The work takes an advantage of the temperature component method for some heat conduction problems. Simplifications of the method for 2D problems are considered. Some examples of calculations are quoted

Jankowski, Janusz

Starosta, Roman

Walczak, Tomasz

English

Messanae Universitas Studiorum

DOI:10.1478/C1S0801021Atti dell’Accademia Peloritana dei PericolantiClasse di Scienze Fisiche, Matematiche e NaturaliVol. LXXXVI, C1S0801021 (2008) - Suppl. 1ACTION OF A HEAT SOURCE AND INFLUENCE OF INITIALCONDITION ON THE PLANE STATE OF TEMPERATUREROMAN STAROSTA, TOMASZ WALCZAK, AND JANUSZ JANKOWSKI*ABSTRACT. The work takes an advantage of the temperature component method for someheat conduction problems. Simplifications of the method for 2D problems are considered.Some examples of calculations are quoted.1. IntroductionThe temperature component method enables solving linear problems of heat conductionfor homogeneous and isotropic bodies. The method required the space value problem to beformulated. Fictitious extension of the body region to the whole space leads to the spaceproblem with known solution. The field obtained this way turn out a particular solution ofthe primary initial-boundary value problem. The temperature component is determined inan approximate form, enabling to satisfy the boundary conditions with imposed accuracy,and, at the same time, exactly observing the governing equations and the initial condition.Intensity values of the fictitious components located beyond the body region are deter-mined in fictitious instants. For this purpose a bounded time interval was introduced inwhich the boundary condition is verified. The testing instants divide the bounded time intoparticular sections. The verifying algorithm of the boundary condition, which is of recur-sive character with regard to the testing time, is based on the properties of the fundamentalsolution for the parabolic operator.The theory of heat conduction relates to a homogeneous and isotropic model of a bodypro-vided that the heat sources are subject to time-changes. The thermal process is de-scribed by the temperature field: T (~(x), t), x ∈ D, t > 0, where x determines the placewith regard to the reference system, and t is the current instant. The interval (0,∞) isreferred to as the time, (0, t] is the current time. A homogeneous and isotropic material isdescribed by Fourier relation(1) qi (~x, t) = −λT,i (~x, t) , ~x ∈ D, t > 0, i = ¯1, 3where qi is the heat flux vector and λ is thermal conductivity. Thermal flow through a facedetermined by a normal versor νi is defined by(2) νq (~x, t) = qj (~x, t) νj (~x, t) .This thermal field illustrates the rate of heat conduction in an arbitrary chosen direction.12 R. STAROSTA ET AL.2. Heat conduction problemLet νi denotes the outward normal to the boundary ∂D. For the process undergoing inthe time (0,∞) in theD domain the limiting condition is considered as an initial conditionfor t = 0 and boundary condition at ∂D. The heat conduction problem includes governingequation(3) T,jj (~x, t) =1κ∂∂tT (~x, t)− 1λW (~x, t) , ~x ∈ D, t > 0,initial condition(4) T (~x, 0) = g (x) , ~x ∈ D,and boundary condition(5) λT,j (~r, t) νj (~r, t) + αT (~r, t) = V (~r, t) + αbT (~r, t) , ~r ∈ ∂D, t > 0.The symbol W is the heat generated per unit volume and time, g stands here for the ini-tial temperature state, thermal diffusivity κ determines the rate of temperature changes,α surface conductance ([1], p. 19), V is the heat generated per unit surface and time bTsurrounding medium temperature. Formulae (3)-(5) are referred to as the heat conductionproblem.The approximate solution of the heat conduction problem obtained by temperature com-ponent method is as follows [2]aT (~x, t) =κλW (D × (0, t)) ∗ F (~x, t)(6) +g(D) ∗ F (~x, t) +p∑l=1n∑m=1w̃ml F (~x− m~y, t− ls).Here F means a fundamental solution to the parabolic operator for the three-dimensionalspace Z [3](7) F (~z, t) =0, t < 0, ~z ∈ Z0, t = 0, ~z 6= ~0(2√πκt)−3exp(− |~z|24κt), t > 0, ~z ∈ ZThe denotation (D)∗ will be called a space convolution multiplication on the set D ×D ,and its result is a space convolution(8) f (D) ∗ F (~z, t) ≡∫Df (~y)F (~z − ~y, t) d~y.Symbol is named a space-time convolution multiplication on the set D × (0, t), and itsresult a space-time convolution (D × (0, t))∗(9) f (D × (0, t)) ∗ F (~z, t) ≡t∫0∫Df (~y, s)F (~z − ~y, t− s) d~yds.ACTION OF A HEAT SOURCE AND INFLUENCE OF INITIAL CONDITION ... 