Mass balances, degrees of freedom and conservation constraints

Include storage and chemical changes correctly

The rate of accumulation of total mass is mass rate in minus mass rate out for the ordinary chemical-process model used here. Chemical reaction rearranges species but does not generate total mass. A species account can include reaction production and consumption; an elemental account follows conserved atoms across the stated reactions.

Steady state sets accumulation to zero; it is an assumption about the quantities within the boundary over time. A separate original tank with 50 kg initially, 40 kg/h in and 30 kg/h out gains (40−30)(0.5)=5 kg in half an hour if those rates stay constant. It ends at 55 kg. A steady-state equation would incorrectly erase that accumulation.

Count genuinely independent equations

A new original mixer receives 40 kg/h containing 0.25 A by mass and an unknown pure B addition F. The product flow P must have A fraction 0.10. Its A balance is 10=0.10 P, giving P=100 kg/h. The total balance 40+F=P then gives F=60 kg/h. Both flows are nonnegative and the separate B account also closes.

Degrees of freedom count unknown variables minus independent equations, including justified specifications and constitutive relationships. Here F and P are two unknowns with two independent balances after the composition target is supplied. The B balance is not a third independent equation: it follows by subtracting the A balance from the total.

A zero count is not sufficient proof

If the product fraction is also unknown, there is an additional variable and a useful specification is missing. If someone additionally requires P=80 kg/h while keeping the 0.10 target and original feed, the A account would demand 8 kg/h out against 10 kg/h in with no declared storage or reaction. That specification set is inconsistent with the model.

An equal number of equations and unknowns does not by itself prove consistency, physical feasibility or a unique nonlinear solution. Check independence, units, compositions, signs and relevant state constraints. A negative calculated stream can indicate an impossible assumed direction or an incompatible target. Do not repair it by changing the reported value without examining the account and assumptions.

Use a redundant balance as a check

The solved product contains 10 kg/h A and 90 kg/h B. Its B input is 30 from the original feed plus 60 from the pure B addition, matching 90 kg/h out. This redundant check can expose an arithmetic or transcription error even though it adds no independent information for solving F and P.

The conflicting 80 kg/h product proposal leaves an A residual of 2 kg/h. That does not establish an actual leak or reaction; it demonstrates that the supplied specifications and assumptions cannot all hold together. Identify whether the model boundary, storage assumption, composition or supplied data needs correction before drawing a physical conclusion.

Try the reasoning

1. Does reaction create total mass in this ordinary chemical-process account?

   Show the answer and reasoning

   No. It changes species amounts while conserving total mass and the relevant elements within the stated model.

2. Why can the B balance be useful even when redundant?

   Show the answer and reasoning

   It checks the calculation and transcription without adding an independent constraint.

3. Does a zero degree-of-freedom count guarantee one physical solution?

   Show the answer and reasoning

   No. The equations can still be inconsistent, physically infeasible or have multiple nonlinear solutions.

What to carry forward

Use independent conservation constraints to solve and diagnose a process, then check the solution’s consistency and physical meaning.