Practical Issues Relating to Tank Volume Determination - American

Ind. Eng. Chem. Res. 2008, 47, 1533-1545

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PROCESS DESIGN AND CONTROL Practical Issues Relating to Tank Volume Determination Robert Binner,† John Howell,*,‡ Greet Janssens-Maenhout,§ Dieter Sellinschegg,† and Ke Zhao† Department of Safeguards, International Atomic Energy Agency, Vienna, Austria, Department of Mechanical Engineering, UniVersity of Glasgow, Glasgow G12 8QQ, United Kingdom, IPSC, Joint Research CentresEuropean Commission, 21020 Ispra, Italy, and Faculty of Engineering, Ghent UniVersity, B-9000 Ghent, Belgium

Issues relating to each element in the determination of a tank’s contents are examined: level/density measurement, calibration, and its reverification. Systematic errors associated with dip tube pressures that are measured to derive level/density are investigated, experimentally. The conversion from these pressures to level/density is then addressed, and the simplest form is found to be adequate. A revised regression model is proposed to obtain a calibration from biased calibration data from one or more calibration runs. Severely biased calibration data can be eliminated, reducing the calibration error due to biases to a minimum. It is shown that a more accurate representation can be obtained using cubic splines. Finally, reverification issues are addressed by proposing a new continuous fill approach. 1. Introduction Conventionally, the volume of a solution contained in a vessel is estimated on the basis of level, density, and temperature measurements. These measurements are combined to form a normalized level, which is input into a calibration equation to obtain a volume, which is subsequently temperature corrected. The calibration equation is obtained by pouring known quantities of calibration liquid into the vessel and measuring the observed liquid level after each increment. Issues relating to the various measurements and to the calibration procedure are of particular importance in nuclear fuel reprocessing plants because of the special nuclear materials involved. Although there is probably less concern in other industries, clearly accuracy might be an issue also. The generation of calibration data, under welldetermined conditions, is a costly and time-consuming exercise because it requires that a specific set of experiments should be performed on each vessel in turn during commissioning. This data must be reverified at intervals to ensure that it has not changed over time. Although, in general, calibrations are particularly important in reprocessing plants, accuracy and precision requirements still vary depending on the vessel’s use. Vessels are installed in reprocessing plants for at least four purposes: to provide temporary storage for solutions that are fed into, and exit out from, unit processes; to act as buffers to decouple, to some extent, the operation of the various stages; to provide long-term storage; and to provide a means of accounting for material that passes through the plant. The term tank is often used instead of vessel, to reflect the fact they can mostly be viewed as relatively large storage devices. Indeed, they can be very large. As an example of scale, the IAEA now * To whom correspondence should be addressed. Tel.: UK 141 330 4070. Fax: UK 141 330 4343. E-mail: [email protected]. † International Atomic Energy Agency. ‡ University of Glasgow. § IPSC, Joint Research Centre and Faculty of Engineering, Ghent University.

monitor the contents in about 60 vessels at the Rokkasho Reprocessing Plant.1 Higuchi et al.2 have described the extensive preparations required to obtain high-quality calibration data in a reprocessing plant to keep the calibration error as low as possible. They recommend that “the calibration liquid should be allowed to equilibrate with the cell temperature”, conclude that this “requirement can only be met during the commissioning period”, then recommend “that the resulting calibration function and associated calibration errors be used for the lifetime of the tank”. DeRidder et al.3 confirm that “other effects ... are negligible in comparison with the uncertainties associated with the temperature corrections”. Unfortunately, calibration stability cannot be guaranteed, necessitating their reverification at regular intervals. Although considerable effort might be devoted to ensuring that an appropriate environment ensues when reverifying the calibrations of those tanks that are critical for accounting materials entering/leaving a specific plant area, there are obvious disincentives for carrying out the same procedures on other tanks. Alternative approaches are desirable. By drawing together the results from experiments performed on facilities in both Italy and Japan, this article examines various practical issues associated with both the calibration of a tank and its reverification. These practical issues are introduced as part of the background, which is given in the next section. 2. Background In theory the volume, V, of solution stored in a tank at any given time, τ, could be estimated by evaluating the following equation:

V ˜ (τ) )

∫0L˜ (τ) A(h) dh

(1)

where L˜ (τ) is a measurement of its level and A(h) is the cross sectional area of the horizontal plane in the tank at the geometric height h. The relationship h.Vs.∫h0A(h) dh is then known as the

10.1021/ie061615t CCC: $40.75 © 2008 American Chemical Society Published on Web 02/05/2008

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Ind. Eng. Chem. Res., Vol. 47, No. 5, 2008

theoretical tank calibration. Assuming that, at any given time, τ, the density at a specific height h is a constant, F(h), across the tank, then the basic equation during a filling operation can be written as

∫0τ Win(t) dt ) ∫0L˜ (τ) A(h)F(h) dh

(2)

where the tank is assumed to be empty at the beginning and then filled at a mass flow rate Win(t). If both A(h) and F(h) are independent of h, then the commonly used undergraduate model is obtained:

AF

dh ) Win dt

(3)

An accurate measurement of the level is usually obtained from a set of instrumented dip tubes. The level estimate, L˜ (τ), is derived from a difference in two pressures, ∆p ) (pL - pR): pL is the pressure required to force air through a dip tube submerged in the liquid, and pR is the pressure required to force air through a dip tube that is inserted into the vapor space at a height hR above the liquid. In such cases, the geometric height, h, is unlikely to be the same as the estimated height, L˜ , because the latter is relative to the height, hL, of the tip of the submerged dip tube, which will not be touching the bottom of the tank. It will be argued in section 3 that, provided that the solution can be assumed to be homogeneous, then

∆p ) (pL - pR) )g

∫hh

R

L

F(h) dh = g

∫hL˜ (τ)+h

L

L

F(h) dh w L˜ (τ) = ∆p/gF

where g is the gravitational constant. If the density of the solution is not known, then one or more dip tubes might be added to the tank: if only one is added, and produces a pressure, pD, then

L˜ (τ) = hD(pL - pR)/(pL - pD) where hD is the true, vertical distance between the two dip tubes. This presupposes that hD, pL, pD, and pR are known accurately. This subject is discussed briefly in section 4. In practice, area A is usually dependent on h, and in ways that are unlikely to be known a priori: there might be inaccuracies in the manufacture of the tank, instrumentation is then inserted into the tank during construction, the inclination that the tank and the dip tubes finally rest at might not be known a priori and so on. Thus, an estimate of the tank calibration must be obtained on site. As will be discussed in section 7, modern flow meters are still not sufficiently accurate for this task, so considerable effort goes into the accurate weighing of relatively small quantities of solution, which are then introduced into the tank as small batches. A table of cumulative batch weights versus level L˜ k is then generated. The solution is homogenized thoroughly to ensure that F is constant and h is replaced by L˜ in the calibration: L˜ k versus Wink. In other words, the tank calibration that is finally adopted is a purely empirical equation, because, for instance, the absolute relationship between h and L˜ is not known. This empirical calibration can be corrected for the volume (the heel) located below the dip tube (i.e., for when L˜ < 0) by appending (L˜ 0, 0) to the table. Volumes at intermediate values in this table are then obtained via interpolation. Interpolation can also introduce errors: section 6 argues for a new approach that is based on cubic-spline regression.

