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Urea release rate from a scoop of coated pure urea

beads: Unified extreme analysis

S.M. Lu, Szu-Lin Chang, Wen-Yu Ku, Hou-Chien Chang,

Jiunn-Yau Wang, Duu-Jong Lee

*

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan Received 26 March 2007; accepted 10 April 2007

Abstract

Urea release from a scoop of coated beads in a given volume of a well stirred liquid has been investigated analytically and experimentally. A method for determining the fractional cumulative release and fractional release rate curves for the scoop without knowing particle number and radii is presented. The representative D/Kbfor a scoop of urea beads spray coated with ethyl cellulose is near 3.5 108cm2/s, and that with cellulose acetate phthalate is near 7 108cm2

/s.

# 2007 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Controlled release; Analysis; Experiment; Spray coated urea beads

1. Introduction

Development of controlled release fertilizers (CRFs) is to design device for preventing nutrient loss and enhancing nutrient utilization efficiency by plants (Trenkel, 1997). Polymer coated fertilizer particles were common CRFs whose release characteristics were not sensitive to soil environments. Mathematical models for single, coated particle were developed for describing release rate of drug with diffusion resistances (Chen and Lee, 2002; Frenning et al., 2003; Li et al., 2006; Liao and Lee, 1997; Lu and Chen, 1993, 1995; Lu, 1994; Shaviv et al., 2003). Kanjickl and Lopina (2004) reviewed some diffusion-controlled models for drug delivery systems. The significant role of diffusional resistances was studied in numerous systems (Chen and Shih, 2006; Huang et al., 2006; Kao and Li, 2006; Kuo and Chung, 2005; Santana and Macias-Machin, 2006; Su et al., 2006).

In fertilizer release experiment or practice from coated beads, it is more often that a scoop of particles, rather than one particle, is used. With particles 1–2 mm in diameter, the number of particles per release experiment can easily exceed several thousands. Small particles could be coated by

spraying a film forming solution in a bed of fluidized core particles. These particles are usually sieved before and after coating and therefore the particles are alike but not exactly the same. Thus, it will be useful to have a method for the analysis of release data obtained from a scoop of such particles.

Fertilizer release from a large coated pure fertilizer beads in a well stirred liquid has been investigated byLu and Lee (1992). The scenario analyzed by Lu and Lee (1992) presented an extreme test for release rate of coated particles in soil with limit moisture and no stirring. Here, their analysis has been extended to the case of fertilizer release from a scoop of particles. The results were then applied to analyze experimental release data obtained from a scoop of particles that had been spray coated in a fluidized bed. The method presented herein can be used as the first and extreme estimate to coated bead design and development.

2. Analysis

Consider a coated non-swellable particle of radius a containing a pure fertilizer bead of radius b as its core. Fertilizer release from this particle in a liquid is considered to proceed in three stages: (1) for time t < t0, liquid infiltration and

concentration build up occur in the coating layer, fertilizer released is negligible; (2) For t0t  ts, solid is present in the

www.elsevier.com/locate/jcice

* Corresponding author. Tel.: +886 2 23625632; fax: +886 2 23623040. E-mail address:djlee@ntu.edu.tw(D.-J. Lee).

0368-1653/$ – see front matter # 2007 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jcice.2007.04.001

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core and fertilizer concentration in the core is at the saturation concentration, Csat, tsis the time at which the solid fertilizer in

the core has just been exhausted; and (3) for t > ts, no more

solid fertilizer is in the core and fertilizer concentration in the core decreases with time.

Fertilizer release from a scoop of coated pure fertilizer beads in a well stirred liquid of volume Ve follows, in general, the

stages described above. In this analysis, the first stage is neglected and only the second and third stages are considered. Details of the analysis are given in Appendix A. In the following, the results that are directly useful in analyzing experimental data are summarized.

1. For t0 t  ts lnð1  FÞ ¼ nN1ðt  t0Þ (1) F KbCe KaCsat (2) nN1 n 4p Ve D Ka ab a b (3) a1 KbVc KaVe (4) Mt Mt1 ¼1þ na1 na1 1 b½1  expðnN1ðt  t0ÞÞ (5) dðMt=Mt1Þ dt ¼ 1þ na1 na1 nN1 b expðnN1ðt  t0ÞÞ (6) 2. At t = ts ts t0¼  1 nN1 ln½1  na1ðb  1Þ (7) b ru Csat (8a) 3. For t > ts Mt Mt1 ¼1þ na1 b  ðb  1Þ þ1 na1ðb  1Þ 1þ na1 ½1  expðN1ðt  tsÞÞ  (9) dðMt=Mt1Þ dt ¼ 1 na1ðb  1Þ b N1exp½N1ðt  tsÞ (10) N1 nN1þ nN1 na1 (11) In the above equations, particle number, n, always appears in product form, that is, as na1 and nN1. Eq.(1) on a semi-log

paper is a straight line with slopenN1. SlopenN1can be

determined by using experimental Ce vs. t data with F

calculated according to Eq.(2)for an assumed Kb/Ka. Eqs.(3),

(4), (8a), and (11) define the parameters as indicated. b, according to Eq. (8a), is a known quantity for a specified system. Eq.(7)calculates time ts t0. Eqs.(5)and(9)calculate

fractional cumulative release before and after ts, and Eqs.(6)

and (10), fractional release rate. In these calculations, only three parameters, b, nN1, and na1, need to be known.

Parameter na1may be expressed as follows:

na1¼ n KbKc KaVe ¼Kb Ka nVcru Veru ¼Kb Ka ðnWuÞ Veru (12) where Veis known experimentally and ru, fertilizer density, is

known. Vcrepresents the volume of one core bead. Product Vcru

is equal to Wu, the initial weight of solid fertilizer in one

particle. In the case that particle number, n, and core radius, b, are known, na1may be calculated by the second term in

Eq.(12)for a given Kb/Ka. However, as are often so in practical

cases, n, and b of a scoop are unknown. In such case, as

nWu¼ ðtotal drug wt: assayedÞ (13)

nWu can be determined experimentally. Manufacturer’s data

may also be consulted. Then, na1may be calculated by the last

term in Eq.(12)for a given Kb/Ka.

3. Method of application

Experimental release data from a scoop of coated pure fertilizer beads may be analyzed by the following trial and error method. First, a value is guessed for Kb/Ka. Then, from

Nomenclature

A radius of a coated pure fertilizer bead (cm) B radius of a coated fertilizer bead (cm)

C fertilizer concentration: Ca;ts, at r = a, t = ts; Cc,

in core; Cc;t1, in core, at t = t1; Ce, in external

liquid; Ce;t1, in external liquid, at t = t1; Csat, at

saturation (g/cm3)

D effective diffusivity of fertilizer in coating layer (cm2/s)

F defined by Eq. (2)

J mass flux of fertilizer (g/(cm2s))

Ki partition constant defined by Eqs. (A-2b)and

(A-2c)for i = a, b

Mt cumulative release at t; Mts at ts; Mt1 at t1 (g)

N number of particles per release experiment N1 defined by Eq. (3), (1/s)

N1 defined by Eq. (11), (1/s)

R radius (cm)

T time: t0, the intercept determined by Fig. 1; ts,

time at which the solid fertilizer in a core dis-appears; t1, infinite time (s)

V volume: Vc, core volume; Ve, volume of external

liquid; Vp, particle volume (cm3)

Wu initial mass of fertilizer per particle (g)

Greek symbols

a1 defined by Eq. (4)

b defined by Eq. (8a)

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experimentally obtained Ce vs. t data, F are calculated

according to Eq. (2) and a plot, (1 F) vs. t, is constructed on a semi-log paper. The data in the second release stage, which can be discerned easily because of the difference in the trend of the data points, is then fitted with a straight line.Fig. 1shows an example with sample EC450R4. Time t0and slopenN1are

thus determined. Next, na1is found by Eqs.(13)and(12).

