UNIVERSITÀ DEGLI STUDI DI PISA
Scuola di Ingegneria Dipartimento di Ingegneria
dell’Energia, dei Sistemi, del Territorio e delle Costruzioni
Corso di laurea magistrale in Ingegneria Energetica
TESI DI LAUREA
Improvement and force transmission error analysis of
two gravimetric sorption analysers, and application to
adsorption measurement
Relatori: Umberto Desideri Markus Richter Candidato: Luca Bernardini Anno accademico 2017/2018I
ABSTRACT
Force transmission error analysis was applied to improve two different gravimetric sorption analysers based on magnetic suspension balance technology. For the setup, improvement and commissioning of the apparatuses the following steps were conducted: installation and modification of mechanical parts and devices, execution of the calibrations required, software development, and data analysing. Force transmission error parameters and density of nitrogen were investigated and compared with reference values available in literature. Three different kind of measurements were conducted: vacuum measurements to calculate the apparatus contribution ϕVac, synthetic air measurements with a mixture of 20,5 ± 0,5 % of O2 in N2 to determine the apparatus specific constant ερ, and pure nitrogen measurements to calculate the correction factor εfse and therefore determine the density of N2. Vacuum measurements were conducted at different temperatures, 20, 50 and 80 °C, and synthetic air and nitrogen measurements were conducted at 20 °C at different pressures, 20, 40, 60 up to 80 bar. Once improved, the second apparatus analysed was applied to sorption equilibrium and kinetic sorption measurements. A sample of synthetic hydrocarbon char (FE/HTC-800) was investigated, measurements conducted in vacuum mode at 25 °C and using CO2 at 25 and 50 °C combined with different pressures, 0,5, 1, 2 up to 4 bar. Equilibrium data were calculated before and after force transmission error correction, and also compared with data carried out with a volumetric apparatus at University of Western Australia in Perth. Kinetic data were analysed using the models of Carslaw and Jaeger and Kociric et al. and the effective diffusion coefficient was carried out.
II
TABLE OF CONTENT
1. INTRODUCTION ... 1
2. THEORETICAL BACKGROUND ... 2
2.1. Adsorption ... 2
2.2. Magnetic Suspension Balance ... 4
2.3. Force Transmission Error Analysis ... 6
3. IMPROVEMENT, CALIBRATION AND SOFTWARE DEVELOPMENT FOR EXPERIMENTAL SETUP ... 10
3.1. ELLI Densimeter Apparatus ... 10
3.1.1. Modifications ... 10 3.1.2. Calibrations ... 13 Mass Calibration ... 13 Volume Calibration ... 15 Temperature Calibration ... 18 3.1.3. Software Development ... 20 LabVIEW ... 20 MATLAB ... 28
3.2. OXYFLAME Sorption Apparatus ... 32
3.2.1. Modifications ... 32 3.2.2. Calibrations ... 34 Mass Calibration ... 34 Volume Calibration ... 35 Temperature Calibration ... 37 Sample Calibration ... 39 3.2.3. Software Development ... 43 LabVIEW ... 43 MATLAB ... 54
4. EXPERIMENTAL DATA AND DISCUSSION ... 58
4.1. FTE Analysis and Density of N2 in ELLI Densimeter Apparatus ... 58
4.2. OXYFLAME Sorption Apparatus ... 65
4.2.1. FTE Analysis and Density of N2 ... 65
4.2.2. Sorption Equilibrium and Sorption Kinetic ... 68
III
REFERENCES ... 76
TABLES AND DATA - ELLI Densimeter Apparatus ... 78
TABLES AND DATA - OXYFLAME Sorption Apparatus ... 84
APPENDIX ... 88
A.1. Mass Calibration ... 88
A.2. MATLAB CODES - ELLI Densimeter Apparatus ... 90
1
1. INTRODUCTION
Very accurate values of properties of state are increasingly important for both, industrial processes and scientific research. Industries use equations of state (EOS) developed by researcher, and nowadays very accurate EOS are available for many pure fluid. For other kind of fluids and for mixtures an accurate EOS is still required and for its development accurate values of density are necessary. The apparatus that carries out most accurate data for fluid densities is a tandem sinker densimeter with magnetic suspension balance[1]. Due to the magnetic coupling of this apparatus, the measurements are influenced by nearby magnetic material, external magnetic fields, and magnetic susceptibility of the fluid used. This error in weight measures is called ‘force transmission error’. The FTE previously studied by McLinden et al.(2007)[2] and Yang et al.(2018)[3], is now an error that can be predicted and corrected. It is characterized by coefficients that, according to the previous studies, can be assumed as reference values. Based on those results, FTE analysis may be used as a valid method for the set up and improvement of new and old apparatuses.
Further, correction of FTE can be applied not only in density measurements but also in other kind of measurements like adsorption, comparing the not corrected data already presented in literature. The study of adsorption behaviour is increasingly important for many processes. In separation of CO2 from flue gas, adsorption-based processes became more interesting than absorption processes due to their smaller energy requirement, and for the stability and non-volatility of the solid sorbent material[4]. Gas diffusion into porous structure and surface of a solid fuel can be described with the effective diffusion coefficients found by adsorption analysis or, at equilibrium, with the loading value. Adsorption measurements are used to understand the behaviour about the kinetic and the thermodynamic equilibrium loading on the surface[5].
In this thesis, the setup and improvement of two gravimetric sorption analysers is presented. In chapter 2, the measuring principle of the magnetic suspension balance is described and short theoretical reviews of adsorption and force transmission error are presented. In chapter 3, the two apparatuses are described, and the modifications, calibrations and software development for the improvements are explained. In chapter 4, the experimental data, collected with those apparatuses, are showed and discussed: the FTE and density of nitrogen for both apparatuses, and the application of the second apparatus to adsorption measurements. Finally in chapter 5, the conclusions about this work are drawn with an eye to future developments.
2
2. THEORETICAL BACKGROUND
2.1. Adsorption
Adsorption describes the attachment of gaseous or liquid molecules on the surface of a solid material[6]. This is very important for many industrial processes, e.g. in the fields of chemical, biochemical and separation processes. The study of separation processes is one of the main task of the energy industries. Separation and capture of CO2 can prevent pipeline corrosion if done on natural gas, and can improve purification of flue gas[7]. Analyse gas adsorption on solid fuels, can improve the knowledge about reaction kinetics during combustion.
Adsorption is a superficial phenomenon, molecules diffuse to the external surface and to the porous surface of a solid. The intensity increase with higher pressures and lower temperatures, and have a kinetic depending by the nature of the gas, the moisture content, and the temperature[8]. It should not be confused with the absorption, which is a volume phenomenon and molecules diffuse into the liquid phase. The desorption is the reverse process where the molecules return to the gaseous phase, it’s characterized by molar enthalpy.
