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Liquid Density Measurement
CHAPTER 5
Liquid Density Measurement 5.1 Introduction The accurate measurement of the density of a liquid or gas is one of the most difficult problems that faces the instrument engineer. Fundamentally the density of a substance is defined as the weight per unit volume, but an equally useful measure is the specific gravity, which is the ratio of weight of the substance to the weight of the equivalent volume of distilled water at a defined temperature, usually 16°C. A choice of measurement techniques is open to the system designer but the type of instrument selected does not automatically ensure that the readings obtained therefrom will necessarily match the calibration performance figures claimed by the manufacturer. There are many problems in obtaining a representative sample of the substance under test, coupled with uncertainties about the physical properties of the substance that make the measurement of density to an accuracy better than 1 part in 1000 extremely difficult. Under ideal laboratory conditions it is possible for accuracies of density measurement to be indicated to 1 part in 10 000 but the question of the ultimate density standard remains. An analysis of the potential sources of error that may occur in measuring a sample by laboratory means, and translating this to an automatic instru ment reading may invalidate claims for higher accuracy than 1 part in 10 000. The various forms of instrument technique used for density measurement may be classified into one of two types, direct and inferred. The only truly direct method of density measurement is 86
LIQUID DENSITY MEASUREMENT
87
to weigh a sample in a container of known volume although certain inferred methods such as measuring the natural frequency of a sprung mass of sample of known dimensions yields an equivalent accuracy. Inferred methods such as those relying on nuclear absorption, or bouyancy of a float are less accurate, particularly for liquids containing solids in suspension. It is proposed to briefly discuss typical problems associated with liquid density measurement, then compare the basic characteristics of two instruments potentially offering the most accurate means of measurement. 5.2 Application Problems Successful continuous measurement of density depends on four factors. Obtaining a representative sample, minimising contamina tion of the instrument, and eliminating of physical disturbances such as mechanical shock and vibration, fluid (or gas) surges, and last but not least, obtaining quantitative data on pressure/tem perature/density relationship of the sample under test. These may be reviewed in turn. Obtaining a representative sample is more difficult than it seems and frequently may involve the use of a bypass system which allows a proportion of the main stream to be routed to the instrument under controlled conditions of flow and pressure. On certain liquids however, such as china clay in suspension, the problem would be to ensure thorough mixing at the point of take off. Equally important is the prevention of air inclusions that are present in most liquids. In certain foodstuffs such as milk-starch compounds it is virtually impossible to preclude air inclusions. Beer in all its various stages of brewing is one liquid that provides a continuous (stimulating!) challenge to instrument designers; in the hot wort stage the contamination risk is high, whereas during fermentation yeast particles and carbon dioxide bubbles almost preclude the reliable measurement of density to better than 1 part in 1000 (one brewers degree). This particular product is subject to H. M. Customs checking by hydrometer whose bouyancy is rightly or wrongly subject to an 'average' sample being adequately mixed. Contamination may take the form of protein build up, precipi tation of solids or other mysterious growths depending on the nature of the sample and/or measuring vessel used. One of the 7*
88
INSTRUMENTATION IN PROCESS CONTROL
worst contaminants is hot wort used in beer making; if allowed to cool before flushing through properly, this sets like varnish and may render the instrument unserviceable. Certain petroleum pro ducts contain impurities such as sand and water that either directly or indirectly invalidate instrument readings. Filtering is very desirable on most density meter installations. The added cost is usually far less than the cost of an instrument overhaul. Physical disturbances occur in most applications because of the presence of machinery associated with any process involved in liquid handling. Certain instruments such as direct weighing types suffer more in this respect than flotation types, and vibrating mass types suffer least. Liquid surges, by virtue of their change in mo mentum, may cause temporary apparent density changes to be detected by the instrument. A preferred installation includes a bypass and bleed system in which a sample is continuously drawn from and returned to the stream at a constant head by means of a separate pump. Referring to quantative data on the pressure/temperature/density relationship of the sample under test—it may appear to be para doxical to ask for the very data that one may in fact be required to measure. But in practice it is almost impossible to define the density of a substance without a simultaneous knowledge or control of the other variables such as temperature or pressure. The rela tionship between temperature and density is seldom linear and a study of tables issued by the Petroleum Institute will illustrate this. The problem becomes particularly acute where mixtures of solids in suspension, based liquid and soluble substances are in volved; the viscosity may change rapidly for small increases in temperature resulting in pressure variations where constant flow rates are being demanded. Such variables as those described can usually only be determined by laboratory tests to enable some secondary form of instrumentation to be added for applying corrections if required. Various forms of density measuring instruments have become available in recent years and the principles of operation are briefly: 1. 'Bubbling tube" and differential pressure transducer. In this instrument density is related to the pressure required to bubble air at two points in the liquid separated by a known vertical height. Accuracy limited by pressure transducer and
LIQUID DENSITY MEASUREMENT
2.
3.
4.
5.
