Equivalent circuit for a system of two heat-conducting rods
Abstract Generated abstract
This paper develops an equivalent thermal circuit for two parallel heat-conducting rods, such as an electrical-machine winding thermally coupled to a core, shaft, or housing, with distributed mutual conductance, cooling to a flow whose temperature varies linearly, and temperature-dependent winding losses. Starting from differential heat-balance equations, the authors derive expressions for the longitudinal temperature fields, relate boundary temperatures to end heat fluxes and mean rod temperatures, and use these relations to construct circuit parameters. Because the exact parameter formulas are cumbersome, series approximations are proposed for small and medium-power machines, with a reported winding-temperature error below 1 percent for rod lengths under 50 cm. The method also provides a convenient parabolic approximation for temperature distribution and is presented as suitable for digital computation of nodal temperatures.
Full Text
Equivalent Circuit for a System of Two Heat-Conducting Rods
V. A. Zhadan, D. I. Sannikov
(Presented by the scientific seminar of the Department of General Electrical Engineering and Electrical Machines)
The solution of a number of problems in calculating the distribution of temperature along the length of the windings of electrical machines, taking into account their thermal connection with other bodies, reduces to the calculation of a system of two parallel rods (winding and core, winding and shaft, winding and housing, etc.) connected with each other and with the cooling medium by uniformly distributed thermal conductances and, in addition, having thermal contact at their ends with other bodies or with the medium.
Let \(\Lambda_{\mathrm{ov}}\) be the mutual conductance per unit length of the rods.
\(\Lambda_{\mathrm{of}}\) and \(\Lambda_{\mathrm{vf}}\) are the conductances between the rods and the cooling medium (per unit length).
\(r_{\mathrm{o}}\) and \(r_{\mathrm{v}}\) are the axial resistances of the rods per unit length.
\(\rho_{\mathrm{o}}\) and \(\rho_{\mathrm{v}}\) are the linear density of losses.
Since the losses in one of the rods (the winding) usually depend on temperature, it is necessary to take
\[ \rho_{\mathrm{o}}=\rho_{\mathrm{oo}}\left[1+\alpha_{\mathrm{o}}\left(\Theta_{\mathrm{o}}-t_{\delta}\right)\right], \tag{1} \]
where \(\rho_{\mathrm{oo}}\) are the losses at the reference temperature \(t_{\delta}\),
\(\alpha_{\mathrm{o}}\) is the temperature coefficient of losses.
In what follows it will be assumed that \(t_{\delta}=0\).
The temperature of the cooling flow \(t_f\) in electrical machines generally varies along the length. This variation may be regarded as practically linear.
It is required to calculate the temperature distribution along the length of the rods, \(\Theta_{\mathrm{o}}(x)\) and \(\Theta_{\mathrm{v}}(x)\), on the interval \(0<x<1\), where
