Approximate Methods for Calculating the Two-Dimensional Temperature Field in the Windings and Cores of Electrical Machines
Abstract Generated abstract
This paper examines approximate methods for calculating two-dimensional steady temperature fields in windings and cores of electrical machines, modeled as bodies with anisotropic thermal conductivity, distributed heat generation, and convective boundary conditions. Using a rectangular bar as the reference problem, it compares a one-dimensional reduction, a method that divides heat flow between two coordinate directions with a constant ratio, and a refined method in which this ratio varies with position while satisfying key boundary conditions more accurately. The analysis shows that the first two approaches can substantially distort the temperature-field shape, especially for large Biot numbers or near boundaries, while the variable-ratio method closely matches the exact series solution in the numerical example. The proposed approximation is therefore recommended for more complex thermal calculations of electrical machine components.
Full Text
APPROXIMATE METHODS FOR CALCULATING THE TWO-DIMENSIONAL TEMPERATURE FIELD IN WINDINGS AND CORES OF ELECTRICAL MACHINES
D. I. SANNIKOV
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
In thermal calculations of electrical machines and transformers, in many cases it is necessary to know the nature of the temperature distribution inside windings and cores. Most such problems can be reduced to the calculation of a two-dimensional temperature field using approximate methods.
Fig. 1. Dimensions and thermal parameters of the body.
Let us consider some of these methods by using as an example the calculation of the temperature \(t(x,y)\) of a bar (Fig. 1) with distributed heat sources \(q\), having different dimensions \((a, b)\), thermal-conductivity coefficients \((\lambda_x, \lambda_y)\), and heat-transfer coefficients from the surface \((\alpha_x, \alpha_y)\) along the coordinate axes, with different temperatures of the cooling medium at the surfaces \(|x| = a\), \(t_{fa}\), and \(|y| = b\), \(t_{fb}\). This problem has an exact solution in the form of a functional series, the calculation of which is considerably labor-intensive.
All approximate methods are based on the introduction of simplifying assumptions in solving Poisson’s equation
\[ \lambda_x \frac{\partial^2 t(x,y)}{\partial x^2} + \lambda_y \frac{\partial^2 t(x,y)}{\partial y^2} + q = 0, \tag{1} \]
with boundary conditions
\[ -\lambda_x \frac{\partial t(a,y)}{\partial x} = \alpha_x \left[t(a,y)-t_{fa}\right], \tag{2} \]
\[ -\lambda_y \frac{\partial t(x,b)}{\partial y} = \alpha_y \left[t(x,b)-t_{fb}\right]. \]
1st method. In the case \(\lambda_x \gg \lambda_y\), the simplest assumption \(\lambda_y=0\) is possible, as a result of which the problem reduces to a one-dimensional one,
\[ \lambda_x \frac{d^2 t(x)}{dx^2} + q = 0, \tag{3} \]
\[ -\lambda_x \frac{dt(a)}{dx} = \alpha_x \left[t(a)-t_{fa}\right], \tag{4} \]
whose solution is the parabola
\[ t(x)=q\left(\frac{a^2-x^2}{2\lambda_x}+\frac{a}{\alpha_x}\right). \tag{5} \]
The same assumption may be adopted in one of the following cases
\[ b \gg a; \]
\[ \alpha_x \gg \alpha_y; \]
\[ t(a)-t_{fa} \gg t(b)-t_{fb}, \]
if the remaining parameters referred to the directions of both axes are of the same order. The computed field pattern differs qualitatively from the actual one; therefore this method permits calculation only of the temperature of the body’s mean point, and its value is overestimated, since one of the possible paths of heat removal is not taken into account. The accuracy of the calculation is the higher, the greater the ratio
\[ \frac{\bar t-t_{fa}}{\bar t-t_{fb}} \cdot \frac{b/\lambda_y+1/\alpha_y}{a/\lambda_x+1/\alpha_x}, \tag{6} \]
where \(\bar t\) is the mean temperature of the body.
