ESDU Copyright material. For current status contact ESDU. ESDU 81038 Endorsed by The Institution of Chemical Engineers The Institution of Mechanical Engineers Heat pipes – performance of two-phase closed thermosyphons Issued October 1981 With Amendment C November 1983
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Dec 11, 2015
Heat pipes – performance of
two-phase closed
thermosyphons
two-phase closed
thermosyphons
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ESDU 81038
Endorsed byThe Institution of Chemical EngineersThe Institution of Mechanical Engineers
Heat pipes – performance oftwo-phase closed
thermosyphons
Issued October 1981With Amendment C
November 1983
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ESDU 81038ESDU 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 The data are founded on an evaluation of all the relevant information, bothpublished and unpublished, and are invariably supported by original work of ESDU staff engineers or consultants.The whole process is subject to independent review for which crucial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 work on this Item was carried out in the Heat Transfer Group of ESDU which is under the supervision of Mr N.Thompson, Group Head. The member of staff who initially undertook the technical work involved was Dr A. Acton(Senior Engineer). The subsequent assessment of the available information and development of the Item wasundertaken by Mr R.A. Smith (Independent) and was sponsored by ESDU.
Dr D. Chisholm — National Engineering LaboratoryMr J.B. Goodacre — Marconi Research Laboratories, ChelmsfordMr G. Rattcliff — Isoterix Ltd Mr D.A.Reay — International Research and Development Co. Ltd Dr G. Rice — Reading University
Chairman Dr G.F. Hewitt — H.T.F.S., Atomic Energy Authority, Harwell
Vice-ChairmanProf. V.Walker — Bradford University
MembersDr T.R. Bott — Birmingham UniversityMr T.D. Hazell — Foster Wheeler Energy LtdMr 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 — IndependentMr M.A. Taylor — B.N.O.C. (Developments) LtdMr G.H. Walter — Imperial Chemical Industries LtdMr N.G. Worley — Babcock Power Ltd.
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ESDU 81038HEAT PIPES – PERFORMANCE OF TWO-PHASE CLOSED THERMOSYPHONS
CONTENTS
Page
1. NOTATION AND UNITS 1
2. INTRODUCTION 32.1 Scope 32.2 Steady-state Performance 42.3 Liquid Fill 5
3. OVERALL THERMAL RESISTANCE 63.1 Network of Resistances 63.2 Internal Resistance 93.3 Boundary Conditions 9
4. INTERNAL THERMAL RESISTANCE 104.1 Internal Resistance With Smooth Vertical Pipes 104.2 Temperature Difference Due to Hydrostatic Head 124.3 Internal Resistance With Smooth Inclined Pipes 144.4 Effect of Turbulence on Internal Resistance 154.5 Effect of Roughness on Internal Resistance 15
5. MAXIMUM RATE OF HEAT TRANSFER 155.1 Vapour Pressure Limit 155.2 Sonic Limit 165.3 Dryout Limit 165.4 Boiling Limit 165.5 Counter-current Flow Limit 165.6 Conclusions on Maximum Heat Transfer 17
6. PROCEDURE FOR PERFORMANCE CALCULATION 17
7. EXAMPLE 18
8. REFERENCES AND DERIVATION 22
APPENDIX A CORRELATION OF THERMAL RESISTANCE DATA 28
A1. CORRELATION OF THERMAL RESISTANCE DATA 28
APPENDIX B CORRELATION OF DATA ON MAXIMUM RATE OF HEAT TRANSFER 31
B1. DESCRIPTION 31
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HEAT PIPES – PERFORMANCE OF TWO-PHASE CLOSED THERMOSYPHONS1. NOTATION AND UNITS
The SI system of units is used throughout.
SI
cross-sectional area, m2
Bond number, defined in Equation (5.4)
constant in Equations (4.1) and (4.2)
specific heat capacity J/kg K
internal diameter m
external diameter m
a function of Bond Number, obtained from Figure 1
a function of , obtained from Equation (5.6)
a function of , obtained from Figure 2
liquid fill (fraction of evaporator covered by static pool),
local gravitational acceleration* m/s2
heat transfer coefficient, referred to inside diameter W/(m2K)
depth m
dimensionless pressure parameter, defined in Equation (5.5)
specific latent heat of vaporisation J/kg
length m
atmospheric pressure† Pa (i.e. N/m2)
pressure at bottom of pool Pa (i.e. N/m2)
vapour pressure Pa (i.e. N/m2)
rate of heat transfer W
maximum rate of heat transfer W
A πD2/4
Bo
C
cp
D
Do
f1
f2 Kp
f3 β
FVl/ Ale( )
g
h
H
Kp
L
l
pa
pp
pv
Q·
Q·
max
Issued October 1981
1
With Amendment A to C, November 1983
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liquid-film Reynolds number in adiabatic section between condenser and evaporator,radius m
surface area, m2
temperature K
saturation temperature K
effective overall temperature difference K
mean temperature difference due to hydrostatic head (Section 4.2)
K
volume m3
volume of liquid in unheated pipe m3
thermal resistance, 1/(hS) K/W
inclination of pipe to horizontal deg
thermal conductivity W/(m K)
porosity, (volume of pores)/(volume of porous material)
dynamic viscosity N s/m2 (i.e. kg/(s m))
density kg/m3
surface tension N/m
Figure of Merit for condensation, (from Reference 9)
kg/(K0.75s2.5)
Figure of Merit for nucleate boiling (Equation (4.5))
Subscripts
refers to adiabatic section
refers to condenser
refers to evaporator
refers to liquid film
refers to hydrostatic head
refers to liquid
mean
Ref4Q· / LµlπD( )
r
S πDl
T
Ts
T∆ Tso Tsi Th∆––( )
Th∆
V
Vl
z
β
λ
ε
µ
ρ
σ
Φ2 Lλ3l ρl
2/µl( )0.25
Φ3
a
c
e
f
h
l
m
2
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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 performance of a two-phase closed thermosyphon, i.e. a device that performs the same functionas a heat pipe, but which does not rely on capillary forces to circulate the liquid; it places the condenserhigher than the evaporator in order to use gravity to drive the liquid from the condenser to the evaporator.A greater rate of flow of liquid is possible if it is physically feasible to use gravity, instead of capillaryforces, to produce this flow. Furthermore the cost of purchasing and installing wicks into the pipes isavoided.
The use of thermosyphons should be considered whenever it is possible to place the source of the heat ator below the level of the sink of the heat. Applications include the recovery of waste heat from the hotexhaust gas leaving an industrial process or furnace (Reference 10); the heat sink may be a steam generator(References 10 and 12), or a preheater for the incoming air or process fluid (Reference 7). They have beenconsidered for the prevention of ice formation on navigation buoys (Derivation 18).
Typical thermosyphons covered by this Data Item are shown in Sketches 2.1 and 2.2. The devices dealtwith have:–
(i) circular tubes with uniform cross-section,
(ii) a single-component working fluid and no non-condensable gas,
(iii) either no wick, or a simple wick or insert in the evaporator section to improve liquid contact withthe wall,
(iv) angle of inclination to the horizontal 5° to 90° , with the evaporator at the lower end.
The gravity-assisted heat pipe, i.e. where capillary and gravity forces combine to drive the liquid from thecondenser to the evaporator, is a special case not covered by this Data Item; this device is dealt with inReference 8.
refers to external dimension
refers to nucleate boiling in the pool
refers to heat sink
refers to heat source
refers to vapour
refers to wick
refers to container wall
* Standard value at sea level = 9.81 m/s2 .† Standard value at sea level = 101 325 Pa.
o
p
si
so
v
w
x
3
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A procedure for calculating the performance of an existing thermosyphon is given in Section 6 and a flowchart is provided in Figure 3; this may also be used to check a design. The procedure is useful for designpurposes, although no method for optimising the design has been included because many optimisingparameters exist and it is difficult to generalise about their relative importance. The details of thecalculational procedure depend on prescribed boundary conditions, as discussed in Section 3.3.The performance calculations consist of two parts.
(1) Any unknown temperatures are calculated, as required; if the rate of heat transfer, , is not given,it must be determined by trial-and-error to satisfy the prescribed temperatures. These calculationsare described in Sections 3 and 4.
(2) The rate of heat transfer is compared with various calculated limits, to ensure that they are notexceeded, as described in Section 5.
An example (Section 7) illustrates the performance calculations.
2.2 Steady-state Performance
Thermosyphons are generally either vertical or inclined at about 5° to the horizontal. Sketch 2.1 illustratesa typical two-phase closed thermosyphon operating vertically; Sketch 2.2 shows a similar item operatingalmost horizontally. Each comprises an evaporator, an adiabatic section (which may be very short) and acondenser, the evaporator being at a lower level than the condenser. It is important to distribute the film ofliquid evenly over the part of the surface of the evaporator not covered by the pool. Thus a vertical itemmust be installed as nearly as possible in the vertical position, otherwise the condensate will be unevenlydistributed around the periphery where it enters the evaporator. When this is not possible, e.g. in a transportapplication, and with inclined items, the distribution of liquid over the heated surface must be encouraged,either by installing a wick, or by cutting spiral grooves inside the pipe, in the evaporator section;alternatively, excess liquid may be used (see Section 2.3).
The advantages of the vertical over the almost horizontal item are that the precautions to ensure good wettingare less important, and the maximum heat flux is higher (see Section 5.5). Almost horizontal pipes are usedwhen it is more convenient to place the heat source beside, rather than below, the heat sink, and when theheat transfer rate is well below the maximum. Intermediate angles may be used; these need the precautionsto ensure good wetting, and if the inclination to the horizontal is less than 50° there may be a reduction inthe maximum heat flux (see Section 5.5).
Q·
4
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2.3 Liquid Fill
Insufficient liquid fill leads to dryout (see Section 5.3). Excess liquid fill should be avoided, because anyliquid carried up into the condenser reduces the performance by rendering the covered area of the condenservirtually useless. Also, the liquids used are often expensive.
The liquid fill, denoted by F, is defined as the ratio of the volume of the liquid in an unheated pipe tothe volume of the evaporator; hence
(2.1)
where is the internal cross-sectional area of the pipe and is the length of the evaporatorsection.
It is recommended in Derivation 19, p. 311, that, for a vertical pipe, F should be more than 50% and also that
(2.2)
where , and are the lengths of the evaporator, the adiabatic, and the condenser sections respectively;this is sufficient to provide a mean film thickness of more than 0.3 mm over the total length.
On the basis of the information available in the Derivations, it is recommended here that, for a pipe withouta wick, Equation (2.2) should be satisfied and the liquid fill (F in Equation (2.1)) should be in the range40% to 60% for vertical pipes and 60% to 80% for inclined pipes. There is some reduction (5% to 10%) inthe maximum heat flux at the upper values of the liquid fill (Derivations 21, 22 and 26).
Sketch 2.1 Vertical thermosyphon
Sketch 2.2 Inclined thermosyphon
Condenser
Evaporator
Adiabaticsection
Heatsink
Heatsource
Vapourflow
Liquidflow
Pool
D
Condenser
Evaporator
Adiabaticsection
Heatsink
Heatsource
VapourflowLiquid
flowPool
D
β
le
la
lc
Vl( )
FVl
Ale
--------=
A πD2/4=( ) le
Vl 0.001D le la lc+ +( )≥
le la lc
5
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When a wick is fitted in the evaporator section, the only requirement is that the minimum liquid fill should be(2.3)
where is the cross-sectional area of the wick and is its porosity (see Reference 8).
3. OVERALL THERMAL RESISTANCE
When operating below its maximum overall rate of heat transfer, , the performance of a two-phaseclosed thermosyphon can be characterised by the overall thermal resistance*, z. The actual overall rate ofheat transfer, , and the effective temperature difference between the heat source and the heat sink
, are then related by
. (3.1)
3.1 Network of Resistances
Section 4.2 deals with the estimation of , the mean temperature difference due to hydrostatic head.
The overall thermal resistance can be represented by the idealised network of thermal resistances to as shown in Sketch 3.1 and explained in this Section. The subscripts are consistent with other Data Itemsin the series.
* In this Data Item, thermal resistance is as defined by Equation (3.1), and has units of K/W.
Vl 0.001D la lc+( ) Awleε+=
Aw ε
Q· max
Q·
T∆ Tso Tsi– Th∆–=( )
Q· ∆T/z=
Th∆
z1 z10
6
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Sketch 3.1 Thermal resistances and their locations
Condenser vapour-liquidinterface
Condenser film(transverse resistance)
Condenser wall(transverse resistance)
Condenser externalsurface-sink
Evaporator liquid-vapourinterface
Evaporator film(transverse resistance)
Evaporator wall(transverse resistance)
Souce-evaporatorexternal surface
Heat source
Vapour pressuredrop
Wall(axial resistance)
Heat sink
Q.
Q
z1
z2
z3
z4
z9
z8
z7
z6z5
z10
.
Tsi
Tco
Tso
Teo
Tv
Tv
Top of condenser
Bottom of evaporator
Heat sourceTso
Tco
Teo
Heat sinkTsi
z5
z2 z1
z4
z3
z8 z9
z6
z7
z10
Vapour
Condensate
7
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The thermal resistances arise as follows.and . (3.2)
If the heat source is a condensing vapour, can be calculated from the normal correlations for condensation(References 1 and 2). The same applies to if the heat sink is a boiling liquid. If heat is removed at thesink by natural convection and radiation, Reference 6 may be useful.
For single-phase forced convection in flow across a bank of plain tubes the heat transfer coefficients, and , can be calculated by the method given in Reference 5. If external transverse fins are attached tothe pipe, Reference 4 may be consulted for the estimation of their efficiency and of the heat transfercoefficient in flow across banks of tubes with transverse fins (correlations or curves of the heat transfercoefficient can often be obtained from the manufacturers of the finned surfaces).
and . (3.3)
(3.4)
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 beknown or the heat flux at one of these surfaces may be prescribed. In these cases one orboth of and need not be calculated. These resistances are given by
and The thermal resistances across the thickness of the container wall in the evaporator andthe condenser respectively. For a tube of wall thickness these resistances are given byEquations (3.3) (Reference 8, p.23):
and The internal thermal resistances of the boiling and condensing fluid in the thermosyphon.This is a complicated function of the fluid properties, the dimensions of thethermosyphon and the rate of heat transfer. Methods of estimating and are given inSection 4.1 for vertical tubes; the possible extension to inclined tubes is discussed inSection 4.3.
and The thermal resistance that occurs at the vapour-liquid interface in the evaporator and thecondenser respectively. These are always neglected, being exceedingly small.
The effective thermal resistance due to the pressure drop of the vapour as it flows fromthe evaporator to the condenser. As a result of this pressure drop, there is a drop in thesaturation temperature (which controls the rate of condensation). Thus is the drop insaturation temperature between the evaporator and the condenser divided by the rate ofheat transfer ( ). When operating well below the maximum heat transfer (see Section 5),
is small compared with and . A method of estimating from the pressure dropof the vapour is given in Reference 8, page 23.
The axial thermal resistance of the wall of the container. As well as conducting heatthrough its thickness, the wall conducts heat axially. This resistance is givenapproximately by
z1 z9
z1 z9
z1 1 / heoSeo( )= z9 1 / hcoSco( )=
z1z9
hehc
z2 z8tx
z2
ln Do/D( )
2π leλx
-------------------------= z8
ln Do/D( )
2π lcλx
-------------------------=
z3 z7
z3 z7
z4 z6
z5
z5
Q·
z5 z3 z7 z5
z10
z10 0.5 le la 0.5 lc+ +( ) / Axλx( )=
8
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For the normal mode of operation (with the evaporator below the condenser) axial conduction in thecontainer contributes very slightly to the performance. When heat is flowing in the reverse direction, thesmall amount of axial conduction is the only mode of heat transfer; thus the thermosyphon may be used toact as a “thermal diode”.
A practical criterion for negligible (less than 5%) axial conduction is
. (3.5)
If Equation (3.5) is satisfied, the overall thermal resistance is approximated by
. (3.6)
Otherwise,
. (3.7)
With reverse heat flow in a thermal diode, there is no internal heat transfer and the overall thermal resistanceis
. (3.8)
3.2 Internal Resistance
The term “internal resistance” is applied to the sum of the resistances inside the pipe, namely. Since and are always negligible and is usually negligible (see Section 5),
it is assumed in the following Sections that the internal resistance is .
