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ANALYSIS OF SUPERCRITICAL START-UP LIMITATIONS FOR CRYOGENIC HEAT PIPES WITH PARASITIC HEAT LOADS

Paulo Couto*

Federal University of Santa Catarina Department of Mechanical Engineering, P. O. Box: 476

Florianopolis, SC, 88040-900 - Brazil

Jay M. Ochterbeck† Clemson University

Department of Mechanical Engineering Clemson, SC, 29631 - USA

Marcia B. H. Mantelli‡

Federal University of Santa Catarina Department of Mathematics, P. O. Box: 476

Florianopolis, SC, 88040-900 - Brazil

Abstract A mathematical model to predict the transient tem-

perature profile of a cryogenic heat pipes during start-up is presented. The improved model accounts for a known time-variable temperature boundary condition at the condenser region, while the boundary condition in the remaining length of the heat pipe is a radiative parasitic heat flux. The vapor pressure is modeled to determine the mass distribution of the saturated liquid inside the heat pipe. An axially grooved aluminum/oxygen cryogenic heat pipe is considered. Theoretical results for the axial temperature profile of the heat pipe for the start-up period are compared to experimental µ-g flight data to validate the new model. Following the valida-tion, the effects of some operational parameters over the axial temperature profile for the start-up period are investigated to determine in which conditions the cryo-genic heat pipe will not start properly.

Nomenclature A area [m2]; also, constant coefficients in Eq. (14) B constant coefficient in Eq. (14) c specific heat [J/kg.K] F view factor; driving forces in Eq. (8) [N] L length [m] h groove depth [m] hfg latent heat [J/kg]

k thermal conductivity [W/m.K] m working fluid mass [kg] P pressure [kPa] qp parasitic heat load [W] Rg gas constant [J/kg.K] s liquid column length [m] T temperature [K] Tcrit critical temperature of the working fluid [K] Tsat saturation temperature of the working fluid [K] t time [s] U average liquid velocity [m/s] V volume [m3] w groove width [m]

Greek Symbols: αeff effective thermal diffusivity [m2/s] β constant coefficient in Eq. (2) δ wall thickness [m] µ dynamic viscosity [N.s/m2] ρ density [kg/m3] σ surface tension [N/m]; also, Stefan-Boltzmann

constant in Eqs. (1) and (13) [W/m2K4]

Subscripts: c condenser section f supercritical/super-heated fluid l liquid layer, liquid properties s solid wall, solid properties v vapor layer, vapor properties

Introduction In 1996, the Satellite Thermal Control Laboratory (NCTS/UFSC) began the development of a Passive Cryogenic Radiator1 in the frame of the University Pro-gram for Space Development (UNIESPAÇO), funded

* Research Assistant, Currently Exchange Visitor, Dept. of Me-chanical Engineering, Clemson University, Member AIAA.

† Associate Professor, Associate Fellow AIAA, ‡

Professor, Currently researcher, Satellite Thermal Control Labo-ratory, Dept. of Mechanical Engineering, Federal University of Santa Catarina, Member AIAA.

8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference24-26 June 2002, St. Louis, Missouri

AIAA 2002-3095

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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by the Brazilian Space Agency (AEB). Passive cryo-genic radiators are used to cool equipment, such as infrared sensors and CCD cameras, to the cryogenic temperature levels required for optimum operation. However, in many satellite designs, the equipment to be cooled cannot be placed near the heat sink. Usually, cryogenic heat pipes are used to transfer heat from the equipment to the heat sink2-5. In addition, cryogenic heat pipes are used in the thermal control of focal plans of infrared sensors6, X-ray telescopes7, and in the cooling of superconducting magnets8. The design of cryogenic heat pipes is now under investigation at the NCTS/ UFSC9 in order to develop a passive cryogenic thermal control device for payloads of the Brazilian satellites, described at the Brazilian Policy for Space Activities10 (PNAE). Cryogenic heat pipes operate at temperatures below 200 K, which makes this system very sensitive to heat leaks from the surrounding environment (e.g., satellite structure). The parasitic heat loads can change signifi-cantly the operational temperature of the cryogenic heat pipe, and may add loads to the heat pipe on the order of the maximum transport capability. In addition to im-posing additional heat load, the parasitic heat loads can adversely affect the transient start-up behavior for the system. Additionally, the operational temperature range of cryogenic heat pipes is relatively narrow11-13. For the development and testing of cryogenic heat pipes, the transient response of the heat pipe temperature must be well known.

Literature Review Unlike low and medium temperature heat pipes, a

cryogenic heat pipe typically starts from a supercritical state. The entire heat pipe must be cooled below the saturation temperature of the working fluid before nominal operation begins. Previously, Colwell14, Bren-nan et al.15, Rosenfeld et al.16, and Yan and Ochter-beck17 have discussed the start-up process of cryogenic heat pipes.

