Transient thermal performance analysis of micro heat pipes

Applied Thermal Engineering 58 (2013) 585e593

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Transient thermal performance analysis of micro heat pipes Xiangdong Liu, Yongping Chen* Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, PR China

h i g h l i g h t s
Transient thermal response of micro heat pipe is simulated by an improved model.
Control theory is introduced to quantify the thermal response of micro heat pipe.
Evaluation criteria are proposed to represent thermal response of micro heat pipe.
Effects of groove dimensions and working fluids on start-up of micro heat pipe are evaluated.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 February 2013 Accepted 22 April 2013 Available online 30 April 2013

A theoretical analysis of transient fluid flow and heat transfer in a triangular micro heat pipes (MHP) has been conducted to study the thermal response characteristics. By introducing the system identification theory, the quantitative evaluation of the MHP’s transient thermal performance is realized. The results indicate that the evaporation and condensation processes are both extended into the adiabatic section. During the start-up process, the capillary radius along axial direction of MHP decreases drastically while the liquid velocity increases quickly at the early transient stage and an approximately linear decrease in wall temperature arises along the axial direction. The MHP behaves as a first-order LTI control system with the constant input power as the ’step input’ and the evaporator wall temperature as the ’output’. Two corresponding evaluation criteria derived from the control theory, time constant and temperature constant, are able to quantitatively evaluate the thermal response speed and temperature level of MHP under start-up, which show that a larger triangular groove’s hydraulic diameter within 0.18e0.42 mm is able to accelerate the start-up and decrease the start-up temperature level of MHP. Additionally, the MHP starts up fastest using the fluid of ethanol and most slowly using the working fluid of methanol, and the start-up temperature reaches maximum level for acetone and minimum level for the methanol. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Micro heat pipe Transient thermal performance System identification theory

1. Introduction Efficient cooling of microelectronics is of significant importance for their safe operation and high performance [1,2]. Recently, due to the powerful heat transfer capability, small size and stable operation, the micro heat pipe (MHP) is introduced as a highly efficient thermal device to fulfill such electronic cooling. The concept of MHP is originally proposed by Cotter [3] as an efficient heat-transfer element in which the mean curvature of the vaporeliquid interface is of the same magnitude as the reciprocal of the hydraulic radius of the total flow groove. The typical MHP has a hydraulic diameter within the range of 10e500 mm and a length up to several centimeters, which is charged with the appropriate

* Corresponding author. Tel.: þ86 25 8379 2483. E-mail address: [email protected] (Y. Chen). 1359-4311/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.04.025

amount of working fluid [4]. Compared with the wick structures such as meshes or grooves in the conventional heat pipes [5e8], the sharp-angled corners of the micro-grooves in the MHPs provide the major capillary pumping pressure for driving the working fluid to circulate from the condenser back to the evaporator. Considerable experimental investigations have already been conducted to study the design, fabrication and actual performances of the MHPs [9e13]. Experimental studies of the MHPs with polygonal cross-section are also performed to investigate the heat transfer characteristics and heat transfer limit [9]. To increase the heat transfer capacity, MHPs are usually implemented in arrays of several tens. Berre et al. [11] experimentally demonstrated the feasibility of integrating high sensitivity thermistors in all-silicon MHP arrays which consist of 27 parallel triangular shaped grooves. The performances of this novel MHP arrays were experimentally tested with various methanol filling ratios and under various experimental conditions. Wu et al. [12] performed an

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X. Liu, Y. Chen / Applied Thermal Engineering 58 (2013) 585e593