3.3 Boundary Conditions
Four common types of boundary condition are considered, the symbols for the temperatures being as shownon Sketch 3.1.
(1) The rate of heat transfer ( ) is specified and it is required to estimate the temperature differenceacross the thermosyphon , to ensure that this is small compared with the overalltemperature difference. One temperature must also be given, so that the temperature of the vapourmay be calculated.
(2) The temperatures of the condenser external surface and of the heat sink are specified;it is required to estimate the rate of heat transfer ( ) and the temperature required at the heat source
. This applies to heating a liquid when there is a limit to the temperature of the heating surface.
(3) The temperatures of the evaporator external surface and of the heat source arespecified; it is required to estimate the rate of heat transfer ( ) and the temperature required at theheat sink . This applies to cooling a gas or liquid when the temperature of the cooling surface
where is the cross-sectional area of the wall of the container and its thermalconductivity.
Ax λx
z10 / z2 z3 z5 z7 z8+ + + +( ) 20>
z z1 z2 z3 z5 z7 z8 z9+ + + + + +=
z z1 z2 z3 z5 z7 z8+ + + +( ) 1–1 /z10+[ ]
1–z9+ +=
z z1 z9 z10+ +=
z3 z4 z5 z6 z7+ + + + z4 z6 z5z3 z7+
Q·
Teo Tco–( )
Tco( ) Tsi( )Q·
Tso( )
Teo( ) Tso( )Q·
Tsi( )
9
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must not fall below a specified minimum.(4) The temperatures of the heat source and of the heat sink are given; it is required toestimate the rate of heat transfer ( ), as illustrated in the example (Section 7).
The external surface temperature of the pipe at the source is given by
. (3.9)
The external surface temperature of the pipe at the sink is given by
. (3.10)
If axial conduction and vapour pressure drop are negligible the vapour temperature is
(3.11)
If the vapour pressure drop is significant, Equation (3.11) gives , the temperature of the vapour in thecondenser. The temperature of the vapour in the evaporator is then
. (3.12)
4. INTERNAL THERMAL RESISTANCE
4.1 Internal Resistance With Smooth Vertical Pipes
The thermal resistance of a film of condensate running down a smooth surface can be calculated fromNusselt’s theory of filmwise condensation, as explained in References 1, 2 and 11. Thus, for a vertical tube,it follows that
(4.1)
where and is a group of thermophysical properties called the “Figure ofMerit”. Values of for common working fluids are given in Reference 9.
Tso( ) Tsi( )Q·
Teo Tso
z1
z-----∆T–=
Tco Tsi
z9
z-----∆T+=
Tv Tsi
z7 z8 z9+ +( )
z------------------------------------∆T+=
Tvc
Tve Tvc z5Q·
+=
z7CQ
· 1 3⁄
D4 3⁄
g1 3⁄
lcΦ24 3⁄
-------------------------------------=
C ¼( ) 3/π( )4/3 0.235= = Φ2Φ2
10
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It is shown in Derivation 16 how an experimental heat pipe (19 mm internal diameter, 585 mm total length)was operated with just sufficient liquid fill for all of the evaporator to operate by evaporation from thesurface of the liquid film formed by the condenser. An extension of film theory shows that the internalresistance, under these idealised conditions, is the same in an evaporating film as in a film of condensate.Thus. (4.2)
In practice it is often difficult to establish a smooth film of liquid on the inner surface of the pipe. Visualobservations of ripples developing in the liquid film in the condenser are reported in Derivation 14, andvisual observations of the film flow in the evaporator breaking up into a series of rivulets with dry patchesin between are reported in Derivation 17. By contrast, the thermal resistance is reduced by the formationof waves on the surface of a film. An examination of experimental data on thermosyphons in Appendix Ashows that the thermal resistance may be appreciably greater than that given by Equation (4.2) when theReynolds number of the liquid film is less than 50 in the adiabatic section; this is given by
(4.3)
where L is the specific latent heat of vaporisation and is the dynamic viscosity of the liquid. The factthat, at low values of , the actual thermal resistance may be considerably greater than the theoretical ispresumably due to incomplete wetting of the heated surface. It is shown in References 2 and 11 thatEquation (4.1) somewhat overestimates the thermal resistance in condensation when . Athigher Reynolds numbers there is an appreciable fall in thermal resistance due to waves on the surface ofthe film and increasing turbulence within the film (see Section 4.4).
Derivation 24 describes measurements of the thermal resistance in nucleate boiling in the pool in atwo-phase thermosyphon and gives the following correlation:
(4.4)
where
(4.5)
where is the specific heat capacity of the liquid and is atmospheric pressure. The physical propertiesare evaluated at the atmospheric boiling temperature. Equation (4.5) applies when . Forexample, substituting into Equation (4.5) the physical properties of ammonia and water at their boilingpoints at atmospheric pressure (101 325 Pa) gives
for ammonia, , (4.6)
for water, . (4.7)
The results in Derivation 25 show that Equations (4.4) and (4.5) may be applied for values of up to20 (see Appendix A).
z3 fCQ
· 1 3⁄
D4 3⁄
g1 3⁄
leΦ24 3⁄
-------------------------------------=
Ref4Q
·
Lµ lπD------------------=
µlRef
100 Ref 1300< <
z3p1
Φ3g0.2
Q· 0.4
πDle( )0.6-----------------------------------------------------=
Φ3 0.32ρl
0.65λl0.3
cpl0.7
ρv0.25
L0.4µl
0.1---------------------------------
pv
pa
-----
0.23
=
cpl pa0.03 pv/pa 2< <
Φ3 56 pv/pa( )0.23=
Φ3 63 pv/pa( )0.23=
pv/pa
11
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It should be noted that the resistances calculated by Equation (4.4) are often much higher than thoseestimated from the more widely used expressions for nucleate boiling.The method recommended for the calculation of an approximate value of the internal thermal resistance ofa thermosyphon with a smooth inner wall, without a wick and located in a vertical position, is that givenin Derivation 24. The steps are as follows.
(4.8)
where F is the liquid fill, which can be determined from Equation (2.1). It is assumed in the derivation ofEquation (4.8) that the level of the liquid where there is nucleate boiling does not change as the rate of heattransfer is increased; in other words, the falling film of evaporating liquid is not significantly disturbed bythe slugs of boiling liquid thrown against it. However, at high rates of heat transfer the falling film becomesso thick that the region of nucleate boiling extends to the top of the evaporator. It is shown in Appendix Athat the experimental results justify this procedure, provided that and .
If there is a wick in the evaporator, should be calculated by the method in Reference 8.
The internal resistance of a vertical thermosyphon can usually be reduced by roughening the inner wall –see Section 4.5.
4.2 Temperature Difference Due to Hydrostatic Head
The experiments described in Derivation 24 showed that when operating at low pressure it is essential toallow for the temperature change due to the hydrostatic head of the liquid in the pool of a verticalthermosyphon. It was found that the change of level in the pool as a result of boiling had a negligible effecton the pressure at the bottom of the pool. Thus the pressure at the bottom of the pool is
. (4.9)
The saturation temperature at the bottom of the pool, , can then be found from the thermodynamic datafor the working fluid as that corresponding to a saturation pressure of . Information is given in Figure3 of Reference 9 for common fluids.
Sketch 4.1 shows a typical relationship between the various temperatures and depth in the evaporator of atwo-phase closed thermosyphon. The temperature of the heat source, , is assumed to be independentof depth. The saturation temperature of the boiling liquid, , follows a curve of the shape indicated; it isroughly equal to the temperature of the vapour in the adiabatic section and condenser, , at the top of the
1. Calculate from Equation (4.1).
2. Calculate from Equation (4.2).
3. Calculate from Equation (4.3). If , the method of calculation is applicable; if, the method is liable to underpredict the thermal resistance, as explained in Appendix A. If
apply the correction factor for turbulence given in Section 4.4.
