3D-model for heat and mass transfer simulation in flat evaporator of copper-water loop heat pipe

Applied Thermal Engineering 33-34 (2012) 124e134

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3D-model for heat and mass transfer simulation in flat evaporator of copper-water loop heat pipe Mariya A. Chernysheva, Yury F. Maydanik* Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, Amundsena St., 106, Yekaterinburg 620016, Russia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 April 2011 Accepted 18 September 2011 Available online 28 September 2011

This paper presents a three-dimension mathematical model of a flat evaporator of a loop heat pipe which takes into account the peculiarities of the evaporator configuration and the specific character of a oneside heat load supply. All the main structural elements of the evaporator, such as its body, wick, vaporremoval grooves, barrier layer and compensation chamber, are included in the model. The intensity of heat-exchange processes during evaporation in the active zone is determined by local drops between the temperature at the wick surface and the vapor temperature. The effects of drying the wick in the evaporation zone are also taken into account. The problem was solved by a numerical method. The results of calculations are presented for a copper evaporator and water as a working fluid in the heat load range from 20 to 1100 W. A comparative analysis of calculated and experimental data has been made.
2011 Elsevier Ltd. All rights reserved.

Keywords: Loop heat pipe Flat evaporator Mathematical model Heat transfer Evaporation

1. Introduction Loop heat pipes (LHPs) are the very efficient two-phase heattransfer devices with capillary pumping of the working fluid. They are actively used in cooling systems of electronic components of ground-based [1,2] and on-board [3e5] equipment for providing a reliable and effective thermal link between the heat source and the heat sink. As shown in Fig. 1, an LHP has an evaporator, a compensation chamber (CC), a condenser, a vapor line and a liquid line. The heat transfer from the heat source to the heat sink is realized in a following way. The heat flow supplied to the evaporator is expended in the liquid evaporation, and the vapor generated moves through the vapor line to the condenser. Here it condenses giving up the thermal energy to the heat sink. The liquid from the condenser returns through the condensate line into the evaporator. Detailed descriptions of the main characteristics and working principles of LHPs are stated in articles [6e10]. The evaporator is a key element of an LHP. It ensures the LHP functioning and largely determines the heat-transfer characteristics of the device. The evaporator contains a fine-pored wick saturated with a liquid. The capillary head created by menisci makes up for the pressure drop in pumping the working fluid trough the loop of a heat-transfer device. Besides, vapor generation is carried out in the active zone of the evaporator. And finally, the

* Corresponding author. Tel.: þ7 343 2678 791; fax: þ7 343 2678 799. E-mail address: [email protected] (Y.F. Maydanik). 1359-4311/$ e see front matter
2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2011.09.025

evaporator is the LHP element which is in thermal contact with the object being cooled. Three types of evaporators have been developed by now: a cylindrical, a disk-shaped, and a flat evaporator, which has a rectangular or a flat-oval cross-section [11]. The first LHPs had a cylindrical form, therefore it is now conventional to call this type classical [12,13]. Cylindrical evaporators for conjugation with flat heat-releasing objects are usually provided with a “cylinder-plane” intermediate element called “a thermal interface”, which increases the thermal resistance of the cooling system and its mass. Two other types of evaporators resulted from searching for alternative designs of evaporators capable of contacting directly a flat thermal contact surface of a heat source. The results of investigating an LHP with disk-shaped evaporators are published in [14,15]. A structural peculiarity of such evaporators is the fact that the volume of the compensation chamber is, as a rule, the determining factor for the evaporator thickness. Since the volume of the CC directly depends on the lengths of the vapor line and the condenser, with an increase in the length of the transportation sections the CC thickness increases accordingly. Therefore for miniature LHPs with a large length of heat transfer the creation of compact disk-shaped evaporators becomes problematic because of the limitations on the decrease of the CC thickness. For rectangular and flat-oval evaporators the problem of thickness is solved much easier as their compensation chamber is usually situated along the evaporator [16e18]. If it is necessary to increase the CC volume, it is done at the expense of changing the length of the CC or its width. Besides, in such evaporators one can

M.A. Chernysheva, Y.F. Maydanik / Applied Thermal Engineering 33-34 (2012) 124e134

Nomenclature

a B Cp H hhv G G* k L, l _ m N n P Q q R R* r T S V u

heat exchange coefficient, W/m2 K width, m fluid specific heat, J/kg K thickness, m latent heat of evaporation, J/kg mass flow rate, kg/s thermal conductance, W/K thermal conductivity, W/m K length, m specific mass flow rate, kg/s m2 iteration number number pressure, Pa heat load, W heat flux, W/m2 thermal resistance, K/W specific thermal resistance, K m2/W radius, m temperature,  C area, m2 volume, m3 velocity along vapor-removal groove, m/s

use the structural pair “copper-water”, which is advantageous enough from the viewpoint of decreasing the thermal resistance. These materials are accessible and fairly cheap. For these reasons they have been long and actively used for conventional heat pipes, whose fabrication practice has been well tested and adjusted [19]. As for copper-water loop heat pipes, their development began comparatively recently [17,20,21]. The main problems here are the high thermal conductivity of copper and the low value of dP/dT of the working fluid at operating temperature of 50e110  C, which are required for the components of electronic equipment being cooled [22]. In total, these factors are a serious obstacle to creating in the wick the temperature drop required for a successful start-up of an LHP and its operation. On the other hand, water has high values of the latent heat of evaporation and the surface tension coefficient, and the heat transfer intensity during the evaporation of water at copper porous surfaces may be quite high. All this can make up for the drawbacks mentioned above, and finally allows regarding copper-water LHPs as promising heat-transfer devices with high heat-transport characteristics in the range of operating

Fig. 1. General view of LHP with flat evaporator.

