Mass-Spring-Damper Model with Steady State Parameters for Predicting the Movement of Liquid Column and Temperature Oscillation in Loop Heat Pipe
Ge Zhang 1, Di Chen1, Ying-ying Hong 1, Li Liu2
(1 School of Mechanical Engineering, University of Science and Technology Beijing, 100083, China
2 School of Advanced Engineer, University of Science and Technology Beijing, 100083, China)
Abstract: This paper investigated the movement of the phase interface as the changed input power by a mass-spring-damper model. The model was solved with MATLAB and was used to explain the high-frequency and low-amplitude temperature oscillation. Temperature variation with the input power from 20 W to 75 W was investigated based on a LHP prototype in a literature. The model agreed well with the experimental data in the literature.The simulated results suggested that the movement of the liquid column was caused by thefluctuation of pressure difference applied on the liquid column and the stiffness coefficients of the vapor springs increasing with the input power.According to parameter analyses, the temperature oscillation at the outlet of the condenser can be weakened by increasing the mass of the liquid column and keeping the temperature at the outlet of the condenser steady.
Keywords:loop heat pipe; temperature oscillation; displacement; mass-spring-damper model; steady state parameters
Chinese library class Number.：TK124
1Introduction
Loop heat pipe (LHP) is known for high heat-transfer efficiency. They are widely used in miniature electric devices, especiallythe devices related to space technology [1-5]. The first LHP was developed and tested successfully by Maydanik in 1972[6]. LHPs are different from their sizes, shapes, materials and working fluids. In general, LHPs are classified into two types by the shape of the evaporator.One type is the LHP with a cylindrical evaporator, which is the most common LHP. The other is the LHP with a flat evaporator. It is recognized as a more advanced LHP in the aspect of the contact area with heat source[7]. The LHP with flat evaporator mainly consists of condenser, vapor pipe, liquid pipe and flat evaporator as shown in Fig. 1.
Fig.1 Schematic of a LHP with flat evaporator
The performance of LHP was investigated with plenty of experiments[8-11], especially the performance of the LHP with a flat evaporator [12]. In the experiments, temperature oscillation was observed frequently during the operation of LHP in certain conditions[13].This phenomenon has negative influence on the performance of LHP. For example, the temperature oscillation appearing during startup process delays the startup of LHP and even makes the startup fail [14]. Huang et al. [15]found that the oscillationin startup process happened when the input power was between critical heat load and minimal heat load. The temperature oscillation during startup is characterized as low-amplitude (no more than 1 K) and high-frequency. It was recognized as a result of the oscillation of the liquid front in the wick. Besides, temperature oscillation also occurred during operation [16-18]. The amplitude of this kind of temperature oscillationis also low, but theperiod and duration is much longer. The temperature oscillation during operation was supposed to be caused by the oscillation of the two-phase interface in the condenser. Temperature oscillation also happens with low-frequency and high-amplitude, which often lasts for hours.It happened in the case thatthe thermal mass evaporator was massive and the sink temperature kept very low.
The performance optimization of LHP has been studied by theoretical models.Someresearches focused on building integrate model for the LHP[19, 20]. Others researched the performance of some key components by building local mathematical models [21]. However, all the known researches ignored the investigation of temperature oscillation.
This paper aims to investigate the movement of the phase interface as the changed input power by a mass-spring-damper model. The model can be used to explain the high-frequency and low-amplitudetemperature oscillation. Parameter analyses were performed to make suggestions for controlling the intensity and frequency of the temperature oscillation. The results were validated by experimental data ina literature [22].
2Model Description
2.1Mass-Spring-Damper Model of LHP
The mass-spring-damper model is based on the force balance of the liquid columnin the sub-cooling region of the condenser and liquid line.The vapor columns in the vapor line and reservoir aretreated as vapor springs respectively because of the compressibility of vapor. The liquid columnis treated as a mass between the two vapor springs for the incompressibility of liquid.The two vapor springspress the liquid columnfrom the opposite directions. The changed input power can be treated as an excitation for breaking the balance of the thermal equilibrium and pressure equilibrium. Fig. 2illustrates the force balance of the column.
Fig.2 Schematic of mass-spring-damper model of LHP
According to the Newton's second law analysis of the liquid column, it yields:
(1)
Where, Mdenotesthemass of the liquid column, xthedisplacement, and tthetime, Fd the force disturbance caused by sudden heat load change, Ff the friction force of the liquid column, and Fk the elastic force of the vapor springs.
According to the law of Newton inner friction and the shear stress expression in laminar pipe flow[23], Ff can be written as
(2)
Where, η denotes the friction coefficient, μlthe dynamic viscosity of liquid, and L the length liquid column.
Assuming the vapor pipe and the reservoir are thermal insulated, the two vapor springs experience isentropic processes. According to the equation of ideal gas state, the elastic force and the stiffness coefficient of the vapor springs can be described as linear[24], when the displacementof the liquid column is in a small range:
Where, κdenotes the stiffness coefficient of the vapor springs,Pthe pressure, Avthe cross-sectional area of the vapor springs, and Vvthe volume of the vapor springs.
