JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 4, pp. 789-804, Warsaw 2004 THE TRANSIENT TEMPERATURE FIELD IN A RECTANGULAR AREA WITH MOVABLE HEAT SOURCES AT ITS EDGE Beata Maciejewska Faculty of Management and Computer Modelling, Kielce University of Technology e-mail: beatam@tu.kielce.pl Thepaperprovidesanexact solution toanonstationarytwo-dimensional heat transfer problem where heat sources move along the edge of the area. Finite Fourier transforms are applied to find the solution. It is given as a sumof four parts.The investigationsaimat the determination of the temperature distribution in a brake drum while the vehicle rolls downa slope at a constant velocity.Brake linings, brought into frictional contact with the drum in braking, constitute moving heat sources. Due to the nature of the process under examination, it is possible to assume that the heat transfer is two-dimensional. The dimensions of the brake drum (the internal radius to external radius ratio is approx. 0.95) and simplifications allow one to model it as a rectangular area. Keywords:movingheat source, 2Dheat conduction, finiteFourier trans- form, brake drum 1. Introduction The problem of the identification of a temperature field generated by a moving heat source has been investigated in numerous papers. In a paper by Grysa (1977a), the author considered the temperature distribution in a long circular cylinder whose lateral surface was affected by temperature being a function of the angular coordinate. The cylinder itself rotated around its axis with a constant angular velocity ω. The problemwas analysed in the cylindri- cal coordinate system r,ϕ,z. Because points were regarded to be located at a sufficient distance fromboth ends of the cylinder, itwas assumed that the tem- perature distribution was a function of time t and spatial variables r and ϕ. The problem was solved by applying Hankel transformations. In a paper by 790 B.Maciejewska Drzewicki et al. (1977), the authors investigated the temperature field in an infinite circular cylinder when its lateral surface was affected by a temperatu- re field distributed according to a function T =T(ϕ,t). The cylinder rotated around its axis with a constant angular velocity ω. The problem was solved by applying Green’s functions. The solution was presented for certain special cases by appropriately specifying the form of the function T = T(ϕ,t). The calculations were made for three forms of the boundary condition: a constant function; a square function of ϕ−ωt at a part of the boundary and zero at the remaining boundarypart; a cosine function at a part of the boundarywith the argument ϕ−ωt and zero elsewhere. In a paper by Grysa and Legutko (1981), the authors determined the intensity of a moving heat source in the contact area between the blade and the grinding detail. The temperature was calculated by applying an analytic method based on inverse heat conduction problems. Two shapes of a grinding object were taken into account: a circular disk and a cubicoid. 2. Mathematical model The aim of the present paper is to provide an exact solution to a transient heat transfer problemwhen a part of the boundary of the area under analysis is heated with amoving heat source. For the sake of calculations, a boundary condition of the second kind is adopted. The condition can be expressed in a form −λ∂T/∂n = qn, where λ denotes the thermal conductivity coefficient [W/mK], ∂T/∂n – derivative in the direction perpendicular to the surface of the body and directed outwards, −qn = qnf(x,t) – normal component of the heat flux density [W/m2], qn – its extreme value at the contact of the source with the body (qn > 0), f(x,t) – polynomial function characterizing the type of the heat flux density distribution (f(x,t)> 0). The determination of the temperature distribution in the brake drum is carried out when velocity of the vehicle riding down a slope is being reduced. It is assumed that the road inclination angle α is constant and velocity of the vehicle is also constant. Brake linings fixed to brake shoes that come into fric- tional contact with the brake drumwhile braking are treated as