CC BY
8
DOI: 10.17277/amt.2019.03.pp.050-055
Determination of the Sensitivity of a Contactless Device for Measuring Viscosity to Influencing Quantities by Measurement Model
S.V. Mishchenko, M.M. Mordasov, A.P. Savenkov*, M.E. Safonova, V.A. Sychev
Tambov State Technical University, 106, Sovetskaya St., Tambov, 392000, Russia * Corresponding author. Tel.: +7 915 884 30 97. E-mail: [email protected]
Abstract
The article describes the contactless aerodynamic method of measuring viscosity with respect to the amplitude of forced oscillations of the surface of a controlled fluid and the device for its realization. Oscillations are excited by a gas stream of varying intensity. The complexity of the function of measuring viscosity by the method under consideration makes it difficult to determine the degree to which the density of a controlled fluid affects the measurement results. The paper presents a method for determining the sensitivity of a measuring device to the influencing quantities from a measurement function that has a complex form.
Keywords
Contactless; influencing quantity; viscosity; density; liquid; measurement; gas jet.
© S.V. Mishchenko, M.M. Mordasov, A.P. Savenkov, M.E. Safonova, V.A. Sychev, 2019
Introduction
The measurement model in the general case is the equation of the relationship between the input and output values of the measurement process, where the output value is measurable [1]. The sensitivity of the measuring device to the influencing quantities can be determined directly from the model or measurement function, however, if they are complex dependencies, then this is difficult to do.
This paper presents an example of determining the degree of influence of density on the viscosity measurement result from the amplitude of forced oscillations of the surface of a controlled fluid excited by the aerodynamic effect of variable intensity [2-5].
Non-contact method for measuring viscosity in amplitude of forced oscillations of the surface of a liquid
Among contactless methods of measuring viscosity based on the force action of a gas jet, there are methods based on self-oscillations in a two-phase system "gas jet - liquid" [6-8], impulse methods [9, 10] and methods based on forced oscillations [2-5]. Determining the degree of influence of the density of a
liquid on the result of measuring the viscosity during the implementation of the latter is of the greatest theoretical interest.
Fig. 1 shows a diagram of a contactless device for measuring viscosity. To determine the amplitude of oscillations in the device, a triangulation detector of the distance to the surface of a liquid with a differential sensitive element was used [11].
Fig. 1. Diagram of a contactless device for measuring viscosity by an amplitude of forced oscillations of the liquid surface
The GG-generator generates harmonic oscillations of pressure P in front of the opening 2 of the outflow of the gas jet [5, 12]. The oscillation frequency f is given by the output signal of the integrator I. The oscillation amplitude AP is independent of frequency. The constant component P0 of the pressure is larger than the amplitude of oscillations; therefore, the pressure P does not fall below the minimum value Pmin (Fig. 2).
Fluctuations of pressure P cause oscillations of the force F of the gas jet action on the surface of the liquid 5 and fluctuations of the height h of the recess 4. As the frequency f increases, the amplitude Ah of the fluctuations of the height h of the recess decreases due to the action of viscous friction (Fig. 2).
The laser diode 1 is supplied from the generator of a high-voltage rectangular pulse (GRP) of stable high-frequency current. Part of the pulsed laser radiation reflected from the surface of the recess 4, falls on the surface of the differential photodetector 3. Because of the reflection from the curved surface of the recess, the laser beam is scattered. High-frequency modulation of laser radiation is necessary to eliminate the effect of external illumination on the device operation due to the low intensity of radiation incident on the surface of detector 3. At the output of the comparator C, a signal Uk = 1 is generated only at times when the same light beams fall on the surface of two parts of the detector 3 (Fig. 2). The signal Uk = 1 corresponds to the given value h1 of the height of the depression, which, in turn, sets the amplitude of oscillations Ah = h0 - h1.
Integrator I works as follows. If during the period of pressure oscillations, at least one impulse Uk = 1 arrived at its input, the integrator increases the output signal which sets the frequency f for the GG-generator. Otherwise (in the absence of pulses at the input), the integrator I reduces the output signal. As the frequency increases, the amplitude of oscillations in the height of
Fig. 2. Graphs of signal changes in the viscosity measuring device
the recess decreases, and when it decreases, it in turn increases (Fig. 2). The device is continuously in search of the boundary state (the presence or absence of pulses Uk = 1), which corresponds to the specified value Ah of the oscillation amplitude. The steady-state value of the frequency f depends on the viscosity of the controlled fluid. The viscosity is determined by the frequency f and shown on the display D.
The model of viscosity measurement by the considered method is represented by the amplitude frequency response (AFC) of the "gas jet - liquid" system [13]
nd
k
A(ra) =
Ah(ra) 2 bp%pgR2
AP(ra)
v bpPgR
A2
(1)
+1
where d and ^ are the diameter and the discharge coefficient of hole 2 (Fig. 1), k is the recess shape factor, bp and b^ are the coefficients responsible for the effect of annexing force and surface tension force on recess 4, p and n are density and viscosity of fluid 5, g is the acceleration of free fall, R is the radius of the recess 4 at half its height, © = 2nf is the circular frequency.
