Calculation of effective area A 0 for six piston cylinder assemblies of pressure balances. Results of the EUROMET Project 740

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1 INSTITUTE OF PHYSICS PUBLISHING Metrologia 42 (25) S197 S21 METROLOGIA doi:1.188/ /42/6/s11 Calculation of effective area A for six piston cylinder assemblies of pressure balances. Results of the EUROMET Project 74 G Molinar 1, M Bergoglio 1, W Sabuga 2, P Otal 3, G Ayyildiz 4, J Verbeek 5 and P Farar 6 1 -CNR, now Istituto Nazionale di Ricerca Metrologica (I.N.RI.M.), Torino, Italy 2 Physikalisch-Technische Bundesanstalt (), Braunschweig, Germany 3 Laboratoire National de Metrologie et d Essais (), Paris, France 4 Ulusal Metroloji Enstitusu (), Gebze Kocaeli, Turkey 5 Nederlands Meetinstituut ( VSL), Delft, The Netherlands 6 Slovensky Metrologicky Ustav (), Bratislava, Slovakia Received 26 August 25 Published 9 November 25 Online at stacks.iop.org/met/42/s197 Abstract Six European metrological institutes compared their calculation methods, and estimated uncertainties, resulting in the derivation of the effective area A (2 C, p atm ) of six piston cylinder assemblies routinely in use in pressure balances that are primary standards of the pressure laboratories, operating in different pressure ranges in gas (up to 1 MPa) and in liquid (up to 1 GPa). The results for A and its standard uncertainty u(a ) are presented and compared. 1. Introduction The derivation of effective area A (t ref,p atm ) of a piston cylinder unit used in pressure balances is of great importance, representing the largest uncertainty contribution to pressure measurements. A CCM key comparison of methods used to derive A was performed (CCM.P-K1a with data available on the KCDB of BIPM and 1) using piston cylinder units of high regularities. Despite the presence of a large diametrical variation in shape for one of the units of the comparison, the calculation results of participants were in excellent agreement. The purpose of this study was to study piston cylinder units with full dimensional characterizations, but with non-ideal shapes of the piston and cylinder with a variety of different diameters for use in pressure balances from a few MPa to 1 GPa full scale; compare the values of A (t ref, p atm ) derived by participants including the evaluation of its uncertainty u(a ); study some key points particularly in the inlet and outlet cylinder zone where dimensional measurements are difficult to make. 2. Piston cylinder units under study and their dimensional measurement results The main characteristics of the six piston cylinder units under study are summarized in table 1. For all the units, the radii of piston and cylinder were averages of values obtained for at least two different azimuthal directions and from repeated measurements in each direction. The value z = is the initial part of the engagement length at the bottom part of the piston cylinder where the maximum pressure is applied. In figure 1 the piston cylinder radial clearance h for all six units is given versus the normalized z value; the normalization is made according to the specific engagement length l value, which is different for each of the units. 3. Calculation methods 3.1. Method used by Two approaches were used to calculate the zero-pressure effective area (A ); in the first one it was calculated as A = πr 2 1+ h + 1 l / (u + U) l 1 dx r r h 3 h dx, (1) /5/6197+5$3. 25 BIPM and IOP Publishing Ltd Printed in the UK S197

2 G Molinar et al Table 1. Details of the six piston cylinder units under study. Name Pressure range Nom. radius /mm Nom. area /mm 2 E p /GPa ν p -DH35 (5 to 35) kpa PG 4 (.2 to 1) MPa DH2L (.4 to 2) MPa NNct (2.5 to 1) MPa DH5L (5 to 5) MPa DH7594 (.1 to 1) GPa E c /GPa ν c Piston cylinder engagement length l/mm Steps of dimensional data/mm Standard uncertainty of piston radius u(r)/nm Standard uncertainty of cylinder radius u(r)/nm from 1.1 to from to z/mm 6+1 z/mm.22 a 9.5 z/mm 9.5 z/mm a A steel sleeve is mounted around the cylinder (E = 2 GPa, ν =.29) DH35.14 PG h / mm DH5L 1NNct DH2L - DH Normalized z according to specific l values Figure 1. Radial clearance h versus the normalized z value for the six piston cylinder units under study. Normalization is obtained for each single unit using the l values as in table 1. (This figure is in colour only in the electronic version) which is valid in the case of measured (gauge) pressure (p) being equal to zero 2. Here x is an axial coordinate with x = ; l corresponding to the piston cylinder engagement length; r is the piston radius at x = ; h is the radial clearance width at x = ; u and U are piston and cylinder shape deviations along the engagement length, u = r r, U = R R ; and r and R are piston and cylinder radii along the engagement length, R = r + h and h = R r. In the second approach, the effective area A p (p) at pressure p was calculated at several pressures close to zero, and then A was determined by a linear extrapolation of A p (p) to p =. A p was calculated as S198 A p = πr 2 1+ h r + 1 r p l p x d (u + U) dx dx, (2) where p x is the gauge pressure distribution along the piston cylinder gap. For gas-operated piston cylinder assemblies p x was calculated from ( x ) 1/2 p x = pamb 2 + p(p +2p amb) 1 dx/h3 l p amb, dx/h3 (3) where p amb is ambient pressure. For oil-operated assemblies p x was calculated from z η(p x + p amb ) dx p x (x) = p 1 ρ f (p x + p amb ) h 3 (x) / l η(p x + p amb ) dx, (4) ρ f (p x + p amb ) h 3 (x) Metrologia, 42 (25) S197 S21

