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Currently, the Norden efficiency factor is the most practical form of specifying the
washing efficiency. It is also the most widely used efficiency parameter in computer
aided mass and energy balancing.
Its mathematical derivation is not well documented in the pulping literature,
perhaps because it is rather lengthy. We will nevertheless derive the Norden efficiency
factor here as this provides us with an opportunity to exercise some mass
balancing. This section, in principle, follows Norden’s original approach [29], but
uses a denotation which is more practical for a pulp engineer. The model is based
on a simplified form of the Kremser equation with n mixing stages in series, as
shown in Fig. 5.24 [30,31].
Pulp feed
P, N0, L0, cL,0
Pulp discharge
P, Nn, Ln, cL,n
Filtrate
F1, cF,1
P, N1,
L1, cL,1
F2, cF,2
F3, cF,3
3 n
Wash water
Fn+1, cF,n+1
m
Fm+1, cF,m+1
P, N2,
L2, cL,2
P, Nm,
Lm, cL,m
Fig. 5.24 Countercurrent cascade of n mixing stages in series
with complete mixing in each stage.
The overall mass balance for a soluble substance over all the mixing stages can
be written in the form:
L 0 cL _0 _ F 1 cF _1 _ Ln cL _ n _ Fn _1 cF _ n _1 _ 0 _23_
where L = flow rate of liquor accompanying the pulp (kg s–1); F = flow rate of filtrate
(kg s–1); cL = concentration of soluble substance in liquor accompanying the
pulp (kg kg–1);and cF = concentration of soluble substance in filtrate (kg kg–1).
Note that this is just a liquor balance. The pulp P is assumed to pass along with
L, not participating in the mass transfer, and is therefore disregarded in the equation.
A similar mass balance can be set up for only a subsection of the total system,
starting from stage 1 and including m stages:
L 0 cL _0__ F 1 cF _1 _ Lm cL _ m _ Fm _1 cF _ m _1 _ 0 _24_
Complete mixing is assumed in each mixing stage i, which results in the same
concentrations of accompanying liquor and filtrate leaving a stage:
cL _ i _ cF _ i _ ci _25_
Furthermore, a constant consistency N i is assumed for all the pulp streams.
This means that
Li _ L _ const _ _26_
With the total liquor balance around a mixing stage
Li _1 _ Fi _ Li _ Fi _1 _ 0 _27_
and consequently
Fi _ Fi _1 _ F _ const _ _28_
the mass balance [Eq. (24)] can be simplified
L c 0 _ F c 1 _ L ci _ F ci _1 _ 0 _29_
and solved for ci
ci _
F
L
ci _1 _ c 0 _
F
L
_ c 1_ _30_
Then, the mass balances for the single subsections are:
c 1 _
F
L
c 2 _ c 0 _
F
L
_ c 1_ _31_
c 2 _
F
L
c 3 _ c 0 _
F
L
_ c 1_ _32_
...
540 5 Pulp Washing
cn _1 _
F
L
cn _ c 0 _
F
L
_ c 1_ _33_
cn _
F
L
cn _1 _ c 0 _
F
L
_ c 1_ _34_
Nesting Eq. (34) into Eq. (33) gives
cn _1 _
F
L
F
L
cn _1 _ c 0 _
F
L
_ _ c 1___ c 0 _
F
L
_ c 1_ _35_
and further on:
cn _1 _
F
L _ _2
cn _1 _
F
_ L _ 1_ c 0 _
F
L
_ c 1_ _36_
This procedure can be repeated until
c 1 _
F
L _ _ n
cn _1 _
F
L _ _ n _1
_
F
L _ _ n _2
____ _
F
_ L _ 1_ c 0 _
F
L
_ c 1_ _37_
Solving for cn +1 and some rearrangement leads to
cn _1 _ c 1 1 _
L
F _
L
F _ _2
____ _
L
F _ _ n _ __ c 0
× L
F _ _ 1 _
L
F _
L
F _ _2
____ _
L
F _ _ n _1 _ _ _38_
Remembering that the sum of a finite geometric progression is calculated by
sn _
n
t _0
qt _
qn _1 _ 1
q _ 1 _39_
Eq. (38) can be re-written as follows:
cn _1 _ c 1
L
F _ _ n _1
_1
L
F _ 1 _ c 0
L
F _ _
L
F _ _ n
_1
L
F _ 1 _40_
cn _1
L
_ F _ 1__ c 1
L
F _ _ n _1
_ _1__ c 0
L
F _ _ L
F _ _ n
_ _1_ _41_
5.5 Washing Efficiency 541
c 1 _ cn _1 _
L
F _ _ c 0 _ cn _1 _ __
L
F _ _ n _1
_ c 0 _ c 1_ _42_
From the mass balance over the complete series of mixing stages [Eq. (23)], we
obtain:
c 1 _ cn _1 _
L
F
_ c 0 _ cn _ _43_
Now. Eq. (42) and Eq. (43) are set equal
L
F
cn _ cn _1 _ __
L
F _ _ n _1
_ c 0 _ c 1_ _44_
and rearranged
F
L _ _ n
_
c 0 _ c 1
cn _ cn _1 _45_
By finally taking the logarithm, the efficiency factor E is defined as the number
of ideally mixed stages in a countercurrent cascade with constant liquor and filtrate
flow rates:
E __ n _
log c 0 _ c 1 cn _ cn _1
log
F
L
_46_
According to the above derivation, non-integer values for the E factor are physically
meaningless. However, they are perfectly suitable to describe washing systems
which perform in between the physically meaningful integer steps.
