pde modeling of a microfluidic thermal process for genetic

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 767853 12 pageshttpdxdoiorg1011552013767853

Research ArticlePDE Modeling of a Microfluidic ThermalProcess for Genetic Analysis Application

Reza Banaei Khosroushahi1 Horacio J Marquez1

Jose Martinez-Quijada1 and Christopher J Backhouse2

1 Electrical and Computer Engineering University of Alberta Edmonton AB Canada T6G 2V42 Electrical and Computer Engineering University of Waterloo Waterloo ON Canada N2L 3G1

Correspondence should be addressed to Reza Banaei Khosroushahi banaeikhualbertaca

Received 12 July 2013 Accepted 29 August 2013

Academic Editor Zhiwei Gao

Copyright copy 2013 Reza Banaei Khosroushahi et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper details the infinite dimensional dynamics of a prototype microfluidic thermal process that is used for genetic analysispurposes Highly effective infinite dimensional dynamics in addition to collocated sensor and actuator architecture require thedevelopment of a precise control framework to meet the very tight performance requirements of this system which are not fullyattainable through conventional lumped modeling and controller design approaches The general partial differential equationsdescribing the dynamics of the system are separated into steady-state and transient parts which are derived for a carefully chosenthree-dimensional axisymmetricmodelThese equations are solved analytically and the results are verified using an experimentallyverified precise finite element method (FEM) model The final combined result is a framework for designing a precise trackingcontroller applicable to the selected lab-on-a-chip device

1 Introduction

Biomedical microdevices usually require the very tight per-formance specification to be satisfied so that they becomeproperly functional Carefully designed controllers can guar-antee certain performance specifications such as robustnessand stability Successful controller design process requires aprecise mathematical model A perfect mathematical modelis the one that captures the highest amount of the dynamicsof the system Likewise numerical tools can provide preciseinsights into a systemrsquos dynamics Finite element method(FEM) is a common numerical tool that can provide precisesimulation of the steady-state and transient behaviour of amicrodevice However FEMmodels are not suitable for con-troller design because they are not in a compactmathematicalform In this work we present a mathematical model thatcaptures infinite dimensional dynamics of the heat distri-bution inside a prototype system The selected system is alab-on-a-chip (LOC) device that is designed and built toperform the Polymerase Chain Reaction (PCR) and capillary

electrophoresis (CE) processes The three-dimensional com-plete FEM model of the system is used as the reference toderive the mathematical model The FEM model is experi-mentally verified with the actual setup

Lab-on-a-chip (LOC) devices are finding growing atten-tion in various applications ranging from chemical analysisand water contamination sensors to genetic analysis [1] ALOC device can be thought of as a miniaturized instrumentcapable of performing single ormultiple laboratory processesthat integrate various functionalities such as sample pretreat-ment sample transportation genetic amplification mixingreaction separation and detection LOC devices offer advan-tages beyond conventional biochemical and clinical labo-ratory settings Dramatic reduction of sample volume andreduced consumption of costly reagents are two distinct ben-efits of miniaturized laboratory processes Most significantlyminiaturization improves analysis characteristics such asshorter analysis time and higher separation performance andcan even enable innovative applications that are not otherwiseattainable

2 Journal of Applied Mathematics

Genetic and molecular analyses are among the majorapplications of LOC devices and have become feasible by theadoption of the Polymerase Chain Reaction (PCR) techniqueUsing PCR specific regions of even a single DNA moleculecan be replicated up to about 107 to 108 times This processis also called selective amplification of DNA The amplifiedDNA segments can be later analyzed and compared to a stan-dard pattern to find out if they correspond to some diseasecondition PCR has been traditionally a lengthy molecularbiology operation but the LOC concept proposed in theearly 1990s [2 3] has pushed this technology to attainever shorter reaction times DNA amplification using PCRinvolves repeatedly cycling the temperature of the reactionchamber in three predefined temperatures Optimum reac-tion temperature and exposure times of the sample to eachtemperature are key factors for successful DNA amplification[4] The PCR-LOC research community emphasizes theimportance of plusmn1∘C temperature precision and rapid transi-tion time for high amplification efficiency At the same timeovershoots and undershoots which are likely to happen athigher speeds have a detrimental impact on the amplificationyield For example overshoots at denaturation temperaturewill reduce the lifetime of replicating enzyme (Taq poly-merase) Undershoots at annealing temperature may causethe amplification of unspecified products Nevertheless rapidthermal cycling of reaction chamber in DNA amplificationnot only reduces the amplification time but it is alsoimportant to ensure the stability of biological reagents duringamplification as the Taq polymerase has a half lifetime thatdecreases rapidly at temperatures close to 100∘C [5] It isimportant to note that about half of the time in a standardPCR process is spent in transitions which are undesirable Asa result researchers have focused their attention to reducetransition time between temperature steps in PCRprocess Toachieve this goal both structural design of the device andthermal control of the process need to be optimized Con-sequently various platforms for PCR-LOC devices have beenreported in the literature [6] which can be roughly dividedinto two major categories continuous-flow PCR and staticPCR [7] Furthermore different heating techniques have beenchosen where thermoelectricmodules (TEMs) and thin-filmresistive heaters [8ndash11] are the most commonly used tech-niques Other techniques such as infrared radiation [12] andmicrowaves [13 14] have also been reported These differentstructural designs are discussed in detail in the excellentreview papers by Zhang et al [7] Roper and coworkers [6]Verpoorte [15] and DeMello [16] The latest achievements inPCR-LOC devices design can be found in the recently pub-lished survey paper by Zhang and Ozdemir [1]

Most of the published work on PCR-LOC devices hasbeen devoted to the structural design aspects and less atten-tion has been paid to optimize the thermal control of the pro-cess As a result there is very little work in the literature on thedynamics and control of these microsystems Although neu-ral network based controllers [17] and hybrid controllers [18]are used for thermal control of PCR-LOC devices PIPIDcontrollers are still the most commonly preferred controllersto deal with the thermal control problem of PCR-LOCs [5 1920]Whatmakes the thermal control ofmost of the PCR-LOC

devices a challenging task is the fact that the temperatureinside the PCR-LOC chips not only depends on time but isalso spatially distributed Thus lumped modeling and con-troller design cannot exactly address the problem funda-mentally because in this approach the distributed parameternature of the system is neglected We observed this criticalproblem to some extent in the formerly designed PCR-LOCchip [21] where switching controllers were employed [8]ThePCR-LOC system studied in this paper is the next generationof the one described in our previous work [22] The newdesign has undergone significant improvement in structuraldesign with respect to its predecessor

While lumped modeling can precisely describe thedynamics of lumped systems it usually cannot provide afair description of the dynamics of the distributed parametersystemsThose systems that can be described by partial differ-ential equations are often referred to as distributed parametersystems or PDE systems which feature distributed dynamicsand are categorized under a general set of the systems knownas complex dynamical systems [23] Distributed parametersystems are subsets of a more general set called infinitedimensional systems because they can bemathematically for-mulated as differential equations on an abstract linear vectorspace of infinite dimension [24] In contrast lumped systemscan be formulated mathematically as ordinary differentialequations on a finite dimensional linear vector space Design-ing controllers for distributed parameter systems is calledinfinite dimensional controller design or PDE controllerdesign [24ndash26]

This paper characterizes the heat distribution dynamicsof a microfluidic thermal process The work begins witha description of the process and experimental setup Thenresults of FEM simulations on the system were analyzed tochoose a suitable structure for themodel Next the PDE equa-tions describing the model in steady-state and transient partswere derived These equations were solved analytically fromwhich the dynamic temperature distribution in the wholedomain can be expressed by an infinite dimensional modelNext we optimize the calculatedmodel tominimize the errordue to the simplification of the actual multidomain system toan abstract single-domain system FEM simulation and lab-oratory experiments are used to verify the model The resultof this work is a framework for boundary control design forthermal control of the aforementioned PCR-LOC device

2 Materials and Methods

21 Chip Configuration Genetic analysis usually consists ofthree steps sample preparation amplification and detectionIn the first step the DNA is extracted from a raw sample suchas blood Then a selected fragment of this DNA is amplifiedthrough the PCR process to obtain sufficient DNA copiesto perform detection Next the amplified region of DNAis labelled with a fluorescent dye and finally it is detectedthrough various analysis techniques [27]

The integrated genetic analysis platform that has beendeveloped in the AppliedMiniaturization Laboratory (AML)at theUniversity of Alberta is capable of performing completegenetic analysis on a singlemicrofluidic chip [28]This system

Journal of Applied Mathematics 3

USB

Power

LaserMicrochip

LensFilter

Electronics andhigh voltage boards

CCDcamera

Figure 1 Integrated genetic analysis platform comprising the PCR-LOC device and the control electronics The platform is operatedthrough a graphical user interface [29]

is depicted in Figure 1 The microchip is composed of twolayers of Borofloat glass with a polydimethylsiloxane (PDMS)layer in the middle Reaction chamber and fluid channels arecarved on one of the glass layers and a Pt heatersensor ispatterned on the bottom glass plate using standard micro-fabrication techniques A close-up view of the microfluidicchip depicted in Figure 2 clearly shows the configuration ofthe platinum heater-sensor ring A schematic representationof the components within the chip is shown in Figure 2(a)Perspective view of the glassPDMSglass chip with hor-izontal laser beam is shown in Figure 2(b) and the chipatop a heatsink and light detector is depicted in Figure 2(c)To improve the cooling rate during thermal cycling themicrochip ismounted on a copper heat sinkTheheat sink hasa circular opening aligned to the center of the PCR chamberand heatersensor ring axis The radius of this opening isoptimized to increase the cooling rate without having neg-ative effect on the heating rate [21] We are mostly interestedin heat transfer dynamics around the reaction chamber andheater-sensor ringThe chamber heatersensor and heat-sinkopening are circular and share the same axis This suggeststhat we should look for an axisymmetric cylindrical structurefor our modelThe half cross-section view of the chip along avertical plane parallel to the smaller side and passing throughthe center of the chip is shown in Figure 3 The irregular andmultilayer multimaterial structure of the PCR-LOC system isnot suitable to form a set of solvable analytic equations addi-tional simplifications are needed To find a valid simplifiedmodel describing the dynamics of the chip where the heaterand chamber are located we first have to analyze the temper-ature profile around the heater and chamber area Figure 4depicts the temperature profile along the chip thickness forsome selected distances from the center of the chip rangingfrom 25mm to 50mm Referring to Figure 4 the tempera-ture profile along the thickness of the chip at the upper glass

layer (1354mm to 2454mm from the bottom of the chip)shows a semilinear trend By choosing the radius of 35mmfrom the center of the chip the slope is negligible as thetemperature varies for less than 08∘C which is even less thanthe acceptable error for the chamber temperature Thereforewith reasonable approximation we can consider a constanttemperature wall as the boundary where this boundary ispositioned at a distance of 35mm from center of the chipThese calculations in addition to several other FEM simula-tions have been performed usingCOMSOLMultiphysics 35aon a comprehensive FEMmodel of the chip that describes itslayered geometry material properties and boundary condi-tions with a high degree of accuracy More FEM simulationsdisclosed some other important facts about this chip whichcan be summarized as follows

(1) The amount of heat going to the upper portion of thechip through the PDMS and top glass layer has anapproximately constant ratio to the total heat beingdissipated by the heater

(2) If we consider a disk-shape bounded volume at thecenter of the chip the amount of heat lost by naturalconvection in air through the upper surface of the diskis negligible compared to the amount of heat lost byconduction through the wall

(3) If the PDMS layer was replaced by a glass layeronly a small percentage of the dynamic temperaturedistribution would be affected due to the alteration ofthermal properties (thermal conductivity and specificheat capacity) This small effect on the dynamics canbe compensated later by using a correction factor

(4) The vertical temperature profile in the top glass layerat a distance of 35mm from the center is approxi-mately constant

22 Model Structure Using these observations we can con-clude that by picking a single layer glass disk we can modelthe temperature distribution at a closed disk in the upperside of the chip with a good level of approximation relativeto the temperature accuracy of plusmn1∘C demanded by PCR-LOCsystemsThis model structure is shown in Figure 5 where theheater is defined by the circular strip at the bottom surfacewith 119896

2width and 119896

1distance to the vertical axis It is worth

noting that we only modelled the area of interest in the chipand we have put aside the rest of the chip Furthermorewe defined a comparison aperture that can be characterizedby a fixed size rectangular area in cross-section view of theactual PCR-LOC microchip and our model The comparisonaperture contains the reaction chamber and it is discussed indetail in the simulation results sectionWe chose this scenarioto arrive at a model structure which is simple enough to findits analytical solution meanwhile it is accurate enough tocharacterize the actual thermal process A simple estimateof the thermal capacity and radial thermal resistance of thePDMS layer shows that they are small compared to those ofthe glass layers It is expected that treating the PDMS layeras if it has the same thermal properties of glass will make


Well 2

Well 1 Well 3

Laserbeam

V valves

Well 4

Pump

Pt heater

PCR chamber

Pump

PCR loadingwell

V-6V-5V-4

V-3V-2V-1

Well 3 PCR unloading wellCE input well

(a)

z

x y

Laserbeam

Flow layer

Control layer

Glass (11 mm)PDMS (025 mm)

(b)

PCR chamberGlass

PDMSGlass

Copperheatsink

Separation (CE) channelMicrofluidic chip

GRIN lensInterferencefilter

Photodiode

Countersinkunder PCR chamber

with radius r

Note figures not drawn to scale

(c)

Figure 2 The modelled PCR-LOC device On (a) a schematic of the chip showing the PCR chamber as a yellow circle The chamber issurrounded by the heater ring The squares in the middle are the heater contact pads The crossing channels form the CE section On (b) a3D view of the chip showing the three layers that comprise the device On (c) a cross-section view of the chip showing the chip placed on theheatsink that provides passive cooling [28]

the systems dynamic behavior 9 too fast (because the radialresistance is predicted to be 10 too low and the thermalcapacity is estimated to be about 1 too high) It is possible tocorrect for this by scaling the thermal parameters of the glassThe assumptionsmade will result in underestimating the ver-tical gradient but that gradient cannot be controlled by con-trol mechanism and must instead be dealt with at the designand simulation level (ie dynamic simulations of the presentdesign ensured that the PCR chamber radius and height weresufficiently small that the vertical gradient could not lead totemperature variations greater thanplusmn1∘C)TheFEMmodel byitself was verified by extensive experiments wherein ther-mochromic liquid crystal (TLC) sensitive to different temper-atures was deposited in the chamber to read the temperaturewithin its volume in steady-state conditions

3 Heat Distribution Model

31 Problem Formulation The partial differential equationdescribing temperature distribution in the chosen domain atany time is given by the heat conduction equation as follows[30]

119870nabla2119906 = 120588C

120597119906

120597119905

(1)

where 119906(120588 120593 119911 119905) is the temperature of every point of thedomain at any given time 119870 is thermal conductivity 120588 isdensity and C is heat capacity Since the chosen modelfeatures axial symmetry the temperature in the cylinder doesnot depend on 120593 This means that 120597119906120597120593 = 0 = 120597

21199061205971205932


Convective heat flux

Chamber

90 120583m15 mm

25 mm02 mm

Thermal insulation

7 mm

MicroheaterBorofloat glass

Borofloat glass

11 mm

PDMS 254 120583m

11 mm

Heat sink

Figure 3 Cross-section view and dimensions of PCR LOC microchip Due to the axisymmetric geometry of the chip only half the chip isshown The axis of symmetry is along the left vertical side A circular hole is cut in the middle of the heatsink to provide thermal insulationunder the heater and chamber Heat is lost by free convection on the top and by conduction through the heatsink [21]

Therefore 120593 can be dropped and the Laplace operator can bewritten in a 2D cross-section domain using 120588 and 119911 as follows

nabla2119906 =

1205972119906

1205971205882+

1

120588

120597119906

120597120588

+1205971199062

1205971199112 (2)

To avoid any confusion between 120588 and the notation usedfor density we introduce here notation 120581 as follows

120581 =119870

120588C(3)

which is thermal diffusivity of the material in 1198982119904 Finally

the partial differential equation giving the temperature119906(120588 119911 119905) at the half part cross-section of our cylindricaldomain with 120588 and 119911 as its horizontal and vertical axes can bewritten as

120581nabla2119906 =

120597119906

120597119905

(4)

The definition of our domain implies that 120588 and 119911 arelimited to 0 le 120588 le 119886 and 0 le 119911 le 119887 The initial and boundaryconditions are given by

119906 (120588 119911 0) = 1199060(120588 119911) (5)

119906 (119886 119911 119905) = 119879119908 (6)

119906120588(0 119911 119905) = 0 (7)

119906119911(120588 119887 119905) = 0 (8)

119906119911(120588 0 119905) = minus119891 (120588) (9)

where 1199060(120588 119911) is the initial distribution of temperature

119879119908is the temperature of the outside wall of the cylinder

The boundary condition (7) comes from the axisymmetricalcylindrical structure of the model The boundary condition(8) is based on the assumption that the upper surface of thechip is isolated 119891(120588) describes the geometric distributionof the control action signal at the bottom surface which is

defined by the heater location A general solution of (4) canbe found by adding the solution of homogeneous and nonho-mogeneous equations describing the steady-state and a tran-sient temperature distribution respectively

119906 (120588 119911 119905) = (120588 119911 119905) + 119906 (120588 119911) (10)

Substituting (10) in (4) we get1

120581

119905= 120588120588

+ 119906120588120588

+1

120588

120588+

1

120588

119906120588+ 119911119911

+ 119906119911119911

= (120588120588

+1

120588

120588+ 119911119911) + (119906

120588120588+

1

120588

119906120588+ 119906119911119911)

(11)

Without the loss of generality we can assume 119906120588120588

+

(1120588)119906120588+119906119911119911

= 0 as we defined 119906(120588 119911) to describe the steady-state temperature By applying (10) to the initial condition (5)and boundary conditions (6) (7) (8) and (9) we arrive at twoPDEs

1

120581

119905= 120588120588

+1

120588

120588+ 119911119911

with

(120588 119911 0) = 0(120588 119911)

(119886 119911 119905) = 0

120588(0 119911 119905) = 0

119911(120588 119887 119905) = 0

119911(120588 0 119905) = 0

(12)

119906120588120588

+1

120588

119906120588+ 119906119911119911

= 0 with

119906 (119886 119911) = 119879119908

119906120588(0 119911) = 0

119906119911(120588 119887) = 0

119906119911(120588 0) = minus119891 (120588)

(13)

The first one is an initial value problem with homoge-neous boundaries while the second one is a nonhomoge-neous boundary value problem

32 Steady-State Part To solve (13) we employ themethod ofseparation of variables [31]

119906 (120588 119911) = 119875 (120588)119885 (119911) + 119879119908 (14)


P = 1967W

0 05 1 15 2 2520

40

60

80

100

120

140

160Vertical temperature profile for radius 120588 = 25 mm to 5 mm

Distance from bottom of the chip (mm)

25262728293031

32333435404550

Tem

pera

ture

T(∘

C)

Figure 4 Steady-state vertical temperature profile predicted byFEM Each curve shows the temperature along a vertical line thatcrosses through the chip Such a line is placed at a defined distancefrom the centreThe legend shows the radial distance inmillimetresThe horizontal axis represents the vertical distance from the bottomto the top of the chip which is 2454mm thick As the distancefrom the center increases from 25 to 5mm the curves flattened outtowards 22∘C The temperatures in the top glass plate (right side ofthe curves) are almost constant

where119875 and 119885 are functions of 120588 and 119911 respectivelyThe con-stant part is used to normalize the boundary conditions andto achieve the homogeneous boundary condition Using (14)in (13) and dividing 119875119885 we arrive to

11987510158401015840

(120588)

119875 (120588)

+1

120588

1198751015840

(120588)

119875 (120588)

= minus11988510158401015840

(119911)

119885 (119911)

(15)

The left side contains functions of 120588 alone while the rightside contains functions of 119911 alone Since this equality musthold for all 120588 and 119911 in the given interval the common value ofthe two sidesmust be a constant sayminus120582

2

varying neither with120588 norwith 119911The constant here was chosen as a negative valueto avoid a trivial solution [31] Nowwe have twoODEs for thetwo factor functions

12058811987510158401015840

+ 1198751015840

+ 120582

2

120588119875 = 0

11988510158401015840

minus 120582

2

119885 = 0

(16)

120588

z

b

a

k1 k2

120593

Figure 5 Geometrical representation of the PDE model Takingadvantage of the axial symmetry of the chip the entire systemis represented as a two-dimensional rectangular region Similarlythe heater is represented as a line instead of being a surfaceThis reduction of one dimension allows the model to be solvedfor the temperature in a three-dimensional space but at a smallfraction of the time and computational resources required by a FEMsimulation

The boundary conditions of (13) can also be stated in theproduct form (14) which results in the following

119875 (119886) = 0 (17)

1198751015840

(0) = 0 (18)

1198851015840

(119887) = 0 (19)

119875 (120588)1198851015840

(0) = minus119891 (120588) (20)

The solutions of (16) are given by [31]

119875 = 11988811198690(120582120588) + 119888

21198840(120582120588) (21)

119885 = 1198883cosh 120582 (119887 minus 119911) + 119888

4sinh 120582 (119887 minus 119911) (22)

where 1198690is the Bessel function of the first kind of order 0 and

1198840is the Bessel function of the second kind of order 0 From

the boundedness condition at 120588 = 0 stated by (18) we musthave 119888

2= 0 because 119884

0is unbounded at zero Thus the solu-

tion (21) becomes

119875 = 11988811198690(120582120588) (23)

From the boundary condition (19) and the fact that

1198851015840

(119911) = minus120582 (1198883sinh 120582 (119887 minus 119911) + 119888

4cosh 120582 (119887 minus 119911)) (24)

we see that

1198851015840

(119887) = minus1205821198884= 0 (25)

which implies 1198884= 0 so the solution (22) becomes

119885 (119911) = 1198883cosh 120582 (119887 minus 119911) (26)

The boundary condition (17) can be employed to deter-mine 120582

119875 (119886) = 11988811198690(120582119886) = 0 (27)


which can be satisfied only if 1198690(120582119886) = 0 so that

119886120582 = 1199031 1199032 1199033 119903

119898 (28)

where 119903119898(119898 = 1 2 3 ) is the119898th positive root of 119869

0(119909) = 0

Thus 120582rsquos can be expressed as

120582119898=

119903119898

119886

119898 = 1 2 3 (29)

So a solution satisfying the boundary conditions is

119906 (120588 119911) minus 119879119908= 119875119898(120588)119885119898(119911)

= 1198601198690(119903119898

119886

120588) cosh119903119898

119886

(119887 minus 119911)

(30)

where 119898 = 1 2 3 and 119860 = 11988811198883 The general solution of

(13) can be obtained by employing the superposition principle[31] By replacing 119860 with 119860

119898and summing up all eigenfunc-

tions it follows that

119906 (120588 119911) = 119879119908+

infin

sum

119898=1

1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911) (31)

From

120597119906

120597119911

=

infin

sum

119898=1

minus 119860119898

119903119898

119886

1198690(119903119898

119886

120588) sinh119903119898

119886

(119887 minus 119911) (32)

and the last boundary condition (20) we can write

infin

sum

119898=1

(119860119898

119903119898

119886

sinh119903119898119887

119886

) 1198690(119903119898

119886

120588) = 119891 (120588) (33)

The right side is the Bessel series expansion of 119891(120588) withcoefficients equal to (119860

119898(119903119898119886) sinh(119903

119898119887119886)) [31] so we get

119860119898=

2

119886119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

int

119886

0

1205881198690(119903119898

119886

120588)119891 (120588) 119889120588

(34)

33 Transient Part We employ the separation of variablesagain to solve (12)

(120588 119911 119905) = (120588)119885 (119911) (119905) (35)

where 119885 and are functions of 120588 119911 and 119905 respectivelyFollowing the same procedure as the steady-state part anddefining 120583 ] and as eigenvalues for 119885 and respec-tively we arrive at the following ODEs

12058810158401015840+ 1015840+ 1205832120588 = 0 (36)

11988510158401015840minus ]2119885 = 0 (37)

1015840+ 1205812 = 0 (38)

where ]2 = 1205832minus 2 By applying (35) into (12) the boundary

conditions become

(119886) = 0 (39)

1015840(0) = 0 (40)

1198851015840(119887) = 0 (41)

1198851015840(0) = 0 (42)

The solutions of (36) (37) and (38) are given by

= 11988811198690(120583120588) + 119888

21198840(120583120588) (43)

119885 = 1198883cosh ] (119887 minus 119911) + 119888

4sinh ] (119887 minus 119911) (44)

= 1198885119890minus1205812119905 (45)

