ISSN 0747�9239, Seismic Instruments, 2009, Vol. 45, pp. 110–114. © Allerton Press, Inc., 2009.Original Russian Text © D.G. Taimazov, 2009, published in Seismicheskie Pribory, 2009, No. 4, pp. 27–35.
110
At present, spring�loaded static gravimeters withthe test mass serving as the sensitive element are mostcommonly used in gravimetric operations [Graviraz�vedka, 1981]. Application of a spring as an elastic ele�ment is justified by the sufficiently high sensitivity ofthe devices, whose operation is based on this principle,in combination with their compactness. The keydrawbacks of such gravimeters are the time instabilityof elastic characteristics of the sensitive element (aspring) yielding zero drift of the device and the nonlin�ear dependence of the elastic characteristics of thespring material (metal or quartz) on temperature,which hinders thermal compensation within the tem�perature range required for practical applications.These drawbacks restrict the accuracy of the relativegravity measurements at the level of ±0.01 mGal in fieldconditions and ±0.001 mGal in stationary conditions.In addition, a gravimeter with zero drift, necessitatingfrequent repetition of measurements at the referencepoint, significantly increases labor expenditures anddecreases the accuracy of areal and profile measure�ments of gravity for prospecting and geodesic purposes.In particular, the accuracy of registering nontidal grav�ity variations (NTGV) drops several tenths of a milligal.
There are two approaches to the solution of the zerodrift problem. The first approach, namely utilization ofabsolute (ballistic) gravimeters (e.g., for NTGV studies)instead of static ones, requires enormous materialexpenses since these devices are bulky and resource�con�suming [High�Accuracy …, 1972]. In addition, the accu�racy of the available ballistic gravimeters does not allowus as yet to reliably determine NTGV [Bulanzhe et al.,1982]. The second approach is to eliminate zero driftin static gravimeters by modifying elastic suspension ofthe test mass, which can be done in two ways: (1) usingan electromagnetic suspension instead of a mechani�
cal spring (superconducting gravimeters) [Gusev,1979] and (2) using monocrystal elastic elementscooled to super�low (helium) temperatures [Bragin�skii, Matyunin, 1977; Matyunin, 1979]. Both of them areassociated with utilization of cryogenic systems and thusare unsuitable for mass field observations. In addition,superconducting gravimeters have not yet justified hopesas far as zero drift elimination is concerned, whereasmonocrystal ones have not yet passed the test stage.
Vibrating�string gravimeters are characterized by asmaller zero drift [Mel’khior, 1975; Mironov, 1980],but their accuracy is limited by the accuracy of mea�suring the frequency of string natural vibrations whichreaches ~10–7 in relative units. This corresponds to theaccuracy of gravity measurements of ±0.2 mGal [Ogor�odova et al., 1978; Yuzefovich, Ogorodova, 1980].
Gas–liquid gravimeters are less widespread inpractice [Mel’khior, 1975; Mironov, 1980; Tsuboi,1982]: though they have been known for quite a while,they have not found wide application due to the fol�lowing objective reasons:
(i) the temperature volumetric expansion coefficientof gas is very large (β = 0.0037), which sets conditionsfor a strong dependence of gas–liquid gravimeter indi�cations on temperature (more than 1000 mGal/°C);
(ii) due to negligible displacements of mercury surfacesin wide vessels, the accuracy of their indication is low;
(iii) application of narrow horizontal capillaries tointensify the displacements as, e.g., in the well�knownHaalck gravimeter [Mel’khior, 1975; Mironov, 1980;Tsuboi, 1982], causes additional errors attributed tothe interaction of liquid with the capillary walls. If alighter liquid (toluene) is poured over mercury to indi�cate the level of the latter, we face additional measure�ment errors attributed to natural thermal expansionand interaction with the capillary walls.
Ways to Improve Metrological and Operational Characteristicsof Liquid Gravimeters
D. G. TaimazovInstitute of Geology, Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala, Russia
Abstract—A gas�liquid gravimeter of the manometric type with a drift�free elastic element, namely gas, pro�posed by the author is described. The possibility to implement an effective self�contained system allowing forcomplete compensation of the temperature and temperature gradient effects, as well as for actual eliminationof the effect of a small inclination on the sensitive element of the gravimeter, is shown. Utilization of a floatconverter equipped with a capacitance transducer of displacements increases the measurement range up to~104 mGal, the calculation error in the entire range being ±1 µGal.