moving heat sources. For the purpose of formulating a mathematical model, the following simplifying assumptions were made: 1. The temperature is constant along the entire width of the drum. It me- ans that the direction of the heat transfer is also assumed to be two- dimensional. The transient temperature field in a rectangular area... 791 2. Due to a large diameter of the drum in relation to its thickness, the annulus is thought to be unrolled into a flat rectangular area. What is investigated is the temperature distributionwithin a rectangular area of the length l and width b. 3. Two heat sources, each of the length a equal to the length of the brake liningandthewidthequal to thedrumwidth,moveata constantvelocity v in a periodic manner. The density distribution of the heat flux at the contact of the brake liningwith the drum ismodelledwith a polynomial function. Figure 1 presents the graph of the function −qn at a fixed time instant. Fig. 1. Distribution of the heat flux on the heated body surface at a fixed time instant 4. Except for the brake lining contact with the brake drum (also on the opposite side), the area is assumed to be thermally insulated. 5. Repeatability of the process resulting from the vehicle wheel rotation is achieved by the adoption of boundary conditions of the fourth kind at the edges of the rectangle, which are perpendicular to the side affected by the heat source. 6. It is assumed that at the initial time instant, the temperature of the brake drum and the environment is constant and equals Θ0. 7. The brake drum is made of a homogenous and isotropic material. 8. It is assumed that the thermal conductivity coefficient κ and thermal diffusivity k of the brake drum do not depend on temperature. The problem will be formulated mathematically in a dimensionless form. Dimensionless (reduced) temperature is defined as follows: T =(Θ−Θ0)/ q n b λ , where Θ denotes actual temperature [K], Θ0 – actual temperature at the initial moment [K], qn – maximum value of the heat flux density resulting from the action of themoving heat source [W/m2], b= rz−rw – thickness of 792 B.Maciejewska thebody [m], rz - the external radiusof thebrakedrum, rw – internal radiusof the brake drum, λ – thermal conductivity coefficient [W/mK]. Dimensionless coordinates are expressed in the following way: x= x/b, y = y/b, t= κt/b 2 where κ is the thermal diffusivity coefficient [m2/s]. We define dimensionless parameters: l = l/b, a = a/b, b = 1, v = vb/κ (l = π(rz − rw) is length of the body [m], a – length of the source [m], v – actual velocity of the source [m/s]). Fig. 2. Amodel of the system for temperature identification in heating with moving heat sources The following dimensionless formof the problemunder analysis is achieved ∂2T ∂x2 + ∂2T ∂y2 − ∂T ∂t =0 (2.1) for (x,y)∈Ω, t> 0, Ω= {(x,y)∈R2 : 0 0 (3.7) Tn1(k,t) =−e −(λ2 n +α2 k )t(Bkncosλnvt−Akn sinλnvt)+Bkn k­ 0 Tn2(k,t) =−e −(λ2 n +α2 k )t(Akncosλnvt+Bkn sinλnvt)−Akn k­ 0 where Akn = (−1)kqn1vλn (vλn)2+(λ2n+α 2 k )2 Bkn = (−1)kqn1(λ 2 n+α 2 k ) (vλn)2+(λ2n+α 2 k )2 The transient temperature field in a rectangular area... 795 Byvirtue of (3.5),wearrive at the following solution to the systemof equations (3.2) T0(y,t)= q0t b + 2 b ∞ ∑ k=1 (−1)kq0 α2 k (1− e−α 2 k t)cosαky Tn1(y,t)= 1 b [(−e−λ 2 n t(B0n cosλnvt−A0n sinλnvt)+B0n]+ (3.8) + 2 b ∞ ∑ k=1 (−e−(λ 2 n +α2 k )t(Bkncosλnvt−Akn sinλnvt)+Bkn)cosαky Tn2(y,t)= 1 b [(e−λ 2 n t(A0n cosλnvt+B0n sinλnvt)−A0n]+ + 2 b ∞ ∑ k=1 (e−(λ 2 n +α2 k )t(Akncosλnvt+Bkn sinλnvt)−Akn]cosαky Then, the problem described by equations (2.1)-(2.3) has the following solution T(x,y,t)= q0t bl + 2 bl ∞ ∑ k=1 (−1)kq0cosαky α2 k (1−e−α 2 k t)+ + 2 bl ∞ ∑ n=1 qn1cosγn0 λ2n { cos[λn(x−vt)+γn0]− e −λ2 n tcos[λnx+γn0] } + (3.9) + 4 bl ∞ ∑ n=1 ∞ ∑ k=1 (−1)kqn1cosγnk cosαky λ2n+α 2 k · · { cos[λn(x−vt)+γnk]− e −(λ2 n +α2 k )t cos(λnx+γnk) } where cosγnk = λ2n+α 2 k √ (vλ2n) 2+(λ2n+α 2 k )2 4. Analysis of the obtained solution Due to the nature of the thermal field changes in time, a division of the transient state is introduced following Kondratiev (1954) into: 796 B.Maciejewska • a purely nonstationary heating process (t< 