4. Calculate from Equation (4.4).
5. If put ; otherwise calculate the mean value of the resistance of the evaporator, ,by interpolation; thus
z7
z3f
Ref 50 Ref 1300< <Ref 50<Ref 1300>
z3p
z3p z3f< z3 z3p= z3
z3 z3pF z3f 1 F–( )+=
Ref 50> F 0.4>
z3
pp pv plgFle βsin+=
Tppp
TsoTs
Tv
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evaporator, and increases with increasing depth till it reaches at the bottom of the evaporator. Forsimplicity it may be assumed that the temperature of the fluid in the evaporator equals above the staticliquid level and increases linearly with depth below this level. Thus the mean temperature of the fluid inthe evaporator is. (4.10)
The mean temperature difference due to the hydrostatic head is then given as
. (4.11)
Sketch 4.1 Typical temperatures in an evaporator
If thermodynamic data are not available for the working fluid, the rate of increase of saturation temperaturewith depth can be determined by combining Equation (4.9) with the Clausius-Clapeyron equation. Thus
. (4.12)
If the variation of properties with depth is negligible, the temperature at the bottom of the pool is
. (4.13)
TpTv
Tme Tv 1 F–( )Tv Tp+
2-------------------F+=
∆Th Tme Tv–Tp Tv–
2------------------- F= =
Top of evaporator
Temperature, T
Depth, H
0
Bottom
Pipe
Liquid beforestart-upTs
(assumedprofile)
Ts (true profile)
Tp
Tv TsoTeo
le
Fle
dTs
dH---------
Tsg
L---------
ρl
ρv
----- 1–=
Tp Tv
dTs
dH--------- leF+=
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For example, Sketch 4.2 gives the change in the saturation temperature of water with depth. By contrast,the value for ammonia at 20°C is 0.2 K/m, compared with 80 K/m for water.Sketch 4.2 Effect of hydrostatic head on saturation temperature
For many applications, is very small, i.e. , but this should be checked, particularly when thepressure of vapour is less than atmospheric.
4.3 Internal Resistance With Smooth Inclined Pipes
Tilting the pipe reduces the thermal resistance of the condensing film but may increase the thermalresistance of the evaporator due to inadequate wetting of the top of the pipe.
The thermal resistance of a film of condensate in a horizontal tube can be calculated from Nusselt’s theoryof filmwise condensation, as explained in References 1 and 2. Hence
. (4.14)
This equation has been found to apply to condensation on the outside of a horizontal pipe, so, if there issufficient slope to avoid a pool , it should also apply to condensation inside a nearly horizontalpipe, as confirmed by results in Derivation 28 (see Appendix A).
20 30 40 50 60 70 80
Water
90 100 1100
10
20
30
40
50
60
70
dTs/dH
(K/m)
Ts (°C)
Th∆ Tme Tv≈
z7( )z3( )
z70.335Q
· 1 3⁄
Dg1 3⁄
lc4 3⁄ Φ2
4 3⁄----------------------------------=
β 5°>( )
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It is shown in Derivation 25 that changing the angle does not significantly alter the internal resistance forangles down to 10° from the horizontal. It is therefore tentatively suggested that the internal thermalresistance of a tilted thermosyphon be estimated from Equations (4.4) and (4.14), provided that the liquidfill is more than 60%; the results reported in Appendix A show that this may lead to overdesign. If there isa wick in the evaporator, should be calculated by the method in Section 8 of Reference 8.4.4 Effect of Turbulence on Internal Resistance
For the following equation, taken from Reference 2 (Figure 13.5) is recommended:
(from Equation (4.1)) . (4.15)
The results in Derivation 25, with the pipe vertical, obtained with Refrigerant 115 near to the criticaltemperature, covered a range of Reynolds numbers from 1000 to 14 000. The experimental values of ,the thermal resistance of the condensate film, in the turbulent region were 70–78% of the predicted values,thus indicating that Equation (4.15) is safe.
4.5 Effect of Roughness on Internal Resistance
The methods of predicting the internal resistance of a thermosyphon given in Sections 4.1 and 4.3 are forsmooth pipes. It is known that artificially roughened surfaces give enhanced heat transfer coefficients innucleate boiling at small temperature differences. It is also known that a two or three-fold enhancement inthe heat transfer coefficient in condensation inside vertical pipes can be achieved by cutting vertical groovesdown the surface.
Tests are reported in Derivation 28 on a thermosyphon inclined at an angle of 5° to the horizontal, for asmooth surface and for a surface with circumferential capillary grooves. The liquid fill (F) ranged from45% to 63%. With water, the grooves enhanced the evaporating coefficient 3 to 4 fold, but there was noenhancement of the condensing coefficient. With Refrigerant 22, the grooves gave no enhancement of theevaporating coefficient but enhanced the condensing coefficient 5 to 6 fold.
Although artificially roughening a surface may be an excellent way of improving an otherwise poor heattransfer coefficient, no method is available for estimating the amount of enhancement.
5. MAXIMUM RATE OF HEAT TRANSFER
As the temperature difference across a heat pipe or a thermosyphon is increased, the overall rate of heattransfer increases until a maximum is reached. This may be due to a boiling crisis (a big increase in asexplained in Section 5.4; it may be due to an excessive pressure drop of vapour (a big increase in , asexplained in Sections 5.1 and 5.2, or it may be due to a failure in the supply of liquid to the heated surface(Sections 5.3 and 5.5). The remarks in this Section apply to vertical pipes, unless otherwise stated.
5.1 Vapour Pressure Limit
When operating a heat pipe or a thermosyphon at a pressure substantially below atmospheric, the pressuredrop of the vapour may be significant compared with the pressure in the evaporator. A method of estimatingthe vapour pressure limit is given in Section 5.1 of Reference 8 (for laminar flow of vapour).
z3
Ref 1300>
z7 Ref 1300>( ) z7= 191Ref0.733–×
z7
z3 )z5 )
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5.2 Sonic LimitAt low operating pressures, the vapour velocity may be appreciable compared with the sonic velocity inthe vapour. A method of estimating the sonic limit is given in Section 5.2 of Reference 8. From this itfollows approximately that the maximum axial vapour mass flux, , is given by
kg/(s m2). (5.1)
It is expected that this formula would apply irrespective of orientation. If the normal working temperatureis well above atmospheric, the sonic limit may apply under start-up conditions, when the working fluid isat the ambient temperature and so the values of and are much less than under operating conditions;this may slow down the start-up procedure. For example, the sonic limit to /A for water at 20°C is7.8 MW/m2 , while at 100°C it is 277 MW/m2 .
5.3 Dryout Limit
As applied to thermosyphons, the term “dryout” implies that the volume of the liquid fill is not sufficientto cover all of the pipe above the pool with a film of liquid. Thus with a vertical pipe, as shown in Sketch 2.1,most of the falling film of liquid would have evaporated before reaching the pool, leaving dry patches, asshown in Derivation 26, with a few rivulets of liquid returning to the pool; with an inclined pipe, as shownin Sketch 2.2, dry patches would appear at the top of the evaporator. The available evidence suggests thatdryout in a vertical thermosyphon is avoided if the volume of liquid fill meets the conditions called for inSection 2.3.
5.4 Boiling Limit
The boiling limit occurs when a stable film of vapour is formed between the liquid and the heated wall ofthe evaporator. This gives the maximum heat flux as
(5.2)
where is the internal surface area of the evaporator, is the surface tension of the liquid andg is the acceleration due to gravity. For example, the boiling limit for water is 200 kW/m2 at 20°C, risingto 1014 kW/m2 at 100°C. Comparing these figures with those at the end of Section 5.2 shows that the soniclimit occurs before the boiling limit if the ratio of the length to the diameter of the evaporator exceeds10, for a working temperature of 20°C, or 68, for a working temperature of 100°C.
Equation (5.2) is recommended in Reference 3, page 853, for the maximum heat flux in pool boiling. Ittherefore applies to the bottom of the pipe, where a pool is formed, as illustrated in Sketches 2.1 and 2.2.It is expected that the maximum heat flux in the region where there is a falling film of liquid is of a similarmagnitude, provided that the requirements to avoid dryout, given in Section 5.3, are met.
If a wick is installed in the evaporator, the boiling limit must then be determined as in a heat pipe (Section 5.4of Reference 8).