89
imperfections in temperature control of long vertical column (3-10 metres). Accuracy in the order of 1 %. Float type densitometer that may use displacement or force balance principles. Satisfactory for liquids of uniform solution (not solids in suspension) and in a static state i.e. not flowing. Ingeneous weir systems have been devised to preserve the datum of the liquid surface. Accuracy of 1 part in 1000 obtainable under laboratory conditions. A typical example is manufactured by Sangamo Weston. Nucleonic radiation. This relies on the use of radio isotopes and the fact that the absorption of radiation by a substance may be related to mass per unit area and hence relative density. For high accuracy, a transmission gauge is used in which the material being checked is interposed between the source and the detector. Accuracies of 1 part in 1000 have been obtained when measuring liquid densities over relatively small spans and under carefully controlled conditions i.e. following on the spot calibration against a laboratory sample. Continuous weighing. Density is obtained by weighing a sample circulated in a U-tube of known volume. Force balance restoration. Accuracy of 1 part in 1000 obtainable under most conditions and 1 part in 10 000 obtainable under controlled flow conditions. Typical examples are the Rotameter Gravitrol and the Sperry Gravitymaster. Vibrating beam. Density is obtained by measuring the natural frequency of mass comprising container of known dimen sions filled with sample liquid. Accuracy of 1 part in 1000 obtainable under most conditions and 1 part in 10 000 under controlled flow conditions. Examples are the Solartron Densitometer, and the Agar vibrating spool fluid density meter.
Advances in the last two forms of instrument have enabled a choice between analogue or digital readout to be obtained from a basic standard instrument. The examples described below have been chosen to illustrate the broad principles involved. Quite apart from their capability of providing either form of output they have been designed to meet similar accuracy requirements and a com parison can be made to illustrate fundamental considerations. In view of their capability to operate in an 'analogue' or 'digital' role,
90
INSTRUMENTATION IN PROCESS CONTROL
it is proposed to discuss the Sperry Gravitymaster and Solatron vibrating tube densitometer in detail. Both meet similar density and associated pressure and temperature specifications. Both instruments resolve density changes of 0.1 kg/m3. (The Gravitymaster is similar to the Rotameter Gravitrol force balance instru ment which was first developed for use on sugar refining in early 1950's). The Gravitymaster, details of the analogue version of which were first published in 1962, is also based on force balance con tinuous weighing principles so that direct calibration in weight per unit volume is achieved. The digital version developed later is based on a form of incremental approximation to produce a parallel binary or bed output. The Solartron densitometer, first shown at the IEA Exhibition in May 1968, relies on an inferred measure of density obtained by first measuring a change in the frequency of a resonant tube filled with the process liquid and relating the frequency to the corresponding change in mass of liquid. The Agar vibrating spool works on a similar principle but is contained within the tube.
5.3 Sperry Gravitymaster The basic Gravitymaster is shown schematically in Figure 5.1 and comprises a U-tube through which the process liquid is circulated. The tube is arranged to pivot horizontally on a cross-leaf suspen sion so as to constitute the weight of a conventional beam balance. A 'suppressed nominal' technique is used in which a calibrated mass is used to counterbalance the tube when filled with a reference liquid such as water of density 103 kg/m3. Any deviation in weight of liquid causes a displacement of the beam; this is detected opti cally by a special pick-off producing a voltage which is then ampli fied to drive a current through a coil suspended in a magnetic field. Movement of the coil returns the beam to horizontal thereby reducing the output of the pick-off to zero i.e. it is a null-seeking servo system. The gain of the amplifier is made high so that the magnitude of the restoring force is effectively equal to the error force. One virtue of the suppressed nominal technique is that a servo loop of only 1% accuracy on a span of 10% change in density will yield an accuracy of density measurement of 0.1 % assuming
91
LIQUID DENSITY MEASUREMENT
that the stability of the counterbalance is better than 0.1%. In practice a figure ten times more accurate is achieved. In the basic analogue version the restoring current is fed through a calibrated readout resistor and the voltage appearing across the resistor is the analogue of the density variation from nominal, positive or negative. PLAN U-TUBE
LIQUID
CROSS LEAF PIVOT
ELEVATION
OPTICAL PICK OFF
=fl ft
BALANCE WEIGHT
FORCE COIL
\s\w\W\ VOLTAGE READOUT TRANSDUCER
AMPLIFIER
Figure 5.1. Sperry Gravitymaster
In the digital version of the Gravitymaster, shown in Figure 5.2, the suppressed nominal technique is retained but instead of meas uring deviations above or below nominal, only positive deviations are measured in order to simplify the decision making logic. This does not limit the span of the instrument because the 'nominal' is offset to the negative end of the scale and the dynamic range is doubled by modification to the force unit circuit. Unlike conven tional analogue to digital converters, the need to generate precision
92
INSTRUMENTATION IN PROCESS CONTROL
voltages is obviated by supplying binary weighted currents to a second force-coil suspended in the same magnetic circuit as the first. The analogue force-balance circuit is used for damping shortPLAN U-TUBE
M A / LIQUID
KsUs ELEVATION
CROSS LEAF PIVOT
PICK OFF
&
=fl
BALANCE WEIGHT
HM
(
o—VWV R
Τ^ΛΛΛ-
J
DIGITAL FORCE COIL ANALOGUE FORCE COIL
fU
l?l
- T~ 0 I
I
J> VOLTAGE <* READOUT TRANSDUCER
I I
r-o ANALOGUE VOLTAGE READOUT IF REQUIRED IN DIGITAL VERSION
AMPLIFIER
INCREASE
BINARY
DECREASE
COUNTER
CENTRE STABLE RELAY
ΠΜ
DIGITAL READOUT
Figure 5.2. Digital Gravitymaster
term disturbances only. The readout resistor is replaced by a centre stable differential relay whose threshold sensitivity is designed to correspond to half the minimum resolution required.
