size distribution (Biggar and Nielsen, 1967) with macropores acting as a passage for the mixing and ... The texture of the soil was clay consisting of 47.7% clay, 19.7% silt and 32.5% sand. ... To ensure a uniform texture of ... smaller porosity than aggregates having diameters of 16 mm and 31 mm. .... (1982) for chalk cubes.
CHAPTER 5 DIFFUSION OF NON-SORBING SOLUTE FROM SPHERICAL AGGREGATES
It has been already described in chapter three that aggregated soils have bi-modal pore size distribution (Biggar and Nielsen, 1967) with macropores acting as a passage for the mixing and flow of the solute while the micropores act as sink or source of the solute (De Smedt and Wierenga, 1979). The removal of solute from micropores depend on the slow process of salt diffusion (Biggar and Nielsen, 1967). Thus, for improving the leaching efficiency it is important that salts which, are held in the micropores of the aggregates diffuse quickly into the flowing macropore water. van Genuchten and Wierenga (1976), Gaudet et al., (1977) and Rao et al., (1980a) introduced a semi-empirical mass transfer model as an alternative of diffusion model. They assumed that solute transfer between the two pore regions was proportional to the difference in the average solute concentrations in these pore regions. In this model they have introduced a mass transfer rate coefficient instead of a diffusion coefficient. Because of the lack of experimental techniques for independently measuring coefficients of diffusion and mass transfer models, these input parameters are often estimated by curve fitting the model to the measured data of salt diffusion in bathing water (Rao et al. 1980a). This chapter describes a study on baked spherical single aggregates with salts diffusing into a well-stirred solution in order to determine these solute transfer parameters needed for modelling in later chapters. Aggregates prepared for experimental work were spherical in shape because boundary conditions are well defined for spheres and because the salt diffusion through spheres is threedimensional as usually occurs under field conditions.
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Materials and methods
5.2.1 The soil
The soil used in this study for preparing the spherical aggregates for measurements was taken from the Cranfield University at Silsoe Farm, Bedfordshire. The soil belonged to the Evesham series calcareous gley (King, 1969 and Verma and Bradley, 1988). The texture of the soil was clay consisting of 47.7% clay, 19.7% silt and 32.5% sand. The dry bulk density of the undisturbed soil was found to be 1.12mg/mm3. The soil contained 12.1% organic matter, it had a sodium adsorption ratio (SAR) of 0.36 and an exchangeable sodium percentage (ESP) of 0.7. The electrical conductivity of the saturated extract was 0.8 dS/m.
5.2.2 Aggregate preparation
The soil taken from the field was flaked and air-dried on trays for two weeks. After the removal of any stones the soil was then ground. To ensure a uniform texture of soil for the preparation of the aggregates, the soil was sieved through a 1 mm mesh. This was then mixed with distilled water in a ratio by weight of 2.5:1 in a 1000 ml plastic beaker. While adding distilled water, the soil was thoroughly mixed using a spatula until a sticky consistency was obtained. The soil was then left for 20 hours. The soil was then divided and moulded by hand into nearly spherical aggregates. With the help of moulds, spherical aggregates of three different diameters (34 mm, 20 mm and 8.8 mm) were prepared. The aggregates were air dried for 24 hours and then oven dried at 105 0C for two days (48 hours). The oven-dried aggregates were then quickly transferred to a furnace so as to avoid any excessive cooling which was observed to cause cracking of aggregates when fired. The furnace temperature, initially set at 105 0
C, was raised to 300 0C after 30 minutes and then in hourly increments it was
increased to 500 0C, 700 0C and finally to 1000 0C following the method of Garnett, (1993). After one hour at 1000 0C, the furnace was turned off and the aggregates were allowed to cool overnight. The aggregates were taken out of the furnace after 12 hours and were allowed to cool further in open air.
