CC BY
24
UDC 519.633.6, 519.688
Model with One Spatial Variable for Design of a Technical
Device
A. S. Ayriyan
Laboratory of Information Technologies Joint Institute for Nuclear Research Joliot-Curie, 6, 141980 Dubna, Moscow Region, Russia
Model with one spatial cylindrical variable for optimization of a technical device is presented. Device is in the form of a multilayer cylindrical sample with a pulse source operating at cryogenic temperatures. The results of numerical experiments are shown.
Key words and phrases: heat equation, partial differential equation, finite-difference scheme, mathematical modeling.
1. Introduction
The main goal of this work is to implement a model of thermal processes for a given object (Fig. 1) for systematic studies in order to find optimal materials and composition of the technical device. The object subject to the investigation is a cryogenic cell pulse (in the millisecond range) feeding the working gases into the working space of charged ion source [1].
Thermal processes occur at periodic passage electric current through conductive layer. Full period of the process is requested to be 10 ms: the heating period is 1 ms (current on, the object surface should be heated up to 60 K), for 9 ms it is cooling (current off, the surface should be cooled down to 4.2 K ~ 10 K).
* z
Figure 1. Schematic view of the object slice. The slice of the object: 0 — cooler (r* = 0.12), 1 — electrical insulator (r* = 0.125), 2 — heat source (conductive layer, r* = 0.13), 3 — external insulator (rmax = 0.1301), 4 — liquid helium temperature
terminal with T = 4.2 K
Due to the cylindrical symmetry of the object, the heat conductivity inside it can be simulated by a model with two spatial cylindrical variables r and z (instead of three) and time variable. This model was presented and discussed in [2] and [3]. Possibility of using model with one spatial cylindrical variable r was proposed and discussed in [3], it is shown that thermal flow in the z direction can be neglected for beginning part of thermal process.
Received 8th February, 2013.
Author thanks Jan Pribis and Hovik Grigorian for useful advices. The work was supported by RFBR grants 11-01-00278.
2. Main Equations and Boundary Conditions
The thermal processes in the object can be described by the following system of parabolic partial differential equations with temperature depended coefficients [4]:
PmCVm(T)^ = 11 (rXm(T)^ + Xm(T), (1)
where r G [0, rmax] and t G [0, ¿max = 10 ms]. For m = 0,1 and 3 the function Xm(T) = 0 (there is no source).
The object involves different materials in construction with different densities and thermal coefficients; thus the index m is introduced for each material (m = 0 — cooler (cooper), m = 1 — electrical insulator, m = 2 — heat source (graphite), m = 3 — external insulator).
Coefficients Cvm and Xm are specific heat capacity and thermal conductivity, respectively for each material m (see Fig. 2 and Fig. 3). It is well known that thermal conductivity and specific heat for many materials could vary up to order of magnitude as temperature varies from T = 4.2 K to T = 60 K. For the chosen materials the corresponding data is obtained from the [5], and then these dependencies were approximated by the least-squares method using the linear and nonlinear analytical functions.
Figure 2. The heat capacities for different materials, C'v (T)
Figure 3. The thermal conductivities for different materials, X(T)
Source dependence was approximated by function X2(T) = %(T)I2/S, where X(T) = 1.8/y/T and constant I2/S = 106. In this formula I means electric current in the graphite slice along direction and S is graphite cross section in the direction orthogonal to the current propagation [2]. The initial condition is given below
T (r ,t = 0) = To,
(2)
where T0 = 4.2 (liquid helium temperature) and the boundary ones are taken as
dT
dr
0.
(3)
r=0 and rm
The functions Cv, A and Xi = X(Ti) have discontinuities of first kind at the following points: r*, r* and r* in the interval [0..rmaxj. Conjugation condition between materials considered to be ideal:
Am(T) f
r=r* — 0
X™+l(T) ^
(4)
r=r* +0
where m = 0,1, 2 and r*m is point of the border between the materials m and m + 1 (discontinuity points).
3. Numerical Algorithm
For numerical solution the initial-boundary-value problem (1)-(3), the equation (1) was approximated by the following explicit finite difference scheme [6]:
P°vi = -Ai [n^iTi] + (5)
ht ri
The spatial finite difference operator is:
1
hr2
Ai [ri\iTi] _ ^ \ri+1 Xi+ 2 (Ti+i - Ti) - r— 2 2 (Ti - T— )_| , (6)
where
Ti _ T (ri,t), Cvi _ Cy (Ti), Xi _ X(Ti), Xi _ X (Ti),
(Ti + Ti±i \ = n ± y, Xi± i = X^-2-) ■
The boundary conditions have been approximated by the following set of formulas:
T0 = 1 (4Ti - T2) ,
! 3 1 ^ . (7)
Tn = 1 (^Tn-! — Tn-^ ,
for r = 0 and r = rmax, respectively.