LIQUID DENSITY MEASUREMENT
93
An incremental form of conversion is used in which successive powers of binary currents are switched through the second force coil by a clock controlled from the relay error detector until equilibrium of the beam is reached, causing the relay to centralize and switch off the clock. The state of the counter register controlling the current feed to the restoring coil is the binary equivalent of density deviation. The clock frequency is relatively slow and chosen to be less than half the normal natural frequency of the analogue restoring loop to render the digital loop relatively insensitive to step disturbances or noise. Additional filtering is attainable by damping the coil of the centre stable relay error detector. The foregoing technique results in an analogue-to-digital con version process that is completed within the closed loop of a force-balance servo system. One attraction of the Sperry Conver sion is the added facility of analogue read-out for conventional display or control, if required, as a back up to a digital system, at relatively low additional cost per instrument. Excellent long term stability is achieved because in the limit the system is dependent on two magnetic circuits. These comprise the centre-stable relay used as an error detector to define the minimum digit level, and the moving-coil transducer used to convert binary related currents into units of force to restore unbalance of the beam. 5.4 Solartron Vibrating Tube Densitometer The new Solartron instrument provides an a.c. voltage output whose frequency is a function of the density of the liquid circulating in a resonant tube. The classic relationship 1 Ί ft Stiffness \ Frequency = - ]/ \ ^ ^ Π is used to derive the value of density. The stiffness term is a constant and obtained from the dimen sions and Young's Modulus of a sampling tube. The inertia term is a function of the volume of the sample and the density of the liquid. In the Solartron instrument great care is exercised to define the length of the tube by machining and mechanical constraint.
94
INSTRUMENTATION IN PROCESS CONTROL
Consequently the only relevant variable in the above relationship is the density of the liquid which provides the inertia term. As shown schematically in Figure 5.3 the transducer consists of a pair of Ni—Span—C 902 tubes welded together to a common support at each end so that they lie parallel to each other. Suspended from the end supports and lying between the two tubes is a drive coil and pick-up coil assembly, inter-connected electrically by means of a maintaining amplifier to drive the tubes MAINTAINING AMPLIFIER
PLAN
FLEXIBLE COUPLINGS
PICK-UPS)
LIQUID
U/xnl
COIL -3
c r iDRIVE COIL
£.
EFFECTIVE LENGTH OF RESONANT TUBES
ELEVATION
V7777777P7777%Z?777777777777777 - A - V MOUNTS
Θ
£It3 CASE
Figure 5.3. Solartron vibrating tube densitometer
at their mean natural frequency. The tubes and coils are mounted in a case on antivibration mountings. The liquid whose density is to be measured flows through two pairs of flexible pipes from the case to the tubes and then out again. A choice of pipe configura tions is available to suit process plumbing requirements. These are: Twin tube straight through bore; Circulating tube by connection of a U-tube link at one end; Single inlet and outlet by connection of Y-tubes at each end. The natural frequency of vibration of the tubes is a function of their mass per unit length and hence the density of the liquid contained in the tubes. This frequency is measured in order to compute the liquid density by simple counting techniques. If only one tube were used there would be considerable movement of the
95
LIQUID DENSITY MEASUREMENT
end mounting and the instrument would be prone to external phys ical disturbance, but by using two tubes which vibrate in opposi tion, a balanced system results in which end movement is reduced to a negligible amount. The operation of the system can be compared to that of a tuning fork. The tube assembly has a high mechanical Q of the order of 3000 which results in a frequency stability of better than 1 part per million under conditions of controlled environment. A change in density of liquid of 1000 kg/m3 corresponds to a 20% deviation in periodic time and the linearity for a span of 50 kg/m3 is better than 0.1 kg/m3; the deviation is in accordance with a precisely defined law of the form mentioned earlier. For this particular instrument the equation reduces to T = To·]/ 11-\ where T = T0 = ρ = Po =
1
periodic time at liquid density periodic time with air at one atmosphere density of measured liquid 2280 kg/m3.