0.5) • a regular heating process (t> 0.5). In the purely nonstationary heating process, the temperature field depends on physical properties of the body, its geometry, dimensions and also initial and boundary conditions. The regular heating process is already awell-established one, in which the time-space temperature distribution depends on the body geometry, its dimensions, physical properties and boundary conditions. The impact of initial conditions on the temperature, however, is negligibly small. For the sake of analysis, relation (3.9) is presented in a form of the sum T(x,y,t)=TK(x,y,t)+TN(x,y,t)+TB(x,y,t)+TS(x,y,t) (4.1) where TK(x,y,t)= q0t bl (4.2) TN(x,y,t)=− 2 bl ∞ ∑ k=1 (−1)kq0cosαky α2 k e−α 2 k t− 2 bl ∞ ∑ n=1 qn1 λ2n e−λ 2 n t cosλnx+ (4.3) − 4 bl ∞ ∑ n=1 ∞ ∑ k=1 (−1)kqn1cosαky λ2n+α 2 k e−(λ 2 n +α2 k )t cosλnx TS(x−vt,y)= 2 bl ∞ ∑ k=1 (−1)kq0cosαky α2 k + 2 bl ∞ ∑ n=1 qn1 λ2n cosλn(x−vt)+ (4.4) + 4 bl ∞ ∑ n=1 ∞ ∑ k=1 (−1)kqn1cosαky λ2n+α 2 k cosλn(x−vt) TB(x,y,t) =− 2 bl ∞ ∑ n=1 qn1 sinγn0 λ2n { sin[λn(x−vt)+γn0]+ −e−λ 2 n t sin[λnx+γn0] } − 4 bl ∞ ∑ n=1 ∞ ∑ k=1 (−1)kqn1 sinγnk cosαky λ2n+α 2 k · (4.5) · { sin[λn(x−vt)+γnk]− e −(λ2 n +α2 k )t sin(λnx+γnk) } and sinγnk = vλn √ (vλ2n) 2+(λ2n+α 2 k )2 The transient temperature field in a rectangular area... 797 By applying formula (5.14) inGrysa (1977b) with x= b, equation (4.4) takes the form TS(x−vt,y)= 2 bl { q0 [η(y− b) 2 (b−y)+ 2+3y2−6b+3b2 12 ] + + ∞ ∑ n=1 qn1 λ2n cosλn(x−vt) } + 4 bl ∞ ∑ n=1 qn1cosλn(x−vt) 2λn sinhλn · (4.6) · { η(y− b)sinhλn sinh[λn(b−y)]+coshλnycosh[λn(1− b)] } where η(z) denotes the Heaviside function. In formula (4.1),its individual terms have the following sense: • TK(x,y,t) is a linear term representing the heat accumulation within the area under analysis • TN(x,y,t) describes temperature changes at individual points of the area,which result fromtheheating, andwhicharenot affectedbymotion of the source • TB(x,y,t) describes the thermal inertia caused by the heating • TS(x−vt,y) is actually a function of two variables: y and the difference x−vt. The change in time t by ∆t leads to a similar change in the value of the function TS as the change in x by ∆x=−v∆t. The temperature distribution expressed by formula (4.3) may be interpre- ted as a temperature field generated by a stationary source after a lapse of very long time described in the system x,y. A characteristic feature of the term is the fact that it does not depend on physical properties of the body, its dimensions and initial conditions. In a purely nonstationary heating process, none of the expressions present in relations (4.1), (4.2), (4.3) and (4.5) can be excluded. In a regular heatingprocess, the function assumes onamuch simpler form. All expressions containing exp[−(λ2n+α 2 k)t], except for exp(−α 2 1t), can be excluded. The expressions highlighted in relation (4.1) take the forms TN(x,y,t) = 2 bl q0cosα1y α21 (e−α 2 1 t) (4.7) TB(x,y,t)=TB(x,y,t)=− 2 bl ∞ ∑ n=1 qn1 sinγn0 λ2n sin[λn(x−vt)+γn0]+ (4.8) − 4 bl ∞ ∑ n=1 ∞ ∑ k=1 (−1)kqn1 sinγnk cosα1y λ2n+α 2 1 sin[λn(x−vt)+γnk] 798 B.Maciejewska Expressions (4.2), (4.6) do not change their forms. Therefore, for t > 0.5, relation (4.1) can be presented as follows T(x,y,t)=TK(x,y,t)+TN(y,t)+TB(x−vt,y)+TS(x−vt,y) (4.9) With time running, the second of the highlighted expressions quickly ap- proaches zero, the third and the fourth are actually functions of two variables; they describe a quasi-steady state. T(x,y,t)→∞, t→∞ because TK(x,y,t)→∞, t→∞ and TN(x,y,t), TB(x,y,t), TS(x,y,t) are limited. The component TK(x,y,t) describes the accumulation of heat collected inside the body. The formulated model and the calculated temperature function are applied to determine the temperature distribution in the brake drum over a short time of the vehicle braking. The velocity v significantly affects the thermal inertia, because the value of the function TB(x−vt,y) depends on sinγnk. For velocities satisfying the condition v� 1,wehave sinγnk � 1 and T B(x−vt,y)≈ 0.The temperature field will be then expressed by the relation T(x,y,t)≈TK(x,y,t)+TN(y,t)+TS(x−vt,y) (4.10) which indicates that the movement of