By conducting adsorption measurements, equilibria and kinetic can be studied. The gas adsorption equilibrium is reached when at constant temperature and pressure the fluid molecules adsorbed by the sample are the same than the released molecules. In this condition the amount of gas adsorbed is constant, m∞ (g). It’s possible to compare this value changing both the temperature or the pressure, or changing the gas used or the solid material used. By testing different adsorber materials for separation of a certain gas, the best combination can be found and the application for industrial use can be developed. For some materials, the data sets of adsorption of pure gases are widely available in literature. These data sets are usually used to develop isothermal adsorption models, which can be combined to describe the adsorption process of a gas mixture. Experimental gas mixture data sets can be used to validate those equations.
Kinetic experiments are important to understand the behaviour of adsorption. The diffusion process in porous structure of a sample determines the kinetics of the adsorption process[5]. The main coefficients used to describe this process are the diffusion coefficient D (cm2s-1) and the effective diffusion coefficient Deff (s-1). D is the diffusion coefficient of gas or liquid filling the pores of the solid. Deff is the effective diffusion coefficient for transport through the pores. To represent experimental data the fractional uptake F is usually used and plotted over the elapsed time t (s). F is described by equation 2.1
3 ( ) m t F m , (2.1)
where m(t) (g) is the time depending adsorbed mass.
From experimental kinetic data, many models can be used to estimate diffusion parameters. Considering isothermal conditions and a constant surface concentration, it’s possible to derive the diffusion equation from the Fick’s second law. For very small values of t, the relationship between the effective diffusivity Deff, time t, gas uptake m(t) and total gas uptake m∞ can be approximated with the isothermal model by Carslaw and Jaeger 1959[8],
eff
( ) 6
m t
D t
m . (2.2)
With equation 2.2 and experimental data the effective diffusion coefficient can be obtained, which is related to the diffusion coefficient according the relation of Smith and Williams 1984[8],
eff 2 0 D D r , (2.3)
where r0 (m) is the diffusion path length also considered equal to the particle radius.
Considering a non isothermal model and trying to fitting all the data set, it’s possible to use the more complex Kocirik model
For the definition of all the parameter please refer to the Kocirik et al.[9].
2 * n 2 eff n 2 2 n p * n =1 n n n * * 2 4 * n n n n n 9 1 exp ( ) 1 1 3 3 cot 1 2 2 D Y q t q r m t F m Y q q q q
. (2.4)4
2.2. Magnetic Suspension Balance
A magnetic suspension balance (MSB) is a balance able to weight samples contactlessly, using a magnetic suspension coupling. Without a mechanical transmission, the balance is isolated from the measuring cell. This property allows the control of the measuring environments (wide ranges of temperature, pressure and density, and corrosive gasses or fluid) as well as getting high accuracy measurement without damaging the balance. The uses of a MSB are various, from the determination of the transport quantity and state quantity, to the investigation of chemical reaction or simulation of processes. The main uses, also the objects of this work, are the determination of the density of a fluid, the calculation of the adsorption of a gas on a solid sample, and the investigation of the force transmission error (FTE) of the balance. For these purposes, the MSB has to be used as a gravimetric sorption analyser, also called “tandem sinker densimeter”.
An overview of the different applications (single sinker or double sinker) and their possible uses can be found in the paper of Wagner and Kleinrahm (2004)[1].
The MSB is a relatively young technology. The first commercialized MSB was developed in 1969 by Gast[10] at the Technical University of Berlin. This one allowed only to measure from vacuum to environment pressure. Afterwards a new MSB for higher pressure were developed in 1984 at NIST in Bolder (US) and at the Ruhr University of Bochum (RUB, Germany). The first application as a gravimetric sorption analyser was suggested by Dreisbach and Lösch (2000)[11].
The MSBs used in this work are advanced instruments developed in RUB[12] and commercialized by the company Rubotherm[13] since 1999. They are able to calculate the adsorption of a gas on the porous sample inside a basket, and at the same time to determine the density of the fluid used. In the configuration with two density sinker (upper sinker, bottom sinker) they are used for the investigation of the FTE and the density measurement.
The main parts of the balance are shown in figure 2.1. An electromagnet is connected to the bottom of a micro-balance, both surrounded by atmosphere environment. A permanent magnet is suspended in the measuring cell and its position is regulated by a position sensor that modulates the voltage. A lifting rod allows to pick up the two sinkers for measuring its weight. Three different weighing position listed below are possible:
5
ZP, zero-point position, allow to weigh only the permanent magnet with its lifting rod.
MP1, measurement-point 1 position, allow to lift and weigh the sinker 1 (bottom sinker) and the lifting rod.
MP2, measurement-point 2 position, allow to add the sinker 2 (upper sinker) at the MP1 measurement.
Figure 2.1. – Two-sinker densimeter and its parts. In yellow are marked the objects lifted in measuring positions MP1 and MP2.
6
2.3. Force Transmission Error Analysis
In gravimetric sorption analyser and magnetic suspension balance in general, a systematic error occurs due to nearby magnetic materials, external magnetic fields and magnetic susceptibility of the fluid used. This error is the so called Force Transmission Error (FTE). Correcting FTE arises to a more accurate measurement in MSB.
FTE in two-sinker densimeter was already investigated in previous works and the order of magnitude of the coefficients that characterize this error are already known[2]. For this reason, the FTE analysis can be a valid method to improve and set up new apparatus. Considering a new tandem-sinker densimeter apparatus, it’s possible to conduct measurements using two density sinkers, one in upper and another in lower positions. With those sinkers the adsorption contributes to weighing can be neglected. The FTE coefficients calculated are useful comparing indexes to understand which improvements the apparatus requires. And the calculation of the corrected value of density can be the index of accuracy of the apparatus. Applying FTE analysis is a relative new challenge. In literature there are not so much data about its application and even less about the application of FTE to other kind of measurements, like adsorption.
This chapter is mainly based on the paper written by Yang et al.[3] where the following conclusions are reached. The FTE analysis must be applied to every apparatus and its effect in density measurements can be calculated and compensated for different pure fluids. Was also verified the hypothesis that FTE can be divided into two components: the apparatus contribution and the fluid-specific contribution. The apparatus contribution is related to the coupling house, and it was investigated with vacuum measurements. The apparatus-specific constant is proportional to the fluid density and to the specific magnetic susceptibility, and it was investigated with synthetic air measurements. Both contributions are independent to the temperature in the already investigated range at least from 293,15 to 333,15 K. According to Yang et al.[3] the expanded uncertainty in density of the measuring system was also investigated, which was 0,0002·ρ.