The conjugation conditions are approximated using one-sided derivatives:
,3T* — 4Ti*-i + T*-2 _ —3T* +4Ti*+! — Ti* +2 ,as
Am(li*)-2!-= )-2!-' (8)
2hr 2hr
where i* is corresponding index of discontinuity point.
Numerical calculations to solve the problem (1)-(3) are carried out on a uniform grid of variables r and t with constant spatial and time steps hr and ht:
h _ ^max
r n ' h2
ht < y min
pCv (T )
max |A(T)|
here n is the number of partitions, and respectively, ri = i ■ hr, for « = 0,..., n.
The scheme (5)-(8) is tree points explicit difference scheme and it approximates the initial-boundary-value problem (1)-(3) with the accuracy 0(|ht| + h2) [4].
4. Results
Set of numerical simulations have been performed using spacial and time steps hr = 10-5 cm and ht = 10-11 s, for which the schema (5) converges. For the fourth material (external insulator) thermal conductivity was taken as constant A4 = 10-3. Other thermal coefficient dependencies for each materials are given on Fig. 2 and Fig. 3. There are T(r) distributions at different time moments on the Fig. 4.
Figure 4. T(r) distribution at different time moments. The solid lines correspond to heating. The dashed lines correspond to cooling
5. Conclusion and Discussion
In this paper the model with one spatial cylindrical variable for optimization of the object design is presented. This model allows modeling the beginning part of the thermal process (about 3 ms); thus it can significantly reduce the study of choosing the optimal design of the object in terms of practical needs and make a lot of preliminary studies.
For example, the results show that the given construction of the object using given materials does not satisfy the requirements (requirements for the thermal process are described in the "Introduction"). Firstly, the object surface (right point in Fig. 4) could not been heated up to 60 K in 1 ms. The heating time takes 1.6 ms. To speed up heating source should be increased. Secondly, for time ~5-10 ms the results for this model will vary slightly disagree with reality, however one can be sure that the object surface could not been cooled down to ~ 4.2-10 K in the remaining time. The electrical insulator between cooler and source does not allow to cool the object down for the requested time. It is needed to "play" with geometrical shape of the insulator or change the insulator material.
It is necessary to do a systematic studies in order to find optimal construction and materials of the object. One of the way of such study can be formulation and solution of the multiparameter optimal control problem [4] for the model with one spatial variable proposed in this paper.
References
1. Электронно-струнные источники многозарядных ионов с линейной и трубчатой геометрией струны / Д. Е. Донец, Е. Д. Донец, Е.Е. Донец и др. // Прикладная физика. — 2010. — № 3. — С. 34-42. [Electron-String Ion Sources of Highly Charged Ions with Linear and Tubular Ge- ometry of String / D. E. Donets, E. D. Donets, E. E. Donets et al. // Applied Physics. — 2010. — No 3. — Pp. 34-42. ]
2. Numerical Simulation of Heat Conductivity in Composite Object with Cylindrical Symmetry / A. Ayriyan, E. Ayryan, E. Donets, J. PribiS // Lecture Notes in Computer Science. — 2012. — No 7125. — Pp. 264-269.
3. Айриян А. С., Прибиш Я. Моделирование процесса теплопроводности в составном образце с цилиндрической симметрией // Математическое моделирование. — 2012. — Т. 24, № 12. — С. 113-118. [Ayriyan A., Jan Pribis. Mathematical Simulation of Heat Conductivity in Compos- ite Object with Cylindrical Symmetry // Mathematical Modelling. — 2012. — Vol. 24, No 12. — Pp. 113-118. ]
4. Samarskii A. A., Vabishchevich P. N. Computational Heat Transfer. — Chichester, England: John Wiley & Sons Ltd., 1995. — Vol. 1, Mathematical Modelling.
5. National Institute Of Standards And Technology. — http://www.nist.com/.
6. Samarskii A. A. The Theory of Difference Schemes. — New York: Marcel Dekker Inc., 2001.
УДК 519.633.6, 519.688
Модель с одной пространственной переменной для проектирования технического устройства
А. С. Айриян
Лаборатория информационных технологий Объединённый институт ядерных исследований ул. Жолио-Кюри, д.6, Дубна, Московская область, 141980, Россия
В работе предложена модель с одной пространственной цилиндрической переменной для проектирования технического устройства. Устройство работает при криогенных температурах и имеет импульсный источник, его можно представить как многослойный цилиндрический образец. Представлены результаты вычислительных экспериментов.
Ключевые слова: уравнение теплопроводности, уравнения в частных производных, разностная схема, математическое моделирование.