For the majority of applications, (when working about a set point for example), the linear range of 50 kg/m3 is more than adequate. An additional feature is the inclusion of platinum resistance thermometers at each end of the tubes to enable the mean liquid temperature to be measured and corrections made to give a liquid density reading referred to a specific temperature. The accuracy of 0.1 kg/m3 is claimed on the basis that a 2°C error in temperature measurement will yield a 0.1 kg/m3 error in the instrument. Taking due note that the density of water varies by 1 kg/m3 for 2°C change, the instrument error is itself an order of magnitude smaller. In practice it is claimed that temperature correction is better than 0.5°C. From the operational point of view two significant advantages are apparent. One is the very high immunity to the vibration that is always present on liquid handling installations, in the building structure, transmitted hydraulically from the pump, or generated by turbulence in the flow. The second is the robust construction of the straight-through pipes used for sampling, enabling the in strument to be easily cleaned in situ and readily inspected visibly for build-up of contaminating precipitants present in some pro cess liquids. The absence of moving parts obviates the need for
96
INSTRUMENTATION IN PROCESS CONTROL
any maintenance whatsoever apart from the occasional visual in spection and cleaning. The instrument can be mounted in any position and at any angle. Consequently it may be secured vertically, thereby minimizing the precipitation of solids and formation of bubbles that might other wise contribute significantly to errors in measurement. The readout of the instrument can be made to display either the actual density or density deviation. A digital binary-coded decimal output is available for transmission purposes. A simple digital-toanalogue converter is available should it be required to connect the instrument to an existing analogue control loop. 5.5 Agar Vibrating Spool Densitometer An interesting development of the frequency modulated trans ducer approach to density measurement is the vibrating spool fluid density meter by Joram Agar1. The principle of operation is shown
Figure 5.4. Principle of operation of the vibrating spool density meter
in Figure 5.4. Here the sensing element is a tube thickened at the two ends to form a spool. In operation it is set in oscillation in a circumferential mode, as a bell rings. Inductive pick off and excita tion coils and an amplifier maintain oscillation.
97
LIQUID DENSITY MEASUREMENT
The fluid to be measured surrounds the tube and is thus similarly set in oscillation. As in the vibrating tube densitometer, the fre quency of oscillation depends on the tube stiffness and the total oscillating mass, which in the case of the spool comprises the tube walls and surrounding fluid. The output of the sustaining amplifier is monitored by a fre quency meter calibrated to read units of density; an increase of density results in a reduced frequency of oscillation. TRICHLOROETHYLENE^o^ 350
-
^ X ° W A T E R 0-92 cp ^ ^ ° A . S . A . 30 OIL 180 cp
300
.•^KEROSENE 1-97cp
Q
2 250 en
THEORETICAL CURVE
200 ^50
y^NITROGEN *AIR i
1
200
1
400
L
600
1
!
1
800 1000 1200 DENSITY-kg/m 3
1
1
1
1400
1600
180C
Figure 5.5. Calibration curve—vibrating spool density meter
An accuracy of one part in 104 of f.s.d. is claimed, with excellent immunity to plant and other noise problems by virtue of the high mechanical Q of the spool (of the order of a few hundred). By suitable selection of spool material the temperature coefficient may approach that of a tuning fork. A typical calibration curve is shown in Figure 5.5. The wide range of change in periodic time from vacuum to trichloroethylene is defined by: ρ = ρο[(Γ.Γ 0 ) 2 -1] where ρ ρο T To
= = = =
measured fluid density scale factor measured time oscillation periodic time at vacuum.
98
INSTRUMENTATION IN PROCESS CONTROL
The advantages of measuring periodic time of frequency modu lated transducers, instead of frequency, are twofold. Firstly the relationship between periodic time and density tends to be more linear thus requiring less correction. Secondly the output may be measured in less time. For example if the carrier frequency is of the order of a few kilohertz (a practical figure is 3 kHz) the measuring time is of the order of one second to resolve 1 part in 3000. However, if the range of frequency is only 10% of the carrier frequency i.e. 300 Hz the measuring time will require to be 3 sec to resolve 1 part in 1000. If periodic time is used, however, this accuracy can be achieved at a fraction of this time. Accordingly, the signal is counted (aver aged) over say, 10 Hz, which takes only 3.3 ms. During this period, a 1 MHz 'clock' will fill up the information storage register with 3300 pulses—3 times more than the required 1000. 5.6 Linearising Problems When linearising density readings from any form of density meter several important factors must be borne in mind to ensure that the meaning of linearising is clearly defined. The relationship between density and temperature of the fluid, the means whereby this is obtained and the accuracy with which it is defined. The relationship between actual density of thefluidand indicated measure of density provided by the instrument at some suitable reference temperature and for the range of operating tempera tures. The indicated density/temperature relationships for the instru ment over the working range. The problem of computing corrections is discussed in detail in Chapter 11. Accurate linearising will involve the continuous calculation of at least two variables and in the limit it may be argued that it is the accuracy obtainable from the computing process that limits the ultimate accuracy of result, or it may be the fundamental limitation of predicted accuracy of density/temperature relationship of the liquid being sampled. Systems based on digital calculation, involving frequency modu-
LIQUID DENSITY MEASUREMENT
99
lated transducers binary rate multipliers and totalising counters would appear to offer a more predictable result than analogue com puting solutions. In operation a digital calculator for processing a frequency mod ulated signal would be based on a binary rate multiplier as de scribed in Chapter 11 in which a 'pulse dropping' technique of computation is performed with binary rate multipliers which op erate on say 1 MHz clock pulses, rather than on the incoming information, which is much slower. Thus by measuring the pe riodic time of the incoming f.m. signal it is possible to linearise and display the information in less than 5 ms. A further application of binary rate multipliers is to combine the value of density measurement with rate of flow of the fluid to obtain true mass flow, one of several techniques to be described in Chapter 7.
