USING THE ADDED MASS EFFECT TO MEASURE THE DENSITY OF SOLUTIONS

M. M. Malovichko, V. A. Anikeev, M. F. Tkachenko

(Presented by the scientific seminar of the Faculty of Physics and Technology)

When technological processes are introduced in many branches of industry, it is necessary to measure individual physicochemical parameters of process media (density, viscosity, etc.) [1].

In addition, measurement of individual parameters of process solutions is necessary when carrying out analysis of multicomponent solutions by the multiparameter method, which is now finding broad application in practice.

This article considers the possibility of creating an automatic densimeter for process solutions based on the use of the added-mass effect of a body moving in a liquid.

It is known that, in unsteady motion of a body in a liquid, the resistance to motion at a given instantaneous velocity will be greater than the resistance at the same velocity but under uniform motion [2].

In unsteady motion it is necessary to overcome the inertia of the body and the inertia of the surrounding mass of liquid, which gives rise to additional resistance.

To analyze the unsteady motion of a solid body in a liquid, the kinetic energy of the liquid \(T\), determined by the formula

\[ T = \frac{1}{2}\rho \iiint\limits_V v^2 dx\,dy\,dz, \tag{1} \]

where
\(V\) is the volume,
\(v\) is the velocity of the liquid,
\(\rho\) is the density of the liquid,

is referred either to the linear velocity in translational motion or to the angular velocity in rotational motion.

\[ T = \frac{1}{2}\rho v_0^2 \iiint\limits_V \frac{v^2}{v_0^2} dx\,dy\,dz, \tag{2} \]

where \(v_0\) is the velocity of the solid body.

The triple integral in expression (2) does not depend on the velocity of motion, but is related only to the shape of the body and the direction of its motion in the liquid medium. Therefore this integral is regarded as a certain independent quantity and is denoted

\[ \lambda = \rho \iiint\limits_V \frac{v^2}{v_0^2} dx\,dy\,dz, \tag{3} \]

\(\lambda\) is the added mass of the body.

For certain body shapes, the values of the added masses have been calculated theoretically. For example, for a sphere of radius \(r\) in translational motion along one of the coordinate axes,

\[ \lambda = \frac{2}{3}\pi \rho r^3, \tag{4} \]

and for a disk of radius \(r\) in translational motion in a direction perpendicular to its plane,

\[ \lambda = \frac{8}{3}\rho r^3. \tag{5} \]

As is seen from formulas (3, 4, 5), for an unchanged direction of motion of a body of a given shape,

\[ \lambda = C f(\rho), \tag{6} \]

where \(C\) is a constant determined by the shape of the body and the direction of motion.

Thus, by measuring the value of the added mass of a body of a given shape in a liquid, one can judge the density of the liquid.

Fig. 1. Device for measuring the added mass of bodies moving in a liquid. 1 — body, 2 — rod, 3 — flat springs, 4 — contact group, 5 — anchor, 6 — electromagnet, 7 — direct-current source, 8 — vessel with the solution under investigation.

There are several methods for determining the added masses of bodies [3]. The simplest method, readily amenable to automation, is the method of small vertical oscillations of a body completely immersed in a liquid.

The added masses of bodies by the method of small vertical oscillations can be determined with the aid of the device shown in Fig. 1.

Body 1, rigidly fastened by a rod to two flat springs 3, which allow only vertical oscillations of the moving system, and to armature 5, is immersed in vessel 8 with the solution under investigation. If the moving system is deflected downward from its equilibrium position, contact group 4 is closed. (The contact group may be replaced by a contactless system based on an inductive, capacitive, or other displacement sensor.) Current flows through the winding of electromagnet 6; the armature, and with it the entire moving system, is deflected upward; the contact opens; under the action of the elastic forces of the springs and of the weight of the body, the moving system is deflected downward; the contact closes, and so on.

A self-oscillatory regime is established in the device. The frequency of the self-oscillations, as shown in [3], is equal to the natural frequency of oscillation of the moving system and is determined by the formula

\[ \omega=\left[\frac{k}{m_0+m_1+\lambda(\rho)}\right]^{1/2}; \tag{7} \]

\(k\) is the stiffness of the springs,
\(m_0\) is the mass of the solid body,
\(m_1\) is the reduced mass of the rod and springs.

In expression (7), the quantities \(k\), \(m_0\), \(m_1\) are constant for the given mechanical system, while \(\lambda(\rho)\) depends on the density of the liquid.

Denoting \(m_0+m_1=m\), we have

\[ \omega=c_1\left[\frac{1}{1+\lambda(\rho)/m}\right]^{1/2}, \tag{8} \]

i.e.,

\[ \omega=c_1 f(\rho). \tag{9} \]

The correctness of the operation of the device is determined by comparing \(\lambda_{\text{theor}}\) and \(\lambda_{\text{exp}}\) for bodies of a definite shape, for example, a disk or a sphere. \(\lambda_{\text{exp}}\) is determined by the formula

\[ \lambda_{\text{exp}}=(m_0+m_1)\left[\left(\frac{\omega_1}{\omega}\right)^2-1\right], \tag{10} \]

where

\[ m_1=\frac{m_0}{\left(\dfrac{\omega_0}{\omega_1}\right)^2-1}, \tag{11} \]

\(\omega_1\) is the frequency of oscillation of the mechanical system in air,
\(\omega_0\) is the frequency of oscillation of the mechanical system without the body in air.

Table 1 gives the theoretically calculated and experimentally determined added masses for disks of Ø 60 mm, Ø 90 mm, Ø 120 mm.

Table 1

In order, when the density changes, to obtain the maximum change in frequency according to formula 9, it is necessary that the ratio \(\lambda \delta m\) be as large as possible. This is achieved by selecting the shape of the body and its position relative to the solid walls, since it is known that the presence of solid walls near an oscillating body causes an increase in the value of the added mass.

Figure 2 shows the experimentally determined relationship between the oscillation frequency of the system with a disk of \(\varnothing\ 120\) mm and the density of a solution of \(\mathrm{Fe}_2(\mathrm{SO}_4)_3\). The frequency was measured by counting pulses in the DC electromagnet circuit over a specified time. The density was determined using a set of densimeters.

Fig. 2. Dependence of the oscillation frequency of the mechanical system on the density of the solution

The experimental results confirm that, on the basis of using the added-mass effect of a body moving in a liquid, a simple automatic device can be created for measuring the density of process solutions.

REFERENCES

1. S. S. Kivilis. Technique for Measuring the Density of Liquids and Solids, Standartgiz, Moscow, 1959.
2. M. Ya. Alfer’ev. Hydromechanics, Moscow, 1961.
3. N. S. Rimskii, R. L. Kreps. Added mass of bodies of various shapes, TsAGI Transactions, No. 635, 1947.