experimental study on the transient thermal performance under start-up process and the rapid changes in the thermal load, and presented the unsteady temperature distribution throughout the longitudinal position as well as the temperature difference between axial locations on the heat pipe. Moreover, the experimental data were compared with the results of a developed analytical model to validate the reasonability of the model. In addition, a silicon MHP array with arteries was fabricated by Launay et al. [13] for increasing the liquid flow cross-sectional area so as to reduce the liquid pressure drop. Compared with a plain silicon wafer, the MHP with such configure realized a great improvement of effective thermal conductivity, which was experimentally verified. Besides the experimental investigations, a lot of numerical and analytical models have been developed to study the steady-state and transient thermal performance as well as the optimal geometric designs of MHPs [13e21]. Babin et al. [9] developed a steady-state, one-dimensional model for a single trapezoidal MHP to examine the effects of the extremely small characteristic dimensions on the conventional steady-state heat pipe modeling techniques. And then a steady-state experiment on copper and silver heat pipes was conducted to validate the theoretical model. Suman and Hoda [14] proposed a detailed simulation of a V-shaped MHP. The model studied the effects of operating parameters on the performance of the MHP, and predicted the dry-out length for different heat inputs. Furthermore, Suman [15] presented a model for fluid and heat transfer in an electrohydrodynamically (EHD) augmented MHP, in which the coulomb and dielectrophoretic forces were considered. It is found that the critical heat input increases and the dry-out length decreases with an increase in the electric field. And the numerical solutions were successfully compared with the experimental results. Khrustalev and Faghri [16] also developed a detailed mathematical model of a MHP with a triangular shaped groove. The model demonstrated that the liquid filling ratio, minimum wetting contact angle, and the shear stresses at the liquidevapor interface play important roles in predicting the maximum heat transfer capacity and thermal resistance. Based on a one-dimensional steady-state model, Qu et al. [17] analyzed the effect of a functional surface with the axial ladder contact angle distribution on the thermal performance of a triangular MHP. The simulation results showed that compared with the traditional MHP with the surface possessing a uniform contact angle distribution, a MHP with a functional surface has a better effective thermal conductivity under the same condition. Sobhan et al. [18] presented a numerical model of the vapor and liquid flow in a micro heat pipe with triangular grooves. The distributions of velocity, pressure, and temperature of the vapor and liquid at transient and steady states were obtained. Suman et al. [19] developed a one-dimensional transient model for fluid flow and heat transfer of heat pipes with axially triangular microgrooves which predicted the steadystate wall temperature distribution. And an improved model considering the shear stress at liquidevapor interface, disjoining pressure and sensible heat of the solid substrate was also presented [20]. To study the effects of geometric design on the thermal performance of a star-groove MHP, a mathematical one-dimensional, steady-state model was developed by Hung et al. [22]. This model can be used to evaluate the heat transport capacity and the corresponding optimal charge level of the working fluid for different geometric designs and operating conditions. The results showed that, with increasing number of corners, the performance of MHP deteriorates. And it is also observed that the increase in the total length of the MHP results in a decrease in its heat transport capacity. Despite there have been a great deal of investigations on MHPs, the transient thermal and hydrodynamic performance of MHPs have not been studied enough. The available theoretical modeling

of the transient pressure and temperature distributions in MHPs [18,19] only consider the coupled heat and mass transfer processes between gas and liquid [18] or solid and liquid [19] rather than the whole coupled processes of heat and mass transfer among all the gas, liquid and solid in MHPs. Additionally, the expression in the available models [18,19] for evaluating the mass flux during the evaporation and condensation is not sufficiently valid [23], especially under the condition of low mass transfer rate. In particular, it is still lack of efficient tools to quantitatively evaluate the transient performance of MHPs. Therefore, the current work develops a transient model and numerically analyzes the transient thermal and hydrodynamic performance of a micro heat pipe with triangular shaped grooves, taking consideration of the whole heat and mass transfer processes among gas, liquid and solid as well as the axial heat transfer in the wall of heat pipe. The calculation method for evaluating the mass flux during the evaporation and condensation is also improved in the model. The transient wall temperature profiles along the MHP, and the unsteady distribution of the capillary radius and liquid velocity in the MHP are presented and discussed. Especially, inspired by the introductions of control theory to realize the quantitative evaluations of transient characteristics (e.g. transient response speed, stability, etc.) of thermodynamic problems such as pool boiling [24], thermal-fluid flow in vascular networks [25] and so on, the system identification theory (an important part of control theory) is introduced here to quantitatively evaluate the transient performance of MHPs. 2. Mathematical model Here, an improved model is developed to numerically simulate the transient performance of MHPs. The model couples heat and mass transfer processes among all the gas, liquid and solid as well as the axial heat transfer in the heat pipe wall. The Kucherove Rikenglaz equation [23] is used to evaluate the mass flux at the vaporeliquid interface. The schematic of the MHP including evaporator section, adiabatic section and condenser section is shown in Fig. 1. A constant heat flux is applied, qin ¼ Qin/Ae (Qin is the input power, Ae is the outer surface area of evaporator section) and the condenser is subjected to convective cooling with a constant temperature, Tcool. The working fluid is evaporated on the evaporator side and condensed on the condenser side, and then returns to the

Convective cooling

Z

Evaporator

Adiabatic

Condenser

qin Condenser L

z

R(z) Vapor

Adiabatic La

d

Evaporator

θ

w

Le

Solid wafer Liquid A-A Fig. 1. Schematic of micro heat pipe with triangular shaped grooves.