The single, definitive calibration table or set of correlations has then to be used for all liquids, even if they have different densities or are at different temperatures. The conventional approach is to obtain the tank calibration while the tank is as close to thermal equilibrium as is practically possible and then to correct for expansion. Liquid expansion/contraction is easily dealt with through correlations in the published literature. The treatment of the thermal contraction/expansion of the tank itself is less obvious. The normal approach is to rewrite eq 1 as

V ˜ (t) ) (1 + 3β∆T)

∫0L˜ (t) A(h) dh

(4)

where β is a linear thermal expansion coefficient and ∆T is the difference between the measured solution temperature, T˜ , and a chosen reference temperature, Tref. Similarly, dip tube expansion is corrected for.

L˜ (t) )

∆p 1 (1 + β∆T) gF

(5)

Section 3 discusses some experiences that have been gained when trying to apply this and other corrections. To keep the temperature of the calibration liquid constant throughout the calibration of a vessel is a common problem as mentioned in ref 3. In this respect, the calibration of large vessels is more demanding than that of small vessels, as the time for calibration may be significantly longer. For very accurate calibrations, the temperature control of the calibration liquid may require that a feeding vessel of the same size as that of the vessel to be calibrated be available in close vicinity to the vessel. Under this condition, an equilibrium temperature can be established before the start-up of the calibration, and this temperature can then be kept throughout the run. Such an ideal situation might not be available in all cases. Fortunately, not all vessels need a very accurate calibration. If the temperature of the calibration liquid cannot be kept constant throughout the calibration run, then the density of the liquid will change during the calibration. Temperature stratification in a vessel is a real possibility because mixing during calibration is avoided due to evaporation. Vessels are not equipped with numerous thermocouples distributed over the height of the vessel, so the temperature and density distribution over the height of the vessel is unknown. Consequently, the volume and height vectors produced by a calibration run may not represent the actual situation of the vessel. This fact will lead to biased calibrations. Section 5 describes a method that utilizes a regression model for fitting biased calibration data for one or more calibration runs. Severely biased calibration data can be eliminated, which reduces the calibration error due to biases to a minimum. In this respect, the proposed method determines the most accurate calibration of a vessel for a given set of biased calibration data. This approach would also be suitable for reverifying the calibrations of those vessels that are so important that they warrant the repetition of the batch-fill approach, for example, for those located at (flow) key measurement points. However, there are obvious disincentives for carrying out the same procedures on other vessels. An alternative, continuous fill, approach is described in section 7. Ideally a tank calibration might be reverified by analyzing normal operational data. More likely, a normal operational procedure might be adapted to facilitate reverification. For instance, appropriate level/temperature data might be collected if the acid or water imported into a tank during normal washouts is metered into a vessel accurately.

Ind. Eng. Chem. Res., Vol. 47, No. 5, 2008 1535 Table 1. Typical Tank-Calibration Values for Calculating L˜ ieb

3. Data Standardization Issues Temperature inhomogeneity of the calibration solution both during a single calibration run and between individual calibration runs will adversely affect the quality of the calibration. Temperature stratification effects are generally undetectable and cannot be corrected for. Measured temperature variations during and between individual calibration runs, however, can be corrected for (eqs 4 and 5). In general, the measured volume and solution height of the tank are standardized to a chosen reference temperature, Tref (25 °C is commonly used). The calibration equation determined on the basis of these standardized volumes and heights then will correctly represent the characteristics of the tank itself and is independent of the characteristics of the calibration solution used. To determine the volume of the solution introduced into a tank during a calibration, the solution increments are generally measured by mass (for the most accurate determination) or by flowmeter. For flowmeter measurements, the total volume of solution measured in the tank is simply

V ) V(flowmeter) × (Fflowmeter/Ftank)

(6)

where Fflowmeter and Ftank are the solution densities determined at the flowmeter and in the tank, respectively. If demineralized water is used as calibration solution, the densities are generally determined using well-established density formulas for water based on measured temperatures. For mass increments, airbuoyancy effects on the volume of solution measured on the scale must be corrected for. The total solution volume in the tank is then

( ) ( ) Fair Fref Fair 1Fprov

M 1V)

Ftank

(7)

where M is the measured mass, Fair is the density of air, Fref is the density of the reference weights used to calibrate the scale, and Fprov is the solution density in the prover (filling tank). An equation is then required to determine the solution height. A number have been proposed,4-6 the most extensive of which is probably that due to Liebetrau.4 This purports to take into consideration a variety of factors that could affect the height and should result in the correct physical height of solution above the tip of the level pressure dip tube in a tank during a calibration,

[

L˜ ieb ) ∆p1 + gE1(Fg,1 - Fa) - gEr(F g,r - Fa) + (δr - δ1) - gλ(Ftank - Fg,1) -

]

2σ /[g(Ftank - Fa)] (8) rb

where ∆p1 is the differential level pressure, g is the local acceleration due to gravity, E1 and Er are the manometer elevations above the level and reference dip tubes, Fg,1 and Fg,r are the densities of the gas in the level and reference dip tubes, Fa is the density of air in the tank, δ1 and δr are the pressure drops in the dip tubes due to flow resistance, λ is distance of the lowest point of the bubble below the tip of the dip tube, σ is the surface tension for the liquid and gas, and rb is the radius of curvature of the bubble at its lowest point. From a practical viewpoint, it is difficult to perceive how all of the factors for the above equation could be determined under

air and solution temperatures (°C) atmospheric pressure (Pa) off-gas pressure (Pa) bubbling rate (L/h) inner diameter of probe (m) relative humidities in bubbling lines/tank air space height of manometer above tip of major probe (m) height of manometer above tip of reference probe (m)