DOI: 10.3103/S0747923909010198
Key words: gas–liquid gravimeter, capacitance transducer, test mass, thermal compensation.
SEISMIC INSTRUMENTS Vol. 45 2009
WAYS TO IMPROVE METROLOGICAL AND OPERATIONAL CHARACTERISTICS 111
There is another drawback which applies to allstatic gravimeters, namely the dependence of thedevice indications on the inclination of the gravimetersensitivity axis. This limits the accuracy of gravitymeasurements by the accuracy of the device levelingover liquid levels and in the case of stationary measure�ments can lead to NTGV imitation by slow variationsin the base inclination.
The advantages of gas–liquid gravimeters are as fol�lows: a linear dependence of the elastic gas propertieson temperature, which facilitates analytical calculationor compensation within a wide temperature range, andconstancy of the elastic gas characteristics in time.However, these advantages can be made use of only whenthe above drawbacks of gas–liquid gravimeters are elimi�nated. This goal has been accomplished to a certainextent in the modifications of gas–liquid gravimetersproposed by the author [Taimazov, 1986a, 1986b] wheremercury columns equalized by gas pressure in closedreservoirs are used as test masses. The modified gas–liquid gravimeter described below [Taimazov, 2006b]is characterized by an optimal combination ofgravimeter advantages: it has a more perfect capaci�tance converter of displacements [Taimazov 2006a],which significantly simplifies the gravimeter designand is equipped with a precision system to compensatethe effect of the working liquid thermal expansion andtemperature gradients in the gravimeter volume andwith a reliable system of sensitive element arrestment.
The profile of the proposed gravimeter is shown inFig. 1. The mercury column 1 occupies the lower partsof the degasified upper reservoir 2 and the lower reser�voir with gas 3, as well as the connector pipe 4. Dis�placements of the upper mercury column level areindicated by a capacitance transducer with a variablearea of facing overlappings [Taimazov 2006a] whosestator ring�shape facings are set on the external sidesurface of a nonconducting cylinder 5, which is sus�pended to the body with a flexible draught 6 and isequipped with a liquid damper, whereas rotor facingsare set on the internal side surface of the float 7 coaxialto it. The capacitance transducer is fed, and the dataare put out via electric terminals 10.
The walls that limit the mercury level in the lowerand upper reservoirs have the shape of concentricspheres with the center at the point of suspension ofcylinder 5. Thanks to it at small inclinations of thegravimeter body, when mercury does not leave the lim�its of the spherical surfaces, the height of the mercurycolumn does not evidently change. The capacitancebetween stator and rotor facings also remainsunchanged since in this case the gravimeter body isinclined at some angle ϕ, and the cylinder 5 and mer�cury surfaces together with the float 7 rotate (withrespect to the body) by the same angle; i.e., their posi�tions with respect to one another are unchanged.
The gravimeter was stopped with the help of thefixers 8 which press the float to the lower rounded partof the walls of the reservoir 2 and simultaneously fixthe cylinder 5. A semipermeable partition 9 in the res�
ervoir 3 hinders mercury penetration into its upperpart. All this makes the requirements on the devicetransportation and storage less strict.
Thermal compensation in the gravimeter is imple�mented as follows.
When the temperature increases, the liquid in thereservoir 11 (toluene) expands and elongates the syl�phon 12 and the sylphon 13, connected to it, thusincreasing the volume of the reservoir 3; when thetemperature decreases, the opposite takes place. Thevolumes of the reservoirs 3 and 11, the diameters of thesylphons 12 and 13, and the coefficient of the liquidvolumetric expansion are chosen so that during tem�perature variations the volume of the reservoir 3changes according to the law
(1)
where V is the reservoir volume at the temperature t; V0
is the reservoir volume at the thermostatting tempera�ture t0; β is the coefficient of gas volumetric expansion(according to the law of isobaric expansion of the gasitself). In this case the gas pressure in the reservoir 3will evidently remain constant irrespective of its initialvalue. The sylphon 13 is covered with the lagging 14. Asealed gap between the silphon 13 and the lagging 14 is
V V0 1 β t t0–( )+[ ],=
1
2
3
4
5
6
7
8
9
10
11
1213
14
15
Fig. 1. Profile of the gas–liquid gravimeter.