3 Effective area A for six piston cylinder assemblies where ρ f and η are the pressure-dependent density and dynamic viscosity of the liquid. In all liquid-operated assemblies di(2)- ethyl-hexyl-sebacate is used, whose density and viscosity were calculated according to 3, 4 for a temperature of 2 C. Three main sources of uncertainty were considered: numerical procedures uncertainty, uncertainty of the dimensional data and uncertainty of the points indicated in the data sets as obtained by extrapolation. The calculating parameters were varied and the corresponding changes in the effective area taken as uncertainties. All A values calculated from equations (1) and (2) are always identical Method used by calculations followed equation (1). Also another formulation was used: A = πr l 1/h2 dz r l 1/h3 dz + 2 l (r r )/h 3 dz r l. (5) 1/h3 dz The results for equation (5) were the same as for equation (1). For the uncertainty evaluation, equation (1) was transformed into a function of r,rand R, given by equation (7) in section 3.4. Equations (1), (5) and (7) are equivalent. In order to calculate u(a ) the following equation is used: u 2 (A ) = ( A / r ) 2 u 2 (r ) +( A / r) 2 u 2 (r) + ( A / R) 2 u 2 (R) +2( A / r )( A / r)u(r )u(r)ρ(r,r) +2( A / r )( A / R)u(r )u(r)ρ(r,r) +2( A / r)( A / R)u(r)u(R)ρ(r, R). (6) Notations are as described in section 3.1; u(r ), u(r) and u(r) are the standard uncertainties of r, r and R, respectively. In equation (6) ρ is the correlation coefficient between the two concerned quantities. All calculations assumed a full correlation (ρ = 1). A special case is represented by the -DH7594 (1 GPa) unit where u(r) and u(r) are functions of z values. Here the approach was similar to that used by. The variation of effective area A was calculated as a consequence of r variation of ±4 nm, R variation of ±6 nm, r variation ±z 5 nm and R variation ±z 1 nm. The largest contribution was selected and the final result was calculated from the root mean square of the squares of each contribution. The final uncertainty value u(a ) is the same as that calculated by Method used by The calculation of A was made following equation (1). The numeric integral value in equation (1) was calculated by Simpson s 3/8 Rule (for piston cylinder units in liquid) and by a method for unequally segmented data (for piston cylinder units in gas). The uncertainty of the effective area is expressed by: u(a ) = 2u(r) A (A /π),.5 where u(r) = (u(r abs )) 2 + (u(r straightness )) 2 + (u(r roundness )) 2.5. Metrologia, 42 (25) S197 S Method used by The effective area at atmospheric pressure A of each piston cylinder assembly is calculated from the equation: L A = πr r + ((r + R)/(R r)3 ) dz L (1/(R. (7) r)3 ) dz The numerical integrations are calculated using Simpson s method with 3 linear interpolations. For more than 3 linear interpolations, relative deviations of the effective areas are less than 1 1 7, except for the PG 4 piston cylinder, where a negative clearance generates difficulties in applying the calculation method. In the same way, nonlinear interpolations (polynomial and FFT interpolations) give insignificant relative deviations. The most significant deviations come from the value of the integration length. The relative deviation of the effective area, considering only 9% of the engagement length, ranges from to For the calculation of A uncertainty the variance V(A ) of each piston cylinder assembly is calculated from equation (6) using correlation coefficients ρ = 1,.8 and.6. For the piston cylinder, the uncertainties of the radius depend on the z axis. In this case the uncertainty of the effective area is estimated as u(a ) = (A max A min)/2 with A max = A (r + u(r); R + u(r)) and A min = A (r u(r); R u(r)) Method used by The effective area was simply calculated as the average of all the dimensional data to determine the neutral surface as an average of the piston and cylinder radius. For the uncertainty of A, the standard deviations of piston and cylinder dimensions were divided by the square root of the number of points; this is taken as the uncertainty due to shape variations. The total uncertainty is the root mean square of the sum of the squares of the shape deviation and uncertainty contributions of u(r) and u(r). When possible, roundness uncertainties were included Method used by The upward force that acts on the floating piston consists of a force due to actual pressure acting on both piston ends, a force due to vertical components of the fluid pressure in the clearance acting on the flanks of the piston that are not vertical, and a frictional force in the clearance. The sum of the corresponding forces divided by the pressure difference is the effective area of the piston gauge: where A = A 1 + A 2 + A 3, π A 1 = (r 2 p p p rl 2 p L), L A 2 = 2π L p(z)r(z) dr(z) p p L dz dz, A 3 = π L r(z)h(z) dp(z) dz, p p L dz S199