For a given release system, b can be calculated by Eq.(8a). (ts t0) is then calculated by Eq.(7). With nN1and na1known,

N1 is calculated by Eq. (11). Fractional cumulative release

curve over the entire release life is then calculated by Eqs.(5)

and(9). This curve may be checked with the experimental data. If the agreement is good and the material balance is correct, the Kb/Kaand the slopenN1are accepted; if not, a new guess is

taken for Kb/Kaand the process is repeated. Fractional release

rate curve over the entire release life is then calculated by Eqs.(6)and(10). The group, n(D/Ka)(ab/(a b)), representing

the mixed effects of variables n, D/Ka, a and b, can be

determined from Eq.(3). Thus, it has been shown that time ts,

fractional cumulative release, and fractional release rate may be calculated without knowing particle number n and particle radii a and b.

One group of variable that may bring further information about the scoop is the D/Kb of the scoop. However, to find

(D/Kb), it is necessary to know a, b, and n. If these are available,

then, from n(D/Ka)(ab/(a b)) which has already been

determined, the D/Kb can be calculated. For particles which

were sieved before and after coating, n can be determined by counting, and radii, a and b, may be evaluated by experimental measurement of the coating thickness and particles size. 4. Experiment

A small Wurster type bottom spray coating unit with a draft tube was designed and setup for coating. Urea beads, 10/12 mesh (ASTM standard), were spray coated respectively with 5% (w/v) acetone solutions of Ethocel 45Std (Dow Chemical) and CAP-482-0.5 (Eastman Chemical). Temperature of the coating solution was kept at 30 8C. Bed temperature, depending on the position, varied from near 51 to 32. Particles were sieved before and after coating.

For release test, 2 g of a sample were placed inside a 125 mL flask containing 100 mL distilled water. The flask was mounted on an orbital shaker at 120 rpm. The release proceeded at 25 8C. The urea released was analyzed by the conductivity method (Chin and Kroontje, 1961).

5. Results and discussion

The results of experiments and analysis are summarized and discussed as follows.

Table 1

The result of application of the analysis to the experiments (Csat= 0.632 g/mL, ru= 1.335 g/mL, Ve= 100 mL)

No. Sample nWu(g) Kb/Ka() Parameters for Eqs.(5),(6),(9),(10) (D/Ka)(nab/

(ab)) (1/s)

Type Wt. (g) Experimental data na1() b() ts-t0(h) ts(h) N1(1/h)

nN1(1/h) t0(h)