125

Greek symbols surface, m2 dynamic viscosity, Pa s ε porosity, m3/m3 r density, kg/m3 s surface tension, N/m

U m

Subscripts and superscripts amb ambient b body (wall) of evaporator cc compensation chamber (or reservoir) coll vapor collector ev evaporation evp evaporator ex external in internal or at the input l liquid n number nucl nucleation out at the output s saturation v vapor vg vapor-removal grooves w wick x, y, z coordinate axes

temperatures of the working fluid from 50 to 110  C. The validity of these statements is demonstrated by the results of testing an allcopper LHP with a flat evaporator, whose scheme is given in Fig. 1, and with the water as a working fluid [17]. According to cited data, a maximum heat flux of 100.1 W/cm2 was achieved at a vapor temperature of 104.4  C and the thermal resistance of the evaporator had a minimum value of 0.012 K/W. As for analytical treatments of heat-transfer processes in the evaporators, all the preceding publications on this subject may be arbitrarily divided into two subgroups. The first includes papers focused on the consideration of the evaporator as a whole [23e25], the second, papers examining the peculiarities of heat transfer only in the evaporation zone [26e33]. The first approach is usually used in constructing a thermal model of an LHP required, for instance, for calculating its temperature characteristics. In this case a thermal model of the evaporator, as one of the main elements, is part of the overall LHP model, and its calculation model is included in the integrated procedure of calculating the thermal state of the device. In creating such models lately an active use has been made of the method of electrical analogies simulating heat and mass transfer processes. In accordance with this method all the main elements of the evaporator, such as evaporator body, wick, compensation chamber, are depicted as nodal elements. The temperature is one of their main parameters. It is also the sought-for quantity of the problem. The thermal links between the elements are determined by the thermal resistances: R (K/W) or thermal conductances: G* ¼ 1/R (W/K). Their values are input ones. Usually they are taken from experimental data of the device which the model is developed, or are determined by fitting calculated temperature values to experimental data. For this reason they do not possess the universality property and are invalid for other LHPs. Owing to such a rigid referencing to fixed input values, the significance of such an approach is considerably reduced, with its help, for instance, it is impossible to predict the thermal state of a device being designed. A disadvantage of this method is also the fact that the thermal state of the evaporator, as a complex object, is described not by

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a continual temperature distribution, but only by its several discrete values, which characterize on the whole the thermal state of nodal elements, as some isolated abstract objects. The second type of problem investigates heat and mass transfer processes only in the LHP evaporation zone. Usually such problems examine the characteristic domain of the active zone which includes a typical segment situated around the vapor-removal groove and including the evaporator body and the wick. The structural parameters of the wick, and also sometimes the nonuniform pore size distribution are taken into consideration. In this case account is taken of the peculiarities of heat transfer during liquid evaporation from fine-pored capillary structures and the effects caused by the pore drying and leading to a shift of the evaporation front deep into the wick. The result of the solution here is the velocity field and the temperature distribution in the domain under consideration, and also the position of the vaporeliquid interface in the wick between the wetted and the dried zone. Problems of this kind are often solved in an attempt to optimize the geometric parameters of the evaporation zone, in particular, the dimensions of vapor-removal grooves and their number, to increase the heat transfer intensity in the evaporation zone or the maximum heat load. As for the processes observed in the other parts of the evaporator, unfortunately, in this case they prove to be beyond the scope of consideration. In comparing the dimensions of a typical segment under investigation, taking into account the scale of the processes inside the wick pore being considered, with the linear dimensions of the whole evaporator, this type of the problems may be characterized as Microscale. The paper presents a complete model of a flat evaporator which describes heat and mass transfer processes in all its main parts, including the wick, active zone, compensation chamber, barrier layer end evaporator body. It also takes into consideration the heat exchange with the outside ambient and the Microscale processes in the evaporation zone connected with the pore drying and the shift of the evaporating front.

active zone and the compensation chamber and spatially separates the evaporation zone from the compensation chamber. This leads to the creation of a temperature and pressure drop between them, which is required for LHP start-up and circulation of the working fluid in the course of operation of a heat-transfer device. Besides it is a hydraulic seal preventing the penetration of the vapor from the active zone into the compensation chamber. In this case the barrier layer is permeable for the liquid, which, coming first from the liquid line into the CC, flows from there to the evaporation zone. Fig. 2B presents a calculated model of the evaporator. Instead of a flat-oval cross-section of the evaporator, for simplicity, use is made of its rectangular analog. In addition the circular cross-section of vapor-removal grooves has been replaced by a square one. All the other peculiarities of the configuration and assembling of the flat evaporator have been accounted for in the model. The evaporator is examined in isolation from the other component parts of the LHP, but a thermal link with them is realized by means of two external, as related to the evaporator, temperature parameters, which are taken as input data. They are the vapor temperature at the evaporator outlet Tvout evp and the liquid temperature at the entrance into the compensation chamber Tlout ll. As is mentioned in [9], exactly these parameters determine considerably the temperature level of the evaporator. The values of the temperatures Tvout evp and Tlout ll are determined by the conditions of heat removal from the condenser to the heat sink, the intensity of heat-transfer processes during vapor condensation in the condenser, and the heat exchange of the transport sections with the outside ambient and are practically independent of the evaporator temperature. The external parameter of the problem which affects the thermal state of the evaporator is the temperature of the outside ambient Tamb. The liquid flow that arrives at the CC from the liquid line ensures a sufficiently intense agitation of the working fluid making its temperature Tcc practically uniform throughout the whole inner space of the CC. In its turn, the value of Tcc depends on the temperature of the condensate at the entrance into the CC, its flow rate, and also the heat flow Qcc penetrating from the active zone into the CC:

2. A physical model of the flat evaporator The flat evaporator of an LHP whose scheme is presented in Fig. 2A consists of a body, a wick, vapor-removal grooves, and a vapor collector. The evaporator is integrated with a compensation chamber. The evaporator’s section including the vapor-removal grooves is usually called the active zone on account of the fact that most of the heat load delivered to the evaporator is taken up here during the liquid evaporation. The resultant vapor is removed from the evaporation zone through the vapor-removal grooves first into a vapor-removal collector, and then to the entrance into a vapor line. The barrier layer of the wick is situated between the

l Tcc ¼ Tout

ll

þ

Qcc Cp ,G

(1)

The heat flow Qcc has two components, which derive from the heat exchange with the inner surface of the evaporator body Ub and butt-end surface of the barrier layer of the wick Uw:

Qcc ¼ Qcc

w

þ Qcc

b

(2)

The temperature field of these surfaces is nonuniform, therefore the heat flux at these surfaces will be determined by a local temperature drop:

Fig. 2. Scheme of flat evaporator (A) and its model (B).

M.A. Chernysheva, Y.F. Maydanik / Applied Thermal Engineering 33-34 (2012) 124e134

qz ¼ ain

cc ,ðT z Tcc Þ

(3)

where Tz is the temperature at the point z at the inner surface that forms the CC (z ˛ Ub or z˛ Uw), ain_cc is the internal heat-exchange coefficient. With allowance for this, the total heat flow is determined as:

Z

Z

qz dU þ

Qcc ¼ Ub

qz dU

(4)

Uw

The local heat exchange with the outside ambient at the outer surface of the evaporator body and the total heat flow Qamb are described in a similar way:

q2 ¼ aex

evp ,ðT z Tamb Þ

Z Qamb ¼

(5)

qz dU

vP vG , dz vG vz

(11)

The model also takes into account the possibility of the liquid evaporation from the butt-end surface of the wick facing the vapor collector (Ucoll). As in the case of evaporation into vapor-removal grooves, the criterion of intensity of the evaporation process Iev at any point of the butt-end surface of the wick is the value of the temperature drop ΔTev between the local value of temperature Tz at its v : surface (z ˛ Ucoll) and the vapor temperature in the vapor collector Tcoll v Iev wTz  Tcoll

(12)

The vapor temperature within the whole inner volume of the collector is believed to be actually the same and differs little from the vapor temperature at the exit from the evaporator:

(6)

(13)

evp

Then the evaporating temperature head ΔTev will be determined as:

where Uamb is the outer surface of the evaporator participating in the heat exchange with the outside ambient, z is the point at this surface (z ˛ Uamb), aex_evp is the external heat-exchange coefficient. Owing to the nonuniformity of the temperature field of the evaporator, the intensity of heat exchange during evaporation in the active zone must be different, depending first of all on the difference ΔTev between the local value of the temperature at the evaporating surface of the vapor-removal grooves Tz (z ˛ Uvg) and the temperature v , which is considered to be saturated: of the vapor in the groove Tvg v DTev ¼ Tz  Tvg

(7)

With allowance for this, the evaporation intensity will be determined as: v Iev wTz  Tvg

(8)

Since the longitudinal dimensions of the grooves are many times larger than its transversal dimensions, changes in the vapor temperature along the length of the groove must be much more considerable than temperature changes at its cross-section: v ðx; y; zÞ ¼ f ðzÞ. Besides the dependence on the longitudinal Tvg coordinate, the vapor temperature in the groove must also depend on the position of the groove in the active zone. Its position is determined by its ordinal number. The vapor temperature has the lowest value equal to Tcoll at the vapor exit from the groove into the vapor collector (l ¼ 0) and increases upstream. At the opposite dead end of the vapor-removal groove (l ¼ Lvg) the vapor has the highest temperature. Variations in the vapor temperature along the length of the groove l with an ordinal number n are described as follows:

Tnv ðlÞ ¼ Tcoll þ

z¼l Z z¼0

v v Tcoll yTout

Uamb

vT  ,DPnv ðlÞ  vP T¼Tcoll

(9)

The pressure drop ΔPv depends on the vapor flow rate, which also varies along the length of the groove:



DPnv ðlÞ ¼

127

DPnv ðlÞ ¼ f Gv ðlÞ



(10)

The tendency of variation of the vapor flow rate is such that it is equal to zero Gv ¼ 0 at the deadlock butt end of the groove (at l ¼ Lvg) and, increasing downstream, reaches its highest value at the exit from the groove (l ¼ 0). Accordingly, the current value of a pressure drop in the vapor groove along the longitudinal coordinate is determined by the expression:

v DTev ¼ Tz  Tout

evp

(14)