Colwell14 presented a numerical analysis of the tran-sient behavior of a nitrogen/stainless steel cryogenic heat pipe with circumferential screen wick structure and composite central slab. The three-dimensional model assumed constant properties, but did not account for the fluid dynamics of the working fluid. Although provi-sions for simulating a supercritical start-up were in-cluded, the author only presented results for the start-up of the heat pipe with an initial temperature already be-low the critical temperature of the working fluid.

A microgravity experiment for two different alumi-num/oxygen axially grooved heat pipes was conducted by Brennan et al.15 The experiment was flown aboard the STS-53 space shuttle mission in December 1992. Reliable start-ups in flight of the two heat pipes were performed, but the start-up process in microgravity was

slower than that obtained in ground tests. This is be-cause in a microgravity environment the condensation of the working fluid develops a liquid slug in the con-denser region. In ground tests the excess of liquid spreads in the wick structure due to the effects of the gravitational forces. In this case, the liquid typically forms a puddle due to the very low surface tension of cryogenic fluids.

Rosenfeld et al.16 presented a study of the supercriti-cal start-up of a titanium/nitrogen heat pipe. The test was performed during mission STS-62 (March, 1994). This heat pipe reached a non-operational steady state thermal condition during microgravity tests. Only 30 % of the heat pipe length cooled below the nitrogen critical point, but the vapor pressure was still above the critical pressure. However, Rosenfeld et al.16 observed that in ground tests, the titanium/nitrogen heat pipe underwent start-up successfully. The authors concluded that, with the addition of parasitic heat loads, the thermal conduc-tion of the titanium/nitrogen heat pipe was insufficient to allow for the internal pressure to decrease below the critical pressure of nitrogen when in microgravity. The successful start-up during ground tests was due to en-hanced thermal transport of the gravity-assisted convec-tion/liquid collection effects. These tests highlighted the significance of the parasitic heat loads, as the heat pipe start-up failure would not occur in microgravity if the heat leaks had been significantly reduced.

Yan and Ochterbeck17 presented a one-dimensional transient model for the supercritical start-up of cryo-genic heat pipes. The start-up process was summarized into two stages. In the first stage, the heat pipe is cooled by pure heat conduction, and the vapor temperature at the condenser and pressure are greater than the critical temperature and pressure (Tc > Tcrit, or P > Pcrit). The cooling effect resulting from the condenser heat rejec-tion is not immediately propagated through the heat pipe, but it is confined to a region extending from the condenser to some penetration depth δ. Beyond δ, the temperature gradient is zero. When the penetration depth equals the heat pipe length, the cooling effect of the condenser has propagated over the entire heat pipe.

In the second stage, the vapor temperature and pres-sure are lower than the critical temperature and pressure (Tc < Tcrit, P < Pcrit). When the condenser temperature is lower than the critical temperature and the internal pres-sure is lower than the critical point, the vapor begins to condense in the condenser section. The advancing liquid layer is subjected to a capillary driving force that is induced by surface tension and opposed by the wall shear stress, as it advances with an average velocity that will vary with respect to the length of the liquid layer. With increasing time, the liquid average velocity in the condenser increases. The liquid front will advance, until the heat pipe reaches its operational steady state, as-suming sufficiently low heat leaks. This model com-

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pared favorably with the microgravity experimental data presented by Brennan et al.15, but it did not include effects of the parasitic heat load over the heat pipe working fluid.

In the current work, provisions to account for the parasitic heat load during the supercritical start-up of cryogenic heat pipes are included in the model of Yan and Ochterbeck17. Also, a methodology to determine the vapor pressure and working fluid mass distribution in the wick and vapor regions are included, which allows the determination of the length of the liquid slug in microgravity.

Analysis The supercritical start-up of an axially grooved cryo-

genic heat pipe is considered. The initial temperature of the heat pipe, T0, is above the critical temperature, and the boundary condition at the condenser region is a specified time-variable temperature [Tc = f(t)]. This con-dition is consistent with most experiments in the litera-ture14-16, which use cryocoolers to provide the required heat rejection at the condenser region. The remaining length of the heat pipe (adiabatic and evaporator re-gions) is considered to be under the influence of a ra-diative parasitic heat load. This parasitic heat load is provided by the radiation heat transfer between the heat pipe and the spacecraft structure, and/or heat loads from the space environment (e.g., direct sun irradiation, earth emission and albedo, etc.). Figure 1 shows the sche-matic of the physical model, as well as the boundary conditions and the coordinate system.

