Controlled-release of fertilizers: modelling and simulation

International Journal of Engineering Science 37 (1999) 1299±1307

Controlled-release of fertilizers: modelling and simulation S.M. Al-Zahrani

*

Chemical Engineering Department, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia Received 13 July 1998; accepted 8 September 1998 _ (Communicated by E.S. S ß UBUHI)

Abstract Explicit mathematical models have been developed to predict the release rate of fertilizers from polymeric membrane. The models can be used to describe the release rate under di€erent operating conditions. Detailed and approximate solution were developed and compared with numerical solution for the fertilizer release rate. These solutions determine a ®nite step for the determination of release characteristics of fertilizer from di€erent type of formulations. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Control-release; Fertilizers; Modelling and simulation

Nomenclature C C0 C1 D k Mt M1 g r R t VS

*

the concentration of the Fertilizer in the sphere initial concentration of the Fertilizer the concentration in the sphere at in®nite time di€usion coecient of the Fertilizer distribution coecient mass di€used up to time t mass di€used after in®nite time dimensionless radius radial distance from the center of the sphere radius of the sphere time volume of well-stirred

Tel.: 00966 1 4676873; fax: 00966 1 4678 770; e-mail: [email protected].

0020-7225/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 8 ) 0 0 1 2 0 - 7

1300

VF u s

S.M. Al-Zahrani / International Journal of Engineering Science 37 (1999) 1299±1307

volume of the sphere dimensionless concentration dimensionless time …Dt=r02 †

1. Introduction and literature review Controlled-release fertilizers have been known for a several years. It has been estimated that the percentage of the fertilizer dose recovered by plants when applied in conventional forms may amount up to only 30±50% [1]. The control of fertilizer release keeps the fertilizer concentration at e€ective levels in the soil solution and releases the fertilizer when the plant most needs it. As a result of this control maximal utilization of the fertilizer from plant systems, remarkable decrease with respect to fertilizer application rate, least possible losses of the fertilizer through volatilization or leaching, prevention of the seedling damage and full protection of the ecosystem in the case of biodegradable carriers [2]. The ®rst study on the application of controlled-release technology to fertilizers are placed in 1962 [3]. Most of the cases cited in literature have to do with reservoir-type formulations: systems where a fertilizing core is encapsulated inside inert carrier, in other words, coated fertilizers. The controlled-release of the fertilizer is controlled by di€usion through the coating. Sulfur-coated urea (SCU) [4], polyethylene-coated urea [5], and coated superphosphate [6] provide typical examples for this class of formulations. Another way of regulating the release of fertilizer is accomplished by means of chemically controlled releasing products, such as urea±formaldehyde condensates [7]. Matrix-type (monolithic) formulations constitute the third major category of controlled-release device. The active agent is dispersed in the matrix and di€uses through the matrix continuum or intergranular openings, that is, through pores or channels in the carrier phase and not through the matrix itself. One of the advantages of monolithic formulations is their simple fabrication. The ®rst comprehensive study on this type of fertilizers appeared in 1987 by Hepburn et al. [8]. In comparison with the previously mentioned categories, matrix-type fertilizer formulations have seldom been studied. Few expressions have been forwarded to describe the release from di€erent sustained-release products. Higuchi [9] was the ®rst to derive an equation describing the drug release from porous matrix when the amount of drug present is greater than its solubility by a factor of 3 or 4. Such treatment was extended to describe the release from planes when the drug is present in low content [10]. Di€usion from plane sheets and cylinders were considered [11]. Higuchi [9] proposed the following relationship for one-dimensional, pseudo-steady state release of active agents dispersed in slabs (planar case) where release occurs through the pores of the matrix Q ˆ ……De=s†…2A ÿ eCs †Cs †1=2 ;

…1†

where Q is the amount of active agent released per unit area exposed, D the di€usion coecient of the active agent in the solution medium, e the porosity of the matrix, s the tortuosity of the matrix, A the concentration of the active agent in the matrix, and the Cs the solubility of the active agent in the dissolution medium.