The difference between the temperature at the wick surface and the corresponding vapor temperature also determines the value of the local liquid superheating in the pores and is used as a condition describing the process of drying of the wick owing to the boiling-up of a superheated liquid. Thus, according to nucleation theory, which is presented in Ref. [34], the formation of a vapor bubble of radius rnucl is possible only if the liquid is sufficiently superheated with respect to its equilibrium state, and the lower level of nucleation superheating ΔTnucl is determined by the formula:

DTnucl ¼

2,s,Ts rv ,rnucl ,hhv

(15)

Using in Eq. (15) the wick pore Rp as a characteristic of a vapor bubble of rnucl which may arise inside a porous structure and occupy the whole internal space of the pore, one can estimate the value of ΔTnucl and compare it with the values of superheatings ΔTev determined by Eq. (7) and (14). With allowance for the aforesaid, the condition of drying of the wick may be formulated as follows:

DTnucl  DTev

(16)

The wick must remain saturated with a liquid in the case if the inverse relation is satisfied:

DTnucl >DTev

(17)

This approach is used in the calculation model for revealing dried sections in the wick. The appearance of such “dry spots” at the surface of the wick in the active zone and at the butt-end surface facing the vapor collector means that there may be no liquid evaporation in these places, and these fragments of the wick surface should be excluded from the process of evaporating heat transfer. Thus, the model developed makes it possible to follow the process of transformation of the wick evaporating surface when “passive” nonevaporating sections appear.

3. A mathematical formulation of the problem The heat transfer equation for the evaporator body with allowance for the three-dimensionality of the mathematical model will look like:

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M.A. Chernysheva, Y.F. Maydanik / Applied Thermal Engineering 33-34 (2012) 124e134

v2 Tb v2 Tb v2 Tb þ þ 2 ¼ 0 vx2 vy2 vz

(18)

For the evaporator being investigated here the liquid velocity at the butt-end surface of the wick on the CC side at a uniform absorption according to evaluations may vary from 3  104 to 2  102 m/s under changes of heat load from 20 to 1200 W. It is well known that at such small filtration rates the temperatures of the wick framework and a liquid in the wick get equalized to the extent that their temperatures may be considered as equal. In such a case, when describing thermal processes in the wick one can use the approach presented in [30,32], in which for a stationary case of heat transfer in a wick the authors used the heat transfer equation:

v2 Tw v2 Tw v2 Tw þ þ ¼ 0 vx2 vy2 vz2

(19)

And the convective component of heat transfer during the liquid filtration through the wick should be allowed for in the energy balance equation. The boundary conditions can be stated as follows. At the heat- supply surface of the evaporator (Uq):

vT kb , b ¼ qload vy

Qload Sq

(29)

kb

Tzþo  Tzo vT  vT  ¼ ¼ kwick    Rc vn zþo vn zo

(30)

The value of the specific contact thermal resistance, according to [35], may vary from 105 to 104 m2 K/W. In calculations use was made of its averaged value: R*c ¼ 5.5$105 m2 K/W. The wick thermal conductivity and effective thermal conductivity are determined by the formula of Odelevski:

kw ¼ kcomp ,ð1  εÞ,ð1 þ εÞ2:1

(31)

keff ¼ kw þ ε,kl

(32)

where kcomp is the thermal conductivity of compact material. The energy equation in the evaporator will look like:

(21)

The first two terms in the right-hand side of the equation are determined according to Eq. (4) and (6), and the third term is the convective component of heat transfer in the wick, which allows for the heating of the liquid moving from the CC to the evaporation surface. Taking into account the nonisothermality of this surface, which is responsible for the nonuniformity of the evaporation process, we have:

(22)

and Lq and Bq are the length and width of the heating area correspondingly. At the evaporating surface of the wick, which includes the surface of the vapor-removal grooves and the butt-end surface of the wick facing the vapor collector (Uvg and Ucoll):

keff ,

Tzþo  Tzo vT  vT  ¼ ¼ keff    vn zþo vn zo Rc

Qload ¼ Qcc þ Qamb þ Qcp þ Qev

where Sq is the heating area:

Sq ¼ Lq ,Bq

kb

(20)

The supplied heat flux is uniform:

qload ¼

dry) and the specific contact thermal resistance R*c we have the following conditions:

  vT   ¼ aev , Tz  T v vn z

Z Qcp ¼ Cp

  _ z , Tz  Tcc dU m

(33)

(34)

Uev

where Tz is the current value of the liquid temperature at the point z at the evaporating surface (z ˛ Uvg or z ˛ Ucoll), the specific mass flow rate of the working fluid at the point z at the evaporating surface is:

(23)

For dried sections, which arise when the condition (16) if fulfilled, instead of (23) use is made of

vT  kw ,  ¼ 0 vn z

(24)

At the boundary between the wet area and the dry area of the wick:

  vT vT   keff , wick  ¼ kw , wick  vn wet vn dry

(25)

_z ¼ m

qz hhv

(35)

The heat flux qz is determined according to the condition (3). The total heat flow expended for vapor generation is determined by the expression

Z Qev ¼ Ucoll

qev dU þ

Z

qev dU

(36)

Uvg

At the outer surface of the evaporator Uamb:

  vT  kb ,  ¼ aamb , Tz  Tamb vn z

4. Solution of the model

(26)

At inner surfaces limiting the compensation chamber ( Ub and

Uw):

vT  kb ,  ¼ ain vn z keff ,



cc ,

vT   ¼ ain vn z

cc ,

Tz  Tcc



  Tz  Tcc

(27)

(28)

At the boundary between the evaporator body and the wick with allowance for the saturation of the porous structure (wet or

