Analysis of Armature Heating in Enclosed Non-Ventilated DC Machines by the Method of the Theory of Thermal Similarity

M. N. Ulyanishkii, V. V. Salomatov

(Submitted by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

As is known, similarity theory makes it possible to combine all parameters characterizing a physical process into unified dimensionless complexes (similarity criteria) and to pass from an analytical dependence in the form of a relation between dimensional quantities to an analytical dependence in the form of a relation between similarity criteria. This transformation, first, reduces the number of variables and thus facilitates the analysis of analytical solutions, and, second, makes it possible to carry out a broad scientific generalization of research results and present them in a form convenient for application in engineering practice.

Using the propositions of the theory of thermal similarity [1], it is possible, in accordance with the calculation scheme of the armature winding adopted in [3], to present the equations of the temperature-distribution curves along the length of individual sections of the winding in dimensionless form¹:

1. For the slot portion

\[ \theta_p = C_1 e^{\xi \sqrt{k_c \frac{\operatorname{Bi}\psi_p}{\varepsilon}}} + C_2 e^{-\xi \sqrt{k_c \frac{\operatorname{Bi}\psi_p}{\varepsilon}}} + \frac{k_c P_0' + P_0}{k_c \frac{\operatorname{Bi}\psi_p}{\varepsilon}}. \]

\[ C_{1(2)} = \frac{ \left( \theta_{2(1)} - \frac{k_c P_0' P_0}{k_c \frac{\operatorname{Bi}\psi_p}{\varepsilon}} \right) e^{\frac{\xi_p}{2}\sqrt{k_c \frac{\operatorname{Bi}\psi_p}{\varepsilon}}} - \left( \theta_{(2)} - \frac{k_c P_0' P_0}{k_c \frac{\operatorname{Bi}\psi_p}{\varepsilon}} \right) e^{-\frac{\xi_p}{2}\sqrt{k_c \frac{\operatorname{Bi}\psi_p}{\omega}}} }{ 2\operatorname{sh}\!\left(\xi_p \sqrt{k_c \frac{\operatorname{Bi}\psi_p}{\xi}}\right) }. \]

1. For the portion of the end connections on the drive side (collector), located under the insulation of the band,

\[ \theta_{\ell(\mathrm{ck})} = C_{1(2)} e^{\xi \sqrt{\gamma \operatorname{Bi}\psi_\ell}} + C_{2(1)} e^{-\xi \sqrt{\gamma \operatorname{Bi}\psi_\ell}} + \frac{P_0}{\operatorname{Bi}\gamma\psi_\ell}; \]

\[ C_1= \frac{ \theta_{4(3)}-\theta_{2(1)} e^{-\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}} -\dfrac{P_0}{\gamma \mathrm{Bi}\psi_l} \left(1-e^{-\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}}\right) }{ 2\,\operatorname{sh}\!\left(\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}\right) e^{\frac{\xi_p}{2}\sqrt{\gamma \mathrm{Bi}\psi_l}} }; \]

\[ C_2= \frac{ \theta_{2(1)} e^{\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}} -\theta_{4(3)} -\dfrac{P_0}{\gamma \mathrm{Bi}\psi_l} \left(1-e^{\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}}\right) }{ 2\,\operatorname{sh}\!\left(\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}\right) e^{-\frac{\xi_p}{2}\sqrt{\gamma \mathrm{Bi}\psi_l}} }. \]

1. For the end part of the frontal connections on the drive side

\[ \theta_t=C_1 e^{\xi\sqrt{\mathrm{Bi}\psi_l}} +C_2 e^{-\xi\sqrt{\mathrm{Bi}\psi_l}} +\frac{P_0}{\mathrm{Bi}\psi_l}; \]

\[ C_{1(2)}= \frac{ \theta_4-\dfrac{P_0}{\mathrm{Bi}\psi_l} }{ 2\,\operatorname{sh}\!\left(\xi_t\sqrt{\mathrm{Bi}\psi_l}\right) } e^{\pm 0.5\sqrt{\mathrm{Bi}\psi_l}}. \]

1. For the end part of the frontal connections on the commutator side

\[ \theta_{tk}=-\frac{P_0}{2}\xi^2+C_1\xi+C_2; \]

\[ C_2=\theta_3-\frac{P_0}{2} \left(\frac{\xi_p}{2}+\xi_n\right) \left(\frac{1}{2}+\xi_t\right); \qquad C_1=-\frac{P_0}{2}. \]

