CMFR Steady Mass Balance

Q C_{in} - QC - k C V = 0 (first-order decay)

For a completely mixed flow reactor (CMFR / CSTR) at steady state with first-order decay, C = C_in / (1 + kθ) where θ = V/Q is the hydraulic detention time. A plug-flow reactor (PFR) follows C = C_in e^{−kθ} for the same k, θ — more efficient for positive-order reactions.

$Q$ (m³/d): Flow rate
$θ$ (d): Hydraulic detention time V/Q
$k$ (1/d): First-order rate constant

Step-by-step solved example

CMFR: C_in = 100 mg/L, k = 0.4 d⁻¹, θ = 2.5 d. Effluent C?

CMFR: accumulation = in − out + generation.

1. Steady CMFR
C = 100 / (1 + 0.4×2.5) = 100 / 2.0 = 50 mg/L.
$C = 50 mg/L$

Answer: 50 mg/L

10 practice questions

0/10 correct

1.Detention time θ is

2.PFR first-order effluent is

3.For the same k, θ > 0, first-order removal is greater in a

4.Two equal CMFRs in series approach a

5.Mass in = 12 kg/d, decay 4 kg/d, steady C_out Q =

6.Conservative tracer (k = 0) at SS has C =

7.Units of k for a first-order reaction are

8.A batch reactor with first-order decay: C(t) =

9.θ = 8 h, Q = 4000 m³/d. V =

10.CMFR, kθ = 3, C_in = 80. C_out =