= Residual correction in tie-line sensitivity
{title2=$\mathbf h=\mathbf r^TW/(\mathbf d^TW\mathbf d)$}
The \b[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 $q=\mathbf d^TW\mathbf d$, residual $\mathbf r=\mathbf y-\mathbf a-\lambda_0\mathbf d$, and row vectors $\mathbf g=\mathbf d^TW/q$, $\mathbf h=\mathbf r^TW/q$, its differential is
$$
\delta\lambda_0=\mathbf g[\delta\mathbf y-(1-\lambda_0)\delta\mathbf a-\lambda_0\delta\mathbf b]
+\underbrace{\mathbf h(\delta\mathbf b-\delta\mathbf a)}_{\text{residual correction}}.
$$
The 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 give
$$
\lambda_0=0.6875,\quad \mathbf r=(0.025,0.025),\quad
\mathbf g=(-1.25,1.25),\quad \mathbf h=(0.078125,0.078125).
$$
Increasing 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>.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1.3–5.1.4, sensitivity coefficients; §5.2.2, correlated-input propagation]{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>.
Back to article page