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The actual diameters of the aggregates to be used in the diffusion experiments were measured with vernier calipers. On air drying, the diameters of the aggregates were reduced to 33.01 mm, 18.00.2 mm and 8.40.4 mm respectively with an average reduction in size of 4%. The diameters of the aggregates after firing were reduced further to 310.49 mm, 160.34 mm and 80.06 mm respectively, with an average reduction in diameter of 8.6%. The aggregates used in the experiments are shown in Plate 5.1 Small spheres (8 mm)
Medium spheres (16 mm)
Large spheres (31 mm)
Baked spherical aggregates used in the experiments
5.2.3 The physical properties of the soil aggregates The dry bulk density of aggregates was calculated from:
d = dry density of aggregate md = dry mass of aggregate V = total volume of sphere
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The porosity of aggregates was determined from:
VA = 1-d /s V
= porosity of aggregates s = density of solids VA = volume of micropores (voids in aggregate) V = total volume of aggregate
The moisture content on a mass basis was calculated from:
m = moisture content on mass basis. mw = mass of water in aggregate
The moisture content on a volume basis was then calculated from that obtained on a weight basis from:
d m w
v = volumetric water content. w = density of water (taken as 1.0 mg/mm3) The physical properties of the aggregates of the three different diameters used in all experiments described in this thesis are given in the Table 5.1. These results are based on the measurements of five randomly taken aggregates from each size. The table shows that the aggregates having a diameter of 8 mm have a larger bulk density and smaller porosity than aggregates having diameters of 16 mm and 31 mm. Similar physical properties of fired spherical aggregates were reported by Cheraghi (1998).
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Physical properties of the baked spherical aggregates used in the experiments.
Aggregate diameter Properties
Dried weight (mg)
Saturated weight (mg)
Total volume of sphere (mm3)
Pore volume (mm3)
Dry bulk density (mg/mm3)
Porosity of the aggregates
Moisture content on weight basis (m) of saturated aggregates Volume of the bathing water (mm3)
Ratio of macro and micropore volumes ( ) No. of aggregates used in the experiment
shows 95% confidence interval (CI) in the data
The initial and boundary conditions for experiments
With the same notations as used in chapter 3, the initial and boundary conditions pertinent to the single spherical aggregate diffusion experiments in a well-stirred bathing solution are:
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CB(t) = CB(0) = 0
at t = 0
CA(r,t) = CA(0) = Co 0 r a
at t = 0
CA(a,t) = CB(t),
at t 0
The diffusion experiments
Before conducting the diffusion experiments, the spherical aggregates were completely washed with de-ionised water to remove any salt already contained in them and then they were oven dried at 105 0C. Three baked aggregates of the same diameter were taken randomly and were weighed. They were placed on three meshes, each on a separate tripod. KCl (0.1 N) solution having an electrical conductivity of 14100 S/cm was placed in a burette and allowed to drip on to the aggregates. Initially all the dripping solute was absorbed by the aggregates. Later on when the aggregates became saturated the solute started draining out of the aggregates. The concentration of the draining solute was measured and was found to be the same as that of the dripping solute from the burette. At this stage a thin film of solute was formed around the aggregates providing enough evidence of saturation of the aggregates. The saturated aggregates were again weighed.
The saturated aggregates were immediately put into three separate beakers, each containing 50 ml of de-ionised water. The bathing water was kept constant (50 ml) for all the aggregate diameters for ease of measurement of the electrical conductivity of the bathing water with the probe. Thus , the ratio of the volume of the solution in the macropores (VB) to that in the micropores (VA) for the three aggregate sizes was 578 for 8 mm, 59 for 16 mm and 8 for 31 mm. Salts in the aggregates started to diffuse out as soon as they were placed in the bathing water. The electrical conductivity (EC) of the bathing water was measured at regular intervals with a conductivity meter, calibrated against KCl solutions of known concentration. At the start of diffusion, the salts diffused very quickly and the readings were taken at very short intervals of time. The temperature was maintained constant (200.5 0C) during the whole experiment because the molecular diffusion rate increases with temperature according to the Nernst-Einsten equation for molecular diffusion coefficient at infinite dilution:
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where R = the universal gas constant (8.134 J mol-1 K-1), T = the absolute temperature, N = Avogadro’s number (6.022 x 1023 /mol), and = the absolute mobility of a particle (Robinson and Stokes, 1959; Shackelford and Daniel, 1991).
The bathing solution was kept well-stirred using a paddle during the whole period of diffusion as shown in Figure 5.1. To avoid any evaporation the beakers were kept covered with glass lids. The experiment was replicated three times.
Experimental set up for single aggregate diffusion
Figure 5.2 shows graphs of the change in the concentration of the bathing water against time for all three aggregate sizes (Appendix-II, file “diffusion data”). The graphs show that the concentration of the bathing water increased more rapidly for large aggregates than for small aggregates. This was expected as the volume of the bathing water was the same (50 ml) for all the aggregates but the volume of the solute in the micropores for each aggregate size was different, the volume of solute in large aggregates was 70 times more than that for small aggregates. All graphs follow the same pattern of initial rapid rise in the concentration of the bathing water followed by slower diffusion.
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Electrical conductivity of the solute concentration in the bathing water plotted against time for initially saturated aggregates. (a) 8 mm, (b) 16 mm, and (c) 31mm.