The main goal of the FTE analysis applied to a tandem-sinker densimeter is to calculate the corrected density of a fluid and its deviation from the density calculated with the equation of state,
ρEOS. The density can be estimated for both the upper sinker and the bottom sinker, ρ (kg·m–3), with
7
MP1 ZP
Vac
MP1 ZP
fluid 1 1 1 1000 W W W W V , (2.5)
MP2 MP1
Vac
MP2 MP1
fluid 2 2 2 1000 W W W W V . (2.6)Where W (g) is the weight acquired during the measurements in the different measurements positions (ZP, MP1, MP2). The subscripts ‘Vac’ and ‘fluid’ have the obvious meaning of vacuum measurement and fluid measurements. The subscript ‘1’ refer to the calculation of difference of weight between MP1 and ZP, and the subscript ‘2’ to the one between MP2 and MP1. V (m3) is volume, α is the balance calibration factor and ϕ is the coupling factor, which accounts for FTE.
The volumes in equations 2.5 and 2.6 are the values corrected considering temperature and pressure of the data set used, according to the following equation
0 sinker 0 L 0 0 1 , 1 3 T T V T p V T T p p K T , (2.7)where V0 is the volume of the sinker at reference state (p0 = 0,10135 MPa and T0 = 293,15 K), 0 L
T T
is the average value of the linear thermal expansion coefficient in the temperature range from T0 to
T, and K(T) is the isothermal compression module. For calculation of V1, the volumes of the bottom sinker Vs1, the lifting rod Vrod and the linked part at the permanent magnet Vp-mag were considered. For calculation of V2 only the volume of the upper sinker Vs2 was considered. Those volumes are the ones obtained from the volume calibration explained forward in chapter 3.1.2 and 3.2.2.
The balance calibration factor, α, is required to supply at the error made during the internal calibration of the balance. Usually the balance is calibrated in air with the calibration masses but the sinkers are weighed in a different fluid. For a typical sea-level air density of ρair ≈ 1,2 kg·m–3 and a density of the stainless steel calibration masses of ρcalib ≈ 8000 kg·m–3, the balance calibration factor is defined by
1 air calib 1 / 1,00015 . (2.8)The coupling factor, ϕ, represents the efficiency of the magnetic coupling and it’s the heart of the FTE analysis. This factor can be separate into two contributions, the apparatus contribution and the fluid-specific contribution. Only the average values of the coupling factors calculated between weighing position have meaning, because the useful information is the weighing difference. So, ϕ1,
8
is the coupling factor considering the difference of weight between MP1 and ZP, and ϕ2, is the one considering the difference of weight between MP2 and MP1. The coupling factor is defined by
S EOS Vac ρ S0 0 X X
, (2.9)where ϕVac is related to the apparatus contribution and is determined with vacuum measurements. The other term, also called εfse, where the subscript ‘fse’ is short for ‘fluid-specific effect’, is related to the fluid-specific contribution and is determined with fluid measurements.
The apparatus contribution of the coupling factor is defined by ϕVac = εVac + 1, where εVac is the first FTE correction factor. This factor usually has a magnitude of about ± 0,000 020 informally wrote as ± 20 ppm. For both the upper and the lower sinkers the factor is defined by
MP1 ZP
Vac 1 Vac,1 1 W W m m , (2.10)
MP2 MP1
Vac 2 Vac,2 2 W W m m . (2.11)The mass m1 is composed by all the masses lifted during the switching from ZP to MP1, mass of the bottom sinker ms1, mass of the lifting rod mrod, and the mass of the linked part at the permanent magnet mp–mag. The mass m2 is equal to the mass of the upper sinker ms2, the only one lifted during the switching from MP1 to MP2. Those masses are the ones obtained from the mass calibration explained forward in chapters 3.1.2 and 3.2.2.
The second correction factor is εfse, the second term of equation 2.9, and is defined by
S EOS fse ρ S0 0 X X
, (2.12)XS is the magnetic susceptibility of the fluid used and XS0 is a reference constant value of 10–8
m3kg–1. ρ0 is a reference constant density of 1000 kg·m–3, and ρ (kg·m–3) is the mean value of the density of the inner parts of the balance considered according to the following equations:
s1 ROD p-mag 1 s1 ROD p-mag 1000m m m V V V , (2.13) s2 2 s2 1000m V
. (2.14)9
In equations 2.9 and 2.12, ερ is the apparatus specific constant and is determined by conducting fluid measurements. Usually to determine this constant the synthetic air is used because of its large value of the magnetic susceptibility that leads to a big sensitivity to determine the value, and because the density of synthetic air is well known. The value of the apparatus specific constant is about ± 40 ppm, and for both the differences of weight is defined by
MP1 ZP fluid
S0 0 ρ,1 Vac,1 1 EOS 1/1000 S EOS W W X m V X , (2.15)
MP2 MP1 fluid S0 0 ρ,2 Vac,2 2 EOS 2/1000 S EOS W W X m V X , (2.16)Once calculated εVac with vacuum measurements and ερ with synthetic air measurements, it’s possible to calculate εfse with fluid measurements to reach the experimental density of this fluid. The verification of the uncertainty of the apparatus is done comparing the density calculated with the FTE analysis and the one obtained by the equation of state, ρEOS. Usually for this verification nitrogen is used because it has a well known equation of state. The deviation of the density and so the accuracy of the apparatus in density measurements is determined by
EOS EOS
. (2.17)Using two density sinkers both in upper and lower position, it’s possible to estimate this value for both the differences of weight. Generally, the one calculated with the difference between MP2 and MP1 is the most accurate.
10
3. IMPROVEMENT, CALIBRATION AND SOFTWARE DEVELOPMENT
FOR EXPERIMENTAL SETUP
3.1. ELLI Densimeter Apparatus
The apparatus described in this chapter is the so called ELLI densimeter. This densimeter was already a working apparatus used for education and experimental practice in density measurements for the students of Ruhr University of Bochum. For this reason, the apparatus was a very simply one with a high uncertainty.
In this chapter, the modifications, the calibrations, and the software development applied to the apparatus to reach a low uncertainty in density measurements are shown. After these works, the ELLI densimeter will be possible to use for the scientific research.
3.1.1. Modifications
A commercially available magnetic suspension balance (MSB, Rubotherm, Bochum, Germany) was used in this apparatus. The measuring cell was completely metal sealed and its temperature was regulated by a heater and a circulating bath (FP40-ME, Julabo, Germany), see figure 3.2. The temperature was measured by a main thermometer (PT-100 4 wires, Ifm electronic, Germany) directly connected to the apparatus. The pressure was measured with a pressure transducer (PMP 50G6, General Electric, USA) with high uncertainty for the control of the system. Two-stage rotary vane pump (vacuum pump, RZ 2.5, vacuubrand, Germany) was used for the vacuum.