5.5 Counter-current Flow Limit
Q· max/ AL( )
Q·
max
AL------------- 0.5 pvρv( )0.5
=
Pv ρvQ·
Q·
max
Se
------------- 0.12L ρv( )0.5 σg ρlRρv–( )[ ]0.25
=
Se πDle=( ) σ
le/D( )
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Even when there is sufficient liquid present in the thermosyphon to prevent dry-out occurring, the overallrate of heat transfer is subject to another limit; this occurs when the rate of entrainment of liquid by thevapour prevents the downward flow of liquid (sometimes known as “flooding” or “entrainment limit”).The correlation derived in Appendix B for the maximum axial vapour mass flux is
(5.3)
where is a function of the Bond number, which is defined as
. (5.4)
The value of can be read from Figure 1, where it is plotted against Bo; when , .
The factor is a function of the dimensionless pressure parameter, , which is defined as
(5.5)
The factor is a function of the inclination of the pipe. When the pipe is vertical, ; when the pipeis inclined, the value of can be read from Figure 2, where it is plotted against , the angle of inclinationto the horizontal, for various values of the Bond number. The product is sometimes called the“Kutateladze number”.
5.6 Conclusions on Maximum Heat Transfer
Whichever of the preceding criteria gives the lowest value of the maximum rate of heat transfer must beassumed to give the true limit to the output of the thermosyphon. It is not advisable to design to this limit,because the methods for estimating the thermal resistance will give underestimates as the maximum massflux for the vapour is approached. It follows from Derivation 27 that when the axial heat flux is less than70% of the counter-current flow limit, vapour shear has no significant effect on the flow of the liquid. It istherefore recommended that thermosyphons be designed to operate at less than 50% of the maximum heatflux.
6. PROCEDURE FOR PERFORMANCE CALCULATION
The flow-chart in Figure 3 has been prepared to assist anyone wishing to predict the performance of anexisting thermosyphon or wishing to consider the possibility of using a thermosyphon for a known duty;for the latter it is necessary to produce a rough design by trial-and-error, using the flow-chart to check eachattempted design. It is assumed that the orientation of the pipe has been chosen from consideration of therequired layout, that the working fluid has been chosen to suit the temperature involved, and that theoptimum fill has been determined from the considerations in Section 2.3. The flow-chart has been preparedto cover the four boundary conditions described in Section 3.3. It is concerned mainly with the estimationof performance, but finishes with a check that the maximum heat transfer rate will not be exceeded.
and if (5.6)and if .
Q·
max
AL------------- f1 f2 f3 ρv( )0.5
g ρl ρv–( )σ[ ]0.25=
f1
Bo Dg ρl ρv–( )
σ---------------------------
0.5=
f1 Bo 11> f1 8.2=
f2 Kp
Kp
pv
g ρl ρv–( )σ[ ]0.5-------------------------------------------=
f2 Kp0.17–
= Kp 4 104×≤
f2 0.165= Kp 4 104×>
f3 f3 1=f3 β
f1f2f3
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7. EXAMPLEHeat is to be transferred from flue gas leaving a furnace through a horizontal duct to the combustion airflowing countercurrent in a duct above the flue gas duct, using vertical thermosyphons. It is required todetermine the rate of heat transfer at the cold end of the exchanger, where the flue gas is at 186°C and theincoming air is at 27°C. The carbon steel pipes are 50 mm external diameter, 2.4 mm thick. The total lengthof each pipe is 4.6 m, split into 1.7 m for the condenser, 0.1 m for the adiabatic section, and 2.8 m for theevaporator. The working fluid is water, and the depth of fill is 1.5 m. Axial conduction is negligible. Thethermal conductivity of the material of the tube is 48 W/(m K). The pipes have external steel fins, the findensity being greater on the shorter condenser section than on the evaportor section. The values of the totalexternal surface area and the effective outside heat transfer coefficient referred to the totalexternal surface area, and including an allowance for the fin efficiency, are
This problem must be solved by trial-and-error, because the heat transfer rate ( ) is not known. This islikely to be controlled by the gas-side heat transfer coefficients. Thus it is possible to make a firstapproximation to the solution of the problem by neglecting the internal thermal resistance
. The first step is to calculate the other resistances, namely , , and ; asaxial conduction is said to be negligible, Equation (3.6) may then be used to estimate the total thermalresistance (z).
(1) From Equation (3.2) the thermal resistance of gas and air are
K/W,
K/W.
(2) The internal diameter (D) is 50 – 2 × 2.4 = 45.2 mm.
The liquid fill (F) is 1.5/2.8 = 0.536.
From Equation (3.3), the wall resistances are
and .
K/W K/W.
(3) Neglecting the internal resistance, Equation (3.6) gives the total thermal resistance:
K/W.
FLUID Hot flue gas Air
Area m23.26 3.08
Coefficient W/m2 K 31 37
So( ) ho( )
So( )
ho( )
Q·
z3 z4 z5 z6 z7+ + + +( ) z1 z2 z8 z9
z1 1 / 31 3.26×( ) 9.896 103–×= =
z9 1 / 37 3.08×( ) 8.775 103–×= =
z250 /45.2( )ln
2π 2.8 48××-----------------------------------= z8
50 /45.2( )ln2π 1.7 48××-----------------------------------=
1.195 104–×= 1.969 10
4–×=
z 9.896 0.1195 0.1969 8.775+ + +( ) 103–× 1.899 10
2–×= =
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From Equation (3.11), the vapour temperature is.
(4) From Reference 9, Tables 6.14 and Figure 3, the following values are obtained for the physicalproperties of water and steam saturated at 102°C.
Pa ;
kg/m3 ; kg/m3 ; J/kg ;
N s/m2 ; N/m ; kg/(K0.75 s2.5).
(5) The pressure at the bottom of the pool is given by Equation (4.9):
Pa.
From the properties of water in Reference 9, .
Alternatively, from Sketch 4.2, the change of boiling temperature with respect to depth, at 102°C,is 2.4 K/m. From Equation (4.13)
.
The mean temperature difference due to hydrostatic head is obtained from Equation (4.11), takingthe mean value of .
.
(6) By definition, the effective overall temperature difference is
.
(7) From Equation (3.1) the rate of heat transfer is
W.
(8) From Equation (4.2), the thermal resistance of the evaporating film is
K/W.
Tv 270.1969 8.775+( ) 10
3–×
1.899 102–×
-------------------------------------------------------------- 186 27–( )+ 102°C= =
pv 1.097 105×=
ρl 956= ρv 0.64= L 2.252 106×=
µl 2.75 104–×= σ 58.5 10
3–×= Φ2 7000=
pp 1.097 105× 956 9.81 1.5 90°sin×××+ 1.238 10
5×= =
Tp 105.4°C=
Tp 102 2.4 1.5×+ 105.6°C= =
Tp 105.5°C=
∆Th105.5 102–( ) 0.536×
2-------------------------------------------------------- 0.9°C= =
∆T 186 27– 0.9 158°C≈–=
Q·
158 /1.899 102–× 8320= =
z3 f0.235 8320
1 3⁄×
0.04524 3⁄
9.811 3⁄
2.8 70004 3⁄××
---------------------------------------------------------------------------------- 3.69 104–×= =
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From Equation (4.3), the Reynolds number in the adiabatic section is.
This is within the range of Reynolds numbers (50 to 1300) for which Equation (4.2) may be used.
(9) The figure of merit for nucleate boiling is obtained from Equation (4.7):
.
The resistance in nucleate boiling is given by Equation (4.4) as
K/W.
(10) As , the mean value of the resistance in the evaporator section is given by Equation (4.8).Hence
K/W.
(11) From Equation (4.1), the thermal resistance of the film of condensate is
K/W.
(12) Adding the internal thermal resistance calculated in steps (8) to (11) to the approximate total thermalresistance calculated in steps (1) to (3) gives
K/W.
From Equation (3.1), the revised rate of heat transfer is
W.
(13) Repeating the calculations in steps (8) to (12) gives
W.
This is very close to the previous value. It is concluded from the calculations of resistance that therate of heat transfer is 7900 W. In these calculations it was assumed that , the effective thermalresistance due to pressure drop of the vapour, was negligible. It is now necessary to check that therewill be no limit on flow and no boiling crisis.