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The graphs in Figure 5.2 do not give a clear description of the results because they do not show the amount of solute remaining in or diffused out from the aggregates. Hence the measured concentration of the bathing water was converted into the mass of the solute diffused and was then plotted in terms of relative solute mass increase in the bathing water with time. The measured increase of the solute mass in the bathing water as a fraction of the solute mass after infinite time when it becomes constant, for the three different aggregate sizes against time is shown in Figure 5.3.
Relative solute mass in the bathing water plotted against time for three different aggregate sizes.
Figure 5.3 shows an initial rapid increase in the solute mass in the bathing solution for all aggregate sizes, due to the initial high concentration gradient between the solute in the aggregates and the bathing solution. This was followed by a more gradual approach to an equilibrium concentration. As the small aggregates have shorter diffusion paths they reached equilibrium much faster than the large aggregates. It took 91, 345 and 1300 minutes for 80% of the initial amount of solute in the spheres to diffuse out of the 8, 16 and 31mm sphere diameters, respectively. Similar curves were also observed by Rao et al. (1980a) for porous ceramic spheres and by Addiscott (1982) for chalk cubes. The breakthrough curves ended with an extended tailing as the solute within the spheres continued slowly to diffuse out and such tailing continued
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longer for the large size aggregates. This was also found by Biggar and Nielsen (1962); Green et al. (1972), and Mac Mahon and Thomas (1974) in diffusion experiments.
Effective diffusion coefficient (De)
The effective diffusion coefficient of the solute in aggregates of all sizes was determined by fitting the experimental diffusion results with the analytical curves given by equation 3.35 (see chapter 3) using the method of least mean squares in the Mathcad programme (see Appendix-II in files “De-8mm, De-16mm and De-31mm”). An excellent agreement (R2 0.98) was found between the experimental results and the analytical curves. The measured and fitted results of the relative solute concentration of the bathing water against time are shown in Figure 5.4.
Experimental data fitted to the diffusion model (equation 3.35) for determining the effective diffusion coefficient of the solute within aggregates. Symbols represent experimental data points while the solid lines show simulation of the data (a) aggregate of size 8 mm; (b) aggregate of size 16 mm and (c) aggregate size of 31 mm.
The curve-fitted values of the effective diffusion coefficient, for all three aggregate diameters are given in Table 5.2. The tortuosity factor ( = De /D0) was calculated by
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taking the molecular diffusion coefficient (Do) for KCl at 20 0C equal to 0.104 mm2/min (Robinson and Stroke, 1959, pp. 513). The values of the effective diffusion coefficient (De) and the tortuosity factor () given in Table 5.2 are smaller than those measured elsewhere on saturated soils (Nye, 1979). It may be because the aggregates used in this study have higher density and more tortuous pathways than those used by Nye (1979). The effective diffusion coefficients and tortuosity factors calculated by Cheraghi (1998) for fired spherical aggregates are smaller than those calculated in this study. This may be because Cheraghi used sorbing solutes for his study. It is seen that the values of De vary with the diameter of the spherical aggregates, perhaps because of the structural differences arising during the preparation of the aggregates which could change the density, porosity and tortuosity of the aggregates. This idea is discussed in section 5.10.1
Effective diffusion coefficients obtained by fitting the experimental results with the analytical curves.
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Mass transfer rate coefficient ( )
The experimental results of the change in concentration of the solute in the bathing water with time were least mean square curve-fitted with the results obtained by using the equation 3.44 for optimizing the measured average values of mass transfer rate coefficient ( ) for all aggregate sizes (see Appendix-II in file “mass transfer coefficient”). The experimental and simulated results gave poor agreement (Figure 5.5). The values of goodness of fit (R2) between the experimental and the simulated results ranged between 0.80 and 0.84. The difference between the measured data and fitted results reflect the inadequacy of the mass transfer model to represent the geometry of the system. The mass transfer model under estimated the data up to the half diffusion period then it over estimated the data. Hence it may be inferred that the mass transfer model (equation 3.44) does not adequately describe the diffusion under batch (stagnant water) conditions. This was also reported by Rao et al. (1982b).
Experimental data fitted to the model (equation 3.44) for determining the mass transfer rate coefficient. Symbols represent data points while solid lines show simulation of the data with the model for (a) aggregate of size 8 mm; (b) aggregate of size 16 mm and (c) aggregate size of 31 mm.