In figure 3.1 the concept scheme of the apparatus is shown. Is possible to observe the already existing pressure control system, in grey, composed by three electric valves and a pressure transducer. At this one was added a system of three manual valves. This modification allows a faster load and unload of the gas, useful in case of kinetic measurements or just for saving time. Then a new comparison thermometer (PT100 Probe, P-M-1/10-3-50-0-TS-2, OMEGA, UK) was installed and its resistance was measured by a data acquisition (DAQ) system (Keysight, 34972A, DataTec, Germany) through a multiplexer board (20 channel multiplexer, 34901A, DataTec, Germany). This thermometer was apply only for an additional indication of the temperature, and not for the control of the apparatus or to analyse the data. An important improvement for the apparatus was the setup of a new pressure transducer (P-30, 0...100 bar, WIKA, Germany) with a lower uncertainty (u(p) = 0,05 % full scale) that measured the value of the pressure inside the measuring cell. This pressure transducer was connected to the DAQ through the multiplexer board.
11 Figure 3.1. – Concept scheme of the apparatus.
12
Another important step in the setup of the densimeter was an accurate alignment of the balance. Not only the balance has to be planar at the floor, but more important, the sinkers and the lifting system must be as centred as possible during the MP1 and MP2 measurements. In this way is possible to avoid most of the bad readings of the weight.
Due to the using of a magnetic coupling, a MSB is also sensitive to external magnetic fields or big metal objects. Hence it's important to take care about the location of the apparatus and the influence that some object have on the reading of the balance. In particular, it is acceptable if something influences the measurement in the same way on the three different positions of the balance. During the analyse of the data only the differences of the weight are considered and so this influence does not have any consequence.
For the ELLI densimeter apparatus only moving the cylinder of gas far away of the balance were necessary, after a notice of a big influence on the measurements.
13
3.1.2. Calibrations
Mass Calibration
An important step for the application of a gravimetric sorption analyser is the mass and volume calibrations of the sinkers; both values are necessary to calculate experimental density.
For mass calibration, an analytical balance (Cubis®, MSE225S-000-DU, Sartorius, Germany) was used. A datalogger (Opus 20 THIP, Lufft, Germany), which is composed of an electronic barometer, a temperature transmitter, and a humidity transmitter, was located next to the balance. The datalogger measured properties of the surrounding environment (temperature T, pressure p, and humidity φ) of the balance; these properties were necessary to calculate the air density of the room. The air density was used for the correction of the mass measured with the balance due to buoyancy effect.
The air density of the room can be calculated by:
s air 3 293 15K 1 188 1 0 378 1 273 15K p T kg p , , , m bar T , p , [14] (3.1)where ps(T) is the vapour pressure of water in room environment, which can be estimated by:
18 285 1 29115K s 20 6 1000 , , T , mbar p T bar e . (3.2)Due to the effect of the buoyancy force of the air, the value read from the balance Wread is different from the real mass mreal of the sinker. The relation between these two values is[15]:
air M
real read calib meas read
calib M air 1 m W W , (3.3)
where the subscript ‘calib’ is short for ‘calibration’, which represents the calibration masses inside the balance, normally made of stainless steel (ρ ≈ 8000 kg/m3). The subscript M represent the material of the object to be measured. Please refer to Appendix A.1 for the detailed derivation of equation 3.3.
The process of mass calibration was done first of all by measuring the properties of the atmosphere to determine the air density. Then, after the balance internal calibration, approximately ten cycles of loading and unloading of the sample were conducted and the average results were
14
obtained. The results of mass calibration of the sinkers, lifting rod, and permanent magnet (p-magnet) are summarized in Table 3.1.
Table 3.1. – Results of the mass calibration.
Object Material ρM Average ρair Average Wread mreal
/[kg/m3] /[kg/m3] /[g] /[g]
sinker 1 Ti 4506 1,17219 32,62381 32,62752
sinker 2 Ti 4506 1,17513 19,63584 19,63808
lifting rod SS 8000 1,17242 1,07688 1,07688
15
Volume Calibration
For the volume calibration, a hydrostatic comparator was used, which is similar to the one used by McLinden and Splett (2008) in NIST[16]. The main component of the comparator is a thermostated fluid bath, which is a triple walled glass cylinder. The outer layer was evacuated for thermal insulation. The middle layer contained water which was circulated by an external circulating bath. The inner volume was filled with the hydrostatic fluid (hydrofluoroether, C7F15OC2H5, 3M Novec High-Tech Flüssigkeit, io·li·tec, Germany) and contained a “stage” for placing the submerged objects. The hydrofluoroether was used as hydrostatic fluid instead of water, because of the following reasons. (1) The relatively high density (1,61 g/cm3) of this fluid increases the buoyancy force on the measured objects, which increases the sensitivity of the volume calibration. (2) The lower surface tension decreases the forces on the suspension wire. (3) Hydrofluoroether has a much higher gas solubility, which reduces the content of air bubbles clinging to the objects. A weighing pan was suspended in the fluid from an analytical balance (Cubis®, MSE225S-000-DU, Sartorius, Germany) via a stainless steel wire. The temperature of the fluid was measured with a standard reference thermometer (Pt25, model 162C, Rosemount, USA) connected to an AC Resistance Bridge (Type F17, ASL, UK). A datalogger (Opus 20 THIP, Lufft, Germany) was located next to the balance to measure the properties of the atmosphere of the room; these properties were used for the calculation of the air density to correct the balance reading, as explained in section Mass Calibration. Detailed information of the calibration procedure and the used equations is presented in the following paragraphs.
Without an object on the pan, the reading of the balance Wpan is:
airpan pan fluid pan
calib 1 W m V / . (3.4)
With an object “1” on the pan (e.g. a sinker made of silica with well calibrated volume), the reading of the balance Wpan+1 becomes:
airpan +1 pan 1 fluid pan 1
calib 1 W m m V V / , (3.5)
where the value of m1 is known or comes from the mass calibration. The (1−ρair/ρcalib) term is the correction due to the air buoyancy. This term is added because the balance is calibrated in air but the objects are suspended inside fluid. ρcalib is the density of the stainless steel (ρ ≈ 8000 kg/m3),
16
which is the material of the internal calibration masses of the balance. The weight of the object “1”,
W1, can be reached by subtracting equation (3.4) from equation (3.5):
air 1 1 fluid 1 calib 1 W m V / . (3.6)The terms of the pan disappear in equation (3.6), therefore, it is not necessary to know the mass and the volume of the pan. Applying the same equations to another object “2” (e.g. the sinker to be calibrated) then the value of V2/V1 can be calculated by:
2 2 air calib 2 1 1 1 air calib 1 1 m W / V V m W / . (3.7)The calibrated volume is normally presented as a value at the reference temperature T = 293,15 K, however, the calibration can hardly be conducted at exactly T = 293,15 K. Therefore, the correction of the volume due to the temperature difference should be applied with
meas 1 3 L
293 15
V T V T , , (3.8)
where, αL is the thermal expansion coefficient. For an object, made of silicon, the thermal expansion coefficient αL is 2,56·10−6 K−1. For a titanium sinker, αL is 8,6·10−6 K−1.