From the boundedness condition at 120588 = 0 expressed by(40) we must have 119888

2= 0 Thus the solution of (43) becomes

(120588) = 11988811198690(120583120588) (46)

From the boundary condition (41) we have

1198851015840(119887) = minus]119888

4= 0 (47)


119885 (119911) = 1198883cosh ] (119887 minus 119911) (48)

From boundary condition (39) we can determine 120583 Con-sider

(119886) = 11988811198690(120583119886) = 0 (49)

so we have

120583119898=

119903119898

119886

(50)


0(119909) = 0

A solution of (36) satisfying the boundary conditions is

119898(120588) = 119888

11198690(119903119898

119886

120588) (51)

where119898 = 1 2 3 From the boundary condition (42) wecan determine ] We have

1198851015840(0) = minus]119888

3sinh ]119887 = 0 (52)

which can be satisfied only if sinh ]119887 = 0 or

] =119896120587119894

119887

119896 = 0 1 2 (53)

Using (53) in (48) and the fact that

cosh 119894119909 = cos119909 (54)

the solution (44) becomes

119885119896(119911) = 119888

3cos 119896120587

119887(119887 minus 119911) (55)


Now from (53) and (50) it follows that

2= 1205832minus ]2 = (

119903119898

119886

)

2

minus (119896120587119894

119887

)

2

=

1199032

119898

1198862+

11989621205872

1198872

(56)

So a solution satisfying all boundary conditions is givenby

(120588 119911 119905) = 119860119890minus120581(1199032

1198981198862+119896212058721198872)1199051198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(57)

where 119860 = 119888111988831198885 119896 = 0 1 2 3 and 119898 = 1 2 3

Replacing 119860 with 119860119896119898

and summing up 119896 and 119898 we obtainthe solution by the superposition principle

(120588 119911 119905) =

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

119890minus120581(1199032

1198981198862+119896212058721198872)119905

times 1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(58)

From the initial condition (120588 119911 0) = 0(120588 119911) we have

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

= 1199060(120588 119911) minus 119906 (120588 119911)

(59)

This can be written asinfin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911) = (1199060(120588 119911) minus 119879

119908)

+

infin

sum

119898=1

minus 1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911)

(60)

where 119860119898

is defined by (34) To simplify the equationwithout the loss of accuracy we consider that 119906

0(120588 119911) = 119879

119908

which is reasonable becausewe start the experimentwhen thesystem is in a uniform temperature distributionWe canwrite(60) in this form

infin

sum

119898=1

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) 1198690(119903119898

119886

120588)

=

infin

sum

119898=1

minus119860119898cosh

119903119898

119886(119887 minus 119911) 119869

0(119903119898

119886

120588)

(61)

Clearly because of orthogonality of the Bessel functionsthese coefficients have to be equal that is

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886(119887 minus 119911) (62)

Written in standard form we have

1198600119898

+

infin

sum

119896=1

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886

(119887 minus 119911) (63)

Recalling the Fourier cosine series expansion [31] the leftside of (63) is the Fourier cosine series expansion of the rightside with respect to 119887 minus 119911 Thus substituting 119909 = 119887 minus 119911 andreplacing 119860

0119898and 119860

119896119898with 119886

02 and 119886

119896 respectively we

have1198860

2

+

infin

sum

119896=1

119886119896cos 119896120587

119887

119909 = minus119860119898cosh

119903119898

119886

119909 (64)

So 119886119896for 119896 = 0 1 2 are given by

119886119896=

2

119887

int

119887

0

(minus119860119898cosh

119903119898

119886

119909) cos 119896120587119887

119909 119889119911 (65)

Equation (65) can be written as

119886119896= minus

2

119887

119860119898int

119887

0

cosh119903119898

119886

119909 cos 119896120587119887

119909 119889119911 (66)

and solving the integral we have

119886119896= minus

2

119887

119860119898

119886119887

119886211989621205872+ 11988721199032

119898

times (119886119896120587 cosh (119903119898

119886

119887) sin (119896120587)

+119887119903119898cos (119896120587) sinh(

119903119898

119886

119887))

(67)

Since 119896 takes only positive integers we can simplify (67)and arrive to

119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (68)

Thus we have

1198600119898

=1198860

2

= minus119860119898

119886

119887119903119898

sinh(119903119898

119886

119887) (69)

and 119860119896119898

is defined by (68) as

119860119896119898

= 119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (70)

where 119896 = 1 2 Hence it follows that the complete solu-tion to the problem is given by (10) (31) and (58) where theircoefficients are defined by (34) (69) and (70)

34 Stability Analysis Stability analysis of the calculatedmodel can be easily performed by studying the transient partof the general solution A quick visit of transient equation (58)reveals that the infinite poles or eigenvalues of the system aregiven by the coefficients of the exponential term in infiniteseries as follows

System Poles minus 120581(

1199032

119898

1198862+

11989621205872

1198872

) (71)

where 119896 = 0 1 2 and 119898 = 1 2 3 Because thermaldiffusivity 120581 takes only positive real values for differentmate-rials all the poles are negative real poles and therefore thesystem is stable Calculating the slowest poles of the systemusing (71) and the parameters given in Table 1 we arrive to thefollowing results

System Poles = minus014 minus076 minus15 minus186 minus211 (72)


Table 1 Parameter values of PCR-LOC chip model

Parameter Symbol Value UnitInner radius of heater 119896

123 mm

Heater trace width 1198962

200 120583mChip radius 119886 494 mmChip height 119887 210 mmHeater power 119875 077 WThermal conductivity of Borofloat glass 119870 111 W(msdotK)Density of Borofloat glass 120588 2200 Kgm3

Thermal capacity of Borofloat glass C 830 JK

4 Simulation Results

The parameters with the values listed in Table 1 are used insimulation We assume that 119879

119908= 22∘C and that an input

power of oneWatt is being applied to the heater The input tothe system is given by the heat flux transferred from the heaterto the systemThis heat flux is considered in definition of119891(120588)which by itself determines the 119860

119898coefficients Definition of

119891(120588) is given based on the chosenmodel configuration by thefollowing equation

119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896

1+ 1198962le 119886 119902 is heat flux from the

heater to chip and 119870 is the thermal conductivity of the chipFor discontinuities we have 119891(119896

1) = 1199022119870 = 119891(119896

1+ 1198962) Now

for 119860119898we can write

119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)

For heat flux 119902 we have

119902 =119875

119860

(75)

where 119875 is the amount of the power transferred from theheater to the chip and119860 is the area of the heater which can becalculated as follows

119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)

Recalling the recurrence relations for the Bessel function[31] we have

119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)

For 119899 = 1 (77) can be written as follows

int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)

Using a change of variable 119909 = (119903119898119886)120588 for the integral

part of 119860119898we can write

int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)

Using (78) in (79) we get

int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)

Taking into account (75) and (80) in (74) we have

119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)

Now we have all the required equations to calculate thetemperature distribution using our analytic solution

The next important step before start running simulationsis how to compensate the error caused due to the simplifi-cation of the actual system into a single-domain model withBorofloat glass as the only material involved Substituting thePDMS layer with a glass layer could result in a significantamount of error andmismatch betweenmodel and the actualsystem To overcome this issue we constructed an optimiza-tion problem to optimize our model parameters arriving to abest fit between single domain analytic model and multido-main actual system Chip radius chip thickness and inputpower in the mathematical model are selected as the opti-mization degrees of freedom We focus our attention on thetwo regions that have the highest importanceThe first regionis a rectangular area which starts from the left bottom sideof the chamber and continues up to the top of the constanttemperature wall and is all made from glass in the analyticalmodel We called this region comparison aperture The sec-ond region which is a subset of comparison aperture and hashigher importance is reaction chamber To get quantitativemeasure of the error in both the chamber and the selectedaperture we define the following parameters in both regionsabsolute average error maximum absolute error and unifor-mity of the errorThe definitions of the first two measures areobvious from their naming and the last measure is defined asfollows

119880ap ≐ max (119890ap (120588 119911)) minusmin (119890ap (120588 119911)) 120588 119911 isin D1

119880ch ≐ max (119890ch (120588 119911)) minusmin (119890ch (120588 119911)) 120588 119911 isin D2

(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100

Max 10438 (∘C)Heat distribution in chip for one-watt input power (FEM)

Figure 6 Steady-state temperature distribution of the cross-section of the chip calculated by the FEMThe axis of symmetry is along the leftvertical side The lines bound four domains assigned to the thermals conductivities of glass PDMS and water The glass plates and PDMSlayer are 11mm thick and 254m thick respectively The thin rectangle on the left defines the PCR chamber The heater is located at the pointof the highest temperature in red

Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35

Heat distribution in chip for one-watt input power (PDE)

Figure 7 Steady-state temperature distribution calculated by the PDE model The geometry of the model is a small portion of the geometryof real system and the FEMmodelThe area shown corresponds to the upper left part of the graph in Figure 6The PDEmodel represents thesystem with a single material The dashed lines indicate the location of the chamber and PDMS layer but do not define domains of differentmaterials

where 119890ap is error at selected aperture indicated byD1 and 119890chis error at chamber region indicated by D

2 Next we define

our weighted-sum objective function including L2norm of

the error at chamber location uniformity measure at cham-ber location and uniformity measure at selected aperture asfollows

119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908

3are weight coefficients and 119860

2is area

of chamber region By selecting proper weights optimizationresults in a maximum absolute error of 014∘C in chambertemperature which is less than 003 of the error when com-pared to the results of a nonoptimized model (119886 = 35mm119887 = 1354mm and 119875 = 1W) satisfy the acceptable error cri-teria for reliable PCR operation Error distribution in wholeaperture area is depicted in Figure 8 It is clear that althoughthemaximumabsolute error reaches 36∘C in the area close tothe heater the error in the chamber area is kept lower than014∘C

The resulted steady-state temperature distributions basedon optimized analytical solution and the FEM simulation areshown in Figures 7 and 6 respectively Comparison apertureis indicated by dash lines The solutions of the two resultsshow an excellent degree of correlation verifying accuracy ofthe steady-state solution

Dynamic simulation of temperature evolution inside thereaction chamber is the next important result that we wouldlike to study about our analytic model Precise FEM simula-tions disclosed that lower center point of the chamber andupper edge of the chamber have the highest and the lowesttemperatures inside the chamber respectively (SW and NEedges of the chamber in Figure 5) The results follow the factthat upper center point of the chamber (NW edge of thechamber in Figure 5) has a temperature close to mean tem-perature of the chamberWe selected upper center point of thechamber to study the dynamic evolution of temperature inour model and in actual system An input power which con-sisted of three predefined values that are required to take thechamber temperature to denaturation annealing and exten-sion temperatures is applied to both models Both systemswere powered up after being settled in ambient temperatureand kept in each power level for 60 seconds The experimentran over a whole PCR cycle in a 240-second interval in open-loop conditionThe results are depicted in Figure 9The opti-mization was performed for the lowest temperature in PCRcycle However the error at the highest temperature is stillunder 1∘C Model dynamics nicely follows the FEM resultsand as we expected the transient response is about 1 fasterthan that of the actual system due to the difference betweenthe thermal properties of PDMS and Borofloat glass


01

23

4

005

115minus4

minus3

minus2

minus1

0

1

Aperture radius (mm)Aperture thickness (mm)

Erro

r (∘ C)

Error of optimized simplified model in selected aperturefor one-watt input power (PDE)

Figure 8 Error distribution in selected aperture after optimizationObviously themaximum absolute error within the selected apertureis kept under 36∘C It reaches to its maximum in the area closeto the heater The error in the chamber area is well kept under014∘CThe chamber area is a rectangular areawith 09mm thicknessand 15mm diameter just on the left bottom corner of the selectedaperture

5 Conclusion

This paper presents the derivation and the solution of theheat distribution equation of amicrofluidic chip designed andmanufactured at the Applied Miniaturization Laboratory atthe University of Alberta Canada The partial differentialequation featuring three-dimensional domain was used todescribe the heat distribution of the chip during a DNAamplification process The axisymmetric architecture of thechip helped reduce the domain to two-dimensional cylindri-cal form The PDE equation describing the heat distributionof the chip was divided into steady-state and transient equa-tions and then solved explicitly Precise FEM simulations inaddition to laboratory measurements were used to verify thedevelopedmodelThemodeling approach is considered to beappropriate for similar thermal processeswith collocated sen-sors and actuators Future work is being carried out towarddeveloping closed-loop tracking control based on boundarycontrol techniques

