Experimental data of solubility at different temperatures: a simple technique

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1 DOI 1.17/s ORIGINAL Experimental data of solubility at different temperatures: a simple technique J. M. P. Q. Delgado Received: 16 May 26 / Accepted: 3 November 26 Ó Springer-Verlag 26 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 K, of 2-naphthol in water, at K, and of salicylic acid in water, at 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 symbols a radius of the active sphere (m) A area of a soluble sphere (m 2 ) c solute concentration (kg/m 3 ) c bulk concentration of solute (kg/m 3 ) c * saturation concentration of solute (kg/m 3 ) c out concentration in the outlet stream (kg/m 3 ) d diameter of inert particles (m) d 1 diameter of active sphere (m) D diameter of test column (m) D L longitudinal dispersion coefficient (m 2 /s) D m effective molecular diffusion coefficient (m 2 /s) D T transverse (radial) dispersion coefficient (m 2 /s) K permeability in Darcy s law (m 3 s/kg) k average mass transfer coefficient (m/s) L length of test column (m) n number of soluble spheres ( ) p pressure (kg/ms 2 ) Pe peclet number based on diameter of active sphere (= u d 1 /D m ) ( ) Q, Q 1 volumetric flowrate (m 3 /s) r spherical radial coordinate (m) s standard deviations (kg/m 3 ) S L surface area per unit length (m) T temperature (K) u interstitial velocity (vector) (m/s) u absolute value of interstitial velocity far from the active sphere (m/s) u r, u h components of fluid interstitial velocity (m/s) Greek letters e bed voidage ( ) / potential function (m 2 /s) h spherical angular coordinate (rad) x cylindrical radial coordinate (distance to the axis) (m) w stream function (m 3 /s) J. M. P. Q. Delgado (&) Departamento de Engenharia Quìmica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, Porto, Portugal jdelgado@fe.up.pt 1 Introduction Solubility is perhaps the most fundamental of all chemical phenomena. The significance of what dissolves

2 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 3 C (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 d 1. If liquid flow is steady, with a uniform volumetric flowrate Q, if the concentration of solute in the liquid fed to the bed is c 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 ¼ ks Lðc c Þ ð1þ where S L 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 = to L and c = c to c. The following equation results, c c c ¼ 1 exp ks L c Q L : ð2þ In order to guarantee that the outlet stream is saturated, it is important to observe the approximate criterion (c c )/(c * c ) >.999 (error less that.1%). The number of soluble spheres presented in a packed bed is given by, n ¼ V column ¼ 3 ð1 eþd 2 L V particle 2 d 3 ð3þ 1 and the general validity of the above theory holds, provided that the approximate criterion 6ð1 eþkl u ed 1 [6:98: 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, u. 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 d 1 (= 2a) buried in a bed of inert particles of diameter d (with d > d 1 ), 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, u,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 =, and Laplace s equation, Ñ 2 / =, is obtained for the flow potential / = Kp. In terms of spherical coordinates (r, h), the potential and stream functions are, respectively (see [2]), / ¼ u 1 þ 1 a 3 r cos h ð5þ 2 r w ¼ u 2 1 a 3 r 2 sin 2 h r and the velocity components are u ¼ u cos h 1 a 3 r u h ¼ ¼ u sin h 1 þ 1 2 a 3 r ð6þ ð7þ : ð8þ

3 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 @/ @w D ð9þ where x is the distance to the flow axis, and D L and D T 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, c, 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 D T = D L = 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 k ¼ e D m d 1 4 þ 4 5 ðpe Þ 2=3 þ 4 1=2 p ðpe Þ ð1þ where Pe = u d 1 /D m is the Peclet number for the soluble sphere. Now, by substituting the average mass transfer coefficient, given by Eq. (1), into Eq. (4), the following expression is obtained for the approximate validity criterion of theory developed above:! 1 Pe 1 þ p Pe þ p d 2 1 [ ð11þ 5ðPe Þ 1=3 ð1 eþl Finally, with Eq. (11) we could predict the volumetric flowrates that guarantee saturation in the outlet stream, Q = Pe p D 2 e D m /(4d 1 ). 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. mm of internal diameter), buried in beds of sand (.496 mm average particle diameter) through which water was steadily forced down, at temperatures in the range K. A stainless steel tube (21 mm i.d. and 2 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 2- naphtol, 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, c out, in the outlet stream (refrigerated to room temperature), as c * = (1 + Q 1 /Q )c out, where Q and Q 1 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

4 Fig. 1 Sketch of experimental set-up liquid had to be observed in the experiments (approximately.15 mm/s). Experiments were performed with spheres 6 mm in diameter, buried in sand with an average grain size of.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 =.4 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 [1], 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, 1] 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 equations were programmed using Microcal Origin software (version 7.). For benzoic acid in water we propose the following equation (R =.99 and p <.1), Table 1 Experimental solubilities at different temperatures and their sample standard deviations (s) System T (K) c* exp (kg/m 3 ) s (kg/mm 3 ) Benzoic acid water Naphthol water Salicylic acid water

5 c * (kg/m 3 ) 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) c* (kg/m 3 ) This work Eisenberg et al. (1955) Seidell (1941) ln( c *) = T (K) ln( c *) = T (K) T (K) Fig. 2 Solubility of benzoic acid in distilled water at different temperatures T (K) Fig. 4 Solubility of salicylic acid in distilled water at different temperatures lnðc Þ¼ 9:971 þ :346 TðKÞ 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 =.995 and p <.1), lnðc Þ¼ 12:355 þ :394 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 =.998 and p <.1), with an error lesser that 5% (see Fig. 4), lnðc Þ¼ 12:2365 þ :432 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. c* (kg/m 3 ) This work Moyle and Tyner (1953) McCune and Wilhelm (1949) Seidell (1941) ln( c *) = T (K) 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/ 22. References 1. Carvalho JRFG, Delgado JMPQ, Alves MA (24) Mass transfer between flowing fluid and sphere buried in packed bed of inerts. AIChE J 5: Currie IG (1993) Fundamental mechanics of fluids. McGraw- Hill, New York 3. Dunker C (1964) Benzoic acid, in Kirk Othmer encyclopaedia of chemical technology, 2nd edn. Interscience Publishers, No. 3, p Eisenberg M, Chang P, Tobias CW, Wilke CR (1955) Physical properties of organic acids. AIChE J 1: Ghosh UK, Kumar S, Upadhyay SN (1991) Diffusion coefficient in aqueous polymer solutions. J Chem Eng Data 36: Kumar S, Upadhyay SN, Mathur VK (1978) On the solubility of benzoic acid in aqueous carboxymethylcellulose solutions. J Chem Eng Data 23: McCune LK, Wilhelm RH (1949) Mass and momentum transfer in solid liquid system fixed and fluidized beds. Ind Eng Chem 41:

6 8. Moyle MP, Tyner M (1953) Solubility and diffusivity of 2-naphtol in water. Ind Eng Chem 45: Sahay H, Kumar S, Upadhyay SN, Upadhyay Y (1981) Solubility of benzoic acid in aqueous polymeric solutions. J Chem Eng Data 26: Seidell A (1941) Solubilities of organic compounds, 3rd edn. D. Van Nostrand, New York 11. Sherwood TK, Pigford RL, Wilke CR (1975) Mass transfer. International Student edn. McGraw-Hill, Kogakusha 12. Steele LR, Geankoplis CJ (1959) Mass transfer from a solid sphere to water in highly turbulent flow. AIChE J 5: Wakao N, Funazkri T (1978) Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds. Chem Eng Sci 33: