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temperature.
As you can see, the obtained relations are quite simple. This is explained by the classical geometry of the configurations under consideration and the stationary nature of the process.
The results presented in the article can be used in further studies and calculations of fuel elements. References:
1. Vlasov N. M., Fedik I. I. Teplovydelyayuschie elements of nuclear rocket engines / M.: TsNII atominform, 2001. 208p.
2. Kanareykin, A. I. Temperature distribution in an elliptical body with an internal heat source under boundary conditions of the first kind // Bulletin of the Kaluga University, series: natural sciences. - 2020, №. 2. - P. 74-76.
3. Kanareykin, A. I. Distribution of the temperature field in a fuel rod with an elliptical cross-section / / Scientific Works of the Kaluga State University named after K. E. Tsiolkovsky, series: natural sciences. - 2016. - P. 230231.
© KaHapeHKHH A.H., 2021
УДК 536.212
Канарейкин А.И.
кан. техн. наук, доцент МГРИ, г. Москва, РФ
СТАЦИОНАРНОЕ РАСПРЕДЕЛЕНИЕ ТЕМПЕРАТУРНЫ В ПРЯМОУГОЛЬНОМ БРУСЕ
Аннотация
В статье производится нахождение распределения температуры в брусе. При этом процесс является стационарным. При этом на каждой грани бруса задана температура. Полученный результат был сопоставлен с решением похожей задачи.
Ключевые слова
Температурное поле, прямоугольный брус, стационарная теплопроводность, граничные условия, метод Фурье.
The problem under consideration is formulated as follows: on two of the side edges of a long rectangular bar (Fig. 1), it is maintained at a given temperature, on the other faces T = 0; find the steady-state temperature at an arbitrary point inside the bar.
z
y
Fig. 1. A bar with the set temperatures on the faces.
From the symmetry of the beam, it is clear that the temperature does not depend on Z and that we can limit ourselves to considering the cross section in the XOY plane. The problem consists in determining the function T = T (x, y) that satisfies the equation of stationary thermal conductivity:
-f+g- (1)
and two pairs of boundary conditions
TL=g (y), TL=f (y) (2)
TU o=0, TU=0 (3)
The solution of this equation is found by the Fourier method [1, p. 199]
T (x, y) = X (x)Y ( y) (4)
Differentiating (4) twice in x and y and substituting in (1), we obtain:
XY + XY" = 0 (5)
separating the variables
X V =_ Y V /X= /Y
(6)
Equating both parts of the constant X2, we arrive at ordinary linear differential equations
X "_A2 X = 0 (7)
Y" + A2Y = 0 (8)
The solution of equation (8) is in the form
Y(x) = C cos Ay + D sin Ay ^
As for equation (8), its solution, as is known, differs only in that, instead of trigonometric, it contains hyperbolic functions
X( x) = AchAy + BshAy ^ q-^
We now choose the constants A, B, C, and X so as to satisfy the boundary conditions (2) and (3). It is more convenient to start with (3) as simpler ones. So,
Y (0) = C = 0 (9)
hence,
Y(x) = D sin Ay (10)
By imposing the second boundary condition on y
Y(b) = D sin Ay = 0 (11)
we conclude that
Ab = — (12)
where n = 1, 2, 3,... From here
—
A-=- - (13)
b
Substituting these discrete values of the parameter X in (10), we get the set of functions X(x) and Y(y)
. , n—x „ , n-x Xn=AnCh — + Bnsh-T~ (14)
b b
Y- =D-sin (15)
b
Multiplying now, Xn(x) by Yn(y), we find the set of functions Tn(x, y) that satisfy equation (1) and boundary conditions (3)
ISSN 2410-6070_MHHOBAUMOHHAH HAyKA_№4 / 2021
Tn ( X, y) =
^ ^ , n—x ^ , n—c^ M nch-+ Nnsh-
v b b
. n—y
sin-
b
Now it remains to satisfy the conditions (2). The first of them will give the dependence
. nny
T =y M
lx=0 /—! n
n=l
sin-
b
= g ( y)
(16)
(17)