Liquid Density Measurement 5.1 Introduction The accurate measurement of the density of a liquid or gas is one of the most difficult problems that faces the instrument engineer. Fundamentally the density of a substance is defined as the weight per unit volume, but an equally useful measure is the specific gravity, which is the ratio of weight of the substance to the weight of the equivalent volume of distilled water at a defined temperature, usually 16°C. A choice of measurement techniques is open to the system designer but the type of instrument selected does not automatically ensure that the readings obtained therefrom will necessarily match the calibration performance figures claimed by the manufacturer. There are many problems in obtaining a representative sample of the substance under test, coupled with uncertainties about the physical properties of the substance that make the measurement of density to an accuracy better than 1 part in 1000 extremely difficult. Under ideal laboratory conditions it is possible for accuracies of density measurement to be indicated to 1 part in 10 000 but the question of the ultimate density standard remains. An analysis of the potential sources of error that may occur in measuring a sample by laboratory means, and translating this to an automatic instru ment reading may invalidate claims for higher accuracy than 1 part in 10 000. The various forms of instrument technique used for density measurement may be classified into one of two types, direct and inferred. The only truly direct method of density measurement is 86
LIQUID DENSITY MEASUREMENT
87
to weigh a sample in a container of known volume although certain inferred methods such as measuring the natural frequency of a sprung mass of sample of known dimensions yields an equivalent accuracy. Inferred methods such as those relying on nuclear absorption, or bouyancy of a float are less accurate, particularly for liquids containing solids in suspension. It is proposed to briefly discuss typical problems associated with liquid density measurement, then compare the basic characteristics of two instruments potentially offering the most accurate means of measurement. 5.2 Application Problems Successful continuous measurement of density depends on four factors. Obtaining a representative sample, minimising contamina tion of the instrument, and eliminating of physical disturbances such as mechanical shock and vibration, fluid (or gas) surges, and last but not least, obtaining quantitative data on pressure/tem perature/density relationship of the sample under test. These may be reviewed in turn. Obtaining a representative sample is more difficult than it seems and frequently may involve the use of a bypass system which allows a proportion of the main stream to be routed to the instrument under controlled conditions of flow and pressure. On certain liquids however, such as china clay in suspension, the problem would be to ensure thorough mixing at the point of take off. Equally important is the prevention of air inclusions that are present in most liquids. In certain foodstuffs such as milk-starch compounds it is virtually impossible to preclude air inclusions. Beer in all its various stages of brewing is one liquid that provides a continuous (stimulating!) challenge to instrument designers; in the hot wort stage the contamination risk is high, whereas during fermentation yeast particles and carbon dioxide bubbles almost preclude the reliable measurement of density to better than 1 part in 1000 (one brewers degree). This particular product is subject to H. M. Customs checking by hydrometer whose bouyancy is rightly or wrongly subject to an 'average' sample being adequately mixed. Contamination may take the form of protein build up, precipi tation of solids or other mysterious growths depending on the nature of the sample and/or measuring vessel used. One of the 7*
88
INSTRUMENTATION IN PROCESS CONTROL
worst contaminants is hot wort used in beer making; if allowed to cool before flushing through properly, this sets like varnish and may render the instrument unserviceable. Certain petroleum pro ducts contain impurities such as sand and water that either directly or indirectly invalidate instrument readings. Filtering is very desirable on most density meter installations. The added cost is usually far less than the cost of an instrument overhaul. Physical disturbances occur in most applications because of the presence of machinery associated with any process involved in liquid handling. Certain instruments such as direct weighing types suffer more in this respect than flotation types, and vibrating mass types suffer least. Liquid surges, by virtue of their change in mo mentum, may cause temporary apparent density changes to be detected by the instrument. A preferred installation includes a bypass and bleed system in which a sample is continuously drawn from and returned to the stream at a constant head by means of a separate pump. Referring to quantative data on the pressure/temperature/density relationship of the sample under test—it may appear to be para doxical to ask for the very data that one may in fact be required to measure. But in practice it is almost impossible to define the density of a substance without a simultaneous knowledge or control of the other variables such as temperature or pressure. The rela tionship between temperature and density is seldom linear and a study of tables issued by the Petroleum Institute will illustrate this. The problem becomes particularly acute where mixtures of solids in suspension, based liquid and soluble substances are in volved; the viscosity may change rapidly for small increases in temperature resulting in pressure variations where constant flow rates are being demanded. Such variables as those described can usually only be determined by laboratory tests to enable some secondary form of instrumentation to be added for applying corrections if required. Various forms of density measuring instruments have become available in recent years and the principles of operation are briefly: 1. 'Bubbling tube" and differential pressure transducer. In this instrument density is related to the pressure required to bubble air at two points in the liquid separated by a known vertical height. Accuracy limited by pressure transducer and
LIQUID DENSITY MEASUREMENT
2.
3.
4.
5.