A dynamic simulation of the system was made using thedevelopedmodel to confirm that the scaling of the glass ther-mal properties could be adjusted to match the temperaturepredicted by the FEM simulations at all points in the PCRchamber with less than 1∘C error at all times during a PCRtemperature cycle

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

0 50 100 150 200 2500

0954

1274

1916

Transient response in chamber for a complete PCR cycle

Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)

Input powerFEM resultsPDE model

Figure 9 Transient response of the PDE model to a predefinedinput power trajectory in comparison to the FEM results The inputpower represents a complete PCR cycle There is less than 1 errorin dynamic response of the derived PDE model and the completeFEM simulation

Acknowledgments

This work was supported by the Natural Sciences and Engi-neering Research Council of Canada (NSERC) The authorswould like to thank S Poshtiban G V Kaigala and J Jiangfor many helpful technical discussions

References

[1] Y Zhang and P Ozdemir ldquoMicrofluidic DNA amplificationmdashareviewrdquo Analytica Chimica Acta vol 638 no 2 pp 115ndash1252009

[2] MNorthrupM Ching RWhite andRWatson ldquoDNAampli-fication with a microfabricated reaction chamberrdquo Transducersvol 93 p 924 1993

[3] P Wilding M A Shoffner and L J Kricka ldquoPCR in a siliconmicrostructurerdquoClinical Chemistry vol 40 no 9 pp 1815ndash18181994

[4] D S Yoon Y-S Lee Y Lee et al ldquoPrecise temperature controland rapid thermal cycling in a micromachined DNA poly-merase chain reaction chiprdquo Journal of Micromechanics andMicroengineering vol 12 no 6 pp 813ndash823 2002

[5] M P Dinca M Gheorghe M Aherne and P Galvin ldquoFastand accurate temperature control of a PCR microsystem witha disposable reactorrdquo Journal of Micromechanics andMicroengi-neering vol 19 no 6 Article ID 065009 2009

[6] M G Roper C J Easley and J P Landers ldquoAdvances inpolymerase chain reaction on microfluidic chipsrdquo AnalyticalChemistry vol 77 no 12 pp 3887ndash3893 2005

[7] C Zhang J Xu W Ma and W Zheng ldquoPCR microfluidicdevices forDNAamplificationrdquoBiotechnologyAdvances vol 24no 3 pp 243ndash284 2006

[8] G V Kaigala J Jiang C J Backhouse and H J MarquezldquoSystem design and modeling of a time-varying nonlinear


temperature controller for microfluidicsrdquo IEEE Transactions onControl Systems Technology vol 18 no 2 pp 521ndash530 2010

[9] M Imran and A Bhattacharyya ldquoThermal response of an on-chip assembly of RTD heaters sputtered sample andmicrother-mocouplesrdquo Sensors and Actuators A vol 121 no 2 pp 306ndash320 2005

[10] R Pal M Yang R Lin et al ldquoAn integrated microfluidic devicefor influenza and other genetic analysesrdquo Lab on a Chip vol 5no 10 pp 1024ndash1032 2005

[11] T Yamamoto T Nojima and T Fujii ldquoPDMS-glass hybridmicroreactor array with embedded temperature control deviceApplication to cell-free protein synthesisrdquo Lab on a Chip vol 2no 4 pp 197ndash202 2002

[12] B C Giordano J Ferrance S Swedberg A F R Huhmer andJ P Landers ldquoPolymerase chain reaction in polymeric micro-chips DNA amplification in less than 240 secondsrdquo AnalyticalBiochemistry vol 291 no 1 pp 124ndash132 2001

[13] J J Shah S G Sundaresan J Geist et al ldquoMicrowave dielectricheating of fluids in an integratedmicrofluidic devicerdquo Journal ofMicromechanics and Microengineering vol 17 no 11 pp 2224ndash2230 2007

[14] S Sundaresan B Polk D Reyes M Rao andM Gaitan ldquoTem-perature control ofmicrofluidic systems bymicrowave heatingrdquoin Proceedings of Micro Total Analysis Systems vol 1 pp 657ndash659 2005

[15] E Verpoorte ldquoMicrofluidic chips for clinical and forensic anal-ysisrdquo Electrophoresis vol 23 no 5 pp 677ndash712 2002

[16] A J DeMello ldquoControl and detection of chemical reactions inmicrofluidic systemsrdquo Nature vol 442 no 7101 pp 394ndash4022006

[17] C-S Liao G-B Lee H-S Liu T-M Hsieh and C-HLuo ldquoMiniature RT-PCR system for diagnosis of RNA-basedvirusesrdquoNucleic Acids Research vol 33 no 18 article e156 2005

[18] Q Xianbo and Y Jingqi ldquoHybrid control strategy for thermalcycling of PCRbased on amicrocomputerrdquo Instrumentation Sci-ence and Technology vol 35 no 1 pp 41ndash52 2007

[19] V P Iordanov B P Iliev V Joseph et al ldquoSensorized nanoliterreactor chamber for DNA multiplicationrdquo in Proceedings ofIEEE Sensors pp 229ndash232 October 2004

[20] I Erill S Campoy J Rus et al ldquoDevelopment of a CMOS-compatible PCR chip comparison of design and system strate-giesrdquo Journal of Micromechanics and Microengineering vol 14no 11 pp 1558ndash1568 2004

[21] V N Hoang Thermal management strategies for microfluidicdevices [MS thesis] University of Alberta 2008

[22] P M Pilarski S Adamia and C J Backhouse ldquoAn adaptablemicrovalving system for on-chip polymerase chain reactionsrdquoJournal of Immunological Methods vol 305 no 1 pp 48ndash582005

[23] Z Gao D Kong andC Gao ldquoModeling and control of complexdynamic systems applied mathematical aspectsrdquo Journal ofApplied Mathematics 2012

[24] R F Curtain and H Zwart An Introduction to Infinite-Dimensional Linear SystemsTheory vol 21 Springer New YorkNY USA 1995

[25] PD ChristofidesNonlinear andRobust Control of PDE SystemsBirkhauser Boston Mass USA 2001

[26] M Krstic and A Smyshlyaev Boundary Control of PDEs ACourse on Backstepping Designs Society for Industrial andApplied Mathematics (SIAM) Philadelphia Pa USA 2008

[27] G V KaigalaM Behnam C Bliss et al ldquoInexpensive universalserial bus-powered and fully portable lab-on-a-chip-based cap-illary electrophoresis instrumentrdquo IET Nanobiotechnology vol3 no 1 pp 1ndash7 2009

[28] G V Kaigala M Behnam A C E Bidulock et al ldquoA scalableand modular lab-on-a-chip genetic analysis instrumentrdquo Ana-lyst vol 135 no 7 pp 1606ndash1617 2010

[29] G V Kaigala V N Hoang A Stickel et al ldquoAn inexpensive andportable microchip-based platform for integrated RT-PCR andcapillary electrophoresisrdquo Analyst vol 133 no 3 pp 331ndash3382008

[30] J H Lienhard IV and J H Lienhard V A Heat Transfer Text-book Phlogiston Press 3rd edition 2008

[31] R V Churchill and J W Brown Fourier Series and BoundaryValue Problems McGraw-Hill NewYork NY USA 7th edition1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Genetic and molecular analyses are among the majorapplications of LOC devices and have become feasible by theadoption of the Polymerase Chain Reaction (PCR) techniqueUsing PCR specific regions of even a single DNA moleculecan be replicated up to about 107 to 108 times This processis also called selective amplification of DNA The amplifiedDNA segments can be later analyzed and compared to a stan-dard pattern to find out if they correspond to some diseasecondition PCR has been traditionally a lengthy molecularbiology operation but the LOC concept proposed in theearly 1990s [2 3] has pushed this technology to attainever shorter reaction times DNA amplification using PCRinvolves repeatedly cycling the temperature of the reactionchamber in three predefined temperatures Optimum reac-tion temperature and exposure times of the sample to eachtemperature are key factors for successful DNA amplification[4] The PCR-LOC research community emphasizes theimportance of plusmn1∘C temperature precision and rapid transi-tion time for high amplification efficiency At the same timeovershoots and undershoots which are likely to happen athigher speeds have a detrimental impact on the amplificationyield For example overshoots at denaturation temperaturewill reduce the lifetime of replicating enzyme (Taq poly-merase) Undershoots at annealing temperature may causethe amplification of unspecified products Nevertheless rapidthermal cycling of reaction chamber in DNA amplificationnot only reduces the amplification time but it is alsoimportant to ensure the stability of biological reagents duringamplification as the Taq polymerase has a half lifetime thatdecreases rapidly at temperatures close to 100∘C [5] It isimportant to note that about half of the time in a standardPCR process is spent in transitions which are undesirable Asa result researchers have focused their attention to reducetransition time between temperature steps in PCRprocess Toachieve this goal both structural design of the device andthermal control of the process need to be optimized Con-sequently various platforms for PCR-LOC devices have beenreported in the literature [6] which can be roughly dividedinto two major categories continuous-flow PCR and staticPCR [7] Furthermore different heating techniques have beenchosen where thermoelectricmodules (TEMs) and thin-filmresistive heaters [8ndash11] are the most commonly used tech-niques Other techniques such as infrared radiation [12] andmicrowaves [13 14] have also been reported These differentstructural designs are discussed in detail in the excellentreview papers by Zhang et al [7] Roper and coworkers [6]Verpoorte [15] and DeMello [16] The latest achievements inPCR-LOC devices design can be found in the recently pub-lished survey paper by Zhang and Ozdemir [1]

Most of the published work on PCR-LOC devices hasbeen devoted to the structural design aspects and less atten-tion has been paid to optimize the thermal control of the pro-cess As a result there is very little work in the literature on thedynamics and control of these microsystems Although neu-ral network based controllers [17] and hybrid controllers [18]are used for thermal control of PCR-LOC devices PIPIDcontrollers are still the most commonly preferred controllersto deal with the thermal control problem of PCR-LOCs [5 1920]Whatmakes the thermal control ofmost of the PCR-LOC

devices a challenging task is the fact that the temperatureinside the PCR-LOC chips not only depends on time but isalso spatially distributed Thus lumped modeling and con-troller design cannot exactly address the problem funda-mentally because in this approach the distributed parameternature of the system is neglected We observed this criticalproblem to some extent in the formerly designed PCR-LOCchip [21] where switching controllers were employed [8]ThePCR-LOC system studied in this paper is the next generationof the one described in our previous work [22] The newdesign has undergone significant improvement in structuraldesign with respect to its predecessor

While lumped modeling can precisely describe thedynamics of lumped systems it usually cannot provide afair description of the dynamics of the distributed parametersystemsThose systems that can be described by partial differ-ential equations are often referred to as distributed parametersystems or PDE systems which feature distributed dynamicsand are categorized under a general set of the systems knownas complex dynamical systems [23] Distributed parametersystems are subsets of a more general set called infinitedimensional systems because they can bemathematically for-mulated as differential equations on an abstract linear vectorspace of infinite dimension [24] In contrast lumped systemscan be formulated mathematically as ordinary differentialequations on a finite dimensional linear vector space Design-ing controllers for distributed parameter systems is calledinfinite dimensional controller design or PDE controllerdesign [24ndash26]

This paper characterizes the heat distribution dynamicsof a microfluidic thermal process The work begins witha description of the process and experimental setup Thenresults of FEM simulations on the system were analyzed tochoose a suitable structure for themodel Next the PDE equa-tions describing the model in steady-state and transient partswere derived These equations were solved analytically fromwhich the dynamic temperature distribution in the wholedomain can be expressed by an infinite dimensional modelNext we optimize the calculatedmodel tominimize the errordue to the simplification of the actual multidomain system toan abstract single-domain system FEM simulation and lab-oratory experiments are used to verify the model The resultof this work is a framework for boundary control design forthermal control of the aforementioned PCR-LOC device

2 Materials and Methods

21 Chip Configuration Genetic analysis usually consists ofthree steps sample preparation amplification and detectionIn the first step the DNA is extracted from a raw sample suchas blood Then a selected fragment of this DNA is amplifiedthrough the PCR process to obtain sufficient DNA copiesto perform detection Next the amplified region of DNAis labelled with a fluorescent dye and finally it is detectedthrough various analysis techniques [27]

The integrated genetic analysis platform that has beendeveloped in the AppliedMiniaturization Laboratory (AML)at theUniversity of Alberta is capable of performing completegenetic analysis on a singlemicrofluidic chip [28]This system


