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Chapter 2 Heat transfer and fluid flow theory c. The pressure due to the gravitational head, ΔPg2.1 INTRODUCTION As discussed in Chapter 6, there are many variants of heat pipe, but in all cases the working fluid must circulate when a temperature difference exists between the evaporator and condenser. In this chapter the operation of the classical wicked heat pipe is discussed. Various analytical techniques are then outlined in greater detail, and these techniques are then applied to the classical heat pipe and the gravity-assisted thermosyphon.2.2 OPERATION OF HEAT PIPES 2.2.1 Wicked heat pipes The overall thermal resistance of a heat pipe, defined by Eq. (2.1), should be low; however, it is first necessary to ensure that the device will function correctly.

The operating limits for a wicked heat pipe, first described in Ref. 1, are illustrated in Fig. R5Thot 2 Tcold Qð2:1ÞEach of these limits may be considered in isolation.

In order for the heat pipe to operate the maximum capillary pumping pressure, ΔPc;max, must be greater than the total pressure drop in the pipe. This pressure drop is made up of three components.which may be zero, positive or negative, depending on the inclination of the heat pipe.

For correct operation,1 ΔPc;max \$ ΔPl 1 ΔPv 1 ΔPgð2:2ÞIf this condition is not met, the wick will dry out in the evaporator region and the heat pipe will not operate. The maximum allowable heat flux for which Eq. (2.2) holds is referred to as the capillary limit. Methods of evaluating the four terms in Eq. (2.2) will be discussed in detail in this chapter. Typically, the capillary limit will determine the maximum heat flux over much of the operating range; however, the designer must check that a heat pipe is not required to function outside the envelope defined by the other operating limits, either at design conditions or at start-up. During start-up and with certain high-temperature liquid metal heat pipes the vapour velocity may reach sonic values.

The sonic velocity sets a limit on the heat pipe performance. At velocities approaching sonic compressibility effects must be taken into account in the calculation of the vapour pressure drop. The viscous or vapour pressure limit is also generally most important at start-up. At low temperature the vapour pressure of the fluid in the evaporator is very low, and, since the condenser pressure cannot be less than zero, the difference in vapour pressure is insufficient to overcome viscous and gravitational forces, thus preventing satisfactory operation.a.

The pressure drop ΔPl required to return the liquidfrom the condenser to the evaporator; b. The pressure drop ΔPv necessary to cause the vapourto flow from the evaporator to the condenser; © 2014 Elsevier Ltd. All rights reserved. DOI: Note that this implies that the capillary pressure rise is considered as positive, while pressure drops due to liquid and vapour flow or gravity are considered positive.15Heat transfer and fluid flow theoryAxial heat fluxEntrainment limitCapillary limitChapter2.3 THEORETICAL BACKGROUND In this section the theory underpinning the evaluation of the terms in Eqs (2.1) and (2.2) and the determination of the operating limits shown in Fig. 2.1 is discussed.2.3.1 Gravitational headSonic limit Boiling limitViscous limitThe pressure difference, ΔPg, due to the hydrostatic head of liquid may be positive, negative or zero, depending on the relative positions of the condenser and evaporator. The pressure difference may be determined from: ΔPg 5 ρl glsinφð2:3Þwhere ρl is the liquid density (kg/m ), g the acceleration due to gravity (9.81 m/s2), l length of the heat pipe (m) and φ the angle between the heat pipe and the horizontal defined such that φ is positive when the condenser is lower than the evaporator. 3Temperature Figure 2.1 Limitations to heat transport in a heat pipe.At high heat fluxes the vapour velocity necessarily increases, if this velocity is sufficient to entrain liquid returning to the evaporator then performance will decline.

Hence the existence of an entrainment limit. The above limits relate to axial flow through the heat pipe. The final operating limit discussed will be the boiling limit. The radial heat flux in the evaporator is accompanied by a temperature difference which is relatively small until a critical value of heat flux is reached above which vapour blankets the evaporator surface resulting in an excessive temperature difference.