3In the formula (6) the first two components depend on heat source and initial thermal state.Component is responsible for fulfill the boundary condition (5). Let us to sign the thirdcomponent by fW . It should to meet the necessary condition(10) fW,jj (~x, t) =1κ∂∂tfW (~x, t) , fW (~x, 0) = 0, ~x ∈ D, t > 0,which is called the component admissibility condition.The boundary condition enables determining of the temperature component(11) λfW,j (~r, t) νj (~r, t) + αfW (~r, t) = f (~r, t) , ~r ∈ ∂D, t > 0.It is here sufficient condition for temperature component. The following denotation is usedhere(12) f (~r, t) ≡ V (~r, t) + αbT (~r, t)− λ[g(D) ∗ F (~r, t)]νj (~r)− αg(D) ∗ F (~r, t).Nevertheless, the temperature component determined this way usually does not satisfy thecomponent admissibility condition (10).Let D̄ = D ∪ ∂D is a body region, inclusive of its boundary. The time (0,∞) willbe contracted to the interval (0,∞) called the current time. An approximate component isintroduced instead of the temperature one(13) afW(~z, t̃)≡p∑l=1n∑m=1wml F(~z − m~y, t̃− ls), ~z ∈ Z̄, t̃ ∈ (0, t] ,m~y ∈ Z \ D̄.The elements in the sum (13) are called fictitious components, my a fictitious place of them-th fictitious component, ls the l-th fictitious instant, wml the component intensity in themy place at the ls instant, the set {m~y : m = ¯1, n} fictitious location, and {l~s : l = ¯1, p}fictitious time. This shape of the temperature component we can find in the solution (6).3. Two - dimensional problemsThe physical processes are three dimensional. But in many cases one could assumean independence of thermal flow from one or two directions. In this way the solutionscan be obtained that have some technical applications. Of course, some calculations areusually much simpler than in a general three dimensional case. We will consider the heatconduction problem with a thermal load depend on time and 2-dimensional position vector2~x = (x1, x2, 0) = ~x(x3 = 0) only. We also will suppose that each cross-section of thebody model perpendicular to the third axis can not be distinguished. One of these cross-sections will be sign by D̃ and called reference section. The region should be long enoughin the third direction (Fig.1).For such assumptions we can speak about two-dimensional heat conduction problem(14) T,jj (~x, t) =1κ∂∂tT (~x, t)− 1λ2W (2~x, t) , ~x ∈ D, t > 0,(15) T (~x, 0) = 2g (2x) , ~x ∈ D,(16) λT,j (~r, t) νj (~r, t) + αT (~r, t) = 2V (~r, t) + α2bT (~r, t) , ~r ∈ ∂D, t > 0.4 R. STAROSTA ET AL.FIGURE 1. Reference cross-section.Here 2W, 2g, 2V, 2bT denotes the two-dimensional thermal load. We will apply the tem-perature component method to derive the two-dimensional heat conduction problem. Todo that we can take the three-dimensional approximate solutiona2T (~x, t) =κλ2W (D × (0, t)) ∗ F (~x, t) + 2g(D) ∗ F (~x, t)(17) +p∑l=1n∑m=1w̃ml F(~x− m~y, t− ls).The thermal load appeared in (17) has two-dimensional character. It leads to the situationthat the integrals along x3 in convolutions contain the fundamental solution only. Theresults of the improper integrations along x3 are constant. The above assumption aboutintegrate limits is justified by the properties of the fundamental solution. In consequencewe havea2T (~x, t) =κλ2W (D̃ × (0, t)) ∗ 2F (2~x, t) + 2g(D̃) ∗ 2F (2~x, t)(18) +p∑l=1n∑m=1w̃ml F(~x− m~y, t− ls).Here 2F means a fundamental solution to the parabolic operator for the two-dimensionalspace(19) 2F (~z, t) =0, t < 0, ~z ∈ R20, t = 0, ~z 6= ~014πκ̃texp(− |~z|24κt), t > 0, ~z ∈ R2.ACTION OF A HEAT SOURCE AND INFLUENCE OF INITIAL CONDITION ... 