the source does not really affect the way in which heat penetrates the inside of the area. Figures 9a,b present the temperature distribution for the dimensionless velocity v =0.001. When the velocity is sufficiently high, then cosγnk ≈ 0 and the terms with cosγnk disappear in expression (3.9), and the temperaturemay be approximatedwith the formula T(y,t)= q0t bl + 2 bl ∞ ∑ k=1 (−1)kq0cosαky α2 k (1− e−α 2 k t) (4.11) It is apparent from formula (4.11) that the temperature depends only on one spatial variable, so the temperature distribution is one-dimensional and heat flow is one-directional. The same solution would be achieved if it was assumedthat the sourcewith theconsistent intensiveness equal to q0/l affected the entire surface of the body, i.e. there was an insulation on the opposite surface and the heat conduction equation had the form ∂2T ∂y2 − ∂T ∂t =0 for y∈ (0,b), t > 0 In formula (4.11), q0 is formulated as follows q0 = l ∫ 0 f(x,t) dx (4.12) The transient temperature field in a rectangular area... 799 where f(x,t) is expressed by relation (2.2)3. Figures 5a,b present the tempe- rature distribution for the dimensionless velocity v=2.1·104. For t> 1.5, the expression TN(y,t) introduces a correction of the order 10−8q0 into formula (4.9), which can be neglected. Formula (4.9) has the form T(x,y,t)≈TK(x,y,t)+TB(x−vt,y)+TS(x−vt,y) (4.13) Upon introductionof anotionof the characteristic time τ0 =1/α 2 1 (it is the time that characterises the heating rate of the brake drum in the regular he- atingprocess), onecannotice that the condition t> 0.5means t> (π/b)2τ0/2, and t> 1.5 forwhich the expression TN(y,t) canbeexcluded in formula (4.9), means t> 3(π/b)2τ0/2.When the source is notmoving (v=0), relation (4.1) is transformed into the form T(x,y,t) =TK(x,y,t)+TN(x,y,t)+TS(x,y) (4.14) where TS(x,t)= 2 bl { q0 [η(y− b) 2 (b−y)+ 2+3y2−6b+3b2 12 ] + ∞ ∑ n=1 qn1 λ2n cosλnx } + + 4 bl ∞ ∑ n=1 qn1cosλnx 2λn sinhλn { η(y− b)sinhλn sinh[λn(b−y)]+ (4.15) +coshλnycosh[λn(1− b)] } TK, TN do not change their forms. 5. Numerical example Let us consider a thermal field in a cast-iron brake drum of a lorry riding down a road inclined at an angle α=10◦ with a constant velocity.We assume the followingnumerical data: λ=50W/mK, κ=0.125·10−4 m2/s, l=1.3m, b=0.013m, a=0.3m, Θ0 =293K, qn =(σGδv sinα)/(2S), δ=0.4 – ratio of braking force acting on front wheels to braking force, σ = 0.95 – heat distribution between brake lining and drum, α=10◦ – road inclination angle, v – vehicle velocity [m/s2], S=0.093m2 – area of contact between two brake linings and brake drum, G=108 ·103 N – vehicle weight. The data and nomenclature were taken from the literature on the subject, see Łukomski et al. (1976), Wrzesiński (1978). 800 B.Maciejewska The temperature field was calculated for a source moving with velocities: v=72 km/h, v=10.8 km/h and v≈ 0 km/h. The qn =7.7 ·10 5 W/m2 was adopted for a source moving with a velocity v = 72 km/h. Figure 3 presents the temperature distribution through the entire drumthickness at themoment t=0.02 s. Figures 4a,b present the temperature distribution in a fragment of the drum close to the heat-affected surface at the same moment. Fig. 3. The temperature distribution at the moment t=0.02 s for the velocity v=72km/h Fig. 4. (a) The temperature distribution in a fragment of the drum close to the heat-affected surface at the moment t=0.02 s for the velocity v=72km/h; (b) a contour diagram Figures 5a,b present the temperature distribution through the entire drum thickness at the moment t = 30 s. Figures 6a,b present the temperature di- stribution in a fragment of the drum close to the heat-affected surface at the same moment. The transient temperature field in a rectangular area... 801 Fig. 5. (a) The temperature distribution at the moment t=30 s for the velocity v=72km/h; (b) a contour diagram Fig. 6. (a) The temperature distribution in a fragment of the drum close to the heat-affected surface at the moment t=30 s for the velocity v=72km/h; (b) a contour diagram The value of qn = 1.2 · 10 5 W/m2 is adopted for a source moving with the velocity v = 10 km/h. Figures 7a,b present the temperature