The mathematical model was solved numerically. To construct a finite-difference analogy of the governing partial differential equations and the boundary conditions, use was made of a control volume method which ensures the conservation of a finitedifference scheme [36]. An explicit method was chosen for obtaining a difference solution. Use was made of an irregular computational grid adapted to the geometric peculiarities of the evaporator and containing 6300 nodes: I (x)  J (y)  K (z) ¼ 28  9  25. The solution algorithm is presented in the Fig. 3. Convergence was assumed to have been reached when the maximum change in temperature between iterations was less than 103:

M.A. Chernysheva, Y.F. Maydanik / Applied Thermal Engineering 33-34 (2012) 124e134

   N N1  maxTi;j;k  Ti;j;k   103

129

(37)

5. Heat-transfer coefficient under evaporation According to experimental data published, the heat-transfer coefficient under evaporation at porous surfaces depends on many parameters, such as the evaporation mode, the thermal

Fig. 3. The overall flowchart of the numerical solution.

Table 1 The main parameter of the flat evaporator. Evaporator Total length, mm Width, mm Thickness, mm Body thickness, mm CC length, mm Active zone length, mm Vapor removal grooves Number Length, mm Diameter, mm Heating zone Length, mm Width, mm Wick Wick length, mm Porosity Breakdown radius, mm

80 42 7 0.5 40 32 12 32 1.8 30 30 40 0.66 20

Fig. 4. Temperature field at Q ¼ 400 W. A e top surface of the body, B e at level of ½ groove depth, C e at level of ½ evaporator thickness, D e evaporator view from above.

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M.A. Chernysheva, Y.F. Maydanik / Applied Thermal Engineering 33-34 (2012) 124e134

Fig. 7. Comparison of the model prediction and the experimental data on evaporator thermal resistance.

Fig. 5. Heat load dependence of the LHP temperature (experimental data). - e temperature at evaporator body. : e vapor temperature at evaporator outlet.  e temperature of compensation chamber.  e liquid temperature at inlet into compensation chamber

properties of the working fluid and the material of the capillary structure, the structural characteristics of the wick, the geometric dimensions and peculiarities of the evaporation zone (an inverted or direct meniscus) etc. The range of heat-transfer coefficients on porous structures saturated with water is sufficiently wide, namely, from 7  103 to 5  104 W/m2K [19]. At the same time, in identifying the value of aev, for solving problems of this kind one usually chooses a simple way and assumed that it is constant and equal to a certain value from the range mentioned above. In the model suggested the value of aev is determined in the process of iteration calculations in accordance with the scheme in Fig. 3. In the first cycle (N ¼ 1) the coefficient is taken to be equal to zero. In each successive iteration N þ 1 use is made of the value of the coefficient calculated on the preceding step N by the formula:

aNev

¼ P  n

Qev N1 Ln ,Sn ,DTev n



(38)

where n is the identification number of the nodal points situated at the evaporating surface of the wick, ΔTev is the temperature drop

Fig. 6. Comparison of the model prediction and the experimental results. exp e experimental results, culc e calculation data.

Fig. 8. Temperature profile on the evaporator body along the symmetry line at e 180,  e 52 (W). different heat loads. : e 900, - e 723,  e 545,  e 370,

between the temperature at the wick surface and the vapor temperature, S is the area of the evaporating surface, L is the factor that allows for the saturation of an elementary cell. If the condition (16) is fulfilled, the wick is dry, evaporation is absent, therefore L ¼ 0. If the condition (17) is fulfilled, L ¼ 1. A stepwise refinement of the value of the coefficient is made in the program body till the fulfillment of one of the convergence criteria established as

Fig. 9. Evaporation intensity in active zone (Q ¼ 400 W).

M.A. Chernysheva, Y.F. Maydanik / Applied Thermal Engineering 33-34 (2012) 124e134

131

   N  2 aev  aN1 ev   10

(39)

    DQ N  DQ N1   103

(40)

N N N N DQ N ¼ Qload  Qcp  Qamb  Qcc  Qcev

where

(41)

6. Thermophysical properties of the working fluid

Fig. 10. Variation of the vapor velocity along vapor-removal grooves (Q ¼ 400 W).

Tabulated data of thermophysical properties of water on the saturation line, such as the latent heat of evaporation, surface tension, thermal conductivity, pressure, temperature, density and viscosity, for liquid and vapor were taken from [19,37]. The calculation program included a module determining the functional temperature dependence of thermophysical properties of the working fluid for the temperature range from 10 to 200  C. Cubic spline interpolation was used as a means of interpolation of temperature dependences J ¼ f (T). The program of temperature calculation at the nodes of the grid was organized in such a way that on every computational step use was made of the current values of thermophysical parameters, which were calculated from the corresponding functional dependences at the temperature of the nodal point on the preceding iteration step. A generalized formation of the scheme described may be presented as follows:



N1 JNi;j;k ¼ f Ti;j;k



(42)

where J is a thermophysical parameter. 7. Modeling results, their analysis and discussion

Fig. 11. Distribution of heat flows in the evaporator under changes of heat load. Qcp, - e Qcc,  e Qev, A e Qamb

:e

Numerical results were obtained for a flat evaporator with the parameters presented in Table 1. The result of the solution is a three-dimensional field of evaporator temperatures. As an example, Fig. 4 presents a layerwise temperature distribution in three positions: at the surface of the evaporator body to which the heat load is supplied, at the cut plane passing through the middle of the depth of the vapor-removal grooves and at that passing through the middle of the evaporator thickness. The same figure, schematically but with the observance of proportions in dimensions, shows a top view of the evaporator and the heating zone layout. Ibidem one can see the location of two thermocouples at the evaporator body, which were used in experiments for control over the temperature in the heating zone and the CC. Fig. 4A demonstrates the contour of a heating spot. In Fig. 4B one can see distinctly enough the vapor-removal grooves and the bridges between them.

