Influence of the Main Dimensions of DC Machines Filled with a Liquid Dielectric on Friction Losses
Abstract Generated abstract
The paper analyzes mechanical friction losses in direct current machines whose rotating parts operate in a liquid dielectric, a condition that substantially increases losses and constrains the choice of armature geometry. It derives formulas for disk, cylindrical, end part, and commutator friction using Reynolds number dependent resistance relations, then combines them into an expression relating total and relative friction losses to machine dimensions, speed, liquid density and viscosity, gaps, roughness, and electromagnetic loading. The analysis indicates that relative losses rise with viscosity, armature diameter, and rotational speed, while higher electromagnetic loading shifts the point of maximum efficiency upward, as illustrated by a comparison of operation in air and kerosene. The paper recommends maximizing the armature length to diameter ratio within commutation limits and maintaining hydrodynamically smooth rotating surfaces to reduce losses.
Full Text
EFFECT OF THE MAIN DIMENSIONS OF DC MACHINES FILLED WITH A LIQUID DIELECTRIC ON FRICTION LOSSES
V. P. Morozkin
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
The emergence of a number of fields of application for DC machines filled with a liquid dielectric requires an assessment of their energy characteristics and a justification of the choice of their main dimensions. A feature of the operation of such machines is the considerable increase in mechanical friction losses when the armature rotates in the liquid. If in submerged AC motors these losses can be reduced by increasing the ratio of the stator length to the diameter, then in DC machines this route is limited by the deterioration of commutation as the ratio \(l/D\) increases. Therefore, when designing such machines it is necessary to take into account the influence of the armature geometry on friction losses. Let us consider these losses.
All mechanical losses of the machine can be divided into two parts:
— friction losses of the lateral surfaces of the armature and the commutator—these are disk-friction losses;
— friction losses of the cylindrical surfaces of the armature and the commutator.
The friction losses of the lateral surfaces of the armature and the commutator may be considered equal to the friction losses of a disk whose diameter is equal to the armature diameter.
The power of disk friction on both sides of the disk is equal to (1):
\[ P = 9.81 \cdot C_{\mathrm{m}} \cdot \rho \cdot \frac{\omega^3}{2} \cdot R^5,\ (\mathrm{W}), \tag{1} \]
where
\(C_{\mathrm{m}}\) — coefficient of the friction moment,
\(R\) — armature radius, \(m\),
\(\omega\) — angular velocity of the armature, \(\dfrac{1}{\mathrm{s}}\),
\(\rho\) — density of the liquid, \(\dfrac{kg}{m^3}\).
The friction coefficient \(C_{\mathrm{m}}\) is a function of the Reynolds number \((\mathrm{Re})\), which for a rotating disk is equal to
\[ \mathrm{Re}=\frac{\omega \cdot R^2}{\nu}, \]
where \(\nu\) is the kinematic viscosity coefficient of the liquid dielectric.
For flow regimes corresponding to \((10^4 < \mathrm{Re} < 10^8)\), the coefficient is equal to
\[ C_{\mathrm{m}} = 0.98(\lg \mathrm{Re})^{-2.58}. \tag{2} \]
For convenience of calculation, formula (2) has been expressed by us as an explicit dependence of \(C_m\) on \(\mathrm{Re}\) with sufficient accuracy:
\[ C_m=\frac{0.139}{\mathrm{Re}^{0.19}}. \tag{3} \]
Substituting the value of \(C_m\) into (1) and expressing \(\mathrm{Re}\) in terms of \(n, D\), we obtain, for the power of disk friction,
\[ P=2.82\cdot 10^{-14}\cdot \rho \cdot n^{2.81}\cdot D^{4.62}\cdot \nu^{0.19}, \tag{4} \]
where \(n\) is the rotational speed, \(\dfrac{\text{rpm}}{\min}\),
\(D\) is the disk diameter,
\(\nu\) is the viscosity, \(\dfrac{m^2}{\sec}\).
The friction losses of the rotating cylindrical parts of the armature and commutator constitute the main part of the friction losses; therefore, we shall consider in greater detail the method for determining these losses.
When a circular cylinder rotates in a viscous incompressible medium, the friction stress \(\tau\) on the surface of the cylinder can be expressed, as in the case of axisymmetric fluid flow, in the form [3]:
\[ \tau=\mu\left(\frac{dv}{dr}-\frac{v}{r}\right) =\mu\cdot r\frac{d}{dr}\left(\frac{v}{r}\right), \tag{5} \]
where \(\mu\) is the dynamic viscosity of the fluid,
\(v\) is the circumferential velocity of the fluid element \(dr\).