X. Liu, Y. Chen / Applied Thermal Engineering 58 (2013) 585e593

evaporator section by the capillary force generated in the sharpangled corners. Table 1 presents the geometry parameters for the investigated micro heat pipe with triangular shaped groove. 2.1. Governing equations 2.1.1. Heat transfer in the solid wall In order to simplify the calculation, lateral heat transport in the heat pipe wall is neglected since it is insignificant compared with the axial heat transport in heat pipe wall. Both the imposed heat (Qin) in the evaporator section and the heat released to cooling fluid in the condenser section are regarded as an inner heat source of solid wall. Therefore, the unsteady energy balance equation of the heat pipe wall can be expressed as

8 hl ðTs  Tl Þwb þ qin lout vTs v2 Ts < rs Cp;s Acs ¼ Acs ks 2 þ hl ðTs  Tl Þwb : h ðT  T Þw  jh l  h vt vz l

s

l

b

fg f

out ðTs


j ¼

b 2s b 2s




M 2pRu

1=2

Pl;sat Pv pffiffiffiffiffi  pffiffiffiffi Tv Tl


 1 1 Pc ¼ Pv  Pl ¼ s þ Rc1 Rc2

(3)

where Rc1 and Rc2 are the radial and axial capillary radius of the meniscus, respectively. Due to that Rc2 is much larger than Rc1, Rc2 can be neglected and the radial capillary radius Rc1 can be assumed as the capillary radius R. Assuming that the vapor pressure keeps constant along the axial direction, the differential form of Eq. (3) can be rewritten as

s dR dPl ¼ 2 dz R dz

(4)

For laminar flow, the liquid pressure distribution along the stream can be calculated by

 Tcool Þlout

where rs is the solid density, Cp,s is the specific heat of the solid, Acs is the cross-sectional area of solid, Ts is the solid wall temperature of heat pipe, Tl is the temperature of liquid in grooves, Tcool is the temperature of cooling water, t is time, ks is the thermal conductivity of solid wall, lf is the fin width, lout is the perimeter of evaporator outer wall, wb is the wetted perimeter of the groove, hl is the convective heat transfer coefficient of the liquid flow in the grooves, hfg is the latent heat and j is the mass flux rate of condensation. Based on the perspective of kinetic theory, Carbajal et al. [23] suggested that the mass flux during evaporation from, and condensation on, the vaporeliquid interface can be evaluated in the form as the KucheroveRikenglaz equation. The expression of the free molecular flow mass flux rate of condensation is [26e30]

587

0  z < Le Le  z < Le þ La Le þ La  z  Le þ La þ Lc

(1)

_ l ðzÞ dPl 2m ðfReÞm ¼  l dz rl Al D2

(5)

h

where rl is the liquid density, ml is liquid dynamic viscosity, fRe is the Poiseuille number. And Dh is the hydraulic diameter, Al is the cross-sectional area of liquid. Combining Eqs. (4)e(5) yields an ordinary differential equation for capillary radius R as

_ l ðzÞ 2 dR 2ml ðfReÞm R ¼ dz Al rl D2h s

(6)

_ l ðzÞ is the mass flow rate of liquid at location z, the where m _ l ðzÞ is described as determination of m

! (2)

s is the accommodation factor, M is the molecular weight, Ru where b is the universal gas constant, Pl,sat is the saturated pressure of the corresponding liquid temperature, Pv is the vapor pressure. 2.1.2. Capillary flow in corners of groove In micro heat pipes, the liquid flows in the corners and vapor flows in the vapor core in a counter-current fashion. The local capillary radius at the liquidevapor interface to be used is calculated using the LaplaceeYoung equation

Table 1 Specification of micro heat pipe. Parameters

Value

Number of grooves, n Vapor core diameter, dv/mm Side length of the groove, w/mm Apex angle, b/degree Evaporator length, Le/mm Adiabatic length, La/mm Condenser length, Lc/mm Initial temperature, TN/ C