\[ \Theta_{\mathrm{o}}(0)=\Theta_{\mathrm{o1}};\quad \Theta_{\mathrm{o}}(1)=\Theta_{\mathrm{o2}};\quad \Theta_{\mathrm{v}}(0)=\Theta_{\mathrm{v1}};\quad \Theta_{\mathrm{v}}(1)=\Theta_{\mathrm{v2}}; \]
\[ t_f(0)=t_{f1};\quad t_f(1)=t_{f2}. \]
On the basis of the heat balance for an element \(dx\), a system of differential heat-conduction equations for the rods is formed:
\[ \frac{1}{r_{\mathrm{o}}}\frac{d^2\Theta_{\mathrm{o}}}{dx^2} +\Lambda_{\mathrm{oo}}\Theta_{\mathrm{o}} +\Lambda_{\mathrm{ov}}\Theta_{\mathrm{v}} +\Lambda_{\mathrm{of}}\left(t_{f1}\frac{1-x}{1}+t_{f2}\frac{x}{1}\right) +\rho_{\mathrm{oo}}=0; \]
\[ \frac{1}{r_{\mathrm{v}}}\frac{d^2\Theta_{\mathrm{v}}}{dx^2} +\Lambda_{\mathrm{ov}}\Theta_{\mathrm{o}} +\Lambda_{\mathrm{vv}}\Theta_{\mathrm{v}} +\Lambda_{\mathrm{vf}}\left(t_{f1}\frac{1-x}{1}+t_{f2}\frac{x}{1}\right) +\rho_{\mathrm{v}}=0, \tag{2} \]
where
\[ \Lambda_{\mathrm{oo}}=-\Lambda_{\mathrm{ov}}-\Lambda_{\mathrm{of}}+\alpha_{\mathrm{o}}\rho_{\mathrm{oo}}, \]
\[ \Lambda_{\mathrm{vv}}=-\Lambda_{\mathrm{ov}}-\Lambda_{\mathrm{vf}}. \]
The solution of the system, expressed in terms of the boundary temperatures, is written in the following form:
\[ \begin{aligned} \Theta_{\mathrm{o}}={}& \theta_{\mathrm{o}1}\left[n_{\Delta} f_{\Sigma}(1-x)+n_{\Sigma} f_{\Delta}(1-x)\right] +\theta_{\mathrm{o}2}\left[n_{\Delta} f_{\Sigma}(x)+n_{\Sigma} f_{\Delta}(x)\right] \\ &+\theta_{\mathrm{v}1} n_{\mathrm{o}}\left[f_{\Delta}(1-x)-f_{\Sigma}(1-x)\right] +\theta_{\mathrm{v}2} n_{\mathrm{o}}\left[f_{\Delta}(x)-f_{\Sigma}(x)\right] \\ &+K_{\mathrm{o}} t_{\mathrm{f}1}\frac{1-x}{1} +K_{\mathrm{o}} t_{\mathrm{f}2}\frac{x}{1} +t_{\mathrm{o}p}; \end{aligned} \tag{3} \]
\[ \begin{aligned} \Theta_{\mathrm{v}}={}& \theta_{\mathrm{o}1} n_{\mathrm{v}}\left[f_{\Delta}(1-x)-f_{\Sigma}(1-x)\right] +\theta_{\mathrm{o}2} n_{\mathrm{v}}\left[f_{\Delta}(x)-f_{\Sigma}(x)\right] \\ &+\theta_{\mathrm{v}1}\left[n_{\Sigma} f_{\Sigma}(1-x)+n_{\Delta}f_{\Delta}(1-x)\right] +\theta_{\mathrm{v}2}\left[n_{\Sigma}f_{\Sigma}(x)+n_{\Delta}f_{\Delta}(x)\right] \\ &+K_{\mathrm{v}}t_{\mathrm{f}1}\frac{1-x}{1} +K_{\mathrm{v}}t_{\mathrm{f}2}\frac{x}{1} +t_{\mathrm{v}p}, \end{aligned} \tag{4} \]
where
\[ \begin{aligned} \theta_{\mathrm{o}1}&=\Theta_{\mathrm{o}1}-K_{\mathrm{o}} t_{\mathrm{f}1}-t_{\mathrm{o}p};\\ \theta_{\mathrm{o}2}&=\Theta_{\mathrm{o}2}-K_{\mathrm{o}} t_{\mathrm{f}2}-t_{\mathrm{o}p};\\ \theta_{\mathrm{v}1}&=\Theta_{\mathrm{v}1}-K_{\mathrm{v}} t_{\mathrm{f}1}-t_{\mathrm{v}p};\\ \theta_{\mathrm{v}2}&=\Theta_{\mathrm{v}2}-K_{\mathrm{v}} t_{\mathrm{f}2}-t_{\mathrm{v}p}, \end{aligned} \tag{5} \]
are the temperature deviations at the ends of the rods under the influence of the cooling conditions at these ends.
The distribution functions of the boundary deviations of the field are:
\[ f_{\Sigma}(x)=\frac{\operatorname{Sh}\beta_{\Sigma}(x)}{\operatorname{Sh}\beta_{\Sigma}l}; \qquad f_{\Delta}(x)=\frac{\operatorname{Sh}\beta_{\Delta}(x)}{\operatorname{Sh}\beta_{\Delta}l} \tag{6} \]
\(f(1-x)\) is obtained by substituting \((1-x)\) instead of \(x\).
