolds number, . Typical velocitydistributions are illustrated in Sketch A1.3.

Sketch A1.3 Typical normalised velocity distributions in condenser

The value of lies between 2.0 and 0, satisfying the equation

. (A1.5)

The static pressure drop for the whole condenser can be calculated from the equation

, (A1.6)

where is a dummy variable of integration and

(A1.7)

With an adiabatic section present, , obtained from equation . If there is no adiabatic section,, obtained from Equation (A1.1).

* No equivalent of Equation A1.4 is given in Derivation 29, but it appears that a term 1/(s+1) was omitted in the analysis and consequentlyFigure 2 in that Derivation is incorrect. The results in that Figure can be obtained by replacing the term [11s+14] / [9(s+1)2] inEquation A1.4 by [2s + 5]/[9(s + 1)2].

∆pva

ρvVv2

------------ 11s 14+

9 s 1+( )2------------------------–

2527------ loge s 1+( )– ½ loge s 2+( ) 23

54------ loge s 2–( )+ +

sa

se

=

∆pva / la*ρvVv

2

la

*

se

s sdsc ReDc

Direction ofaxial velocity

sc

ReDc

2 sc2

4–( ) sc 1+( )2

sc sc2

2sc 5––( )------------------------------------------------=

∆pvc

ρvVv2

------------sd 2+

ReDc

----------------sd 2+

sd 1+---------------- 1

ReDc

------------- 2– f s ,ReDc( ) sdsd

s ′∫

exp s ′dsd

sc

∫+–=

s ′

f s ,ReDc( ) s2

6s 7+ +

s 1+( )3 s 2+( )s

22s– 5–

s 1+( )2-----------------------------

2 s2

4–( )sReDc

------------------------–

------------------------------------------------------------------------------------------------------------=

sd sa=sd se=

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Figure 5 shows as a function of , obtained from Equations (A1.5), (A1.6) and (A1.7)*,for the complete range of values of . Equation (A1.6) cannot be evaluated analytically, and it is necessaryto perform the double integration in this equation numerically. It can be seen from Figure 5 that

is positive for low values of , corresponding to a decrease in pressure in the flowdirection, but that for greater than about 3, the pressure increases in the flow direction. This increasein pressure occurs because, at high flow rates, recovery of the dynamic pressure as the flow decelerates inthe condenser more than offsets the decrease in pressure due to friction.

A numerical solution of the governing equations for laminar, incompressible vapour flow but with noboundary layer approximations is given in Derivations 15 and 21. The solution is for heat pipes withoutadiabatic sections. Pressure drops in the evaporator from this solution agree to within 2 per cent withpressure drops calculated from Equations (A1.1) and (A1.2). The numerically obtained data for thecondenser are incomplete, and comparison over the full range of inlet conditions is not possible. However,data for the overall pressure drop in the evaporator and the condenser show good agreement with data fromEquations (A1.1), (A1.2), (A1.5), (A1.6) and (A1.7) for all up to 30. For , thepressure “drops” in the condenser are likely to be significantly affected by reverse flow close to theliquid-vapour interface. Equations (A1.5), (A1.6) and (A1.7) do not take account of reverse flow, andconsequently tend to predict values of that are too large, although there are insufficient data inDerivations 15 and 21 to quantify the error in most cases. However, for the purposes of this Data Item aconservative estimate of the overall pressure drop, , will be obtained by taking equal to zerowhenever exceeds 30.

The numerically obtained data agree with theoretical and experimental data obtained by other workers andreferred to in Derivations 15 and 21. However, no experimental data for the pressure drops under actualoperating conditions in heat pipes have been found.

An alternative analysis in Derivation 9 also leads to equations for the pressure drop in the vapour inevaporator, adiabatic and condenser sections. However, the analysis from Derivation 29 is recommendedin this Data Item because, out of the two, it gives the best agreement with the numerically obtained data inDerivations 15 and 21.

* No equivalents of these equations are given in Derivation 29. However, Figure 5 in that Derivation gives static pressure drops that agreeover the range with data computed from Equations A1.5, A1.6 and A1.7 within the accuracy to which that Figure can beread.

∆pvc / ρvVv2( ) ReDc

1 ReDc 30<<

sd

∆pvc / ρvVv2( ) ReDc

ReDc

ReDe ReDc= ReDc 30>

∆pvc

∆pv ∆pvcReDc

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ESDU 79012KEEPING UP TO DATE

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ESDU 79012Heat pipes – performance of capillary-driven designsESDU 79012

ISBN 978 0 85679 256 4, ISSN 0141-0141-402X

Available as part of the ESDU Series on Heat Transfer. For informationon all ESDU validated engineering data contact ESDU International plc,27 Corsham Street, London N1 6UA.

ESDU 79012 is one of a group of five on heat pipe performance. Itprovides information from which the thermal resistance of the pipe canbe estimated and the overall rate of heat transfer predicted. That overallheat transfer rate must then be compared with various calculated limitsthat must not be exceeded. They are the circulation limit that defines themaximum pressure difference that the wick can support, the vapourpressure (or viscous) limit which constrains the pressure drop in thevapour duct, the sonic limit for vapour sonic flow, and the entrainmentlimit above which the vapour will tear liquid droplets off the wick. Acomprehensive practical worked example shows how a calculationflowchart is applied in design. ESDU 79013 gives wick properties,ESDU 81038 treats two-phase closed thermosyphons, ESDU 80013introduces the use of heat pipes and includes practical designinformation, and ESDU 80017 gives properties of fluids relevant to heatpipe operation at temperatures of 210 to 570 K.