The measurement scheme started with the internal calibration and the tare of the balance. Then the environment properties were measured and used for the air density calculation with equation (3.1). Later, a series of measurements were conducted, in the order of:
A – B – A – C – A – B – A – C – A.
Letter A indicates the measurement of the pan, i.e. without any object on it. Letter B indicates the measurement of the standard silicon object located on the pan. The volume and the mass of the standard sinker were known. Letter C includes the weight of the unknown object and the pan.
Since the measuring environment is not much stable, for each weighing (A, B, or C), after a waiting of approximately 8 minutes, 5 values of weighing were recorded in an interval of 5 seconds and averaged. Meanwhile, the temperature of the hydrostatic fluid was recorded. At the end of a measurement, the environment properties were measured again to update the air density. All these data were collected in an excel file.
Using the mass calibration data was possible to obtain a well approximated value of the mass (m0) of a sinker, assuming a reference value of the density (ρ0). Using the volume calibration data was possible to obtain a well approximated value of the volume of a sinker (V0), considering the
17
founded value m0. By the ratio m0/V0 was therefore possible to calculate a better value of the density (ρ1), which substituted at the one assumed in the mass calibration file allowed to find a better second approximation of the mass (m1). A second approximation of the volume (V1) could be also found. It is possible to repeat this iteration until the change in density is less than 1,0 kg/m3. Hence the final values founded are the best approximation of the volume and the mass of a sinker.
The volume calibration was conducted with both upper sinker and bottom sinker. The calibrated results and information about the standard sinker are listed in table 3.2. The volume calibration was not conducted on the other components of the MSB. The volumes of these components were calculated by the ratio between mass and density. The uncertainty due to this rough estimation is negligible.
Table 3.2. – Results of the volume calibration and the correction of densities and masses.
sinker material m V ρ
/[g] /[cm3] /[kg/m3] standard silicon 98,627306 42,345925 2329 bottom titanium 32,627518 7,238767 4507 upper titanium 19,638088 4,367250 4497
18
Temperature Calibration
The temperature calibration was conducted with the temperature measurements of a thermostat with both the thermometer to be calibrated, Tmeas, and a standard reference thermometer (Pt25, model 162C, Rosemount, USA) Tstd. The measurement results are listed in table 3.3. The thermostat is a heating circulator (F25-ME, Julabo, Germany). The standard reference thermometer was connected to an AC Resistance Bridge (Type F17, from ASL, UK) to measure the resistance. The temperature measurements were conducted at T = (increasing from 0, 20, 40, 60, to 80 °C and then decreasing from 80, 70, 50, 30, 10, to 0 °C) considering the mean value of a time interval of about 3-4 minutes. In this manner, the reproducibility of the measurement was checked. The relation of Tmeas-Tstd vs. Tmeas was correlated at a quadratic equation by Matlab, and the equation will be used to correct the measurement of the thermometer.
Table 3.3. – Data and results of the temperature calibration.
Tstd /[°C] Tmeas /[°C] Tmeas−Tstd /[°C] −0,011 0,99 1,001 19,679 20,75 1,071 39,667 40,81 1,143 59,873 61,08 1,207 79,844 81,10 1,256 79,812 81,06 1,248 69,640 70,88 1,240 49,678 50,86 1,182 29,689 30,80 1,111 9,701 10,73 1,029 −0,019 0,98 0,999
Figure 3.3. – Behaviour of the error of measurement with the temperature.
19
20
3.1.3. Software Development
LabVIEW
The software development environment LabVIEW (LabVIEW 2014) by National Instrument was used to control the ELLI densimeter apparatus. LabVIEW is intuitive and user friendly as it is based on a graphical programming interface which is commonly used for data acquisition, instrument control, and industrial automation. Programming in LabVIEW means to use function-nodes, displayed as graphic symbols, and connect each other with wires. These wires propagate the variables and the nodes can only be executed when all required input variables are available. The code is then executed in order of the data flow and not from the first line to the last line like in text-based programming language. So, it is possible to have a clean programming interface where the data flow is clear to the user.
Programs encoded in LabVIEW are called Virtual Instruments (VIs). Other programs can be encoded and recalled in the main VI to keep the main working panel cleaner, to separate different functions and to minimize recurring code. These other programs are called subVIs. Each VI consists of a block diagram and a front panel. The front panel is the graphical user interface built using controls (inputs) and indicators (outputs). These objects will also appear in the block
diagram as terminals. The block diagram contains the graphical source code which consists of the
structures and functions performing operations on the controls, supplying of data to indicators, and communicating with connected devices.
For full automation, the various devices connected to the apparatus must be controlled with LabVIEW. First, the connection of each device with the PC has to be tested by using a command line program like NI–MAX (National Instrument – Measurement & Automation Explorer). Second, communication is implemented in LabVIEW with steps as such initializing, configuring, sending commands and analysing responses, and closing of communication. These steps can be done with the proper commands available in the driver of the instrument, or with the commands of NI–VISA (National Instrument – Virtual Instrument Software Architecture), a universal software standard for programming and configuring of serial interface.
For the ELLI densimeter the already existing program was based on an Event Driven State
Machine architecture. In this kind of architecture the sequence of operations is generally predefined
and enclosed in the state machine queue. A queue is a list of elements of the same data type to be executed by the program. However, it is possible to interrupt the data flow and execute the user inputs immediately. In programming languages a state machine is a case structure inside a while
21
control flow statement that allows code to be execute continuously based on a given Boolean condition like a stop button. The case structure contains the states of the state machine and executes one of these when the structure executes according to the value wired to the selector terminal. The event structure can be programmed to capture defined user interaction like selecting a button or a value changing in a numeric control. When the event structure notices this interaction, it appends or removes the appropriate elements to the beginning or the end of the queue of the state machine.
Figure 3.4. – Block diagram of the main VI. The upper part is the event structure inside the while loop. The bottom part is the state machine, a case structure inside a while loop.
22
Block diagram
In the following paragraphs the structure of the LabVIEW code, and the changes needed to integrate the modification and calibration, described in chapters 3.1.1 and 3.1.2, are presented. The
block diagram of the main VI contains the structure of the event driven state machine, see figure
3.4.
The upper part of the block diagram is the ‘event driven’ control or producer loop, and the
event structure contains the events that could happen during execution of the code. The “Timeout”
event occurs when nothing happens and so the queue of the state machine remains empty. In this case, the “Read system status” state is added continuously to the queue and then executed in the
consumer loop until an event is registered. The other events are value changes in the flowsheet on
the front panel. In all these cases the event structure appends the corresponding status to the queue to be execute by the consumer loop. For example, if the set point of the temperature is changed the
event structure appends the “Flowsheet – Control” state to the queue. If the value of the stop button
is changed the event structure appends “Stop” state to the queue.
