System dynamics model and startup behavior of loop heat pipe

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System dynamics model and startup behavior of loop heat pipe

B.J. Huang

*

_{, H.H. Huang, T.L. Liang}

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan

a r t i c l e

i n f o

Article history:

Received 28 February 2008 Accepted 21 March 2009 Available online 28 March 2009

Keywords: Loop heat pipes Startup of loop heat pipe Transient performance of a LHP

a b s t r a c t

The purpose of this article is to study the system dynamics model and startup behavior of loop heat pipe. A mathematical model in normal operation was derived from the experimental data. The model is pre-sented as a matrix of second-order transfer functions. It is found that the system dynamics of a LHP is a variable-structure system that changes with operating conditions. The startup phenomena are studied experimentally. The startup phenomena of a LHP can be classiﬁed into four modes, based on the heat load: (1) failure mode: _Qin< _Qmin, (2) oscillating mode: _Qmin< _Qin< _Qcrit, (3) overshoot mode:

_

Qcrit< _Qin< _Qs, and (4) normal mode: _Qs< _Qin. For heat load _Qin<40 Wð _QsÞ, the overshoot phenome-non is observed. For 20 Wð _QcritÞ < _Qin<40 Wð _QsÞ, the oscillation phenomenon is observed. For

_

Qin<5 Wð _QminÞ, the startup failure is observed. All the startup behavior is of second- order.

1. Introduction

A loop heat pipe (LHP) is a passive heat-transfer device that uses micro-scale wick technology which can transfer a large amount of heat over a long distance through a ﬂexible pipe and has a good anti-gravity performance[1].

Fig. 1shows the schematic of a LHP. A LHP consists of an evap-orator, a condenser, a compensation chamber, and liquid and vapor lines. The evaporator is made of a wick (capillary material) that is inserted in a case and has a liquid channel in the core. The liquid line directly connects to the compensation chamber and transfers working ﬂuid to the liquid channel of the wick. Heat is absorbed at the outer surface of the evaporator. There are several vapor channels (grooves) at the outer surface of the wick which admit va-por ﬂowing out of the evava-porator and into the vava-por line[2].

There is a startup condition for the LHP which requires a tem-perature difference across the wick structure according to the fol-lowing equation[3]:

D

Ptot

D

Pw¼ ðdP=dTÞðTe TccÞ ð1Þ

The relationship of Eq.(1)is important for minimal startup heat load and temperature overshoot. The startup of a LHP is a behav-iour of dynamic system that is very complex. The factors affecting the startup of a LHP include the design of compensation chamber and evaporator, the initial state of working ﬂuid in the evaporator, the charge volume of working ﬂuid, the attitude of LHP and the ini-tial state of a LHP before heat loads are applied. Two phenomena

were observed very often in the startup of a LHP: (1) temperature overshoot and (2) minimal heat load to startup[4].

The understanding of system dynamics behaviour of a LHP is important for predicting its performance during transient opera-tion, including startup. Experimental analysis of the transient per-formance at differently operating conditions is carried out in the present study.Table 1lists the design speciﬁcation of a LHP that is used in the present study.

The design of a LHP for the present experiment is shown in

Fig. 2. A heating block was attached to the bottom of the evapora-tor to provide the heat load for test. The power input of the heating block was controlled by an AC power supply. Two DC fans were used to cool the condenser plate in the experiment.

Six T-type thermocouples with uncertainty of ±0.5 °C were used to measure the temperatures. The thermocouples were connected to data logger (YOKOGAWA MV200) that transfers the data to a computer through ethernet at a rate of one recording every 1 s. An-other recorder WM-01 was used to measure the heating power every 1 s.

2. Derivation of system dynamics model of a LHP

From system control theory, the voltage of DC fan and the heat load are the controllable inputs. The system outputs are the temperature of the heating block (Tb) and the temperature of

the compensation chamber (Tcc). A LHP is thus a multiple-input–

multiple-output (MIMO) system, as shown in the block diagram ofFig. 3. Since a LHP is a nonlinear system, we derive the linearly perturbed model as an approximation, using perturbed variables from a steady state, e.g. ~GLHPðsÞ .

The four transfer functions of a LHP model can be identiﬁed sep-arately by isolation method. For example, ~G11ðsÞ is identiﬁed with a

* Corresponding author. Tel.: +886 2 23634790; fax: +886 2 23640549. E-mail address:[email protected](B.J. Huang).

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Applied Thermal Engineering

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constant fan speed, i.e. ~Vfan¼ 0, while applying a step input of ~Qin

and measuring the response of ~Tb. The operating conditions of the

experiments are shown inTable 2.