4 G Molinar et al DH35 PG 4 -DH2L -1NNct -DH5L -DH7594 (A,i -A )/A Piston-cylinder unit / Institute Figure 2. Deviation of A,i values as determined by each laboratory from the average A value, for the six piston cylinder units under study. The average A value for each unit is calculated as the arithmetic mean of the determinations of all the six laboratories. and r, r L and p, p L are the piston radii and corresponding pressures at the lower and upper ends of the piston, r(z) is the piston radius at the z coordinate along the piston engagement, h(z) represents the piston cylinder clearance at the z coordinate. The pressure distribution along the piston cylinder engagement is calculated using a formula similar to equation (3). The effective area calculation is performed iteratively starting with a linear pressure distribution. In all subsequent iterations the actual crevice h(z) is replaced with the effective crevice that accounts for the effect of effective area gas species and mode of operation dependences. A second order slip flow model is used for modelling this effect and the effective crevice h ef (z) is included in the calculations. The uncertainty calculations were numerically performed and the overall effective area uncertainties were calculated as the square root sum of squares of the individual uncertainty contributions in the numerical integration of the above integral equations. For gauges operating with incompressible pressure-transmitting media, the pressure distribution along the engagement length was calculated using a similar approach but without using slip coefficients. 4. Results and analysis In figure 2, for each unit, the results obtained by each laboratory are given. For each unit the results obtained by each laboratory are expressed as a deviation from the average value. The bars in the graphs represent the standard uncertainty u(a )/A obtained by each laboratory. On A values it can be observed that the agreement of A values between // ranges (considered as a maximum to minimum difference) from to ; the agreement of A values between all participants for the two units in gas ranges from to ; S2 the agreement of A values between all participants for the four units in liquid ranges from to ; the simplified method based on the averaging of diameters produces results that are strictly related to the geometrical quality of the units. For this reason this calculation method has to be avoided. On A uncertainty it can be observed that the derived estimated uncertainties range from to more than depending on the specific unit and on the method used to calculate uncertainty; only for piston cylinders of large effective area (typically >5 mm 2 ) and of high geometrical regularity is it possible to obtain uncertainties of A below ; there is a full agreement in the uncertainties calculated by, and as the only difference is the assumption of the degree of correlation between u(r) and u(r). has assumed no correlation, while and assume they are correlated. 5. Conclusions On the determination of A there is, in some cases, excellent agreement within ; if extremely simplified methods (e.g. by overall averaging of the piston cylinder diameters) are used there is a deterioration in this agreement. It is absolutely necessary to treat all available data in such a way as to avoid any conflict of diameter value (e.g. the unit PG 4), otherwise a logical and unique solution cannot be obtained. There is a need for a more detailed analysis of uncertainty. Some methods are not well adapted if the radius Metrologia, 42 (25) S197 S21

5 Effective area A for six piston cylinder assemblies uncertainty is a function of the vertical coordinate z. In such cases simplified methods to derive uncertainties cannot be used. The two methodologies used by are a good approach to give a solution to the problem of checking a single derivation of A (for example by equations (1) or(5) or (7) that are equivalent between them) with a method, still related to the Dadson theory, in which the value of A is obtained by an extrapolation of a linear fit using, e.g., equations (2) (4). Measurements of the cylinder in a position close to the entrance and exit zones are difficult (e.g. below 1 mm); in such a case it is advisable to extrapolate values using those closest to the specific points in consideration. It is crucial to have a sufficient number of diameters. Piston or cylinder generatrices should be measured in millimetre increments or less. For some of the units if only 9% of the engagement length between the piston cylinder is considered, this will produce a change in the calculated effective area between and u(r) and u(r) are the dominant uncertainty components in u(a ). The good agreement in A determination with the use of the Dadson theory 2 by equations (1), (2), (5) or (7) tells only that the different numerical calculations are equivalent. This is not a full verification of the Dadson model itself; more detailed studies will be necessary to test the eventual limitations in the Dadson theory. Even if the shapes of the piston and cylinder enter in the evaluation of the uncertainty of the effective area A via sensitivity coefficients that relate to h 3 or h 4, it is still a matter of investigation to better understand if this is enough as an evaluation of the influences of the shape in the estimation of A uncertainty of the piston cylinder unit. It is confirmed that only for a piston cylinder of a large effective area, and having a shape not worse than the one described here and having standard uncertainty of diameters of the order of 3 nm to 4 nm, is it possible to obtain standard uncertainties of A below An extensive report with more details on all calculation methods of all participants is given in 5. References 1 Molinar G et al 1999 Metrologia Dadson R S, Lewis S L and Peggs G N 1982 The Pressure Balance: Theory and Practice (London: HMSO) p 29 3 Vergne P 199 High Temp. High Pressures Molinar G 1998 EUROMET Project 463. Density and dynamic viscosity versus pressure, up to 1 GPa, for the di(2)-ethyl-hexyl-sebacate fluid at 2 C Internal Report R486,, Torino 5 Molinar G, Bergoglio M, Sabuga W, Otal P, Ayyildiz G, Verbeek J and Farar P 25 Calculation of effective area A for six piston cylinder assemblies of pressure balances. Results of the EUROMET Project 74 (Final report) -CNR Technical Report 131 Metrologia, 42 (25) S197 S21 S21