Residual correction in tie-line sensitivity ($\mathbf{h}={\mathbf{r}}^{T}W/({\mathbf{d}}^{T}W\mathbf{d})$)

The residual correction in tie-line sensitivity accounts for rotation of a fitted liquid-liquid tie line when the measured overall composition does not lie exactly on it. In independent mole fraction coordinates, set $\mathbf{d}=\mathbf{b}-\mathbf{a}$, hold a symmetric positive-definite matrix $W$ fixed, and use the interior unconstrained tie-line phase-fraction fit. With $q={\mathbf{d}}^{T}W\mathbf{d}$, residual $\mathbf{r}=\mathbf{y}-\mathbf{a}-{\lambda }_0\mathbf{d}$, and row vectors $\mathbf{g}={\mathbf{d}}^{T}W/q$, $\mathbf{h}={\mathbf{r}}^{T}W/q$, its differential isThe full derivation is in tie-line phase-fraction sensitivity. The normal equation makes $\mathbf{r}$ orthogonal to $\mathbf{d}$ in the chosen metric; it does not make the residual zero.

For an original example, take $\mathbf{a}=(0.60,0.10)$, $\mathbf{b}=(0.20,0.50)$, $\mathbf{y}=(0.35,0.40)$ and $W=I$. The third mole fraction is fixed by closure. These choices giveIncreasing only the first coordinate of $\mathbf{b}$ by $0.001$ predicts a phase fraction change $0.000859375+0.000078125=0.0009375$. Omitting the correction misses the second contribution even though the fit satisfies its normal equation. A centered finite-difference recalculation as the perturbation shrinks tests this derivative directly.

This is a local sensitivity, not an uncertainty until combined with the joint input covariance matrix.^[ref] The formula assumes distinct endpoints and an interior fit with fixed $W$. A changing metric requires sensitivity of a tie-line fit to its weight matrix; a constrained boundary requires a separate local analysis. A nonzero residual can reflect composition noise or model mismatch and does not independently establish liquid-liquid equilibrium.

Sensitivity of a tie-line fit to its weight matrix describes how changing the metric changes an inferred phase fraction, even when the measured compositions are held fixed. For distinct endpoints $\mathbf{a},\mathbf{b}$, define $\mathbf{d}=\mathbf{b}-\mathbf{a}$, $q={\mathbf{d}}^{T}W\mathbf{d}$ and the interior unconstrained tie-line phase-fraction fitDifferentiating the normal equation ${\mathbf{d}}^{T}W\mathbf{r}=0$, with $\mathbf{a},\mathbf{b},\mathbf{y}$ fixed, givesThis term supplements the composition and endpoint derivatives in tie-line phase-fraction sensitivity; it is not included in their fixed-metric formulas.

For an original example, use independent mole fraction coordinates $\mathbf{a}=(0.70,0.10)$, $\mathbf{b}=(0.30,0.50)$ and $\mathbf{y}=(0.45,0.38)$. With $W=I$, the fitted fraction is $0.6625$, $q=0.32$ and $\mathbf{r}=(0.015,0.015)$. Increasing the second diagonal weight from $1$ to $1.02$ predictsDirect refitting gives $0.21424/0.3232\approx 0.6628713$, a change of $0.0003713$. The small difference is the neglected higher-order term.

Two invariances clarify the result. An exactly balanced composition has $\mathbf{r}=0$ and the same fraction under every positive-definite metric. Also, replacing $W$ by $cW$ for positive $c$ leaves the fraction unchanged: the scalar rescaling invariance of a tie-line weight matrix follows from cancellation in its ratio. The differential agrees because ${\mathbf{d}}^{T}W\mathbf{r}=0$.

If $W={\Sigma }^{-1}$ for a composition covariance matrix, its perturbation is $\delta W=-W(\delta \Sigma )W$ to first order. Uncertain fitted weights can therefore add sensitivity, with correlations to the composition measurements retained in the measurement model. This is an application of the guide's general sensitivity-coefficient framework, not a tie-line formula quoted from it.^[ref] Boundary-constrained fits and coincident endpoints need separate treatment.

Scalar rescaling invariance of a tie-line weight matrix means that multiplying every fit weight by the same positive number leaves the fitted phase fraction unchanged. Let endpoint compositions $\mathbf{a},\mathbf{b}$ and overall composition $\mathbf{y}$ be expressed in independent mole fraction coordinates. With $\mathbf{d}=\mathbf{b}-\mathbf{a}$, a fixed symmetric positive-definite matrix $W$ defines the tie-line phase-fraction fit by minimizingFor $c>0$, $Q_{cW}(\lambda )=c{Q}_W(\lambda )$. Multiplication by a positive constant preserves every comparison between objective values, so the sets of minimizers agree. The proof also applies when the same constraint $0\le \lambda \le 1$ is imposed in both fits.

For distinct endpoints, the unconstrained formula makes the cancellation visible:The differential in sensitivity of a tie-line fit to its weight matrix agrees: for a perturbation $\delta W=\epsilon W$, its numerator is $\epsilon {\mathbf{d}}^{T}W\mathbf{r}=0$ by the fit's normal equation. Thus the scalar direction is insensitive even when the residual is nonzero.

In an original example, take $\mathbf{a}=(0.65,0.10)$, $\mathbf{b}=(0.25,0.50)$, $\mathbf{y}=(0.42,0.36)$ and $W=\mathrm{diag}(1,4)$. The remaining composition coordinate is fixed by closure. These values giveReplacing $W$ by $7W$ gives the same fraction, although the minimum objective increases from0.00072to0.00504. Changing only the second diagonal weight from4to7 instead changes the fitted fraction to0.640625. Relative weights determine the geometry; their common multiplier does not.

Point-estimate invariance does not by itself establish invariance of physical measurement uncertainty. Merely rewriting the same objective with $cW$ leaves propagation through the same measured-input covariance matrix unchanged. If instead $W={\Sigma }^{-1}$ represents absolute precision and $cW$ asserts a genuinely different covariance $\Sigma /c$, that is a different noise model. With exact endpoints, the linear-model variance $1/({\mathbf{d}}^{T}W\mathbf{d})$ then decreases by a factor $c$. The measurement model and covariance interpretation must be stated.^[ref]