Here:\(^1\)

\[ \theta_p=\frac{\vartheta_p}{t_k}; \qquad \theta_{l(lk)}=\frac{\vartheta_{l(lk)}}{t_k}; \qquad \theta_{t(tk)}=\frac{\vartheta_{t(tk)}}{t_k}; \]

\[ \xi=\frac{x}{l}; \qquad \xi_p=\frac{l_p}{l}; \qquad \xi_n=\frac{l_n}{l}; \qquad \xi_t=\frac{l_t}{l}; \qquad P_0=\frac{q_m l^2}{\lambda_m f_p t_k}; \qquad P_0'=\frac{q_c l^2}{\lambda_m f_p t_k}; \]

\[ \mathrm{Bi}=\frac{\alpha l}{\lambda_m}; \qquad \varepsilon=\frac{\Pi}{b_k}; \qquad \psi_p=\frac{\Pi l}{f_p}; \qquad \psi_l=\frac{b_l l}{f_p}; \]

\[ \kappa_c=\frac{1}{1+\dfrac{\alpha \Delta_p b_k}{\Pi\lambda_p}}; \qquad \gamma=\frac{1}{1+\dfrac{\alpha \Delta_l}{\gamma_l}}; \]

\[ \theta_1= \frac{ (P_2P_5-1)(P_3+P_4)+(P_1+P_2P_3)a_1 }{ (P_2P_5-1)P_5a_2-P_1a_1^2 }; \qquad \theta_2=\frac{\theta_1P_5a_2-P_3-P_4}{a_1}; \]

\[ \theta_3=\frac{\theta_1}{\operatorname{ch}\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}}-P_4; \qquad \theta_4=\frac{\theta_2+P_1}{P_2}; \]

\[ a_1= \frac{ \operatorname{sh}\!\left(\xi_n\sqrt{\gamma \mathrm{Bi}\psi_p}\right) \sqrt{\dfrac{\kappa_c\psi_p}{\xi_t\psi_l}} }{ \operatorname{sh}\!\left(\xi_p\sqrt{\kappa_c\dfrac{\mathrm{Bi}\psi_p}{\varepsilon}}\right) }; \qquad a_2= \frac{ \operatorname{ch}\!\left(\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}\right)-1 }{ \operatorname{ch}\!\left(\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}\right) }; \]

\[ a_3=\frac{1}{\sqrt{\gamma}}\, \operatorname{th}\!\left(\xi_t\sqrt{\mathrm{Bi}\psi_l}\right)\cdot \left(\operatorname{sh}\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}\right); \]

\[ P_1= \left[ a_3+\frac{a_2}{\gamma} \operatorname{ch}\!\left(\xi_n\sqrt{\gamma \mathrm{Bi}\psi_p}\right) \right] \frac{P_0}{\mathrm{Bi}\psi_l}; \qquad P_2=a_3+\operatorname{ch}\!\left(\xi_n\sqrt{\gamma \mathrm{Bi}\psi_l}\right). \]

\[ P_3=\frac{a_2P_0}{\gamma \mathrm{Bi}\psi_l}\operatorname{ch}\left(\xi_i\sqrt{\gamma \mathrm{Bi}\psi_l}\right) +\frac{\left(k_cP_0' + P_0\right)}{k_c \dfrac{\mathrm{Bi}\psi_p}{\varepsilon}} \left(\operatorname{ch}\left(\xi_p\sqrt{k_p\dfrac{\mathrm{Bi}\psi_p}{\varepsilon}}\right)-1\right), \]

\[ P_4=\frac{a_2P_0}{\gamma \mathrm{Bi}\psi_l} +\frac{P_0\xi_t}{\sqrt{\gamma \mathrm{Bi}\psi_l}}\operatorname{th}\left(\xi_i\sqrt{\gamma \mathrm{Bi}\psi_l}\right); \quad P_5=\operatorname{ch}\left(\xi_i\sqrt{\gamma \mathrm{Bi}\psi_l}\right)+ \]

\[ +\,a_1\operatorname{ch}\left(\xi_p\sqrt{k_c\dfrac{\mathrm{Bi}\psi_p}{\varepsilon}}\right). \]

Thus, the current temperature is represented as a function of various combinations of eleven dimensionless complexes:

\[ \theta=f(P_0,P_0',\mathrm{Bi},\psi_p,\psi_l,k_c,\gamma,\Sigma \xi_p,\xi_i,\xi_t). \]

Similar phenomena, as is known, are characterized by equality of all like-named dimensionless complexes or their combinations. However, in direct-current machines this condition is not satisfied. For example, the ratio \(P_0'/P_0=q_c/q_m\) in the general case has different numerical values. As a consequence, the heating processes of armatures of direct-current machines do not constitute a group of similar phenomena, although they do belong to phenomena of one class [2].