Ref4 8320×
2.252 106
2.75 104– π 0.0452×××××
------------------------------------------------------------------------------------------------------- 378= =
Φ3 63 1.097 /1.013( )0.2364= =
z3p1
64 9.810.2
83200.4 π 0.0452 2.8××( )0.6×××
----------------------------------------------------------------------------------------------------------------------- 4.65 104–×= =
z3p z3f>
z3 4.65 104–× 0.536 3.69 10
4–1 0.536–( )×+× 4.20 10
4–×= =
z70.235 8320
1 3⁄×
0.04524 3⁄
9.811 3⁄
1.7 70004 3⁄×××
----------------------------------------------------------------------------------------- 6.05 104–×= =
z 4.20 104–
6.05 104–
1.899 102–×+×+× 2.002 10
2–×= =
Q·
158 /2.002 102–× 7892= =
z3 f 3.62 104–× z3p; 4.75 10
4–× z3; 4.23 104–
z7;× 5.97 104;–×= = = =
z 2.000 102–
Q·
;× 158 /2.000 102–× 7900= = =
z5
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(14) As the pressure is above atmospheric, it is not necessary to estimate the vapour pressure limit. Thesonic limit is given by Equation (5.1). This requires a knowledge of the cross-sectional area. Thus
m2 .
Equation (5.1) gives
,
whence W.
This is well above the required rate of heat transfer 7900 W, so there is no danger of sonic limitationunder design conditions. Under start-up conditions the sonic limit will be lower; it is stated inSection 5.2 that the sonic limit for water at 20°C is 7.8 MW/m2 . For the pipe in this example themaximum rate of heat transfer at start-up is 7.8 × 106 × 1.605 × 10–3 = 12 520 W. The rate of heattransfer at start-up depends on how quickly the furnace is heated; in this application it is not likelyto exceed the normal rate, so sonic conditions will be approached, but not reached.
(15) The function of physical properties required for determining both the boiling limit andcounter-current flow limit is
.
(16) The boiling limit depends on the heat flux in the evaporator. The surface area for heat transfer inthe evaporator is
m2.
The maximum heat flux is given by Equation (5.2) as
W/m2,
giving W.
This also is well above the required rate of heat transfer (7900 W).
(17) The counter-current flow limit can be determined by the procedure described in Section 5.5. Thefirst step is to calculate the Bond number. From its definition in Equation (5.4):
.
As this is greater than 11, .
A π 0.0452( )2/4 1.605 10 3–×= =
Q·
max
1.605 103–
2.252 106×××
---------------------------------------------------------------------- 0.5 1.097 105
0.64××( )0.5
=
Q·
max 4.79 105×=
L ρv( )0.5 σg ρl ρv–( )[ ]0.252.252 10
6× 0.64( )0.558.5 10
3–9.81 956 0.64–( )×××[ ]
0.258.71 10
6×= =
Se π 0.0452 2.8×× 0.398= =
Q·
max
Se
------------- 0.12 8.71 106×× 1.045 10
6×= =
Q·
max 1.045 106× 0.398× 4.15 10
5×= =
Bo 0.04529.81 956 0.64–( )
58.5 103–×
-------------------------------------------0.5
18.1= =
f1 8.2=
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The dimensionless pressure parameter is obtained from Equation (5.5) as.
The function is given by Equation (5.6). As
.
The pipe is vertical, so the function is unity.
The maximum axial heat flux can now be calculated from Equation (5.3)
W/m2
W.
(18) Of the three values of calculated in steps (14), (16) and (17), the lowest is 2.73 × 104 Wcalculated in step (17) for the counter-current flow limit. This will therefore determine themaximum rate of heat transfer that could be obtained in the thermosyphon. The required rate ofheat transfer, 7900 W, is only 29% of the maximum. This is well below the recommended amountof 50%, so the thermosyphon should achieve its required output and the calculations of internalresistance should not be affected by vapour shear.
Note. The reason for the rate of heat transfer being well below the maxima set by the various limits in thisexample is that the rate of heat transfer is controlled by the low heat transfer coefficients from the hot gasto the pipe and from the pipe to the air. In other applications it is likely that the required rate of heat transferwill be nearer to the maximum.
8. REFERENCES AND DERIVATION
References
The References given are recommended sources of information supplementary to that in this Data Item.
1. KERN, D.Q. Process heat transfer. McGraw-Hill, 1950.
2. McADAMS, W.H. Heat transmission, 3rd Edition. McGraw-Hill, 1954.
3. OLLIER, J.G.WALLIS, G.B.see also:LIENHARD, J.H. DHIR, V.K.
Lecture notes for course on two-phase flow and heat transfer, Glasgow,1967.
Extended hydrodynamic theory of the peak and minimum pool boilingheat fluxes. NASA CR-2270, 1973.
4. KERN, D.Q.KRAUS, A.P.
Extended surface heat transfer. McGraw-Hill, 1972.
Kp1.097 10
5×
9.81 956 0.64–( ) 58.5 103–××[ ]
0.5-------------------------------------------------------------------------------------------- 4685= =
f2 Kp 4 104×<
f2 46850.17–
0.238= =
f3
Q·
max/A 8.2 0.238 1.0 8.71 106×××× 1.70 10
7×= =
Q·
max 17.0 106× 1.605 10
3–×× 2.73 104×= =
Q·
max
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Derivation
The Derivation lists selected sources that have assisted in the preparation of this Data Item.
5. ESDU Convective heat transfer during crossflow of fluids over plain tubebanks. Data Item No. 73031, ESDU International plc, London, 1973.
6. ESDU Heat transfer by free convection and radiation–simply shaped bodies inair and other fluids. Data Item No. 77031, ESDU International plc,London, 1977.
7. BARRATT, R.O. HENDERSON, G.O.
The heat pipe air preheater. Vol. 41, Proc. Am. Power Conf., 1979.
8. ESDU Heat pipes – performance of capillary-driven designs. Data Item No.79012, ESDU International plc, London, 1979.
9. ESDU Thermophysical properties of heat pipe working fluids: operating rangebetween –60°C and 300°C. Data Item No. 80017, ESDU Internatonalplc, London, 1980.
10. GROLL, M. NGUYEN-CHI, H. KRAHLING, H.
Heat recovery devices employing gravity supported heat pipes oncomponents. Paper presented to International Energy AgencyConference on Energy Conservation Technology, West Berlin, 1981.
11. HIRSCHBURG, R.I. FLORSCHUETZ, L.W.
Laminar wave-film flow: Part II – condensation and evaporation.ASME Publication 81-HT-14, presented at 20th National Heat TransferConference, Milwaukee, Wisconsin, August 1981.
12. LITTWIN, D.A. McCURLEY, J.
Heat pipe waste heat recovery boilers. Advances in Heat PipeTechnology: IV International Heat Pipe Conference, London,September 1981. Pergamon Press, Oxford, 1982.
13. FREA, W.J. Two-phase heat transfer and flooding in counter-current flow. PaperB5.10, International Heat Transfer Conference, Paris, 1970.
14. LARKIN, B.S. An experimental study of the two-phase thermosyphon tube. CanadianSoc. mech. Engrs, Paper 70-CSME-6. The Engng Jl, EIC Vol. 14, No.B-6, 1971.
15. LEE, V.MITAL, U.
A two-phase closed thermosyphon, Int. J. Heat Mass Transfer, Vol. 15,pp.1695-1707, 1972.
16. STREL’TSOV, A.I. Theoretical and experimental investigation of optimum filling for heatpipes. Heat Transfer – Sov. Res., Vol. 7, No. 1, pp. 23-27, 1975.
17. ANDROS, F.E. FLORSCHUETZ, L.W.
The two-phase closed thermosyphon: an experimental study with flowvisualization. Proceedings of the Symposium Workshop on Two-phaseFlow and Heat Transfer, Fort Lauderdale, Florida, USA, pp. 247-250,1976.
18. LARKIN, B.S.DUBUC, S.
Self de-icing navigation buoys using heat pipes. 2nd International HeatPipe Conference, Bologna, Italy, 1976.
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19. BEZRODNYI, M.K.ALEKSEENKO, D.B.Investigation of the critical region of heat and mass transfer inlow-temperature wickless heat pipes. High Temperature, Vol. 15,pp. 309-313, 1977.
20. UNK, J. Ammoniak-Gravitationswarmerohre zur Warmeruckgewinnung inluftungstechnischen Anlagen, Ki Klima-Kalte-Heizung 10/1979,pp. 399-405, 1979. (In German.)
