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8.7. Solid – liquid extraction
8.7.1. Theoretical background
Solid – liquid extraction (or leaching) is the separation of a solid solute from a mixture of solids by dissolving it in a liquid phase. Basically, there are three com- ponents in leaching: solid solute, insoluble solids and solvent. In most cases, the diffusion of intra-particle soluble component(s) controls the extraction rate. There- fore the process is often called as diffusion extraction. Solid – liquid extraction is widely used in food (e.g. extraction of sugar from sug- ar cane or sugar beet; isolation of vegetable oils from different seeds) and pharma- ceutical industries (e.g. extraction of active components from medicinal plants), and in hydrometallurgy (e.g. leaching of metal ions from ores).
For mathematical modeling of leaching an ideal solid – liquid extraction is de- fined. The solvent added to the dry raw material is partly taken up by the solid ma- terial and the soluble ingredients are instantaneously dissolved. Often all the so- lute is already dissolved (e.g. sugar in sugar beet). The solution is then split into two parts. The external solution is called extract and the internal one raffinate (in the case of plant materials this is cell liquor). The simplifying assumptions are defined as follows:
- For batch extraction the mass ratio of extract ( B ) and raffinate ( A ) does not change during extraction (liquid ratio: f B A =constant). In a continuous system the mass flow rate ratio of extract and raffinate remains constant throughout the equipment.
- The densities of solutions remain practically constant during extraction.
- The solid matrix has no absorptive properties. Hence, after infinite time the concentrations in the extract and raffinate phases are the same.
- The surface of solid material available for mass transfer does not change with time.
- The concentration of the raffinate (cell liquor) is initially the same throughout the solid matrix: x (^) 0 (kg solute/ kg raffinate).
- The solid phase consists of regular particles of the same shape and size: plates (leaves of plants), infinitely long cylinders (e.g. stems), or spheres (e.g. ground plant material). The shape and size of particles does not change during extrac- tion.
- The interior of the solid matrix is isotropic in all direction of diffusion, thus a constant „apparent or effective” diffusion coefficient ( D^ eff ) can be defined for the given material.
- In a batch system the extract phase is mixed, hence, the concentration gradient is zero in every direction in the system. If the extract is circulated or continuous system is used this gradient is zero in the direction perpendicular to the fluid flow.
For the calculation of any of the operations mentioned we may start from Fick’s equation of diffusion.
8.7.1.1. Batch operation mode
If the diffusion coefficient ( D (^) eff ) of the solutes inside the solid particles can be
considered as constant, Fick’s second law may be written as
D
x r
C r
x r
x eff t
2 2
1
^ (8.7-1)
Where (^) x concentration of raffinate (kg/kg), r coordinate relative to the axis of symmetry (m), t time (s), C factor of geometrical symmetry ( C =1 for plates/planes, C =2 for cyl- inders, C =3 for spheres).
The surface to volume ratio of particles can be expressed as:
F V
C R
p p
Where^ Fp surface of particle (m^2 ),
Vp volume of particle (m
R half thickness or half diameter of solid particles (m).
The boundary conditions:
- At the phase boundary the concentration in the raffinate phase is equal to con- centration of the extract phase, since the resistance of mass transfer in the ex- tract phase may be neglected (the mass transfer coefficient in the extract phase is much more higher than that in raffinate phase).
x r R t , y t (8.7-3)
F F
C R
(^) p Vp (8.7-8)
where liquid hold-up inside the particles (m^3 /m^3 ):
A A Vp
substituting into equation (8.7-8) gives
F
A C AR
From equations (8.7-3), (8.7-7) and (8.7-10) we obtain
fR C D
x t
x eff (^) r R r r R
The solution of equation (8.7-11) can be given as 2 :
x t x x x
y y t y y
P t
0 0
^
P t f f C
F i
f f C
j j j
1
2 (^2 ) 1
exp
where (^) Fi D (^) eff t R^2 t tD FICK-number,
tD R Deff (^2) t ime constant of diffusion (s),
(^) j j-th root of the characteristic equation relevant to the given geomet- rical shape planes : tg j f j
cylinders:
J
J
j f
j
(^1) j
0 2
J (^) 0 and J (^) 1 are the Bessel-functions 3 .
spheres: tg j f
j j
3 3 2
The values of
y y t ^ y y
(^0)
term as a function of Fick-number for spherical particles
are presented in Figure 8.7-2.