The parameters changed depend on the different operating points. The frequency response of ~G11ðsÞ at various operating points

is presented inFig. 4. The same method is applied to ~G21ðsÞ; ~G12ðsÞ;

and ~G22ðsÞ .

The identiﬁed parameters of ~G11ðsÞ; ~G21ðsÞ; ~G12ðsÞ and ~G22ðsÞ are

listed inTables 3 and 4. RSS is deﬁned as RSS ¼X

N t¼1

½TmeasðtÞ TcalðtÞ2 ð2Þ

where Tmeasis the measured temperature and Tcalis the calculated

temperature from the model.

Fig. 5shows the comparison between the average model and

the experimental data at the different operating points. We may use an average model to represent the dynamics models of LHP that are expressed in the following equations:

~ G11ðsÞ ¼ ~ TbðsÞ ~ QinðsÞ ¼ K1ðs þ Z1Þ ðs þ P1Þðs þ P2Þ ¼ 0:00448ðs þ 0:00151Þ ðs þ 0:00553Þðs þ 0:00138Þ ð3Þ ~ G21ðsÞ ¼ ~ TccðsÞ ~ QinðsÞ ¼ K1ðs þ Z1Þ ðs þ P1Þðs þ P2Þ¼ 0:00139ðs þ 0:00097Þ ðs þ 0:00458Þðs þ 0:00098Þ ð4Þ ~ G12ðsÞ ¼ ~ TbðsÞ ~ VfanðsÞ ¼ K2ðs þ Z1Þ ðs þ P1Þðs þ P2Þ ¼ 0:00693ðs þ 0:00093Þ ðs þ 0:005Þðs þ 0:00095Þ ð5Þ ~ G22ðsÞ ¼ ~ TccðsÞ ~ VfanðsÞ ¼ K2ðs þ Z1Þ ðs þ P1Þðs þ P2Þ¼ 0:01408ðs þ 0:00157Þ ðs þ 0:00945Þðs þ 0:00148Þ ð6Þ From the identiﬁed model (Eqs.(3)–(6)), LHP is a stable second-order system with one zero that can be used to predict the system dynamic behaviour of a LHP. It can be noted that a model reduction can be applied to ~G12and ~G21, since a pole and the zero are very

close. However, the system dynamics model of a plant has to be consistent for all of its components according to the fundamental concept of system science. Besides, it will be easier for a control system design if the system dynamics models of all the compo-nents are of the same form.

dP/dT the slope of the pressure–temperature, Pa/°C G Laplace transfer function

K gain

Mo the maximum overshoot, Mo¼ epn= ffiffiffiffiffiffiffiffi 1n2

p

P pole

_

Q heat transfer rate, W _

Qin head load applied to heating block, W RSS root sum square

Vfan voltage of fan, V

T temperature, °C

s complex variable in Laplace transform

t time, s

DPtot total pressure drop in a LHP, Pa

DPw pressure drop in wick, Pa

Z zero

Greek

x

n natural frequency, rad/s

n damping ratio Subscripts a ambient b heating block cc compensation chamber cal calculated

crit critical heat load

e evaporator

LHP loop heat pipe meas measured min minimal heat load s overshoot of heat load Over line

perturbation

Table 1

Speciﬁcation of a LHP used in the present study.

Evaporator Material Nickel powder

Capillary force 5 cm Hg

Porosity 60%

Pore size 5–10lm Outer diameter 14 mm Total length 90 mm Connecting tube Material Copper (C12200)

Outer diameter 4 mm Inner diameter 2.3 mm Condenser plate Material Copper (C12200)

Dimension 200 mm 300 mm 0.5 mm Cooling device A fan on top of the plate, with

adjustable fan speed by changing input voltage Heating block Material Aluminum alloy 6063

Dimension 45 mm 40 mm 20 mm Working ﬂuid Acetone

Compensation Chamber Wick Liquid Line Vapor Line Heating Block Condenser Evaporator Fig. 1. Schematic of a LHP.

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3. Startup behaviour of a LHP

The dynamic model identified above is obtained by analyzing the dynamic test data of a LHP which has been started success-fully. A successful startup is defined as the case that the tempera-ture of a LHP will reach steady state. However, the startup of a LHP may fail when the heat load is small. For a LHP used in the present study, the overshoot and the oscillation phenomena are observed at the heat load lower than 40 W. It is found that the startup phe-nomena of a LHP can be classified into four modes based on the heat loads:

(1) failure mode: _Qin< _Qmin;

(2) oscillating mode: _Qmin< _Qin< _Qcrit;

(3) overshoot mode: _Qcrit< _Qin< _Qs;

(4) normal mode: _Qs< _Qin.

