Available online at ScienceDirect. Procedia Engineering 177 (2017 )

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1 Available online at ScienceDirect Procedia Engineering 177 (2017 ) XXI International Polish-Slovak Conference Machine Modeling and Simulations 2016 A finite element multi-mesh approach for heat transport between disconnected regions Tomasz Skrzypczak* Czestochowa University of Technology, Institute of Mechanics and Machine Design Foundations, Dąbrowskiego 73, Częstochowa, Poland Abstract The paper presents a method of numerical modeling of transient heat transport between separated areas in two-dimensional space. The gap of variable width filled with heat conducting material separates considered regions wherein the case of ideal contact is also possible. The mathematical model is based on the partial differential equation of heat conduction supplemented by appropriate initial and boundary conditions of I-IV kind. The numerical model uses finite element method (FEM) with independent spatial discretization of interacting regions resulting in multi-mesh problem. The movement of spatially discretized regions relative to each other is also possible. The results of numerical simulations for the two and three areas that were in thermal interaction are presented and discussed The Tomasz Authors. Skrzypczak. Published by Published Elsevier Ltd. by Elsevier This is an Ltd. open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the organizing committee of MMS Peer-review under responsibility of the organizing committee of MMS 2016 Keywords: computational mechanics; heat trasport; thermal contact problem; finite element method; multi-mesh approach; 1. Introduction Heat transfer does not always occur in a single, continuous area. The heat exchange between two or more physical regions can take place by perfect contact or through the gap. Various kinds of machine parts work together going into the mechanical and thermal interactions. Mechanical contact is the result of direct impacts of their external boundaries. The heat flow from one region to another can also happens through the medium in which they are located. The mathematical model describing the process is based in this case on the equation of heat conduction supplemented by appropriate boundary and initial conditions. A fourth type of boundary condition plays a crucial * Corresponding author. Tel.: ; fax: address: t.skrzypczak@imipkm.pcz.pl The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the organizing committee of MMS 2016 doi: /j.proeng

2 Tomasz Skrzypczak / Procedia Engineering 177 (2017 ) role in the mathematical description of such process. One can distinguish two kinds of such condition describing the perfect contact or the contact through a gap of variable width, filled with heat conducting medium. One of the first descriptions of the thermal contact problem can be found in the work of Cooper et al. [1]. The results of numerical solution of the process of transient heat conduction between two half-infinite regions using finite difference method (FDM) was discussed in details in the work of Schneider et al. [2]. In addition to numerical solutions of the problem there are also analytical solutions, which can be found in [3]. At the present time many scientists from various branches of science are interested in the ideal and non-ideal thermal contact. Taking into account thermal conductance between partially separated or adjoining areas is necessary in modeling of heat flow in biological tissues [4] and also during the investigation of solidification process [5-9]. 2. Mathematical description The governing equation of presented model is the heat conduction equation describing transient heat transfer in the two-dimensional area: i i i T T T i i c i (1) x x y y t where λ (i) [J m -1 K -1 s -1 ] is the coefficient of thermal conductivity, (cρ) (i) [J m -3 K -1 ] denotes volumetric heat capacity, T (i) [K] - temperature, t [s] time, x, y [m] Cartesian coordinates, while i=1, 2,..., N is the subscript relating to the particular region in the set of N interacting regions. Fig. 1. Scheme of the contact between three two-dimensional regions. Fig. 1 shows a scheme of contact for three separated regions Ω 1, Ω 2, Ω 3. There is some kind of thermal interaction between them. In general, there is a gap of varying width, thus the parameter h may be the function of spatial coordinates and time. The gap is not a physical feature of the model, thus its influence on the heat exchange between regions must be taken into account by the suitable boundary conditions: 1 2 T g 1 2 T 1 Tb Tb 2 (2) n h n 12 21

3 206 Tomasz Skrzypczak / Procedia Engineering 177 ( 2017 ) T g 1 3 T 1 Tb Tb 3 (3) n h n T g 2 3 T 2 Tb Tb 3 (4) n h n where h [m] is the local width of the gap, λ g [J m -1 K -1 s -1 ] - thermal conductivity of the medium filling the gap, T b (1) - T b (3) [K] are the local temperatures of the regions 1-3 at the boundary, while n 1-2, n 2-1, n 1-3, n 3-1, n 2-3, n 3-2 are the local directions of the vectors normal to the contact boundaries. At the non-contact boundaries the following conditions are assumed T T T 1 q1, 2 q2, 3 q3 (5) n n n where n 1 - n 3 are the directions of the vectors normal to the boundaries Γ 1 - Γ 3, q 1 - q 3 [J s -1 m -2 ] - heat fluxes normal to the boundaries Γ 1 - Γ 3. It should be noted that q 1 - q 3 may be known a priori or calculated with the following formulas 1 2 T T q T T q T 3, T (6) q1 b amb 2 b amb, 3 where α [J s -1 m -2 K -1 ] is known as the heat transfer coefficient at the boundary and T amb [K] is the ambient temperature. Both of them may depend on spatial coordinates and time. Dirichlet boundary conditions at the non-contact boundaries are also acceptable. Because the heat transfer is transient the initial temperatures of the regions Ω 1 - Ω 3 must be introduced: T b t 0 Tinit, T t 0 Tinit, T t 0 T init (7) Equation (1) complemented by additional conditions (2-7) form a mathematical model of the considered problem. 3. Numerical model The numerical model uses FEM [10] and is derived from the criterion of the method of weighted residuals [11]. Equation (1) is multiplied by the weighting function w and integrated over the region Ω i : i i i i T T T w 0 i i c i d i (8) x x y y t The form of the weighting function w depends on the method used in the process of spatial discretization of equation (8). As the result of using Green's theorem, Galerkin method and backward Euler's time-integration scheme which were described in details in the work [12] the following matrix equation is obtained: amb 1 f 1 1 f K MT B MT t t (9) where K is the thermal conductivity matrix, M heat capacity matrix, B vector associated with the boundary conditions, Δt [s] - time step, f - time level.