)(tfTc =

x = 0

Transport sectionCondenser Evaporator

x L =

q = AF T - T σε ( )4HP,

4

cLx −= eLx =

Figure 1. Physical model and coordinate system

(1st stage of start-up).

The supercritical start-up process of a cryogenic heat pipe can be represented generically in a pressure-spe-cific volume diagram, shown in Fig. 2. At the beginning of the process the heat pipe is considered to be isother-mal at T0, where the initial condition of the heat pipe is represented by point 1 in the supercritical region. As the start-up proceeds, the state of the heat pipe is repre-sented by a horizontal line, where the pressure is as-sumed essentially uniform in the heat pipe at any point in time. This line lowers as the pressure of the vapor

decreases (line 2a-2b). In Fig. 2, point 2a represents the specific volume at the condenser end, and point 2b rep-resents the specific volume at the evaporator end. The condensation process will begin only when the tem-perature and pressure of the working fluid at the con-denser end are below the critical temperature and pres-sure, and when the condenser reaches the saturation zone (line 3a-3b-3c), at temperature T*

sat. At this point, saturation condition will exist (points 3a-3b) within the condenser region. The remaining length of the heat pipe will contain superheated or supercritical working fluid (line 3b-3c). As the temperature of the condenser de-creases, more liquid is condensed in the wick structure (line 4a-4b-4c). When the liquid column reaches the evaporator end, the heat pipe is completely primed, and the state of the saturated liquid in the wick is given by point 5a, and the saturated vapor by point 5b. 1st stage of the start-up process (Tcond > T*

sat) The temperature in which the condensation process

starts is defined as T*sat. Another condition that must be

satisfied for the condensation to occur is that the vapor pressure must be equal to the saturation pressure at T*

sat. The methodology for determining T*

sat and the vapor pressure will be explained later in this paper.

In the 1st stage, the heat pipe contains only vapor (supercritical or superheated), and the temperature at the condenser is specified. As the condenser temperature is considered to be uniform, the origin of the coordinate system used for the modeling of the supercritical start-up of the cryogenic heat pipe is located at the interface between the condenser and the transport sections. The heat conduction at the dry region of the heat pipe wall and wick structure is one-dimensional, and the con-ductivity of the working fluid is negligible when compared to the conductivity of the heat pipe wall.

4

5

x

x

TT

T

T

crit

sat*

op

0

Pcrit

Saturationzone

Pres

sure

Specific volume

1

2a 2b

3c3b

3a

4a

5a 5b

4b 4c

Figure 2. Pressure-specific volume diagram.

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The one-dimensional heat conduction equation for the physical model presented in Fig. 1 is given by:

Vk

TTAF

x

T

t

T

s

HP

eff

)(144

,

2

2 −+

∂∂

=∂∂ ∞∞σε

α (1)

where αeff is the effective thermal diffusivity of the heat pipe, T is the temperature of the heat pipe wall, σ is the Stefan-Boltzmann constant, ε is the emissivity of the external wall of the heat pipe, A is the external area of the heat pipe from x = 0 to x = L, FHP,∝ is the view factor between the heat pipe and the surrounding environment, T∝ is the temperature of the surrounding environment, ks is the conductivity of the heat pipe wall, and V is the volume of the heat pipe wall. The second term on the right hand side of Eq. (1) accounts for the radiative parasitic heat load applied over the heat pipe external wall. To account for the heat capacity of the working fluid in the effective thermal diffusivity, Yan and Ochterbeck17 added to the thermal diffusivity of the wall a constant coefficient that accounts for the total change of the internal energy from the initial state to the final state for the heat pipe wall and the working fluid.

The initial condition for Eq. (1) is considered to be a constant temperature T0 for the entire heat pipe length. The boundary conditions at the interface between the condenser and the transport section is a time-variable temperature given by the cryocooler, Tc(t), while the evaporator end is considered to be insulated:

T = T0; for t = 0 and 0 ≤ x ≤ L (2)

)(tTT c= ; for x = 0 and t > 0 (3)

0=∂∂

x

T; for x = L and t > 0 (4)

2nd stage of the start-up process (Tcond ≤≤ T*

sat) When the temperature of the condenser decreases to

a value below T*sat, and the vapor pressure is equal to

the saturation pressure at T*sat, the vapor begins to con-

dense at the wick structure, and a liquid column devel-ops inside the groove channels. Once the groove is filled with saturated liquid, the liquid column will advance in the direction of the evaporator end with a rewetting velocity Ur. At the leading edge of the liquid column, liquid is vaporized because the dry region of the heat pipe is at a higher temperature. Also, the parasitic heat load will vaporize some liquid along the column. If the summation of the parasitic heat load and the heat sup-plied by the dry region is greater than that heat needed to vaporize the entire advancing liquid column, the rewetting process is stagnated. The fluid flow in the liquid column is considered to be laminar, and Newto-

nian, with an average velocity U. The total length of the liquid column is s, and its temperature is considered to be uniform and equal to the temperature of the wall of the heat pipe. At the condenser, this temperature is equal to the cryocooler temperature. Figure 3 shows the physical model for the 2nd stage of the start-up.

