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This manuscript presents a discussion on the algebraic state update step performed during recursive filtering of the differential–algebraic equation (DAE) systems. Existing DAE state estimation approaches follow a two-step state update procedure at each sampling instant. In particular, they first estimate the differential states using the Kalman update, and then update algebraic states by explicitly solving the algebraic equations. Specifically, for the case of DAE systems involving linear algebraic equations though the differential equations are nonlinear, we show that when appropriately initialized, this two-step state update procedure is not needed. It can instead be replaced with a one-step state update procedure that computes the differential and algebraic state estimates simultaneously through the Kalman update. The satisfaction of algebraic equations is guaranteed by this one-step update without it being explicitly enforced. Towards this end, we show that the error covariance matrix of augmented states, when properly initialized, satisfies a null-space property after prediction and update step at each sampling instant. This property ensures that the state estimates obtained using the proposed one-step update approach, satisfy the algebraic equations. This holds for both analytical linearization based extended Kalman filtering and statistical linearization based sigma-point filtering approaches. We also propose a heuristic-based update procedure for state estimation of DAE systems that involve nonlinear algebraic equations. This procedure draws out inferences from the case of DAE systems involving linear algebraic equations and is based on the analysis of algebraic equations residuals obtained from the updated differential and algebraic state estimates with a one-step state update. The efficacy of the proposed state update procedures is demonstrated by performing simulation studies on a benchmark drum boiler system case study. Results demonstrate that the proposed update procedures satisfactorily estimate the differential and algebraic states of a DAE system when compared to the traditional two-step update procedure. • One-step state update proposed for DAE state estimation. • Approach exact for linear algebraic equations coupled with nonlinear differential equations. • Null-space property satisfaction by augmented prediction-error covariance matrix established. • Valid for both analytical linearization, and statistical linearization based approaches. • Heuristic one-step state update for DAEs comprising nonlinear algebraic equations proposed. [ABSTRACT FROM AUTHOR]

Niobium and tantalum are valued for their uses in nuclear, steel, aviation and electronics industries. This paper elucidates the development of a multistage counter-current separation process for obtaining pure niobium and tantalum from the hydrofluoric acid solution of Coumbite-Tantalite ore (leach liquor), with trioctyl amine (TOA) as the extractant solvent. Contrary to the conventionally used multistage process this novel route does not use any other acid in addition to hydrofluoric acid, resulting in the elimination of the generation of mixed fluoride waste. Reaction equilibrium constants for the reactions involving the extraction of hydrofluoric acid, niobium and tantalum have been determined and were used to generate equilibrium distribution curve of niobium and tantalum in the organic and aqueous phase. The number of counter-current stages required for an appropriate organic to aqueous phase volume (O/A) ratio was determined using McCabe-Thiele calculations. Selective extraction of 99% of tantalum from the leach liquor was achieved using 5 v / v % TOA and two counter-current stages for an O/A ratio of four. The tantalum lean raffinate was treated with commercial grade hydrofluoric acid to form the niobium feed solution having 16 M free acidity. Near complete extraction of niobium was achieved using a three stage counter-current process with O/A ratio of two and 10 v /v% TOA as the organic extractant. The calculations for the stage-wise variation of concentrations were validated by batch experimental simulation of counter-current processes. • A trioctyl-amine based continuous multistage counter-current process for niobium-tantalum separation has been developed. • Multiple numbers of HF molecules combines with one molecule TOA depending on the free acidity of the aqueous phase. • Selective extraction of tantalum from the leach liquor is achieved with low TOA concentration in the organic phase. • The stage-wise variation of metal concentrations was elucidated by batch experimental studies. [ABSTRACT FROM AUTHOR]