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The time-averaged values of of all three aggregate sizes were also estimated by using equations 3.49 and 3.50 (see chapter 3). The measured average values of obtained by curve fitting were compared with those estimated by using equations 3.49 and 3.50 (see Appendix-II in file “estimation of alpha”) and are given in Table 5.3. The measured and estimated values of were statistically not different. Results showed that increased with the diameter of the aggregate as also reported by Rao et al. (1980a). The results showed that was not constant but changed with time, as also reported by Rao et al. (1980a and b). It decreased with increasing time of diffusion as shown in Figure 5.6. The experiments results showed that the depended on the size of the spherical aggregate (see Table 5.3), time of the diffusion, volumes of macro and micropores, and effective diffusion coefficient (De), as also reported by (Rao et al. 1980 a and b).
Change in plotted against time of diffusion (size of aggregate 8 mm).
The graph shows decreases with increasing time of diffusion with a rapid decrease during an initial period of diffusion. This means that in the beginning there was a rapid solute transfer from the aggregate into the bathing water compared with that at
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the end of the period. Thus only a single value of (time-averaged) cannot describe the solute transfer for the whole period of the diffusion.
Measured and calculated values of .
Relation between effective diffusion coefficient (De) and mass transfer rate coefficient ( )
A theoretical relation between the effective diffusion coefficient of the solute within the aggregate and the mass transfer rate coefficient was determined by using the diffusion equation 3.35 and the mass transfer rate equation 3.44 (Appendix-II in file “De-alpha”). An aggregate of size 16 mm with a micropore volume of 870 mm 3 bathing in 870 mm3 of water was considered for this theoretical calculation. The values of qn’s used in equation 3.35 were first calculated for (the ratio of macro and micropore volumes) equal to 1. Assuming the effective diffusion coefficient (De) of
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the solute in the sphere equal to 0.01 mm2/min, the relative change in the concentration of bathing water with time was calculated with equation 3.35. Then this relative change in concentration of bathing water was least mean square curve fitted with the mass transfer equation 3.44 (chapter 3) for optimizing the average mass transfer rate coefficient ( ) at ( = (B+A)/ BA) equal to 6.98. By choosing suitable values of De between 0.01 and 0.07 mm2/min, which are usually observed for porous materials, different values of were calculated from equation 3.44. These calculated values of are plotted against De in Figure 5.7. The graph shows a linear relation between De and with intercept 0 and slope equal to 450 as also reported by Coats and Smith, (1964) cited by van Genuchten et al., (1977). Based on these calculations the following theoretical relation between and De was obtained:
Theoretical relation between the effective diffusion coefficient (De) and mass transfer rate coefficient ( ) (aggregate size 16 mm, ratio of macro and micropore volumes =1).
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Statistical analysis of the effect of diameter and moisture content on the rate of diffusion
The results of the concentration in the bathing water show that the rate of the diffusion depended on the size of the aggregate. To determine whether this variation in concentration with size of aggregate was statistically significant, a statistical hypothesis was set that aggregate diameter had no effect on the rate of diffusion. To test the hypothesis a statistical test (ANOVA) was carried out. The test showed that the size of aggregate had a significant (p 0.05) effect on the rate of diffusion. The larger the aggregate diameter the slower the rate of diffusion. To find out which aggregate size had a significantly rapid change in concentration, another statistical test ‘Least Significance Difference’ (LSD) was carried out. The LSD test showed that the change in concentration of the aggregate of diameter 8 mm was significantly more rapid (p 0.05) compared with change in the other two diameters. Theoretically rate of diffusion for all sizes should be significantly different for same material as shown in equation 3.35. But in our case rate of diffusion from small size is only significantly different. It may be due to variability in measuring data during the experiment. 5.10
Application to the real soil
The numerical solution of the diffusion equation, in chapter 4, showed that the diffusion equation for a well-stirred solution could be applied for non-stirred solutions. Hence we propose that the diffusion model can be applied to real soil if the effective diffusion coefficient of that soil is known. Experiments were thus conducted to determine the effective diffusion coefficient of a real soil so that the diffusion model could be applied to field conditions. The undisturbed samples, from a soil which was used previously in preparing spherical aggregates, were collected for this experiment.
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Physical properties of the soil used for the experiment.
