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ARTICLE IN PRESSG ModelOCS-249; No. of Pages 7

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Contents lists available at ScienceDirect

Journal of Computational Science

journa l h om epage: www.elsev ier .com/ locate / jocs

athematical modelling and parameter optimization of pulsatingeat pipes

in-She Yanga,∗, Mehmet Karamanoglua, Tao Luanb, Slawomir Koziel c

School of Science and Technology, Middlesex University, London NW4 4BT, UKSchool of Energy and Power Engineering, Shandong University, Jinan, ChinaSchool of Science and Engineering, Reykjavik University, IS-103 Reykjavik, Iceland

r t i c l e i n f o

rticle history:eceived 4 October 2012eceived in revised form5 November 2013ccepted 13 December 2013vailable online xxx

eywords:symptotic

a b s t r a c t

Proper heat transfer management is important to key electronic components in microelectronic appli-cations. Pulsating heat pipes (PHP) can be an efficient solution to such heat transfer problems. However,mathematical modelling of a PHP system is still very challenging, due to the complexity and multiphysicsnature of the system. In this work, we present a simplified, two-phase heat transfer model, and our anal-ysis shows that it can make good predictions about startup characteristics. Furthermore, by consideringparameter estimation as a nonlinear constrained optimization problem, we have used the firefly algo-rithm to find parameter estimates efficiently. We have also demonstrated that it is possible to obtaingood estimates of key parameters using very limited experimental data.

irefly algorithmeat transferathematical modellingptimizationscillations

© 2013 Elsevier B.V. All rights reserved.

ulsating heat pipewo-phase flow

. Introduction

A pulsating heat pipe is essentially a small pipe filled with bothiquid and vapour and the internal diameter of the heat pipe is athe capillary scale [1,2]. The liquids in the pipe can form segmentsr plugs, between vapour segments. When encountering heat, partf the liquid many evaporate and absorb some heat, thus causing aifferential pressure and driving the movement of the plugs. Whenapour bubbles meet a cold region, some of the vapour may con-ensate, and thus releasing some heat. The loop can be open orlosed, depending on the type of applications and design. This con-inuous loop and process will form an efficient cooling system ifesigned and managed properly for a given task. Therefore, suchystems have been applied to many applications in heat exchanger,pace applications and electronics, and they can potentially haveven wider applications [43]. On the other hand, the emergencef nanotechnology and the steady increase of the density of thearge-scale integrated circuits have attracted strong interests in

Please cite this article in press as: X.-S. Yang, et al., Mathematical mComput. Sci. (2014), http://dx.doi.org/10.1016/j.jocs.2013.12.003

odelling heat transfer at very small scales, and the heat manage-ent of microdevices has become increasingly important for next

eneration electronics and miniaturization.

∗ Corresponding author. Tel.: +44 2084112351.E-mail address: x.yang@mdx.ac.uk (X.-S. Yang).

877-7503/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.jocs.2013.12.003

Both loop heat pipes (LHP) and pulsating heat pipes (PHP)may provide a promising solution to such challenging prob-lems, and thus have attracted renewed attention in recent years[1,4,7,12,13,24,25,42]. In many microelectronic applications, con-ventional solutions to heat management problems often use fans,heat exchangers and even water cooling. For examples, the fan-driven air circulation system in desktop computers and manylaptops have many drawbacks such as bulky sizes and poten-tial failure of mechanical, moving parts. In contrast, loop heatpipes have no mechanical driving system, and heat circulationis carried out through the pipes, and thus such LHP systemscan be very robust and long-lasting. In addition, miniaturizationand high-performance heat pipes systems are being developed[11,15,18,26,39,43]. Simulation tools and multiphase models havebeen investigated [16,17]. All these studies suggested that LHP sys-tems can have some advantages over traditional cooling systems.

A PHP system may often look seemingly simple; however,its working mechanisms are relatively complex, as such systemsinvolve multiphysics processes such as thermo-hydrodynamics,two-phase flow, capillary actions, phase changes and others.Therefore, many challenging issues still remain unsatisfactorily

odelling and parameter optimization of pulsating heat pipes, J.

modelled. There are quite a few attempts in the literature to modela PHP system with various degrees of approximation and success.

In this paper, we will use a mathematical model based on oneof the best models [24,25], and will carry out some mathematical

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ARTICLEOCS-249; No. of Pages 7

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nalysis and highlight the key issues in the state of the art models.n this paper, we intend to achieve two goals: to present a simpli-ed mathematical model which can reproduce most characteristicsf known physics, and to provide a framework for estimating keyarameters from a limited number of measurements. The rest of theaper is organized as follows: we first discuss briefly design opti-ization and metaheuristic algorithms such as firefly algorithm,e then outline the main multiphysics processes in the mathemat-

cal models. We then solve the simplified model numerically andompared with experimental data drawn from the literature. Afteruch theoretical analysis, we then form the parameter estimations an optimization problem and solve it using the efficient fireflylgorithm for inversely estimating key parameters in a PHP system.inally, we highlight the key issues and discuss possible directionsor further research.

. Design optimization of heat pipes

The proper design of pulsating heat pipes is important, so thathe heat in the system of concern can be transferred most effi-iently. This also helps to produce designs that use the least amountf materials and thus cost much less, while lasting longer withouthe deterioration in performance. Such design tasks are very chal-enging, practical designs tend to be empirical and improvementsend to be incremental. In order to produce better design options,e have to use efficient design tools for solving such complexesign optimization problems. Therefore, metaheuristic algorithmsre often needed to deal with such problems.

.1. Metaheuristics

Metaheuristic algorithms such as the firefly algorithm and batlgorithm are often nature-inspired [28,31], and they are nowmong the most widely used algorithms for optimization. Theyave many advantages over conventional algorithms, such as sim-licity, flexibility, quick convergence and capability of dealing with

diverse range of optimization problems. There are a few recenteviews which are solely dedicated to metaheuristic algorithms28,29,34]. Metaheuristic algorithms are very diverse, includingenetic algorithms, simulated annealing, differential evolution,nt and bee algorithms, particle swarm optimization, harmonyearch, firefly algorithm, cuckoo search, flower algorithm and oth-rs [29,30,33,36].

In the context of heat pipe designs, we have seen a lot of interestsn the literature [5,7]. However, for a given response, to identify theight parameters can be considered as an inverse problem as wells an optimization problem. Only when we understand the rightorking ranges of key parameters, we can start to design bettereat-transfer systems. To our knowledge, this is the first attempt tose metaheuristic algorithms to identify key parameters for givenesponses. We will use the firefly algorithm to achieve this goal.

.2. Firefly algorithm

Firefly algorithm (FA) was first developed by Xin-She Yang in008 [28,30] and is based on the flashing patterns and behavior ofreflies. In essence, FA uses the following three idealized rules:

. Fireflies are unisex so that one firefly will be attracted to otherfireflies regardless of their sex.

. The attractiveness is proportional to the brightness and they

Please cite this article in press as: X.-S. Yang, et al., Mathematical mComput. Sci. (2014), http://dx.doi.org/10.1016/j.jocs.2013.12.003

both decrease as their distance increases. Thus for any two flash-ing fireflies, the less brighter one will move towards the brighterone. If there is no brighter one than a particular firefly, it willmove randomly in the form of a random walk.

PRESSonal Science xxx (2014) xxx–xxx

3. The brightness of a firefly is determined by the landscape of theobjective function.

As a firefly’s attractiveness is proportional to the light inten-sity seen by adjacent fireflies, we can now define the variation ofattractiveness with the distance r by

= ˇ0e−�r2, (1)

where ˇ0 is the attractiveness at r = 0.The movement of a firefly i attracted to another more attractive