Since the vapor springs are in adiabatic condition, the coefficient γcan be taken as 1.4. As a disturbance term, Fd can be determined by calculating the pressure difference ΔPd between the twosurfaces of the wick[25].
Where, Aldenotes the cross-sectional area of the liquid column. Substitutingthe expression of Fkand Ffin Eq. 1, it yields:
(3)
(4)
(5)
Where, βdenotes the damping factor, and ω0 the natural frequency.
Solving Eq. 3with Laplace transform and Inverse Laplace transform, the analytical solution can be achieved as:
Where, φ denotes the initial displacement.
2.2Determination of State Parameters and Properties
The state parameters and properties of LHP are determined based on energy balance and thermodynamic relationships. In the LHP, pressure disturbance transports at local sound speed, which is much quicker than the heat transfer. It means the thermal state changes little when the liquid column reacts to the pressure disturbance, i.e. excitation. Therefore, the state parameters and properties involved in the mass-spring-damper model are all calculated in steady state.
The following assumptions are made to simplify the mathematical model:
(i)The LHP model is in adiabatic condition, i.e. no heat releases from the profile of the LHP to the ambient. The error produced from the adiabatic assumption can be ignored according to the experimental data of Singh et al. [22].
(ii)The heat transfer inside the casing of the evaporator and the reservoir is ignored.
(iii)The evaporation happens at the wick-groove interface.
(iv)The superheat is ignored. The evaporating temperature is the same as condensing temperature.
(v)The water is treated as incompressible fluid in the LHP.
Fig.3 Thermal network of the LHP model
The simplified thermal network of the LHP model was shown as Fig. 3. For the evaporator, the total heat load Qin produced by the heat source dissipates in two ways: one is to heat the subcooled liquid Tco to the saturated temperature Ts, the other is to cause the evaporation at the wick-groove interface. The heat transfer process can by described by
(6)
Where, Qsen is the sensible heat given to the subcooled liquid from the liquid pipe, Qeva is the latent heat for evaporation, Qsr is the heat transferred from the groove to the reservoir, λsr is the thermal conductivity from the groove to the reservoir, zwick is the thickness of the wick, is the mass flow rate, Tris the temperature of the reservoir, Cplis the fluid specific heat, and the coefficient ζ is the ratio of subcooled liquid entering the wick without having heat transfer to the total.
The coefficient ζ is surely influenced by the mass flow rate [26]. The coefficient ζ can be determined by a polynomial fitted by the data of Singh et al. [22]
Table 1 Coefficients of polynomial
Coefficient / C0 / C1 / C2 / C3 / C4 / C5 / C6Value / -4.8269 / 6.1772e-1 / -2.8101e-2 / 6.5490e-4 / -8.1663e-6 / 5.1904e-8 / -1.3227e-10
Since the liquid line is thermal isolated to the environment, the temperature of the water at the outlet of condenser Tco is equal to the temperature at the inlet of reservoir. The vapor in the reservoir is considered to be saturated. The saturated temperature can be described by Antoine equation [27] as follow:
Where, a, b, c are coefficients for the Antoine equation, Ps is the saturate pressure, and ψ is the coefficient for unit conversion.
The pressure drops in the vapor line ΔPv and liquid line, ΔPv and ΔPl, can be calculated by [28]:
Where, the friction factor f can be evaluated as:
The condenser is usually a combination of two-phase region and sub-cooling region. Since the vapor pipe is considered as adiabatic, the vapor enters the two-phase region at the evaporating temperature Ts and condenses to saturated water in the two-phase region. The energy balance can be described as follow:
Where, λ is the latent heat, L2p is the length of the two-phase region in the condenser, Tsinkis the heat sink temperature, h2p is the heat transfer coefficient of the two phase region, hsinkis the heat transfer coefficient of the heat sink,
In the sub-cooling region, the saturated water is cooled to a lower temperature Tcoand release sensible heat. The energy balance can be described by
Where, hl is the heat transfer coefficient of the liquid, Do is the outer diameter of the condenser, and Lcis the length of the condenser.
To calculate the liquid temperature in the compensation chamber, Valeri et al. [29] divided the liquid entering the chamber into two parts.One part of the liquid enters the wick without having heat transfer with the water retained in the reservoir. The parameter ζ is used to represent the ratio of this part to the whole, and the rest is represented by (1-ζ). For the rest part, the energy balance can be described as follow:
Where, λsris the heat conductance from the groove to the reservoir. The heat convection inside the wick is neglected since it does not play a major role comparing with heat conduction. The heat conductance can be described as follow [30]:
Where, λland λwmrepresents the heat conductance of the liquid and the wick material, respectively, ε is the porosity of the wick.
