Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 129 (2015) 317 – 320
International Conference on Industrial Engineering
Modeling a rotating circle thermal field with a thermal source on the edge Herreinstein A.V.a, Herreinstein E.A.b, Mashrabov N.a,b* a
ab
South Ural State University, 76, Lenin Avenue, Chelyabinsk, 454080, Russian Federation South Ural State Agricultural University, Institute of Agricultural Engineers, Chelyabinsk, Russia, 454080
Abstract The task of circle heating by a thermal source moving along the circle edge with a constant angular velocity is considered. The circle is a cylinder in which the temperature is the same on each straight line parallel to a cylinder axis, i.e. at each point the temperature is a function of time, distances from a cylinder axis and a polar angle. The computing schemes based on numerical methods are usually used for the similar tasks solution. Explicit schemes solve one equation for each time step of each spatial knot. Implicit schemes solve a system of linear algebraic equations for each time step. The number of equations is connected with the number of spatial knots. In the first case the process is rather simple, but the scheme is not absolutely steady. For this scheme the step order by time should not exceed the square of step order by coordinate. In the second case the scheme is absolutely steady, but it is necessary to solve a complex system of equations, especially for a multidimensional case. The purpose of this work is to use absolutely steady differential-difference explicit scheme based on linear equation application with partial derivatives of the first order which analytical solution (an explicit formula) is known. © 2015 The Authors. Published by Elsevier Ltd. © 2015 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 the International Conference on Industrial Engineering (ICIE(http://creativecommons.org/licenses/by-nc-nd/4.0/). 2015). Peer-review under responsibility of the organizing committee of the International Conference on Industrial Engineering (ICIE-2015) Keywords: heat conductivity; heat conductivity equation; differential-difference scheme; polar coordinate system; the equation of the first order partial derivatives.
* Corresponding author. Tel.: +7-9517856571. E-mail address:
[email protected]
1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the International Conference on Industrial Engineering (ICIE-2015)
doi:10.1016/j.proeng.2015.12.068
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1.Research foundation The differential schemes of the heat conductivity equation approximate solution are divided into explicit and implicit ones [1,2,3]. Implicit schemes are absolutely steady, but they require solution of complex equation system, storing of big data array. Moreover, it results in rounding errors accumulation. Explicit schemes don't demand storing of big arrays, but these schemes are relatively steady (the step size maximum of time is determined by the step of coordinate; violation of this condition results in instability) [3]. This explicit scheme drawback can be eliminated by application of the explicit differential-difference scheme [4,5,6,7,8,9]. 2.Materials and methods We will use the following designations: t' - time, r' - distance from this point to the center of a circle, Q' - heat source power density on the circle edge, Z' - the angular speed of circle rotation, P ' - heat-transfer coefficient on the circle edge, R - circle radius, q - heat conductivity coefficient, c - thermal capacity per the unit of volume, T ambient temperature, I - polar angle. Let’s introduce nondimensional variables and parameters [10,11] r , t , Q, Z, P using the formulas:
r ' rR, t ' tR2 c / q, Z ' Z q / cR2 , Q ' Qq / R, P ' P q / R .
(1)
Dimensionless radius of a circle in this case will be equal to 1. The heat source affects the part of the circle edge (for example, heat emitted when polishing a circle surface). Other part of the edge is cooled according to the Newton law. The equation of heat conductivity has the form [12,13]: wu wu Z wt wI
1 wu w 2 u 1 w 2 u . r wr wr 2 r 2 wI 2
Instead of the variable r we will introduce the variable
s
r2 .
(2)
s by the formula: (3)
The equation of heat conductivity will take the form: wu wu Z wt wI
4
wu w 2u 1 w 2u . 4s 2 s wI 2 ws ws
Condition on the edge ( s wu ws
(4)
1 ):
0.5(Q P (u T )) .
(5)
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j
un,j
un+1,j
i
Fig. 1. The grid used
We break the radius (length 1) into n identical elementary parts of h 1 / n length, a circle – on m identical sectors of D 2S / m length. When replaced (1) elementary segments turn into rectangles of the same (radius independent) sizes. To provide edge conditions we will add the so-called fictitious layer [14,15,16] to the external part of the circle, i.e. we will increase the radius for the value h . We place knots of a grid in the center ( i, j ) of an elementary rectangle, where i – the number of radius part, j – the number of sector by angle. Throughout each time interval of the dt length we will designate temperature as uij at the beginning of this time interval and as U ij – the current temperature. We will designate also: pi
fij
8i 2i , h D 2 ih ui 1, j ui 1, j ui 1, j ui 1, j ui , j 1 ui , j 1 2 4i . h h ihD 2
(6) (7)
As a result we will obtain the differential-difference scheme with partial derivatives of the 1st order: wU ij wt
Z
wU ij wI
piU ij
fij .
(8)
We will choose a dt time step so that the circle turns on D angle for this time. Then at the end of a time interval [0, dt ], solving the equation (8), we will obtain [17]:
U ij
e pi dt ui , j 1
fij e pi dt fi , j 1 pi
fi , j 1 fij pi2 dt
(1 e pi dt ) .