(brighter) firefly j is determined by

xt+1i

= xti + ˇ0e

−�r2ij (xt

j − xti ) + �t

i , (2)

where the second term is due to the attraction. The third term israndomization with being the randomization parameter, and �t

iis

a vector of random numbers drawn from a Gaussian distribution oruniform distribution at time t. If ˇ0 = 0, it becomes a simple randomwalk. Furthermore, the randomization �t

ican easily be extended to

other distributions such as Lévy flights.The Lévy flight essentially provides a random walk whose ran-

dom step size s is drawn from a Lévy distribution

Lévy∼s−� (1 < � ≤ 3), (3)

which has an infinite variance with an infinite mean. Here the stepsessentially form a random walk process with a power-law step-length distribution with a heavy tail. Some of the new solutionsshould be generated by Lévy walk around the best solution obtainedso far; this will speed up the local search. Lévy flights are moreefficient than standard random walks [29].

Firefly algorithm has attracted much attention [3,10,23,35].A discrete version of FA can efficiently solve non-deterministicpolynomial-time hard, or NP-hard, scheduling problems [23], whilea detailed analysis has demonstrated the efficiency of FA for awide range of test problems, including multiobjective load dis-patch problems [3,32]. Highly nonlinear and non-convex globaloptimization problems can be solved using firefly algorithm effi-ciently [9,35]. The literature of firefly algorithms have expandedsignificantly, and Fister et al. provided a comprehensive literaturereview [8].

3. Mathematical model for a PHP system

3.1. Governing equations

Mathematical modelling of two-phase pulsating flow inside apulsating heat pipe involves many processes, including interfa-cial mass transfer, capillary force, wall shear stress due to viscousaction, contact angles, phase changes such as evaporation and con-densation, surface tension, gravity and adiabatic process. All thesewill involve some constitutive laws and they will be coupled withfundamental laws of the conservation of mass, momentum andenergy, and thus resulting in a nonlinear system of highly coupledpartial differential equations. Consequently, such a complex modelcan lead to complex behaviour, including nonlinear oscillations andeven chaotic characteristics [14,19,20,24,27,41].

A mathematical model can have different levels of complexity,and often a simple model can provide significant insight into the

odelling and parameter optimization of pulsating heat pipes, J.

working mechanism of the system and its behaviour if the modelis constructed correctly with realistic conditions. In most cases, fullmathematical analysis is not possible, we can only focus on someaspects of the model and gain some insight into the system.

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ARTICLEOCS-249; No. of Pages 7

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A key relationship concerning the inner diameter of a typicalulsating heat pipe is the range of capillary length, and many designptions suggest to use a diameter near the critical diameter [5]

c = 2

√�

g(�l − �v), (4)

here � is the surface tension (N/m), while g is the accelerationue to gravity, which can be taken as 9.8 m/s2. �l and �v are theensities of liquid and vapour, respectively. In the rest of the paper,e will focus on one model which may be claimed as the state-of-

he-art, as it is based on the latest models [24,25], with a simplifiedodel for a system of liquid plugs and vapour bubbles as described

n [37].

.1.1. Temperature evolutionThe temperature Tv in a vapour bubble is governed by the energy

alance equation

vcvvdTv

dt= −hlfv(Tv − �)L�(di − 2ı) − rmhvL�(di − 2ı) − pv

dV

dt,

(5)

here

m = 2�0

(2 − �0)1√2�R

(pv√

Tv− pl√

�). (6)

ere L is the mean plug length, di is the inner diameter of the pipe,nd ı is the thickness of the thin liquid film. h corresponds to theeat transfer coefficient and/or enthalpy in different terms. Here0 is a coefficient and R is the gas constant.

In addition, the volume in the above equation is given by

= �d2i

4(L + Lv), (7)

here the length of the vapour bubble is typically Lv = 0.02 m, and ≈ 1.

The temperature � in the liquid film is governed by

f cvld�

dt= −hlfw(� − Tw)L�di + hlfv(Tv − �)L�(di − 2ı)

+ rmhvL�(di − 2ı), (8)

here Tw is the initial wall temperature.