)(tfTc =

x = 0 x s t = ( )

Dry region

Liquid column

x L =

q = A F T - T σε ( )4HP,

4

cL−

Liquidslug

Figure 3. Physical model and coordinate system (2nd stage of start-up).

The one-dimensional heat transfer equation for the

dry region of the physical model presented in Fig. 3 is equal to Eq. (1). The initial condition for this equation is the last temperature profile obtained with the 1st stage modeling. Now, as the liquid column eventually moves towards the evaporator end, the boundary condition for this stage is considered to be a moving boundary condition at the position x = s(t) – Lc:

)(tTT c= ; for x = s(t) – Lc and t > 0 (5)

The position of the liquid column s can be obtained as a function of t by a heat balance at the leading edge of the liquid column, as follows:

−=

∂∂

−= dt

dsUhA

x

TAk fg

Lsxss

c

llρ (6)

where ρl is the density of the saturated liquid, Al is the groove cross sectional area, hfg is the latent heat of vaporization of the working fluid, and ds/dt is the velocity that the liquid column advances, i.e., the rewetting velocity Ur. The initial condition is given as s = Lc at the time that the condensation process of the working fluid begins in the wick structure. Liquid column modeling

The liquid column is modeled according to the methodology presented by Yan and Ochterbeck17. In this case, the rewetting process is considered to be the rewetting of a rectangular groove. A momentum balance is performed at the liquid column, and the total liquid column is taken as the control volume. The advancing liquid column is subjected to a capillary driving force

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induced by the surface tension, and is opposed by the wall shear stress:

dt

mUdFF friccap

)(=− (7)

where m is the mass of the liquid column, and U is the average velocity of the liquid in the column. The mathematical solution of Eq. (7) provides an expression for the length of the liquid column and the velocity U as a function of time. As the velocity U is related with the length of the liquid column by U = ds/dt, the velocity of the liquid inside the column for an unheated surface can be expressed as follows:

2

2

)2(2 whs

hwU

+=

lµσ

(8)

where details about the derivation and solution can be found in Yan and Ochterbeck17. The average liquid velocity U obtained from Eq. (8) refers to an unheated groove. The parasitic heat load applied to the heat pipe wall will cause some working fluid to evaporate from the liquid column. Therefore, a mass balance in the liquid column between x = 0 and x = s(t) – Lc is per-formed to obtain the corrected value for the velocity U at the column leading edge:

evxmm

dt

dm&& −=

=0 (9)

The evaporation mass flux evm& in Eq. (9) is a func-

tion of the parasitic heat load, and it provides a relation between the velocity U and the parasitic heat load that affects the liquid column:

fgff

p

c hA

q

whL

hwU

ρµσ

−+

=2

2

)2(2 l

(10)

Vapor pressure modeling

The vapor pressure can be obtained by using the following thermodynamic relation:

TZRP gfρ= (11) where P is the pressure, ρf is the density of the fluid, Z is the compressibility factor, Rg is the gas constant and T is the wall temperature. As the vapor pressure is con-stant, and as the temperature of the heat pipe varies axially, the specific volume must vary axially so that the conservation of the working fluid mass in the heat pipe is satisfied. Thus, the process to determine the vapor

pressure is iterative. First, the heat pipe length is divided into J finite volumes, and a vapor pressure is guessed for each time step. Equation (11) is then solved for each one of the finite volumes to determine the density of each volume, including the condenser. Then, the mass of fluid in each volume is obtained from mj = Vjρj. The total mass is given by the summation of the mass of each volume:

∑= jf mm (12)

If the mass obtained from Eq. (12) is different than

the real fluid mass of the heat pipe, another pressure is guessed until the mass conservation inside the heat pipe is reached. When the temperature of the condenser, Tc, is below the critical temperature, the pressure of the heat pipe, P, is compared to the saturation pressure at the condenser temperature, Psat(Tc). If P = Psat(Tc), then the condensation process begins, and the temperature in which it begins is T*

sat = Tc. The compressibility factor in Eq. (11) is given by18:

∂∂

+==δα

δρ

1TR

PZ

g

(13)

where δ is the reduced density (ρ/ρcrit), and α is the residual Helmoltz energy, determined by least square fitting of experimental data. The procedure for the cal-culation of α is fully described by Jacobsen et al.18 and Stewart et al.19