(9)
On the elementary part containing the center of a circle, the formula (9) for temperature U 0 in the center of the circle will be the following: 1 e p0 dt m 1 U 0 e p0 dt u0 (10) ¦ u1 j . m j 0 Having obtained values of temperature U ij in the circle (where i 0,1,, n 1 ), on a fictitious layer for edge conditions satisfaction we determine U nj temperatures from the equation: 1 P ( )U nj h 4
1 P Q PT . ( )U n 1, j 2 h 4
(11)
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Obtaining such scheme is due to the results of research work [4,6,7]. Stability substantiation of the scheme offered is similar to the one given in the following works [18,19]. The numerical method was considered under the initial condition (23) and with the radius step of 1/10 and with the S / 5 angle and was compared to the exact solution (with the first lena of Fourier’s method solution). The relative error didn't exceed 6%. The schemes obtained were used for a blank temperature field modeling in the process of flute grinding of spiral drills [20,21].
3.Conclusions
The explicit differential-difference scheme offered is absolutely steady in comparison with purely differential scheme and allows calculating the thermal field of a circle (cylinder) beyond the differential scheme stability. References [1] V.M. Verzhbitsky, Fundamentals of numerical methods, Higher school, Moscow, 2002.. [2] A.A. Samarskiy, Theory of differential schemes, Science, Moscow, 1989. [3] N.S. Bakhvalov , N.P. Zhidkov, G.M. Kobelkov, Numerical methods, Science, Moscow, 1987. [4] A.V. Herreinstein, N. Mashrabov, E.A. Herreinstein, Steady explicit schemes for the heat conductivity equation, Bulletin of SUSU, a "Mathematical modeling and programming" series. 1-15(115) (2008) 9–11. [5] A.V. Herreinstein, N. Mashrabov, Heating of a circle by moving heat source. - Applied and industrial mathematics review. 15(5) (2008) 870– 871. [6] A.V. Herreinstein, E.A. Herreinstein, N. Mashrabov, Steady explicit schemes of the heat conductivity equation for an axially symmetric problem . SUSU Bulletin, "Mathematics, mechanics, physics» Series. 2,9(185) (2010) 4–8. [7] A.V. Herreinstein, E.A. Herreinstein, Steady explicit schemes of the heat conductivity equation SUSU Science: materials of the 62nd scientific conference. Sections of natural sciences, Chelyabinsk: SUSU Publishing center. (2010) 22–26. [8] A.V. Herreinstein, N. Mashrabov, E.A. Herreinstein, The state registration certificate of the computer program 2008612210. (2008). [9] A.V. Herreinstein, A.A. Dyakonov, A.A. Koshin, The state registration Certificate of the computer program 2010610052. [10] L.I. Sedov, Methods of similarity and dimension in mechanics. eigth ed., Science, Moscow, 1977. [11] M.G. Ivanov, Dimensions and similarity, Dolgoprudny, 2013. [12] A.N. Tikhonov, A.A. Samarskiy, Equations of mathematical physics, Science, Moscow, 1972. [13] I.G. Aramanovich, V.I. Levin, Equations of mathematical physics, Science, Moscow, 1969. [14] N. Mashrabov, A.V. Herreinstein, E.A. Herreinstein, Steady explicit schemes of the heat conductivity equation for a one-dimensional problem. CSAA Bulletin. 67(1) (2014) 50–54. [15] A.V. Herreinstein, N. Mashrabov, E.A. Herreinstein, Modeling of thermal fields with variable heat physical properties of a detail, Materials of the LIII international scientific and technical conference Part III, CSAA, Chelyabinsk. (2014) 31–38. [16] M.Z. Hayrislamov, A.V. Herreinstein, The explicit scheme of the third mixed problem solution for the quasilinear equation of heat conductivity, SUSU bulletin "Mathematics, mechanics, physics" series. 5(2) (2013) 174–177. [17] V.V. Stepanov, The differential equations course, Fizmatgiz, Moscow, 1959. [18] A.V. Herreinstein, M.Z. Hayrislamov, Explicit differential scheme of the one-dimensional quasilinear heat conductivity equation solution. SUSU bulletin , "Mathematics, mechanics, physics" series. 5(1) (2013) 12–17. [19] A.V. Herreinstein, N. Mashrabov, E.A. Herreinstein, Certificate on registration of an electronic resource 19347 [20] A.V. Herreinstein , A.A. Koshin, Yu.S. Khudyakova, D.V. Vostroknutov, Mathematical problem definition of blank temperature field modeling in the process of flute grinding of spiral drills, Progressive technologies in mechanical engineering : collected articles, Chelyabinsk: SUSU Publishing house. (2008) 206–211. [21] A.V. Herreinstein, A.A. Koshin, Yu.S. Khudyakova, D.V. Vostroknutov, The numerical problem solution of a blank temperature field modeling in the process of flute grinding of spiral drills, collected articles, Chelyabinsk: SUSU Publishing house. (2008) 211–215.