2.3Solving Process
The mass-spring-damper model is solved with MATLAB. The complete processis showed in Fig. 4. An iterative procedure is used to solve the state parameters by initializing an assumed mass flow rate m0. The deduced mass flow rate is corrected by Eq. 6. The iteration terminates when the relative error of mass flow rate is less than 0.1%.
The disturbance of the input heat flux Qin can lead to the change of state parameters and the thermal conductivity of the working mass. Therefore, the mass-spring-damper model is re-solved when the input heat flux Qin is changed. At the end of the procedure, an array is used to record and output the results.
Fig.4 Thermal network of the LHP model
3Model validation
The model was validated with the structural parameters and experimental data of a LHP prototype designed by Singh et al. [22]. The LHP prototype with a flat-disk shaped evaporator is operated in horizontal orientation. The wick is made of nickel, the porosity of which is 75%.The evaporator is made of stainless steel, and the working fluid is water. For the condenser, the heat sink temperature is 295.15 K, and the heat transfer coefficient of the heat sink is 2000 W/ (m2·K). Table 2 shows the detail parameters of the LHP prototype.
Table 2 Parameters of the LHP prototype
Component / Parameter / Value (mm)Wick / Diameter / 40
Thickness / 3
Evaporator / Diameter / 41
Reservoir depth / 10
Vapor groove width / 1
Condenser / (Inner/Outer) diameter / 2/2.4
Length / 100
Line / (Inner/Outer) diameter of transport line / 2/2.4
Length of vapor line / 150
Length of liqui line / 290
Fig. 5 presents the comparison between the model results and experimental data of the saturation temperature Ts in the groove. Good agreement can be seenin the figure. Generally, the results of the model were a little bit higher than the experimental data. It is because that the model is built based on the assumption of adiabatic condition to the ambient, i.e. no heat releases from the profile of the LHP to the ambient.
Fig.5 Comparison between the model and the data from Singh [22]
4Results and discussion
Fig. 6shows the change of the input power.The power is increased from 20 W to 70 W in increments of 5 W, which is corresponded with the condition of the reference[22]. The time interval of the power change was set as 5 minutes.
Fig.6Step change of the input power
4.1Temperature variation
The variation of Ts, Tr and Tcoin the power range from 20 W to 70 W is plotted in Fig. 7. In this figure, the three temperatures showed nonlinear growths as the input powerQin increased. The temperature of the condenser outlet Tco kept steadyand was equal to the temperature of cooling water serving as the heat sink when the input power was lower than 40 W.For higher input power, Tco started to increase with it. Correspondingly, Ts started to increase substantially either.
Usually, the cooled objects are supposed to work in a narrow temperature range. Theabove results suggests that to keep the temperature of the subcooled fluid at the outlet of the condenser stable is the premise to avoid substantial fluctuation of the cooled objects. This goal can be achieved by adjusting the amount of cooling water.
For all the input heat loads, the saturated temperature in the groove Tsand the temperature of reservoir Trare always very close to each other as seen in Fig.5. It suggests that the temperature in the reservoir is dominated by the heat leak through the wick. The result agrees well with the experimental data of Singh et al [22]. Some researchers [32] calculated Tras the local saturated temperature. In this way, Tr was either close to Ts since the high latent heat of water leads to small mass flow rate and tiny pressure drop in the LHP.
Fig.7 Temperature distribution in different input powers
4.2Oscillation of theliquid column
Different from solid mass-spring model, the size of the liquid column and the vapor springs changes during the step changes of the input power and makes the model complex. Therefore, the position of the phase interface in the condenser was evaluated in each step change.
The displacement of the liquid column caused by the step change of input power is shown in Fig. 8.When the input power increased from 20 W to 35 W, the liquid column moved to the reservoir. When the input power increased more, the liquid column tended to move far from the reservoir. It should be noticed that the increment of displacement was not uniform during the input power change.
Fig.8 Displacement of the liquid column caused by input power
In this study, the LHP was made of opaque material, so that the liquid column oscillation could not be observed. Eric K.Begg[31] ever researched the phase interface movement of a loop thermosyphon, which structure and size was similar to the LHP studied by this paper. He changed the heat load and recorded the phase interface movement photographically. Although the magnitude was different, the liquid column oscillation was also observed when the heat load was increased as shown in Fig. 9.
Fig.9 Photographic record of phase interface movement by Eric K.Begg[31]
The displacement of the liquid column is decided not only by the pressure difference on the two opposite sides, but also by the stiffness coefficient of the two vapor springs. When the input power stepped up, the pressure difference on the two opposite sides fluctuated as shown in Fig. 10, and the stiffness coefficient was raised consistently as presented in Fig. 11. The increased stiffness coefficient improved the elastic force of the vapor spring and made it harder to be pressed by the pressure difference. It is easy to know that the size of the vapor spring in the reservoir reduces as the liquid ratio in the reservoir increases, which makes the stiffness coefficient increased. The inference of the liquid ratio is also discussed in other researches [32]. After the input power reached over 35 W, thestiffness coefficient increased sharply while the pressure difference decreased.In result, the position of the liquid column moved back to the condenser.