The function of measuring the viscosity we obtain from the AFC (1) in the form
n( f)
bppgR
2fJ
^ nd
k AP
\2
2 bpnpgR2 Ah
-1.
(2)
Determination of the density impact degree of the controlled liquid on the viscosity measurement result
It can be seen from function (2) that the result of measuring viscosity n depends on the density p of the controlled liquid, but it is difficult to determine the extent of this influence directly from this function. Function (2) includes the value Ah = h0 - hi, in which ho also depends on the density.
The constant component h0 of the recess height is determined from the frequency response (1) provided © = 0 by the formula
nd ^ k
2 bpnpgR7
-Po.
(3)
h
o
The change in density at constant viscosity has an effect on the output signal of the device - the frequency f In determining the effect of density p on the result of measuring viscosity n, we use two values: the initial po and the current p. These values of density correspond to two values of frequency f0 and f We assume that all values in expressions (2) and (3), including viscosity n, remain constant when the density changes. The only exceptions are the frequency f and the amplitude Ah, and h0 is a variable, and h1 is a constant.
We introduce the coefficient A0 = Ah / h0, taking into account which for the density p0 from formula (3) we determine the value
h = (1 - Ao)
nd p k 2 bpnpogR2
-P0
(4)
substituting which in function (2) after transformations we get
bD gR
fo
2nnbn f AP > 2
P o ! -1
V V Ao po J
(5)
For density p from expressions (3) and (4), we define
Ah =
1 1
—(1 - Ao)— P Po.
nd p k
-Po.
2 bpKgR2
(6)
To simplify the final expression, we take P0 = AP. Then after substitution (5) and (6) in (2) we get
f = _P
fo Po'
1
[1 -(1 - Ao )p/po ]]
-1
1
--1
Ao2
(7)
Expression (7) shows how much the output signal of a viscosity measuring device varies with density changes.
Fig. 3 shows the dependence of the frequency f change on the parameter Ao. If the value of Ao is chosen to be o.17, then for the initial density h\ = o.83ho. If the density of the controlled liquid increases by 2o %, the value of ho will also be o.83 from the initial value, and the specified amplitude Ah of the oscillations of the recess depth will be zero. To achieve such an amplitude, an increase in the oscillation frequency to infinity is required. In this case, the density impact on the viscosity measurement is the greatest.
Fig. 3. The dependence of the change in the output signal of the device for measuring viscosity under the influence of the density of the controlled liquid on the parameter A0:
solid lines - p / p0 = 0.8; stroke - p / p0 = 1.2
The left part of the graph f / f0 for p / p0 = 1.2 theoretically represents the dependence (7). Under real conditions, with p / p0 = 1.2 and A0 < 0.17, a device for measuring the viscosity of liquids, the scheme of which
is shown in Fig. 1, is inoperable. With a decrease in the density of the liquid, the dependence of f / f0 on A0 has no discontinuities; however, at small values of A0, the density impact on the result of viscosity measurement increases significantly.
From Fig. 3 follows that in order to reduce the influence of density one should choose larger values of the parameter A0. Despite this, the choice of A0 values close to 1 is impractical. For A0 ~ 1, Ah ~ h0 and h1 ~ 0. With small deviations from the constant component h0, the amplitude Ah weakly depends on the frequency ro. Therefore, at A0 ~ 1, the error in viscosity measurement significantly increases. In this connection, it is of interest to determine the influence of the height h1 error signal on the viscosity measurement result.
Determination of the impact degree of the error signaling of the recess height on the viscosity measuring result
To determine the density impact on the viscosity measurement result, it is necessary to know the value of the parameter A0 (Fig. 3), the choice of which, in turn, is influenced by the signaling error of a given recess height h1. Let us determine the dependence of the impact degree of the error signaling of the h1 value of the recess height on the output signal of the viscosity
measuring device (frequency f) on the parameter A0. For P0 = AP from expression (5) we get
fo =
bpPo gR05
2%nh \
f 1 >2
A
-1.
(8)
Determination of the value of h1 in the device, the scheme of which is shown in Fig. 1, is produced with absolute error Ah1. And the amplitude Ah is determined with the same error. Therefore, for the parameter A0, is valid the expression
Ao =
Ah ±Ah1 h0
= A ± Y h,
(9)
where yh = Ah1/h0 is the error in determining the given value of h1, reduced to the value of the average value h0 of the recess depth. After substituting the expression (9) in the formula (8) we get
fo + Af =
bpP0 gR05
2nnb \
1
A0 ± Y h.
-1.
(10)
where Af is the change in frequency f due to the error in determining the given value of h1. Dividing formula (10) by (8), we find the dependence of the change in the output signal of the viscosity measuring device under the influence of the error yh on the parameter A0
1
Af
8 f ( A0) = —= 1 -f f0
A +Y h.
-1
f 1 ^2
(11)
V A0 J
-1
Due to the fact that the frequency f is the output signal of the device in question, the value 5f is the relative error in measuring viscosity due to the error in determining the given value of h1. After normalizing expression (11) with the error Yh, we obtain the dependence
Z h ( A0) =
8 f ( A0) = _1_
Yh Yh
1 -
r 1 ^ 2
-1
1 A) +Y h
f 1 ^2
V A0 J
-1
(12)
showing how many times the relative error of viscosity measurement exceeds the error yh of determining the
Fig. 4. Dependence of the ratio Zh of errors 5// yh in viscosity measurements and determination of the recess height from the parameter A0:
solid and dashed lines correspond to | Yh | equal to 1 and
10 %; curves 1 - positive values of Yh; 2 - negative
set value h1 of the depth of the depression for different values of the parameter A0 (Fig. 4).
From Fig. 4, it can be seen that the error in measuring the viscosity of a liquid, due to the error in determining the given value h1 of the recess depth, significantly increases both small and large values of the parameter A0. In the range of A0 from 0.42 to 0.74 with | Yh | < 1 %. The error in measuring viscosity is no more than three times the value of Yh.
Determination of the cumulative effect of the density of the fluid and the error signaling the height of the recess on the viscosity measurement
The influence of the fluid density on the result of viscosity measurement can lead to both a reduction or increase in the error influence in determining the recess height. Let us define the marginal value of the error in measuring viscosity, due to the influence of the liquid density and the inaccuracy of determining the recess height. To do this, we use the f / f0 dependence on A0 for p / p0 > 1. For p / p0 < 1, the error in measuring viscosity is smaller.
Let us introduce the error Yh into expression (7), as a result of which we obtain
f = _P
f0 P0
1
[1 -(1 - A0 )P/P0-Y h ]2
-1
1
--1
(13)
2
2
where the sign of yh is chosen to coincide with the sign of the term (1 - A0)p / p0. Laser triangulation distance detectors provide an alarm when the specified value is achieved with an absolute error of no more than ± 0.05 mm [11]. For h0 = 3 mm, we obtain the value of yh not exceeding 2 %. Fig. 5 shows the graphs of function (13) for yh = 2 %. For comparison, the dashed lines show the graphs of function (7).
It follows from Fig. 5 that in order to reduce the impact of fluid density on the viscosity measurement result, one should select the value of the parameter A0 equal to 0.9 (solid curve 4). However, in this case, in the absence of density effects, the relative error in measuring viscosity increases to 10 %, while at A0 = 0.707, this error does not exceed 6 % (curve 1).
The value A0 = 0.707 corresponds to the first mating frequency of the system "gas jet - liquid" (L(ro) = -3 dB), in which the influence of viscosity and density of the liquid on the amplitude of oscillations of the recess height is the same (Fig. 5) [4, 13]. At larger values of A0, the influence of density prevails, therefore we finally choose A0 = 0.707. For this value, the error in measuring viscosity with a density change of 10 % (solid curve 2) will be 26 %, and with a change of 20 % (solid curve 3) - 49 %.
A direct analysis of the functions (1) and (2) shows that a change in viscosity leads to a proportional change in the output signal of the device - the frequency f. Consequently, in the method under consideration, the influence of density on the
0 0,25 0,5 0,75 A,
Fig. 5. Dependence of changes in the output signal of the viscosity measuring device under the influence
of the controlled fluid density from the parameter A0
at Yh = 2 % (solid lines, function (13)) and yh = 0% (dashed lines - function (7)):
1 - 4 correspond to p / p0 equal to 1.0; 1.1; 1.2; 1.4
measurement result exceeds the influence of the measured quantity - viscosity. This is its main disadvantage.
Such a method could be recommended for measuring the viscosity of liquids with a known density, but in the presented theoretical studies the effect of density also includes the effect of surface tension on the measurement result. This is due to the fact that to ensure the non-contact determination of the recess height, a small degree of the liquid surface deformation is used, in which the actions of the annexing force and the surface tension force are comparable. The effect of surface tension on the viscosity measurement result is a greater disadvantage than the effect of density, since the surface tension varies depending on the surface purity of the controlled fluid.
Results and discussion
The results of theoretical studies presented in this study made it possible to establish that in the non-contact aero-hydrodynamic method of measuring viscosity by the amplitude of forced oscillations of the surface of a controlled fluid, density has a greater effect on the measurement result than the measured value. When choosing a non-contact method for measuring viscosity, one should give preference to other aero-hydrodynamic methods, for example, pulsed, presented in [9, 10].
Conclusion
The theoretical analysis of the measuring device sensitivity to the action of the influencing quantities made it possible to identify a serious drawback of the measurement method without sophisticated experimental studies. These studies ensured a reduction in the time and material resources spent on creating prototypes of measuring devices and obtaining experimental data. The method for determining the measuring device sensitivity to the affecting value discussed in the article can be used to analyze measuring instruments based on physical effects of any nature.
References
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