Setup
Active species $i\in\mathcal{A}$, reactions $j=1,\dots,R$. Net stoichiometry matrix
\[N\in\mathbb{R}^{|\mathcal{A}|\times R},\qquad N_{ij}=\nu^{\mathrm{prod}}_{ij}-\nu^{\mathrm{reac}}_{ij}.\]
Feed vectors $Y^{(1)},Y^{(2)}\in\mathbb{R}^{|\mathcal{A}|}$ (stream 1 at $f=1$, stream 2 at
$f=0$). The no-reaction line at mixture fraction $f\in[0,1]$ is
\[M_i(f)=f\,Y^{(1)}_i+(1-f)\,Y^{(2)}_i.\]
Let $y$ be the current ODE state, $y^0$ its initial value, and $\langle\,\cdot\,\rangle$
the expectation under the step’s $\mathrm{Beta}(\alpha_t,\beta_t)$ pdf.
1. Selectivity ray
The accumulated change $\Delta:=y-y^0$ lies in $\operatorname{range}(N)$. The
selectivity ray is taken directly as this accumulated extent:
\[d = \Delta = y - y^0.\]
This preserves the true reaction direction without distortion from a nonnegativity clip —
important when products form by more than one route (parallel reactions, intermediates).
Near $y^0$ (where $\Delta\approx 0$ and the direction is ill-conditioned) the code falls
back to the mass-action projection $d = N\,r(y)$, where $r(y)$ is the vector of
instantaneous mass-action rates.
The selectivity ray $d=\Delta$ records the net moles
of each species consumed or produced per unit of accumulated reaction progress. Its
components $d_i$ are the net stoichiometric coefficients of a single effective lumped
reaction encoding the current mixture of all underlying reaction pathways.
The ray $d$ encodes the current selectivity: which species have been consumed or
produced and in what ratio. It rotates as the ODE state evolves, tracking the shift
between competing pathways.
2. The complete-reaction limit $B(f)$
$B_i(f)$ is built by reacting the mixing line $M(f)$ to completion along the ray $d$
— advancing as far as possible without any species going negative.
Feeds are pure. Every reactant is fed by exactly one stream — $Y^{(1)}_i=0$ or
$Y^{(2)}_i=0$. This is a required assumption, checked at setup, and cannot be
violated.
Maximum extent. Split the consumed species ($d_i<0$) by which stream feeds them:
\[\kappa_1 = \min_{\substack{i:\,d_i<0\\ Y^{(2)}_i=0}} \frac{Y^{(1)}_i}{-d_i},
\qquad
\kappa_2 = \min_{\substack{i:\,d_i<0\\ Y^{(1)}_i=0}} \frac{Y^{(2)}_i}{-d_i}\]
($\kappa_1$: the extent achievable from the stream-1 side alone; $\kappa_2$: from
stream 2; a side with no consumed species imposes no constraint there). The
complete-reaction kink and its peak extent are
\[f_s^{\star} = \frac{\kappa_2}{\kappa_1+\kappa_2},
\qquad c^{\star} = f_s^{\star}\kappa_1 = (1-f_s^{\star})\kappa_2,\]
giving the two-segment tent:
\[B_i(f) = \begin{cases}
M_i(f) + \dfrac{f}{f_s^{\star}}\,c^{\star}\,d_i & f \le f_s^{\star}\\[8pt]
M_i(f) + \dfrac{1-f}{1-f_s^{\star}}\,c^{\star}\,d_i & f > f_s^{\star}
\end{cases}\]
with control-point peak value $v^{\mathrm{fs}}_i = M_i(f_s^{\star})+c^{\star}d_i$ and
pure-feed endpoints $B_i(0)=Y^{(2)}_i$, $B_i(1)=Y^{(1)}_i$.
Intermediates are safe. A species absent from both feeds has $y^0_i=0$, and since
concentrations are non-negative, $d_i = y_i - y^0_i = y_i \ge 0$. So an intermediate
can never have $d_i<0$; it is never a candidate for $\kappa_1$ or $\kappa_2$.
3. Closure profile $C_i(f)$ and scalar $\lambda$
The closure profile interpolates between the complete-reaction limit $B_i$ and the
no-reaction line $M_i$:
\[\boxed{C_i(f) = B_i(f) + \bigl[M_i(f)-B_i(f)\bigr]\,\lambda_i}\]
$\lambda_i=0$ gives the pure complete-reaction limit; $\lambda_i=1$ gives pure
no-reaction mixing. $\lambda_i$ is set so that $\langle C_i\rangle = y_i$:
\[\lambda_i = \operatorname{clip}_{[0,1]}\frac{y_i - \langle B_i\rangle}{\langle M_i\rangle - \langle B_i\rangle}.\]
Segment integrals use the regularised incomplete Beta function $I_x(\alpha,\beta)$:
\[\int_{f_a}^{f_b}(sf+b)\,p_\beta\,df
= s\,\frac{\alpha_t}{\alpha_t+\beta_t}
\bigl[I_{f_b}(\alpha_t{+}1,\beta_t)-I_{f_a}(\alpha_t{+}1,\beta_t)\bigr]
+ b\,\bigl[I_{f_b}(\alpha_t,\beta_t)-I_{f_a}(\alpha_t,\beta_t)\bigr].\]
Global $\lambda$ for pure cross-stream. Since $M_i$ is linear,
$\langle M_i\rangle = M_i(\bar{f}) = y^0i$, and $B_i - M_i = c{\max}\,d_i$
by construction, so
\[\lambda_i = \frac{y_i - \langle B_i\rangle}{y^0_i - \langle B_i\rangle}
= \frac{d_i(1-\langle c_{\max}\rangle)}{-\langle c_{\max}\rangle\,d_i}
= 1 - \frac{1}{\langle c_{\max}\rangle}\]
— the same scalar for every species — and $\langle C_i\rangle = y_i$ exactly
(no clamp needed). Per-species $\lambda_i$ diverge from this only when the clamp
$[0,1]$ binds.