Fig. 12. Effect of the wick saturation on the evaporation process.

132

M.A. Chernysheva, Y.F. Maydanik / Applied Thermal Engineering 33-34 (2012) 124e134

The CC temperature, according to the mathematical model, is uniform, which is reflected in the graph. Fig. 4C shows that the temperature field deep in the evaporator becomes more uniform. And nevertheless, the locations of the active zone, the barrier layer and the CC as well as their edges and the counter of the body are distinctly seen here. To evaluate the quality of the model, numerical predictions have been compared with the experimental data for a copperwater LHP published in Ref. [17]. Fig. 5 shows experimental dependences of operating temperatures under changes of heat load. In experiments the LHP was positioned horizontally, the cooling temperature was equal to 20  C, and the ambient temperature was 20  C. In calculations the ambient temperature Tamb was the same, and ambient heat-exchange coefficient aamb was taken equal to 5 W/m2K. A typical value for the convection heat transfer coefficient at the internal surface of the compensation chamber ain_cc was 100 W/m2K. The values of the input parameters of the problem Tvout evp and Tlout ll were taken in accordance with the data obtained in experiment (see Fig. 5). Calculated heat-load dependences of the temperature are presented in the Fig. 6. The indicated values of Tb and Tcc correspond to the location of thermocouples. Experimental points are also present in the graph. It can be seen that there is a sufficiently good agreement between experimental and calculated results especially in the region of medium and high heat loads. The same conclusion is confirmed by the results for the thermal resistance of an evaporator Revp shown in Fig. 7. The values of Revp were calculated by the formula (Fig. 6):

Revp ¼

v Tb  Tout

Q

evp

the evaporation zone are absent. Also no signs of crisis phenomena in the wick are observed on experimental curves. It should be noted that the reason for limitation on the heat load of the experimental LHP was the vapor temperature, which at Q ¼ 900 W reached 105  C. To avoid the deformation of the evaporator body, which has thin-walled flat surfaces, at a high internal vapor pressure the heat load was not raised above 900 W. At the same time, in analyzing the results of calculations it was developed that at this heat load under the heating spot center one could observe local superheatings close to the limiting values at which dry sections can be formed. A simulation experiment of thermal processes in the evaporator was

(43)

As for low heat loads, evidently there are processes affecting the evaporator temperature which require a closer examination in a mathematical model. This is a subject for further investigations. Fig. 8 shows variations in the temperature profile at the outer surface of the evaporator body along the symmetry line (see the scheme in Fig. 4D) at different heat loads. It can be seen that with increasing Q the temperature under the heater changes more profoundly than that in compensation chamber. The evaporation intensity in the active zone is not uniform. As an example, Fig. 9 presents a diagram of evaporation intensity at Q equal to 400 W. It should be mentioned that at heat loads below the critical ones such a kind is topical. It can be seen that the peripheral regions of the active zone located outside the heating spot function less efficiently. The reason is that they are colder than the central part. The variable evaporation intensity tells on the workload of the vapor-removal grooves. The vapor flow rate in them varies, as is shown in Fig. 10. The graph demonstrates changes in the vapor flow along the grooves depending on their positions. The vapor-removal grooves at the center of the active zone are subjected to the highest load. According to data, the vapor velocity in them may exceed by an order the vapor velocity in the extreme grooves. The relation between the heat flows of the main heat sinks in the evaporator is presented in Fig. 11. It can be seen that under changes in the heat load supplied the relation between Qev, Qamb, Qcc and Qcp changes, although this change is not considerable. Most of the heat supplied is expended in the liquid evaporation. The fraction of Qev is no less than 90% of the value of Q. The heat flows into the compensation chamber are small. They do not exceed 2% of the heat load. The liquid heating requires from 2 to 12%, depending on the value of Q. No more than 2% is lost during the heat exchange with the outside ambient. Calculations have shown that in the heat load range from 20 to 900 W the effects connected with the origination of dry regions in

Fig. 13. Variation of the vapor velocity along vapor-removal grooves under changes in the wick saturation.

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Fig. 14. Temperature profile along the evaporator body under changes of heat load and wick saturation. 1 e 900, 2 e 1000, 3 e 1100 (W). d x ¼ 0.5$Bevp; —— x ¼ Bevp.