The bottom part of the block diagram is the consumer loop, and the case structure contains the possible states to be executed. The “initialize” state contains a subVI for starting the communication with all the devices connected to the apparatus. The “Read system status” state reads the status of the system and the response of the devices, and updates the front panel with the last values read, see figure 3.4. The “Flowsheet – Control” state applies changes to the corresponding components if something is changed in the inputs on the flowsheet of the front
panel. It also checks the state of the set points and the temperature and pressure control. The “Stop”
state closes the communication with all the devices and then stops the software.
Front panel
The front panel of the main VI contains all controls and indicators useful to control the apparatus including a graph and a log, see figure 3.5. In this window is possible to change the set points to control temperature and pressure. A 2D array and a radio button allow to control the timing and the position of the balance. There are also buttons to evacuate the measuring cell, to save a data file with a name given by the user, to clear the graph, to tare the balance, and to stop the software. These are all the events that could be happen and change the software status.
The numeric indicators allow to check the current values of temperature and pressure, for both the thermometers and both the pressure transducers. Another numeric indicator show the weight measurement of the balance. A graph plots the behaviour of the three main variables of the system, and a log lists the actions made by the user.
23
Figure 3.5. – Complete front panel of the main VI. Controls, buttons and the balance timing and positioning are the inputs of the system. Indicators, graph and the log are the outputs of the system.
Modifications
As explained in chapter 3.1.1, some modification are required to reduce the uncertainty of the apparatus. These modification needs to be implemented also in LabVIEW.
The communication with the DAQ serial interface was added using the commands of the driver
Agilent 34970 available publicly by Keysight[17]. The initialization and the closing of the communication were implemented both in a subVI, see figure 3.7, within the “Initialize” and the “Stop” states of the state machine, see figure 3.6, with the commands of all the other devices.
Figure 3.6. – “Initialize” and “Stop” states in the the stopping of the devices are marked with a red circle.
24
“Initialize” and “Stop” states in the case structure of the state machine. The subVIs for the initialization and the stopping of the devices are marked with a red circle.
(a)
(b)
Figure 3.7. – SubVIs for the initializing with a red circle.
The control of the DAQ was added as a separate VI, status” state in the subVI, see
3.8(b). To integrate the new comparison thermometer the command of the used. The value obtained was then converted in a temperature value using both a
the calibration coefficients. To integrate the low uncertainty pressure transmitter, the command of the direct current reading was used. The value obtained was then converted in a pressure value using both a formula node and the calibration coefficients provided by the company
25
SubVIs for the initializing (a) and the stopping (b) of the devices. The commands of the DAQ are marked
The control of the DAQ was added as a separate VI, see figure 3.8(c), inside the “Read system see figure 3.8(a), where all the other devices are controlled,
. To integrate the new comparison thermometer the command of the resistance reading used. The value obtained was then converted in a temperature value using both a
To integrate the low uncertainty pressure transmitter, the command of was used. The value obtained was then converted in a pressure value and the calibration coefficients provided by the company
of the devices. The commands of the DAQ are marked
, inside the “Read system devices are controlled, see figure
resistance reading was
used. The value obtained was then converted in a temperature value using both a formula node and To integrate the low uncertainty pressure transmitter, the command of was used. The value obtained was then converted in a pressure value and the calibration coefficients provided by the company WIKA.
Figure 3.8. – (a) “Read system status” state in the devices and the new two outputs are marked. and the new outputs are marked. (c) resistance are marked.
(a
)
(b
)
26
“Read system status” state in the case structure of the state machine. The subVI for the control of the devices and the new two outputs are marked. (b) SubVI for the control of the devices. The VI for the control of the DAQ (c) VI for the control of the DAQ. The commands used to read the current and the
(c
)
of the state machine. The subVI for the control of the SubVI for the control of the devices. The VI for the control of the DAQ VI for the control of the DAQ. The commands used to read the current and the
As it is possible to observe in figure outputs. These numeric indicators from the communication with the DAQ.
Figure 3.9. – Numeric indicators added to the
The data file writing system was then modified to allow writing of these new parameter text file with all other parameters. So, the header of the file was changed,
correspond with the order of data acquisition, analysis of the experiments data, using the program
(a)
(b)
Figure 3.10. – (a) Data file header. The new data acquired are marked. acquired are marked.
27
possible to observe in figure 3.8(a), two numeric indicators were linked at the two new
ators appear also in the front panel and show temperature and pressure
from the communication with the DAQ.
Numeric indicators added to the front panel.
The data file writing system was then modified to allow writing of these new parameter text file with all other parameters. So, the header of the file was changed, see
correspond with the order of data acquisition, see figure 3.10(b). This modification permits the analysis of the experiments data, using the program MATLAB.
Data file header. The new data acquired are marked. (b) Data file writing system. The new data were linked at the two new appear also in the front panel and show temperature and pressure
The data file writing system was then modified to allow writing of these new parameters in a see figure 3.10(a), to . This modification permits the
28
MATLAB
The software MATLAB (Matrix Laboratory, R2017b) by MathWorks was used to analyse the data collected with LabVIEW during the measurements. MATLAB is a numerical computing environment based on matrix and array manipulation. Its own text programming language allows to do numerical analysis, to plot functions and data, to implement algorithms, and to interface with programs written in other languages.
MATLAB is mainly composed by a user interface and a text editor. The text editor allows to write and run the code, and to visualize tables of data from different kind of files. The text code is executed from the first line to the last line according to the functions, the instructions, and the loops contained in it. The user interface is composed by a workspace, where all the variables used are shown, and a command window, where the results of the numerical analysis are shown. The graphics are plotted in other temporary windows.
For the analysis of the data of the ELLI densimeter apparatus four main codes were written, based on the FTE equations shown in the chapter 2.3 and on the parameters calculated by the calibrations in chapter 3.1.2. These codes allow to read a set of data collected with LabVIEW, and to calculate the FTE parameters and the corrected value of the density of a fluid. In the following figure 3.11 the concept scheme of the MATLAB codes is shown, and in the following paragraphs these codes, that can be found in appendix A.2, are explained.
29
Figure 3.11. – Concept scheme of the MATLAB codes for the ELLI densimeter apparatus.
The first code (MSB_MEAS.m) analyses the sets of data collected with LabVIEW and creates a data file useful for the others codes. It is encoded in two versions, the first one was able to manipulate the data before the modification of the LabVIEW code, and so with the measure of the old, high uncertainty, pressure transducer. The second one is able to analyse data from the upgraded and completed LabVIEW code. The output data file from LabVIEW contains many arrays. The main ones used in those MATLAB codes are the temperature, the pressure, the weight measured by the MSB, and the measuring position of the MSB. This position defines the cyclic repetition of the measurements and it’s collected as a number (ZP = 1, MP1 = 2, MP2 = 3).