.1.2. Mass transferFor a vapour bubble, rate of change in mass is governed by the

onservation of mass

dmv

dt= −�(di − 2ı)rm. (9)

s mv = �vV , we have

vdV

dt= �(di − 2ı)rm − rv�diLv, (10)

here �v ≈ 1 kg/m3 and the transfer rate rv = 0 if Tw > Tv.

.1.3. Motion of a plugThe position xp of the liquid plug is governed by the momentum

quation

pd2xp

dt2= �

4(di − 2ı)2(pv1 − pv2) − �diLpsw + mpg, (11)

here2

Please cite this article in press as: X.-S. Yang, et al., Mathematical mComput. Sci. (2014), http://dx.doi.org/10.1016/j.jocs.2013.12.003

p = �l

�di

4(L0 − xp), (12)

ith L0 ≈ 25di and �l = 1000 kg/m3. Here the term pv1 − pv2 is theifferential pressure between the two sides of the plug.

PRESSnal Science xxx (2014) xxx–xxx 3

In the above equations, we also assume the vapour acts as anideal gas, and we have

pv = mvRTv

V. (13)

The shear stress sw between the wall and the liquid plug is givenby an empirical relationship

sw = 12

Cf �lv2p, (14)

where Cf ≈ 16/Re when Re = �lvpdi/�l ≤ 1180; otherwise,

Cf ≈ 0.078Re−1/4. (15)

Here, we also used that vp = dxp/dt.The mathematical formulation can include many assumptions

and simplifications. When writing down the above equations, wehave tried to incorporate most the constitutive relationships suchcontact angles, capillary pressures and viscous force into the equa-tions directly so that we can have as fewer equations as possible.For details, readers can refer to [5,24,25]. This way, we can focus ona few key equations such as the motion of a plug and temperaturevariations, which makes it more convenient for later mathematicalanalysis.

3.2. Typical parameters

The properties of the fluids and gas can be measured directly,and typical values can be summarized here, which can be relevantto this study. Most of these values have been drawn from earlierstudies [6,24,38].

Typically, we have L = 0.18 m, di = 3.3 × 10−3 m, ı = 2.5 × 10−5 m.The initial temperature is Tv ≈ � = 20 ◦C, while the initial walltemperature is Tw = 40 ◦C. The initial pressure can be takenas pv ≈ 5.5816 × 104 Pa. hlfw = 1000 W/m2 K, hv = 10 W/m2 K.cvl = 1900 J/kg ◦C, R = 8.31 and cvv = 1800J/kg ◦C.

In addition, we have the initial values: mv0 = pv0V0/(Rv(Tv0 +273.15) where Tv0 = 20 ◦C, pv0 = 105 Pa, Rv = 461 J/kg K, and mf 0 ≈mv0/10.

4. Nondimensionalization, analysis and simulation

After some straightforward mathematical simplifications, thefull mathematical model can be written as the following equations:

mvcvvdTv

dt= −hlfv(Tv − �)L�(di − 2ı) − rmhvL�(di − 2ı)

−pv�

�v[(di − 2ı)rm − diLvrv],

mf cvld�

dt= −hlfw(� − Tw)L�di + hlfv(Tv − �)L�(di − 2ı) + rmhvL�(di − 2ı),

dmv

dt= −�(di − 2ı)rm,

mpd2xp

dt2= �

4(di − 2ı)2(pvk − pvk) − �diLpsw + mpg.

(16)

Though analytically intractable, we can still solve this full modelnumerically using any suitable numerical methods such as finitedifference methods, and see how the system behaves under var-ious conditions with various values of parameters. Preliminarystudies exist in the literature [6,24], and we have tested our sys-tem that it can indeed reproduce these results using the typical

odelling and parameter optimization of pulsating heat pipes, J.

parameters given in Section 3.2 [37]. For example, the mean tem-perature in a plug is consistent with the results in [41], while theoscillatory behaviour is similar to the results obtained by others[5,12,25,40,41].