112
SEISMIC INSTRUMENTS Vol. 45 2009
TAIMAZOV
filled with the same gas as the reservoir 3, the pressurebeing the same. The spacing between the lagging 14and the body 15 is degassed as the rest volume.
Let us differentiate Eq. (1):
On the other hand, from the thermal compensationrequirement it follows that
(2)where q is the ratio of areas of bases of the sylphons 13and 12; βT is the coefficient of toluene volumetricexpansion; and VT is the toluene volume. Thus, thecondition of thermal compensation has the form
(3)
In the case when q = 5 and the thermal coefficientsof liquid and gas expansion are ~0.001 (toluene) and~0.0037, respectively, the liquid volume makes upnearly 0.75 of the gas volume. If we take into accountthat, physically, the reservoir with a thermocompensat�ing liquid will be coil�shaped (which will be consideredbelow in greater detail), the payload volume (gas vol�ume) will make up nearly half of the reservoir volume,the volume of the coil tube wall being take into account.
The key advantages of the above thermocompensa�tor are attributed to the fact that its parameters do notdepend on stabilized pressure and it is self�containedin its design; i.e., this thermocompensator can be pro�duced as an independent unit—a manostat—andused in different devices where gas pressure is requiredto be constant. Thus, it is still possible to perform ther�mocompensation within a wide temperature rangewith subsequent thermostatting and to spatially matchthe thermocompensator and the gas to be thermosta�
dV βV0dt.=
dV qβTVTdt,=
βV0 qβTVT.=
bilized thus avoiding a temperature drop betweenthem. All this brings the temperature variations of thegas pressure to naught.
In the case of a bimetallic thermocompensator[Taimazov, 1986b], the effect of the main error source,namely, a temperature drop between the gas and the ther�mocompensating liquid, is negligible since thermosensi�tive elements are initially distributed uniformly in the gasreservoir volume which occupies the spaces betweendilatometric pairs. In the case of a liquid thermocompen�sator, the effect of this drop can be avoided in a similar way,namely by uniform distribution of the thermocompensat�ing liquid in the gas reservoir. From a design point of view,this problem is solved by making the reservoir with thethermocompensating liquid in the form of a long capillarytube which can easily be distributed uniformly over thereservoir volume.
The second�in�significance source of errors (afterthermal gas expansion) is the dependence of mercurydensity ρ on temperature t, which makes t be depen�dent of the mercury column pressure P = ρgH. In thefirst approximation (without T gradients taken intoaccount), the problem of compensating for thisdependence is solved by a sufficient increase in thethermocompensating liquid volume (since the errorsattributed to temperature expansion of the gas andmercury are of the same sign).
In the presence of temperature gradients, the problemof compensation becomes more complex, but within theframework of the approach considered below, there is asolution which meets the metrological requirementsspecified by us. It lies on the development of self�con�tained thermocompensation systems for each of the threemercury column fragments which are located in the reser�voir 1, in the connector 2, and in the lower reservoir 3(Fig. 2). Introduction of self�contained thermocompen�sation systems that react at the integral temperatures ofthese fragments (which saves us the necessity of takinginto account temperature gradients) significantly simpli�fies thermocompensation itself, reducing it to matchingthe thermal–physical parameters of individual elementsof the thermodynamic system with their geometricaldimensions. The only design innovation is a sealed tube 4filled with additional toluene volume which is placedco�axially with the tube 2 and communicates with thesylphon 12 (see Fig. 1).
To search for a correlation between the above�men�tioned parameters which meet the requirement of tem�perature stability of the measurement system, let us
introduce the following notations: H1, , and t1 are thecolumn height, the volume, and the integral temperature
of mercury in the reservoir 1; H2, , and t2 are the col�umn height, the volume, and the integral temperature of
mercury in the connecting tube 2; and t3 are the vol�ume and the integral mercury temperature in reservoir 3;βHg and βT are the coefficients of thermal expansion of
mercury and toluene; is the volume of toluene
V1Hg
V2Hg
V3Hg
V2T
1
2
3
4
5
6
7
H1�h H1
H1
Fig. 2. Scheme of compensation of temperature gradienteffects.