21. GROLL, M. NGUYEN-CHI, H. KRÄHLING, H.
Wärmerückgewinnungsanlagen mit Reflux-Wärmerohren alsBauelemente. IKE, Stuttgart, Report No. IKE 5TF-363-80, March1980. (In German.)
22. GROLL, M. NGUYEN-CHI, H. KRÄHLING, H.
Heat recovery units using reflux heat pipes as components. IKE,Stuttgart, Report No. IKE 5TF-381-80, April 1980.
23. ANDROS, F.E. Heat transfer characteristics of the two-phase closed thermosyphon(wickless heat pipe) including direct flow observation. Arizona StateUniversity, PhD Thesis, 1980.
24. SHIRAISHI, M. KIKUCHI, K. YAMANISHI, T.
Investigation of heat transfer characteristics of a two-phase closedthermosyphon. Advances in Heat Pipe Technology: IV InternationalHeat Pipe Conference, London, September 1981. Pergamon Press,Oxford, 1982.
25. HAHNE, E.GROSS, U.
The influence of the inclination angle on the performance of a closedtwo-phase thermosyphon. Advances in Heat Pipe Technology: IVInternational Heat Pipe Conference, London, September 1981.Pergamon Press, Oxford, 1982.
26. NGUYEN-CHI, H. GROLL, M.
Entrainment or flooding limit in a closed two-phase thermosyphon.Advances in Heat Pipe Technology: IV International Heat PipeConference, London, September 1981. Pergamon Press, Oxford, 1982.
27. BEZRODNYI, M.K. VOLKOV, S.S.
Study of hydrodynamic characteristics of two-phase flow in closedthermosyphons. Advances in Heat Pipe Technology: IV InternationalHeat Pipe Conference, London, September 1981. Pergamon Press,Oxford, 1982.
28. LARKIN, B.S. An experimental study of the temperature profiles and heat transfercoefficients in a heat pipe for a heat exchanger. Advances in Heat PipeTechnology: IV International Heat Pipe Conference, London,September 1981. Pergamon Press, Oxford, 1982.
24
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FIGURE 1 VARIATION OF IN EQUATION (5.3) WITH BOND NUMBER
FIGURE 2 VARIATION OF IN EQUATION (5.3) WITH ANGLE OF INCLINATION AND BOND
NUMBER
Bond Number, B0
0 2 4 6 8 10 12
f1
4
5
6
7
8
f1
β°
0 10 20 30 40 50 60 70 80 90 100
f3
0.2
0.4
0.6
0.8
1.0
1.2
1.4
≥ 4
2
1
Bo
f3 β
25
ESDU 81038
ESDU
Cop
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ater
ial.
For
cur
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cont
act E
SDU
.
FIGURE 3a FLOW CHART FOR PERFORMANCE CALCULATION
START
Input data: either plusone temperature or twotemperatures; the amount ofliquid fill; the orientation ofthe pipe.
Q·
Calculate external thermalresistances(Eqns. (3.2) and (3.3))
IsQ known
?
Are and
given?Tco Tsi
No NoAre and
given?Tso Tsi
Are and
given?Teo Tso
No
Yes
Calculate Q from, and z1 Teo Tso
Calculate Q from, and z9 Tco Tsi
Set Calculate from Equation (3.6)
z3 z4 z5 z6 z7 0= = = = =z
From Fig. 3b
Calculate andfrom (if notgiven)
TsoTsi
Calculate Tso Calculate Tsi
Calculate (Equation (3.11))Determine physical properties of operating fluid at (Reference 9)Calculate (Equation (4.11))Calculate effective (from definition)
TνTν
∆Th∆T
Yes Yes Yes
From Fig. 3b
Calculate Q·
Calculate Reynolds number of film from Equation (4.3)Ref( )
26
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ESDU 81038
FIGURE 3b CONTINUATION OF FLOW CHART
Is50 <
?Ref
No suitable methodavailable; look forsimilar problem inAppendix B orDerivations.
Calculate (Section 4.1 or Reference 8, Section 8)Calculate (Section 4.1 or 4.3)Recalculate (Equation (3.6) including and
z3z7
z z3 z7
Recalculate and (ifnot given)
TsoTsi
Recalculate TsoRecalculate Tsi Recalculate Q·
No
Are newvalues same
as approx. oneswithin accuracy
of data?
Yes
No
Calculate by all methods in Section 5Set equal to lowest value
QmaxQ· max
It may be impossibleto attain . Internalresistances may exceed estimated values.
Q·
cannot beattainedQ·
Yes
Is <
?Q· Qmax
Is < 0.5
?Q· Qmax
Yes
Finish
From Fig. 3aFrom Fig. 3a
From Fig. 3a
27
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APPENDIX A CORRELATION OF THERMAL RESISTANCE DATAA1. CORRELATION OF THERMAL RESISTANCE DATA
The method of estimating the internal thermal resistance of a two-phase closed thermosyphon recommendedin this Data Item is that proposed by Shiraishi and his co-workers in Derivation 24. Table A1.1 gives asummary of their experimental work on thermal resistance, together with a summary of other experimentalwork, the results of which have been compared with the recommended method of estimation. The finalcolumn of the table gives the range of values obtained for the ratio of the experimental to the predictedvalues of z, the thermal resistance, defined in Equation (3.1). This final column has four sub-divisions: thefirst gives the ratio for , the thermal resistance in the pool (see Sketches 2.1 and 2.2); the second dealswith the upper region of the evaporator; the third deals with the condenser; finally the fourth column givesthe ratio for the internal thermal resistance, which is the sum of (from step (5) of Section 4.1) and .
Shiraishi et al. measured all three resistances separately and compared them with their correlations. It isseen from Table A1.1 that the measured values of , with F = 0.5, agree to within a factor of 1.4 withthe values predicted by the correlation that they developed for boiling in the pool; they said that a goodagreement was also obtained with F = 1.0, but these results are not given. At high heat fluxes, they foundthat the resistance in the upper region of the evaporator (with F = 0.5) also agreed with that predicted bytheir equation for boiling in the pool; this they attributed to nucleate boiling, as observed in Derivation 23(Section 5.1.2). At lower heat fluxes, the resistance in the upper region of the evaporator generally agreedwith that predicted for film evaporation, using Nusselt’s equation for filmwise condensation. The exceptionwas water, where the agreement was good at the highest flux, but at the lowest flux the measured resistancewas 12 times that predicted, presumably due to the formation of rivulets. They found that the transitionfrom film evaporation to nucleate boiling occurred roughly at the point where the two different mechanismscrossed. The thermal resistance in condensation agreed with, or was up to twice that predicted from Nusselt’sequation, except for water at low Reynolds numbers, where the resistance ranged from 0.5 to 3.5 times thatpredicted. The overall thermal resistance agreed mostly to within % of that predicted, there being somecancellation of the discrepancies with water.
Frea (Derivation 13) studied flooding in boiling in an open pipe. His measured values of the thermalresistance in boiling agree quite well with predictions.
The results in Derivations 14, 15, 18 and 20 all show that at low fills and low Reynolds numbers the observedinternal resistance is many times greater than that predicted, presumably due to incomplete wetting. Forpractical reasons it is recommended in this Data Item that the fill should be more than 40%; furthermore itis likely that Reynolds numbers in industrial applications will be higher than those in the experimentsanalysed. It was therefore decided not to try to derive a correlation for a correction factor to allow forincomplete wetting. Anyone wishing to operate a thermosyphon at low fill and a low Reynolds numbershould consult the papers referred to in the hope of finding results under similar conditions.
Andros in Derivation 23 describes extensive research in two devices: his “visual” device consisted of aheated stainless steel pipe inside a glass container, thus giving flow in an annulus; his “opaque” device wasa pipe. Table A1.1 gives details of his two devices and a comparison between some of his results andpredictions. The method of prediction is confirmed; the results on evaporation are of interest in confirmingthe method of interpolating between film evaporation and nucleate boiling when the estimated resistancewith the latter exceeds that with the former. The results with the opaque device show that when the heatflux is so high that the resistance from film theory exceeds that calculated from the boiling correlation, theexperimental values of the resistance are even lower than is predicted from the boiling correlations (thereare no other results in this region).
z3p
z3 z7
z3p
30±
28
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The results of Hahne and Gross in Derivation 25 are of interest in that they show that varying the angle ofinclination does not alter the performance very much. Also their Reynolds numbers were much higher thanany others and show the transition from laminar to turbulent film condensation. Unfortunately they do notgive the temperature of the vapour; to estimate this, it was necessary to guess the comparatively smallresistance of the coolant.The results of Larkin in Derivation 28 give good data on the performance of an almost horizontal pipe.They confirm that Equation (4.14) gives a safe estimate of the resistance in the condenser. Equation (4.4)somewhat underpredicts the resistance in the evaporator; however it confirms that there is nucleate boilingall round the tube (film theory would give a very much lower resistance).