25 101325 500 7 0.01 20%/50% 10 8

anything but laboratory conditions. Moreover, it is questionable whether the equation gives the true physical height of the solution above the tip of the dip tube, because Figure 1 shows that a simple application of the equation using fairly typical values (Table 1) for the factors results in a negative height for all of the positive differential pressures up to 50 Pa. The objective of determining a tank calibration equation lies in the requirement to correctly calculate the amount of solution volume contained in the tank based on a differential pressure measurement. It does not depend on determining the correct physical height of solution. For all practical purposes, the above equation for L˜ ieb may be viewed as a simple linear function of the common relationship height ) differential pressure/(density * g), that is, L˜ ieb ) a + bL˜ , where a and b are constants, and L˜ ) ∆p1/(g * Ftank). Mathematically, it makes no difference whether L˜ or (a + bL˜ ) is mapped onto V to determine the calibration equation, therefore it is much more practical to use the simpler model L˜ . To show that all of the factors in L˜ ieb except ∆P1 may be considered constants, the values of the individual factors in the equation were calculated for a typical tank, with water as the calibration liquid (assumed density 1 g/cm3), and for differential pressures of 1 and 10 kPa, over a temperature range of 10 to 40 °C, and with all of the other values as given in Table 1. Figure 2 shows the variation in L˜ ieb to the value determined at 20 °C. Because temperature is the only possible factor influencing the values of the factors in the equation, it is seen that any temperature change during a typical calibration (usually not more than 5 °C) has a minuscule influence on the value determined for L˜ ieb, far less than 0.1 mm, or 1 Pa. Similar plots for the individual factors of L˜ ieb show that none of the factors varies by more than the sum plotted in Figure 2. Similar calculations performed for a higher-density solution (F ) 1.5 g/cm3) show similar results. Tank calibrations are often performed using water as calibration solution. Because the process solutions subsequently measured in the tank are usually of significantly different (higher) density, the differences of the factors of L˜ ieb for solutions of density 1.00 and 1.50 g/cm3 were also compared. The differences of the four factors [gE1...] in L˜ ieb between the two solutions at pressures of 1/1.5 kPa and 10/15 kPa over the temperature range 10 to 40 °C were calculated and likewise seen to be less than 1 Pa, showing that these factors are for all practical purposes constant for the two different densities. There is one factor in L˜ ieb that does give a significant difference between the two different densities, however, namely [2σ/rb]/[g(Ftank - Fa). This factor shows a constant difference of around 0.7 mm between the two liquids. It should be noted, however, that there is a large uncertainty associated with this term, which is dependent on σ, the surface tension between the liquid and the bubbling gas. For these calculations, the surface tension was assumed to be equal for the two liquids, which most likely is an unwarranted assumption. A 10% difference in the value of the surface tension for one of the liquids would already cause a difference in the heights of between 1 and 2 mm, far greater than any difference observed in any of the other factors

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Figure 1. L˜ ieb vs differential pressure.

Figure 2. L˜ ieb vs temperature.

in this study. It is unlikely that any exact values for the surface tension of process solutions encountered in reprocessing plants can be determined. In conclusion, there is no practical reason not to use the simple relationship L˜ for determining the solution height and applying the thermal expansion correction to it to arrive at the standardized solution height to which to fit the calibration equation. Aside from the correct standardization of the raw measurement data, possible effects relating to the bahavior of bubble formation under various conditions of solution density and hydrostatic pressure may become evident. This could especially become an issue for calibrations performed under conditions requiring the highest possible degree of accuracy; for example, for carefully temperature-controlled calibrations performed on accountancy tanks. The effects described in the following section highlight these effects, and efforts to correct for them may under certain conditions become necessary. 4. Systematic Errors in Density Measurement It is often observed that dip-tube-derived density measurements deviate with increasing level height. Typical of these are the results from a tank experiment that are shown in Figure 3: each plot pertains to measurements taken when the air is forced

through the dip tubes at a different hourly flow rate. It is clear from these that an appropriate air flow rate must be chosen and then maintained. Even if this is done, there will still be a heightrelated error in the level measurements and so in the volume determination. Reasons for these systematic deviations include: the stratification of the liquid with layers of different temperature and hence density, (including Marangoni convection effects); the change in air-filled bubble formation at the tip of the dip tube under different hydrostatic pressures (including the transiently moving interface); the decrease in the height of the air-filled bubble at the detachment with an increasing hydrostatic head and so increasing level (including the change in RayleighTaylor instability point); the change in bubble frequency at different liquid levels, which is a consequence of the change in Rayleigh-Taylor instability point; if the dip tubes are located in a confined channel, the bubbles might produce a significant void fraction. Being empirical, tank calibrations will automatically accommodate the inherent systematic error relating to the repetitive dip tube action involving the build-up, then collapse, of a bubble volume. Unfortunately, unless experiments are repeated, in a controlled manner, at different temperatures and with solutions

Ind. Eng. Chem. Res., Vol. 47, No. 5, 2008 1537

Figure 3. Differential density pressure measurements at various tank fill levels and dip-tube airflow rates.

Figure 4. Pressure and two-volume measurements for one bubble cycle.

of different densities, it is unlikely that the error will be accommodated accurately. This section describes an experiment that was performed to investigate how bubble formation might affect the systematic error. The focus is on the periodicity of the bubble behavior without making any assumptions about the bubble’s shape. The bubble shape depends directly on the surface tension of the solution, which for representative solutions of plutonium nitrate, is given by Hosoma et al.7 4.1. Bubble Cycle Visualization. A small-scale experiment was setup with a rectangular, isothermal water bath in which one stainless steel dip tube was inserted with a 6.0 mm inner, and 10.0 mm outer, diameter ((0.1 mm). Dry air was supplied with flow rates varying from 0.5 L/h to 14 L/h, controlled with a mass flow controller of 0.07 L/h precision. The pressure was measured by means of a vibrating cylinder type transducer with a data rate of 10 kHz. Bubble formation was photographed by two cameras, located at 90° to each other, triggered by the pressure signal, and images were taken at 100 frames/s. The bubble height and diameter in each image were measured by counting pixels: assuming axial symmetry, the volume was calculated by summing circular segments of one pixel height. The images of each camera were treated separately, leading to two different volumes. Figure 4 gives the results obtained for one bubble cycle, from formation until detachment, for a dip tube immersed 2 cm in a salt water bath of 1.15 g/cm3 density, at 22 °C, and with a dry air flow rate of 1 L/h dry air. Different regions in the bubble formation can be identified during the 1.38 s cycle: large pressure oscillations, due to the swapping of the liquid/gas

Figure 5. Bubble height vs time (dip-tube submersion 2 cm).

interface inside/outside the dip tube; vertical growth of the bubble with pressure buildup; circular growth of the bubble at constant pressure; lateral expansion of the bubble (with lateral displacement) at constant height (decreasing pressure); and detachment of the bubble with rupture of the liquid/gas interface (steep pressure drop). Because variations in the measured pressure can be directly coupled to variations in the height of the displaced water column, the bubble height was defined as the maximum vertical distance between the tip of the dip tube and the bubble/gas interface. The bubble height shows, in contrast to the bubble volume, not a linear but a second-order increase (Figure 5). 4.2. Systematic Errors. To assess global effects such as the density under different conditions of hydrostatic head and in particular the decrease in density with increasing level, all of the signals of pressure amplitude, volume variation, and height have to be averaged over time. The sensitivity of these timeaveraged parameters on temperature variations (22-38 °C), solution density (1 g/cm 3 - 1.29 g/cm3), and dip-tube insertion depth (2 cm-12 cm) are analyzed in.8 With increasing air flow rate, the bubble frequency increases. At high air flow rates (above 13 L/h at a hydrostatic pressure of 600 Pa), the bubble cycles become very short and the bubbles very deformed until an air jet is formed. At higher air flow rates, a larger variation on the bubble volume and height can be expected. If it is assumed that the dip tube measurements correspond to the pressure at the tip of the dip tube, the bubble growth will introduce a systematic error with periodic dynamics. As shown in Figure 6, the maximum bubble volume increases for

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Figure 6. Bubble volume just before detachment vs flow rate and the insertion depth.