89
imperfections in temperature control of long vertical column (3-10 metres). Accuracy in the order of 1 %. Float type densitometer that may use displacement or force balance principles. Satisfactory for liquids of uniform solution (not solids in suspension) and in a static state i.e. not flowing. Ingeneous weir systems have been devised to preserve the datum of the liquid surface. Accuracy of 1 part in 1000 obtainable under laboratory conditions. A typical example is manufactured by Sangamo Weston. Nucleonic radiation. This relies on the use of radio isotopes and the fact that the absorption of radiation by a substance may be related to mass per unit area and hence relative density. For high accuracy, a transmission gauge is used in which the material being checked is interposed between the source and the detector. Accuracies of 1 part in 1000 have been obtained when measuring liquid densities over relatively small spans and under carefully controlled conditions i.e. following on the spot calibration against a laboratory sample. Continuous weighing. Density is obtained by weighing a sample circulated in a U-tube of known volume. Force balance restoration. Accuracy of 1 part in 1000 obtainable under most conditions and 1 part in 10 000 obtainable under controlled flow conditions. Typical examples are the Rotameter Gravitrol and the Sperry Gravitymaster. Vibrating beam. Density is obtained by measuring the natural frequency of mass comprising container of known dimen sions filled with sample liquid. Accuracy of 1 part in 1000 obtainable under most conditions and 1 part in 10 000 under controlled flow conditions. Examples are the Solartron Densitometer, and the Agar vibrating spool fluid density meter.
Advances in the last two forms of instrument have enabled a choice between analogue or digital readout to be obtained from a basic standard instrument. The examples described below have been chosen to illustrate the broad principles involved. Quite apart from their capability of providing either form of output they have been designed to meet similar accuracy requirements and a com parison can be made to illustrate fundamental considerations. In view of their capability to operate in an 'analogue' or 'digital' role,
90
INSTRUMENTATION IN PROCESS CONTROL
it is proposed to discuss the Sperry Gravitymaster and Solatron vibrating tube densitometer in detail. Both meet similar density and associated pressure and temperature specifications. Both instruments resolve density changes of 0.1 kg/m3. (The Gravitymaster is similar to the Rotameter Gravitrol force balance instru ment which was first developed for use on sugar refining in early 1950's). The Gravitymaster, details of the analogue version of which were first published in 1962, is also based on force balance con tinuous weighing principles so that direct calibration in weight per unit volume is achieved. The digital version developed later is based on a form of incremental approximation to produce a parallel binary or bed output. The Solartron densitometer, first shown at the IEA Exhibition in May 1968, relies on an inferred measure of density obtained by first measuring a change in the frequency of a resonant tube filled with the process liquid and relating the frequency to the corresponding change in mass of liquid. The Agar vibrating spool works on a similar principle but is contained within the tube.
5.3 Sperry Gravitymaster The basic Gravitymaster is shown schematically in Figure 5.1 and comprises a U-tube through which the process liquid is circulated. The tube is arranged to pivot horizontally on a cross-leaf suspen sion so as to constitute the weight of a conventional beam balance. A 'suppressed nominal' technique is used in which a calibrated mass is used to counterbalance the tube when filled with a reference liquid such as water of density 103 kg/m3. Any deviation in weight of liquid causes a displacement of the beam; this is detected opti cally by a special pick-off producing a voltage which is then ampli fied to drive a current through a coil suspended in a magnetic field. Movement of the coil returns the beam to horizontal thereby reducing the output of the pick-off to zero i.e. it is a null-seeking servo system. The gain of the amplifier is made high so that the magnitude of the restoring force is effectively equal to the error force. One virtue of the suppressed nominal technique is that a servo loop of only 1% accuracy on a span of 10% change in density will yield an accuracy of density measurement of 0.1 % assuming
91
LIQUID DENSITY MEASUREMENT
that the stability of the counterbalance is better than 0.1%. In practice a figure ten times more accurate is achieved. In the basic analogue version the restoring current is fed through a calibrated readout resistor and the voltage appearing across the resistor is the analogue of the density variation from nominal, positive or negative. PLAN U-TUBE
LIQUID
CROSS LEAF PIVOT
ELEVATION
OPTICAL PICK OFF
=fl ft
BALANCE WEIGHT
FORCE COIL
\s\w\W\ VOLTAGE READOUT TRANSDUCER
AMPLIFIER
Figure 5.1. Sperry Gravitymaster
In the digital version of the Gravitymaster, shown in Figure 5.2, the suppressed nominal technique is retained but instead of meas uring deviations above or below nominal, only positive deviations are measured in order to simplify the decision making logic. This does not limit the span of the instrument because the 'nominal' is offset to the negative end of the scale and the dynamic range is doubled by modification to the force unit circuit. Unlike conven tional analogue to digital converters, the need to generate precision
92
INSTRUMENTATION IN PROCESS CONTROL
voltages is obviated by supplying binary weighted currents to a second force-coil suspended in the same magnetic circuit as the first. The analogue force-balance circuit is used for damping shortPLAN U-TUBE
M A / LIQUID
KsUs ELEVATION
CROSS LEAF PIVOT
PICK OFF
&
=fl
BALANCE WEIGHT
HM
(
o—VWV R
Τ^ΛΛΛ-
J
DIGITAL FORCE COIL ANALOGUE FORCE COIL
fU
l?l
- T~ 0 I
I
J> VOLTAGE <* READOUT TRANSDUCER
I I
r-o ANALOGUE VOLTAGE READOUT IF REQUIRED IN DIGITAL VERSION
AMPLIFIER
INCREASE
BINARY
DECREASE
COUNTER
CENTRE STABLE RELAY
ΠΜ
DIGITAL READOUT
Figure 5.2. Digital Gravitymaster
term disturbances only. The readout resistor is replaced by a centre stable differential relay whose threshold sensitivity is designed to correspond to half the minimum resolution required.
