Heat Mass Transfer
DOI 10.1007/s00231-006-0209-4
ORIGINAL
Experimental data of solubility at different temperatures: a
simple technique
J. M. P. Q. Delgado
Received: 16 May 2006 / Accepted: 3 November 2006
Springer-Verlag 2006
Abstract This article describes a simple and inexpensive experimental technique, easy to set-up in a
laboratory, for the measurement of solute solubilities
in liquids (or gases). Experimental values of solubility
were determined for the dissolution of benzoic acid in
water, at 293–338 K, of 2-naphthol in water, at
293–373 K, and of salicylic acid in water, at 293–343 K.
The experimental results obtained are in good agreement with the theoretical values of solubilities
presented in literature. Empirical correlations are
presented for the prediction of solubility over the entire range of temperatures studied, and they are shown
to give the solubility value with very good accuracy.
List of
a
A
c
c0
c*
cout
d
d1
D
DL
D¢m
symbols
radius of the active sphere (m)
area of a soluble sphere (m2)
solute concentration (kg/m3)
bulk concentration of solute (kg/m3)
saturation concentration of solute (kg/m3)
concentration in the outlet stream (kg/m3)
diameter of inert particles (m)
diameter of active sphere (m)
diameter of test column (m)
longitudinal dispersion coefficient (m2/s)
effective molecular diffusion coefficient
(m2 /s)
J. M. P. Q. Delgado (&)
Departamento de Engenharia Quı̀mica,
Faculdade de Engenharia da Universidade do Porto,
Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
e-mail: jdelgado@fe.up.pt
DT
K
k
L
n
p
Pe¢
Q, Q1
r
s
SL
T
u
u0
ur, uh
transverse (radial) dispersion coefficient
(m2/s)
permeability in Darcy’s law (m3s/kg)
average mass transfer coefficient (m/s)
length of test column (m)
number of soluble spheres (–)
pressure (kg/ms2)
peclet number based on diameter of active
sphere (= u0 d1 /D¢m) (–)
volumetric flowrate (m3/s)
spherical radial coordinate (m)
standard deviations (kg/m3)
surface area per unit length (m)
temperature (K)
interstitial velocity (vector) (m/s)
absolute value of interstitial velocity far from
the active sphere (m/s)
components of fluid interstitial velocity (m/s)
Greek letters
e bed voidage (–)
/ potential function (m2/s)
h spherical angular coordinate (rad)
x cylindrical radial coordinate (distance to the axis)
(m)
w stream function (m3/s)
1 Introduction
Solubility is perhaps the most fundamental of all chemical phenomena. The significance of what dissolves
123
Heat Mass Transfer
what, to what extent, at what temperature and pressure,
and the effects of other species, was recognized at a very
early stage. In more recent times the importance of
solubility phenomena has been acknowledged
throughout science. For example, in the environment,
solubility phenomena influence the weathering of rocks,
the creation of soils, the composition of natural water
bodies and the behaviour and fate of many chemicals.
The characteristic ability of water to behave as a
polar solvent changes when water is subjected to high
temperatures and pressures. As water becomes hotter,
its molecules seem much more likely to interact with
nonpolar molecules. For example, at 300C (and high
pressure) water has dissolving properties very similar
to acetone, a common organic solvent.
Also, solid–liquid and solid–gas mass transfer
investigations with Newtonian or non-Newtonian fluids
are frequently made by following the rate of dissolution of a low solubility solute. In all researches, accurate solubility data are required.
On mass transfer investigations in porous media, as
in studies of dispersion coefficients and solute transport, the most common solutes used are benzoic acid,
2-naphthol, naphthalene, salicylic acid and succinic
acid with water or air (see [13]). In these experiments,
knowledge of accurate solubility data at different
temperatures is very important, i.e., for low solubility
solutes.
The experiment proposed is simple and inexpensive,
and it provides an accurate method for the measurement of solubilities of solid solutes in liquids and gases.