Figure 7. Maximum bubble height vs flow rate.

increasing flow rate from 2 L/h to 12 L/h (at 29 °C), and varies more at smaller insertion depths (2 cm) than at larger insertion depths (12 cm). At higher air flow rates, the larger amount of air has to pass through the liquid by creating bubbles, not only in a faster way, but also of larger size. The bubbles can expand more when the pressure of the surrounding liquid is lower, that is, at smaller insertion depths. Stagnation can be seen for the 14 L/h data points, because a deviation from regularly formed bubbles occurs (an air jet is formed). At very small flow rates (around 1 L/h), the bubble detachment period becomes longer and less predictable. This might explain the exceptionally large bubble volumes at large insertion depths. The hydrostatic head measured with the bubbling dip tube corresponds directly to the total height of the immersed dip tube, corrected with the addition of the mean bubble height. The bubble height is another systematic error, which varies for different flow rates, different insertion depths, and different liquids. Figure 7 shows an increase in bubble height of 10% when the flow rate is changed from 2 L/h to 12 L/h (29 °C, insertion depth of 6 cm). The combined effect of different insertion depths and different liquids with different densities and different surface tensions can be reasonably simplified to the change in pressure of the surrounding liquid. Bubbles formed in sugar water (density 1.29 g/cm3) experience a surrounding liquid overpressure of 710 Pa, whereas bubbles formed in water (density 1.00 g/cm3) only have to withstand a liquid overpressure of 551 Pa. Bubble height in sugar water is therefore almost 0.2 mm smaller than that in water. 5. Calibration Error Due to Biased Calibrations As discussed previously, the variation of liquid temperature during calibration runs generates biased calibration data. A segmented polynomial regression model that includes biases is proposed in this section to estimate the calibration error due to

such biases. An example of calibrating two tanks illustrates the analysis of biased calibration data, the calculation of the overall calibration error, and the judgment of whether a calibration is accurate. 5.1. The Design Matrix. The internal shape of a tank influences the number of volume increments needed for calibration because smaller increments are required at those levels where the cross-sectional area of the tank changes significantly. The pouring schedule of a calibration experiment can then be represented by a volume vector, v: Vi ) ∑ji ) 1 ∆Vj where ∆Vj is the jth volume increment. Each volume increment produces a new volume measurement and a corresponding new level measurement. The calibration equation of a tank is often represented as a set of equations: usually the tank’s range of heights is segmented, and each segment is represented by its own polynomial, which might be of a different degree. The design matrix of a tank9 specifies the segmentation and the degree of polynomials of each segment. For example, the design matrix, DR, of a cylindrical tank with three segments and linear polynomials is, for calibration run R and n increments, given by

[ ]

1 1 .. 1 DR ) 1 .. 1 .. 1

L˜ 1,R L˜ 2,R .. c1 c1 .. c1 .. c1

0 0 .. L˜ j+1,R - c1 L˜ j + 2,R - c1 .. c2 - c1 .. c 2 - c1

0 0 .. 0 0 .. L˜ k + 1,R - c2 .. L˜ n,R - c2

(9)

where c1, c2, and c3 are break points: 0 < L e c1 defines segment 1, c1 < L e c2 defines segment 2, and so on. Equation 9 then has four fitting parameters. Though the basic structure of the design matrix is the same for different calibration runs with the same pouring schedule, the observed design matrices of different calibration runs are different due to measurement errors. 5.2. Calibration Based on one Calibration Run. If a segmented polynomial regression model (SPRM) fits the observed calibration data, then the regression model of run R is VR ) DRβ + R, R ) 1, 2, ..., m, where VR is an (nR × 1) vector of observed volumes, DR is an (nR × k) design matrix, β is a (k × 1) vector of true fitting parameters, and R is an (nR × 1) vector of observed measured errors. Vector R is a realization of random variable (RV) ER and vector VR is a realization of RV VR. Thus, the SPRM in terms of RVs is given by

VR ) DRβ + ER

(10)

with mean E(ER) ) 0, and variance Var(ER) ) diag(σ2R). For calibration run R, the minimum variance unbiased estimate (mvue) of β is given by10

βˆ R ) (D′RDR)-1 D′RVR

(11)

The estimate βˆ R is a realization of RV BR with mean E(BR) ) β:

BR ) (D′RDR)-1D′RVR ) β + (D′RDR)-1D′RER

(12)

The dispersion matrix of BR, representing the error due to fitting,


is given by

Table 2. Temperature Variations (°C) during Calibration of Tank 1

ΣBR ) σR2(D′RDR)-1

(13)

An unbiased estimate of the variance of the measurement error is given10 by

σˆ R2 )

1 (V - DRβˆ R)′(VR - DRβˆ R) nR - k R

(14)

run

max ∆Tp

max ∆Tv

max (Tp - Tv)

1 2 3 4 5

3.1 5.3 2.2 2.0 12.9

7.0 4.0 3.0 7.0 8.5

5.5 0 0 11.5 0

h , that is, the bias Let ηˆ R denote the difference between βˆ R and β of calibration run R with respect to β h:

Thus, an estimate of the dispersion matrix of the error due to fitting is given by

ηˆ R ) βˆ R - β h , R ) 1, 2, ...., m

Σˆ BR ) σˆ R2(D′RDR)-1

Then ηˆ R is a realization of RV H, and therefore an unbiased estimate of the variance of H is given by the sample varianceThis matrix represents the error due to biased calibration data.

(15)

5.3. Calibration Based on Several Calibration Runs. Usually, a number of calibration runs are performed to improve the accuracy of the calibration by reducing the fitting error. Calibration data from more than one calibration run also brings run-to-run differences to light, which most likely result from liquid temperature variations. The following SPRM is used for biased calibration data: VR ) DR(β + ηR) + R, where ηR is a (k × 1) vector representing the bias of that calibration run. Liebetrau9 used the same model to account for run-to-run fluctuations. His approach, however, leads to unrealistically small run-to-run errors. In terms of RVs, the SPRM for biased data is given by

VR ) DR(β + H) + ER

(16)

where the RV H represents the error due to biased data with H ∼ N(0, ∑H). The problem is then to determine an appropriate estimate of β, which essentially represents the tank’s calibration, and to determine an estimate of ∑H. For each fitting parameter estimate βˆ R, volume predictions V˜ R can be calculated for any grid of liquid levels by

V˜ R ) Dβˆ R, R ) 1, 2, ...., m

(17)

Σˆ H )

E(V ˜ ) ≈ Vj )

m

∑ V˜ R

(18)

m R)1

The variance of V ˜ provides an estimate of the error due to biased calibration data because volume predictions are not affected by measurement errors. If V ˜ ∼ N(Vj, ∑V˜ ), then an unbiased estimate of Σˆ V˜ is given as the sample variance.11

Σˆ V˜ )

Σˆ V˜ ) DΣˆ HD′

∑ (V˜ R - Vj)(V˜ R - Vj)′ m - 1 R)1

(19)

Vj ) D

(

1

∑ βˆ R

m R)1

)

The variance of the predicted volume V ˆ (L˜ ) is of interest in the following as this represents the overall calibration error. According to eq 20, B h is given by

B h)

Substituting from eq 20 into eq 19 gives

[

Σˆ V˜ ) D

1

m

(20)

]

∑ (βˆ R - βh )(βˆ R - βh )′ D′

m - 1 R)1

1

m

∑ BR

m R)1

(24)

If we substitute from eq 16 into eq 12, then BR for biased calibration data is given by

BR ) β + H + (D′RDR)-1D′RER

(25)

Substituting eq 25 into eq 24 gives

B h )β+H+

1 [(D′1D1)-1D′1E1 + ... + (D′mDm)-1D′mEm] m

The mean of B h is calculated as E(B h ) ) β and the dispersion matrix of B h is calculated as

h - β)(B h - β)′] ΣBh ) E[(B

) Dβ h

(23)

V ˆ (L˜ ) ) D(L˜ )B h

Substituting from eq 17 into eq 18 gives m

(22)

This matrix represents the volume error (calibration error) due to biased calibration data. Equation 20 indicates that β h is the best estimate of the true fitting parameter β in case more than one calibration run is used for calibration. That is, the calibration equation for determining the predicted volume for any measured level L˜ , denoted as Vˆ (L˜ ), is given by Vˆ (L˜ ) ) D(L˜ )β h , where D(L˜ ) is the row of the design matrix representing level L˜ . If Vˆ (L˜ ) and β h are considered as realizations of RVs V ˆ (L˜ ) and B h, respectively, then the calibration equation in terms of RVs is given by

m

1

∑ ηˆ Rηˆ ′R m - 1 R)1

Substituting eq 22 into 21:

where the design matrix D has the same structure as DR but corresponds to any grid of liquid levels. Volume predictions V˜ R are considered as realizations of RV V ˜ , where an estimate of the mean of V ˜ is given by the sample mean

1

m

1

(26)

Substituting eq 24 into eq 26, and noting that covariance terms are zero,

ΣBh ) ΣH +

1 m

2

[σ12(D′1D1)-1 + ... + σm2(D′mDm)-1]

(21) Substituting eq 13 into eq 27 gives

(27)

1540


ΣBh ) ΣH +

1 (ΣB1 + ... + ΣBm) m2

(28)

The first term of ∑Bh represents the error due to biased data, and the second term represents the fitting error of m calibration runs. Though more calibration runs reduce the fitting error, biased calibration data introduce an additional calibration error. That is, more calibration runs will yield a more accurate calibration only if the calibration runs are performed with small liquid temperature variations (small error due to biased data). The differences between the observed volume and the volume predicted with respect to β h represent the fluctuations due to measurement errors. For calibration run R, this difference is given by VR - DR(β h + ηˆ R) ) VR - DRβˆ R. Thus, σˆ R2, the variance of the measurement error, is the same for biased and unbiased calibration runs and is given by eq 14. An estimate of ∑Bh is obtained from eq 28 if ∑H is replaced with its estimate given by eq 22, and ∑BR is replaced with its estimate given by eq 15. Then an estimate of ∑Bh is given by

∑ B ) ∑H +

1 (∑B1 + ... + ∑Bm) m2

Figure 8. Tank 1 calibration based on runs 1, 2, 3, 4, and 5.

(29)

and an estimate of the variance of the predicted volume V ˆ (L˜ ) is given by

σˆ V2ˆ (L˜ ) ) D(L˜ )∑ ˆ BD(L˜ )′ Note that this variance represents the total calibration error. That is, almost all observed volume residuals calculated with respect to β h , defined as the difference between the measured and predicted volumes, should be within a 2-sigma bound if the calibration error is limited to fitting errors and errors due to biases. If many residuals happen to be outside error bounds, then the calibration has a lack of fit error, which is caused by the use of an inappropriate regression model. This stresses the importance of using an appropriate regression model for calibration. 5.4. Validation. The only requirement of volumetric calibration is to produce an estimate of true volume for any given level observation. The proposed calibration model is validated here by applying it to actually observed calibration data of two tanks, which were modified in their geometry. The variation of liquid temperature during the calibration of tank 1 is characterized by the maximum difference of the observed liquid temperature in the prover (max ∆Tp), the maximum difference of the observed liquid temperature in the tank (max ∆Tv), and the maximum difference between the observed liquid temperature in the prover and in the tank (max (Tp - Tv)). The temperature variation observed for tank 1 is given in Table 2. Tank 2 was calibrated with nearly perfect temperature control where the liquid temperature increased during the calibration by about 0.5 °C. Tank 2 was calibrated with a larger number of volume increments because it was significantly larger than tank 1. All five calibration runs were initially used to calibrate tank 1. The volume residuals of the five runs are depicted in Figure 8 together with two 2-sigma error bounds. The dotted error bounds represent the fitting error, and the solid error bounds represent the overall calibration error, that is, the fitting error and the error due to biased calibration data. The errors due to biases dominate all of the segments of tank 1 except segment one. At maximum level, the error due to biases is more than 10 times the fitting error. The lack of fit error is negligible because almost all of the residuals are within the 2-sigma solid error

Figure 9. Tank 1 calibration based on runs 1, 2, and 3.

bounds. The temperature variations indicate that run 4 (marked with an x), and run 5 (marked with a 0), have the highest temperature variations, whereas the remaining runs have significantly lower temperature variations. The biases of these runs confirm the hypothesis that a large temperature variation leads to a large bias. Therefore, runs 1, 2, and 3 were selected to calibrate tank 1. Figure 9 depicts the result. Elimination of the highly biased calibration runs 4 and 5 reduced the overall calibration error, at a maximum level, by about two thirds. More biased calibration runs do not necessarily improve the accuracy of the calibration, which is a different conclusion to performing unbiased calibration runs. Runs 1, 2, and 3 were carried out on three consecutive days with moderate ambient temperature changes over the day, whereas runs 4 and 5 were carried out with large temperature changes over the day; run 4 was performed on a relatively cool day and run 5 on a relatively hot day. Only two calibration runs were available to calibrate tank 2. The volume residuals of the two runs are depicted in Figure 10, together with two 2-sigma error bounds as used for the calibration of tank 1. Although the fitting errors and the errors due to biases are small, the lack of a fit error is significant as many residuals are outside the 2-sigma solid error bounds. This lack of fit arises because of the particular geometry of tank 2, the representation of which requires a large number of volume increments. A more detailed regression model is needed, which is difficult to find by trial and error. Note that the lack of fit error, for the middle part, is about three times the fitting error plus the error due to biases. An improved regression model for this vessel will be obtained in the next section. The proposed regression model for fitting biased calibration data provides an estimate of the calibration error due to biases. The calculation of this error does not only enable the elimination


Figure 11. Tank 2 calibration, direct cubic-spline regression. Figure 10. Tank 2 calibration based on runs 1 and 2.

of heavily biased calibration runs, thereby reducing this error to a minimum, but it also enables the calculation of the overall calibration error, consisting of the fitting error and the error due to biases. For a good calibration, almost all of the observed volume residuals should be within 2-sigma error bounds. If the error bounds do not represent the total calibration error, then too many residuals will be outside these error bounds, indicating that the lack of fit error is significant and cannot be ignored. This should trigger a modification of the regression model. This is discussed in the next section. 6. Spline Regression Modeling of Tank Calibration Equations The selection of the segmentation breakpoints and the choice of polynomial degrees are very difficult to establish by trial and error. To improve on this situation, cubic B-spline regression12 is proposed to model the nonlinear relationship between liquid volume and level for tank calibration. The underlying function is approximated by cubic B-spline basis functions over m knots, {z0, z1, ..., zm}, as m+3

Vˆ (L˜ i) )

cjNj(L˜ i) ∑ j)1

(30)

where Nj(L˜ i) are linearly independent normalized cubic B-spline vectors in m + 3-dimensional space. The corresponding m + 3 expansion coefficients cj can be easily obtained by least-square solution. The approximation function Vˆ (L˜ i) defined in eq 30 is a polynomial with at most 3 degrees in each of the intervals defined by two neighboring knots. The function values, its first derivatives, and second derivatives are normally continuous across the m knots. For this reason, it can be shown that only {Nj(L˜ i), Nj+1(L˜ i), Nj+2(L˜ i), Nj+3(L˜ i)} have nonzero values in the interval [L˜ j-1, L˜ j]. However, if repeated knots are placed on certain locations, the approximation function can also handle isolated discontinuities and discontinuous derivatives. The advantage of using cubic B-spline regression in tank calibration is its simplicity within the framework of least-square solution and its flexibility of adjusting model complexity by increasing the number of knots and inserting repeated knots at those locations where discontinuities occur at internal structures in the tank. In fact, the performance of B-spline regression is sensitive to the number of knots and not sensitive to their placements. From a practical point of view, operators have some rough information about the calibrated tank, so they usually take more volume increments at those locations where significant changes in cross sections occur. This information can be automatically captured by a spline regression approach if the

placement of a given number of knots is based on the quantiles of calibration level points. To reiterate, it is necessary to have the regression error significantly smaller than the measurement error to have a correct estimate of the measurement error from the residuals. Conversely, it is also true that the regression error is small enough when the estimated variance of the residuals is close to a known measurement error. These statements have two implications for tank calibration: the measurement error can be estimated approximately by comparing the difference between two calibration runs when the calibration level points are quite close together, and model prediction capability can be validated using a dataset different from that used to obtain the regression model. The former is especially useful when insufficient calibration data are available. When sufficient calibration data are available, the odd-even method can be used to separate the modeling dataset and the validation dataset. A procedure can also be provided to convert obtained spline regression models to the segmented polynomial format, which is more familiar to plant operators. Model derivatives can indicate where breakpoints should be placed: derivatives up to second order can be obtained from eq 30. If this approach is adopted, then the degree of polynomial for each segment needs to be determined to obtain the SPRM. The best model can be selected, among the limited number of models that are possible, using information criteria that penalize both lack of fit and model complexity. The criterion, ICOMP, is appropriate here,13

ICOMP ) 2 log(L(βˆ )) + 2C(F -1)

(31)

where βˆ is the vector of polynomial regression coefficients, L(βˆ ) is the likelihood function, and C(F-1) is the complexity function of the inverse Fisher matrix F-1. The dataset of tank 2 is particularly suited to demonstrating this approach, because the SPRMs obtained by trial and error were not able to capture the relationship between the liquid volume and level correctly. Too many volume residual points were outside the 2-sigma bounds at approximately the same locations for both runs. Figure 11 shows the calibration results that were now obtained. An automatic algorithm was used to determine the placement of the knots for the two runs. Because the placements are slightly different, it is difficult to characterize the calibration errors in terms of regression parameters directly. However, as shown in section 5, the covariance matrix of calibration error due to biased calibration data for a given liquid level is equivalent to the covariance matrix of the predicted volumes of individual regression models. Therefore, this calibration error can be established by calculating the covariance of the volume predictions. Similarly, the fitting error can be established when an averaged model prediction is used as the calibration model. The plotted calibration errors for both runs,

1542


this conversion: the bias between the two runs in Figure 12 is not as clean as that in Figure 11. The measurement error bounds are also different because these are calculated through error propagation of the estimated parameters (section 5). 7. Continuous Reverification

Figure 12. Cubic-spline-model-based segmented polynomial regression results.

Figure 13. Bias-corrected test statistic for 100kh/hr experiments.

the difference between the measured volume and predicted value from the obtained calibration model, behave quite close to normal distribution, which indicates that the underlying relationship between liquid volume and level is well decoupled from the random measurement noise. Note also that a significant bias between the two runs exists especially at locations above 500 mm. If the regression models are not detailed enough, then this bias would be confused with the random fitting error. The many bumps in the calibration error bounds are an indication of the model complexity used for the calibration: 60 knots represented more than 200 data points. Finally, the difference between the error bounds of the total calibration error and those of the measurement error is small. The regression models obtained for individual runs are indeed modeling the true relationship. The cubic-spline regression model has better performance than the calibration model obtained from trial-and-error-based SPRMs. Figure 12 shows the calibration results of tank 2 when spline regression models are converted into SPRMs. Examination of the second derivatives of the obtained cubic-spline regression models for individual runs lead to 69 segments. The corresponding degrees of polynomials within each interval were determined by information-based model selection criteria. After the breakpoints and the degree of polynomials were known, the calibration model and calibration errors were calculated using the approach of section 5. Figure 12 shows a similar performance to that of Figure 11. The total calibration error is less than 1 L for all of the level points, demonstrating that converting the spline regression models to SPRMs is successful. There is a slight sacrifice in

Tank calibrations must continue to be valid during the lifetime of a plant. Although, specifically in reprocessing plants, certain key tanks might be recalibrated at regular intervals, the intention might be to keep the original calibrations for others. Thus, it is important to reverify their validities at intervals. Ideally, a tank calibration would be reverified by analyzing normal operational data. More likely, a normal operational procedure might be adapted to facilitate reverification. For instance, the acid or water imported into a tank during washouts might be metered accurately at a constant flow rate, monitored, and dip tube/ temperature trends evaluated. Both approaches involve continuous (as opposed to discrete,) filling. Monitoring during an emptying operation is not considered to be practicable because of the complications that might have to be overcome to meter solution out of a radioactive tank. This section examines the possibility of metering at a constant flow rate using a conventional flow rate regulated loop consisting of a digital mass flow instrument connected to a proportional plus integral control system. A sufficiently slow fill rate is assumed so that level measurement can be assumed to be in steady-state, that is, nonsteady disturbances on bubble formation are minimized. The extent to which filling might increase, artificially, the measured dip tube pressure, and hence measured level, might require further investigation. 7.1. The Reverification Equations. The following three equations would have to be satisfied:

tank calibration, f(): V ) f(L)

