Uncertainty estimates for experimental density measurements: Effects of temperature, pressure and sample preparation Diego O. Ortiz-Vega a , Ivan D. Mantilla a , Hugo Y. Acosta a , Martin A. Gomez-Osorio a , James C. Holste a , Kenneth R. Hall a,⇑ , Gustavo A. Iglesias-Silva b a Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX 77843-3122, USA b Departamento de Ingeniería Química, Instituto Tecnológico de Celaya, 38010 Celaya, Guanajuato, Mexico article info Article history: Received 30 April 2012 Received in revised form 25 September 2012 Accepted 26 September 2012 Available online 8 November 2012 Keywords: Density measurement Uncertainty Composition effect Gravimetric effect abstract A complete description of the combined uncertainty for experimental measurements contains effects from multiple sources. This paper presents an analysis that uses density measurements as examples for which the sources are the measurement uncertainties associated with density, pressure, temperature, composition, gas constant, component masses and molar masses. Applying the analysis to two gas mix- tures reveals the largest sources of error as the apparatus effect followed by the pressure effect. All other effects are minor by comparison in state-of-the-art measurements, but they are easy to include. It is also possible to determine how the accuracy of the balance used in a gravimetric preparation of the mixture affects the density uncertainty employing this technique. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The fluid density is a crucial property for the design and opera- tion of industrial processes. However, knowing the density value is incomplete without having an associated uncertainty. The uncer- tainty plays an important role practically and when developing equations of state [1]; the higher the uncertainty, the smaller the weight for the data during fitting procedures. Determination of the uncertainty of a measurement is a math- ematical description of error propagation during the experiment [2]. Errors propagate in two principal ways, via linear combination, which applies when the major contribution of the error is essen- tially constant (systematic error, bias) or via a square rule combi- nation, which arises when the error is essentially random [3]. The square rule combination is usually applicable for high-quality density measurements [4,5]. It is necessary to specify the variables involved in the uncertainty determination. For example, past work with magnetic suspension densimeters has emphasized three ef- fects [6]: the density measurement (apparatus) effect, which is a quantitative determination of the error coming from the balance accuracy and resolution as well as the force transmission effect [7–10] (the apparatus effect is complicated and extensive descrip- tions of its determination appear in these references); and temper- ature and pressure effects, which describe the accuracy of the thermometry and pressure measurement respectively. However, other properties could be important for uncertainty calculations, such as the gas constant, component mass, molar mass and compo- sition. This paper examines all these potential contributions to the combined uncertainty as well as considering sample preparation: either gravimetric or based upon composition measurements. 2. Theory The mass density as a function of observables with gravimetric sample preparation is q ¼ qðp; T ; R; x i fm i ; M i gÞ: ð1Þ Approximating the relative uncertainty of the mass density re- quires adding the apparatus contribution uðqÞ q  uðqÞ app q þ @q @p   T;R;m i ;M i p q uðpÞ p þ @q @T   p;R;m i ;M i T q uðT Þ T þ @q @R   p;T;m i ;M i R q uðRÞ R þ X N i¼1 @q @m i   p;T;R;m j–i ;M i m i q uðm i Þ m i þ X N i¼1 @q @M i   p;T;R;m i ;M j–i M i q uðM i Þ M i : ð2Þ Squaring equation (2)and neglecting covariance terms 0021-9614/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jct.2012.09.030 ⇑ Corresponding author. E-mail address: krhall@tamu.edu (K.R. Hall). J. Chem. Thermodynamics 58 (2013) 14–19 Contents lists available at SciVerse ScienceDirect J. Chem. Thermodynamics journal homepage: www.elsevier.com/locate/jct