Hence, it can be seen that the constant factors Mn must be the coefficients of the Fourier series expansion of the function g (y) in terms of the sine
2b n
Mn = ^ | g (y)sin dy
To satisfy the last boundary condition = f (y), we also make an infinite sum
lx=a
Tn ( x, y) = Z\Mnch^ + Nnsh
n—x
b
b
. nny
sin-
b
n=1 V b b J
and select the coefficients Mn in such a way that the series for x ^ a converges to the function f(y)
A
vl ^ ; nna ,r , n—a
T|x=a =El MnCh— + N~Sh-
=1
b
b
J
■ —
sin—f- = f ( y) b
(18)
(19)
(20)
nna n—a
From this it can be seen that the constant factors Mnch--h N nsh-must be the coefficients fn of the
b b
Fourier series expansion of the function f(y) by sin-sam
, _ , n— ^ , n— Mch-+ N„sh-= f
n T n T J n
b b
(21)
from here
fn - Mnch
n—a
N =
sh
n—a
(22)
where
O b
N„ = 2 \f ( y)sin ^dy
(23)
Substituting the values of the coefficients Nn in the series (16), we obtain finally
T ( x, y) = £
n=1
M„
ch
n—x
, n—a , n—x ch-sh
b
b
sh
n—a
sh
+ fn
n—x
sh
n—a
. n—y sin—— b
(24)
v b J
We will analyze the resulting solution. Let the temperature on the left side face be T=0, then Mn =0. In this case, we obtain the solution given in [2, p. 121]
nnx
T ( x, y) = X fn
sh
n=1
sh
b . n—y —sin—— n—a b
(25)
In this paper, we present a solution for the distribution of the temperature field in a beam under certain boundary conditions. The resulting result was analyzed and compared with a similar task. References:
1. Kanareykin A. I. Application of the Poisson equation in thermophysics // Scientific Works of the Kaluga State
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n
b
0
ISSN 2410-6070_ИННОВАЦИОННАЯ НАУКА_№4 / 2021
University named after K. E. Tsiolkovsky, series: natural sciences. - 2016. - P. 199-200. 2. Nesis E. I. Methods of mathematical physics / M.: Prosveshchenie, 1977. 199 p.
© KaHapeHKHH A.H., 2021
УДК 620.92:620.97
Канарейкин А.И.
кан. техн. наук, доцент МГРИ, г. Москва, РФ
ОБ ЭФФЕКТИВНОСТИ ИСПОЛЬЗОВАНИЯ СОЛНЕЧНОЙ ЭНЕРГИИ В МАРТЕ МЕСЯЦЕ В СРЕДНЕЙ ПОЛОСЕ РФ
Аннотация
Работа посвящена оценке эффективности использования солнечных батарей для автономной работы частного дома в марте месяце в средней полосе РФ. Проведен анализ проблемы использования солнечной энергии. Представлены как теоретические оценки выработки электроэнергии, так и практические данные, полученные в ходе измерения сгенерированной и потреблённой электроэнергии. В результате полученных измерений дано заключение об экономической целесообразности использования солнечной энергии данном месяце.
Ключевые слова
Солнечная энергетика, солнечная батарея, солнечная электростанция, солнечные фотоэлектрические станции, инсоляция.
In recent decades, the global solar energy industry has been developing rapidly, and solar power plants are becoming part of the energy infrastructure of many countries. The development of solar technologies has a significant impact on the economy. It can be expected that in the coming decades, solar energy will become an incentive for the economic development of countries and regions that have the maximum "solar" resource [1, p.81].
Solar photovoltaic power plants are a type of power plant that generates electricity by directly converting the energy of solar radiation into electricity.
In addition to solar panels, an autonomous solar power plant usually contains rechargeable batteries (AB) and a charge-discharge controller (fig. 1). If it is necessary to supply electricity to consumers requiring a standard voltage of 220/380V AC, it is necessary to include it in the FES. switch on the inverter.
Figure 1 - Autonomous photovoltaic power supply system installed on a private house
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