Model Solution As Eq. (1) is non-linear in temperature due to the

parasitic heat load term, a numerical solution based on the finite volume method20 was used. The heat pipe length was divided into J volumes, and Eq. (1) was integrated inside each volume to provide implicit equa-tions for the temperature. The general form of the equa-tion is given by:

jjjjjjjj BTATATAT +++= ++−− 11110 (14)

where Aj , Aj-1 , Aj+1 and Bj are constant coefficients, T0

j is the temperature in the last time step, Tj , Tj-1 and Tj+1 are the unknown temperatures. A Taylor Series was used to linearize the source term of Eq. (1)20. Equation (14) allows the problem to be written in a matrix form as follows:

[T0] = [A][T] + [B] (15) where the matrix [T] is unknown. The solution is found from:

[T] = [A]-1{[T0]+[B]} (16)

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The numerical solution allows the thermophysical properties of the heat pipe wall to be considered as a function of the local wall temperature. The thermo-physical properties of the saturated working fluid are considered to be a function of the condenser tempera-ture and are found in Jacobsen et al18.

For the 1st stage of the start-up process, Eq. (16) is solved together with boundary conditions (3) and (4) until the temperature at the condenser reaches T*

sat, when the condensation process begins (2nd stage). For the 2nd stage of the start-up process, Eq. (10) is used to determine U. Then, Eqs. (6) and (16) are solved iteratively to determine the temperature profile and the position of the liquid column. The solution of the model is performed until the liquid column reaches the evaporator end (s = L), or until the temperature at the evaporator end varies less than 1 K/h (steady-state condition). Once the temperature profile of the heat pipe is determined, Eqs. (11) and (12) are solved to determine the pressure and the mass distribution of the working fluid.

For the validation of the model, the experimental data of an aluminum/oxygen cryogenic heat pipe presented by Brennan et al.15 (TRW heat pipe) were used. The parameters are shown in Table 1.

For an accurate solution of the model, the εFHP,∝ term to be used during the evaluation of the parasitic heat load in Eq. (1) must be determined. Brennan et al.15 estimated the parasitic heat load to be from 1.1 W to 1.9 W, when the heat pipe was isothermalized around 60 K. Therefore, the term εFHP,∝ can be found by using the relation:

)( 44HPeHPp TTAFq −= ∞σε (17)

Table 1. Aluminum/Oxygen Heat Pipe Design

Summary (Brennan et al.15)

Tube material: Aluminum Tube dimensions:

Outer diameter 11.2 mm Vapor diameter 7.37 mm Wall thickness 1.02 mm

Lengths: Evaporator 0.15 m Condenser 0.15 m Transport section 1.02 m Effective 1.17 m

Working Fluid and Charge Oxygen/10.3 g Number of Grooves 17

Groove width 0.445 mm Fin Fip Radius 0.102 mm Wetted Perimeter (1 Groove) 2.09 mm Total Groove Area 6.07 mm2

The following assumptions are considered: 1. The heat pipe is isothermalized at THP = 60 K; 2. The surrounding temperature T∝ is constant with

time and it is equal to the initial temperature of the heat pipe T0 (280 K);

3. The term εFHP,∝ is considered to be constant with temperature. The εFHP,∝ was obtained for the two extreme para-

sitic heat loads estimated by Brennan et al.15 An average value of 1.5 W was also considered. Table 2 presents these estimated values.

Table 2. Estimated values for εFHP,∝.

qp εεFHP,∝∝ 1.1 Watt 0.077 1.5 Watt 0.104 1.9 Watt 0.134

Results and Discussion

Figure 4 shows the comparison between the theo-retical model and the experimental flight data for the TRW aluminum/oxygen cryogenic heat pipe start-up. The temperature at the condenser (boundary condition) was obtained by fitting a 6th order polynomial to the experimental data of the cryocooler. The overall agreement between the model and the experimental data is good. However, the model slightly underestimates the temperatures at the evaporator end. Some possible reasons for this deviation are listed below: 1. Inadequate estimation of the parasitic heat load.

Brennan et al.15 had not specified whether the parasitic heat load was radiative or (and) conductive;

2. The capillary radius of the liquid column was considered only for the calculation of the capillary driving force on Eq. (7), but not considered for the calculation of the mass of liquid. If the curvature of the free surface was taken into account, the area of the liquid would be smaller, and consequently, the mass of the liquid column would decrease.