(32)

total volume input history, g(): V(t) ) g(t)

(33)

level history, h(): L(t) ) h(t)

(34)

where V(t) and L(t) are the true volume and level at time t. If the tank contents are in thermal equilibrium with the tank structure, which is itself in equilibrium with its environment, all would be at the same temperature θ. If in an ideal world there are no random and/or systematic errors and the tank calibration has not changed, then

f(L˜ (t)) ) gh-1(L˜ (t))

(35)

If f() is not the true tank calibration, then there will be a misalignment, x(t), between the two volume estimates:

x(L˜ (t)) ) f(L˜ (t)) - gh-1(L˜ (t))

(36)

A null hypothesis could be tested, based on eq 36, to determine if the tank calibration has changed significantly. Unfortunately there would still be errors associated with all three functions so that the test statistic x would also be influenced by the compounded effect of the various errors. As was seen in section 3, errors are inherent in the tank calibration equations themselves. It is argued here that, as far as reverification is concerned, it is best to avoid these errors by only focusing on the table of raw values (L/i , V/i ) (* is used to denote the fact that these relate to the calibration experiments). Thus, the revised test would be: ∀i, determine the time, τi, at which for each L/i ) L˜ (τi), then compare the total volume input at that time with V/i . The test statistic can then be reformulated as


x(L/i ) ) V/i - gh-1(L/i )

(37)

Unfortunately, and as highlighted previously, there would be errors associated with the determination of both L˜ and τi. Their net effect can be modeled as small displacements in equation tank calibration,

V(L˜ (τi)) ) V(L/i + ∆Li) ≈ V/i +

|

dV ∆L dL i i

(38)

that is, dV/dL|i ∆Li will be one component of x(L/i ). Turning to eq 33, fill times are such that those errors that arise whilst the flow loop settles down should be negligible, so errors of any significance will derive from steady-state filling: current metering devices typically have a repeatability of 0.1% of rate and an accuracy of 0.5% of rate. Total volume input history can be predicted from

V(τi) )

∫0

τi

q + N(∆q, σr2) dt F

(39)

where parameter q is the setpoint flow rate, ∆q relates to the accuracy, σr2 relates to the repeatability, and F is the density of the solution. It is assumed that the term repeatability relates to the performance of the specific instrument at a specified flow rate, whereas accuracy relates to inter-instrument and interflow rate variations. If the density is constant, the total volume input, Vˆ (τi), by time τi is

Vˆ (τi) )

q + ∆q τi F

(40)

The test statistic (eq 37) can then be reformulated as

x(L/i ) ) V/i - Vˆ (τi)

(41)

Combining eqs 38, 39, and 41 leads to

V/i + i ) Vˆ i +

∆q τ + νi + γi w V/i - Vˆ i F i ∆q τ + γi + νi - i (42) ) F i

where i relates to dV/dL|i∆Li, νi relates to the repeatability and γi is included to accommodate other, unspecified error sources. Thus, the test statistic will be biased in time by ∆q. Reverification is unlikely to be performed under the ideal conditions that are sought during calibration. Thus, the errors are likely to be significantly larger and even less predictable. The test statistic will be affected by various compound influences that include: nonequilibrium thermal expansion of the structure, temperature stratification of the solution in the tank, a lack of knowledge about the temperature of the solution entering the tank, and physical movements in the tank structure. It is difficult to elaborate further here without detailed understanding of the extent of these various conditions. To gain some insight, suppose that the tank metal is of a uniform differential temperature of ∆θ relative to its contents, and in the extreme, this differential is maintained during the filling operation. Ignoring random noise errors, eq 40 can be equated to a volume expression that incorporates this thermal expansion,

(q + ∆q) τi ) F

∫0L Ao(L)[1 + κ∆θ] dL

where κ is a thermal expansion coefficient. Thus,

(43)

(q + ∆q) τ ) F[1 + κ∆θ] i

∫0L Ao(L) dL = F1 (qκ∆θ - ∆q)τi

(44)

It can be seen that this differential would be observed as a bias, which is additional to the flow meter bias. Equation 42 should be revised because of this: V/i - Vˆ i ) βτi + γi + νi i where β ) 1/F(qκ∆θ - ∆q). As in the previous section, it is suggested that the bias, β, should be estimated using least squares to produce the corrected test statistic (xc(L/i ) : xc(L/i ) ) V/i - Vˆ i - βτi) and tolerances should be superimposed, which are based on variance estimates of νi, that is ∑ ˆ V, and i, that is ∑ ˆ e. However, it will be shown in next subsection that the considerable uncertainties described above produce considerable variations in βˆ R that are estimated from repeated experiments. Thus, a slightly different approach is proposed. 7.2. Experimental Confirmation. These were performed on the 450-L output accountancy D tank installed in the Joint Research Centre’s TAME laboratory. This tank is composed of sections of pipe of the same diameter, which are formed and connected together so that its profile takes the shape of a D in two dimensions. It is essentially planar. Uprights are appended at the top and bottom, and two vertical ties enable its suspension from the ceiling. The tank swings very easily on these two ties, and it is suspected that slight, in-plane rotation might occur as the tank is filled. It is arguably the most difficult of tanks to calibrate because dV/dL varies by a factor of 10 between the uprights and the ‘flat’ sections of the ‘D’. In addition the tank calibration table (L/i , V/i ) contains only 42 points. Five 100 kg/h experiments were performed over a period of 1 year to accommodate seasonal variations. During this period, and as part of a feasibility study to move the D tank to another location, the D tank was taken off of its supports, then rehung, without recalibration. From the experiments described here, it is clear that this must have resulted in an in-plane rotation of the D tank. One experiment (N100-1) was carried out before rehanging, and four (N100-2 to N100-5) were carried out afterward. Water was metered into the D tank from a 200-L header tank, located some height above. The metering device had a repeatability of 0.1% of rate and an accuracy of 0.5% of rate. Flow rates used were in the range 75 to 125 kg/h, which meant that a single experiment took typically 4 or more hours to complete, and only one experiment was performed in any 1 day, with the tank emptied at the end of the experiment. The solution temperature in the tank was recorded occasionally: it was not found to change by more than a degree or so, so an average was used in the calculations. The bias estimates, βˆ R, were as follows: -0.1676, 0.1677, -0.1171, -0.6164, and -0.2271. The fourth, βˆ 4, is clearly an outlier: this experiment was performed on a Monday when the building’s air-conditioning system was malfunctioning; not only did the temperature of the environment surrounding the metering station increase significantly during that day, but also the temperature of the environment surrounding the D tank itself would have decreased considerably during the previous weekend when the building was closed. Thus, the temperature of the D tank structure would have been uniform, but relatively cold, whereas the solution entering the D tank toward the end of the experiment might well have been in excess of 35 °C. This represents a valuable, extreme case from a thermal nonequilibrium perspective. The standard deviation of the other estimates is 0.175, which is significantly larger than the 0.1 that would be expected from a flowmeter repeatability, but only slightly too large if thermal effects are also accommodated. Referring to the previous section, it would not be sensible to

