The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 64
Текст из файла (страница 64)
The composite curve enclosing the shaded region inFigure 4.8.13 gives the maximum heat transfer rate of the heat pipe as a function of the operationaltemperature. The figure shows that as the operational temperature increases, the maximum heat transferrate of the heat pipe is limited by different physical phenomena. As long as the operational heat transferrate falls within the shaded region, the heat pipe will function properly.FIGURE 4.8.13 Heat transfer limitations in heat pipes.The vapor-pressure limitation (or viscous limitation) in heat pipes develops when the pressure dropin the vapor core reaches the same order of magnitude as the vapor pressure in the evaporator. Underthese conditions, the pressure drop due to flow through the vapor core creates an extremely low vaporpressure in the condenser preventing vapor from flowing in the condenser.
A general expression for thevapor-pressure limitation is (Dunn and Reay, 1982)Qnp,max =prn4 h fg r v,e Pn,e12m v,e leff(4.8.24)where rv is the cross-sectional radius of the vapor core (m), hfg is the latent heat of vaporization (J/kg),rv,e is the vapor density in the evaporator (kg/m3), Pve is the vapor pressure in the evaporator (Pa), andmv,e is the vapor viscosity in the evaporator (N sec/m2).
leff is the effective length of the heat pipe (m)equal to leff = 0.5(le + 2la + lc). The vapor-pressure limitation can occur during the start-up of heat pipesat the lower end of the working-fluid-temperature range.© 1999 by CRC Press LLC4-264Section 4The sonic limitation also can occur in heat pipes during start-up at low temperatures. The lowtemperature produces a low vapor density, thereby reducing the speed of sound in the vapor core. Thus,a sufficiently high mass flow rate in the vapor core can cause sonic flow conditions and generate a shockwave that chokes the flow and restricts the pipes ability to transfer heat to the condenser.
Dunn and Reay(1982) give an expression for the sonic limitation that agrees very well with experimental data,12Qs,max = 0.474 An h fg (rv Pv )(4.8.25)where Av is the cross-sectional area of the vapor core (m2). The sonic limitation should be avoidedbecause large temperature gradients occur in heat pipes under choked-flow conditions.The entrainment limitation in heat pipes develops when the vapor mass flow rate is large enough toshear droplets of liquid off the wick surface causing dry-out in the evaporator.
A conservative estimateof the maximum heat transfer rate due to entrainment of liquid droplets has been given by Dunn andReay (1982) asQe,maxér s ù= An h fg ê v l úêë 2rc,ave úû12(4.8.26)where sl is the surface tension (N/m) and rc,ave is the average capillary radius of the wick. Note that formany applications rc,ave is often approximated by rc,e.The capillary limitation in heat pipes occurs when the net capillary forces generated by the vaporliquid interfaces in the evaporator and condenser are not large enough to overcome the frictional pressurelosses due to fluid motion. This causes the heat pipe evaporator to dry out and shuts down the transferof heat from the evaporator to the condenser.
For most heat pipes, the maximum heat transfer rate dueto the capillary limitation can be expressed as (Chi, 1976).öé rl s l h fg ù é Aw K ù æ 2 é rl ù- ê ú gLt cos Y÷Qc,max = êúçúêøë m l û ë leff û è rc,e ë s l û(4.8.27)where K is the wick permeability (m2), Aw is the wick cross-sectional area (m2), rl is the liquid density(m3), ml is the liquid viscosity (N sec/m2), rc,e is the wick capillary radius in the evaporator (m), g is theacceleration due to gravity (9.8 m/sec2), and Lt is the total length of the pipe (m).