1 EC150R1 2 1.932 2.5 2.17E-3 1.0 0.03619 2.1112 18.9 19.9 0.06214 4.797E-6

2 EC150R2 2 1.912 3.5 2.96E-3 1.0 0.05014 2.1112 19.4 20.4 0.06200 6.543E-6

3 EC150R3 2 1.920 4 2.89E-3 1.0 0.05754 2.1112 22.9 23.9 0.05312 6.388E-6

4 EC300R1 2 1.189 2.5 4.64E-4 4.5 0.02226 2.1112 54.0 58.5 0.02131 1.026E-6

5 EC300R2 2 1.177 5 8.15E-4 8.9 0.04408 2.1112 61.7 70.6 001930 1.802E-6

6 EC300R3 2 1.201 3 5.19E-4 6.4 0.02698 2.1112 58.7 65.1 0.01975 1.147E-6

7 EC450R4 2 0.5566 2 9.22E-5 108 0.008339 2.1112 101.1 209.2 0.01110 2.038E-7

8 EC450R5 2 0.5686 2 1.10E-4 87.0 0.008518 2.1112 86.5 173.5 0.01297 2.432E-7

9 EC450R6 2 0.5725 2 1.17E-4 82.9 0.008577 2.1112 81.9 164.8 0.01357 2.586E-7

10 CAP150R1 2 1.873 2.5 4.88E-3 0.0 0.03507 2.1112 8.2 8.2 0.14403 1.079E-5

11 CAP150R2 2 1.861 3 5.53E-3 0.0 0.04182 2.1112 8.6 8.6 0.13778 1.222E-5

12 CAP150R3 2 1.849 3 5.51E-3 0.0 0.04155 2.1112 8.6 8.6 0.13813 1.218E-5

13 CAP300R1 2 1.101 2 7.23E-4 2.5 0.01650 2.1112 25.6 28.1 0.04454 1.598E-6

14 CAP300R2 2 1.093 2 7.01E-4 3.0 0.01638 2.1112 26.2 29.2 0.04350 1.550E-6

15 CAP300R3 2 1.093 2.5 7.98E-4 3.0 0.02048 2.1112 28.9 31.9 0.03977 1.764E-6

Fig. 1. Fitting experimental data to Eq.(1), for EC450R4. Slope and intercept t0

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5.1. Experiments and analysis

Application of the analysis to the results of experiments is shown step by step inTable 1. Columns 1 and 2 show sample number and sample type. For the later, the first two or three letters show the coating material, and the next three digits show a number which is proportional to the mass f coating material sprayed. The last two letters show run number. Column 3 shows that 2 g of a sample was used in each release experiment. Column 4 shows nWu, the weight of urea in 2 g of

sample, evaluated as shown by Eq.(13). Column 5 shows the magnitudes of Kb/Ka used. Columns 6 and 7 show,

respectively, slope nN1 and intercept t0 determined from

experimental data by a plot likeFig. 1. Columns 8–12 show na1, b, ts t0, ts and N1 determined as indicated. The last

column lists n(D/Ka)(ab/(a b)) calculated using Eq. (3).

Thus, the parameters needed for computing a fractional

cumulative release curve and a fractional release rate curve have been obtained as shown without requiring information for particle number and radii.

Experimental and calculated fractional cumulative release for EC and CAP coatings are shown inFig. 2. Good agreement has been obtained. The calculated fractional release rate curves are shown inFig. 3. The t0of most samples were small and were

ignored except EC450R4–6 for which t0 exceeded 81 h.

Generally speaking, the rate rises from zero to a fairly constant rate within time t0, maintains a fairly constant rate for time

length ts t0, and falls after ts. Particles with EC coating

resulted in lower fairly constant rate over a longer time than particles with CAP coating which gave a fairly high constant rate over a shorter time. Urea release is faster by EC coating than by CAP coating. Particles with either coating gave long tailing time after ts. This is not due to the coatings but to the

system (urea and water) which is a system of high solubility

Fig. 2. Comparison of the experimental and calculated fractional cumulative release for scoops of spray coated urea beads releasing in water. For each sample, the solid line is calculated by Eq.(5)(t < ts) and Eq.(9)(t > ts), where

tsare listed inTable 1. (a) EC coating and (b) CAP coating.

Fig. 3. Fractional release rate curves calculated for scoops of spray coated urea beads releasing in water, by Eq.(6)(t < ts) andTable 1. (a) EC coating and (b)

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(Lu, 1994).Figs. 2 and 3may thus be obtained without knowing particle number and radii of the scoop.

5.2. D/Kbof a scoop

The representative D/Kb of a scoop of particles may be

determined as shown inTable 2. Column 3 ofTable 2shows the number of particles counted in 2 g of sample. Column 4 shows nWuwhich has already been shown inTable 1. As n is

also the number of core beads, thus, Wu, the representative

weight of one urea bead for the scoop, may be found by dividing column 4 by column. Column 5 shows the result. As the density of urea is known, b can be calculated as shown in column.

It should be noted that b may also be estimated from mesh size and sieve openings. However, for thinly coated particles, like sample no. 1, 2, 3 and 10, 11,12, most coated particles still fall in 10/12 mesh, the same as for the core beads. Thus, b is estimated as has already been explained. The percentage relative difference between the b in Table 2 and the average sieve openings (10/12 mesh) is 3.5%, and with the average of direct measurement, is 2.1%.

