IDEAL PLUG FLOW REACTOR
Characteristics of ideal plug flow
PERFECT MIXING IN THE RADIAL DIMENSION (UNIFORM
CROSS SECTION CONCENTRATION)
NO MIXING IN THE AXIAL DIRECTION, OR NO AXIAL
DISPERSION (SEGREGATED FLOW)
TRACER PULSE INPUT AT t = 0 TRANSLATED TO EQUAL PULSE
OUTPUT AT t = τ = L/v (L = PFR length, v = average velocity)
COMPARE WITH CSTR RESPONSE TO TRACER PULSE DISPERSION
In an ideal PFR, concentration is a function of both distance along the flow
path, x, and time, t:
C = C(x,t)
For a mass balance on a reacting compound, take mass balance on
differential axial element with uniform reaction potential (concentration),
where
dV = differential volume
A = cross sectional area
dx = differential distance
and
dV = Adx
Mass balance over differential element on a reactant, C
In = QCx
Out = QCx+dx
Generation = dVrC = AdxrC
Accumulation = dV ∂Cx/∂t = Adx ∂Cx/∂t
QCx – QCx+dx + dVrC = dV ∂Cx/∂t
Cx+dx = Cx + dCx
Q(Cx – Cx – dCx) + dVrC = dV ∂Cx/∂t
−Q ∂Cx/∂V + rC = ∂Cx/∂t = − ∂Cx/∂(V/Q) + rC since Q is constant
(V/Q) =
Cx
rC
Cx
t
is the non-steady state ideal PFR mass balance for a reactant.
At steady state,
∂Cx/∂t = 0
And the ordinary differential can be substituted for the partial differential
dCx/dτ = rC
Comments
1. At steady-state, the concentration of a reactant at any single point
along the PFR is constant at Cx. Overall a stable concentration profile
is obtained at steady state, with the concentration varying in space as
the reaction occurs along the flow path.
2. In an ideal PFR, τ is the absolute residence time for mass flowing
through the reactor, not the average residence time as in a CSTR.
3. Compare ideal batch and ideal PFR mass balances:
Ideal PFR:
dC/dτ = rC
Ideal batch:
dC/dt = rC
Position in a PFR is equivalent to time in a batch reactor
For a 1st order reaction, r = -kC, in a PFR at steady state
dC/dτ = -kC
∫ dC/C = ∫ -k dτ
C_L = C_0 exp(-kτ)
Ideal PFR, steady-state 1st order reaction profile
Example:
Chlorine contact basin for disinfection
Q,
C0
X0
Q,
Ce,
Xe
Where
Q = 0.25 m3/s
A = channel cross section between baffles = 18 m2
rd = rate of microorganism kill in presence of chlorine = -kdX
X = concentration of microorganisms at any point in contact reactor
Xo = influent concentration of microorganisms = 106
E. coli/100 ml
kd = 5 hr-1
rc = rate of chlorine decay (from microorganism Cl-demand) = -kcX
kc = 10-5
(mg-chlorine/L)(#/100mL)-1
hr-1
2 rate expressions, 2 constituents, 2 coupled mass balances
find:
1. reactor volume and flow path length, L, such that XL < 103
cells/100 ml
2. chlorine concentration which must be added to insure that there is
detectable chlorine at PFR exit (detection level = CL = 0.05 mg/L)
1. Steady-state mass balance on cells
XL = Xo exp(-kdτ)
τ = (1/kd) ln(Xo/XL) = (1/5)(hr) ln(106/103) = 1.4 hr
V = Qτ = 0.25 m3/s * 3600 s/hr * 1.4 hr = 1,260 m3
L = V/A = 1,260 m3 / 18 m2 = 70 m
3. Steady state mass balance on chlorine
dCc/dτ = -kcX = -kcXo exp(-kdτ)
∫ dCc = -kcXo ∫ exp(-kdτ) dτ
C_L = Cco − (kcXo/kd) + kcXo exp(-kdτ)/kd
C_L = Cco − (kcXo/kd)(1 - exp(-kdτ))
CCO = 0.05 + (10-5(106)/5)(1-exp(-5(1.4)) = 2.05 mg/L
Chlorine contact PFR
E. coli (#/100 mL)
Cc (mg-chlorine/L)
τ = x/v (hr)