S.M. Al-Zahrani / International Journal of Engineering Science 37 (1999) 1299±1307

1301

Sinclair and Peppas [12] provided a very simpli®ed ``empirical'' equation, which may serve as a potential tool in revealing the mechanism that governs the di€usion of active agents from nonswellable devices Qt ˆ k  tn ;

…2†

where Qt is the fraction of active agent released at time t, k a constant incorporating the characteristics of the carrier-active agent system, and n the di€usional exponent, indicative of the transport mechanism. When the release of the active agent is di€usion controlled and expressed by Fick's laws, the exponent is equal to 0.5. Schwartz [13] reported the ®rst order kinetics release of active agent by the following expression Qt ˆ 1 ÿ eÿkt ;

…3†

where Qt is the fraction of the active agent released at time t and k a release rate constant. Baker and Losdale [14] found that the amount of dissolved active agent (which di€use through the polymer membrane) released at any given time per total amount of the active agent initially Mt =M1 for a spherical particle can be approximated by:  0:5 Mt Dt 3Dt ˆ6 ÿ 2 ; for Mt =M1 < 0:4; M1 pr2 r    2  Mt 6 ÿp Dt exp ; for Mt =M1 > 0:6; ˆ1ÿ 2 M1 p r2

…4† …5†

where D is the di€usion coecient in polymer and r the radius of the spherical particle. The main objective of this work is to develop mathematical models which can describe the release rate of fertilizers from spherical polymeric membrane under di€erent operating conditions. 2. Mathematical modelling For a spherical fertilizer particle of radius R and volume VF . The concentration (C0 ) of the solute is uniformly distributed all over the sphere and di€uses into a well-stirred solution of zero fertilizer concentration and volume VS . The fertilizer concentration (C) in the sphere particle is uniform at all times. Therefore, the unsteady state mass balance on a spherical shell is given by Eq. (1): oC o2 C 2D oC ˆD 2 ‡ : ot or r or

…6†

This mass balance equation is subject to the following boundary conditions: at r ˆ 0 and t ˆ 0

oC ˆ 0; or

…7†

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at r < R and t ˆ 0

C ˆ C0 ;

…8†

Vs oC ˆ ÿAJ ; k ot   Vs oC oC 2 ; ˆ ÿ4pR D k ot or rˆR

at r ˆ R and t > 0

at r ˆ R and t ˆ 0

…9†

C ˆ 0:

…10†

2.1. The analytical solution It is more convenient to rewrite the above equations in terms of the following dimensionless variables: uˆ

C ÿ C1 ; C0 ÿ C1

…11†

sˆ

Dt ; R2

…12†

g ˆ r=R:

…13†

Therefore the above mass balance equation and the boundary conditions can be rewritten in terms of these dimensionless variable to have the following form ou o2 u 2 ou ˆ 2‡ ; os og g og

…14†

while the boundary conditions will have the following forms: g ˆ 0 and s > 0

ou ˆ 0; og

g < 1 and s ˆ 0 u ˆ 1; g ˆ 1 and s > 0

ou ÿ3 3 k ou 3 k ou ˆ ÿ4pR ˆ 4pR os Vs og 3 Vs og gˆ1 ou ˆ ÿ3a ; og

g ˆ 1;

sˆ0

then u ˆ ÿa;

…15† VF k ou ˆ ÿ3 V1 og

…16†

gˆ1

…17†

…18†

where aˆ

C1 VF K ˆ : C0 ÿ C1 Vs

…19†

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The general solution of the system (Eqs. (14)±(18)) will have the following form: (see Appendix A for the details) uˆ

1 ÿ6a…1 ‡ a† X sin kn gexp…ÿk2n s† ; 2 g nˆ1 sin kn 9a…1 ‡ a† ‡ kn

…20†

where the kn are the nonzero positive roots of tan kn ˆ

3akn : 3a ‡ k2n

…21†

The mass balance on a spherical particle at any time can be calculated by: Zs Mt ou ˆ ÿ3 ds; M1 og gˆ1

…22†

0

where Mt is the mass di€used up to time t and M1 the corresponding amount after in®nite time (see Appendix A for the details) 1 X Mt u…1; t† 6…1 ‡ a† ˆ1‡ exp…ÿk2n s†: ˆ1ÿ 2 M1 a nˆ1 9a…1 ‡ a† ‡ kn

…23†

This mass balance equation can be used to calculate the release of fertilizer at short or long times for any number of spherical particles and for the external solution as well. The in®nite series given by Eq. (23) converges rapidly at long times but slowly at small times. The approximation for short time of Eq. (23) have the following form p Mt …1 ‡ a† 2s ˆ ‰1 ÿ e9a erf…3a s†Š M1 a