IN PRESSG ModelJ

4 putational Science xxx (2014) xxx–xxx

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ARTICLEOCS-249; No. of Pages 7

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.1. Non-dimensionalized model

For the current purpose of parameter estimation, the system cane considered stationary under appropriate conditions, and thusany terms can be taken as constants. Then, the full mathematicalodel (16) can be non-dimensionalized and written as

udTv

dt= a(Tv − �) − ˛1 − ˛2,

d�

dt= b(Tv − �) − � + ˛3,

du

dt= −,

d2xp

dt2= 1

(ˇ1 − ˇ2xp)

[ − �

(dxp

dt

)2]

,

(17)

here

= −hlfvL�(di − 2ı)cvv

, ˛1 = rmhvL�(di − 2ı)cvv

, (18)

2 = pv�

�vcvv[(d − 2ı)rm − diLvrv], b = −hlfvL�(di − 2ı)

mf cvl, (19)

= hlfwL�di

mf cvl, ˛3 = rmhvL�(di − 2ı)

mf cvl, (20)

= �(di − 2ı)rm

mv0, = �(di − 2ı)2(pv1 − pv2)

4+ g, (21)

= �diLpCv�l

2, ˇ1 = �lL0�d2

i

4, ˇ2 = �l�d2

i

4. (22)

In the above formulation, u is the dimensionless form of mv, thats u = mv/mv0.

.2. Simplified model

Under the assumptions of stationary conditions and xp/L0 � 1for example at the startup stage), the last equation is essentiallyecoupled from the other equations. In this case, we have anpproximate but simplified model

d�

dt= b(Tv − �) − � + ˛3,

d2xp

dt2= 1

ˇ1

[ − �

(dxp

dt

)2]

,

(23)

ow the second equation can be rewritten as

d2xp

dt2= A − B

(dxp

dt

)2

, A = ˇ

ˇ1, B = �

ˇ1. (24)

For the natural condition xp = 0 at t = 0, the above equation has ahort-time solution

p = 1B

ln[cosh(√

AB t)], (25)

hich is only valid when xp/L0 is small and/or t is small. Indeed, itives some main characteristics of the startup behaviour as showns the dashed curve in Fig. 1.

Furthermore, the first equation for the liquid plug temperatures also decoupled from the PDE system. We have

d�

Please cite this article in press as: X.-S. Yang, et al., Mathematical mComput. Sci. (2014), http://dx.doi.org/10.1016/j.jocs.2013.12.003

dt= Q1 − Q2�, Q1 = bTv + ˛3, Q2 = b + . (26)

or an initial condition � = 0 at t = 0, we have the following solution

= Q1(1 − e−Q2t), (27)

Fig. 1. Comparison of the full numerical results with approximations and experi-mental data [25].

which provides a similar startup behaviour for small times as thatin [41].

The approximation solutions are compared with the full numer-ical solutions and experimental data points, and they are all shownin Fig. 1. Here the experimental data points were based on [5,24,25].It is worth pointing that the approximation can indeed providesome basic characteristics of the fundamental behaviour of thestartup, and the full numerical results can also give a good indi-cation of the final steady-state value.

This simplified model also confirms that some earlier studiesusing mainly xp as the dependent variable can indeed give goodinsight into the basic characteristics of the complex system. Forexample, Wong et al. used an equivalent viscous damping systemwithout considering the actual heat transfer system [27], and theirmain governing equation can be written as

d2x

dt2+ a

dx

dt+ b(k + t)x = 0, (28)

where a, b, k are constants. This system has typical features of a vis-cous damping system. On the other hand, Yuan et al. used a systemwith primarily a second-order nonlinear ODE for xp

d2xp

dt2+ 2C

di

(dxp

dt

)2

+ 2g

Lxp = (pv1 − pv2)

L�l, (29)

where C is the friction coefficient. This system can have even richercharacteristics of oscillatory behaviours [40].

In comparison with earlier simpler models, our simplified modelcan produce even richer dynamic features of the system withoutusing much complex mathematical analysis, thus provides a basisfor further realistic analysis and simulations.

