Degree-of-Freedom Analysis for the Capstone
Count unknowns and independent equations for a two-component recycle flowsheet with one reaction.
Before you solve a recycle flowsheet with reaction and separation, you can predict whether you have enough information — DOF tells you.
What Changes When Streams Have Composition?
In a two-component system (A, B), each stream typically needs two component flow rates (e.g., ṅA and ṅB) to be fully defined. You can equivalently use total flow + one independent composition, but DOF counting is usually cleanest with component flow rates.
In recycle problems, the added complexity is not just “more algebra” — it's feedback: separator performance (especially VLE) depends on composition, which depends on the recycle, which depends on the separator.
If you write component balances for A and B on a unit, the total balance is usually not independent (it's the sum of the component balances). Count independent equations only.
Capstone DOF Scope (Simplified for Counting)
For DOF bookkeeping, we'll focus on the three main balance blocks:
- Mixer: fresh + recycle → reactor feed
- Reactor: A → B with a specified conversion (or extent)
- Separator (flash): splits into vapor recycle + liquid product
We'll treat the cooler as an energy unit (handled in later lessons). Here we only track material variables.
- Define streams and unknowns using component flow rates (A and B).
- Streams with unknown component flows: recycle vapor R (n_A,R, n_B,R), mixed feed M (n_A,M, n_B,M), reactor effluent E (n_A,E, n_B,E), separator vapor V (n_A,V, n_B,V), separator liquid L (n_A,L, n_B,L).
- Unknown count: 5 streams × 2 components = 10 unknowns. (Fresh feed F is specified and pure A, so it adds no unknown.)
- Mixer equations (component balances): n_A,M = n_A,F + n_A,R; n_B,M = n_B,F + n_B,R. Since fresh is pure A: n_A,F = F and n_B,F = 0. → 2 independent equations.
- Reactor equations: single-pass conversion X_sp specifies consumption of A: n_A,E = (1 − X_sp) n_A,M. → 1 equation.
- Stoichiometry A → B with 1:1 means B produced equals A consumed: n_B,E = n_B,M + (n_A,M − n_A,E). → 1 equation.
- Separator equations using given recoveries to vapor: n_A,V = r_A n_A,E and n_B,V = r_B n_B,E. → 2 equations.
- Liquid is the remainder: n_A,L = n_A,E − n_A,V and n_B,L = n_B,E − n_B,V. → 2 equations.
- Recycle identification: recycle stream equals separator vapor: n_A,R = n_A,V and n_B,R = n_B,V. → 2 equations.
- Total independent equations: mixer (2) + reactor (2) + separator split (4) + recycle identity (2) = 10.
- DOF = unknowns − equations = 10 − 10 = 0 → exactly specified (for material variables) under the given conversion and recovery specs.
- Unknowns remain 10 (component flows for R, M, E, V, L).
- Without separator performance (no r_A, r_B, and no VLE model), you cannot compute V from E, so the two 'liquid remainder' relations depend on V and are not independently useful.
- Count independent equations without separator performance: mixer (2) + reactor (2) + recycle identity (2) = 6.
- DOF = 10 − 6 = 4 → under-specified; you need four more independent relations (e.g., separator model/specs).
In later lessons we replace “given recoveries” with physics: a flash model (VLE) at specified T and P. That supplies the missing independent equations — but it also couples the mass and energy solution because T can become an unknown.
Check your understanding
- For binary systems, each stream carries multiple material variables; DOF accounting must reflect that
- Only independent equations count; totals are often redundant with component balances
- A recycle flowsheet can be exactly determined (DOF = 0) once conversion and separator relations are specified
- Replacing fixed recoveries with VLE adds realism but often increases coupling and the need to iterate