27 0.8485 0.3, 0.5, 0.7 71 5 15 5 40

_ l ðzÞ ¼ m

Zz

Zz jRm dz ¼

0

jðp  qÞRdz

(7)

0

where Rm is the meniscus surface area per unit length of the heat pipe, q is the apex angle. 2.1.3. Heat and fluid flow in sharp-angled corners The evaporation phase change occurs at the liquidevapor interface while the condensation of vapor occurs at the liquide vapor interface and it also takes place on the thin liquid film. And in the adiabatic section, no mass transfer occurs at the liquidevapor interface. So the transient mass balance can be expressed as

8 vððrl Al Vl Þ > >  jRm > > vz > > > < vðrl Al Þ vðrl Al Vl Þ ¼ > vt vz > > > >   > r vð > l Al Vl Þ :  j lf þ Rm vz

0  z < Le Le  z < Le þ La Le þ La  z  Le þ La þ Lc (8)

588

X. Liu, Y. Chen / Applied Thermal Engineering 58 (2013) 585e593

where Vl is the liquid velocity, jRm represents the mass of the liquid evaporated, and j (lf þ Rm) represents the mass of the liquid condensed. For the liquid flow within the microgrooves, the unsteady-state momentum balance of liquid can be expressed as

vðrl Al Vl Þ vV vP ¼ rl Al Vl l þ Al l  wb sw  Rm slv vt vz vz

vTl v2 T vT ¼ Al kl 2l þ rl Cp;l Vl Al l þ hl ðTs  Tl Þwb  jhfg Rm vt vz vz (10)

where jhfgRm represents the heat current induced by condensation/ evaporation at the liquidevapor interface. 2.1.4. Vapor flow in vapor core For the temperatureepressure relationship in the vapor, the ideal gas equation of state is introduced:

Pv ¼ rv Ru Tv =M

(11)

In general, the vapor in the vapor core is at the saturation condition, so the vapor temperature can be evaluated by the corresponding vapor pressure which is a function of vapor density and temperature as presented by Eq. (11), hence,

Tv ¼ Tðrv Þ

At t ¼ 0; Ts ¼ TN ; Tl ¼ Tr ; Vl ¼ 0; Pl ¼ Pv0 

sl R0

; R ¼ R0 ; for all z: (17)

(9)

where sw is the wall shear stress, and slv is the shear stress on the liquidevapor interface. The unsteady energy conservation of the liquid flow in the microgrooves for the evaporator, adiabatic and condenser section can be expressed in the same format

rl Cp;l Al

location of the channel is equal to the radius of vapor core (R0 ¼ rv). In addition, the temperature of the whole heat pipe is the same as the ambient temperature, Tr. Therefore, the initial conditions:

2.3. Numerical solution Eqs. (1) and (10) are a set of partial differential equations (PDEs), which can be solved by means of control volume finite-difference technique to obtain the axial distribution of solid wall temperature and liquid temperature. The ordinary differential equation Eq. (6) governing the capillary flow is solved by using fourth-order RungeeKutta method. The thermophysical properties of the solid wall are assumed to be constant, but those of the working fluid change with the temperature in the grooves. The calculation procedure can be summarized as follows: (1) Eqs. (1) and (10) are discretized first, and then solved by using GausseSeidel method to determine the transient distribution of axial temperature for both the solid wafer and liquid in the groove; (2) Calculate the evaporation/condensation mass flux rate by Eq. (2); (3) From the liquid continuity equations, Eq. (8), and momentum equation, Eq. (9), obtain the transient distribution of axial liquid velocity; (4) Use fourth-order RungeeKutta method to calculate axial distribution of capillary radius by Eq. (13);

(12)

The vapor volume is assumed to be unchanged during the transient process, so the variation of rv is induced by the change of vapor mass mv

drv 1 dmv ¼ Vv dt dt

(a) 2.0

(13)

1.5

mv ¼ m0 þ Dme  Dmc

ΔTe / K

Δx = Lt / 160

where Vv is the volume occupied by vapor. It is assumed that the initial vapor mass as m0, the expression of the vapor mass at any subsequent instant of time t ¼ t0 þ Dt can be presented as

Δx = Lt / 250

1.0

Δx = Lt / 320

0.5

Δx = Lt / 400

(14)

where Dme is the increment in the vapor mass due to liquid evaporation, Dmc is the mass of the vapor condensed.