5The temperature component method requires of wrapping the region by some regular facesparallel to the boundary. So, we have to take a bounded region in two-dimensional prob-lems. But, following the popularly two-dimensional problems, the boundary conditionwill be defined on reference section only. In consequence, the temperature field does notdepend on the third variablea2T (~x, t) =κλ2W (D̃ × (0, t)) ∗ 2F (2~x, t) + 2g(D̃) ∗ 2F (2~x, t)(20) +p∑l=1n∑m=1w̃ml F(2~x− m~y, t− ls).Thanks to three-dimensional function F in (20) the admissibility condition is fulfilled.In (20) the intensity values w̃ml were calculated in the same way as in three-dimensionalcase [2]. One of the ways of approximate satisfaction of boundary conditions is providedby the method of boundary collocation ([5], p. 8). We shall apply a modification of thismethod named the method of boundary collocation in the least-squares approach ([5], p.13). For two-dimensional problems the body model region in boundary condition is rep-resented by the reference section. Now the boundary of the reference section is a closedcurve ∂D̃ , which should be split into regular arches(21) ~r = ~̄r(ω).A close interval in the set of the real parameters ω is a domain of the function, whosederivative is denoted by ~̄r,ω . Unit normal to the regular arch for the fixed 0ω is defined bythe formula(22) ν̃i = ± (h)−1 hi, hi ≡ εijkajbk, ~a ≡ ~r,ω (0ω) , ~b ≡ (0, 0, 1).Let us choose the sign of the normal ν̃ placed outside the regular arch cut out from theclosed curve ([4], p. 108). Relation h 6= 0 is always valid for the regular arch ([4], p.96). A parallel arch located in the distance d from the regular arch (21) is defined by theformula(23) ~y = ~̄r(ω) + dν̃(~̄r(ω)).Let us defined the collocation nodes h~r, h = ¯1, 1N on the first regular arch (21) cut outfrom ∂D̃. The fictitious places m~y, m = ¯1, 1n, 1n < 1N shall be determined on the partof the first parallel arch (23) in the distance 1d, that does not intersect ∂D̃. Let us definedthe collocation nodes h~r, h = ¯1N + 1, 2N on the second arch (21) cut out from theboundary ∂D̃. The next fictitious places m~y, m = ¯1n+ 1, 2n, 2n < 2N corresponding toit will be selected at the part of the second parallel arch (23) in the distance 2d, that does notintersect the curve ∂D̃ nor the first parallel arch. All the collocation nodes h~r, h = ¯1, 1Nand the fictitious location {m~y : m = ¯1, n} may be obtained by a recurrent method withregard to the other regular arches on ∂D̃, assuming fictitious locations for the parts ofparallel arches that do not intersect ∂D̃ nor previous parallel arches. Similar to the three-dimensional approach verifying the boundary condition leads to a set of linear equationsdefined at nodes on ∂D̃ at the chosen test instant.The Fourier law (1) allows to write the heat flux vector from (20)a2qi (~x, t) = −κ2W (D̃ × (0, t)) ∗ 2F,i(2~x, t)− λ2g(D̃) ∗ 2F,i(2~x, t)6 R. STAROSTA ET AL.(24) λp∑l=1n∑m=1w̃ml F,i(2~x− m~y, t− ls), i = ¯1, 2, a2q3 (~x, t) = 0.The thermal flux changes in two dimensions only.The thermal flow (2) through a face determined by a normal versor νi isνqi (~x, t) = −κ2W (D̃ × (0, t)) ∗ 2F,ν(2~x, t)− λ2g(D̃) ∗ 2F,ν(2~x, t)(25) −λp∑l=1n∑m=1w̃ml F,ν(2~x− m~y, t− ls).As we can see the third component of a unit normal has no influence on the thermal flow.4. Influence of the initial condition on heat conduction solutionsLet us admit of 2W = 0 in (14)(26) a2T (~x, t) = 2g(D̃) ∗ 2F (2~x, t) +p∑l=1n∑m=1w̃ml F(2~x− m~y, t− ls).The expression (26) represents the approximate solution to the problem (14 - 16). It ischaracterized by local dependence of the heat generated per unit surface and time andsurrounding medium temperature, the dependence being intermediated by the boundarycondition. The solution meets the boundary condition only in collocation nodes and in testinstants. Thus, we