The coefficients of propagation of the boundary deviations are:
\[ \beta_{\Sigma}=\sqrt{\frac{c_{\Sigma}+c_{\Gamma}}{2}}; \qquad \beta_{\Delta}=\sqrt{\frac{c_{\Sigma}-c_{\Gamma}}{2}}, \tag{7} \]
where
\[ \begin{aligned} c_{\Sigma}&=-(r_{\mathrm{o}}\Lambda_{\mathrm{oo}}+r_{\mathrm{v}}\Lambda_{\mathrm{vv}});\\ c_{\Delta}&=r_{\mathrm{o}}\Lambda_{\mathrm{oo}}-r_{\mathrm{v}}\Lambda_{\mathrm{vv}};\\ c_{\Gamma}&=\sqrt{C_{\Delta}^{2}+4r_{\mathrm{o}}r_{\mathrm{v}}\Lambda_{\mathrm{ov}}^{2}}. \end{aligned} \tag{8} \]
The relative shares of both deviation functions in the total deviation of the field are:
\[ n_{\Delta}=\frac{c_{\Gamma}-c_{\Delta}}{2c_{\Gamma}}; \qquad n_{\Sigma}=\frac{c_{\Gamma}+c_{\Delta}}{2c_{\Gamma}}, \tag{9} \]
\[ n_{\mathrm{o}}=\frac{r_{\mathrm{o}}\Lambda_{\mathrm{ov}}}{c_{\Gamma}}; \qquad n_{\mathrm{v}}=\frac{r_{\mathrm{v}}\Lambda_{\mathrm{ov}}}{c_{\Gamma}}. \]
The conditional excesses of the rod temperatures over the temperature of the cooling medium, i.e., the excesses in the absence of axial heat conduction of the rods, are:
\[ t_{\mathrm{o}p}=\frac{-p_{\mathrm{oo}}\Lambda_{\mathrm{vv}}+p_{\mathrm{v}}\Lambda_{\mathrm{ov}}}{D_{\Lambda}}; \qquad t_{\mathrm{v}p}=\frac{p_{\mathrm{oo}}\Lambda_{\mathrm{ov}}-p_{\mathrm{v}}\Lambda_{\mathrm{oo}}}{D_{\Lambda}}, \tag{10} \]
where
\[ D_{\Lambda}= \left| \begin{matrix} \Lambda_{\mathrm{oo}} & \Lambda_{\mathrm{ov}}\\ \Lambda_{\mathrm{ov}} & \Lambda_{\mathrm{vv}} \end{matrix} \right|. \]
The coefficients taking into account the influence of heating of the air flow when the losses depend on temperature are:
\[ k_{\mathrm{o}}= \frac{-\Lambda_{\mathrm{of}}\Lambda_{\mathrm{vv}}+\Lambda_{\mathrm{vf}}\Lambda_{\mathrm{ov}}}{D_{\Lambda}}; \qquad k_{\mathrm{v}}= \frac{\Lambda_{\mathrm{of}}\Lambda_{\mathrm{sv}}-\Lambda_{\mathrm{vf}}+\Lambda_{\mathrm{oo}}}{D_{\Lambda}} \tag{11} \]
To use expressions (3) and (4), it is necessary to know the temperatures at the ends of the rods, which can be found by taking into account the conditions of their connection with neighboring bodies. For this purpose, on the basis of (3) and (4), equations are drawn up for the relation between the boundary temperatures and the heat fluxes directed from the end faces of the rods to the neighboring bodies,
\[ Q_{\mathrm{o}1}=-\frac{1}{r_{\mathrm{o}}}\frac{d\Theta_{\mathrm{o}}}{dx}\quad (x=0), \tag{12} \]
\[ Q_{\mathrm{o}2}=+\frac{1}{r_{\mathrm{o}}}\frac{d\Theta_{\mathrm{o}}}{dx}\quad (x=1), \]
and similarly for \(Q_{\mathrm{v}1}\) and \(Q_{\mathrm{v}2}\).
Next, the mean temperatures of the rods are determined
\[ \overline{\Theta}_{\mathrm{o}}=\frac{1}{l}\int_{0}^{1}\Theta_{\mathrm{o}}\,dx; \qquad \overline{\Theta}_{\mathrm{v}}=\frac{1}{l}\int_{0}^{1}\Theta_{\mathrm{v}}\,dx, \tag{13} \]
and, with their aid, the conditional temperatures \(t_{\mathrm{op}}\) and \(t_{\mathrm{vp}}\) are eliminated from (12).