conducted at Q exceeding 900 W and made it possible to observe crisis phenomena in the wick connected with the drying of the evaporation zone. Such investigations are interesting because they simulate processes observed during the LHP operation at heat loads close to limiting ones. The results presented in Fig. 12 demonstrate transformation of the evaporating front with the corresponding changes in the evaporation intensity. The first diagram has a form typical of evaporation with a wetted wick. In the second, one can see a flexure in the central part, which is a result of a shift of the evaporating front deep into the wick. The results of calculations show that in this situation the dried regions are the upper parts of the bridges between the vapor-removal grooves situated close to the evaporator body, and the evaporating front is located at the bottom of the grooves. One can also observe activation of the peripheral sections of the active zone, which had been passive before that. The results of calculations presented in Fig. 13 demonstrate the involvement of the lateral vapor-removal grooves in the process of active vapor removal. With a further increase in the heat load up to 1100 W crisis phenomena are aggravated. One can observe a complete drying of the porous structure close to the vapor eremoval grooves situated under the heating spot. For this reason the central part of the active zone does not participate in vaporization processes, and the evaporating front moves to the peripheral sections, which are still wetted by a liquid. In such a situation the whole load connected with the vaporremoval lies on the lateral grooves. An external manifestation of crisis phenomena in the wick is an abrupt temperature increase at the evaporator body. Fig. 14 shows the dynamics of temperature variation on the evaporator wall at heat loads close to critical ones. Two curves are given for every heat load. The first shows the temperature variation along the length of the evaporator along the symmetry line, and the second at a distance of 20 mm from this line, i.e. at the edge of the evaporator. It can be seen that a heatexchange crisis during evaporation is caused by an abrupt rise of temperature on the evaporator body, and especially in the central part of the heating spot. 8. Conclusion A three-dimensional mathematical model has been developed for numerical simulation of a thermal state of the flat evaporator of a copper-water LHP. It takes into account the specific character of a one-sided heat load supply, the peculiarities of the configuration and geometric parameters of the evaporator and all its structural elements. Calculations have been made of evaporator temperature fields in the heat load range from 20 to 1100 W. The paper presents

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a comparative analysis of experimental data and results of calculation, which has shown that the model adequately describes thermal processes in the evaporator and may be used for analyzing the performance characteristics of LHPs. The results obtained have demonstrated that with increasing heat load the rate of temperature rise in the heating zone exceeds the rate of its rise in the compensation chamber. The active zone exhibits a nonuniform evaporation rate. In normal operating conditions, owing to the insufficient heating of its peripheral sections, there are lowevaporation zones. For this reason the main load in vaporremoval falls on the vapor-removal grooves located in the centre of the active zone. A different picture is observed at high heat loads close to maximum ones. In this case high superheats are responsible for the drying of the wick in the central part of the active zone and activation of evaporation processes in the peripheral sections. Under changes in the supplied heat load Q the relation between the heat flows Qev, Qamb Qcc and Qcp changes only slightly. Most of the heat supplied goes into the liquid evaporation. The fraction of Qev is no less than 90% of the value of Q. The heat flows into the compensation chamber Qcc are inconsiderable, viz. less than 2% of the heat load supplied. Acknowledgements This work was supported by the Russian Foundation for Basic Research, Grant No. 11-08-00369-a. References [1] L. Vasiliev, D. Lossouarn, C. Romestant, A. Alexandre, Y. Bertin, Y. Piatsiushyk, V. Romanenkov, Loop heat pipes for cooling of high-power electronic components, International Journal of Heat and Mass Transfer 52 (2009) 301e308. [2] V. Pastukhov, Yu. Maydanik, Low-noise cooling system for PC on the base of loop heat pipe, Applied Thermal Engineering 25 (2007) 894e901. [3] C. Figus, A. Larue de Tournemine, N. Queruel, W. Supper, T. Tjiptahardja, Long term life test & in orbit test of miniaturized loop heat pipe, in: Proceedings of the 14th Int. Heat Pipe Conference, Florianopolis, Brazil (2007), pp. 196e201. [4] M.L. Parker, B.L. Drolen, P.S. Ayyaswamy, Loop heat pipe for spacecraft thermal control, Part 1: vacuum chamber tests, Journal of Thermophysics and Heat Transfer 18 (4) (2004) 417e429. [5] M.L. Parker, B.L. Drolen, P.S. Ayyaswamy, Loop heat pipe for spacecraft thermal control, Part 2: ambient Conditions Tests, Journal of Thermophysics and Heat Transfer 19 (2) (2005) 129e136. [6] J.T. Ku, Operating characteristics of loop heat pipe, SAE (1999) paper No. 199901-2007. [7] Y.F. Maydanik, Loop heat pipes, Applied Thermal Engineering 25 (2005) 635e657. [8] Yu.F. Maydanik, Yu.G. Fershtater, Theoretical Basis and Classification of Loop Heat Pipes and Capillary Pumped Loops, Preprint of 10th Int. Heat Pipe Conference, Stuttgart, Germany, 1997, Keynote lecture X-7. [9] M.A. Chernysheva, S.V. Vershinin, Yu.F Maydanik, , Operating temperature and distribution of a working fluid in LHP, International Journal of Heat and Mass Transfer 50 (13e14) (2007) 2704e2713. [10] A.A. Adoni, A. Ambirajan, V.S. Jasvanth, D. Kumar, P. Dutta, K. Srinivasan, Thermohydraulic modeling of capillary pumped loop and loop heat pipe, Journal of Thermophysics and Heat Transfer 21 (2) (2007) 410e421. [11] Yu. Maydanik, Miniature loop heat pipes, Proceedings of 13th Int. Heat Pipe Conference, Shanghai, China, 2004, pp. 23e35. [12] Yu.E. Dolgirev, Yu.F. Gerasimov, Yu.F. Maydanik, V.M. Kiseev, Calculation of heat pipe with separate channels for vapor and liquid, Journal of Engineering Physics and Thermophysics 34 (6) (1978) 988e993 (in Russian). [13] V.M. Kiseev, V.A. Nouroutdinov, N.P. Pogorelov, Analysis of Maximal Heat Transfer Capacity of capillary Loops, Proceedings of 9th Int. Heat Pipe Conference, Albuquerque, NM, 1995, pp. 1007e1014. [14] R. Singh, A. Akbarzadeh, C. Dixon, M. Mochizuki, Operational characteristics of a miniature loop heat pipe with flat evaporator, International Journal of Thermal Sciences 47 (2008) 1504e1515. [15] A.A.M. Delil, V. Baturkin, Yu. Friedrichson, Yu. Khmelev, S. Zhuk, Experimental Results of Heat Transfer Phenomena in a Miniature Loop Heat Pipe with a Flat Evaporator, Proceedings of the 12th Int. Heat Pipe Conference, Moscow, Russia, 2002, pp. 126e133. [16] J.H. Boo, W.B Chung, E.G. Jung, A study on the alternative capillary structure and vapor passage for a loop heat pipe with a flat evaporator, Proceedings of the 8h Int. Heat Pipe Symposium, Kumanato, Japan, 2006. pp. 21e30.

