Chemical Engineering August 2021 - 29

conversion in a CSTR [3]. For a firstorder
reaction, the conversion (X)
can be calculated using Equation (1):
NOMENCLATURE
(2)
(1)
This is also a key assumption in
the design of pH-control systems.
Figure 1 also shows a feedback
pH-control scheme with the pH of
the fluid leaving the vessel being
measured and with its value fed
back to adjust the position of the
reagent feed valve. If the assumption
of perfect back-mixing is not
valid, the system will be out of control
and the pH will oscillate around
the desired setpoint. This is particularly
difficult if a strong acid is being
neutralized with a strong base, and
vice versa [4].
For this assumption to be true,
the incoming feed must be instantaneously
blended with the fluid inside
the vessel, which is physically impossible.
But even though instantaneous
blending is not realistic, under
certain circumstances, the assumption
of perfect back-mixing can be
made valid.
A rule of thumb has been proposed
in Ref. 5 relating the mean
residence time to the batch blend
time, θ. This rule of thumb states
that the contents of the continuous
reactor can be assumed to be
perfectly back-mixed if the following
statement is true:
And the momentum generated
by the impeller is defined
in Equation (3):
(3)
where Mo is the impeller's Momentum
number. The Momentum
number can be calculated
from Equation (4) in terms of
the impeller's Flow number [8]:
(4)
Jones and others [7] concluded
that in cases where
Mimp ≥ 100Mjet, flow generated
by the impeller dominates
the flow within the vessel and
perfect CSTR behavior could
be achieved. If bypassing is
being observed in an existing
CSTR, the residence-time distribution
(RTD) can be moved
toward " perfect behavior " by
reducing the momentum of
the feed jet. Since the flowrate,
Qfeed, is fixed by process
requirements, the jet momentum
can be reduced by increasing
the diameter of the
feed pipe, Dfeed.
Ref. 6 provides measurements of
residence-time distribution for three
impellers over a range of flowrates
and, hence, residence times. The
authors concluded that the degree
of bypassing and stagnation (dead
zone behavior) increased as the
feedrate increased, but were unable
to correlate their conclusions to any
ratio of timescales.
The authors of Ref. 7 also carried
out similar experiments and concluded
that the degree of bypassing
observed was dependent on the
ratio of the momentum generated
by the impeller to the momentum of
the feed jet, Mjet. The momentum of
the feed jet is given in Equation (2):
Another consideration in the
design of CSTRs is the location
of the inlet and outlet to the
vessel. The general rule is that
the outlet should be located
far from the inlet [9]. Time-dependent
computational fluiddynamics
(CFD) techniques,
such as large-eddy simulation,
have been successfully applied
to investigate issues relating to
both the momentum ratio and
inlet and outlet positions in
CSTR operation [10, 11].
A
CA
D
Dfeed
Fl
H
kR
Mo
Mimp
Mjet
N
Po
Qfeed
Qimp
Re
ReB
S
T
UBF
Ufeed
Utip
V
VEQ
X
α
β
Ε
-
ε
Γ
ζ
η
μ
θ
ρ
τCirc
τRes
Φ
Ψ
Velocity and circulation time
One of the common scaling
rules for sizing agitated vessels
is constant torque per
unit volume. Torque is the product
of a force and a distance, so it has
the same units of measurement as
CHEMICAL ENGINEERING WWW.CHEMENGONLINE.COM
AUGUST 2021
Vessel cross-sectional area m2
Concentration of A
Diameter of impeller
Diameter of feed pipe in CSTR m
Flow number (= Qimp / ND3)
Liquid depth
Reaction rate constant
Momentum number (= Mimp /
ρN2D4)
Momentum of impeller discharge
flow
m
s-1
-
kg m s-1
Momentum of feed jet in CSTR kg m s-1
Impeller rotational speed
Power number (= P/ρN3D5)
Flow rate of feed in CSTR
Impeller pumping capacity
Reynolds number (= ρND2µ)
Reynolds number at boundary
between the turbulent and transitional
regimes
Fluid specific gravity
Diameter of vessel
Bulk fluid velocity
Velocity of feed jet in CSTR
impeller tip speed
Vessel volume
Equivalent volume
Conversion
Constant in correlation for blend
time
Exponent in correlation for blend
time
Energy input per mass
Power input per mass
Agitator shaft torque
Degree of homogeneity
Impeller pumping efficiency
Fluid viscosity
Blend time in batch system
Liquid density
Mean circulation time (V/Qimp)
Mean residence time in CSTR
(V/Qfeed)
Torque per mass
Torque per volume
rps
-
m3 s-1
m3 s-1
-
-
-
m
m s-1
m s-1
m s-1
m3
m3
-
-
-
J kg-1
W kg-1
Nm
%
kg J-1
kg m-1 s-1
s
kg m-3
s
s
Nm kg-1
Nm-2
kmol m-3
m
work or energy. This means that
torque per volume can be thought of
as the work that is done, or energy
27
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Chemical Engineering August 2021

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