As a result, a system of six equations is obtained, relating the temperatures \(\Theta_{\mathrm{o}1}\), \(\Theta_{\mathrm{o}2}\), \(\overline{\Theta}_{\mathrm{o}}\), \(\Theta_{\mathrm{v}1}\), \(\Theta_{\mathrm{v}2}\), \(\overline{\Theta}_{\mathrm{v}}\), the heat sources \(F'_{\mathrm{oo}}=p_{\mathrm{oo}}l\), \(P'_{\mathrm{vv}}=p_{\mathrm{v}}l\), equal to the total losses in the rods, and the boundary fluxes \(Q_{\mathrm{o}1}\), \(Q_{\mathrm{o}2}\), \(Q_{\mathrm{v}1}\), \(Q_{\mathrm{v}2}\). On the basis of the system obtained, an equivalent circuit can be constructed (Fig. 1).
The circuit parameters:
the total conductances between the rods and the cooling air are
\[ G_{\mathrm{of}}^{-}=\Lambda_{\mathrm{of}}l, \qquad G_{\mathrm{vf}}^{-}=\Lambda_{\mathrm{vf}}l; \tag{14} \]
the conductance that takes into account the increase of losses in the winding with temperature (negative) is
\[ G_{\mathrm{ot}}=-\alpha_{\mathrm{o}}p_{\mathrm{oo}}l. \tag{15} \]
The remaining expressions for the parameters become extremely cumbersome; therefore, for machines of small and medium power it is expedient to expand them in power series and take one or two terms of the expansion, depending on the required accuracy.
\[ G_{\mathrm{fo}}=\frac{1}{12}\overline{G}_{\mathrm{of}}; \qquad G_{\mathrm{fv}}=\frac{1}{12}\overline{G}_{\mathrm{vf}}. \tag{16} \]
the conductances that take into account the asymmetric character of the cooling caused by heating of the flow,
\[ g_{\mathrm{fo}}=\frac{6}{r_{\mathrm{o}}l}-\frac{\Lambda_{\mathrm{oo}}l}{10}; \qquad g_{\mathrm{fv}}=-\frac{6}{r_{\mathrm{v}}l}-\frac{\Lambda_{\mathrm{vv}}l}{10}; \tag{17} \]
\[ g_{1\mathrm{o}}=-\frac{2}{r_{\mathrm{o}}l}-\frac{\Lambda_{\mathrm{oo}}l}{30}; \qquad g_{1\mathrm{v}}=-\frac{2}{r_{\mathrm{v}}l}-\frac{\Lambda_{\mathrm{vv}}l}{30}; \]
are the equivalent axial conductances of the rods.
The mutual conductances of the rods are
\[ \overline{G}=36\frac{\Lambda_{\mathrm{ov}}l}{30}; \qquad \overline{G}'=4\frac{\Lambda_{\mathrm{ov}}l}{30}; \tag{18} \]
\[ \overline{G}''=-3\frac{\Lambda_{\mathrm{ov}}l}{30}; \qquad \overline{G}'''=-\frac{\Lambda_{\mathrm{ov}}l}{30}. \]
The sum of all mutual conductances
\[ \overline{G}+2\overline{G}^{\prime}+4\overline{G}^{\prime\prime}+2\overline{G}^{\prime\prime\prime} =\Lambda_{\mathrm{ov}}\,l. \]
is equal to the total conductance between the rods.
Fig. 1. Equivalent circuit of two rods.
The error in calculating the winding temperature when using approximate formulas for the thermal conductances does not exceed 1 percent for \(l<50\) cm. Thus, the proposed method for calculating the parameters of the circuit is quite suitable for the active length of electrical machines of low and medium power.
When calculating the temperature distribution along the length of a rod, instead of the exact formulas (3) and (4), it is more convenient to use the formula of the equivalent parabola,
\[ \Theta_o = \Theta_{o1}\frac{1-x}{l} + \Theta_{o2}\frac{x}{l} + 3(2\overline{\Theta}_o-\Theta_{o1}-\Theta_{o2}) \frac{x}{l}\frac{1-x}{l} \tag{19} \]
The proposed type of equivalent circuit is especially convenient when using an electronic digital computer, owing to the fairly large number of equations that must be solved in determining the nodal temperatures of the circuit and, at the same time, owing to the simplicity of calculating the coefficients of these equations.