5. Inverse parameter identification and optimization

The aim of an inverse problem is typically to estimate impor-tant parameters of structures and materials, given observed datawhich are often incomplete. The target is to minimize the differ-ences between observations and predictions, which is in fact anoptimization problem. To improve the quality of the estimates, wehave to combine a wide range of known information, including anyprior knowledge of the structures, available data. To incorporate

odelling and parameter optimization of pulsating heat pipes, J.

all useful information and carry out the minimization, we have todeal with a multi-objective optimization problem. In the simplestcase, we have to deal with a nonlinear least-squares problem withcomplex constraints [21,22].

IN PRESSG ModelJ

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Table 1Least-squares parameter estimation.

Parameters Estimates True values

L 0.15–0. 24 0.18 mdi 0.001–0.005 0.0033 m

◦

ARTICLEOCS-249; No. of Pages 7

X.-S. Yang et al. / Journal of Com

The constraints for inversion can include the realistic ranges ofhysical parameters, geometry of the structures, and others. Theonstraints are typically nonlinear, and can be implicit or evenlack-box functions. Under various complex constraints, we haveo deal with a nonlinear, constrained, global optimization problem.n principle, we can then solve the formulated constrained prob-em by any efficient optimization techniques [37,29]. However,s the number of the degrees of freedom in inversion is typicallyarge, data are incomplete, and non-unique solutions or multipleolutions may exist; therefore, metaheuristic algorithms are par-icularly suitable.

.1. Parameter estimation as an inverse problem

Inverse problems in heat transfer and other disciplines tendo find the best parameters of interest so as to minimize the dif-erences between the predicted results and observations. In theimplest case, a linear inverse problem can be written as

= Kq + w, (30)

here u is the observed data with some noise w, q is the parametero be estimated, and K is a linear operator or mapping, often writ-en as a matrix. An optimal solution q* should minimize the norm|u − Kq||. In case when K is rectangular, we have the best estimate

∗ = (KT K)−1

KT u. (31)

But in most cases, K is not invertible, thus we have to usether techniques such as regularization. In reality, many inverseroblems are nonlinear and data are incomplete. Mathematicallypeaking, for a domain such as a structure with unknown butrue parameters q* (a vector of multiple parameters), the aim isnd the solution vector q so that the predicted values yi (i = 1, 2,

. ., n), based on a mathematical model y = �( x, p) for all x ∈ ˝, arelose to the observed values d = (d1, d2, . . ., dn) as possible.

The above inverse problem can be equivalently written as aeneralized least-squares problem [21,37]

in f = ||d − �(x, q)||2, (32)

r

in∑

i

[di − �(xi, q)]2, (33)

Obviously, this minimization problem is often subjected to a setf complex constraints. For example, physical parameters must beithin certain limits. Other physical and geometrical limits can also

e written as nonlinear constraints. In general, this is equivalent tohe following general nonlinear constrained optimization problem

in f (x, q, d) (34)

ubject to

j(x, q) = 0, (j = 1, ..., J), gk(x, q) ≤ 0, (k = 1, ..., K). (35)

here J and K are the numbers of equality and inequality con-traints, respectively. Here all the functions f, hj, and gk can beonlinear functions. This is a nonlinear, global optimization for q. Inhe case when the observations data are incomplete, thus the sys-em is under-determined, some regularization techniques such asikhonov regularization are needed. Therefore, the main task nows to find an optimal solution to approximate the true parameteret q*.

From the solution point of view, such optimization can be in

Please cite this article in press as: X.-S. Yang, et al., Mathematical mComput. Sci. (2014), http://dx.doi.org/10.1016/j.jocs.2013.12.003

rinciple solved using any efficient optimization algorithm. How-ver, as the number of free parameters tends to be very large,nd as the problem is nonlinear and possible multimodal, conven-ional algorithms such as hill-climbing usually do not work well.

Tv 10–27 20 CTw 20–49 40 ◦Cpv 40–129 100 kPa

More sophisticated metaheuristic algorithms have the potential toprovide better solution strategies.