Commercial washing equipment only seldom fulfils the postulate of constant
flows. In particular, the feed liquor flow L 0 is mostly higher than the liquor flow leaving
the washer with the discharged pulp L n, and the wash liquor flow F n+1 is lower
than the filtrate flow F 1. This fact is considered in the E factormodel by assuming constant
flow rates in all stages except in stage 1. The mass balances around stage 1
L 0 _ F 1 _ L _ F _ 0 _47_
L 0 c 0 _ F 1 c 1 _ L c 1 _ F c 2 _ 0 _48_
give
c 1 _ c 2 _
L 0
F
_ c 0 _ c 1_ _49_
which can be inserted into the equation for the E factor for stages 2 through n
542 5 Pulp Washing
F
L _ _ n _1
_
c 1 _ c 2
cn _ cn _1 _50_
to give
F
L _ _ n _1
_
L 0
F
c 1 _ c 2
cn _ cn _1 _51_
and finally
E _
log L 0
L
_ cnc 0__ cnc _1 1_
log FL
_52_
This is probably the most important equation in pulp washing. Returning to the
more descriptive original denotation (as per Fig. 5.23), Eq. (52) reads:
E _
log
Lin
Lout
cin _ cF
cout _ cWL _ _
log WL
Lout
_53_
The E factor depends on the type of washing equipment, on the mode of operation
of the equipment, on the pulp furnish to be washed, on the temperature and
substance loading of the involved liquors and, of course, also on diffusion and
sorption phenomena. This was originally observed by Norden for rotary drum
washers [29,32], but it holds true also for contemporary washing equipment.
In contrast to the displacement ratio, the E factor shows only a limited dependence
on the dilution factor over the industrially relevant dilution factor range of
maybe }1 t odt–1. This is what makes the E factor a very useful measure for washing
efficiency. In most industrial washing applications, it is acceptable to regard
the E factor as a constant over the range of reasonable dilution factors and also
over the range of reasonable production capacities.
If two or more pieces of washing equipment with the same inlet and outlet consistencies
are arranged in countercurrent mode, the E factors of the individual
machines can be summed to give the E factor of the whole system.
When the performance of a washer is determined in the field, the number of
measurements is limited for practical reasons. The measurements normally
needed as a minimum for determination of the E factor are three of the concentrations,
the feed and discharge consistencies, the wash liquor flow rate, as well as
the pulp production. The fourth concentration comes from the overall mass balance,
Eq. (23). In most cases this is the filtrate concentration:
cF _
F
_ Lin cin _ Lout cout _ WLcWL _ _54_
5.5 Washing Efficiency 543
544 5 Pulp Washing
Example: Determination of the E factor
The measurements around a multi-stage brownstock washer have given
the following results:
COD concentration in pulp feed, c in= 155 000 mg kg–1
COD concentration in pulp discharge c out = 6400 mg kg–1
COD concentration in wash liquor, c WL= 2600 mg kg–1
Wash liquor flow rate, WL = 345 t h–1
Feed consistency, N in= 4%
Discharge consistency, N out = 13%
Pulp production capacity, P = 40 odt h–1
First, we calculate the flow rates of liquor accompanying the pulp. These
are for the pulp feed
Lin _ P
Nin _ _ 1__ 40
_0_ 04 _ 1__ 960 t h_1
and for the discharged pulp:
Lout _ P
Nout _ _ 1__ 40
_0_ 13 _ 1__ 268 t h_1
The filtrate flow rate and filtrate COD concentration are:
F _ Lin _ Lout _ WL _ 960 _ 268 _ 345 _ 1 037 t h_1
cF _
F
_ Lin cin _ Lout cout _ WLcWL _
_
1 037
_960 _ 155 000 _ 268 _ 6 400 _ 345 _ 2 600__ 142 700 mg_kg_1
Finally, the E factor on a COD basis is:
E _
log
Lin
Lout
cin _ cF
cout _ cWL _ _
log
WL
Lout
_
log
268 _
155 000 _ 142 700
6 400 _ 2 600 _ _
log
_ 9_ 7
5.5.5
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