[12]

Experimental data

0.0

(b) 2.0

2.2. Boundary and initial conditions 1.5

The boundary conditions: At the evaporator end (z ¼ 0):

for all t:

(15)

1.0

s

-5

s

-6

s

-6

s

Δt = 1 × 10 Δt = 5 × 10

0.5

At the condenser end (z ¼ L):

  s vTs  vTl  R ¼ R0 ; Pl ¼ Pv0  l ; ¼ 0; ¼ 0; R0 vz z¼L vz z¼L

ΔTe / K

  vTs  vTl  ¼ 0; ¼ 0; Vl ¼ 0; vz z¼0 vz z¼0

-5

Δt = 5 × 10

Δt = 1 × 10

for all t: (16)

At the initial stage (t ¼ 0), there is no flow in the sharp-angled channels, the effective capillary pressure is equivalent to the static capillary pressure, i.e. the capillary radius at the arbitrary

[12]

Experimental data

0.0 0

25

50

75

100

t/s Fig. 2. Convergence tests and experimental validation: (a) grid independence test (Dt ¼ 5  106 s), and (b) time step independence test (Dx ¼ Lt/320).

X. Liu, Y. Chen / Applied Thermal Engineering 58 (2013) 585e593

Condenser

Adiabatic

Evaporator

Adiabatic

Evaporator

5

1.44335

2 - t = 0.5 s

4

3-t=1s

60

1.44320 2

o

2 - t = 0.5 s 4

2 50

3-t=1s

1

4-t=2s

5

5 - steady state

55

1 - t = 0.2 s

3

1.44290

4-t=2s

3 Ts / C

1.44305

Condenser

1 - t = 0.2 s

65

1

R / mm

589

45

5 - steady state

1.44275 0

5

10 z / mm

15

20

40 0

5

10

15

20

z / mm

Fig. 3. Transient distribution of the capillary radius. Fig. 5. Transient wall temperature profiles.

(5) The time stepping is continued until the thermal performance of the MHP reaches the steady state. 2.4. Convergence study Herein, a spatial and temporal convergence study is conducted to ensure that the main results of current study are insensitive to the computational parameters choices. As shown in Fig. 2, the numerical results on the time-domain evaporator temperature difference of the MHP, DTe ¼ Te,t  Te,N, are compared under various grid sizes (Dx ¼ Lt/160, Lt/250, Lt/320, Lt/400) and time steps (Dt ¼ 5  105 s, 1  105 s, 5  106 s, 1  106 s), where Te,t is the temperature of evaporator end at time of t and Te,N is the initial temperature of evaporator end. It is indicated that the numerical results of the time-domain evaporator temperature difference of the MHP (DTe) converge with grid refinement from Lt/ 160 to Lt/400. In addition, with the decrease in time step from 5  105 s to 1  106 s, the results become insensitive to the choice of time step. Therefore, based on an overall consideration of computational efficiency and the accuracy of calculation results, the grid size of Lt/320 and the time step of 5  106 s is adopted for the current study.

Adiabatic

Evaporator

5

3-t=1s

-3

Vl / 10 m/s

4-t=2s

    1 X DTe; exp  DTe; pred  MAEðDTe Þ ¼  100% DTe; exp n

(18)

is 18.09% for the number of data n ¼ 55. This acceptable agreement between the numerical results and experimental data verifies that the present model is reasonable. 3. Results and discussion Based on the presented mathematical model, the transient performance of the silicon MHP with triangular shaped grooves is numerically simulated, and its detailed geometry parameters are listed in Table 1. The transient performance starts once the input power, Qin ¼ 6 W, is imposed on the evaporator section, and the condenser section of heat pipe is cooled by water with a constant temperature of Tcool ¼ 40  C.

c(t)

2 - t = 0.5 s

0.12

The present mathematical model is validated by the transient experimental investigation of MHP by Wu et al. [12]. Both numerical results and the corresponding experimental ones on the DTe are compared in Fig. 2. In addition, the total mean absolute error (MAE) for the numerical prediction, defined as [31]

Condenser

1 - t = 0.2 s 5 - steady state

0.15

2.5. Experimental validation

1.000

4

0.09

3

0.06

0.632

2 63.2%

0.03

1

0.00 0

5

10 z / mm

15

Fig. 4. Transient distribution of the liquid velocity.

20

0

tp

t

Fig. 6. Typical response for a typical first-order LTI control system to a unit step input [31].

590

X. Liu, Y. Chen / Applied Thermal Engineering 58 (2013) 585e593

w = 0.3 mm 66

w = 0.5 mm w = 0.7 mm

3.2. Transient performance 54

T

es

o

/ C

60

the wall, an approximate linear decrease in the wall temperature arises along the axial direction. In addition, the wall temperature rises drastically during the initial transient period. As the process continues, the rise of wall temperature becomes smaller.

48

Tes - numerical results 42

Tes- control theroy 0

3

6

12

9

t /s Fig. 7. Effect of groove dimension on the transient start-up performance of MHPs.

3.1. Dynamic thermal response of MHP under uniform constant input power 3.1.1. Capillary flow The fluid flow inside MHP is driven by the capillary force generated in the sharp-angled corners. The axial distribution of capillary radius plays an important role in the transient performance of MHPs. Fig. 3 represents the transient distributions of the capillary radius along the axial direction. As shown in the figure, there is a drastic decrease of capillary radius along the axial direction at the early transient period and the radius of curvature along the axial direction is monotonically decreased from the cold end to the hot one. Therefore, the rapid response of capillary radius generates enough driving force to push the liquid from the cold end to the hot one during the start-up operation. In addition, the decrease trend of capillary radius slows down as the process continues. The axial distribution of the liquid velocity at various times is shown in Fig. 4. As shown, the liquid is first accelerated and decelerated later from the cold end to the hot end for both transient and steady states. There is a steep increase in velocity from condenser section to adiabatic one due to the mass addition in the liquid caused by condensation. Furthermore, it is interesting to note that the maximum value of liquid velocity occurs in the adiabatic section, which implies that condensation is extended into the adiabatic section due to the heat conduction along the wall. Likewise, the evaporation-induced decrease of liquid velocity starts in the adiabatic section where the external heating is provided. 3.1.2. Temperature distribution The transient wall temperature distribution along the axial direction is the primary indicator of thermal response performance for a MHP [23]. Fig. 5 shows the transient axial distributions of wall temperature on the flat micro heat pipes. As shown, during the whole start-up process, there is a quick wall temperature rise in the evaporator section while the wall temperature increases gently in the condenser section. Since the evaporation and condensation extend into the adiabatic section, as well as the heat conduction of

In control theory [32,33], several evaluation methods have been proposed to quantitatively analyze the transient characteristics of a control system, such as the response speed, relative stability, etc. In the current study, it is interesting to note that the transient thermal response of MHP can be regarded as the dynamic response of a ’control system’ (i.e. MHP here) corresponding to the ’step input’ (i.e. input power here). Therefore, the control theory is introduced here to quantitatively analyze the transient start-up performance of MHPs. Since the evaporator wall temperature is the most important characteristic parameter which determines whether the working temperature of MHP satisfies the safe working temperature of the cooled substrate [27e30], it is regarded as a representative ’output’ of the ’control system’ with respect to the ’step input’. In addition, as shown in Figs. 6 and 7, an analogy is made between the transient responses of evaporator wall temperature in MHP and that of a typical first-order LTI (linear time-invariant) system corresponding to a step input [32]. Then, in order to validate such an analogy, the system identification theory (an important part of control theory) [33] is utilized here. According to the system identification theory, the transfer function for the typical first-order LTI system is introduced to obtain its typical transient responses corresponding to a step input which is expressed as

HðSÞ ¼

YðSÞ 1 ¼ kp $ XðSÞ 1 þ tp S

(19)

where X(S) and Y(S) are correspondingly Laplace transforms of the input signals x(t) (here input power, Qin(t)) and output signal y(t) (here average evaporator wall temperature, T es ), S is a complex variable, kp is the gain of transfer function, tp is the time constant of first-order inertia link. Accordingly, based on the transient numerical simulation results of T es during the start-up process of MHPs with different groove dimension, the coefficients of Eq. (19), kp and tp, corresponding to the cases illustrated in Fig. 7, are determined by the least square method, which are listed in Table 2. According to the obtained transfer function, the dynamic responses of the ’output’ (evaporator wall temperature) defined by control theory are determined and shown in Fig. 7. In addition, the root mean square error (RMSE) is utilized to quantitatively measure the difference between the control theoretical values of ’output’ ðb x i Þ and the numerical simulation results of the ’output’ (xi), which is computed from [34]

E ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP un 2 u ðb x i  xi Þ t i n

ði ¼ 1; 2; 3:::Þ

(20)

Table 2 Coefficients of transfer functions, corresponding root mean square errors (RMSEs), and some evaluation criteria for start up performance under different groove dimension (methanol). Transfer functions

w/mm

Dh/mm

kp

tp/s

E

T0/ C

DT sd ¼ ðT es;sd  T cs;sd Þ= C

1 kp $ 1 þ tp S

0.3 0.5 0.7

0.18 0.30 0.42

4.435 3.981 3.733

0.9823 0.9681 0.9434

0.2685 0.4424 0.7954

57.1 55.3 54.3

25.5 23.5 21.7

X. Liu, Y. Chen / Applied Thermal Engineering 58 (2013) 585e593

Furthermore, a larger temperature difference between evaporator and condenser (DT sd as shown in Table 2) is needed for a smaller groove to transfer the same amount of heat due to a larger friction loss under the same amount of evaporation. In the same way, the numerically simulated start-up thermal responses of the MHPs with different working fluids and the corresponding values of evaluation criteria are obtained, which are indicated in Fig. 8 and Table 3, respectively. As shown, under the present investigated conditions, the MHP starts up fastest with the fluid of ethanol and most slowly by using the working fluid of methanol. In addition, the start-up temperature reaches the maximum level for the working fluid of acetone and minimum level for the methanol.

66

o

T es / C

60

54

48

42

0

Acetone

Ethanol

Methanol

Ammonia

3

6 t/s

9

591

12

4. Conclusions

Fig. 8. Effect of working fluid on the transient start-up performance of MHPs.

where n is sample size (here n ¼ 1000), and they are also given in Table 2. Based on the aforementioned method, the maximum RMSE is obtained as 0.7954  C for current study, implying that the MHP can be regarded as a ‘first-order LTI system’ with uniform constant input power as its ’step input’ and evaporator wall temperature as its ’output’. Therefore, two evaluation criteria for evaluating the dynamic characteristics of first-order LTI system are introduced to quantitatively evaluate the start-up performance of MHPs: (a) Time constant (tp). Time constant is the main characteristic unit of a first-order LTI system, which can be modeled by a single first order differential equation in time. In this paper, it is defined as the time from the application of input power till when the rise magnitude of mean wall temperature of evaporator reaches 63.2% of temperature difference between its initial temperature (TN) and the final steady temperature ðT es;sd Þ as shown in Fig. 6. It is utilized to valuate the thermal response speed of MHP during the start-up operation with a constant input power suddenly applied. (b) Temperature constant (T0). Temperature constant is defined as the mean wall temperature of the evaporator corresponding to the time constant tp as indicated in Fig. 6. It is a valuable reference to reflect the temperature level of MHP under startup operation. Accordingly, the evaluation criteria included in Table 2 enable the quantitative evaluation of effects of groove dimension on transient start-up performance of the MHPs, which show that the small groove leads to slow start up and large start-up temperature level within the w ¼ 0.3e0.7 mm (i.e. hydraulic diameter Dh ¼ 0.18e0.42 mm) under the same filling ratio and input power. It is understandable that a small groove induces large friction loss and capillary pumping pressure for working fluid to circulate in the grooves. Therefore, the above results illustrate that the increase of friction loss owning to the smaller groove is more effective than that of driving capillary for slowing down the start-up process.

Table 3 Coefficients of transfer functions, corresponding root mean square errors (RMSEs), and some evaluation criteria for start up performance under different working fluids (w ¼ 0.5 mm). Transfer functions