The three heat loads characterizing the four modes ( _Qmin; _Qcrit; _Qs)

can be determined experimentally. 3.1. Failure mode: _Qin< _Qmin

The minimum heat load _Qminis deﬁned as a condition in which

a LHP can start up successfully. When the heat load is smaller than the minimum value ( _Qmin), the temperature at the evaporator will

continue to increase and cannot reach a steady state. The startup fails. The experimental result shows that the minimum heat load is 5 W as shown inFig. 6.

3.2. Oscillating mode: _Qmin< _Qin< _Qcrit

Startup experiments reveal that there is an oscillating phenom-enon for _Qmin< _Qin< _Qcrit. At this mode, the vapor ﬂow to the

con-denser is not stable but in oscillating or pulsating state. The test results ofFig. 7show that the temperature exhibits an oscillating phenomenon at _Qin¼ 20 W.

In this oscillating startup mode, the temperature of the evaporator starts to increase and the vapor starts to ﬂow into the condenser. The vapor speed response is always faster than the temperature response at the condenser. Due to the time lag of vapor condensation in the condenser, the temporary vapor accumulation in the condenser occurs and causes heat accumulation. This leads to a constant rise in the temperature of the evaporator. Once the vapor inside the condenser starts to condense, the sub-cooled liquid will start to return to the compensation chamber and the temperature of the evaporator will start to decrease. Consequently, the vapor ﬂow rate decreases due to the decrease in the temperature of the evaporator.

In this mode, the vapor flow is high enough to push the liquid segments existing in the connecting pipes and condenser forward. Since the vapor flow is not high enough due to low heat load, once the vapor flows to some distance, vapor condensation will create a retardation force to the flow. This results in the phenomenon of

Condenser

Evaporator

Compensation

Chamber

Heating block

s

G

)

(

~

s

Q

_in

)

(

~

s

T

b

)

(

~

s

T

cc ) ( ~ s Vfan 1

K

2

K

+ + + +

Fig. 3. The block diagram of the MIMO system.

Table 2

The operating conditions of the experiments.

Operating point Qin(W) Vfan(V)

1 60 ? 80 9 2 80 ? 100 9 3 100 ? 80 9 4 80 ? 60 9 5 80 6 ? 9 6 80 9 ? 12 7 80 12 ? 9 8 80 9 ? 6

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ﬂow oscillation or pulsation as well as temperature oscillation observed in the condenser as shown inFig. 7. From the test results ofFig. 7, we can derive a second-order system dynamics model, Eq.

(7), with oscillation period at 218 s and natural frequency 0.029 rad/s. ~ GðsÞ ¼ ~ TbðsÞ ~TaðsÞ ~ QinðsÞ ¼

x

2 n s2_þ

_x

2 n ¼ 0:029 2 ðs þ 0:029iÞðs 0:029iÞ ð7Þ It is shown that a LHP is an undamped system with two poles. 3.3. Overshoot mode: _Qcrit< _Qin< _Qs

As the heat load is higher than _Qcrit, a LHP enters into the

over-shoot mode startup and an overover-shoot of temperature is observed as shown inFig. 8.

In this mode, the vapor flow is higher than the oscillation mode and is able to push the liquid segments existing in the connecting pipes and condenser forward. A retardation force to the vapor flow is, however, generated at the beginning stage by liquid segments collision and merging with vapor condensation. The vapor pressure continues to rise with the heat load and finally reaches a critical value which is able to push the liquid to return to the evaporator to complete the fluid circulation along the loop. A stable flow then starts to develop. ~ GðsÞ ¼ð~Tb ~TaÞðsÞ _~ Qin ¼

x

2 n s2_{þ 2n}

_x

ns þ

x

2n ð8Þ According to the test results ofFig. 8, the maximum overshoot ðMo¼ epn=

ffiffiffiffiffiffiffiffi

1n2

p

Þ is 0.1127,which is the maximum peak value of the response curve, and the rise time is 139 s, which is the time -50 -40 -30 -20 -10 0

Magnitude (dB)

10-1 100 101 102 103 -90 -45 0

Phase (deg)

Frequency (rad/sec)

point2 point3 point4 average point1 point2 point3 point4 average

Fig. 4. Model comparison of ~G11ðsÞ at the different operating points.

Table 3

The identiﬁed parameters of ~G11ðsÞ and ~G21ðsÞ.

Model Operating point Heat load (W) RSS K1 Z1 P1 P2

~ G11ðsÞ 1 60 ? 80 5.8712 0.0041 0.0022 0.0054 0.002 2 80 ? 100 8.3497 0.0041 0.0018 0.0045 0.0016 3 100 ? 80 7.9801 0.0044 0.0095 0.0052 0.0009 4 80 ? 60 21.5688 0.0053 0.0011 0.007 0.001 ~ G21ðsÞ 1 60 ? 80 12.7164 0.00099 0.0008 0.0032 0.0009 2 80 ? 100 8.9805 0.000962 0.0014 0.003 0.0015 3 100 ? 80 10.3471 0.0015 0.0005 0.0045 0.0005 4 80 ? 60 18.5976 0.0021 0.0012 0.0076 0.001 Table 4

The identiﬁed parameters of ~G12ðsÞ and ~G22ðsÞ.