The discrepancy between the computed and actual field patterns decreases as the Biot criterion for the coordinate \(y\) decreases,
\[ Bi_y=\frac{\alpha_y b}{\lambda_y}. \]
2nd method. An approximate accounting for the heat flux removed in the direction of the \(y\)-axis gives a field pattern closer to the real one. In the simplest case it is assumed that
\[ \frac{ \lambda_x \dfrac{\partial^2 t(x,y)}{\partial x^2} }{ \lambda_y \dfrac{\partial^2 t(x,y)}{\partial y^2} } = \frac{q_x}{q_y} = \xi = \mathrm{const}, \tag{8} \]
\[ q_x+q_y=q, \]
that is, the heat losses released in each unit volume are divided into two parts, one of which, \(q_x\), is removed in the direction parallel to the \(x\)-axis, and the other, \(q_y\), along the \(y\)-axis; moreover, the ratio of these parts \(\xi\) is the same for all points of the body. In this case the problem reduces to the calculation of two one-dimensional fields. The solution has the form—for heat transfer in the \(x\) direction,
\[ t(x,y)=q_x\,\frac{a^2-x^2}{2\lambda_x}+t(a,y), \tag{10} \]
for heat transfer in the \(y\) direction,
\[ t(x,y)=q_y\,\frac{b^2-y^2}{2\lambda_y}+t(x,b). \]
Both solutions must be identical; therefore,
\[ t(a,y)=q_y\,\frac{b^2-y^2}{2\lambda_y}+t(a,b), \tag{11} \]
\[ t(x,b)=q_x\,\frac{a^2-x^2}{2\lambda_x}+t(a,b). \]
The calculated field pattern in the sections \(x=\mathrm{const}\) has the form of parabolas of identical curvature; the same is also true for the sections \(y=\mathrm{const}\). This is its essential difference from the actual field pattern, for which the mean curvature of the profiles \(t(x)\) as \(y\to b\) and \(t(y)\) as \(x\to a\) decreases. The distortion obtained is caused mainly by violation of the boundary conditions. In the calculation, the heat-flux density through the outer surfaces of the body is assumed constant:
\[ w_x=q_x a, \tag{12} \]
\[ w_y=q_y b. \]
On the other hand, on the basis of the boundary conditions,
\[ w_x(y)=\alpha_x\,[t(a,y)-t_{fa}]\ne \mathrm{const}, \tag{13} \]
and analogously for \(w_y(x)\). Thus, a discontinuity of the heat-flux density is obtained at the surfaces of the body.
The simplification of the boundary conditions in order to resolve this contradiction is carried out in two ways:
a) according to [1], the heat-flux density is determined from the maximum temperature on the surface
\[ q_x a=\alpha_x[t(a,0)-t_{fa}], \tag{14} \]
\[ q_y b=\alpha_y[t(0,b)-t_{fb}]; \]
b) according to [2]—from the mean temperature
\[ q_x a=\alpha_x\left[\frac{1}{b}\int_0^b t(a,y)\,dy-t_{fa}\right], \tag{15} \]
\[ q_y b=\alpha_y\left[\frac{1}{a}\int_0^a t(x,b)\,dx-t_{fb}\right]. \]
On the basis of expressions (11), (12), (14), and (15), a formula for calculating \(\xi\) is derived: for the first method,
\[ \xi= \frac{ q\left(\dfrac{b^{2}}{2\lambda_y}+\dfrac{b}{\alpha_y}\right)+t_{fb}-t_{fu} }{ q\left(\dfrac{a^{2}}{2\lambda_x}+\dfrac{a}{\alpha_x}\right)+t_{fa}-t_{fb} }, \tag{16} \]
for the second method, instead of 2 in the denominators, 3 is obtained. Further, having determined
\[ q_x=q\,\frac{\xi}{\xi+1}, \]
\[ q_y=q\,\frac{1}{\xi+1}, \tag{17} \]
one can calculate the temperature field
\[ t(x,y)=t_0-q_x\frac{x^2}{2\lambda_x}-q_y\frac{y^2}{2\lambda_y}, \tag{18} \]
where \(t_0\) is the temperature at the center of the body.
For the first method,
\[ t_0=q_x\left(\frac{a^2}{2\lambda_x}+\frac{a}{\alpha_x}\right)+t_{fa}, \tag{19} \]
for the second method \(t_0\) is greater by the amount \(q_y b^2/6\lambda_y\).
The error of the method based on the assumption \(\xi=\mathrm{const}\) is smaller than with the assumption \(\lambda_y=0\) or \(q_y=0\); however, it is excessively large for large values of the Biot criterion. Calculation of heat transfer from the surface according to the maximum temperature (the first method) gives underestimated results; calculation according to the mean temperature (the second method) gives overestimated results. The greatest distortion of the field is obtained in the peripheral parts of the body.
3rd method. A considerable increase in the accuracy of the approximate calculation is achieved under the assumption
\[ \frac{ \lambda_x\dfrac{\partial^2 t(x,y)}{\partial x^2} }{ \lambda_y\dfrac{\partial^2 t(x,y)}{\partial y^2} } = \frac{q_x(y)}{q_y(y)} =\xi(y), \tag{20} \]
under the condition
\[ Bi_y=\frac{\alpha_y b}{\lambda_y}>Bi_x=\frac{\alpha_x a}{\lambda_x}. \tag{21} \]
Thus, it is assumed that the ratio between the heat fluxes in the \(x\) and \(y\) directions is the same for all points of the straight line \(y=\mathrm{const}\), drawn in the direction of greatest thermal conductivity, but is different for different positions of this straight line.