Working fluid

kp

tp/s

E

T0/ C

1 kp $ 1 þ tp S

Methanol Ethanol Acetone Ammonia

3.981 4.169 4.355 3.903

0.9681 0.5377 0.6331 0.7026

0.4424 0.2909 0.4522 0.5549

55.3 55.7 56.5 54.1

A numerical simulations is conducted to investigate the thermal response of a triangular shaped micro heat pipe based a developed transient model. Additionally, the system identification theory is introduced into the quantitative analysis of the transient thermal performance of MHP. The effects of groove dimension and working fluid on transient start-up performance are discussed. The conclusions can be summarized as: (1) During the initial start-up stage, the capillary radius along axial direction of MHP decreases drastically while the liquid velocity increases quickly. As the process continues, the changing trend of these two parameters slows down. (2) The evaporation and condensation processes are both extended into the adiabatic section rather than restricted to the evaporator and condenser sections due to the heat conduction along the wall. (3) During the whole start-up process, there is a quick rise in the evaporator in contrast to the gradual wall temperature rising in the condenser section. Since the evaporation and condensation extend into the adiabatic section, as well as the heat conduction of the wall, an approximative linear decrease in the wall temperature arises along the axial direction. (4) For the start-up operation, the MHP behaves as a first-order LTI control system with the constant input power as the ’step input’ and the evaporator wall temperature as the ’output’. Two corresponding dynamic performance evaluation criteria derived from the control theory, time constant and temperature constant, are able to quantitatively evaluate the thermal response speed and temperature level of MHP during the startup process. It is indicated that a larger triangular groove’s hydraulic diameter within 0.18e0.42 mm is able to accelerate the start-up and decrease the start-up temperature level of MHP. In addition, the MHP starts up fastest using the fluid of ethanol and most slowly using the working fluid of methanol, as well as the start-up temperature reaches maximum level for the working fluid of acetone and minimum level for the methanol. Acknowledgements The authors gratefully acknowledge the supports provided by National Natural Science Foundation of China No. 51222605, No. 11190015 and Research Fund for the Doctoral Program of Higher Education of China No. 20110092110049. Nomenclature Acs Ae Al

cross-sectional area of solid, m2 outer surface area of evaporator section, m2 cross-sectional area of liquid in the groove, m2

592

c(t) Cp,l Cp,s dv Dh E fRe hfg hl hout H(S) j kl kp ks lout lf L La Lc Le Lt m m0 Dmc Dme _ l ðzÞ m mv M MAE n Pl,sat Pc Pl Pv Pv0 qin Qin rv R R0 Rc1 Rc2 Rm Ru S t tp DTe T0 Tcool T es Tl Tr Ts Tv TN Vl Vv w wb xi

X. Liu, Y. Chen / Applied Thermal Engineering 58 (2013) 585e593

typical dynamic value versus time of a control system specific heat of liquid, J kg1 specific heat of solid, J kg1 vapor core diameter, m hydraulic diameter, m root mean square error (RMSE) Poiseuille number latent heat, J/kg convective heat transfer coefficient of the liquid flow in the grooves, W m1 convective heat transfer coefficient of the outer wall of condenser, W m1 transfer functions for typical first-order LTI system mass flux rate of evaporation/condensation, kg m2 s1 thermal conductivity of liquid, W m1 gain of transfer function thermal conductivity of solid, W m1 perimeter of evaporator outer wall, m length of thin liquid film, m heat pipe length, m adiabatic length, m condenser length, m evaporator length, m total length of MHPs, m number of grooves vapor mass under initial condition, kg mass of the vapor condensed, kg mass of the liquid evaporated, kg mass flow rate of liquid at location z, kg/s vapor mass, kg molecular weight, kg/mol mean absolute error, % sample size saturated liquid pressure for the corresponding liquid temperature, Pa capillary pressure, Pa liquid pressure, Pa vapor pressure, Pa initial vapor pressure, Pa input heat flux, W/m2 input power, W vapor core radius, m capillary radius, m initial capillary radius, m radial capillary radius of the meniscus, m axial capillary radius of the meniscus, m meniscus surface area per unit length, m universal gas constant, J kg1 mol1 complex variable time, s time constant of first-order inertia link, s time-domain evaporator temperature difference,  C temperature constant,  C temperature of cooling water,  C mean wall temperature of the evaporator,  C temperature of liquid in grooves,  C ambient temperature,  C solid wall temperature of heat pipe,  C vapor temperature,  C initial temperature of whole heat pipe,  C liquid velocity, m/s vapor volume, m3 side length of the groove, m wetted perimeter, m values of numerical results being estimated

b xi

Dx X(S) Y(S) z

values of control theory being estimated grid size, mm the transforms of the input signals the transforms of the output signals axial coordinate, m

Greek symbols b apex angle,  q included angle between solid and liquid, rad ml liquid dynamic viscosity, Pa s rl liquid density, kg/m3 rs solid density, kg/m3 rv vapor density, kg/m3 s surface tension coefficient, N/m sw wall shear stress, N/m2 slv shear stress on the liquidevapor interface, N/m2

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