distribution through the entire drum thickness at the moment t = 30 s. Figures 8a,b present the temperature distribution in a fragment of the drum close to the heat-affected surface at the same moment. The qn = 0.04W/m 2 is adopted for a source moving with the velocity v≈ 0 km/h (v=10−7 km/h). Figures 9a,b present the temperature distribu- tion at the moment t=20 s. 802 B.Maciejewska Fig. 7. (a) The temperature distribution at the moment t=30 s for the velocity v=10km/h; (b) a contour diagram Fig. 8. (a) The temperature distribution in a fragment of the drum close to the heat-affected surface at the moment t=30 s for the velocity v=10km/h; (b) a contour diagram Fig. 9. (a) The temperature distribution at the moment t=20 s for the velocity v≈ 0 km/h; (b) a contour diagram The transient temperature field in a rectangular area... 803 6. Conclusions It is apparent fromFig.1 to fig.9 that thepostulates adopted in the second section of the paper have been satisfied, i.e. at the beginning and at the end of the unrolled drum the temperatures and heat fluxes are equal (the tangents to isotherms are parallel). The perpendicularity of isotherms to the surface y = 0.013 beyond the area affected by the sources indicates the satisfaction of insulation that was assumed to be present there. For a source moving with the velocity v=10 km/h, the predicted effect of inertia caused bymovement of the sources can be observed, and with the velocity v = 72 km/h, for the purpose of better visualisation of that effect, some fragments of the figures had to be enlarged. As it was also expected, that effect is not present at a velocity of the sources close to zero, which can be observed in Fig.9. The heat penetration depth inside thedrumdependsmostly onvelocity and the number of cycles. The slower the velocity is, the deeper the heat penetrates the inside. The opposite holds true for the number of consecutive passes of the sources – the greater it is, the deeper the heat penetrates the inside and is accumulated there, which is represented by the expression TK expressed by formula (4.2). References 1. Drzewiecki A., Grysa K., Jankowski J., 1977, The field of temperature in a long cylinder due to heating its lateral surface with temperature descri- bed by the function T = T(ϕ,t), Zeszyty Naukowe Politechniki Poznańskiej, Mechanika, 20, 27-39 2. Grysa K., 1977a, Non-steady state of temperature in a rotating circular cy- linder due to piece-wise constant temperature on its surface, Mech. Teoret. i Stos., 15, 2, 215-225 3. Grysa K., 1977b, Sumation of certain Fourier-Bessel series, Mech. Teoret. i Stos., 15, 2, 205-214 4. Grysa K., Legutko S., 1981, The problem of temperature identification in grinding process, PAN-Oddział w Poznaniu, 37, 245-262 5. Kondratiev G.M., 1954, Regular Heating Process, Gos. Izd. Tech.-Teoret., Moskva 6. Łukomski Z., Pałacha R., Kukliński Z., Zapłotyński W., 1976, Car Repairs Star 266, WydawnictwaKomunikacji i Łączności,Warszawa 7. Sneddon I.N., 1951,Fourier Transforms, McGraw-Hill Book Company, Inc. 804 B.Maciejewska 8. WrzesińskiT., 1978,AutomotiveVehicleBraking,WydawnictwaKomunikacji i Łączności,Warszawa Nieustalone pole temperatury w obszarze prostokąta z ruchomymi źródłami ciepła na jego brzegu Streszczenie W pracy rozwiązane zostało, w sposób ścisły, niestacjonarne dwuwymiarowe za- gadnienie przepływu ciepła z poruszającymi się źródłami ciepła wzdłuż brzegu ob- szaru. W celu znalezienia rozwiązania zastosowana została skończona transformata Fouriera. Rozwiązanie podane zostało w postaci sumy czterech składników. Zastoso- wane zostało do wyznaczenia rozkładu temperatury w bębnie hamulcowym podczas utrzymywania stałej prędkości samochodu zjeżdżającego z pochyłości. Okładziny ha- mulcowe trąc o bęben hamulca w trakcie hamowania stanowią poruszające się źródła ciepła. Ze względu na charakter badanego procesumożna przyjąć, że wymiana ciepła jest dwuwymiarowa.Wymiary bębna hamulcowego (stosunek promieniawewnętrzne- godo zewętrznegowynosi około0.95) i poczynioneuproszczeniapozwalająmodelować go obszarem o kształcie prostokąta. Manuscript received November 25, 2003; accepted for print April 5, 2004