In a more general case, equation (5) may be written in the form [1]:
\[ \frac{\tau}{\rho}=-\varepsilon\cdot r\cdot \frac{d}{dr}\left(\frac{v}{r}\right), \tag{6} \]
where \(\varepsilon\) is the kinematic coefficient of turbulent viscosity. Equation (6), in the form given, cannot have practical significance, since the law of velocity distribution near the cylinders is unknown.
It has been established by numerous experimental data [1] that the character of the flow in the wall regions near cylinders is similar to the character of the velocity distribution in the laminar sublayer on the surface of a flat plate. Therefore, the problem of determining the friction stress on the cylinder surface reduces to determining the velocity gradient in the laminar boundary layer. This problem has been solved for the case of turbulent motion in pipes, and all regularities of the relation between tangential stresses and the velocity profile in a pipe have been obtained. The tangential stress at the pipe wall in turbulent flow is obtained in the form [2]:
\[ \tau_0=\frac{\lambda}{8}\cdot \rho \cdot v_{\mathrm{avg}}, \tag{7} \]
where
\(\lambda\) is the resistance coefficient for turbulent motion of a liquid in pipes,
\(v_{\mathrm{avg}}\) is the mean flow velocity over the cross section.
The coefficient \(\lambda\) is a function of the Reynolds number:
a) for laminar motion \((\mathrm{Re}<\mathrm{Re}_{\mathrm{cr}})\)
\[ \lambda=\frac{64}{\mathrm{Re}}, \tag{8} \]
b) for turbulent motion (Nikuradse formula)
\[ (\mathrm{Re}_{\mathrm{cr}}<\mathrm{Re}<10^7) \]
\[ \lambda = 0.0032 + \frac{0.221}{\mathrm{Re}^{0.237}} . \tag{9} \]
The Reynolds number \(\mathrm{Re}\) for the cylindrical surface of the active part of the armature is equal to
\[ \mathrm{Re}=\frac{v}{2}\cdot\frac{2\delta}{\nu}=\frac{\omega\cdot r\cdot\delta}{\nu}, \tag{10} \]
where \(\delta\) is the gap between the pole and the armature.
The friction power of cylindrical surfaces is determined by the formula
\[ P=9.81\cdot \omega\cdot M=9.81\cdot \tau_0\cdot 2\pi r^2\cdot l\cdot \omega \quad (\mathrm{W}). \tag{11} \]
On the basis of relations (7–11), the power of losses of the active part is determined:
a) for the laminar regime
\[ P=8.45\cdot 10^{-8}\cdot \rho\cdot D^3\cdot l\cdot n^2\cdot \nu\cdot \delta^{-1}\quad (\mathrm{W}), \tag{12} \]
b) for the turbulent regime
\[ P=4.42\cdot 10^{-17}\cdot \rho\cdot l\cdot n^3\cdot D^4 +5.55\cdot 10^{-14}\cdot \rho\cdot l_0\cdot n^{2.75}\cdot D^{3.763}\cdot \nu^{0.237}\cdot \delta^{-0.237},\ \mathrm{W}. \tag{13} \]
For the end parts of the armature, the friction losses can be determined from (13), as for the active part of the armature, by introducing an equivalent gap \(\delta_{\mathrm{e}}\) between the end parts and the frame elements (Fig. 1). The surface of the collector
Fig. 1. Sketch of the machine
with collector grooves is a rough surface with relative roughness equal to
\[ \overline{\Delta}=\frac{k}{R_k}, \tag{14} \]
where \(k\) is the depth of the collector grooves,
\(R_k\) is the radius of the collector.
In this case the resistance coefficient \(\lambda\) does not depend on the Reynolds number \(\mathrm{Re}\) and is determined by formula (15), [2]:
\[ \lambda=\frac{1}{\left(2\lg\frac{R_k}{k}+1.74\right)^2}, \tag{15} \]
and the friction power, in accordance with (11) and (15), is
\[ P=1.38\cdot 10^{-14}\cdot \rho\cdot \frac{D_k^4\cdot l\cdot n^3} {\left(2\lg\frac{R_k}{k}+1.74\right)^2},\ (\mathrm{W}). \tag{16} \]
To express the sum of the friction losses as a function of the main dimensions of the machine, let us introduce the following generally accepted dimensions for a machine according to Fig. 1:
$D$ — diameter of the armature and of the end parts, cm,
$l$ — length of the armature, cm,
$k \cdot l$ — total length of the end parts, cm,
$c \cdot D$, $m \cdot l$ — diameter and length of the commutator, cm,
$n$ — armature rotational speed (rpm),
$\delta$, $\delta_{\mathrm{e}}$ — gaps, cm.