USB

Power

LaserMicrochip

LensFilter


CCDcamera









2width and 119896




Well 2

Well 1 Well 3

Laserbeam

V valves

Well 4

Pump

Pt heater

PCR chamber

Pump

PCR loadingwell

V-6V-5V-4

V-3V-2V-1


(a)

z

x y

Laserbeam

Flow layer

Control layer


(b)

PCR chamberGlass

PDMSGlass

Copperheatsink



Photodiode


with radius r


(c)





119870nabla2119906 = 120588C

120597119906

120597119905

(1)


21199061205971205932



Chamber

90 120583m15 mm

25 mm02 mm

Thermal insulation

7 mm


Borofloat glass

11 mm

PDMS 254 120583m

11 mm

Heat sink



nabla2119906 =

1205972119906

1205971205882+

1

120588

120597119906

120597120588

+1205971199062

1205971199112 (2)


120581 =119870

120588C(3)



120581nabla2119906 =

120597119906

120597119905

(4)


119906 (120588 119911 0) = 1199060(120588 119911) (5)

119906 (119886 119911 119905) = 119879119908 (6)

119906120588(0 119911 119905) = 0 (7)

119906119911(120588 119887 119905) = 0 (8)

119906119911(120588 0 119905) = minus119891 (120588) (9)





119906 (120588 119911 119905) = (120588 119911 119905) + 119906 (120588 119911) (10)


120581

119905= 120588120588

+ 119906120588120588

+1

120588

120588+

1

120588

119906120588+ 119911119911

+ 119906119911119911

= (120588120588

+1

120588

120588+ 119911119911) + (119906

120588120588+

1

120588

119906120588+ 119906119911119911)

(11)


+

(1120588)119906120588+119906119911119911


1

120581

119905= 120588120588

+1

120588

120588+ 119911119911

with

(120588 119911 0) = 0(120588 119911)

(119886 119911 119905) = 0

120588(0 119911 119905) = 0

119911(120588 119887 119905) = 0

119911(120588 0 119905) = 0

(12)

119906120588120588

+1

120588

119906120588+ 119906119911119911

= 0 with

119906 (119886 119911) = 119879119908

119906120588(0 119911) = 0

119906119911(120588 119887) = 0

119906119911(120588 0) = minus119891 (120588)

(13)



119906 (120588 119911) = 119875 (120588)119885 (119911) + 119879119908 (14)


P = 1967W

0 05 1 15 2 2520

40

60

80

100

120

140



25262728293031

32333435404550

Tem

pera

ture

T(∘

C)



11987510158401015840

(120588)

119875 (120588)

+1

120588

1198751015840

(120588)

119875 (120588)

= minus11988510158401015840

(119911)

119885 (119911)

(15)


2


12058811987510158401015840

+ 1198751015840

+ 120582

2

120588119875 = 0

11988510158401015840

minus 120582

2

119885 = 0

(16)

120588

z

b

a

k1 k2

120593



119875 (119886) = 0 (17)

1198751015840

(0) = 0 (18)

1198851015840

(119887) = 0 (19)

119875 (120588)1198851015840

(0) = minus119891 (120588) (20)


119875 = 11988811198690(120582120588) + 119888

21198840(120582120588) (21)

119885 = 1198883cosh 120582 (119887 minus 119911) + 119888

4sinh 120582 (119887 minus 119911) (22)




2= 0 because 119884


tion (21) becomes

119875 = 11988811198690(120582120588) (23)


1198851015840

(119911) = minus120582 (1198883sinh 120582 (119887 minus 119911) + 119888

4cosh 120582 (119887 minus 119911)) (24)

we see that

1198851015840

(119887) = minus1205821198884= 0 (25)


119885 (119911) = 1198883cosh 120582 (119887 minus 119911) (26)


119875 (119886) = 11988811198690(120582119886) = 0 (27)



119886120582 = 1199031 1199032 1199033 119903

119898 (28)


0(119909) = 0


120582119898=

119903119898

119886

119898 = 1 2 3 (29)


119906 (120588 119911) minus 119879119908= 119875119898(120588)119885119898(119911)

= 1198601198690(119903119898

119886

120588) cosh119903119898

119886

(119887 minus 119911)

(30)





119906 (120588 119911) = 119879119908+

infin

sum

119898=1

1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911) (31)

From

120597119906

120597119911

=

infin

sum

119898=1

minus 119860119898

119903119898

119886

1198690(119903119898

119886

120588) sinh119903119898

119886

(119887 minus 119911) (32)


infin

sum

119898=1

(119860119898

119903119898

119886

sinh119903119898119887

119886

) 1198690(119903119898

119886

120588) = 119891 (120588) (33)


119898(119903119898119886) sinh(119903

119898119887119886)) [31] so we get

119860119898=

2

119886119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

int

119886

0

1205881198690(119903119898

119886

120588)119891 (120588) 119889120588

(34)


(120588 119911 119905) = (120588)119885 (119911) (119905) (35)


12058810158401015840+ 1015840+ 1205832120588 = 0 (36)

11988510158401015840minus ]2119885 = 0 (37)

1015840+ 1205812 = 0 (38)


conditions become

(119886) = 0 (39)

1015840(0) = 0 (40)

1198851015840(119887) = 0 (41)

1198851015840(0) = 0 (42)


= 11988811198690(120583120588) + 119888

21198840(120583120588) (43)

119885 = 1198883cosh ] (119887 minus 119911) + 119888

4sinh ] (119887 minus 119911) (44)

= 1198885119890minus1205812119905 (45)



(120588) = 11988811198690(120583120588) (46)


1198851015840(119887) = minus]119888

4= 0 (47)


119885 (119911) = 1198883cosh ] (119887 minus 119911) (48)


(119886) = 11988811198690(120583119886) = 0 (49)

so we have

120583119898=

119903119898

119886

(50)


0(119909) = 0


119898(120588) = 119888

11198690(119903119898

119886

120588) (51)


1198851015840(0) = minus]119888

3sinh ]119887 = 0 (52)


] =119896120587119894

119887

119896 = 0 1 2 (53)


cosh 119894119909 = cos119909 (54)


119885119896(119911) = 119888

3cos 119896120587

119887(119887 minus 119911) (55)



2= 1205832minus ]2 = (

119903119898

119886

)

2

minus (119896120587119894

119887

)

2

=

1199032

119898

1198862+

11989621205872

1198872

(56)


(120588 119911 119905) = 119860119890minus120581(1199032

1198981198862+119896212058721198872)1199051198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(57)

where 119860 = 119888111988831198885 119896 = 0 1 2 3 and 119898 = 1 2 3

Replacing 119860 with 119860119896119898


(120588 119911 119905) =

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

119890minus120581(1199032

1198981198862+119896212058721198872)119905

times 1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(58)


infin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

= 1199060(120588 119911) minus 119906 (120588 119911)

(59)


sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911) = (1199060(120588 119911) minus 119879

119908)

+

infin

sum

119898=1

minus 1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911)

(60)

where 119860119898


0(120588 119911) = 119879

119908


infin

sum

119898=1

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) 1198690(119903119898

119886

120588)

=

infin

sum

119898=1


119903119898

119886(119887 minus 119911) 119869

0(119903119898

119886

120588)

(61)


infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886(119887 minus 119911) (62)


1198600119898

+

infin

sum

119896=1

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886

(119887 minus 119911) (63)


0119898and 119860

119896119898with 119886

02 and 119886


have1198860

2

+

infin

sum

119896=1

119886119896cos 119896120587

119887

119909 = minus119860119898cosh

119903119898

119886

119909 (64)


119886119896=

2

119887

int

119887

0


119903119898

119886

119909) cos 119896120587119887

119909 119889119911 (65)


119886119896= minus

2

119887

119860119898int

119887

0

cosh119903119898

119886

119909 cos 119896120587119887

119909 119889119911 (66)


119886119896= minus

2

119887

119860119898

119886119887

119886211989621205872+ 11988721199032

119898

times (119886119896120587 cosh (119903119898

119886

119887) sin (119896120587)

+119887119903119898cos (119896120587) sinh(

119903119898

119886

119887))

(67)


119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (68)

Thus we have

1198600119898

=1198860

2

= minus119860119898

119886

119887119903119898

sinh(119903119898

119886

119887) (69)

and 119860119896119898


119860119896119898

= 119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (70)




1199032

119898

1198862+

11989621205872

1198872

) (71)






123 mm










119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896



1) = 1199022119870 = 119891(119896

1+ 1198962) Now


119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)


119902 =119875

119860

(75)


119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)


119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)


int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)



int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)


int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)


119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)





(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










USB

Power

LaserMicrochip

LensFilter


CCDcamera









2width and 119896




Well 2

Well 1 Well 3

Laserbeam

V valves

Well 4

Pump

Pt heater

PCR chamber

Pump

PCR loadingwell

V-6V-5V-4

V-3V-2V-1


(a)

z

x y

Laserbeam

Flow layer

Control layer


(b)

PCR chamberGlass

PDMSGlass

Copperheatsink



Photodiode


with radius r


(c)





119870nabla2119906 = 120588C

120597119906

120597119905

(1)


21199061205971205932



Chamber

90 120583m15 mm

25 mm02 mm

Thermal insulation

7 mm


Borofloat glass

11 mm

PDMS 254 120583m

11 mm

Heat sink



nabla2119906 =

1205972119906

1205971205882+

1

120588

120597119906

120597120588

+1205971199062

1205971199112 (2)


120581 =119870

120588C(3)



120581nabla2119906 =

120597119906

120597119905

(4)


119906 (120588 119911 0) = 1199060(120588 119911) (5)

119906 (119886 119911 119905) = 119879119908 (6)

119906120588(0 119911 119905) = 0 (7)

119906119911(120588 119887 119905) = 0 (8)

119906119911(120588 0 119905) = minus119891 (120588) (9)





119906 (120588 119911 119905) = (120588 119911 119905) + 119906 (120588 119911) (10)


120581

119905= 120588120588

+ 119906120588120588

+1

120588

120588+

1

120588

119906120588+ 119911119911

+ 119906119911119911

= (120588120588

+1

120588

120588+ 119911119911) + (119906

120588120588+

1

120588

119906120588+ 119906119911119911)

(11)


+

(1120588)119906120588+119906119911119911


1

120581

119905= 120588120588

+1

120588

120588+ 119911119911

with

(120588 119911 0) = 0(120588 119911)

(119886 119911 119905) = 0

120588(0 119911 119905) = 0

119911(120588 119887 119905) = 0

119911(120588 0 119905) = 0

(12)

119906120588120588

+1

120588

119906120588+ 119906119911119911

= 0 with

119906 (119886 119911) = 119879119908

119906120588(0 119911) = 0

119906119911(120588 119887) = 0

119906119911(120588 0) = minus119891 (120588)

(13)



119906 (120588 119911) = 119875 (120588)119885 (119911) + 119879119908 (14)


P = 1967W

0 05 1 15 2 2520

40

60

80

100

120

140



25262728293031

32333435404550

Tem

pera

ture

T(∘

C)



11987510158401015840

(120588)

119875 (120588)

+1

120588

1198751015840

(120588)

119875 (120588)

= minus11988510158401015840

(119911)

119885 (119911)

(15)


2


12058811987510158401015840

+ 1198751015840

+ 120582

2

120588119875 = 0

11988510158401015840

minus 120582

2

119885 = 0

(16)

120588

z

b

a

k1 k2

120593



119875 (119886) = 0 (17)

1198751015840

(0) = 0 (18)

1198851015840

(119887) = 0 (19)

119875 (120588)1198851015840

(0) = minus119891 (120588) (20)


119875 = 11988811198690(120582120588) + 119888

21198840(120582120588) (21)

119885 = 1198883cosh 120582 (119887 minus 119911) + 119888

4sinh 120582 (119887 minus 119911) (22)




2= 0 because 119884


tion (21) becomes

119875 = 11988811198690(120582120588) (23)


1198851015840

(119911) = minus120582 (1198883sinh 120582 (119887 minus 119911) + 119888

4cosh 120582 (119887 minus 119911)) (24)

we see that

1198851015840

(119887) = minus1205821198884= 0 (25)


119885 (119911) = 1198883cosh 120582 (119887 minus 119911) (26)


119875 (119886) = 11988811198690(120582119886) = 0 (27)



119886120582 = 1199031 1199032 1199033 119903

119898 (28)


0(119909) = 0


120582119898=

119903119898

119886

119898 = 1 2 3 (29)


119906 (120588 119911) minus 119879119908= 119875119898(120588)119885119898(119911)