Figure 5 shows the pressure-specific volume dia-

gram for the supercritical start-up of the TRW cryogenic heat pipe. The specific volume is shown at the interface between the condenser and transport section (x = 0) and for the evaporator end (x = L). The condensation process will begin at T*

sat = 153.6 K. Also, the condensation begins from the saturated liquid line. It means that for a short period of time (for less than 2 min), the condensed liquid in the condenser region will be in a sub cooled condition. After the condensation process begins, the quality at the condenser remains closer to zero until the liquid column starts to move towards the evaporator end. This is because more liquid is being condensed as the temperature of the condenser decreases.

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Brennan et Al. (1993)Flight experiment (STS 53 - Dec. 92)

(1.1 W < qp < 1.9W - estimated) Condenser

x = 0.15 m x = 0.72 m Evaporator end

Condenser

x = 0.15 m

x = 0.72 m

Evaporator end

Present model:εF = 0.077 (qp = 1.1 W)εF = 0.104 (qp = 1.5 W)εF = 0.134 (qp = 1.9 W)

0 1 2 3 4 5 6 7 850

100

150

200

250

300

Time [hours]

Tem

pera

ture

[K

]

Figure 4. Comparison between the theoretical model and the experimental data.

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10Specific volume [m /kg]3

Pres

sure

[M

Pa]

x = 0

εF q = 0.077 ( = 1.1 W)p

εF q = 0.104 ( = 1.5 W)p

εF q = 0.134 ( = 1.9 W)p

Satu

rate

d li q

u id

line Saturated vapor line

Tcrit

Pcrit

Tcrit

x L=

Figure 5. Pressure-specific volume diagram (TRW heat pipe).

The liquid fill as a function of the temperature of the

condenser is shown in Fig. 6. The liquid fill is defined as the ratio between the condensed liquid mass ml and the mass that the wick structure can hold mw:

l

l

vsA

vsAx

m

mN

w

cc

w /

/)1( −== (18)

where x is the quality of the saturated fluid in the wetted region, Ac is the cross sectional area of the condenser

(liquid + vapor), s is the position of the liquid column for a given time, Aw is the cross sectional area of the grooves, and vc and vl are the specific volumes of the saturated fluid in the wetted region and saturated liquid, respectively. If N = 1, the mass of liquid is enough to fill the grooves with no excess liquid. For N > 1 there is excess liquid. As seen, there is a large excess of liquid when the start-up process begins. This is because the oxygen enters the saturation zone from a sub-cooled condition. Therefore, the condenser is flooded with saturated liquid. The fill rate starts to decrease as the liquid column begins to advance, and as the liquid specific volume decreases with continued decrease in temperature.

Figure 7 shows the position of the liquid column in-side the axial grooves as a function of time. Also the position of the liquid slug is shown. This slug was ob-served by Brennan et al.15 and only occurs in micrograv-ity environment. On the ground, the capillary forces cannot support a slug across the vapor region, and ex-cess of liquid is spread as a puddle15. This effect facili-tated the priming of a heat pipe in ground tests.

As shown in Fig. 7, when the start-up process begins, the liquid column remains stagnated at the interface between the condenser and the transport region for a few minutes (12.5 min.). This is because the liquid velocity in the liquid column is not sufficient to provide the cooling to the dry region of the heat pipe. As the temperature of the condenser decreases, the surface tension of the working fluid increases, thus, increasing the capillary driving force, causing the liquid velocity to increase. So, the liquid column will eventually advance towards the evaporator end. For a lower parasitic heat load (1.1 W), the start-up process was faster, because less fluid was evaporated from the liquid column. For a higher parasitic heat load (1.9 W), the

εF q = 0.077 ( = 1.1 W)p

εF q = 0.104 ( = 1.5 W)p

εF q = 0.134 ( = 1.9 W)p

160 140 120 100 80 600

1

2

3

4

5

6

7

8

9

Temperature of the condenser [K]

Liq

uid

fill

rat

e -

/m

mli

quid

wic

k

Figure 6. Liquid fill.

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analytical model shows that the heat pipe would not prime completely, with the liquid column stagnating at about 0.21 m from the evaporator end.

The position of the slug was determined from the data shown in Fig. 6, for N > 1. The position of the liquid slug increases with time until it reaches a maximum at about 0.40 m from the condenser end, and then it decreases. For the lowest parasitic heat load (1.1 W) the position of the slug when the heat pipe primed is at 0.20 m from the condenser end. This because the mass of 10.3 g of oxygen on the TRW heat pipe was designed to fill the wick structure with saturated liquid and no excess at 60 K. The heat pipe is completely primed at t ≅ 4.6 hours, and its temperature is 90 K. The data presented in Fig. 6 shows that

there is an excess of liquid of 1.8 g at 90 K, while the mass of liquid inside the wick structure is 8.1 g. The remaining 0.4 g is the mass of saturated vapor. For the average parasitic heat load of 1.5 W, the heat pipe primed at about 75 K, and the slug was at 0.14 m from the condenser end. It is important to say that if the temperature of the heat pipe continues to decrease until 60 K, there will be no liquid excess, and therefore, no slug. For the highest parasitic heat load, the liquid slug was stagnated at 0.15 m, as the heat pipe did not primed completely. Figure 8 shows the predicted vapor pressure as a function of time.