Column 7 ofTable 2shows the averaged coating thickness measured by a 60 precision magnifier. Column 8 shows mesh size of coated particles. Column 9 shows radius of coated particle a obtained by adding columns 6 and 7. a may also be estimated by the average sieve opening. The percentage difference between these two a is less than about 9%. It is recommended that coating thickness be determined by direct measurement and not by subtraction of b from a, or vice versa. The later can result in large errors, particularly for thin coating. Column 10 shows radius ratio. Columns 11 and 12 show D/Ka

and D/Kb, calculated as indicated.

The D/Kb evaluated as above is plotted against coating

thickness, as shown inFig. 4. The magnitudes of D/Kbobtained

for thick coatings are more easily consistent with each other than those for thin coatings. Care must be taken for coating thickness below 0.01 cm.Fig. 4shows that the representative

D/Kbfor urea in spray coating layer is around 3.5 108cm2/s

for EC and 7 108cm2/s for CAP. 5.3. General discussion

Inspection of Eqs.(5),(9),(6),(10), and(A-8)and(A-19)

show that in the analysis, fertilizer release from a scoop of particles is equivalent to each particle releasing in a liquid volume Ve/n. That is, fertilizer release from n particles in a

liquid of volume Ve is equivalent to fertilizer release from a

single particle in a liquid of volume Ve/n. When Ve/n is large, the

condition of a perfect sink is justified, the effect of particle number on the release may be neglected, and relatively constant release rate is maintained from t0to ts. However, when Ve/n is

small, the effect of n appears and the release rate from t0to ts

may show considerable changes.

For small Ve/n, there are other complications. Besides the

increased interference among the particles, other practical Table 2 Calculation for D/Kb No. Sample n () nWu(g) Wu 100 (g) b 100 (cm) (ab)  100 (cm) mesh a (cm) a/b () D/Ka 108 (cm2/s) D/Kb 108 (cm2/s) 1 EC150R1 486 1.932 0.3976 8.925 0.889 10/12 0.098 1.100 1.0017 0.4007 2 EC150R2 481 1.912 0.3976 8.925 0.889 10/12 0.098 1.100 1.3805 0.3944 3 EC150R3 483 1.920 0.3976 8.925 0.889 10/12 0.098 1.100 1.3422 0.3356 4 EC300R1 299 1.189 0.3976 8.925 2.642 8/10 0.116 1.296 0.8780 0.3512 5 EC300R2 296 1.177 0.3976 8.925 2.642 8/10 0.116 1.296 1.5577 0.3115 6 EC300R3 302 1.201 0.3976 8.925 2.642 8/10 0.116 1.296 0.9718 0.3239 7 EC450R4 140 0.5566 0.3976 8.925 4.953 7/8 0.139 1.555 0.5821 0.2910 8 EC450R5 143 0.5686 0.3976 8.925 4.953 7/8 0.139 1.555 0.6800 0.3400 9 EC450R6 144 0.5725 0.3976 8.925 4.953 7/8 0.139 1.555 0.7181 0.3590 10 CAP150R1 471 1.873 0.3976 8.925 0.711 10/12 0.096 1.080 1.8943 0.7577 11 CAP150R2 468 1.861 0.3976 8.925 0.711 10/12 0.096 1.080 2.1591 0.7197 12 CAP150R3 465 1.849 0.3976 8.925 0.711 10/12 0.096 1.080 2.1659 0.7220 13 CAP300R1 277 1.101 0.3976 8.925 2.489 8/10 0.114 1.279 1.4095 0.7048 14 CAP300R2 275 1.093 0.3976 8.925 2.489 8/10 0.114 1.279 1.3771 0.6886 15 CAP300R3 275 1.093 0.3976 8.925 2.489 8/10 0.114 1.279 1.5673 0.6269

Fig. 4. D/Kbobtained for scoops of spray coated urea beads releasing in water

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problems, like methods for good mixing and rupture of particles by friction and collision, arise. In the case that the above problems may be ignored, there is the limiting Ve/n at which the

system ends release at t = ts. Below this limiting Ve/n, fertilizer

release ends with solid fertilizer still present in the core. In laboratory work, these situations are easily avoided by using sufficient amount of liquid. In applications, such situation could be encountered if particles are applied in a very limited environment.