= 1198601198690(119903119898

119886

120588) cosh119903119898

119886

(119887 minus 119911)

(30)





119906 (120588 119911) = 119879119908+

infin

sum

119898=1

1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911) (31)

From

120597119906

120597119911

=

infin

sum

119898=1

minus 119860119898

119903119898

119886

1198690(119903119898

119886

120588) sinh119903119898

119886

(119887 minus 119911) (32)


infin

sum

119898=1

(119860119898

119903119898

119886

sinh119903119898119887

119886

) 1198690(119903119898

119886

120588) = 119891 (120588) (33)


119898(119903119898119886) sinh(119903

119898119887119886)) [31] so we get

119860119898=

2

119886119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

int

119886

0

1205881198690(119903119898

119886

120588)119891 (120588) 119889120588

(34)


(120588 119911 119905) = (120588)119885 (119911) (119905) (35)


12058810158401015840+ 1015840+ 1205832120588 = 0 (36)

11988510158401015840minus ]2119885 = 0 (37)

1015840+ 1205812 = 0 (38)


conditions become

(119886) = 0 (39)

1015840(0) = 0 (40)

1198851015840(119887) = 0 (41)

1198851015840(0) = 0 (42)


= 11988811198690(120583120588) + 119888

21198840(120583120588) (43)

119885 = 1198883cosh ] (119887 minus 119911) + 119888

4sinh ] (119887 minus 119911) (44)

= 1198885119890minus1205812119905 (45)



(120588) = 11988811198690(120583120588) (46)


1198851015840(119887) = minus]119888

4= 0 (47)


119885 (119911) = 1198883cosh ] (119887 minus 119911) (48)


(119886) = 11988811198690(120583119886) = 0 (49)

so we have

120583119898=

119903119898

119886

(50)


0(119909) = 0


119898(120588) = 119888

11198690(119903119898

119886

120588) (51)


1198851015840(0) = minus]119888

3sinh ]119887 = 0 (52)


] =119896120587119894

119887

119896 = 0 1 2 (53)


cosh 119894119909 = cos119909 (54)


119885119896(119911) = 119888

3cos 119896120587

119887(119887 minus 119911) (55)



2= 1205832minus ]2 = (

119903119898

119886

)

2

minus (119896120587119894

119887

)

2

=

1199032

119898

1198862+

11989621205872

1198872

(56)


(120588 119911 119905) = 119860119890minus120581(1199032

1198981198862+119896212058721198872)1199051198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(57)

where 119860 = 119888111988831198885 119896 = 0 1 2 3 and 119898 = 1 2 3

Replacing 119860 with 119860119896119898


(120588 119911 119905) =

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

119890minus120581(1199032

1198981198862+119896212058721198872)119905

times 1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(58)


infin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

= 1199060(120588 119911) minus 119906 (120588 119911)

(59)


sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911) = (1199060(120588 119911) minus 119879

119908)

+

infin

sum

119898=1

minus 1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911)

(60)

where 119860119898


0(120588 119911) = 119879

119908


infin

sum

119898=1

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) 1198690(119903119898

119886

120588)

=

infin

sum

119898=1


119903119898

119886(119887 minus 119911) 119869

0(119903119898

119886

120588)

(61)


infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886(119887 minus 119911) (62)


1198600119898

+

infin

sum

119896=1

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886

(119887 minus 119911) (63)


0119898and 119860

119896119898with 119886

02 and 119886


have1198860

2

+

infin

sum

119896=1

119886119896cos 119896120587

119887

119909 = minus119860119898cosh

119903119898

119886

119909 (64)


119886119896=

2

119887

int

119887

0


119903119898

119886

119909) cos 119896120587119887

119909 119889119911 (65)


119886119896= minus

2

119887

119860119898int

119887

0

cosh119903119898

119886

119909 cos 119896120587119887

119909 119889119911 (66)


119886119896= minus

2

119887

119860119898

119886119887

119886211989621205872+ 11988721199032

119898

times (119886119896120587 cosh (119903119898

119886

119887) sin (119896120587)

+119887119903119898cos (119896120587) sinh(

119903119898

119886

119887))

(67)


119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (68)

Thus we have

1198600119898

=1198860

2

= minus119860119898

119886

119887119903119898

sinh(119903119898

119886

119887) (69)

and 119860119896119898


119860119896119898

= 119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (70)




1199032

119898

1198862+

11989621205872

1198872

) (71)






123 mm










119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896



1) = 1199022119870 = 119891(119896

1+ 1198962) Now


119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)


119902 =119875

119860

(75)


119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)


119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)


int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)



int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)


int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)


119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)





(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Well 2

Well 1 Well 3

Laserbeam

V valves

Well 4

Pump

Pt heater

PCR chamber

Pump

PCR loadingwell

V-6V-5V-4

V-3V-2V-1


(a)

z

x y

Laserbeam

Flow layer

Control layer


(b)

PCR chamberGlass

PDMSGlass

Copperheatsink



Photodiode


with radius r


(c)





119870nabla2119906 = 120588C

120597119906

120597119905

(1)


21199061205971205932



Chamber

90 120583m15 mm

25 mm02 mm

Thermal insulation

7 mm


Borofloat glass

11 mm

PDMS 254 120583m

11 mm

Heat sink



nabla2119906 =

1205972119906

1205971205882+

1

120588

120597119906

120597120588

+1205971199062

1205971199112 (2)


120581 =119870

120588C(3)



120581nabla2119906 =

120597119906

120597119905

(4)


119906 (120588 119911 0) = 1199060(120588 119911) (5)

119906 (119886 119911 119905) = 119879119908 (6)

119906120588(0 119911 119905) = 0 (7)

119906119911(120588 119887 119905) = 0 (8)

119906119911(120588 0 119905) = minus119891 (120588) (9)





119906 (120588 119911 119905) = (120588 119911 119905) + 119906 (120588 119911) (10)


120581

119905= 120588120588

+ 119906120588120588

+1

120588

120588+

1

120588

119906120588+ 119911119911

+ 119906119911119911

= (120588120588

+1

120588

120588+ 119911119911) + (119906

120588120588+

1

120588

119906120588+ 119906119911119911)

(11)


+

(1120588)119906120588+119906119911119911


1

120581

119905= 120588120588

+1

120588

120588+ 119911119911

with

(120588 119911 0) = 0(120588 119911)

(119886 119911 119905) = 0

120588(0 119911 119905) = 0

119911(120588 119887 119905) = 0

119911(120588 0 119905) = 0

(12)

119906120588120588

+1

120588

119906120588+ 119906119911119911

= 0 with

119906 (119886 119911) = 119879119908

119906120588(0 119911) = 0

119906119911(120588 119887) = 0

119906119911(120588 0) = minus119891 (120588)

(13)



119906 (120588 119911) = 119875 (120588)119885 (119911) + 119879119908 (14)


P = 1967W

0 05 1 15 2 2520

40

60

80

100

120

140



25262728293031

32333435404550

Tem

pera

ture

T(∘

C)



11987510158401015840

(120588)

119875 (120588)

+1

120588

1198751015840

(120588)

119875 (120588)

= minus11988510158401015840

(119911)

119885 (119911)

(15)


2


12058811987510158401015840

+ 1198751015840

+ 120582

2

120588119875 = 0

11988510158401015840

minus 120582

2

119885 = 0

(16)

120588

z

b

a

k1 k2

120593



119875 (119886) = 0 (17)

1198751015840

(0) = 0 (18)

1198851015840

(119887) = 0 (19)

119875 (120588)1198851015840

(0) = minus119891 (120588) (20)


119875 = 11988811198690(120582120588) + 119888

21198840(120582120588) (21)

119885 = 1198883cosh 120582 (119887 minus 119911) + 119888

4sinh 120582 (119887 minus 119911) (22)




2= 0 because 119884


tion (21) becomes

119875 = 11988811198690(120582120588) (23)


1198851015840

(119911) = minus120582 (1198883sinh 120582 (119887 minus 119911) + 119888

4cosh 120582 (119887 minus 119911)) (24)

we see that

1198851015840

(119887) = minus1205821198884= 0 (25)


119885 (119911) = 1198883cosh 120582 (119887 minus 119911) (26)


119875 (119886) = 11988811198690(120582119886) = 0 (27)



119886120582 = 1199031 1199032 1199033 119903

119898 (28)


0(119909) = 0


120582119898=

119903119898

119886

119898 = 1 2 3 (29)


119906 (120588 119911) minus 119879119908= 119875119898(120588)119885119898(119911)

= 1198601198690(119903119898

119886

120588) cosh119903119898

119886

(119887 minus 119911)

(30)





119906 (120588 119911) = 119879119908+

infin

sum

119898=1

1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911) (31)

From

120597119906

120597119911

=

infin

sum

119898=1

minus 119860119898

119903119898

119886

1198690(119903119898

119886

120588) sinh119903119898

119886

(119887 minus 119911) (32)


infin

sum

119898=1

(119860119898

119903119898

119886

sinh119903119898119887

119886

) 1198690(119903119898

119886

120588) = 119891 (120588) (33)


119898(119903119898119886) sinh(119903

119898119887119886)) [31] so we get

119860119898=

2

119886119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

int

119886

0

1205881198690(119903119898

119886

120588)119891 (120588) 119889120588

(34)


(120588 119911 119905) = (120588)119885 (119911) (119905) (35)


12058810158401015840+ 1015840+ 1205832120588 = 0 (36)

11988510158401015840minus ]2119885 = 0 (37)

1015840+ 1205812 = 0 (38)


conditions become

(119886) = 0 (39)

1015840(0) = 0 (40)

1198851015840(119887) = 0 (41)

1198851015840(0) = 0 (42)


= 11988811198690(120583120588) + 119888

21198840(120583120588) (43)

119885 = 1198883cosh ] (119887 minus 119911) + 119888

4sinh ] (119887 minus 119911) (44)

= 1198885119890minus1205812119905 (45)



(120588) = 11988811198690(120583120588) (46)


1198851015840(119887) = minus]119888

4= 0 (47)


119885 (119911) = 1198883cosh ] (119887 minus 119911) (48)


(119886) = 11988811198690(120583119886) = 0 (49)

so we have

120583119898=

119903119898

119886

(50)


0(119909) = 0


119898(120588) = 119888

11198690(119903119898

119886

120588) (51)


1198851015840(0) = minus]119888

3sinh ]119887 = 0 (52)


] =119896120587119894

119887

119896 = 0 1 2 (53)


cosh 119894119909 = cos119909 (54)


119885119896(119911) = 119888

3cos 119896120587

119887(119887 minus 119911) (55)



2= 1205832minus ]2 = (

119903119898

119886

)

2

minus (119896120587119894

119887

)

2

=

1199032

119898

1198862+

11989621205872

1198872

(56)


(120588 119911 119905) = 119860119890minus120581(1199032

1198981198862+119896212058721198872)1199051198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(57)

where 119860 = 119888111988831198885 119896 = 0 1 2 3 and 119898 = 1 2 3

Replacing 119860 with 119860119896119898


(120588 119911 119905) =

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

119890minus120581(1199032

1198981198862+119896212058721198872)119905

times 1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(58)


infin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

= 1199060(120588 119911) minus 119906 (120588 119911)

(59)


sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911) = (1199060(120588 119911) minus 119879

119908)

+

infin

sum

119898=1

minus 1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911)

(60)

where 119860119898


0(120588 119911) = 119879

119908


infin

sum

119898=1

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) 1198690(119903119898

119886

120588)

=

infin

sum

119898=1


119903119898

119886(119887 minus 119911) 119869

0(119903119898

119886

120588)

(61)


infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886(119887 minus 119911) (62)


1198600119898

+

infin

sum

119896=1

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886

(119887 minus 119911) (63)


0119898and 119860

119896119898with 119886

02 and 119886


have1198860

2

+

infin

sum

119896=1

119886119896cos 119896120587

119887

119909 = minus119860119898cosh

119903119898

119886

119909 (64)


119886119896=

2

119887

int

119887

0


119903119898

119886

119909) cos 119896120587119887

119909 119889119911 (65)


119886119896= minus

2

119887

119860119898int

119887

0

cosh119903119898

119886

119909 cos 119896120587119887

119909 119889119911 (66)


119886119896= minus

2

119887

119860119898

119886119887

119886211989621205872+ 11988721199032

119898

times (119886119896120587 cosh (119903119898

119886

119887) sin (119896120587)

+119887119903119898cos (119896120587) sinh(

119903119898

119886

119887))