Start-up Limitations The start-up limitation for an axially grooved

cryogenic heat pipe is now discussed based on the theoretical results given by the model presented. The analysis will focus on the effects of the parasitic heat load over the temperature profile and liquid column length, as well as the effects of some geometrical parameter. The TRW heat pipe15 is considered here. All the geometric parameters were kept equal to those shown in Table 1, except when specified.

Figure 9 shows the effect of the parasitic heat load over the temperature of the heat pipe. The figure shows different transient temperatures for different values of εF. The number inside the brackets in the legend refers to the corresponding parasitic heat load, for a heat pipe isothermal at 60 K, with the same geometry of the TRW heat pipe. Temperatures are shown for three different positions: x = 0 (condenser), x = 0.15 m (inside the transport section), and x = L (evaporator end).

As it was expected, the priming of the heat pipe is slower as the parasitic heat load increases. It appears that the TRW heat pipe is almost at the edge of the para-sitic heat load. It primed for a parasitic heat load of 1.5 W, but did not for a parasitic heat load of 1.9 W and above (see Figs. 4 and 7). It is important to note that the TRW heat pipe was specifically designed to have a low transport capability such that data could be obtained for analytical model validation15.

The position of the liquid column as a function of the start-up time is shown in Fig. 10. If there is no para-sitic heat load, the heat pipe will prime completely in 3.67 h. The time that the liquid column stays stagnated in the interface between the condenser and the transport section is 11 min for no parasitic heat load, up to 15 min. for 2.9 W. In the later case, only 0.82 m of the heat pipe could be primed.

Figure 11 presents the effect of the condenser length on the transient temperature of the heat pipe during the start-up. The time for the complete priming of the heat pipe decreases as the relative length of the condenser increases. This occurs because a longer condenser cor-responds to a larger heat pipe length being cooled di-rectly. The position of the liquid column as a function of time is presented at Fig. 12.

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Condenser

Liquid slug

Liquid column

Time [hours]

Axi

al p

osit

ion

[m]

εF q = 0.077 ( = 1.1 W)p

εF q = 0.104 ( = 1.5 W)p

εF q = 0.134 ( = 1.9 W)p

Figure 7. Predicted liquid front and liquid slug position.

0.001

0.01

0.1

1

10

Pres

sure

[M

Pa]

εF q = 0.077 ( = 1.1 W)p

εF q = 0.104 ( = 1.5 W)p

εF q = 0.134 ( = 1.9 W)p

0 1 2 3 4 5 6 7 8Time [hours]

Figure 8. Predicted vapor pressure.

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9

The effect of the cross sectional area of the groove on the temperatures of the heat pipe is given in Fig. 13. The aspect ratio for all groove areas presented in this figure is constant and equal to the aspect ratio of the TRW heat pipe (h/w = 1.8).

Figure 14 shows the position of the liquid column for the three groove areas presented in Fig. 13. For the largest groove area, the start-up time was 4.17 h. For the smallest groove area, steady state was not reached even after 8 h from the beginning of the start-up.

The effect of the external diameter of the heat pipe container on the transient temperature is presented in Fig. 15. The groove area and the vapor area were kept

constant for the three diameters shown in this figure, and their values are the same as those in the TRW heat pipe. A larger external diameter will provide a larger external area of the heat pipe, which will increase the incidence of parasitic heat load. More importantly, the larger diameter corresponds to an increased wall thick-ness, which increases the thermal mass of the heat pipe. Additionally, the greater wall thickness will increase the heat transfer from the dry region to the liquid column (see Eq. 6). Thus, more liquid from the leading edge of the liquid column will be evaporated, decreasing the rewetting velocity. Figure 16 shows the liquid front position for the three diameters presented in Fig. 15.

0 1 2 3 4 5 6 7 850

100

150

200

250

300

Condenser

x = 0.15 m

Evaporator end

εF

0.01 ( = 0.15 W)0.05 ( = 0.72 W)0.104 ( = 1.5 W)0.2 ( = 2.9 W)

0 ( = 0)qp

qqq

q

p

p

p

p

Time [Hours]

Tem

pera

ture

[K

]

Tcrit = 154 K

Figure 9. Effects of the parasitic heat load.

