Chemical Engineering July 2020 - 38

Table 1. example calculaTion parameTers for an evaporaTive
crysTallizer
Parameter
Concentration of solute in feed
Solubility of solute in ML at separation
temperature
Concentration factor [marker(ML) / marker(F)]
Feed
Water produced from subsaturation of feed
Saturated feed
Mother liquor
Water produced at crystallization
Total water
Crystal
Natural slurry density at evaporator(s)
Mother liquor removal
Crystal recycle
Slurry density at evaporator(s)
Crystal recycle fraction
Mother liquor removal fraction
Water balance:
W = W1 + W2
Solubility equation:
W2 = (100 - S) / S ✕
crystal
Crystallizer balance:
SF = W2 + crystal + ML
Nomenclature
C
S
CF
F
W1
SF
ML
W2
W
crystal
NSD
MLR
cR
SD
cR#
MLR#
Unit
wt.%
wt.%
#
arbitrary units
(a.u.)
a.u.
a.u.
a.u.
a.u.
a.u.
a.u.
wt.%; g%
a.u.
a.u.
wt.%; g%
#
#
(5)
(6)
(7)
Equations (1), (3), (5) and (7) are straightforward mass
balances. Equations (2), (4) and (6) are concentrationbased.
Because Equation (7) can be derived from Equations
(1), (3) and (5), this equation is not independent and
can be used as a check. Mathematical stability demands
that S ≠ 0.
The above equation set consists of six independent
equations and seven unknown mass flowrates (F, SF,
W1, W2, W, crystal and ML). Consequently, only one
mass flowrate can be chosen freely and fixes the capacity
of the crystallization process. This is fundamental, as
after all, the above given set of equations is valid
irrespective the measurement unit selected for the
mass flows (meaning that ton/s is as justifiable
as mg/yr).
If, for example, a feed flowrate of 100 arbitrary
units (a.u.) is chosen as in Figure 1, all other mass
flows are fixed, assuming that C, S and CF are
known. Of course, if preferred, one of the other
mass flows can be freely chosen, but then the feed
flowrate is fixed.
Consequently, one needs to measure only one
mass flowrate to access all other mass flowrates,
which can be quite convenient, especially at large
production plants. The most suitable mass flowrates
to measure are either the condensate (W) or
the feed (F).
38
Equation (9) shows that the natural slurry density is independent
of mass flowrate and can easily be calculated
from C, S and CF. Also, a simple graph can be constructed
that gives NSD as a function of CF, as shown in
Figure 2, whereby C = S.
The physical reason for the direct causality between
NSD and CF is due to the evaporation of water, as illustrated
in Figure 3. If water evaporates from a saturated
solution, both CF and NSD increase.
Glass and weight percentage
In Table 1, NSD is presented one time as a weight percentage
(wt.%) and one time as a glass percentage (g%).
NSD is first defined as wt.% in Equation (8), which is convenient
for mass-balance calculations.
80
70
60
50
40
30
20
10
1
2
3
4
5
6
Concentration factor (CF)
FIGURE 2. Natural slurry density is a function of concentration factor, but is independent
of mass flowrate
ChemiCal engineering www.Chemengonline.Com July 2020
7
8
Example value
45
50
2
100
10
90
50
20
30
20
28.6; 30.8
15
4
40.7; 47.3
20%
30%
Equation (9):
NSD = 100 ✕
Measurement of the crystal mass flow
or the ML mass flow is more difficult, as
at the point of separation, some mother
liquor remains attached to the crystals,
and tiny crystals may persist in the
mother liquor.
It is evident that the saturated-feed
mass flowrate (SF) is technically inaccessible
for measurement.
An additional feature of this approach
is that measuring two mass flows (for example,
W and F) provides a convenient
method to check the performance of the
two flowmeters involved. Furthermore,
note that C can potentially be larger than
S, resulting in a negative W1 value.
Natural slurry density
The natural slurry density (NSD; wt.%) of a
crystallization process is given in Equation
(8) below:
NSD = 100 ✕
crystal / (crystal + ML)
(8)
Substitution and rearrangement using
Equations (1) through (8) shows that this
NSD equation can be converted into
(C x CF - S) / (C ✕
CF - S + 100)
(9)
NSD (wt.%)
9
10
Natural slurry density (NSD, wt.%)
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