Though above inverse problems may be solved if they arewell-posed, however, for most inverse problems, there are manychallenging issues. First, data are often incomplete, which leads tonon-unique solutions; consequently, the solution techniques areoften problem-specific, such as Tikhonov regularization. Secondly,inverse problems are highly nonlinear and multimodal, and thusvery difficult to solve. In addition, problems are often large-scalewith millions of degrees of freedom, and thus requires efficientalgorithms and techniques. Finally, many problems are NP-hard,and there is no efficient algorithms of polynomial time exists. Inmany of these cases, metaheuristic algorithms such as genetic algo-rithms and firefly algorithm could be the only alternative. In fact,metaheuristic algorithms are increasingly popular and powerful[28,37]. In the rest of this paper, we will use the firefly algorithm dis-cussed earlier to carry out the parameter estimations for pulsatingheat pipes.

5.2. Least-squares estimation

Now we try to solve the following problem for parameter esti-mation: Suppose we measure the response (i.e., the location xp of aplug) of a pulsating heat pipe, as the results presented in Fig. 1, canwe estimate some key parameters using these results through oursimplified mathematical model? One way to deal with this prob-lem is to consider it as a nonlinear, least-squares best-fit problemas described earlier in Eq. (33).

Let us focus on the key parameters L, Tv, Tw , di, and pv. Whenwe carry out the estimation using the least squares methods, wecan only get crude estimates as shown in Table 1. Here we haveused 25 data points to establish estimates for 5 key parameters.The wide variations of these parameters, though near the true val-ues, suggest that there are insufficient conditions for the inverseproblem to have unique solutions. This can be attributed to the fac-tors that the measured data points were sparse and incomplete,no constraints were imposed on the ranges of the parameters, andadditional constraints might be needed to make the inverse prob-lem well-posed.

5.3. Constrained optimization by the firefly algorithm

In order to get better and unique solutions to the problem, toincrease of the number of measurements is not the best choice, asexperiments can be expensive and time-consuming. Even with thebest and most accurate results, we may still be unable to get uniquesolutions. We have to make this problem well-posed by imposingenough conditions.

First, we have to impose more stringent bounds/limits. Thus,we apply L ∈ [0.15, 0.22] m, di ∈ [0.002, 0.004] m, Tv ∈ [15, 25] ◦C,Tw ∈ [35, 45] ◦C, and pv ∈ [80, 120] kPa. Then, we have to minimizethe parameter variations, in addition to the residual sum of squares.Thus, we have

odelling and parameter optimization of pulsating heat pipes, J.

Minimize∑

i

[di − �(xi, q)]2 +5∑

k=1

�2k , (36)

ARTICLE ING ModelJOCS-249; No. of Pages 7

6 X.-S. Yang et al. / Journal of Computati

Table 2Parameter estimation by constrained optimization.

Parameters Estimates (mean ± standard deviation) True values

L 0.17 ± 0.1 0.18 mdi 0.0032 ± 0.0004 0.0033 mTv 20.4 ± 0.9 20 ◦C

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Tw 39.2 ± 1.1 40 ◦Cpv 98 ± 8.8 100 kPa

here �2k

is the variance of parameter k. Here we have 5 parameterso be estimated. Now we have a constrained optimization problemith imposed simple bounds.

By using the firefly algorithm with n = 20, � = 1, = 1 and 5000terations, we can solve the above constrained optimization. Weave run the results 40 times so as to obtained meaningful statistics,sing the same set of 25 observed data points, and the results areummarized in Table 2.

We can see clearly from the above results that unique parameterstimates can be achieved if sufficient constraints can be imposedealistically, and the good quality estimates can be obtained evenor a number of sparse measurements. This implies that it is fea-ible to estimate key parameters of a complex PHP system fromxperimental data using the efficient optimization algorithm andhe correct mathematical models.