(67)


119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (68)

Thus we have

1198600119898

=1198860

2

= minus119860119898

119886

119887119903119898

sinh(119903119898

119886

119887) (69)

and 119860119896119898


119860119896119898

= 119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (70)




1199032

119898

1198862+

11989621205872

1198872

) (71)






123 mm










119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896



1) = 1199022119870 = 119891(119896

1+ 1198962) Now


119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)


119902 =119875

119860

(75)


119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)


119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)


int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)



int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)


int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)


119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)





(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Chamber

90 120583m15 mm

25 mm02 mm

Thermal insulation

7 mm


Borofloat glass

11 mm

PDMS 254 120583m

11 mm

Heat sink



nabla2119906 =

1205972119906

1205971205882+

1

120588

120597119906

120597120588

+1205971199062

1205971199112 (2)


120581 =119870

120588C(3)



120581nabla2119906 =

120597119906

120597119905

(4)


119906 (120588 119911 0) = 1199060(120588 119911) (5)

119906 (119886 119911 119905) = 119879119908 (6)

119906120588(0 119911 119905) = 0 (7)

119906119911(120588 119887 119905) = 0 (8)

119906119911(120588 0 119905) = minus119891 (120588) (9)





119906 (120588 119911 119905) = (120588 119911 119905) + 119906 (120588 119911) (10)


120581

119905= 120588120588

+ 119906120588120588

+1

120588

120588+

1

120588

119906120588+ 119911119911

+ 119906119911119911

= (120588120588

+1

120588

120588+ 119911119911) + (119906

120588120588+

1

120588

119906120588+ 119906119911119911)

(11)


+

(1120588)119906120588+119906119911119911


1

120581

119905= 120588120588

+1

120588

120588+ 119911119911

with

(120588 119911 0) = 0(120588 119911)

(119886 119911 119905) = 0

120588(0 119911 119905) = 0

119911(120588 119887 119905) = 0

119911(120588 0 119905) = 0

(12)

119906120588120588

+1

120588

119906120588+ 119906119911119911

= 0 with

119906 (119886 119911) = 119879119908

119906120588(0 119911) = 0

119906119911(120588 119887) = 0

119906119911(120588 0) = minus119891 (120588)

(13)



119906 (120588 119911) = 119875 (120588)119885 (119911) + 119879119908 (14)


P = 1967W

0 05 1 15 2 2520

40

60

80

100

120

140



25262728293031

32333435404550

Tem

pera

ture

T(∘

C)



11987510158401015840

(120588)

119875 (120588)

+1

120588

1198751015840

(120588)

119875 (120588)

= minus11988510158401015840

(119911)

119885 (119911)

(15)


2


12058811987510158401015840

+ 1198751015840

+ 120582

2

120588119875 = 0

11988510158401015840

minus 120582

2

119885 = 0

(16)

120588

z

b

a

k1 k2

120593



119875 (119886) = 0 (17)

1198751015840

(0) = 0 (18)

1198851015840

(119887) = 0 (19)

119875 (120588)1198851015840

(0) = minus119891 (120588) (20)


119875 = 11988811198690(120582120588) + 119888

21198840(120582120588) (21)

119885 = 1198883cosh 120582 (119887 minus 119911) + 119888

4sinh 120582 (119887 minus 119911) (22)




2= 0 because 119884


tion (21) becomes

119875 = 11988811198690(120582120588) (23)


1198851015840

(119911) = minus120582 (1198883sinh 120582 (119887 minus 119911) + 119888

4cosh 120582 (119887 minus 119911)) (24)

we see that

1198851015840

(119887) = minus1205821198884= 0 (25)


119885 (119911) = 1198883cosh 120582 (119887 minus 119911) (26)


119875 (119886) = 11988811198690(120582119886) = 0 (27)



119886120582 = 1199031 1199032 1199033 119903

119898 (28)


0(119909) = 0


120582119898=

119903119898

119886

119898 = 1 2 3 (29)


119906 (120588 119911) minus 119879119908= 119875119898(120588)119885119898(119911)

= 1198601198690(119903119898

119886

120588) cosh119903119898

119886

(119887 minus 119911)

(30)





119906 (120588 119911) = 119879119908+

infin

sum

119898=1

1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911) (31)

From

120597119906

120597119911

=

infin

sum

119898=1

minus 119860119898

119903119898

119886

1198690(119903119898

119886

120588) sinh119903119898

119886

(119887 minus 119911) (32)


infin

sum

119898=1

(119860119898

119903119898

119886

sinh119903119898119887

119886

) 1198690(119903119898

119886

120588) = 119891 (120588) (33)


119898(119903119898119886) sinh(119903

119898119887119886)) [31] so we get

119860119898=

2

119886119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

int

119886

0

1205881198690(119903119898

119886

120588)119891 (120588) 119889120588

(34)


(120588 119911 119905) = (120588)119885 (119911) (119905) (35)


12058810158401015840+ 1015840+ 1205832120588 = 0 (36)

11988510158401015840minus ]2119885 = 0 (37)

1015840+ 1205812 = 0 (38)


conditions become

(119886) = 0 (39)

1015840(0) = 0 (40)

1198851015840(119887) = 0 (41)

1198851015840(0) = 0 (42)


= 11988811198690(120583120588) + 119888

21198840(120583120588) (43)

119885 = 1198883cosh ] (119887 minus 119911) + 119888

4sinh ] (119887 minus 119911) (44)

= 1198885119890minus1205812119905 (45)



(120588) = 11988811198690(120583120588) (46)


1198851015840(119887) = minus]119888

4= 0 (47)


119885 (119911) = 1198883cosh ] (119887 minus 119911) (48)


(119886) = 11988811198690(120583119886) = 0 (49)

so we have

120583119898=

119903119898

119886

(50)


0(119909) = 0


119898(120588) = 119888

11198690(119903119898

119886

120588) (51)


1198851015840(0) = minus]119888

3sinh ]119887 = 0 (52)


] =119896120587119894

119887

119896 = 0 1 2 (53)


cosh 119894119909 = cos119909 (54)


119885119896(119911) = 119888

3cos 119896120587

119887(119887 minus 119911) (55)



2= 1205832minus ]2 = (

119903119898

119886

)

2

minus (119896120587119894

119887

)

2

=

1199032

119898

1198862+

11989621205872

1198872

(56)


(120588 119911 119905) = 119860119890minus120581(1199032

1198981198862+119896212058721198872)1199051198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(57)

where 119860 = 119888111988831198885 119896 = 0 1 2 3 and 119898 = 1 2 3

Replacing 119860 with 119860119896119898


(120588 119911 119905) =

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

119890minus120581(1199032

1198981198862+119896212058721198872)119905

times 1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(58)


infin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

= 1199060(120588 119911) minus 119906 (120588 119911)

(59)


sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911) = (1199060(120588 119911) minus 119879

119908)

+

infin

sum

119898=1

minus 1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911)

(60)

where 119860119898


0(120588 119911) = 119879

119908


infin

sum

119898=1

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) 1198690(119903119898

119886

120588)

=

infin

sum

119898=1


119903119898

119886(119887 minus 119911) 119869

0(119903119898

119886

120588)

(61)


infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886(119887 minus 119911) (62)


1198600119898

+

infin

sum

119896=1

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886

(119887 minus 119911) (63)


0119898and 119860

119896119898with 119886

02 and 119886


have1198860

2

+

infin

sum

119896=1

119886119896cos 119896120587

119887

119909 = minus119860119898cosh

119903119898

119886

119909 (64)


119886119896=

2

119887

int

119887

0


119903119898

119886

119909) cos 119896120587119887

119909 119889119911 (65)


119886119896= minus

2

119887

119860119898int

119887

0

cosh119903119898

119886

119909 cos 119896120587119887

119909 119889119911 (66)


119886119896= minus

2

119887

119860119898

119886119887

119886211989621205872+ 11988721199032

119898

times (119886119896120587 cosh (119903119898

119886

119887) sin (119896120587)

+119887119903119898cos (119896120587) sinh(

119903119898

119886

119887))

(67)


119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (68)

Thus we have

1198600119898

=1198860

2

= minus119860119898

119886

119887119903119898

sinh(119903119898

119886

119887) (69)

and 119860119896119898


119860119896119898

= 119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (70)




1199032

119898

1198862+

11989621205872

1198872

) (71)






123 mm










119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896



1) = 1199022119870 = 119891(119896

1+ 1198962) Now


119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)


119902 =119875

119860

(75)


119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)


119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)


int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)



int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)


int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)


119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)





(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










P = 1967W

0 05 1 15 2 2520

40

60

80

100

120

140



25262728293031

32333435404550

Tem

pera

ture

T(∘

C)



11987510158401015840

(120588)

119875 (120588)

+1

120588

1198751015840

(120588)

119875 (120588)

= minus11988510158401015840

(119911)

119885 (119911)

(15)


2


12058811987510158401015840

+ 1198751015840

+ 120582

2

120588119875 = 0

11988510158401015840

minus 120582

2

119885 = 0

(16)

120588

z

b

a

k1 k2

120593



119875 (119886) = 0 (17)

1198751015840

(0) = 0 (18)

1198851015840

(119887) = 0 (19)

119875 (120588)1198851015840

(0) = minus119891 (120588) (20)


119875 = 11988811198690(120582120588) + 119888

21198840(120582120588) (21)

119885 = 1198883cosh 120582 (119887 minus 119911) + 119888

4sinh 120582 (119887 minus 119911) (22)




2= 0 because 119884


tion (21) becomes

119875 = 11988811198690(120582120588) (23)


1198851015840

(119911) = minus120582 (1198883sinh 120582 (119887 minus 119911) + 119888

4cosh 120582 (119887 minus 119911)) (24)

we see that

1198851015840

(119887) = minus1205821198884= 0 (25)


119885 (119911) = 1198883cosh 120582 (119887 minus 119911) (26)


119875 (119886) = 11988811198690(120582119886) = 0 (27)



119886120582 = 1199031 1199032 1199033 119903

119898 (28)


0(119909) = 0


120582119898=

119903119898

119886

119898 = 1 2 3 (29)


119906 (120588 119911) minus 119879119908= 119875119898(120588)119885119898(119911)

= 1198601198690(119903119898

119886

120588) cosh119903119898

119886

(119887 minus 119911)

(30)





119906 (120588 119911) = 119879119908+

infin

sum

119898=1

1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911) (31)

From

120597119906

120597119911

=

infin

sum

119898=1

minus 119860119898

119903119898

119886

1198690(119903119898

119886

120588) sinh119903119898

119886

(119887 minus 119911) (32)


infin

sum

119898=1

(119860119898

119903119898

119886

sinh119903119898119887

119886

) 1198690(119903119898

119886

120588) = 119891 (120588) (33)


119898(119903119898119886) sinh(119903

119898119887119886)) [31] so we get

119860119898=

2

119886119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

int

119886

0

1205881198690(119903119898

119886

120588)119891 (120588) 119889120588

(34)


(120588 119911 119905) = (120588)119885 (119911) (119905) (35)


12058810158401015840+ 1015840+ 1205832120588 = 0 (36)

11988510158401015840minus ]2119885 = 0 (37)

1015840+ 1205812 = 0 (38)


conditions become

(119886) = 0 (39)

1015840(0) = 0 (40)

1198851015840(119887) = 0 (41)

1198851015840(0) = 0 (42)


= 11988811198690(120583120588) + 119888

21198840(120583120588) (43)

119885 = 1198883cosh ] (119887 minus 119911) + 119888

4sinh ] (119887 minus 119911) (44)

= 1198885119890minus1205812119905 (45)



(120588) = 11988811198690(120583120588) (46)


1198851015840(119887) = minus]119888

4= 0 (47)


119885 (119911) = 1198883cosh ] (119887 minus 119911) (48)


(119886) = 11988811198690(120583119886) = 0 (49)

so we have

120583119898=

119903119898

119886

(50)


0(119909) = 0


119898(120588) = 119888

11198690(119903119898

119886

120588) (51)


1198851015840(0) = minus]119888

3sinh ]119887 = 0 (52)


] =119896120587119894

119887

119896 = 0 1 2 (53)


cosh 119894119909 = cos119909 (54)


119885119896(119911) = 119888

3cos 119896120587

119887(119887 minus 119911) (55)



2= 1205832minus ]2 = (

119903119898

119886

)

2

minus (119896120587119894

119887

)

2

=

1199032

119898

1198862+

11989621205872

1198872

(56)


