IDEAL REACTORS

Definition: a reactor is a system (volume) with boundaries. Mass may enter and leave across boundary.

Characteristics:

System:

1. Closed or intermittent: no mass enters or leaves during reaction(s) are batch or semi-batch reactors

2. Open (control volume): mass enters/leaves during reaction(s) are continuous flow reactors

Mixing:

1. Completely mixed: mass is homogeneous throughout system

Batch/semi-gatch

Continuous: Continuous stirred tank reactor (CSTR)

2. Completely segregated: mass does not mix, no dispersion with heterogeneous conditions

Plug flow reactor (PFR)

NON-IDEAL REACTORS

Definition: reactors do not meet ideal conditions of flow and mixing due to:

Dispersion deviates from ideal plug flow conditions

Short-circuiting and dead spaces deviate from ideal mixing and plug flow conditions

Filling and drawing deviate from ideal batch conditions

MASS BALANCE

Mass Inflow + Mass generated = Mass outflow + Mass accumulated

Inflow and outflow terms are associated with mass crossing the system (reactor) boundary

Generation term is associated with reactions (chemical or biological)

Accumulation term is associated with the actual accumulation (or loss) of mass from the system resulting from combined effects of inflow, outflow and reaction.

APPLICATION OF MASS BALANCE

Ideal Batch Reactor, volume = V, reactant concentration = C

mass balance with inflow = out flow = 0

d(VC)/dt = VrC

for constant volume

V dC/dt = VrC

dC/dt = rC

for a first-order reaction where C is consumed from an initial concentration of Co:

rC = -kC

and

dC/dt = -kC

∫ from Co to C dC/C = -k ∫ from 0 to t dt

C = Co exp(-kt)

Ideal Continuous Stirred Tank Reactor (CSTR)

Q = fluid flowrate (m3/d)

V = volume (m3)

Co = influent concentration of C (g/m3)

C = reactor and effluent concentration of C (g/m3)

Steady-flow of water conditions: Qin = Qout = Q and dV/dt = 0

QCo + VrC = QC + V dC/dt

÷ Q

Co - C + (V/Q) rC = (V/Q) dC/dt

Quantity V/Q is defined as the hydraulic residence time (HRT) denoted with the symbol, τ.

For a conservative tracer, rC = 0

Restate mass balance:

Co - C = τ dC/dt

Integrate for CSTR with a step input of tracer, Co beginning at t = 0

∫ from 0 to C dC/(Co - C) = (1/τ) ∫ from 0 to t dt

ln((Co - C)/Co) = -t/τ

C = Co(1-exp(-t/τ))

Step tracer input

CSTR response to step tracer

Note asymptote, as t → ∞, C → C0, which is equivalent to dC/dt → 0, which defines the steady state condition (accumulation = 0)

Example:

Calculate time to reach 95% of the steady-state condition in a CSTR:

C/C0 = 0.95 = (1-exp(-t/τ))

exp(-t/τ) = 1 - 0.95

-t/τ = ln(0.05) = -3

t95% = 3τ

This is characteristic of CSTR flow, and also can be shown to be true in a CSTR with a reaction.

C = C0/(1+kτ) (1-exp(-t/τ))

CSTR with first order reaction and steady-state conditions:

C0 - C + τ(-kC) = 0

C0 - C(1 + kτ) = 0

C = C0/(1 + kτ)