2 Theory
Consider a vertical column of length L, containing a
packed bed of soluble spherical particles of diameter
d1. If liquid flow is steady, with a uniform volumetric
flowrate Q, if the concentration of solute in the liquid
fed to the bed is c0 and the solubility of the solid particle is c*, a mass transfer boundary layer will develop.
In the analysis of results of experiments of dissolution of soluble spherical particles in liquid flow, the
equation for dissolution rate is given by,
@c
¼ kSL ðc c0 Þ
@x
c c0
kSL
¼
1
exp
L
:
c c0
Q
In order to guarantee that the outlet stream is
saturated, it is important to observe the approximate
criterion (c – c0)/(c* – c0) > 0.999 (error less that
0.1%). The number of soluble spheres presented in a
packed bed is given by,
n¼
Vcolumn 3 ð1 eÞD2 L
¼
Vparticle 2
d31
6ð1 eÞkL
[6:908:
u0 ed1
ð4Þ
If the criterion of Eq. (4) is to be satisfied, it is
important to know the value of the average mass
transfer coefficient, k, so as to be able to estimate the
interstitial velocity of liquid, u0.
2.1 Mass transfer around a buried soluble sphere
For the propose of analysis, let as consider the situation
of a slightly soluble sphere of diameter d1 ( = 2a)
buried in a bed of inert particles of diameter d (with d
> d1), packed uniformly (void fraction e) around the
spheres. The packed bed is assumed to be ‘‘infinite’’ in
extent and a uniform interstitial velocity of liquid, u0, is
imposed, at a large distance from the spheres.
In order to obtain the flow field in the vicinity of the
buried sphere, Darcy’s law, u = – K grad p, is coupled
with the continuity equation, div u = 0, and Laplace’s
equation, 2 / = 0, is obtained for the flow potential
/ = Kp.
In terms of spherical coordinates (r, h), the potential
and stream functions are, respectively (see [2]),
1 a3
/ ¼ u0 1 þ
r cos h
2 r
a3 u0
1
w¼
r2 sin2 h
r
2
and the velocity components are
where SL is the active surface area per unit length and
k is the average mass transfer coefficient. Given
constant flowrate, uniformly distributed particles and
isothermal conditions, Eq. (1) is integrated between
the inlet and outlet conditions of the bed, x = 0 to L
and c = c0 to c. The following equation results,
a3 @/
ur ¼
¼ u0 cos h 1 @r
r
1 @/
1 a3
¼ u0 sin h 1 þ
:
uh ¼
r @h
2 r
123
ð3Þ
and the general validity of the above theory holds,
provided that the approximate criterion
ð1Þ
Q
ð2Þ
ð5Þ
ð6Þ
ð7Þ
ð8Þ
Heat Mass Transfer
Making use of the potential and stream lines, it is
possible to perform a material balance on the solute
in a differential element of a ‘‘stream tube’’ to obtain
(see [1])
@c
@
@c
@
2 @c
¼
DL
DT x
þ
@/ @/
@/
@w
@w
ð9Þ
where x is the distance to the flow axis, and DL and DT
are the longitudinal and transverse dispersion coefficients, respectively.
The boundary conditions to be observed in the
integration of Eq. (9) are: (1) the solute concentration
is equal to the background concentration, c0, far away
from the sphere; (2) the solute concentration is equal
to the equilibrium concentration, c = c*, on the surface
of the sphere and (3) the concentration field is symmetric about the flow axis.
For very low fluid velocities, dispersion is the direct
result of molecular diffusion, with DT = DL = D¢m, and
the numerical solution presented by Carvalho et al. [1]
applies. Those authors suggest that their results are
well approximated (with an error of less than 1%) by
1=2
D0m
4
4
0 2=3
0
k¼e
4 þ ðPe Þ þ ðPe Þ
5
p
d1
ð10Þ
where Pe¢ = u0 d1/D¢m is the Peclet number for the
soluble sphere.