It should also be pointed out that for very small core beads, coating thickness should be kept thin relative to the core radius if a fairly constant release rate is to be obtained. A very small bead with very thick coating may lead to the situation that exhaustion of solid fertilizer in a core occurs before the development of a pseudo-steady state concentra-tion profile.

6. Conclusion

The fractional cumulative release and fractional release rate curves of a scoop of spray coated pure fertilizer beads releasing in a well stirred liquid may be determined without knowing the particle number and radii. The representative D/Ka for the

scoop may also be determined once the representative coating thickness and core radius of the scoop are estimated.

Acknowledgement

This project (NSC 85-2214-E002-026) was supported by National Science Council, ROC.

Appendix A

A.1. Derivation of the equations

The basic equation for fertilizer concentration in the coating layer, C, and the conditions are

@ðrCÞ @t ¼ D @2ðrCÞ @r2 ; b< r < a (A-1) Cðb; tÞ ¼Csat Kb ; t0 t  ts (A-2a) Cðb; tÞ ¼CcðtÞ Kb ; t> ts (A-2b) Cða; tÞ ¼CeðtÞ Ka ; t> 0 (A-2c)

t0 is assumed to be the time that the external liquid has

infiltrated the coating layer and that the pseudo-steady state concentration profile is established in the coating layer. Thus, C(b, t < t0) = 0 is assumed.

With the assumption that pseudo-steady state exists for t t0, the pseudo-steady state concentration profile for t t0is

Cðr; tÞ ¼aCða; tÞ  bCðb; tÞ a b þ 1 r ab a b½Cðb; tÞ  Cða; tÞ (A-3) With Fick’s law, then the rate of diffusion is

Jð4pr2Þ ¼ D  @C @r  ð4pr2Þ ¼ 4pD ab a b½Cðb; tÞ  Cða; tÞ (A-4) The time derivatives of fertilizer concentration in the external liquid and in the core may be obtained, respectively, by material balance and by Eq. (A-4). The results are

dCe dt ¼ nJð4pr2Þ Ve ¼ n4pD Ve ab a b½Cðb; tÞ  Cða; tÞ (A-5a) dCc dt ¼ 4pD Ve ab a b½Cðb; tÞ  Cða; tÞ (A-5b) 1. When t0 t  ts

Notice that t0 t  ts, Cc(t) = Csat. Eq.(A-5a)then becomes

dCe dt ¼ nN1  Ka Kb Csat Ce  (A-6) where nN1 n 4p Ve D Ka ab a b

Eq. (A-6), after separation of variables, is integrated from t = t0, Ce= 0 to t = t, Ce= Ce, the result is ln  1 KbCe KaCsat  ¼ nN1ðt  t0Þ (A-7) or Ce Csat ¼Ka Kb ½1  expðnN1ðt  t0ÞÞ (A-8)

Eq.(A-7) may be written as lnð1  FÞ ¼ nN1ðt  t0Þ

where F KbCe

KaCsat

on a semi-log plot, Eq.(1)represents a straight line with slope (nN1).

To fine cumulative release, Mt, as

Mt¼ VeCe (A-9)

by substituting the Ce in Eq. (A-8) into the above equation,

obtain Mt¼

VcCsat

a1

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where a1

KbVc

KaVe

The fractional cumulative release can be obtain by dividing the above equation by Mt1, i.e., Eq.(A-21)which is derived later. The result is

Mt Mt1 ¼1þ na1 na1 1 b½1  expðnN1ðt  t0ÞÞ where b is defined by Eq.(8a).