After reading the data file, the code selects the useful data, usually the last 30% of the total, and counts how many values are in each measuring position. Then the measured weighs are analysed and plotted in a graph, so the user can see and select if there are some bad points. Bad points occur when there is a disturb during the measurements, or the alignment of the balance is not perfect. These are easily recognizable on a graphic because they hugely deviate from the other measurements, see figure 3.12. A good set of data should have a deviation of weigh measurements around 10 ppm.
30
Figure 3.12. – Bad point in MP1 measurements. Data set ‘2018-02-21 12.12.22 - FTE_nitrogen_Ta80b .txt’, nitrogen at environment temperature and pressure of 80 bar.
Figure 3.13. – Behaviour of the temperature and the pressure. Data set ‘2018-02-21 12.12.22 - FTE_nitrogen_Ta80b .txt’, nitrogen at environment temperature and pressure of 80 bar.
31
Once the bad points were selected, the code analyses the data again without bad points. The values of the temperature array is then corrected with an external function according to the temperature calibration showed in chapter 3.1.2. The weigh arrays are plotted again to see the difference after the correction of the bad points. Also a temperature and pressure graphic is plotted, see figure 3.13, to check the constancy of these variables. A good set of data should have a deviation of temperature measurements less than 0,01.
At the end of the first code a data file is written. This one contains the arrays with the mean value per cycle of the temperature, the pressure, and the weigh at each measurements position (ZP, MP1, MP2). This file will be used in the other codes to calculate the FTE parameters and the density of the fluid used.
The second MATLAB code (EPS_VAC.m) is dedicated to the vacuum measurements and it calculates the vacuum parameters of the FTE, like εVac and φVac. After reading the data file generated by the previous code, the mass parameters from the mass calibration are specified. Then the equations to calculate the FTE vacuum parameters are defined, both for the upper sinker and the bottom sinker. At the end, a text file is written with those results.
The third MATLAB code (EPS_FLUID.m) is dedicated to the fluid measurements and it calculates the fluid parameters of the FTE, like ερ and εfse. After the reading of the data file, the pressure is corrected with the offset found in vacuum measurements. Then the volume parameters from the volume calibration are specified. Also other parameters required for the next steps are specified, like the magnetic susceptibility, the molecular weight, and the mole composition, depending to which fluid is used. With those parameters the density from the equation of state is calculated by using an external function called ‘Trend’. Trend is a free library of equations of state for many pure fluids and mixtures developed by the chair of thermodynamic of the Ruhr University of Bochum. The next step of the code is to correct the volume of the sinkers with the temperature of the set of data considered, this using an external function. Then the equations to reach the FTE fluid parameters are defined, both for the upper sinker and the bottom sinker. At the end, a text file is written with those results.
The last MATLAB code (DEN.m) collects all the parameters found in the other codes, and read both a vacuum and a fluid set of data. With those it is able to calculate the density of the considered fluid according to the FTE equations, and to compare it with the ρEOS through calculating the deviation. At the end, a text file is written with those results.
32
3.2. OXYFLAME Sorption Apparatus
The apparatus described in this chapter is the so called OXYFLAME sorption apparatus. This densimeter was a new apparatus based on a standard gravimetric sorption analyser model. It was born for the purpose to do kinetic measurements for the investigation of adsorption.
In this chapter, the modifications, the calibrations, and the software development applied to the apparatus to reach a low uncertainty in density and sorption measurements are shown. After these works, the OXYFLAME densimeter will be possible to use for the scientific research.
3.2.1. Modifications
The OXYFLAME sorption apparatus was similar to the ELLI densimeter apparatus and it used the same model of a MSB by Rubotherm, see figure 3.2. The temperature inside the measuring cell was regulated by a heater and a circulating bath (Unistat 510, 1005.0082.01, Huber, Germany), and was measured by a main thermometer (PT-100 4 wires, TMH, Germany) directly connected to the apparatus. Another circulating bath (Ministat, 28985/94, Huber, Germany) was used and settled constantly to 20 °C only for the cooling of the bottom part of the micro-balance. This allow to adopt high temperature inside the measuring cell avoiding that the heat reaches and damages the balance.
A vessel for the storage of the gas was installed between the gas cylinder and the cell, see figure 3.14. The gas flow from the cylinder was regulated by a manual valve, an uncalibrated high uncertainty pressure transducer (KTE/KTU 6000 series, KTE6300GQ0, First Sensor, Germany) detected its pressure. The value obtained from this pressure transducer needed only for a manual control of the apparatus and was not used for the analysis of the data. A pneumatic valve allowed to open instantaneously the connection between the vessel and the measuring cell. It was controlled with compressed air by a relay (Relay Card, 393905, Conrad, Germany). This kind of pneumatic valve is used for kinetic measurements when a fast increase of pressure is required. A low uncertainty pressure transmitter (P-30, 0...100 bar, WIKA, Germany) measured the pressure directly inside the cell. Both the pressure transducers were connected to a DAQ system (Keysight, 34972A, DataTec, Germany) and their value measured through a multiplexer board (20 channel multiplexer, 34901A, DataTec, Germany). Two-stage rotary vane pump (vacuum pump, RZ 2.5, vacuubrand, Germany) was used for the vacuum.
33 Figure 3.14. – Concept scheme of the apparatus.
Like in the apparatus described in the previous chapters, was necessary to pay attention at the alignment of the balance and at the magnetic influence of the neighbour objects. So, before the closing of the measuring cell was checked that the basket in lower position and the sinker in upper position were as centred as possible during MP1 and MP2 measurements. Because the apparatus was located inside a ventilated box was also checked the influence of the position of the vertical door of the box. The result was that the position of the door really have some influence on weight measurements, but there was a fixed position where the influence was close to zero and equal for all the measurement positions. The successive measurements were, hence, conducted with this fixed position, paying attention to do not move the door during the collection of data.
34
3.2.2. Calibrations
As in the ELLI densimeter apparatus, calibrations were required to obtain more accurate measurements. In this chapter the results for the mass and volume calibrations of the objects weighed inside the measuring cell, and for the temperature calibration of the main thermometer are shown. The calibration of the pressure transducer was not necessary because it has been well calibrated by the producer company, same as that of the previous apparatus.
Once the FTE method was applied, the OXYFLAME apparatus was used for kinetic sorption measurements. This required the calibrations of the mass and the volume of the carbon sample inside the basket.
Mass Calibration
The mass calibration procedure was exactly the same as described in chapter 3.1.2 about the ELLI densimeter apparatus. Also the instrumentations and the equations used are the same, see Appendix A.1. The density sinker used in the previous apparatus was used also in the upper position of the measuring cell in the OXYFLAME apparatus. So, it was already calibrated, with mass and volume known.