Dry weight of the soil (mg)
Saturated weight (mg)
Pore volume (mm3)
Dry density of soil (mg/mm3)
Volume of bathing water (mm3)
Macro and micropore volume
ratio () *
in Table shows the confidence interval in data
5.10.1 Methodology and Results Three undisturbed soil samples were collected from the Cranfield University’s Farm using circular iron coring rings of 60 mm diameter and 20 mm thickness. Samples contained in the rings were carefully taken to the laboratory in plastic bags. Three soil samples in the rings were oven dried at 105 0C for 24 hours. The weight of the dried soil samples was measured, then they were partially submerged in KCl (0.1N) solution in trays. The samples were then left overnight to saturate in the trays. The saturated samples in the rings were gently placed individually in beakers containing 150 ml of de-ionised water for one-dimensional diffusion to take place.
The solute in the soil started to diffuse out in the bathing water resulting in an increased concentration of the bathing water. The change in concentration of the bathing water was recorded using a conductivity meter (see Appendix-II in file “real soil”) as shown in Figure 5.1. From this data for the concentration in bathing water, the change in mass of the solute in the soil sample was determined. By fitting the change in concentration of the solute in the bathing water with equation 3.35, the effective diffusion coefficient of the real soil in the shape of a slab was determined
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(see Appendix-II in file “De-real soil”). The experimental results and fitted results with the diffusion model (equation 3.35) showed a good agreement as shown in Figure 5.8. The measured values of De for the real soil are given in Table 5.5
Effective diffusion coefficient of the real soil
Density (mg/ mm3)
De (mm2 /min)
* Do = 0.104 mm2/min for KCl at 20 0C (Robinson and Stokes, 1959, p. 513)
Relative solute concentration in the bathing water against time which is fitted to the diffusion model (equation 3.35) for determining the effective diffusion coefficient of the solute within real soil. Symbols represent experimental data points while the solid line shows fitting of the data
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The effective diffusion coefficient of the real soil shown in Table 5.5 is much higher than that of the baked aggregates given in Table 5.2. This may be due to the higher density and lower porosity of the baked aggregates. To find the effect of porosity and density of aggregate/soil on the effective diffusion coefficient, the effective diffusion coefficients of the solute in baked aggregates and in the real soil are plotted against their porosity and density in Figure 5.9 a and b.
These graphs show a linear relationship between the effective diffusion coefficient and the density and porosity of the soil. As the porosity increases the pathways in the soil become less tortuous resulting in a higher effective diffusion coefficient. On the other hand diffusion pathways in the soil become more tortuous with the increasing density of the soil. This suggests that during leaching, a soil that is porous with good structure and has low density will experience faster diffusion resulting in higher efficiency of the leaching.
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Effective diffusion coefficient plotted against (a) porosity (b) density of the aggregates. The symbol ( ) represents real soil, while symbol (
represents baked aggregates.
Discussions and Conclusions
In these experiments the solute in saturated aggregates was initially uniformly distributed throughout its volume. When the aggregate was placed in the bathing water, solute near the interface diffused faster than the solute in the centre of the aggregate. Hence if the diameter of the aggregate is small, salts in the centre have only a small distance to cover to reach the surrounding water. Therefore the solute in small aggregates diffused faster compared with the large aggregate. However, in large aggregates the amount of solute in the micropores was more and the diffusion pathways were longer compared to those in small aggregates, resulting in slow diffusion and breakthrough curves with longer tailing (Biggar and Nielsen, 1962; Green et al., 1972 and Mac Mohan and Thomas, 1974). This is shown in the significant differences in the rate of diffusion for the three diameters used in the experiments.
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The effective diffusion coefficient (De) increased from 0.025 mm2/min to 0.035 mm2/min when the diameter of the aggregates increased from 8 to 31 mm. This may be due to differences in density, porosity and tortuosity of the aggregates. The large aggregates were less dense but more porous resulting in less tortuous diffusion pathways and larger effective diffusion coefficient. The results of De showed higher values for real soil than that for baked spherical aggregates, due to the larger porosity and lower density. As the density of real soil was less compared with baked aggregates the real soil had less tortuous pathways, which resulted in higher effective diffusion coefficient of real soil.
There was a poor agreement between the experimental data and that estimated by the mass transfer model using single averaged mass transfer rate coefficient ( ). The single average value of could not properly simulate the solute transfer from the micropore (stagnant) region into the bathing water as also reported by Rao et al. (1982). Therefore the assumption made by van Genuchten and Wierenga (1976) that
compensates the inadequate description geometry of micropore regions in mass transfer model appears to be incorrect.
In this chapter experiments on single aggregates in well-stirred bathing water have been reported. During leaching water is flowing around the aggregates and in the following chapter the theoretical considerations of diffusion from aggregates are taken into consideration with different leaching techniques. The effective diffusion coefficients that have been determined here are used in the following chapter for modelling of the leaching processes.
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