For most practicaloperating conditions, this limitation can be used to determine maximum heat transfer rate in heat pipes.The boiling limitation in heat pipes occurs when the degree of liquid superheat in the evaporator islarge enough to cause the nucleation of vapor bubbles on the surface of the wick or the container. Boilingis usually undesirable in heat pipes because local hot spots can develop in the wick, obstructing the flowof liquid in the evaporator. An expression for the boiling limitation is (Chi, 1976)Qb,max =4 pleff keff Tv s v æ 11 öçh fg rl ln(ri rv ) è rn rc,e ÷ø(4.8.28)where keff is the effective thermal conductivity of the composite wick and working fluid (W/m K), Tv isthe vapor saturation temperature (K), ri is the inner container radius (m), rn is the nucleation radius(equal to 2.00 ´ 10–6 m in the absence of noncondensable gas).© 1999 by CRC Press LLC4-265Heat and Mass TransferEffective Thermal Conductivity and Heat Pipe Temperature DifferenceOne key attribute of the heat pipe is that it can transfer a large amount of heat while maintaining nearlyisothermal conditions.
The temperature difference between the external surfaces of the evaporator andthe condenser can be determined from the following expressionDT = Rt Q(4.8.29)where Rt is the total thermal resistance (K/W) and Q is the heat transfer rate (W).
Figure 4.8.14 showsthe thermal resistance network for a typical heat pipe and the associated thermal resistances. In mostcases, the total thermal resistance can be approximated byRt = R1 + R2 + R3 + R5 + R7 + R8 + R9FIGURE 4.8.14 Thermal resistance network in a heat pipe.© 1999 by CRC Press LLC(4.8.30)4-266Section 4The reader is referred to Peterson (1994) for the specific mathematical relationships used to calculateeach thermal resistance. The effective thermal conductivity of the heat pipe is defined as the heat transferrate divided by the temperature difference between the heat source and heat sink,keff =LtRt At(4.8.31)where At is the overall cross-sectional area of the pipe (m2).
Under normal operating conditions, the totalthermal resistance is relatively small, making the external surface temperature in the evaporator approximately equal to that in the condenser. Thus, the effective thermal conductivity in a heat pipe can bevery large (at least an order of magnitude larger than that of aluminum).Design ExampleDesign a water heat pipe to transport 80 W of waste heat from an electronics package to cooling water.The heat pipe specifications are1.
Axial orientation — complete gravity-assisted operation (condenser above the evaporator; y =180°)2. Maximum heat transfer rate — 80 W3. Nominal operating temperature — 40°C4. Inner pipe diameter — 3 cm5. Pipe length — 25 cm evaporator length, 50 cm adiabatic section, and 25 cm condenser lengthThe simplest type of wick structure to use is the single-layer wire mesh screen wick shown in Table4.8.5.
The geometric and thermophysical properties of the wick have been selected as (this takes someforethought)dw12Ne= 2.0 ´ 10–5 m= 6.0 ´ 10–5 m= rc = 1/2(2.0 ´ 10–5 + 6 ´ 10–5) = 4.0 ´ 10–5 m=1WmKtw = 1.0 ´ 10–3 mkeff = k1 = 0.630K=(1 ´ 10 -3t w2=1212)2= 8.33 ´ 10 -8 m 2The other heat pipe geometric properties arerv = ri – tw = 0.015 – 0.001 = 0.014 m0.25 + 0.25+ 0.5 = 0.75 m2Lt = 0.25 + 0.50 + 0.25 + 1.0 mleff =Aw = p(ri2 - rv2 ) = p[(0.015)2 – (0.014)2] = 9.11 ´ 10–5 m2Av = prv2 = p(0.014)2 = 6.16 ´ 10–4 m2The thermophysical properties of water at 40°C are (see Table 4.8.4):rl = 992.1 kg/m3rv = 0.05 kg/m3© 1999 by CRC Press