Introducing the above dimensions and summing the losses of the individual parts, we obtain the dependence of the total friction losses on the main dimensions of the machine and the properties of the liquid dielectric for a turbulent flow regime in the form:
\[ \begin{aligned} P_{\Sigma} ={}& 10^{-14}\cdot \rho \cdot D^{3.5}\cdot l \cdot n^{2.75}\cdot \nu^{0.2} \Bigl[ 2.82\cdot n^{0.06}\cdot D^{1.1}\cdot l^{-1}\cdot \nu^{-0.01} \\ &+ 4.42\cdot 10^{-3}\cdot n^{0.25}\cdot D^{0.5}\cdot \nu^{-0.2}(1+k) + 5.55\cdot n^{0.013}\cdot D^{0.263}\cdot \nu^{0.037} \to \\ &\to \delta^{-0.237}\left(1+k\left(\frac{\delta}{\delta_{\mathrm{e}}}\right)^{0.227}\right) + \frac{1.38\cdot D^{0.5}\cdot n^{0.25}} {\left(2\lg \frac{R_{\mathrm{k}}}{k}+1.74\right)^{2}} \cdot c^{4}\cdot m\cdot \nu^{-0.2} \Bigr]\;(\mathrm{W}). \end{aligned} \tag{17} \]
It should be noted that formula (17) is valid only in the case where all rotating surfaces are hydrodynamically smooth, i.e., the surface roughness is less than the thickness of the laminar boundary layer, which usually amounts to tenths of a millimeter. Increasing the roughness of the surfaces leads to an increase in friction losses; in this case the drag coefficient $\lambda$, as for the commutator, ceases to depend on the Reynolds number and is determined only by the roughness of the surfaces (15). In formula (17), the expression enclosed in brackets has a weak dependence on the machine parameters; in particular, for a liquid-dielectric viscosity of $10^{-6}\div 20\cdot 10^{-6}\ \mathrm{m}^{2}/\mathrm{s}$, for machines with an armature diameter from 10 to 50 cm in the speed range $n=1000\div 4000$ rpm, it lies within the limits $(40\pm 4)$. Denoting this expression by the symbol $s$ and taking, for DC machines,
\[ P_{\mathrm{e}} = \frac{D^{2}\cdot l\cdot n\cdot \alpha\cdot A\cdot B_{\delta}} {6.1\cdot 10^{8}}, \quad (\mathrm{W}) \tag{18} \]
we obtain the formula for the relative armature friction losses of DC machines filled with a liquid dielectric:
\[ \frac{\Delta P}{P_{\mathrm{e}}} = \frac{P_{\Sigma}}{P_{\mathrm{e}}} = \Delta\eta = \frac{6.1\cdot 10^{-6}\cdot \rho\cdot D^{1.5}\cdot n^{1.75}\cdot s\cdot \nu^{0.2}} {\alpha\cdot A\cdot B_{\delta}}, \tag{19} \]
where $\alpha$ is the pole-arc coefficient,
$A$ — linear loading, A/cm,
$B_{\delta}$ — induction in the air gap, G.
As follows from (19), the relative armature friction losses, other conditions being equal, decrease as the electromagnetic loadings of the machine increase; therefore the maximum efficiency of DC machines operating in a liquid-dielectric medium shifts toward higher loads. Figure 2 gives the dependence of the efficiency of a real machine on load, recorded during operation in air and in kerosene (at a kerosene temperature of $+20^\circ\mathrm{C}$). In air, the maximum efficiency occurs at $P=9.5$ kW ($A=250$ A/cm), while in kerosene it occurs at $P=13$ kW ($A=385$ A/cm).
Fig. 2. Dependence of the efficiency of a DC machine on power and linear loading. 1 — in air, 2 — in kerosene
Conclusions
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The relative friction losses of DC machines filled with a liquid dielectric increase with increasing viscosity of the liquid dielectric, and also with the armature diameter and rotational speed, i.e., with the power of the machine.
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The maximum efficiency of such machines shifts toward higher electromagnetic loadings.
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To reduce friction losses, it is necessary to choose the ratio \(l/D\) to be as large as possible under the conditions of commutation, and to make all rotating surfaces hydrodynamically smooth.
References
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L. A. Dorfman. Hydrodynamic Resistance and Heat Transfer of Rotating Bodies. Fizmatgiz, 1960.
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I. L. Povkh. Technical Hydromechanics. “Mashinostroenie,” 1964.
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P. G. Loitsyanskii. Laminar Boundary Layer. Moscow, Fizmatgiz, 1962.