1544


examining the effect of different, physically acceptable values of ∆Li. Figure 14 shows plots of the corrected test statistic for N100-3, N100-4, and N100-5 when taking N100-2 as the reference. Two sigma error bounds, based on ∑ ˆe + ∑ ˆ V, are shown for a very plausible, dynamic variation ∆Li of 0.25 mm, which is perhaps twice the size of likely static, level measurement errors. Perhaps the bounds are a little tight and could be increased to accommodate other sources of error, γi. However, these narrow bounds do provide for sensitive detection. To return to the original objective of reverifying a tank calibration, because only one set of data prior to rehanging was available, and this is insufficient to estimate ∑ ˆ V empirically, the error bounds estimated above were used on the grounds that it was the same tank, flowmeter, data processing, and instrumentation. The results are shown in Figure 15: it can be seen that bounds based on a ∆Li of 0.25 mm are far too narrow and even those for 0.75 mm are slightly too tight. Once again, this raised doubts about the possibility of slight in-plane rotation taking place even prior to rehanging.

Figure 14. Bias-corrected test statistics relative to N100-2.

8. Conclusions

Figure 15. Calibration reverification based on N100-1 data with different bounds.

consider an average bias, β h , because each βˆ R indicates that the flow meter metered a slightly different quantity and the tank was subject to slightly different thermal conditions. Figure 13 shows the bias (βˆ R) corrected, test statistic, evaluated at each calibration point and for all five experiments. It can be seen that the N100-1 plots are considerably different to the others, supporting the view that there was considerable in-plane rotation when the tank was rehung. This meant that the tank calibration was invalid for experiments N100-2 to N100-5. Note also the very distinctive two-hump shapes, which correlate closely with the change in horizontal cross-sectional area. To continue, the corresponding points in the N100-2 data were now taken as the new calibration. The variance of the true repeatability, ∑V, was now estimated by determining the variance between the n plots N100-3, N100-4, and N100-5 at each calibration point:

Σˆ V|i )

1

Literature Cited

R)5

∑

n - 1 R)3

Nuclear material accountancy measures require an accurate estimate of volume. Issues arise at every stage of the determination of this quantity. Dip-tube bubble formation can introduce a small systematic error, which varies with air flow rate and solution density. Solution height determination is contentious: it is argued here that the simple relationship height ) differential_pressure/(density*g) is sufficient, with an air-buoyancy correction to solution volume, and a thermal-expansion correction to both dip-tube heights and tank volumes. Up until now, it has not been possible to decide whether the calibration of a vessel satisfies the requirements of an accurate calibration or not. The reason is that several mistakes could be made during the calibration of a vessel without receiving any feedback. A further error will result if the underlying structure of the regression model is inappropriate, for instance because the segmentation of the vessel and/or the degree of polynomials for each segment is not properly selected. Up until now, neither the error due to biases nor the lack of fit error could be quantified. Calibration data from different calibration runs usually exhibit a significant run-to-run variability: application of the standard regression method therefore leads to an unrealistically small calibration error, because only the fitting error is taken into account. A larger number of biased calibration runs do not necessarily lead to a more accurate calibration, which is the case when the standard regression model is used for unbiased calibration data. On the contrary, the accuracy of a calibration can be significantly improved by eliminating calibration runs that are heavily biased. A proper regression model always avoids the lack of fit error: a methodology has been proposed, which avoids difficulties encountered by traditional trial and error based segmented polynomial regression in finding appropriate breakpoints and degrees of polynomials. Finally, reverification has been addressed by proposing a new method for analyzing continuous fill situations.

(xcR(L/i ) - xjcR(L/i ))2

(45)

Variance estimate ∑ ˆ e was derived by differentiating, numerically, either the tank calibration (when determining bounds for N100-1) or indirectly from N00-2 data (when determining bounds on the other experiments) to obtain dV/dL|i and then

(1) Ehinger, M.; Chesnay, B.; Creusot, C.; Damico, J.; Johnson, S.; Wuester, J.; Masuda; S.; Kajii, M. Solution Monitoring Applications for the Rokkasho Reprocessing Plant. In International Conference on Facilities Operations-Safeguards Interface, Charleston, SC, 2004. (2) Higuchi H.; Takeda, S.; Uchikoshi, S.; Watanabe, Y.; Kaieda, K.; Sellinschegg W. D.; Binner R. High Quality Tank Calibration Study. In IAEA Nuclear Safeguards Symposium, Vienna, IAEA-SM-367/8/02/P, 2001.

Ind. Eng. Chem. Res., Vol. 47, No. 5, 2008 1545 (3) De Ridder, P.; Binner, R.; Richet, S.; Peter, N. Tank Calibration Data Evaluation. Presented at the 45th INMM Conference, Orlando, FL, 2004. (4) Liebetrau, A. M. Volume Calibration for Nuclear Materials Control: ANSI N15.19-1989 and Beyond. In IAEA Nuclear Safeguards Symposium, Vienna, 1994, Vol. 1, pp 379-91. (5) Caviglia, M.; Foggi, C.; Hunt, B.; Pestana, S. Aerostatic Effects in Volume Measurement. In 36th INMM Meeting, Palm Desert, CA, 1995. (6) Cauchetier, P. Etalonnage d’une CuVe de Comptabilitie par Mesure de Pression de Bullage. In 15th ESARDA Symposium, Rome, Italy, 1993. (7) Hoshoma, T.; Aritomi, M.; Kawa, T. Formulas to Correct Excess Pressure and Pressure Shift to be used in Volume Measurement for Plutonium Nitrate Solution. Nucl. Technol. 2000, 129 (2), 218-235. (8) Vandekendelaere, S. Investigations on Bubble Behaviour at a Dip Tube for Enhanced Densitometry and Level Measurements, M. Eng. Thesis, University Ghent, 2006. (9) Liebetrau, A. M. A Single Model Procedure for Estimating Tank Calibration Equations, Pacific Northwest National Laboratory Report, PNNL-11760, 1997.

(10) Kendall, M.; Stuart, A. The AdVanced Theory of Statistics, Vol. 1; Charles Griffin & Company Limited: London, 1977. (11) Anderson T. W. An Introduction to MultiVariate Statistical Analysis; John Wiley & Sons, Inc.: New York, 1958. (12) Fan, J.; Gijbels, I. Local Polynomial Modelling and Its Applications; Chapman & Hall: London, 1996. (13) Bozdogan, H. On the Information-Based Measure of Covariance Complexity and its Application to the Evaluation of Multivariate Linear Models, Commun. Stat., Theory Methods 1990, 19 (1), 221-278.

ReceiVed for reView December 15, 2006 ReVised manuscript receiVed November 6, 2007 Accepted November 16, 2007 IE061615T

Practical Issues Relating to Tank Volume Determination - American

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