Model Operating point Voltage of fan (V) RSS K2 Z1 P1 P2

~ G12ðsÞ 5 6 ? 9 12.975 0.009 0.00088 0.0048 0.0008 6 9 ? 12 5.5623 0.004 0.00085 0.0032 0.001 7 12 ? 9 3.5448 0.0065 0.0011 0.0085 0.001 8 9 ? 6 9.4071 0.0082 0.0009 0.0035 0.001 ~ G22ðsÞ 5 6 ? 9 14.9401 0.023 0.00079 0.0102 0.0008 6 9 ? 12 8.3551 0.0075 0.0016 0.0055 0.0019 7 12 ? 9 4.079 0.0092 0.001 0.0108 0.0011 8 9 ? 6 4.9071 0.0166 0.0029 0.0113 0.0021

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required for a system step response to rise from 0% to 90%. The nat-ural frequency is thus 0.013 and the damping ratio (n) is 0.0573 and the poles are P ¼ n

x

n

x

n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2

p

i ¼ 0:00745 0:01065i. The system dynamics model of this mode is also of second-order which follows Eq.(8).

3.4. Normal mode: _Qs< _Qin

During the heat load _Qs< _Qin, a smooth startup is observed.

Smooth transient curves which indicate a rise in the temperature of the condenser are observed. This is supposed to be the normally operated mode of a LHP which is a second-order system with one zero.

FromFig. 9, it is seen that the system dynamics model identiﬁed previously, Eqs.(3)–(6), represents the normal startup mode per-formance of a LHP with _Qs< _Qin.

4. Conclusion

This paper studies the dynamic behaviour of a LHP. The system dynamics model of a LHP is derived from the experimental results and the startup behavior is studied. It is found that the system dynamics of a LHP is a variable-structure system that changes with the operating conditions. The system dynamics in normal mode can be described by the second-order transfer functions. It is also found that the startup phenomena of a LHP can be classiﬁed into four modes, based on the heat load: (1) failure mode: _Qin< _Qmin,

(2) oscillating mode: Q_min< _Qin< _Qcrit, (3) overshoot mode:

_

~

22

s

G

)

(

~

21

s

G

)

(

~

12

s

G

)

(

~

11

s

G

Fig. 5. Comparison of ~G11ðsÞ; ~G21ðsÞ; ~G12ðsÞ and ~G22ðsÞ between experiment and model.

0 500 1000 1500 2000 2500 3000 3500 4000 20 25 30 35 40 45

Time, sec

Temperature,

o

C

Heating Block Outlet of Evaporator Compensation Chamber Inlet of Condenser Ambient

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0 500 1000 1500 2000 2500 3000 3500 20 25 30 35 40 45

Time, sec

Temperature,

o

C

Outlet of Evaporator Compensation Chamber Inlet of Condenser Ambient Condenser

Fig. 7. Temperature response at _Qin¼ 20 W.

0 500 1000 1500 2000 2500 3000 3500 20 25 30 35 40 45 50 Time, sec

Temperature,

o

C

Heating Block Outlet of Evaporator Compensation Chamber Inlet of Condenser Ambient Condenser

Fig. 8. Temperature response at heat load _Qin¼ 30 W.

0 500 1000 1500 2000 2500 3000 3500 4000 20 40 60 80 100

Time, sec

T

emperature,

o

C

120W 110W 90W 100W 50W 80W 70W 60W 40W

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_

Qin<40 Wð _QsÞ, overshoot phenomenon is observed. For

20 Wð _QcritÞ < _Qin<40 Wð _QsÞ, oscillation phenomenon is observed.

For _Qin<5 Wð _QminÞ, the startup failure is observed.

References

[1] Yu.F. Maydanik, Review loop heat pipes, Applied Thermal Engineering 25 (2005) 635–657.

[2] R.R. Riehl, T. Dutra, Development of an experimental loop heat for applica-tion in future space missions, Applied Thermal Engineering 25 (2005) 101– 112.

[3] J. Ku, Operation Characteristics of Loop Heat Pipes, SAE-Society of Automotive Engineers, Paper # 1999-01-2007, 1999.

[4] H. Nagano, J. Ku, Start-up Behavior of a Miniature Loop Heat Pipe with Multiple Evaporators and Multiple Condensers, in: 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2007, AIAA 2007-1213.