The fractional release rate is obtained by differentiating Eq.(5) with respect to time, i.e.

dðMt=Mt1Þ dt ¼ 1þ na1 na1 nN1 b expðnN1ðt  t0ÞÞ 2. At t = ts

At ts, the amount of fertilizer left in a core is VcCsatand the

cumulative release is Mts. As Wu is the initial amount of

fertilizer in a core, Mts is expressed by

Mts ¼ nðWu VcCsatÞ (A-11)

Equate Eq.(A-11)to Eq. (A-10)at ts and solve the resulting

equation for tsto find

ts t0¼  1 nN1 ln½1  na1ðb  1Þ where b¼ Wu VcCsat ¼ ru Csat (8b) 3. For t > ts

By material balance for the fertilizer and by Eq. (A-5a), obtain nVc dCc dt ¼ Ve dCe dt ¼ n  4pD ab a b  ½Cðb; tÞ  Cða; tÞ (A-12) By using Eqs. (A-2b), (3) and (4), the above equation becomes dCðb; tÞ dt ¼ N1 a1 ½Cðb; tÞ  Cða; tÞ (A-13)

Similarly, by using Eqs. (A-2c), (3) and (4), Eq. (A-12)

becomes dCða; tÞ

dt ¼ nN1½Cðb; tÞ  Cða; tÞ (A-14)

Addition of the above two equations results in d½Cðb; tÞ  Cða; tÞ dt ¼ N1½Cðb; tÞ  Cða; tÞ (A-15) where N1 nN1þ nN1 na1

Eq.(A-15), after separation of variables, is integrated from t = tsto t = t, the result is

Cðb; tÞ  Cða; tÞ ¼ ½Cðb; tsÞ  Cða; tsÞ exp b  N1ðt  tsÞ c

(A-16) substitute the above equation into Eq. (A-5a)and integration the resulting equation from t = ts to t = t. With

Cðb; tsÞ ¼ Csat Kb (A-17) Cða; tsÞ ¼ nðWu VcCsatÞ VeKa (A-18) the result of integration is rearranged as follows:

Ce Csat ¼ nVc Ve  ðb  1Þ þ1 na1ðb  1Þ 1þ na1 ½1  expðN1ðt  tsÞÞ  (A-19) cumulative release is obtained from Eqs.(A-19)and(A-9), i.e.,

Mt¼ nVcCsat  ðb  1Þ þ1 na1ðb  1Þ 1þ na1 ½1  expðN1ðt  tsÞÞ  (A-20) As t! 1, the above equation becomes

Mt1¼

nWu

1þ na1

(A-21) Fractional cumulative release is then obtained as

Mt Mt1 ¼1þ na1 b  ðb  1Þ þ1 na1ðb  1Þ 1þ na1 ½1  expðN1ðt  tsÞÞ 

Fractional release rate is dðMt=Mt1Þ

dt ¼

1 na1ðb  1Þ

b N1exp½N1ðt  tsÞ

The concentration of fertilizer in the core is obtained by substituting Eq. (A-16), with Eqs. (A-17) and (A-18), into Eq.(A-5b)and integrating the resulting equation from t = tsto

t = t. The result is Cc Csat ¼ 1 1 na1ðb  1Þ 1þ na1 ½1  exp½N1ðt  tsÞ (A-22)

As t! t1, the above equation becomes

Cc;t1

Csat

¼ na1b 1þ na1

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As Ce;t1 ¼ Mt1=Ve, then with Eq.(A-23), obtain Cc;t1 Ce;t1 ¼Kb Ka (A-24) References

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Chen, P. H. and H. C. Shih, ‘‘A Chemical Kinetics Model for Oxide Chemical Mechanical Polishing,’’ J. Chin. Inst. Chem. Engrs., 37, 401 (2006). Chin, W. T. and W. Kroontje, ‘‘Conductivity Method for Determination of

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數據

Fig. 1. Fitting experimental data to Eq. (1), for EC450R4. Slope and intercept t 0
Fig. 3. Fractional release rate curves calculated for scoops of spray coated urea beads releasing in water, by Eq
Table 2 Calculation for D/K b No. Sample n () nW u (g) W u  100 (g) b  100 (cm) (ab)  100(cm) mesh a (cm) a/b () D/K a  10 8(cm2/s) D/K b  10 8(cm2/s) 1 EC150R1 486 1.932 0.3976 8.925 0.889 10/12 0.098 1.100 1.0017 0.4007 2 EC150R2 481 1.912 0.39

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