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Transportsection

Evaporator

Condenser

Time [hours]

Liq

uid

fron

t pos

ition

[m

]

εF

0.01 ( = 0.15 W)0.05 ( = 0.72 W)0.104 ( = 1.5 W)0.2 ( = 2.9 W)

0 ( = 0)qp

qq

qq

p

p

p

p

Figure 10. Liquid column length.

0 1 2 3 4 5 6 7 850

100

150

200

250

300

Condenser

x = 0.15 m Evaporator end

Condenser length 0.10 m 0.15 m (TRW) 0.20 m 0.30 m

Time [Hours]

Tem

pera

ture

[K

]

Tcrit = 154 K

Figure 11. Effects of the condenser length.

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Transportsection

Evaporator

Condenser

Time [hours]

Liq

uid

fron

t pos

ition

[m

]

Condenser length 0.10 m 0.15 m (TRW) 0.20 m 0.30 m

Figure 12. Liquid column length.

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10

Summary An improved model for the supercritical start-up of

cryogenic heat pipes was presented. The model included the effects of parasitic heat loads for the entire length of the heat pipe. Also, a methodology for determining the vapor pressure and the mass distribution in the heat pipe was presented. This methodology allowed the determi-nation of the length of the liquid column and the length of the liquid slug.

The presented model compared well with micro-gravity experimental data available in the literature for axially grooved cryogenic heat pipes. An analysis of the supercritical start-up limitation of an axially grooved cryogenic heat pipe in a microgravity environment was

performed based on the results obtained from the improved model. The results are summarized below: • The parasitic heat load plays an important hole in the

start-up process of cryogenic heat pipes. The para-sitic heat loads increases the temperature gradient in the dry region, increasing the vaporization of liquid at the leading edge of the liquid column. Also, the parasitic heat load will vaporize liquid along the length of the liquid column once it starts to advance towards the evaporator;

• A longer condenser will facilitate the priming proc-ess as it allows a larger condensation mass rate;

• Larger cross sectional areas of the grooves also helps the priming process to occur.

Groove cross sectional area

4.50 x 10 m-6 2

6.02 x 10 m (TRW)-6 2

9.11 x 10 m-6 2

Condenser

x = 0.15 m Evaporator end

0 1 2 3 4 5 6 7 850

100

150

200

250

300

Time [Hours]

Tem

pera

ture

[K

]

Tcrit = 154 K

Figure 13. Effects of groove cross sectional area.

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Transportsection

Evaporator

Condenser

Time [hours]

Liq

uid

fron

t pos

ition

[m

]

Groove cross sectional area

4.50 x 10 m-6 2

6.02 x 10 m (TRW)-6 2

9.11 x 10 m-6 2

Figure 14. Liquid column length.

0 1 2 3 4 5 6 7 850

100

150

200

250

300

External diameter 9.5 mm 11.2 mm (TRW) 15.0 mm

Condenser

x = 0.15 m

Evaporator end

Time [Hours]

Tem

pera

ture

[K

]

Tcrit = 154 K

Figure 15. Effects of the heat pipe external diameter.

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Transportsection

Evaporator

Condenser

Time [hours]

Liq

uid

fron

t pos

ition

[m

]

External diameter 9.5 mm 11.2 mm (TRW) 15.0 mm

Figure 16. Liquid column length.

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11

Acknowledgements The authors would like to acknowledge the help of

Dr. Y. H. Yan during the development of this work. Couto and Mantelli would like to acknowledge the Brazilian Space Agency – AEB, CAPES Foundation, and the Brazilian Council of Research and Development – CNPq for supporting this project.

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13 Chi, S. W., and Cygnarowicz, T. A., “Theoretical Analyses of Cryogenic Heat Pipes”, ASME Paper No. 70-HT/SpT-6, 1970.

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17 Yan, Y. H. And Ochterbeck, J. M., “Analysis of Supercritical Start-Up Behavior for Cryogenic Heat Pipes”, AIAA Journal of Thermophysics and Heat Transfer, Vol. 13, No. 1, pp. 140 – 145, Jan.-Mar. 1999.

18 Jacobsen, R. T., Penoncello, S. G., and Lemmon, E. W., Thermodynamic Properties of Cryogenic Fluids, Plenum Press, New York, 1997.

19 Stewart, R. B., Jacobsen, R. T., and Wagner, W., “Thermodynamic Properties of Oxygen from the Triple Point to 300 K with Pressures to 80 MPa”, Journal of Physical and Chemical Reference Data, Vol. 20, No. 5, pp. 917-1021, 1991.

20 Patankar, S. V., Numerical Heat transfer and Fluid Flow, Hemisphere Pub. Corp., Washington D.C., 1980.