4 Tomasz Skrzypczak / Procedia Engineering 177 (2017 ) The solution of (9) is obtained step-by-step in the form of temporary temperatures, independently in each region. Each step requires a sequential solution of (9) for Ω 1 - Ω 3. It is important to properly introduce the conditions (2-4) to the solution process [13]. If Ω 1 - Ω 3 are in motion the calculation of local gap width h between regions staying in contact is necessary in each time step. In the case of unmovable regions with fixed boundaries this operation is only needed at the start of the calculation process. Because the global set of equations is built and solved independently for each region such approach is based on the set of finite element meshes where each of them has its own set of nodes. There are no nodal connections between meshes. 4. Examples of calculation In order to simulate considered phenomenon, the appropriate software was written in C++. At first, test calculations have been carried out and the results were compared with analytical solution [3]. Heat transfer between two rectangular regions with different material properties and initial temperatures was examined. Portions of areas with the mesh of finite elements and parameters used in the calculations are shown in Fig. 2. At the external boundaries thermal insulation was adopted. Both regions were fixed and the gap width was constant during entire process. Fig. 2. Scheme of the test problem. Simulation of heat transfer was carried out with a time step Δt=1 s. Fig. 3a-b show temporary temperature distributions obtained in the analytical and numerical ways. The comparison between them shows very good agreement and confirms the high accuracy of numerical solver. a) b) Fig. 3. Temperature distribution in the copper-steel system in the horizontal direction at time a) t=100 s, b) t=300 s.

5 208 Tomasz Skrzypczak / Procedia Engineering 177 ( 2017 ) The next task involved the heat transfer between the three regions made of steel (Fig. 4). Regions Ω 1, Ω 2 were fixed while Ω 3 was rotating around its centroid at constant angular velocity ω=π/150 s -1. External boundaries Γ 1, Γ 2 were thermally insulated. Material properties, initial temperatures and gap conductivities are shown in Fig. 4. Fig. 4. Scheme of the thermal interaction between three regions. Figures 6-7 show temporary temperature fields at 50, 250, 650, 850 s. The effects of heat exchange are best seen in the vicinity of moving rounded corners of the region Ω 3. a) b) Fig. 5. Temperature distribution at time a) t=50 s, b) t=250 s.

6 Tomasz Skrzypczak / Procedia Engineering 177 (2017 ) c) d) Fig. 6. Temperature distribution at time a) t=650 s, b) t=850 s. 5. Conclusions Presented approach of numerical modeling of heat transfer between separated two-dimensional regions may be useful in simulation of thermal processes occurring in objects composed of multiple parts. For example, the solidification of the metal in the mold is a process where the gap of various width separating two regions is observed. Hot embossing is also a potential area of application of the presented model. It should be noted that in mentioned cases one should also take into account the mechanical contact and the associated possible deformations of interacting regions. References [1] M.G. Cooper, B.B. Mikic, M.M. Yovanovich, Thermal contact conductance, Int. J. Heat. Mass. Transf. 12 (1969) [2] G.E. Schneider, A.B. Strong, M.M. Yovanovich, Transient thermal response to two bodies communicating through a small circular contact area, Int. J. Heat. Mass. Transf. 20 (1977) [3] W. Longa, Krzepnięcie odlewów w formach piaskowych, Śląsk, Katowice, [4] M. Ciesielski, B. Mochnacki, Numerical simulation of the heating process in the domain of tissue insulated by protective clothing, J. Appl. Math. Comput. Mech. 13(2) (2014) [5] E. Majchrzak, J. Mendakiewicz, A. Piasecka-Belkhayat, Algorithm of the mould thermal parameters identification in the system castingmould-environment, J. Mater. Process. Technol. 164 (2005) [6] R. Dyja, E. Gawrońska, N. Sczygiol, The effect of mechanical interactions between the casting and the mold on the conditions of heat dissipation: a numerical model, Arch. Metall. Mater. 60(3A) (2015) [7] L. Sowa, Mathematical model of solidification of the axisymmetric casting while taking into account its shrinkage, J. Appl. Math. Comput. Mech. 13(4) (2015) [8] L. Sowa, Effect of steel flow control devices on flow and temperature field in the tundish of continuous casting machine, Arch. Metall. Mater. 60(2A) (2015) [9] Z. Saternus, W. Piekarska, M. Kubiak, T. Domański, L. Sowa, Numerical analysis of deformations in sheets made of x5crni18-10 steel welded by a hybrid laser-arc heat source, Procedia. Eng. 136 (2016) [10] O.C. Zienkiewicz, Metoda elementów skończonych, Arkady, Warszawa, [11] H. Grandin, Fundamentals of the Finite Element Method, Waveland Press, Long Grove, [12] E. Majchrzak, B. Mochnacki, Metody numeryczne. Podstawy teoretyczne, aspekty praktyczne i algorytmy, Wyd. Pol.Śl., Gliwice, [13] T. Skrzypczak, E. Węgrzyn-Skrzypczak, Modeling of thermal contact through gap with the use of finite element method, J. Appl. Math. Comput. Mech. 14(4) (2015)