Now, by substituting the average mass transfer
coefficient, given by Eq. (10), into Eq. (4), the following expression is obtained for the approximate
validity criterion of theory developed above:
1
Pe0
! 2
p
p
d1
1þ 0þ
[
ð1 eÞL
Pe 5ðPe0 Þ1=3
ð11Þ
Finally, with Eq. (11) we could predict the volumetric
flowrates that guarantee saturation in the outlet
stream, Q = Pe¢p D2 e D¢m/(4d1). However, an
important aspect to consider is the dependence of Q on
the effective molecular diffusion coefficient, D¢m.
Fortunately, values of D¢m increase with temperature,
and the value of D¢m, at room temperature or lower, is
a good estimate.
3 Experimental setup
Experiments were performed on the dissolution of
spheres of benzoic acid, 2-naphthol and salicylic acid
(6.0 mm of internal diameter), buried in beds of sand
(0.496 mm average particle diameter) through which
water was steadily forced down, at temperatures in the
range 293–373 K.
A stainless steel tube (21 mm i.d. and 200 mm long)
was used to hold the bed of soluble solid spheres in an
upright position while a metered stream of distilled
water was fed to the top of the column, as sketched in
Fig. 1. Near the bottom of the stainless steel column, a
perforated plate, covered with fine wire mesh, was used
to support the bed.
The distilled water was initially deaerated, under
vacuum, to avoid liberation of gas bubbles in the rig, at
high temperature. The test column was immersed in a
silicone oil bath kept at the desired operating temperature by means of a thermosetting bath head (not
represented in the figure). The copper tubing feeding
the distilled water to the column at a constant metered
rate was partly immersed in a pre-heater and it had a
significant length immersed in the same thermosetting
bath as the test column; the copper tubing leaving the
test column was immersed in a chillier to cool the
outlet stream before reaching the UV analyser.
The water flowrate was then adjusted to the required value, Q, and the concentration of solute in the
outlet stream was continuously monitored by means of
a UV/VIS Spectrophotometer (set at 274 nm, for 2naphtol, at 226 nm, for benzoic acid and 292 nm, for
salicylic acid).
The solubility of the solutes studied in water was
calculated from the steady state average concentration
of solute, cout, in the outlet stream (refrigerated to
room temperature), as c* = (1 + Q1 /Q )cout, where Q
and Q1 were the measured volumetric flowrates.
The spheres of solutes studied were prepared from
p.a. grade material, which was molten and then poured
into moulds made of silicone rubber. Wherever any
slight imperfections showed on the surface of the
spheres, they were easily removed by rubbing with fine
sand paper. Using callipers three measurements were
made of the diameter of each sphere along three perpendicular directions.
4 Results and discussion
The reproducibility of the experiments was tested by
independently repeating the measurement of solubility
under identical operating conditions, and in the vast
majority of cases the results of repeated measurements
of solubility did not differ by more than 8%.
Each experimental run gives a value of c* if the
condition represented by Eq. (11) is observed. This
requirement meant that extremely low velocities of
123
Heat Mass Transfer
Fig. 1 Sketch of
experimental set-up
liquid had to be observed in the experiments
(approximately 0.15 mm/s).
Experiments were performed with spheres 6 mm in
diameter, buried in sand with an average grain size of
0.496 mm. As the experiments were carried out with
pressurized and deaerated water, it was possible to
work at temperatures of up 373 K. The value e = 0.40
was taken from references in the literature (see [11]).
Table 1 summarizes the experimental solubilities
obtained, as well as the corresponding sample standard
deviations (s). The experimental solubility values obtained for the three solutes in water are plotted as a
function of temperature in Figs. 2, 3 and 4.
The values of c* obtained for 2-naphthol in water are
in good agreement with the values of Moyle and Tyner
[8], McCune and Wilhelm [7] and Seidell [10], and
those for benzoic acid in water are in good agreement
with the values proposed by Ghosh et al. [5], Sahay
et al. [9] and others. For salicylic acid, only a few
previous values [4, 10] were found and they are shown
in Fig. 4. This highlights the usefulness of the present
work.