. Discussions and conclusions

Though a pulsating heat pipe system can be hugely complexs it involves multiphysics processes, we have shown that it isossible to formulate a simplified mathematical model under one-imensional configuration, and such a model can still be capableo reproduce the fundamental characteristics of time-dependenttartup and motion of liquid plugs in the heat pipe. By using scalingariations and consequently non-dimensionalization, we have ana-ysed the key parameters and processes that control the main heatransfer process. Asymptotic analysis has enabled us to simplifyhe model further concerning small-time and long-time behavior,o that we can identify the factors that affect the startup charac-eristics. Consequently, we can use the simplified model to predictnd then compare the locations of a plug with experimental data.he good comparison suggests that a simplified model can workery well for a complex PHP system.

Even though we have very good results, however, there are stillome significant differences between the full numerical model andxperimental data, which can be attributed to unrealistic param-ter values, incomplete data, oversimplified approximations andnaccounted experimental settings. Further work can focus on thextension of the current model to 2D or 3D configuration withealistic geometry and boundary conditions.

By treating parameter estimation as an inverse problem andubsequently a nonlinear constrained optimization problem, weave used the efficient metaheuristic algorithms such as fireflylgorithm to obtain estimates of some key parameters in a 1D PHPystem. We have demonstrated that a least-squares approach isot sufficient to obtain accurate results because the inverse prob-

em was not well-posed, with insufficient constraints. By imposingxtra proper constraints on parameter bounds and also minimizinghe possible variations as well as minimizing the best-fit errors, weave obtained far more accurate estimates for the same five keyarameters, and these parameter estimates are comparable withheir true values.

Please cite this article in press as: X.-S. Yang, et al., Mathematical mComput. Sci. (2014), http://dx.doi.org/10.1016/j.jocs.2013.12.003

There is no doubt that further improvement concerning theathematical model and parameter estimation will provide further

nsight into the actual behaviour of a PHP system, and subsequentlyllows us to design better and more energy-efficient PHP systems.

[

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PRESSonal Science xxx (2014) xxx–xxx

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Xin-She Yang received his DPhil in Applied Mathematicsfrom the University of Oxford. He worked at CambridgeUniversity and then at National Physical Laboratory as aSenior Research Scientist. He is Reader in Modelling and

Please cite this article in press as: X.-S. Yang, et al., Mathematical mComput. Sci. (2014), http://dx.doi.org/10.1016/j.jocs.2013.12.003

Simulation at Middlesex University, UK and an AdjunctProfessor at Reykjavik University, Iceland. He is the IEEECIS Task Force Chair on Business Intelligence and anHonourary Fellow of Australia Institute of High EnergyMaterials.

PRESSnal Science xxx (2014) xxx–xxx 7

Mehmet Karamanoglu is Professor in Design Engineer-ing at Middlesex University. He graduated with a BEngdegree in Mechanical Engineering and followed onto tocomplete his PhD in numerical methods, supported byBritish Aerospace. He is currently heading the departmentof Design Engineering and Mathematics. His expertiseincludes manufacturing automation, CAD, design engi-neering, modelling and robotics. His research interestsinclude numerical analysis, process simulation, designstrategies and robotic systems.

Tao Luan is Professor in Energy and Power Engineering atShandong University. He graduated from Shandong Uni-versity, China, in 1983, and obtained his PhD from theUniversity of Leeds, UK, in 1997. He has served in bothindustry and research institutes for many years in China,Europe and Canada. His main research areas are in the fieldof heat transfer, combustion, energy utilisation, environ-mental protection, and heat exchanger/condenser design.

Slawomir Koziel received the MSc and PhD degrees inelectronic engineering from Gdansk University of Tech-nology, Poland, in 1995 and 2000, respectively. He alsoreceived the MSc degrees in theoretical physics and inmathematics, in 2000 and 2002, respectively, as well asthe PhD in mathematics in 2003, from the Universityof Gdansk, Poland. He is currently a Professor with the

odelling and parameter optimization of pulsating heat pipes, J.

School of Science and Engineering, Reykjavik University,Iceland. His research interests include CAD and mod-elling of microwave circuits, simulation-driven design,surrogate-based optimization, evolutionary computationand numerical analysis.