The position of the curves and shape of the operating envelope shown in Fig. 2.1 depends upon the wick material, working fluid and geometry of the heat pipe.2.2.2 Thermosyphons The two-phase thermosyphon is thermodynamically similar to the wicked heat pipe but relies on gravity to ensure liquid return from the condenser to the evaporator. A wick or wicks may be incorporated in at least part of the unit to reduce entrainment and improve liquid distribution within the evaporator.2.2.3 Loop heat pipes and capillary pumped loops As will be seen later, the characteristics which produce a wick having the capability to produce a large capillary pressure also lead to high values of pressure drop through a lengthy wick. This problem can be overcome by incorporating a short wick contained within the evaporator section and separating the vapour flow to the condenser from the liquid return.162.3.2 Surface tension and capillarity 2.3.2.1 Introduction Molecules in a liquid attract one another. A molecule in a liquid will be attracted by the molecules surrounding it and, on average, a molecule in the bulk of the fluid will not experience any resultant force. In the case of a molecule at or near the surface of a liquid the forces of attraction will no longer balance out and the molecule will experience a resultant force inwards. Because of this effect the liquid will tend to take up a shape having minimum surface area; in the case of a free falling drop in a vacuum this would be a sphere.

Due to this spontaneous tendency to contract a liquid surface behaves rather like a rubber membrane under tension. In order to increase the surface area work must be done on the liquid. The energy associated with this work is known as the free surface energy, and the corresponding free surface energy/unit surface area is given the symbol σl. For example, if a soap film is set up on a wire support as in Fig. 2.2(a) and the area is increased by moving one side a distance dx, the work done is equal to F dx, hence the increase in surface energy is 2σl ldx. The factor 2 arises since the film has two free surfaces.

Hence if T is the force/unit length for each of the two surfaces, 2Tl dx 5 2σl l dx or T 5 σl. This force/unit length is known as the surface tension.

It is numerically equal to the surface energy/unit area measured in any consistent set of units, e.g. Values for surface tension for a number of common working fluids are given in Appendix 1. Since latent heat of vaporisation, L, is a measure of the forces of attraction between the molecules of a liquid we might expect surface energy or surface tension σl to be 2 Chapter 2 Heat transfer and fluid flow theory(a)(b)θθFl(c)σl R P2P1ΔP = P1–P2 = 2σl/Rσl Figure 2.2 Representation of surface tension and pressure difference across a curved surface.related to L. This is found to be the case. Solids also will have a free surface energy and in magnitude it is found to be similar to the value for the same material in the molten state. When a liquid is in contact with a solid surface, molecules in the liquid adjacent to the solid will experience forces from the molecules of the solid in addition to the forces from other molecules in the liquid.

Depending on whether these solid/liquid forces are attractive or repulsive, the liquid solid surface will curve upwards or downwards, as indicated in Fig. The two best-known examples of attractive and repulsive forces respectively are water and mercury. Where the forces are attractive the liquid is said to ‘wet’ the solid. The angle of contact made by the liquid surface with the solid is known as the contact angle, θ. For wetting θ will lie between 0 and π=2 radians and for non-wetting liquids θ. The condition for wetting to occur is that the total surface energy is reduced by wetting.abWetting σSL+ σLV σSV θ=πFigure 2.3 Wetting and non-wetting contact.σlP1σlP2R σlσsl 1 σlv, σsv where the subscripts, s, l and v refer to solid, liquid and vapour phases, respectively, as shown in Fig.

Wetting will not occur if σsl 1 σlv. Σsv as in Fig. 2.3(c) while the intermediate condition of partial wetting σsl 1 σlv 5 σsv is illustrated in Fig. 2.3(b).cσl Figure 2.4 Pressure difference across a curved liquid surface.balanced by the net force on the surface due to the pressures which is ðP1 2 P2 ÞπR2 or ΔPπR2. Thus, ΔP 52σl Rð2:4ÞIf the surface is not spherical, but can be described by two radii of curvature, R1 and R2 at right angles, then it can be shown that Eq.

(2.4) becomes: 0001 0003 1 1 ΔP 5 2σl 1 ð2:5Þ R1 R217Heat transfer and fluid flow theory Due to this pressure difference, if a vertical tube, radius r, is placed in a liquid which wets the material of the tube, the liquid will rise in the tube to a height above the plane surface of the liquid as shown in Fig. The pressure balance gives, for a tube radius r: 2σl cos θ 0002 ρl gh ðρl 2 ρv Þgh 5 ð2:6Þ r For the case of a non-circular tube, 0001 0003 1 1 1 5 1 ð2:7Þ r R1 R2 where ρl is the liquid density, ρv is the vapour density and θ is the contact angle.

This effect is known as capillary action or capillarity and is the basic driving force for the wicked heat pipe. For non-wetting liquids cos θ is negative and the curved surface is depressed below the plane of the liquid level. In heat pipes wetting liquids are always used, the capillary lift increasing with liquid surface tension and decreasing contact angle.2.3.2.3 Change in vapour pressure at a curved liquid surface From Fig. 2.5 it can be seen that the vapour pressure at the concave surface is less than that at the plane liquid surface by an amount equal to the weight of a vapour column of unit area, length h. Pc 2 Po 5 gρv h Assuming that ρv is constant. Combining this with Eq.