For all points at \(y=\mathrm{const}\),
\[ q_x(y)=\mathrm{const}. \]
Taking exact account of the boundary conditions on the surface \(x=a\), we have
\[ t(x,y)=q_x(y)\cdot\left(\frac{a^2-x^2}{2\lambda_x}+\frac{a}{\alpha_x}\right)+t_{fa}. \tag{22} \]
— a parabolic distribution along the direction \(x\), the curvature of the parabolas being different for different values of \(y\).
On the basis of the energy balance for an elementary layer \(dy\), we obtain
\[ q_y(y)=-\lambda_y\frac{d^2\overline{t}(y)}{dy^2}, \tag{23} \]
where
\[ \overline{t}(y)=\frac{1}{a}\int_0^a t(x,y)\,dx \]
is the mean temperature along the line \(y=\mathrm{const}\).
From (22)
\[ \overline{t}(y)=q_x(y)\cdot r+t_{fa}, \tag{24} \]
where
\[ r=\frac{a^2}{3\lambda_x}+\frac{a}{\alpha_x}. \tag{25} \]
Hence
\[ q_y(y)=q-q_x(y)=q-\frac{\overline{t}(y)-t_{fa}}{r}. \tag{26} \]
On the basis of expressions (23) and (26), the problem is reduced to the calculation of the heating of a homogeneous rod with heat sources distributed along its length, cooled along its length and from its ends,
\[ \lambda_y\frac{d^2\overline{t}(y)}{dy^2} -\frac{\overline{t}(y)}{r} +\left(q+\frac{t_{fa}}{r}\right)=0, \tag{27} \]
\[ -\lambda_y\frac{d\overline{t}(y)}{dy} =\alpha_y\left[\overline{t}(b)-t_{fb}\right]. \tag{28} \]
The solution of the problem is
\[ \overline{t}(y)=qr\left[1-\left(1+\frac{t_{fa}-t_{fb}}{qr}\right)\frac{\operatorname{ch}\beta y}{N}\right]+t_{fa}. \tag{29} \]
Here
\[ \beta=\frac{1}{\sqrt{r\lambda_y}}, \tag{30} \]
\[ N=\operatorname{ch}\beta b+\frac{\lambda_y\beta}{\alpha_y}\operatorname{sh}\beta b. \tag{31} \]
Determining \(q_x(y)\) from (29) and (24) and substituting into (22), we obtain the complete approximate solution
\[ t(x,y)=q\left[1-\left(1+\frac{t_{fa}-t_{fb}}{qr}\right)\frac{\operatorname{ch}\beta y}{N}\right] \cdot\left(\frac{a^2-x^2}{2\lambda_x}+\frac{a}{\alpha_x}\right)+t_{fa}. \tag{32} \]
In contrast to the two preceding methods, the boundary conditions are satisfied exactly on the surfaces \(|x|=a\), and, under the condition \(t_{fa}=t_{fb}\), also on the surfaces \(|y|=b\).
Figure 2 gives the results of calculating the temperature field in the cross section of a bar for \(a=15\ \mathrm{cm}\), \(b=10\ \mathrm{cm}\), \(\lambda_x=20\ \mathrm{W}/(\mathrm{cm}\cdot\mathrm{deg})\), \(\lambda_y=0.014\ \mathrm{W}/(\mathrm{cm}\cdot\mathrm{deg})\), \(\alpha_x=\alpha_y=0.01\ \mathrm{W}/(\mathrm{cm}^2\cdot\mathrm{deg})\), \(q=0.02\ \mathrm{W}/\mathrm{cm}^2\), \(t_{fa}=t_{fb}=0\).
The solid curves represent the result of the exact solution. The results of the approximate calculation are shown by points.
Fig. 2. Results of calculating the temperature field.
— exact solution;
× — 1st method (assuming \(\lambda_y = 0\));
△ — 2nd method (assuming \(\xi = \mathrm{const}\); boundary conditions according to [1]), \(y = 0\) and \(10\);
● — 2nd method (boundary conditions according to [2]), \(y = 0\) and \(10\);
○ — 3rd method (assuming \(\xi = f(y)\)).
The third approximate-calculation method—by formula (32)—provides the greatest accuracy in comparison with the others. The maximum deviation from the exact-calculation data in the present example is \(\pm 0.7\%\, t_0\); the shape of the temperature-field profiles is not distorted.
Thus, the principle of approximate calculation of the temperature field based on assumptions (20) and (23) is the most effective and may be recommended for calculations in more complex cases.
REFERENCES
- G. Gotter. Heating and Cooling of Electrical Machines. GEI, 1961.
- G. P. Boikov, Yu. A. Korolenko. On the question of the temperature field of bodies with internal heat generation. Izvestiya TPI, vol. 110, Tomsk, 1962.