can introduce the measurement data for the thermal load on this pointsonly. Thermal flow through a face determined by a normal versor νi is(27) νqi (~x, t) = −λ2g(D̃) ∗ 2F,ν(2~x, t)− λp∑l=1n∑m=1w̃ml F,ν(2~x− m~y, t− ls).In (26) formula the global dependence of the initial state is of direct character. On the otherhand, the indirect local dependence on the boundary condition adjusted by the intensities offictitious components enables the surroundings temperature and the boundary heat sourceto affect the distribution heat flow in the region D in any instant t > 0.5. ExampleLet us consider a rectangular parallelepiped body, with heat generated per unit volumeand time equal zero. The initial condition is given by the function (Fig. 2).g(x, y) = 4− 0.1x2 − 0.125y2.The boundary condition is assumed different on the various parts of the boundary. On thecross-section various kinds of boundary conditions are presented (Fig.3). Solution obtainedby the Temperature Component Method is presented below (Fig.4).ACTION OF A HEAT SOURCE AND INFLUENCE OF INITIAL CONDITION ... 7FIGURE 2. Graph of the initial condition.FIGURE 3. Mixed boundary conditions on the reference section.8 R. STAROSTA ET AL.FIGURE 4. Temperature distributions for 0.5, 1.5 and 2.5 seconds.6. Action of a Heat SourceThe method could be useful to observe the evolution of the temperature field causedby some heat source. Let us consider a rectangular parallelepiped with a height, load onthe boundary, and the heat generated per unit volume and time that allows us to treat theheat conduction as a two-dimensional problem. At the moment the body has a uniformtemperature distribution compatible with surround temperature. On the boundary we usethe condition of the third kind. The heat generated per unit volume is time independent.For the case we intend to ob-serve changes of temperature field and thermal flows in chosendirections. The reference section is a rectangle of dimensions .FIGURE 5. The reference section with source W.ACTION OF A HEAT SOURCE AND INFLUENCE OF INITIAL CONDITION ... 9Solution.FIGURE 6. Temperature distribution for 3, 4 and 5 seconds.7. ConclusionsThe temperature component method elaborated for three-dimensional problems was ap-plied to solve some problems with a plane state of temperature. The reference cross-sectionwas introduced in order to take into consideration two-dimensional heat conduction and fic-titious components were located at the arches parallel to the boundary curve. Temperaturedistributions through the body region caused by the initial condition and action of the heatsource were illustrated. The method is effective and numerically stable. Computationaltime for presented examples is comparable with other well-known methods such as FEM,BEM [5]. The method is quite easy to implement.Acknowledgement The paper was supported by the Project 21-212/2006 DSReferences[1] H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids (Oxford University Press, London, 1959)[2] J. Jankowski, ”Temperature component method for heat conduction”, Thermocon’05, Messina (2005)[3] H. Marcinkowska, Introduction to partial differential equations (Polish Scientific Publishers, Warszawa,1972)[4] A. Goetz, Differential geometry (Polish Scientific Publishers, Warszawa, 1965)[5] J.A. Kolodziej, Zastosowanie metody kollokacji brzegowej w zagadnieniach mechaniki(WPP, Poznan, 2001)10 R. STAROSTA ET AL.Roman Starosta, Tomasz Walczak, Janusz JankowskiPoznan University of TechnologyInstitute of Applied MechanicsUl. Pietrowo 360 - 965 Poznań, Poland* E-mail: janusz.jankowski@put.poznan.plPresented: September 27, 2005Published on line: February 01, 2008

Action of a heat source and influence of initial condition on the plane state of temperature

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Action of a heat source and influence of initial condition on the plane state of temperature

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