(120588 119911 119905) = 119860119890minus120581(1199032

1198981198862+119896212058721198872)1199051198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(57)

where 119860 = 119888111988831198885 119896 = 0 1 2 3 and 119898 = 1 2 3

Replacing 119860 with 119860119896119898


(120588 119911 119905) =

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

119890minus120581(1199032

1198981198862+119896212058721198872)119905

times 1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(58)


infin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

= 1199060(120588 119911) minus 119906 (120588 119911)

(59)


sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911) = (1199060(120588 119911) minus 119879

119908)

+

infin

sum

119898=1

minus 1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911)

(60)

where 119860119898


0(120588 119911) = 119879

119908


infin

sum

119898=1

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) 1198690(119903119898

119886

120588)

=

infin

sum

119898=1


119903119898

119886(119887 minus 119911) 119869

0(119903119898

119886

120588)

(61)


infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886(119887 minus 119911) (62)


1198600119898

+

infin

sum

119896=1

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886

(119887 minus 119911) (63)


0119898and 119860

119896119898with 119886

02 and 119886


have1198860

2

+

infin

sum

119896=1

119886119896cos 119896120587

119887

119909 = minus119860119898cosh

119903119898

119886

119909 (64)


119886119896=

2

119887

int

119887

0


119903119898

119886

119909) cos 119896120587119887

119909 119889119911 (65)


119886119896= minus

2

119887

119860119898int

119887

0

cosh119903119898

119886

119909 cos 119896120587119887

119909 119889119911 (66)


119886119896= minus

2

119887

119860119898

119886119887

119886211989621205872+ 11988721199032

119898

times (119886119896120587 cosh (119903119898

119886

119887) sin (119896120587)

+119887119903119898cos (119896120587) sinh(

119903119898

119886

119887))

(67)


119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (68)

Thus we have

1198600119898

=1198860

2

= minus119860119898

119886

119887119903119898

sinh(119903119898

119886

119887) (69)

and 119860119896119898


119860119896119898

= 119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (70)




1199032

119898

1198862+

11989621205872

1198872

) (71)






123 mm










119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896



1) = 1199022119870 = 119891(119896

1+ 1198962) Now


119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)


119902 =119875

119860

(75)


119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)


119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)


int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)



int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)


int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)


119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)





(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886120582 = 1199031 1199032 1199033 119903

119898 (28)


0(119909) = 0


120582119898=

119903119898

119886

119898 = 1 2 3 (29)


119906 (120588 119911) minus 119879119908= 119875119898(120588)119885119898(119911)

= 1198601198690(119903119898

119886

120588) cosh119903119898

119886

(119887 minus 119911)

(30)





119906 (120588 119911) = 119879119908+

infin

sum

119898=1

1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911) (31)

From

120597119906

120597119911

=

infin

sum

119898=1

minus 119860119898

119903119898

119886

1198690(119903119898

119886

120588) sinh119903119898

119886

(119887 minus 119911) (32)


infin

sum

119898=1

(119860119898

119903119898

119886

sinh119903119898119887

119886

) 1198690(119903119898

119886

120588) = 119891 (120588) (33)


119898(119903119898119886) sinh(119903

119898119887119886)) [31] so we get

119860119898=

2

119886119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

int

119886

0

1205881198690(119903119898

119886

120588)119891 (120588) 119889120588

(34)


(120588 119911 119905) = (120588)119885 (119911) (119905) (35)


12058810158401015840+ 1015840+ 1205832120588 = 0 (36)

11988510158401015840minus ]2119885 = 0 (37)

1015840+ 1205812 = 0 (38)


conditions become

(119886) = 0 (39)

1015840(0) = 0 (40)

1198851015840(119887) = 0 (41)

1198851015840(0) = 0 (42)


= 11988811198690(120583120588) + 119888

21198840(120583120588) (43)

119885 = 1198883cosh ] (119887 minus 119911) + 119888

4sinh ] (119887 minus 119911) (44)

= 1198885119890minus1205812119905 (45)



(120588) = 11988811198690(120583120588) (46)


1198851015840(119887) = minus]119888

4= 0 (47)


119885 (119911) = 1198883cosh ] (119887 minus 119911) (48)


(119886) = 11988811198690(120583119886) = 0 (49)

so we have

120583119898=

119903119898

119886

(50)


0(119909) = 0


119898(120588) = 119888

11198690(119903119898

119886

120588) (51)


1198851015840(0) = minus]119888

3sinh ]119887 = 0 (52)


] =119896120587119894

119887

119896 = 0 1 2 (53)


cosh 119894119909 = cos119909 (54)


119885119896(119911) = 119888

3cos 119896120587

119887(119887 minus 119911) (55)



2= 1205832minus ]2 = (

119903119898

119886

)

2

minus (119896120587119894

119887

)

2

=

1199032

119898

1198862+

11989621205872

1198872

(56)


(120588 119911 119905) = 119860119890minus120581(1199032

1198981198862+119896212058721198872)1199051198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(57)

where 119860 = 119888111988831198885 119896 = 0 1 2 3 and 119898 = 1 2 3

Replacing 119860 with 119860119896119898


(120588 119911 119905) =

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

119890minus120581(1199032

1198981198862+119896212058721198872)119905

times 1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(58)


infin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

= 1199060(120588 119911) minus 119906 (120588 119911)

(59)


sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911) = (1199060(120588 119911) minus 119879

119908)

+

infin

sum

119898=1

minus 1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911)

(60)

where 119860119898


0(120588 119911) = 119879

119908


infin

sum

119898=1

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) 1198690(119903119898

119886

120588)

=

infin

sum

119898=1


119903119898

119886(119887 minus 119911) 119869

0(119903119898

119886

120588)

(61)


infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886(119887 minus 119911) (62)


1198600119898

+

infin

sum

119896=1

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886

(119887 minus 119911) (63)


0119898and 119860

119896119898with 119886

02 and 119886


have1198860

2

+

infin

sum

119896=1

119886119896cos 119896120587

119887

119909 = minus119860119898cosh

119903119898

119886

119909 (64)


119886119896=

2

119887

int

119887

0


119903119898

119886

119909) cos 119896120587119887

119909 119889119911 (65)


119886119896= minus

2

119887

119860119898int

119887

0

cosh119903119898

119886

119909 cos 119896120587119887

119909 119889119911 (66)


119886119896= minus

2

119887

119860119898

119886119887

119886211989621205872+ 11988721199032

119898

times (119886119896120587 cosh (119903119898

119886

119887) sin (119896120587)

+119887119903119898cos (119896120587) sinh(

119903119898

119886

119887))

(67)


119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (68)

Thus we have

1198600119898

=1198860

2

= minus119860119898

119886

119887119903119898

sinh(119903119898

119886

119887) (69)

and 119860119896119898


119860119896119898

= 119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (70)




1199032

119898

1198862+

11989621205872

1198872

) (71)






123 mm










119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896



1) = 1199022119870 = 119891(119896

1+ 1198962) Now


119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)


119902 =119875

119860

(75)


119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)


119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)


int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)



int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)


int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)


119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)





(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2= 1205832minus ]2 = (

119903119898

119886

)

2

minus (119896120587119894

119887

)

2

=

1199032

119898

1198862+

11989621205872

1198872

(56)


(120588 119911 119905) = 119860119890minus120581(1199032

1198981198862+119896212058721198872)1199051198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(57)

where 119860 = 119888111988831198885 119896 = 0 1 2 3 and 119898 = 1 2 3

Replacing 119860 with 119860119896119898


(120588 119911 119905) =

infin

sum

119896=0

infin

sum

119898=1

119860119896119898

119890minus120581(1199032

1198981198862+119896212058721198872)119905

times 1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

(58)


infin

sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911)

= 1199060(120588 119911) minus 119906 (120588 119911)

(59)


sum

119896=0

infin

sum

119898=1

119860119896119898

1198690(119903119898

119886

120588) cos 119896120587119887

(119887 minus 119911) = (1199060(120588 119911) minus 119879

119908)

+

infin

sum

119898=1

minus 1198601198981198690(119903119898

119886

120588) cosh119903119898

119886(119887 minus 119911)

(60)

where 119860119898


0(120588 119911) = 119879

119908


infin

sum

119898=1

infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) 1198690(119903119898

119886

120588)

=

infin

sum

119898=1


119903119898

119886(119887 minus 119911) 119869

0(119903119898

119886

120588)

(61)


infin

sum

119896=0

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886(119887 minus 119911) (62)


1198600119898

+

infin

sum

119896=1

119860119896119898

cos 119896120587119887

(119887 minus 119911) = minus119860119898cosh

119903119898

119886

(119887 minus 119911) (63)


0119898and 119860

119896119898with 119886

02 and 119886


have1198860

2

+

infin

sum

119896=1

119886119896cos 119896120587

119887

119909 = minus119860119898cosh

119903119898

119886

119909 (64)


119886119896=

2

119887

int

119887

0


119903119898

119886

119909) cos 119896120587119887

119909 119889119911 (65)


119886119896= minus

2

119887

119860119898int

119887

0

cosh119903119898

119886

119909 cos 119896120587119887

119909 119889119911 (66)


119886119896= minus

2

119887

119860119898

119886119887

119886211989621205872+ 11988721199032

119898

times (119886119896120587 cosh (119903119898

119886

119887) sin (119896120587)

+119887119903119898cos (119896120587) sinh(

119903119898

119886

119887))

(67)


119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (68)

Thus we have

1198600119898

=1198860

2

= minus119860119898

119886

119887119903119898

sinh(119903119898

119886

119887) (69)

and 119860119896119898


119860119896119898

= 119886119896= 2119860119898

119903119898

119886119887

(minus1)119896+1

1199032

1198981198862+ 119896212058721198872sinh(

119903119898

119886

119887) (70)




1199032

119898

1198862+

11989621205872

1198872

) (71)






123 mm










119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896



1) = 1199022119870 = 119891(119896

1+ 1198962) Now


119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)


119902 =119875

119860

(75)


119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)


119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)


int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)



int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)


int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)


119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)





(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












123 mm










119891 (120588) =

119902

119870

if 1198961lt 120588 lt (119896

1+ 1198962)

0 otherwise(73)

where 0 lt 1198961 1198962le 119886 0 lt 119896



1) = 1199022119870 = 119891(119896

1+ 1198962) Now


119860119898=

2119902

119886119870119903119898sinh (119903

119898119887119886) 119869

2

1(119903119898)

times int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

(74)


119902 =119875

119860

(75)


119860 = 120587(1198961+ 1198962)2

minus 1205871198962

1= 1205871198962(21198961+ 1198962) (76)


119889

119889119909

[119909119899119869119899(119909)] = 119909

119899119869119899minus1

(119909) (119899 = 1 2 ) (77)


int1199091198690(119909) 119889119909 = 119909119869

1(119909) + 119888 (78)



int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588 = (119886

119903119898

)

2

int

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

1199091198690(119909) 119889119909

(79)


int

1198961+1198962

1198961

1205881198690(119903119898

119886

120588) 119889120588

= (119886

119903119898

)

2

[1199091198691(119909)]

100381610038161003816100381610038161003816100381610038161003816

(119903119898119886)(1198961+1198962)

(119903119898119886)1198961

=119886

119903119898

((1198961+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962)) minus 119896

11198691(119903119898

119886

1198961))

(80)


119860119898= 2119875((119896

1+ 1198962) 1198691(119903119898

119886

(1198961+ 1198962))

minus11989611198691(119903119898

119886

1198961))

times (1205871198962(21198961+ 1198962)1198701199032

119898sinh(

119903119898119887

119886

) 1198692

1(119903119898))

minus1

(81)





(82)


0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7minus1

minus05

0

05

1

Chip radius (mm)Ch

ip th

ickn

ess (

mm

)

30405060708090100



Chip radius (mm)

Chip

thic

knes

s (m

m)

30405060708090100

108060402

0minus02

Max 10889 (∘C)

0 05 1 15 2 25 3 35




2 Next we define



119869 = 1199081

1

1198602

int

1198602

119890ch21198891198602+ 1199082119880ch + 119908

3119880ap (83)

where 1199081 1199082 and 119908


2is area





01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01

23

4

005

115minus4

minus3

minus2

minus1

0

1


Erro

r (∘ C)



5 Conclusion





0 50 100 150 200 2500

0954

1274

1916


Inpu

t pow

er (W

)

Time (s)

22

58

70

94

Tem

pera

ture

(∘C)



Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









pde modeling of a microfluidic thermal process for genetic

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