The results presented in Figs. 2, 3 and 4 suggest that
our data are consistent and accurate; they were taken
as reference data to help identify mathematical
expressions for the prediction of c*. The regression
123
equations were programmed using Microcal Origin
software (version 7.0). For benzoic acid in water we
propose the following equation (R = 0.990 and
p < 0.0001),
Table 1 Experimental solubilities at different temperatures and
their sample standard deviations (s)
System
T (K)
c*exp (kg/m3)
s (kg/mm3)
Benzoic acid–water
293.15
303.15
313.15
323.15
333.15
338.15
293.15
303.15
313.15
323.15
333.15
343.15
353.15
363.15
368.15
293.15
303.15
313.15
323.15
333.15
343.15
2.86
4.14
5.94
8.42
11.87
13.76
0.61
0.93
1.32
2.01
3.01
4.32
6.45
9.05
11.22
1.50
2.40
3.80
5.80
8.70
13.00
0.15
0.22
0.14
0.06
0.08
0.14
0.03
0.07
0.16
0.14
0.10
0.02
0.14
0.11
0.19
0.08
0.12
0.10
0.09
0.13
0.08
2-Naphthol–water
Salicylic acid–water
Heat Mass Transfer
14
16
This work
Ghosh et al. (1991)
Sahay et al. (1981)
Kumar et al. (1978)
Dunker (1964)
Steele and Geankoplis (1959)
Eisenberg et al. (1955)
Seidell (1941)
8
6
4
10
8
6
4
2
2
ln( c*) = −9.0971 + 0.0346 × T (K)
0
0
270
280
290
300
310
320
330
340
270
280
Fig. 2 Solubility of benzoic acid in distilled water at different
temperatures
lnðc Þ ¼ 9:0971 þ 0:0346 TðKÞ
lnðc Þ ¼ 12:0355 þ 0:0394 TðKÞ
ð13Þ
and the data shown in Fig. 3 are within 7% of this line.
Finally, for salicylic acid in water we propose the
following equation (R = 0.998 and p < 0.0001), with
an error lesser that 5% (see Fig. 4),
lnðc Þ ¼ 12:2365 þ 0:0432 TðKÞ
ð14Þ
The agreement between our results and other published data suggests that the proposed method is
accurate. However, it is important to remember that,
for higher solubility solutes, natural convection near
the surface of the dissolving solids may become significant, thus invalidating the method.
This work
Moyle and Tyner (1953)
McCune and Wilhelm (1949)
Seidell (1941)
14
12
3
ln( c*) = −12 .0355 + 0.0394 × T (K)
10
8
6
4
2
0
275
285
295
305
315
325
335
300
310
320
330
340
350
Fig. 4 Solubility of salicylic acid in distilled water at different
temperatures
ð12Þ
and the data shown in Fig. 2 are within 5% of this line.
For 2-naphthol in water we propose the following
equation (R = 0.995 and p < 0.0001),
16
290
T (K)
T (K)
c* (kg/m )
ln( c*) = −12 .2365 + 0.0432 × T (K)
12
3
3
c * (kg/m )
10
c* (kg/m )
12
This work
Eisenberg et al. (1955)
Seidell (1941)
14
345
355
365
375
T (K)
Fig. 3 Solubility of 2-naphthol in distilled water at different
temperatures
5 Conclusions
The main conclusion from this work is that the
experimental technique described for measuring the
solubilities of a solute in a liquid, at different temperatures, is perfectly suitable and easy to use. The results
show that it is possible to obtain good results for solubility values, using simple procedures.
The experimental values obtained for c* of 2-naphtol in water, benzoic acid in water and salicylic acid in
water, over a range of temperatures above ambient,
are in good agreement with the values given in the
literature.
Acknowledgments The author wishes to thank Fundação para
a Ciência e a Tecnologia for the Grant N SFRH/BPD/11639/
2002.
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123
Heat Mass Transfer
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