The mass calibration was applied to two different baskets and on the other inner parts of the balance (the stainless steel lifting rod and stainless steel part connected at the permanent magnet). The first basket calibrated (basket 1) is simple, made of stainless steel, and used for the determination of the force transmission error and for the calculation of the density. The second basket (basket 2) have also a cap to cover it and a little blocking ring, and was calibrated all together. This one was used to keep the carbon sample during the adsorption measurements.
The complete set of data collected during the mass calibration are shown in Tables and Data chapter, while the results of the mass calibration are shown below in table 3.4.
Table 3.4. – Results of the mass calibration.
Object Material ρM Average ρair Average Wread mreal
/[kg/m3] /[kg/m3] /[g] /[g]
basket 1 SS 8000 1,19488 2,42113 2,42113
basket 2 SS 7896 1,19197 7,68241 7,68243
lifting rod SS 8000 1,19342 1,07760 1,07760
35
Volume Calibration
The volumes of the inner parts and the basket 1 didn’t require a calibration but were calculated with the simple ratio between the mass and the density, see table 3.5. The uncertainty due to this rough estimation is negligible considering the small value of the masses.
Table 3.5. – Volume calculated with the ration between mass and density.
Object Material ρM m V
/[kg/m3] /[g] /[cm3]
Basket 1 SS 8000 2,42113 0,30264
lifting rod SS 8000 1,07760 0,13470
p-magnet SS 8000 1,93996 0,24250
The hydrostatic comparator calibration showed in the previous chapter for the sinkers of the ELLI densimeter apparatus wasn’t applicable to the basket 2. This because there were difficulties to let the calibration fluid into the closed basket and difficulties to calibrate its parts separately. Another calibration technique was applied to the basket 2. It consists in conducting measurement with the apparatus itself and calculating the same equations for FTE and density determination. But instead of calculating the density of a gas using a known volume of the basket, the volume of the basket VB2 was calculated according to equation 3.9, using a well-known density of the fluid through the equation of state ρEOS. For this reason, it was decided to conduct measurement at a pressure of 80 bar and a temperature of 20 °C with nitrogen, where the EOS gives a low uncertainty value of the density.
B2 TOT ROD P-MAGN
V V V V . (3.9)
2MP1 ZP Vac MP1 ZP N TOT
EOS Vac,1 fse,1 1000 1 W W W W V
. (3.10)It’s important to note that εfse,1 is a function of VB2 itself, see equations 3.11-3.12. It necessary to assume a starting value for the volume and then conduct some iteration to reach the more accurate value searched.
36
S EOS 1
B2 fse,1 B2 1 S0 0 , V X f V X . (3.11) ROD P-MAGN B2 1 ROD P-MAGN B2 1000m m m V V V
. (3.12)For more accuracy, when a value of the volume is found, is possible to calculate a new value of the density of the basket 2 and substitute this one in the mass calibration file. A new value of the mass is so found and can be used to repeat the iteration for the volume calibration. It is possible to go on with this loop until the value of the mass change at a relative deviation of less than 1.0 ppm and so the values found are the more accurate possible, table 3.6.
Table 3.6. – Iterations and result of the volume calibration of the basket 2.
Iteration ρ /[kg/m3] m /[g] V /[cm3] σ(V) [%] 8000 7,682413 0,96030163 1 0,97300920 1,3294 2 0,97301010 0,0062 3 0,97301010 4 7895,5120 7,682428 0,97301200 5 0,97301199 0,0061 6 0,97301199 7 7895,5123 7,682428 0,97301199
To verify the goodness of the volume found with the calibration it’s possible to calculate the density and its deviation from EOS using this value. Another measurement was done with nitrogen at a pressure of 60 bar and a temperature of 20 °C. As shown in table 3.7 the result of this verification was satisfying.
Table 3.7. – Verification of the volume of the basket 2.
T mean /[K] p mean /[Mpa] ρEOS /[kg/m3] ρ10 /[kg/m3] σ(ρ) [%] 292,991 59,369 68,6375 68,6281 -0,0136
37
Temperature Calibration
The same method showed in chapter 3.1.2 was used for the temperature calibration of the main thermometer, such as the same instrumentations, and the same equations. Further were tested the difference of measurement of a complete sunk position and a not complete sunk position of the thermometer at 0 °C, see first value and last two values of the table 3.8. This because also in the apparatus the thermometer was not completely inserted.
As it’s possible to see in the following table and in the figure 3.15 that the difference between the two temperature measurements is very small and the trend it’s not regular. For this reason, it was decided to take into consideration a linear correlation for the trend of the data used for the small correction of the temperature.
Table 3.8. – Data and results of the temperature calibration.
Tstd Tmeas Tmeas−Tstd /[°C] /[°C] /[°C] −0,020 0,15 0,166 39,550 39,70 0,153 39,547 39,70 0,155 59,635 59,82 0,181 59,633 59,82 0,183 79,751 79,96 0,206 79,771 79,98 0,210 79,782 80,00 0,221 79,781 80,00 0,221 69,664 69,87 0,205 69,660 69,87 0,205 49,558 49,75 0,188 29,357 29,55 0,193 29,343 29,53 0,188 29,343 29,53 0,190 19,398 19,59 0,191 9,741 9,95 0,205 9,721 9,93 0,207 −0,019 0,13 0,146 −0,020 0,13 0,149
Figure 3.15. – Behaviour of the error of measurement with the temperature.
38
39
Sample Calibration
The calibrations of mass and volume of the sample were done in two different ways. Both the methods required measurements with the already settled up apparatus. Before the kinetic adsorption application of the apparatus, the two results were compared and the best one was chosen.
First Method
The first method required two measurements, a vacuum one at a T = 25 °C, and a helium one at
p = 80 bar and T = 25 °C. The vacuum measurement needed to calculate the mass of the sample,
mSAM, with the equations
SAM TOT B ROD P-MAGN
m m m m m , (3.13)
MP1 ZP
Vac TOT Vac W W m . (3.14)Where the parameters mB, mROD, mP-MAGN are the ones collected with the mass calibration, and α and ϕVac are the FTE parameters explained in chapter 2.3. Then, a first approximation value of the density of the system, ρ1, is calculated using the parameters found in the volume calibration sections (VB, VROD, VP-MAGN).
B ROD P-MAGN
1 B ROD P-MAGN 1000 m m m V V V . (3.15)It is possible to calculate the volume of the sample applying the same equations 3.9-3.10 used in the volume calibration section. Where the density of the fluid used was assumed equal to the one calculated with the equation of state, ρEOS. Differently from that case, the volume of the basket is known and in the weight measured by the balance is enclosed also the weight of the sample.
SAM TOT B ROD P-MAGN
V V V V V . (3.16)
MP1 ZP Vac MP1 ZP He TOT
EOS Vac,1 fse,1
1000 1 W W W W V . (3.17)
