Tie-line phase-fraction sensitivity ($\delta \lambda$)

Tie-line phase-fraction sensitivity measures how an inferred phase fraction changes when the overall composition or the measured liquid-liquid tie line endpoints change. A sensitivity coefficient is a local partial derivative of the inference formula, rather than an uncertainty by itself. It becomes an uncertainty contribution only after combination with the input covariance matrix.^[ref]

For one composition coordinate of a binary mixture, write the lever rule aswhere $a,b$ are endpoint mole fractions, $z$ is the overall mole fraction and $\lambda$ is the mole-based amount of phase $\beta$. Differentiating the mass balance givesThus a measurement error in the overall composition and an equal measurement error in the phase-$\alpha$ endpoint have different effects. The conditioning of tie-line phase-fraction inference deteriorates as the endpoint separation becomes small: the same composition measurement error then produces a larger change in the inferred phase fraction.

For an original example, take $a=0.20$, $b=0.80$, $z=0.44$, giving $\lambda =0.40$. Increasing only $a$ by $0.003$ gives a first-order change $\delta \lambda =-0.003$. Direct substitution gives $\lambda =0.237/0.597\approx 0.396985$, a change of $-0.003015$. For the closer endpoints $a=0.20$, $b=0.26$ and balanced composition $z=0.224$, the local sensitivity to the same endpoint error is ten times larger. This is a property of the inference geometry; it does not establish whether either hypothetical pair is in liquid-liquid equilibrium.

The multidimensional tie-line phase-fraction fit has an additional detail away from an exactly balanced composition. On independent mole fraction coordinates, hold a symmetric positive-definite matrix $W$ fixed and definewith distinct endpoints. Define row vectors $\mathbf{g}={\mathbf{d}}^{T}W/q$ and $\mathbf{h}={\mathbf{r}}^{T}W/q$. Differentiating the generalized least squares normal equation ${\mathbf{d}}^{T}W\mathbf{r}=0$ givesThe residual correction in tie-line sensitivity vanishes when $\mathbf{r}=0$. Otherwise, moving an endpoint also rotates the fitted line relative to the nonzero ternary tie-line consistency residual. Omitting that contribution is an additional approximation. If $W$ itself changes with the measured inputs, its derivative must also enter the calculation.

These formulas describe the unconstrained fit locally. Imposing $0\le \hat{\lambda }\le 1$ introduces boundary effects, and coincident endpoints make this inference undefined. A practical sensitivity check recomputes the fit after small positive and negative perturbations of one measured coordinate, keeping composition closure and the stated weight matrix. Agreement as the perturbation shrinks tests the derivative calculation; it does not validate the underlying equilibrium model.

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]

The conditioning of tie-line phase-fraction inference describes how strongly composition perturbations change an inferred phase fraction. For a binary mixture with distinct endpoints, the lever rule gives $\lambda =(z-a)/(b-a)$. Its absolute sensitivity to $z$, with endpoints fixed, is $1/|b-a|$. This is an absolute amplification coefficient; a relative condition number additionally depends on the chosen input and output scales.

For an interior $0\le \lambda \le 1$, the differential in tie-line phase-fraction sensitivity yieldsto first order. If each composition perturbation is bounded by $\epsilon$, this becomes $2\epsilon /|b-a|$. For original stipulated $\epsilon =0.001$, endpoint separations $0.50$ and $0.05$ give local worst-case bounds $0.004$ and $0.040$. Tenfold closer endpoints make the same composition precision tenfold less informative about phase amount. Correlations can change the attainable combinations of errors, as common affine calibration cancellation in the lever rule illustrates.

For fixed multicomponent endpoints and overall-composition covariance matrix $\Sigma$, the tie-line phase-fraction uncertainty isThe denominator is endpoint separation after a whitening transformation: separation must be compared with measurement precision, not only plotted distance. This follows from covariance-based uncertainty propagation and the linear fit.^[ref]

Near coincident endpoints the inverse problem becomes poorly informative, and at coincidence it is undefined. Near the physical bounds, a large propagated uncertainty also warns that the interior linear approximation may fail. More numerical digits in the fitted answer do not resolve this measurement limitation.