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M.A. Chernysheva, Y.F. Maydanik / Applied Thermal Engineering 33-34 (2012) 124e134

[17] Yu. Maydanik, S. Vershinin, Development and investigation of copper-water LHP with high operating characteristics, Heat Pipe Science and Technology 1 (2) (2010) 151e162. [18] W. Joung, T. Yu, J. Lee, Experimental studies on the operating characteristics of a flat bifacial evaporator, International Journal of Heat and Mass Transfer 52 (2010) 276e285. [19] D. Reay, P. Kew, Heat Pipes, fifth ed. Linacre House, Jordan Hill, Oxford, 2006. [20] S. Becker, S. Vershinin, V. Sortre, E. Laurien, J. Bonjour, Yu.F. Maydanik, Steady state operation of a copper-water LHP with a flat-oval evaporator, Applied Thermal Engineering 31 (2011) 686e695. [21] J. Li, D. Wang, G.P. Peterson, Experimental studies on a ligh performance compact loop heat pipe with a square flat evaporator, Applied Thermal Engineering 30 (2010) 741e752. [22] Y. Maydanik, S. Vershinin, M. Chernysheva, S. Yushakova, Investigation of a compact copper-water loop heap pipe with a flat evaporator, Applied Thermal Engineering (2011), doi:10.1016/j.applthermaleng.2011.07.008, in press. [23] T. Kaya, R. Perez, C. Gregori, A. Torres, Numerical simulation of transient operation of loop heat pipes, Applied Thermal Engineering 28 (8e9) (2008) 967e974. [24] L. Bai, G. Lin, H. Zhang, D. Wen, Mathematical modeling of steady-state operation of a loop heat pipe, Applied Thermal Engineering 29 (2009) 2643e2654. [25] V.V. Vlassov, R.R. Riehl, Mathematical model of loop heat pipe with cylindrical evaporator and integrated reservoir, Applied Thermal Engineering 28 (2008) 942e954. [26] A.S. Demidov, E.S. Yatsenko, Investigation of heat and mass transfer in the evaporation zone of a heat pipe operating by the ‘inverted meniscus’ principle, International Journal of Heat and Mass Transfer 37 (14) (1994) 2155e2163.

[27] C. Ren, Q. Wu, M. Hu, Heat transfer with flow and evaporation in loop heat pipe’s wick at low or moderate heat fluxes, International Journal of Heat and Mass Transfer 50 (11e12) (2007) 2296e2308. [28] D. Khrustalev, A. Faghri, Heat transfer in the inverted meniscus type evaporator at high heat fluxes, Int. Journal of Heat Mass Transfer 38 (16) (1995) 3091e3101. [29] T. Kaya, J. Goldak, Numerical analysis of heat and mass transfer in the capillary structure of a loop heat pipe, International Journal of Heat and Mass Transfer 49 (2006) 3211e3220. [30] E.I. Altman, M.Ia. Mukminova, H.F. Smirnov, The Loop Heat Pipe Evaporators Theoretical analysis, Proceedings of the 12th Int. Heat Pipe Conference, Moscow, Russia, 2002, pp. 159e164. [31] T. Coquard, M. Prat, A.L de Tournemine A. Larue de Tournemine, C. Figus, Porenetwork models as a tool for the analysis of heat and mass transfer with phase change in the capillary structure of loop heat pipe, Proceedings of the 14h Int. Heat Pipe Conference, Florianopolis, Brazil, 2007, pp. 353e358. [32] C. Figus, Y. Le Bbray, S. Bories, M. Prat, Heat and mass transfer with phase change in a porous structure partially heated: continuum model and pore network simulations, International Journal of Heat and Mass Transfer 42 (1999) 2557e2569. [33] T.S. Zhao, Q. Liao, On capillary-driven flow and phase-change heat transfer in a porous structure heated by a finned surface: measurements and modeling, International Journal of Heat and Mass Transfer 43 (2000) 1141e1155. [34] V.P. Skripov, Metastable Liquids. Halsted Press, New York, 1974. [35] S.V. Vershinin, Yu.G. Fershtater, Yu.F. Maydanik, On the effect of contact thermal resistance on the heat exchange during evaporation from fine-pored capillary structures, High Temperature 30 (1992) 811e817. [36] D.A. Anderson, J.C. Tannehill, R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Vol.1e2, Hemisphere Publishing Corporation, New York, 1990. [37] webbook.nist.gov, 10.08.2011.