SEISMIC INSTRUMENTS Vol. 45 2009
WAYS TO IMPROVE METROLOGICAL AND OPERATIONAL CHARACTERISTICS 113
between coaxial tubes 2 and 4; h is the gap between thelower base of cylindrical float 5 and the bottom of res�
ervoir 1; is the toluene volume in reservoir 7.
The equation of the gas�mercury column systemequilibrium has the form
(4)
where P is the gas pressure in the lower reservoir. Thecondition of the system temperature stability in theabsence of gradients will be as follows:
(5)
The last term in the middle part of this equationtakes into account gas pressure variations due tochanges in its volume: dVt = SdH2.
Taking into account the gradients t, i.e., at t1 ≠ t2 ≠ t3,
(6)
Both theoretical calculations and practical imple�mentation of thermocompensation are significantlysimplified on the whole when autonomous thermo�compensation of the mercury column H1 is intro�duced; the latter comes to the requirement of the con�stancy of its contribution to the pressure of the mer�cury column H, i.e.,
where ρ1 is the mercury density at the temperature t1.Then
(7)
Substituting into these equations the values of
(8)
(9)
after elementary transformations we get the relation
(10)
which is easily met by selection of the parameters.When this condition is satisfied, g = const – P1 =const. In other words, P1 depends only on the gravityacceleration and variations in t1 do not dynamicallyaffect the mercury column H2.
On the other hand, to provide for independentcounting over the device converter on t1, the conditionof the independence of the Archimedian force actingon float 5 at h = const (in the case of a string converter)or at h depending only on g (in the case of a capaci�tance converter) on t1 should be met. This conditioncan be expressed by the following apparent relation:
V3T
P ρgH ρg H1 H2+( ),= =
dP g dρ H1 H2+( ) ρdH1 ρdH2+ +[ ]=
+SdH2
V�����������P 0.=
dP dP1 dP2SdH2
V�����������P+ +=
= g dρ1H1 ρ1dH1 dρ2H2 ρ2dH2+ + +[ ]SdH2
V�����������P+ = 0.
P1 ρ1gH1 const,= =
dP1 g dρ1H1 ρ1dH1+( ) 0.= =
dρ1 ρ1βHgdt1,–=
dH1 βHgV1
Hgdt1/S,=
V1Hg H1S,=
FAr V flρ1 const= =
(at constant g). Here Vfl is the volume of the sub�merged part of the float. Hence,
(11)
It is also evident that
(12)
where Q is the float cross section over the mercury line.Substituting this equation and the values of dρ1 anddH1 from Eqs. (8) and (9) into Eq. (10), we get
(13)
In this case the variations in the Archimedian forcedue to mercury density variations are exactly compen�sated by changes in the volume of the submerged partof the float. This principle of thermocompensationwas first proposed by A.M. Lozinskaye for stringgravimeters with liquid damping (see [Yuzefovich,Ogorodova, 1980]).
It can seen be easily that, to provide for equality ofEq. (13) in the case when the float is cylinder�shaped,it is necessary to add the additional volume ∆Vfl = Qhto its submerged part so that Vfl = Q(H1 – h) + Qh =QH1 in full agreement with Eq. (13). This additionalvolume is shown in Fig. 2 as the ring 6.