Scheme: R1: $A{+}B\to R$, R2: $A{+}R\to S$, R3: $A{+}B\to T$, R4: $A{+}T\to S$, R5: $A{+}C\to Q$.
Feeds: $Y^{(1)}=15\,A$, $Y^{(2)}=1.2\,B+1.2\,C$; $\bar{f}=0.0625$.
The effective single reaction. The selectivity ray $d=\Delta$ records the net moles
of each species consumed or produced per unit of accumulated reaction progress. Its
components $d_i$ are the net stoichiometric coefficients of a single effective lumped
reaction encoding the current mixture of all five pathways:
\[|d_A|\,A + |d_B|\,B + |d_C|\,C \;\longrightarrow\; d_R\,R + d_T\,T + d_S\,S + d_Q\,Q\]
The coefficients rotate as the ODE state evolves. The table gives them at each
trajectory’s output-grid time-midpoint ($t=t_{\mathrm{end}}/2$) for three dissipation
rates $\varepsilon$ — the actual $A$ conversion reached at that instant is $77\%$,
$70\%$, and $92\%$ respectively, not a fixed $50\%$; the extent of reaction still
differs substantially at the same relative point because the trajectories are far
from self-similar across $\varepsilon$:
| |
$\varepsilon=10^{-6}$ |
$\varepsilon=1$ |
$\varepsilon=10^{6}$ |
| $(\alpha_t,\beta_t)$ |
$(0.311,\,4.66)$ |
$(0.372,\,5.58)$ |
$(14.5,\,218)$ |
| kink $f_s^{\star}$ |
$0.1927$ |
$0.1366$ |
$0.0781$ |
| $\lambda$ (global) |
$0.000^{\dagger}$ |
$0.0008$ |
$0.0541$ |
| $d_A$ |
$-0.7225$ |
$-0.6556$ |
$-0.8616$ |
| $d_B$ |
$-0.2422$ |
$-0.3314$ |
$-0.8135$ |
| $d_C$ |
$-0.2421$ |
$-0.2868$ |
$-0.0470$ |
| $d_R$ |
$+0.0040$ |
$+0.2832$ |
$+0.7560$ |
| $d_T$ |
$+0.0000$ |
$+0.0108$ |
$+0.0564$ |
| $d_S$ |
$+0.2382$ |
$+0.0374$ |
$+0.0011$ |
| $d_Q$ |
$+0.2421$ |
$+0.2868$ |
$+0.0470$ |
$^{\dagger}$At $\varepsilon=10^{-6}$ the mean has reached the complete-reaction average, so $\lambda$ clamps to $0$ and $\langle C_i\rangle=\langle B_i\rangle$.
Building the limit at $\varepsilon=1$. The effective reaction at this step is
\[0.6556\,A + 0.3314\,B + 0.2868\,C \;\longrightarrow\; 0.2832\,R + 0.0108\,T + 0.0374\,S + 0.2868\,Q.\]
The complete-reaction limit $B(f)$ is found by advancing $M(f)$ along this reaction as
far as possible without a reactant going negative:
\[c_{\max}(f) = \min\!\left(\frac{15f}{0.6556},\;\frac{1.2(1-f)}{0.3314},\;\frac{1.2(1-f)}{0.2868}\right).\]
$A$ is the only stream-1 reactant, so $\kappa_1=15/0.6556=22.88$. Of the stream-2
reactants, $B$ is tighter than $C$ (smaller value), so
$\kappa_2=\min(1.2/0.3314,\,1.2/0.2868)=\min(3.621,\,4.183)=3.621$, from $B$.
The kink is $f_s^{\star}=\kappa_2/(\kappa_1+\kappa_2)=3.621/26.50=0.1366$, with peak
extent $c^{\star}=f_s^{\star}\kappa_1=3.127$. Peak values $v^{\mathrm{fs}}_i=M_i(f_s^{\star})+c^{\star}d_i$
(e.g.\ $v^{\mathrm{fs}}_R = 3.127\times0.2832=0.886$) and resulting averages and closure scalars:
| $\varepsilon=1$ |
$\langle M_i\rangle=y^0_i$ |
$v^{\mathrm{fs}}_i$ (mol/m³) |
$\langle B_i\rangle$ (mol/m³) |
$y_i$ (mol/m³) |
$\lambda_i$ |
$\langle C_i\rangle$ |
| $A$ |
0.9375 |
0.000 |
0.2814 |
0.2819 |
0.0008 |
0.2819 |
| $B$ |
1.1250 |
0.000 |
0.7934 |
0.7936 |
0.0008 |
0.7936 |
| $C$ |
1.1250 |
0.139 |
0.8379 |
0.8382 |
0.0008 |
0.8382 |
| $R$ |
0.0000 |
0.886 |
0.2835 |
0.2832 |
0.0008 |
0.2832 |
| $T$ |
0.0000 |
0.034 |
0.0108 |
0.0108 |
0.0008 |
0.0108 |
| $S$ |
0.0000 |
0.117 |
0.0374 |
0.0374 |
0.0008 |
0.0374 |
| $Q$ |
0.0000 |
0.897 |
0.2871 |
0.2868 |
0.0008 |
0.2868 |
Every $\lambda_i$ equals the global scalar and $\langle C_i\rangle=y_i$ to the quoted digits.
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