LLC4-267Heat and Mass Transfersl = 2.402 ´ 106 J/kgml = 6.5 ´ 10–3 kg/m secmv = 1.04 ´ 10–4 kg/m secPv = 7000 PaThe various heat transfer limitations can now be determined to ensure the heat pipe meets the 80 Wheat transfer rate specification. The vapor-pressure limitation is4Qnp,max =(())p(0.014) 2.402 ´ 10 6 (0.05) (7000)12 1.04 ´ 10 -4 (0.75)= 1.08 ´ 10 5 W(4.8.32)The sonic limitation is()(Qs,max = 0.474 6.16 ´ 10 -4 2.402 ´ 10 612)[(0.05)(7000)](4.8.33)= 1.31 ´ 10 4 WThe entrainment limitation is(Qe,max = 6.16 ´ 10-4)(é (0.05) (0.07) ùú2.402 ´ 10 ê-5êë 2 4.0 ´ 10 úû6)(12)(4.8.34)= 9.79 ´ 10 3 WNoting that cos y = –1, the capillary limitation is(é (992.1) (0.07) 2.402 ´ 10 6Qc,max = ê6.5 ´ 10 -3êë) ùú éê (9.11 ´ 10 ) (8.33 ´ 10 ) ùú é-50.75úû êë-82992.1+9.8(1.0)ùú-5ê0.07ûúû ë 4.0 ´ 10(4.8.35)= 4.90 ´ 10 4 WFinally, the boiling limitation isQb,max =(4 p(0.75) (0.63)(313)(0.07) é11ù0.015 ö êë 2.0 ´ 10 -6 4.0 ´ 10 -5 úûæ62.402 ´ 10 (992.1) lnè 0.014 ø)(4.8.36)= 0.376 WAll of the heat transfer limitations, with the exception of the boiling limitation, exceed the specifiedheat transfer rate of 80 W.
The low value of 0.376 W for the boiling limitation strongly suggests thatthe liquid will boil in the evaporator and possibly cause local dry spots to develop. The reason the liquidboils is because the effective thermal conductivity of the wick is equal to the conductivity of the liquid,which is very low in this case.
Because the liquid is saturated at the vapor-liquid interface, a low effectivethermal conductivity requires a large amount of wall superheat which, in turn, causes the liquid to boil.This problem can be circumvented by using a high conductivity wire mesh or sintered metal wick, whichgreatly increases the effective conductivity. It should be noted, however, that because porous wicks have© 1999 by CRC Press LLC4-268TABLE 4.8.5Physical Properties of Wick StructuresWick TypeaThermal ConductivityPorosityMinimumCapillaryRadiuskeff = kee=1rc = 1/(2N)Estimatedfrome = 1 Ð (pNd)/4rc = 1/(2N)PermeabilityK = t w2 12Single-layer wire meshscreens (heat-pipe axis inthe plane of the paper inthis sketch)1/N = d + wN = number ofaperatures per unit lengthMultiple wire mesh screens,bplain or sintered (screendimensions as for singlelayers illustrated above)k eff =[]ke ke + k s - (1 - e ) (ke - k s )ke + k s + (1 - e ) (ke - k s )k=d 2e22122(1 - e )Section 4© 1999 by CRC Press LLCPhysical Properties of Wick Structures (continued)Wick TypeaUnconsolidated packedsphericalparticles (d= averageparticlediameter)PlainSinteredThermal Conductivityk eff =Porosity[]ke 2 ke + k s - 2(1 - e ) (ke - k s )2 ke + k s + (1 - e ) (ke - k s )k eff =[MinimumCapillaryRadiusEstimated from(assuming cubicpacking) e =0.48rc = 0.21d]Permeabilityk=k e 2 k s + k e - 2 e( k s - k e )2 k s + k e + e( k s - k e )d 2e22150(1 - e )k = C1Sinteredmetal Þbers(d = Þberdiameter)k eff = e 2 ke (1 - e ) k sy2 - 1y2 - 1where24e(1 - e ) ke k s+ke + k sHeat and Mass TransferTABLE 4.8.5Usemanufacturersdatadrc =2(1 - e )y = 1+C2 d 2 e 3(1 - e) 2C1 = 6.0 ´ 10 -10 m 2C2 = 3.3 ´ 10 7 1/ m 2aThe axis of the pipe and direction of ßuid ßow are normal to the paper.These wicks are positioned so that the layers follow the contours of the iner surface of the pipe wall.Revised from Peterson, G.P., An Introduction to Heat Pipes Modeling, Testing, and Applications, John Wiley & Sons, New York, 1994.b4-269© 1999 by CRC Press LLC4-270Section 4lower permeabilities, the capillary limitation should be lower as well.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
