Monte Carlo uncertainty estimation for an oscillating-vessel viscosity measurement

Abstract

This paper discusses the initial design and evaluation of a high temperature viscosity measurement system with the focus on the uncertainty assessment. Numerical simulation of the viscometer is used to estimate viscosity uncertainties through the Monte Carlo method. The simulation computes the system response for a particular set of inputs (viscosity, moment of inertia, spring constant and hysteretic damping), and the viscosity is calculated using two methods: the Roscoe approximate solution and a numerical-fit method. For numerical fitting, a residual function of the logarithmic decay of oscillation amplitude and oscillation period is developed to replace the residual function of angular oscillation, which is mathematically stiff. The results of this study indicate that the method using computational solution of the equations and fitting for the parameters should be used, since it almost always out-performs the Roscoe approximation in uncertainty. The hysteretic damping and spring stiffness uncertainties translate into viscosity uncertainties almost directly, whereas the moment of inertial and vessel-height uncertainties are magnified approximately two-fold. As the hysteretic damping increases, so does the magnification of its uncertainty, therefore it should be minimized in the system design. The result of this study provides a general guide for the design and application of all oscillation-vessel viscosity measurement systems.

Sensitivity analysis of the transient torque viscosity measurement method

“…Also easily observed is the regression failure present in the results when the moment of inertia (I) is too far off the actual value. This is caused by the change in oscillation period cause by variation in I which cannot be accounted for by other variables, especially the viscosity and body force constant, and has been observed in previous analysis [17]. This problem can be remedied by allowing I to vary during the fitting process and then verifying that the fitted moment of inertia matches the measured value afterwards.…”

“…Since previous studies of the uncertainty in oscillating cup viscometer measurements noted a linear relationship between the uncertainty of input parameters and the resultant uncertainty of the viscosity, a full Monte Carlo simulation was deemed unnecessary for the present work [17]. Since the experimental data from the only RMF viscometer experiment is available for use, the parameters of that physical experiment were used as the baseline values for the simulations, and deviations from this baseline were simulated.…”

Monte carlo uncertainty analysis for photothermal radiometry measurements using a curve fit process

“…The first method finds primary use due to its simplicity, both theoretical and practical. The second method, that of Monte Carlo uncertainty propagation, has found favor more recently as personal computers have become sufficiently powerful to handle the computational costs associated with it [17,18,19]. As the name implies, this method is a stochastic estimation of the uncertainty through repeated application of the data reduction process.…”

“…These two groups are defined for each layer in the sample, and are so generally well understood as to require no additional comment here. Equation (18) defines π 1 which is the ratio of effusivities and has no commonly used name, but appears as b i, j in the one-dimensional model. This value was found by Almond and Patel to quantify the thermal mismatch between the two materials, and determines the magnitudes of the reflection and transmission of thermal waves [2].…”

“…Monte Carlo uncertainty propagation has recently found favor as personal computers have become sufficiently powerful to handle the computational costs associated with it; the method follows a similar procedure to that used for sensitivity analysis, but adds stochastic characteristics to reflect the uncertain combination of errors likely to occur during actual experiments [18][19][20]. The Monte Carlo method is ideally suited for measurement procedures with arbitrarily complex calculations used to compute the final measurands from other measured quantities, such as the curve fit used in PTR.…”