LIQUID DENSITY MEASUREMENT
93
An incremental form of conversion is used in which successive powers of binary currents are switched through the second force coil by a clock controlled from the relay error detector until equilibrium of the beam is reached, causing the relay to centralize and switch off the clock. The state of the counter register controlling the current feed to the restoring coil is the binary equivalent of density deviation. The clock frequency is relatively slow and chosen to be less than half the normal natural frequency of the analogue restoring loop to render the digital loop relatively insensitive to step disturbances or noise. Additional filtering is attainable by damping the coil of the centre stable relay error detector. The foregoing technique results in an analogue-to-digital con version process that is completed within the closed loop of a force-balance servo system. One attraction of the Sperry Conver sion is the added facility of analogue read-out for conventional display or control, if required, as a back up to a digital system, at relatively low additional cost per instrument. Excellent long term stability is achieved because in the limit the system is dependent on two magnetic circuits. These comprise the centre-stable relay used as an error detector to define the minimum digit level, and the moving-coil transducer used to convert binary related currents into units of force to restore unbalance of the beam. 5.4 Solartron Vibrating Tube Densitometer The new Solartron instrument provides an a.c. voltage output whose frequency is a function of the density of the liquid circulating in a resonant tube. The classic relationship 1 Ί ft Stiffness \ Frequency = - ]/ \ ^ ^ Π is used to derive the value of density. The stiffness term is a constant and obtained from the dimen sions and Young's Modulus of a sampling tube. The inertia term is a function of the volume of the sample and the density of the liquid. In the Solartron instrument great care is exercised to define the length of the tube by machining and mechanical constraint.
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INSTRUMENTATION IN PROCESS CONTROL
Consequently the only relevant variable in the above relationship is the density of the liquid which provides the inertia term. As shown schematically in Figure 5.3 the transducer consists of a pair of Ni—Span—C 902 tubes welded together to a common support at each end so that they lie parallel to each other. Suspended from the end supports and lying between the two tubes is a drive coil and pick-up coil assembly, inter-connected electrically by means of a maintaining amplifier to drive the tubes MAINTAINING AMPLIFIER
PLAN
FLEXIBLE COUPLINGS
PICK-UPS)
LIQUID
U/xnl
COIL -3
c r iDRIVE COIL
£.
EFFECTIVE LENGTH OF RESONANT TUBES
ELEVATION
V7777777P7777%Z?777777777777777 - A - V MOUNTS
Θ
£It3 CASE
Figure 5.3. Solartron vibrating tube densitometer
at their mean natural frequency. The tubes and coils are mounted in a case on antivibration mountings. The liquid whose density is to be measured flows through two pairs of flexible pipes from the case to the tubes and then out again. A choice of pipe configura tions is available to suit process plumbing requirements. These are: Twin tube straight through bore; Circulating tube by connection of a U-tube link at one end; Single inlet and outlet by connection of Y-tubes at each end. The natural frequency of vibration of the tubes is a function of their mass per unit length and hence the density of the liquid contained in the tubes. This frequency is measured in order to compute the liquid density by simple counting techniques. If only one tube were used there would be considerable movement of the
95
LIQUID DENSITY MEASUREMENT
end mounting and the instrument would be prone to external phys ical disturbance, but by using two tubes which vibrate in opposi tion, a balanced system results in which end movement is reduced to a negligible amount. The operation of the system can be compared to that of a tuning fork. The tube assembly has a high mechanical Q of the order of 3000 which results in a frequency stability of better than 1 part per million under conditions of controlled environment. A change in density of liquid of 1000 kg/m3 corresponds to a 20% deviation in periodic time and the linearity for a span of 50 kg/m3 is better than 0.1 kg/m3; the deviation is in accordance with a precisely defined law of the form mentioned earlier. For this particular instrument the equation reduces to T = To·]/ 11-\ where T = T0 = ρ = Po =
1
periodic time at liquid density periodic time with air at one atmosphere density of measured liquid 2280 kg/m3.