29
ESDU Copyright material. For current status contact ESDU.
30
ES
DU
81038TABLE A1.1 Correlation of Thermal Resistance Data (smooth tubes)
(experimental z)/(predicted z)
0.8 – 1.1––––
*
––––
–†2.95–0.57†2.93–0.97†5.56–0.67†3.5–0.68
and reduces the maximum value of this
––––
––––
†4.3–1.61†3.1–2.07†3.0–2.2†2.0
– – †4.0–1.2
1.0–15‡ 1.8–0.7 –
0.7 – 1.0‡3–1.1–
––
0.5–1.5‡ 0.78 –
0.9–1.40.8–0.90.7–1.1–––
1–120.6–1.20.6–1.3***
1.2–0.52–0.91.5–1.13.5–11.4–0.91.5–1.1
0.7–1.50.8–1.50.8–1.30.9–1.20.8–1.90.9–1.1
0.76–0.310.81– 0.55
**
0.60–0.780.86– 0.46
––
1.1–1.50.5–0.8
**
0.5–0.70.8–0.9
––
z3p z3f z7 z3 z7+
.4 Rel 50>
DerivationDimensions (mm)
FLUID FILLTemp
°C (watts)Re
D
13. FREA14. LARKIN15. LEE
MITAL
2.518.727.0
88457610
––
914762
90°90°90°
waterwaterwaterwaterR11
128%50%40.5%11.5%24.5%
10027 – 1245858, 6629
15 – 38100 – 1990192 – 2492196 – 361439 – 525
12 – 313 – 2808 – 1009 – 17024 – 330
* Not applicable for nearly horizontal pipes or when .† These poor performances were obtained at low fills and/or low Reynolds numbers: applying the limits given in Section 4.1 of
ratio to 1.08.
18. LARKINDUBUC
72.9 2180 190 1140 90° ammoniaammoniaammoniaR12
3.4%6.9%21%21%
0000
225–520139–999102–183195
17–417–788–1490
20. UNK 12 to 29 90° ammonia 14–24% 20–31 100–535 35–240
23. ANDROS 50.8 (o.d.) 152 – up to 90° R113 61% 25 25–410 11–300
‡ Combined (predicted from Equation (4.8)).
34.9 (heated i.d.)
191 ethanolwater
61%61%
2537
25–40070–1400
2–303–67
26.6 610 – 760 90° R113 6–62% 25 50–1000 30–700
24. SHIRAISHIKIKUCHIYAMANISHI
37 280 500 450 90° waterethanolR113waterethanolR113
50%50%50%100%100%100%
32, 45, 6032, 453232, 45, 6032, 4532
33–114033–114033–114033–114033–114033–1140
0.6–362–6010–3000.6–362–6010–300
25. HAHNEGROSS
40 510 1016 483 90°10°
R115 176% 37–76 500–2850500–2850
1000–14 000280 – 1700
28. LARKIN 24.2 550 20 580 5° waterR22
56%85%
38–9520–70
500–1100500–1100
17–81700–2800
β Q·
le la lc
F 1.0≥
F 0>
z3
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ESDU 81038
APPENDIX B CORRELATION OF DATA ON MAXIMUM RATE OF HEAT TRANSFERB1. DESCRIPTION
Section 5 describes five mechanisms whereby the rate of heat transfer may be limited to a maximum. Thefirst and second (vapour pressure and sonic limits) have been dealt with in Reference 8. The available dataon dryout, which is discussed in Section 5.3, were obtained in electrically heated equipment, usuallyoperating with a low fill (F). It is recommended in this Data Item that dryout should be avoided by usingfills of more than 40%. The equation given in Section 5.4 for the boiling limit occurs in many textbookson heat transfer, but with different values of the constant, ranging from 0.12 to 0.18; Reference 3 showsthat the experimental values of the constant lie in this range and recommends the use of the lowest valuefor safety. This Appendix is therefore concerned only with the derivation of the correlation given in Section5.5 for the counter-current flow limit.
Most of the work on the counter-current flow limit (sometimes called “flooding” or “entrainment” limit)has been done by Bezrodnyi (Derivations 19 and 27) and Groll (Derivations 21, 22 and 26). The correlationderived here is based on these Derivations and on Derivation 13. It is found to predict the results reportedin Derivation 25 within the accuracy of the data. Groll presented all his results in the form of graphs of
, the maximum rate of heat transfer, plotted against , the angle of inclination to the horizontal; inhis experiments ranged from 5° to 86°. The curves through his results have been extrapolated to .In Figure B1 the experimental values of the function fl in Equation (5.3) are plotted against Bo, the Bondnumber, which is defined in Equation (5.4). This curve is for vertical pipes. The vertical crosses have beentaken direct from Derivation 27 (Figure 5); the other points have been calculated from the experimentalvalues of the maximum rate of heat transfer and the properties of water (from Reference 9). The extrapolatedvalues from the results of Groll are plotted along with the experimental points from Derivations 13 and 27.The upper curve is the recommended curve from Derivation 27; it is seen to be rather optimistic, particularlywith respect to the results from Derivation 26. It is recommended that the lower curve should be used; thisis reproduced in Figure 1.
Figure B2 shows the effect of varying the angle of inclination to the horizontal. Each experimental pointshows the ratio of the experimental value of at the angle to the experimental value at (plotted in Figure B1). The recommended curves, reproduced in Figure 2, are also shown.
Q·
max ββ β 90°=
Q·
max β β 90°=
31
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ESDU 81038
FIGURE B1 VARIATION OF IN EQUATION (5.3) WITH BOND NUMBER
Bond Number, B0
0 2 4 6 8 10 12 14 16 18 20
f1
3
4
5
6
7
8
9
10
X
Curve from Derivation 27Recommmended curve
Symbol Derivation D(mm) Fluid
1321
26
27
2.557
10
17
10-36
water
Water
R11R113
Ethanol
21
21
water
waterwaterwater
x
+ {
f1
32
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FIGURE B2 VARIATION OF IN EQUATION (5.3) WITH ANGLE OF INCLINATION AND BOND
NUMBER
β°0 30 60 90
f3
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Recommended curves
1
2
>4B0
f3 β( )
Symbol Derivation D (mm) Bond No
1321212126
2.557
1017
1.01.92.63.86.4
2
,
)
'
+
33
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ESDU 81038KEEPING UP TO DATE
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All rights are reserved. No part of any Data Item may be reprinted, reproduced, ortransmitted in any form or by any means, optical, electronic or mechanical includingphotocopying, recording or by any information storage and retrieval system withoutpermission from ESDU International plc in writing. Save for such permission allcopyright and other intellectual property rights belong to ESDU International plc.
© ESDU International plc, 2007
ESDU 81038Heat pipes – performance of two-phase closed thermosyphonsESDU 81038
ISBN 978 0 85679 371 4, ISSN 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.
The physical processes involved in a thermosyphon, whereby high ratesof heat transfer can be obtained between surfaces that have only a smalltemperature difference between them, are described. Heat is transferredby means of evaporation and condensation, and gravity is used to returnthe liquid film to the evaporator as compared with capillary-driven designswhich use a wick as described in ESDU 79012. ESDU 81038 relates tothermosyphons having (i) circular tubes of uniform cross section, (ii) asingle component working fluid and no non-condensable gas, (iii) eitherno wick or a simple wick or insert in the evaporator wall and (iv) anglesof inclination to the horizontal of 5 degrees to 90 degrees. The maximumoverall rate of heat transfer depends on the overall temperature differenceand the sum of the thermal resistances of the various solid, liquid andvaporous media and interfaces involved. Methods are given forcalculating each thermal resistance. Advice and expressions are alsogiven for the limits of vapour pressure, sonic velocity in the vapour,dry-out, boiling limit, and the counter-current flow limit.
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