…24†

and for small values of a this reduces to Mt 6…1 ‡ a† ˆ p s1=2 : M1 p

…25†

2.2. The numerical solution The major diculty in the analytical solution lies on the estimation of the roots of Eq. (21) and the large number of terms needed for the series which need a long time for the series terms to be calculated. Therefore one can approximate solution numerically using the DPDES subroutine available in the IMSL library [15]. DPDES used to solve a system of partial di€erential equations of the form UT ˆ FCN (X, T, U, UX, UXX) using the method of lines with cubic hermite polynomials. It has been used here to solve the system of Eq. (14)±(18). Using a method of lines, in which the solution is expanded in a series of cubic hermite basis functions. The unknown coecients, which represent the values of u and ux at the knots, are functions of time. When the boundary conditions are imposed at the two end points and the

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di€erential equations are required to be satis®ed at two Gaussian quadrature points between each pair of knots (this is known as collocation) there results a system of 2 * NPDES * NX ordinary di€erential equations for the 2 * NPDES * NX unknown coecient functions. This ODE system is solved by calling a nucleus DPDEV which is based on DGEAR, but modi®ed to handle the generalized problem Ay ˆ f …t; y†. 3. Results and discussion The roots of Eq. (21) have been calculated numerically using bisectional method. The reduction of the general release equation (i.e., Eq. (23)) into the special form Eq. (25) at small values of a proves the generalization of Eq. (23) which can be approximated to provide the especial form where most of the researchers used for the prediction of the release rate due to its simplicity (i.e., Eq. (25)). The numerical full-range solution is shown in Fig. 1 as the fraction of mass transfer against the time for di€erent values of a. Any increase in the value of a will lead to an increase in the release rate. The value of a can be increased either by increasing the volume of the sphere or by decreasing the solution volume. It is obviously clear that the fertilizers which have a higher distribution coecient will have a higher release rate. The numerical solution is compared to the analytical solution when the value of a ˆ 1 for the release rate as a function of the square root of time is shown in Fig. 2. This ®gure re¯ects that the numerical solution of the system is almost identical to the analytical solution. 4. Conclusions It is very necessary to control the transfer of a fertilizer in agriculture. The thickness of the polymeric membrane plays an important role for retarding the fertilizer transfer. The mathematical

Fig. 1. Release pro®les from spherical particle at di€erent values of a.

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Fig. 2. Comparison of the release pro®les calculated by numerical and analytical techniques.

model developed can be used to predict release rates from di€erent polymeric membranes. Eq. (23) predicts the mass transfer rate, over the full range of time, from spherical particles into ®nitevolume ¯uids, with resistance being considered in the solid phase only. The model is based on both analytical and numerical methods is tested for describing the release of fertilizer. It is able to determine the pro®le of fertilizer concentration throughout the polymeric membrane at any time. Mathematical simulation can be made with the help of the model in order to de®ne exactly the role played by the various parameters characterizing the fertilizer and the polymeric membrane. These calculations can be of great importance in the selection of the polymer necessary for preparing the polymeric membrane. Appendix A The mass balance equation is ou o2 u 2 ou ˆ 2‡ : os og g og

…A:1†

By taking the Laplace transform for the mass balance equation, we have  2 d d2 u u  ‡ 1 ˆ 0; ‡ ÿ Su 2 g dg dg

…A:2†

where S is the transform variable. To solve this second order di€erential equation analytically let: ˆ u

U ; g

…A:3†

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then d u du U ˆ ÿ 2; dg g dg g

…A:4†

 d2 u d2 U 2dU 2U ˆ ÿ 2 ‡ 3 : 2 2 g dg g dg g dg

…A:5†

Now by substituting Eqs. (A.4) and (A.5) into Eq. (A.2), we have d2 U 2 dU 2U 2 dU 2U U ‡ 3 ‡ 2 ÿ 3 ÿS ‡1ˆ0 ÿ 2 2 g dg g dg g g dg g g

or

d2 U ÿ SU ˆ ÿg: dg2

…A:6†

The solution of this simpli®ed form of the di€erential equation can have the following form ˆ u

p p A B 1 sinh S g ‡ cosh S g ‡ : g g S

…A:7†

Eliminating A and B using the boundary conditions: B ˆ 0; Aˆ

…A:8†

1‡a p p : …3a ÿ S† sinh S ÿ 3a cosh S

…A:9†

Subsituting Eqs. (A.8) and (A.9) into Eq. (A.7) p …1 ‡ a† sinh S g 1 p p ‡ : ˆ u g …3a ÿ S† sinh S ÿ 3a cosh S S This equation may be inverted to the time domain by the method of residues to uˆ