Then from Eqs. (6), (7) there follows the conditionof temperature stability for the gas�mercury columnsystem which arises from the requirement of theabsence of dynamic effect of the mercury column H2on the mercury column H1 during variations in tem�peratures t2 and t3:
(14)
In our further calculations, we will proceed fromthe evident fact that though we compensated for theconsequences of temperature variations, the compen�sation itself, i.e., variations in the gas volume accord�ing to the temperature variations, is an isothermal pro�cess which is described by the equation
where from
If Eq. (14) is substituted instead of dP, we get varia�tions in the volume V due to thermal expansion ofmercury V2 and V3:
(15)
By analogy with Eq. (2), the following equation isthe condition of thermocompensation
(16)
Taking Eqs. (14) and (15) into account, it follows fromit that
dFAr dρ1V fl V flρ1+ 0.= =
dV fl QdH1,=
V fl H1Q.=
dP dP2SdH2
V�����������P+=
= g dρ2H2 ρ2dH2+( )SdH2
V�����������P.+
d VP( ) PdV VdP+ 0,= =
dV VP��dP.–=
dV ' VP��g dρ2H2 ρ2dH2+( )– SdH2.–=
dV ' qβT V2Tdt2 V3
Tdt3+( ),=
114
SEISMIC INSTRUMENTS Vol. 45 2009
TAIMAZOV
(17)
Here P is determined by Eq. (4).Temperature variations in the height of the mercury
column H2 are attributed to variations in the mercurylevel in the lower part of the reservoir due to tempera�ture variations in the mercury volumes V2 and V3, i.e.,
(18)
Substituting this equation as well as dρ2 =⎯ρ2βHgdt2, P = ρgH into Eq. (17) and taking intoaccount that substitution of ρ2 by ρ doesnot yield a sig�nificant contribution to thermocompensation accu�racy, after elementary transformations we get
(19)
It is apparent that a stable dynamic equilibrium ofthe system in the presence of temperature gradients ispossible only in the case of independent compensationof the effect of local temperature variations dt2 and dt3.It implies the partitioning of Eq. (19) into two inde�pendent equations
which after transformations take their final form
(20)
(21)
Together with Eqs. (3), (10), and (13), these equa�tions make an independent set of easily satisfied condi�tions necessary and sufficient for providing for completetemperature stability of the described liquid gravimeter aswell as of its string modification [Taimazov, 1986a]. Theabsence of the cross�effect and additivity of these condi�tions is also attributed to the fact that, in the presence ofthermostatting at the level of ±0.001°C (which is foreseenand easily implemented), the maximum compensatingtemperature errors do not exceed 10–6 of the measuredvalue, namely of the acceleration of gravity g.
Thus contrary to spring gravimeters, precisioncompensation of the effect of device sensitivity to the
qβT V2Tdt2 V3
Tdt3+( )
= VP��g dρ2H2 ρ2dH2+( )– SdH2.–
dH2dH2
Hg dV3Hg+
S�������������������������–=
= β
HgV2Hgdt2
S��������������������–
βHgV3
Hgdt3
S��������������������.–
qβT V2Tdt2 V3
Tdt3+( )β
HgVH2
H���������������dt2=
+ VH���β
HgV2Hg
S��������������dt2
VH���β
HgV3Hg
S��������������dt3 β
HgV2Hgdt2 β
HgV3Hgdt3.+ + +
qβTV2Tdt2
βHgVH2
H���������������dt2
VH���β
HgV2Hg
S��������������dt2 β
HgV2Hgdt2;+ +=
qβTV3Tdt3
VH���β
HgV3Hg
S��������������dt3 β
HgV3Hgdt3,+=
qβTV2T
βHgV2
Hg��������������
VH2
V2HgH
���������� VHS������ 1,+ +=
qβTV3T
βHgV3
Hg�������������� V
HS������ 1.+=
axes inclinations as well as the temperature and its gra�dients in the device volume to the results of measure�ments is possible in gas–liquid gravimeters. In combi�nation with the absence of zero drift, it opens newopportunities for using it for registering NTGV in sta�tionary conditions, those attributed to vertical dis�placements of the Earth’s crust included, thus allow�ing us to avoid repeated labor�consuming and expen�sive geometric leveling.
In conclusion we would like to note that for the rea�sons of safety other heavy liquids, e.g., Thule liquid(ρ = 3.2 g/cm3) and Klerichi liquid (ρ = 4.9 g/cm3), canbe used in the proposed gravimeter instead of mercury,though its design characteristics in this case becomeworse. In particular, at the same pressure of the workinggas (~20 Torr) the device dimensions increase by severaltimes. Keeping this in mind and taking into accountthat the working liquid is contained in a double�wallreservoir, mercury is given preference.
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