Figure 8.7-2. Batch extraction of spherical particles
P ( t )
Fi
P t
y y t y y
0
F i
t t (^) D
(^) *
Figure 8.7-3. Continuous extraction of spherical particles Fi
8.7.2. Objectives
Laboratory batch extraction is carried out to determine the time constant of diffu-
sion t (^) D^ R^2 Deff . From the experimental t (^) D^ value the effective diffusion coef-
ficient ( Deff ) can be calculated, if the geometry and size of the particles are
known.
8.7.3. Equipment
The experimental set-up is presented in Figure 8.7-4.
Figure 8.7-4.
- Thermostated extraction column
- Vessel for solvent feed
- Conductivity cell
- Connection to conductivity meter
- Three-way cock
- Thee-way cock
- Throttle valve
- Liquid level
- Pump
- Rotameter
- Liquid inflow
Figure 8.7-4. Flow diagram of a laboratory solid – liquid extractor
8.7.5. Data analysis
The extract concentration ( y ) can be read directly from the conductivity meter.
The density of the sodium chloride solution can be taken as 1 g/cm^3 -nek through- out. The measured (^) y (^) value is the last reading. The measured final extract con-
centration should be compared with that calculated from the component balance equation (8.7-6). Plot the extract concentrations against time ( y - t diagram). Plot the calculated
time constant of diffusion versus time ( t D - t diagram), and determine the mean
value in the range where it is really constant. Calculate the effective diffusion co-
efficient from the mean value of t D (please note, that R 4 mm). Compare the
effective diffusion coefficient with the molecular diffusion coefficient of NaCl, measured in pure water at 20 °C ( D = 1_._ 39 10 ^5 cm^2 s).
The following Table should be completed and attached to the report:
Dry mass of tablets: g Mass of tablets saturated with NaCl: g Mass of raffinate ( A ): g Mass of extract ( B ): g Phase ratio ( (^) f B A ):
Rotameter scale: dm^3 /h Temperature: °C Initial concentration of raffinate ( x (^) 0 ): g/kg
Measured final extract concentration y (^) : g/kg
Calculated final extract concentration (Eq. 8.7-6) (^) y (^) : g/kg
D (^) eff D :
Set up a Table with the headings for determination of the time constant ( t D )!
Time (min)
y g k g
y y y
Fi
t (^) D^ (^) (min)
8.7.6. Design of continuous extraction
On the basis of batch experiment the mean residence time in a continuous ex-
traction unit can be designed. The steady value of t D is determined from batch
laboratory experiment. The relative loss and extract concentration are given by the instructor. If neat solvent ( (^) y (^) 0 0 ) is used for extraction, the phase rate ratio can
be calculated from the material balance of continuous extraction
x (^) 0 xout fyout (8.7-14)
where yout and xout are the outlet concentrations in the extract and raffinate, re-
spectively. The Fick-number can be read in Figure 8.7-3.
The results should be summarized in the following Table:
Given Determined Relative loss ( xout x 0 ): Phase flow rate ratio ( f ):
Relative extract concentration ( y (^) out/ x 0 ): Fick - number ( t (^) extr tD^ ):
Time constant of diffusion ( t D ) Mean extraction time ( t ):
References
1 Fourier, J. B. J.: Théorie analytique de la chaleur , 1808. 2 Akszelrud, G. A.: Zsurnal Fizicseszkoj Himii, 33, 2316 (1959) 3 Crank J.: The mathematics of diffusion, Clarendon Press, Oxford, 1956.