In accordance with the theory of thermal similarity, the results of a separate experiment may legitimately be extended only to processes and phenomena that are similar to one another. Consequently, the armature of each specific machine in each fixed operating mode must serve as an independent object of investigation. In this connection, obtaining general conclusions characterizing the temperature fields in the armatures of machines of the type under investigation requires analysis of a large number of thermal systems that are not similar.

To solve this problem with the aid of the “Minsk-1” electronic digital computer, a large number of computational variants were carried out with variation of the dimensionless complexes over the range of their values encountered in practice.

Taking into account the results of the analytical studies presented in [2], certain simplifications were introduced into the calculations. The values of the complexes \(\xi_p\), \(\xi_i\), and \(\xi_t\) were taken as constant and equal to their mean values encountered in practice, respectively \(0.49\), \(0.183\), and \(0.0745\). With this assumption, the error introduced into the calculated determination of the armature-winding temperature does not exceed \(1\%\).

According to the results of the calculations performed, when the dimensionless complexes vary within the following ranges:

\[ \mathrm{Bi}\le 0.04;\quad P_0=0.1\text{—}2.5;\quad \frac{P_0'}{P_0}=1.0\text{—}5.0;\quad k_c=0.85\text{—}0.98;\quad \gamma=0.7\text{—}0.91; \]

\[ \varepsilon=2.5\text{—}5.0;\quad \psi_l=40\text{—}110;\quad \psi_l/\psi_p=0.17\text{—}0.4, \]

the values of the maximum and average temperature rises of the armature winding above the frame differ by no more than \(8\text{—}10\%\).

According to experimental data, the average temperature rise of the armature winding above the frame in enclosed non-ventilated direct-current machines is approximately \(40\text{—}60\%\) of the average temperature rise of the winding above the ambient air. Accordingly, within the indicated range of values of the dimensionless complexes, the maximum and average temperature rises of the winding above the ambient air will differ by no more than \(5\text{—}6\%\). Thus, if, under weak thermal coupling of the armature winding with the commutator, the values of the dimensionless complexes lie within the indicated limits, then, with accuracy quite sufficient for practical calculations, the armature may be regarded as a single body and सीमited

…to reduce the problems of thermal calculations to the determination of only a single average temperature rise of the winding.

The complex $\psi_l$ has the most significant effect on the nature of the temperature distribution along the length of the armature winding. When this complex is varied within the range encountered in practice, the value of the temperature nonuniformity coefficient $(\mathrm{KN}=\vartheta_{a\max}/\vartheta_{a\mathrm{av}})$ changes by a factor of 2–3. As the complex $\psi_l$ increases, the nonuniformity of the temperature field increases.

The nature of the temperature distribution along the length of the armature winding does not depend on the magnitude of the criteria $P_0'$ and $P_0$, and is determined only by their ratio. When the ratios $P_0'/P_0$ and $\psi_l/\psi_p$, as well as the complex $\varepsilon$, are varied, the values of $\mathrm{KN}$ change by a factor of 1.5–2.5. The complexes $K_c$ and $\gamma$ do not have a significant effect on the shape of the curve $\vartheta_a=f(l)$; when they are varied, the values of $\mathrm{KN}$ differ by no more than 0.8–1.2%. As the criterion $\mathrm{Bi}$ increases, the nonuniformity of the temperature field increases.

Of the dimensionless complexes characterizing the geometry of the armature, the value of the complex $\psi_l$ has the greatest influence on $\vartheta_{a\mathrm{av}}$. The average temperature rises of the armature winding corresponding to different values of this complex may differ by a factor of 2.0–2.8. The difference between $\vartheta_{a\mathrm{av}}$ for different values of the complexes $K_c$ and $\gamma$ does not exceed 2.5 and 10%, respectively.

The minimum value of $\vartheta_{a\mathrm{av}}$ corresponds to the upper limit of the values of the complexes $\psi_l$, $\gamma$, $\psi_l/\psi_p$ and to the lower limit of the values of the complex $\varepsilon$.

REFERENCES

1. L. S. Eigenson. Modeling, Sovetskaya Nauka Publishing House, 1952.
2. A. V. Lykov. Theory of Thermal Conductivity, Gostekhizdat, 1952.
3. M. N. Uyanchik and V. V. Salomatov. Analytical study of the influence of various parameters on the level and distribution of temperature along the length of the armature winding of enclosed non-ventilated direct-current machines, article in the present collection.

1. In passing to dimensionless parameters, the average frame temperature \(t_k\) and the length of a half-turn of the armature winding \(l\) were taken as the determining temperature and determining dimension. ↩