1 ÿ6a…1 ‡ a† X sin kn g exp…ÿk2n s† : 2 g nˆ1 sin kn 9a…1 ‡ a† ‡ kn

…A:10†

The mass balance on a spherical particle up to time t gives Zs Mt ou ˆ ÿ3 ds; M1 og gˆ1 0

where uˆ

1 ÿ6a…1 ‡ a† X sin kn g exp…ÿk2n s† g sin kn 9a…1 ‡ a† ‡ kn2 nˆ1

…A:11†

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at g ˆ 1 Zs 0

" # 1 1 X X ou ‡6a…1 ‡ a†exp…ÿk2n s† ÿ3a k2n  ds ˆ ÿ ‡1 2 2 3a og 3a nˆ1 9a…1 ‡ a† ‡ kn …ÿkn † nˆ1 ˆ

1 X 2a…1 ‡ a†exp…ÿk2 s† n

nˆ1

9a…1 ‡ a† ‡ k2n

at g ˆ 1 ! 1 1 X X Mt 2…1 ‡ a† 6a…1 ‡ a† 2 ˆ ÿ3 exp…ÿk s† ˆ 1 ÿ exp…ÿk2n s†: n 2 2 M1 nˆ1 9a…1 ‡ a† ‡ kn nˆ1 9a…1 ‡ a† ‡ kn

…A:12†

References [1] R. Prasad, G. Rajale, B. Lacakhdive, Nitri®cation retarders and slow-release nitrogen fertilizers, Adv. Agron. 23 (1971) 337. [2] S. Allen, D. Mays, Sulfur-coated urea for controlled release: agronomic evaluation, J. Agric. Food Chem. 19 (5) (1971) 809. [3] J. Ortli, J. Lunt, Controlled release of fertilizer minerals by encapsulating membranes: I. Factors in¯uencing the rate of release, Soil Sci. Soc. Am. Proc. 26 (6) (1962) 579. [4] D. Rindt, G. Blouim, J. Gestsinger, Sulfur coating on nitrogen fertilizer to reduce the dissolution rate, J. Agric. Food Chem. 16 (5) (1968) 773. [5] O. Salman, Polyethylene-coated urea: 1. Improved storage and handling properties, Ind. Eng. Chem. Res. 28 (5) (1989) 630. [6] K. Subrahamanyan, L. Dixit, E€ect of di€erent coating materials on the pattern of phosphorous release from superphosphate, J. Indian. Soil Sci. Soc. 36 (3) (1988) 461. [7] P. Stajer, V. Glasser, J. Vosolsobe, J. Vidensky, Review of methods for production of slow-release nitrogen fertilizers, I. Chem. Prum. 36 (11) (1986) 577. [8] C. Hepburn, S. Young, R. Arizal, Rubber matrix for the slow release of urea fertilizer, Am. Chem. Soc., Div. Polym. Chem. 28 (28) (1987) 94. [9] T. Higuchi, Mechanism of sustained-action medication: theoretical analysis of rate of release of solid drug dispersed in solid matrices, J. Pharm. Sci. 52 (12) (1963) 1145. [10] S.J. Desai, P. Singh, A.J. Simonelli, W.I. Higuchi, Investigation of factors in¯uencing release of solid drug dispersed in inert matrices: III. Quantitative studies involving the polyethylene plastic matrix, J. Pharm. Sci. 55 (1966) 1230. [11] S.J. Desai, P. Singh, A.J. Simonelli, W.I. Higuchi, Investigation of factors in¯uencing release of solid drug dispersed in inert matrices: IV. Quantitative studies involving the polyvinyl chloride matrix, J. Pharm. Sci. 55 (1966) 1935. [12] G. Sinclair, N. Peppas, Analysis of non ®ckian transport in polymers using simpli®ed exponential expression, J. Membr. Sci. 17 (1984) 329. [13] J. Schwartz, A.J. Simonelli, W.I. Higuchi, Drug release from wax matrices: I. Analysis of data with ®rst order kinetics and with di€usion controlled model, J. Pharm. Sci. 57 (2) (1968) 274. [14] R. Baker, H. Losdale, Controlled release: mechanisms and rates, in controlled release of biologically active agents, in: A.C. Tanquary, R.E. Lacy (Eds.), Advances in Experimental Biology and Medicine, vol. 47, Plenum Press, New York, 1974, p. 15. [15] IMSL, MATH/PC-Library, 1985.