(2.6): 2σl ρv cos θ Pc 2 Po 5 r ðρl 2 ρv Þð2:8ÞThis pressure difference Pc 2 Po is small compared to the capillary head ð2σl =rÞcos θ and may normally be neglected when considering the pressures within a heat pipe. It is, however, worth noting that the difference in PcρlρvChaptervapour pressure between the vapour in a bubble and the surrounding liquid is an important phenomena in boiling heat transfer.2.3.2.4 Measurement of surface tension There is a large number of methods for the measurement of surface tension of a liquid, and these are described in the standard texts 2,3. For our present purpose we are interested in the capillary force, σl cos θ, which is dependent on both liquid and surface properties.

The simplest measurement is that of capillary rise h in a tube, which gives: ρl ghr ð2:9Þ 2 In practical heat pipe design it is also necessary to know r, the effective pore radius. This is by no means easy to estimate for a wick made up of a sintered porous structure or from several layers of gauze. By measuring the maximum height the working fluid will attain it is possible to obtain information on the capillary head for fluid wick combinations. Several workers have reported measurements on maximum height for different structures and some results are given in Chapter 3. The results may differ for the same structure depending on whether the film was rising or falling; the reason for this is brought out in Fig. Another simple method, for the measurement of σl, due to Ja¨ger, and shown schematically in Fig.

2.7 is sometimes employed. This involves the measurement of maximum bubble pressure. The pressure is progressively increased until the bubble breaks away and the pressure falls. When the bubble radius reaches that of the tube pressure is a maximum Pmax and at this point, 2σl Pmax 5 ρl hg 1 ð2:10Þ r This method has proved appropriate 4 for measurement of the surface tension of liquid metals.

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The surface tension of two liquids can be compared by comparing the mass of the droplets falling from a narrow σl cos θ 0002hPo hFigure 2.5 Capillary rise in a tube (wetting and non-wetting fluids).18Interface with rising columnInterface with falling columnFigure 2.6 Rising and falling column interface. 2 Chapter 2 Heat transfer and fluid flow theoryvertical tube.

If the mass of the droplets for liquids 1 and 2 are m1 and m2, respectively, then: m1 ρl1 σf1 5 ð2:11Þ m2 ρl2 σf22.3.2.5 Temperature dependence of surface tension Surface tension decreases with increasing temperature, it is therefore important to take temperature effects into account if using the results of measurements at typical ambient temperatures. Values of surface tension for over 2000 pure fluids have been tabulated by Jasper 5 and temperature corrections of the form σ 5 a 1 bT are suggested. For water the following interpolating equation 6 gives good values of surface tension: σl 5 Bð12Tr Þμ ð1 2 bð1 2 Tr ÞÞð2:12Þwhere Tr is the reduced temperature 5 T=Tc, Tc 5 647.096 K, B 5 235.8 mN/m, b 5 0.625 and μ 5 1.256. Equation (2.12) is valid between the triple point (0.010003 C) and the critical temperature, and is in agreement with measured data to within experimental uncertainty. Eo¨tvo¨s proposed a relationship which was later modified by Ramsay and Shields to give: 0001 00032=3 M σl 5 HðTc 2 6 2 TÞ ð2:13Þ ρl where M is the molecular weight, Tc is the critical temperature (K), T the fluid temperature (K) and H is a constant, the value of which depends upon the nature of the liquid. The Eo¨tvo¨s0004Ramsay0004Shields equation does not give agreement with the experimentally observed behaviour of liquid metals and molten salts. Bohdanski and Schins 7 have derived an equation which applies to the alkali metals.

While Fink and Leibowitz 8 recommended: 0001 0003 T n σl 5 σo 12 ðmN=mÞ ð2:14Þ Tc for sodium, where σo 5 240.5 mN/m, n 5 1.126 and Tc 5 2503.7 K. Alternatively, the surface tension of liquid metals may be estimated from the data provided by Iida and Guthrie 9 and summarised in Table 2.1. The value of surface tension may then be calculated:Pmax hσl 5 σlm 1 ðT 2Tm Þdσ=dT ρlA method of evaluating surface tension based upon the number and nature of chemical bonds was first suggested by Walden 10 and developed by Sugden 11 and Quale 12.

### Fluid Flow And Heat Transfer In Wellbores Pdf Download Windows 10

P ðρ 2 ρv Þ ð2:15Þ M l where P was defined by Walden as a parachor. Values of the increments to be used in evaluating the parachor, adapted to give values of surface tension in N/m are given in Ref.