For the majority of applications, (when working about a set point for example), the linear range of 50 kg/m3 is more than adequate. An additional feature is the inclusion of platinum resistance thermometers at each end of the tubes to enable the mean liquid temperature to be measured and corrections made to give a liquid density reading referred to a specific temperature. The accuracy of 0.1 kg/m3 is claimed on the basis that a 2°C error in temperature measurement will yield a 0.1 kg/m3 error in the instrument. Taking due note that the density of water varies by 1 kg/m3 for 2°C change, the instrument error is itself an order of magnitude smaller. In practice it is claimed that temperature correction is better than 0.5°C. From the operational point of view two significant advantages are apparent. One is the very high immunity to the vibration that is always present on liquid handling installations, in the building structure, transmitted hydraulically from the pump, or generated by turbulence in the flow. The second is the robust construction of the straight-through pipes used for sampling, enabling the in strument to be easily cleaned in situ and readily inspected visibly for build-up of contaminating precipitants present in some pro cess liquids. The absence of moving parts obviates the need for
96
INSTRUMENTATION IN PROCESS CONTROL
any maintenance whatsoever apart from the occasional visual in spection and cleaning. The instrument can be mounted in any position and at any angle. Consequently it may be secured vertically, thereby minimizing the precipitation of solids and formation of bubbles that might other wise contribute significantly to errors in measurement. The readout of the instrument can be made to display either the actual density or density deviation. A digital binary-coded decimal output is available for transmission purposes. A simple digital-toanalogue converter is available should it be required to connect the instrument to an existing analogue control loop. 5.5 Agar Vibrating Spool Densitometer An interesting development of the frequency modulated trans ducer approach to density measurement is the vibrating spool fluid density meter by Joram Agar1. The principle of operation is shown
Figure 5.4. Principle of operation of the vibrating spool density meter
in Figure 5.4. Here the sensing element is a tube thickened at the two ends to form a spool. In operation it is set in oscillation in a circumferential mode, as a bell rings. Inductive pick off and excita tion coils and an amplifier maintain oscillation.
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LIQUID DENSITY MEASUREMENT
The fluid to be measured surrounds the tube and is thus similarly set in oscillation. As in the vibrating tube densitometer, the fre quency of oscillation depends on the tube stiffness and the total oscillating mass, which in the case of the spool comprises the tube walls and surrounding fluid. The output of the sustaining amplifier is monitored by a fre quency meter calibrated to read units of density; an increase of density results in a reduced frequency of oscillation. TRICHLOROETHYLENE^o^ 350
-
^ X ° W A T E R 0-92 cp ^ ^ ° A . S . A . 30 OIL 180 cp
300
.•^KEROSENE 1-97cp
Q
2 250 en
THEORETICAL CURVE
200 ^50
y^NITROGEN *AIR i
1
200
1
400
L
600
1
!
1
800 1000 1200 DENSITY-kg/m 3
1
1
1
1400
1600
180C
Figure 5.5. Calibration curve—vibrating spool density meter
An accuracy of one part in 104 of f.s.d. is claimed, with excellent immunity to plant and other noise problems by virtue of the high mechanical Q of the spool (of the order of a few hundred). By suitable selection of spool material the temperature coefficient may approach that of a tuning fork. A typical calibration curve is shown in Figure 5.5. The wide range of change in periodic time from vacuum to trichloroethylene is defined by: ρ = ρο[(Γ.Γ 0 ) 2 -1] where ρ ρο T To
= = = =
measured fluid density scale factor measured time oscillation periodic time at vacuum.
98
INSTRUMENTATION IN PROCESS CONTROL
The advantages of measuring periodic time of frequency modu lated transducers, instead of frequency, are twofold. Firstly the relationship between periodic time and density tends to be more linear thus requiring less correction. Secondly the output may be measured in less time. For example if the carrier frequency is of the order of a few kilohertz (a practical figure is 3 kHz) the measuring time is of the order of one second to resolve 1 part in 3000. However, if the range of frequency is only 10% of the carrier frequency i.e. 300 Hz the measuring time will require to be 3 sec to resolve 1 part in 1000. If periodic time is used, however, this accuracy can be achieved at a fraction of this time. Accordingly, the signal is counted (aver aged) over say, 10 Hz, which takes only 3.3 ms. During this period, a 1 MHz 'clock' will fill up the information storage register with 3300 pulses—3 times more than the required 1000. 5.6 Linearising Problems When linearising density readings from any form of density meter several important factors must be borne in mind to ensure that the meaning of linearising is clearly defined. The relationship between density and temperature of the fluid, the means whereby this is obtained and the accuracy with which it is defined. The relationship between actual density of thefluidand indicated measure of density provided by the instrument at some suitable reference temperature and for the range of operating tempera tures. The indicated density/temperature relationships for the instru ment over the working range. The problem of computing corrections is discussed in detail in Chapter 11. Accurate linearising will involve the continuous calculation of at least two variables and in the limit it may be argued that it is the accuracy obtainable from the computing process that limits the ultimate accuracy of result, or it may be the fundamental limitation of predicted accuracy of density/temperature relationship of the liquid being sampled. Systems based on digital calculation, involving frequency modu-
LIQUID DENSITY MEASUREMENT
99
lated transducers binary rate multipliers and totalising counters would appear to offer a more predictable result than analogue com puting solutions. In operation a digital calculator for processing a frequency mod ulated signal would be based on a binary rate multiplier as de scribed in Chapter 11 in which a 'pulse dropping' technique of computation is performed with binary rate multipliers which op erate on say 1 MHz clock pulses, rather than on the incoming information, which is much slower. Thus by measuring the pe riodic time of the incoming f.m. signal it is possible to linearise and display the information in less than 5 ms. A further application of binary rate multipliers is to combine the value of density measurement with rate of flow of the fluid to obtain true mass flow, one of several techniques to be described in Chapter 7.