Mathematical Model for Potassium Release from Polymer-coated Fertiliser

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1 ARTICLE IN PRESS Biosystems Engineering (2004) 88 (3), Avaiabe onine at doi: /j.biosystemseng SW}Soi and Water Mathematica Mode for Potassium Reease from Poymer-coated Fertiiser C. Du 1 ; J. Zhou 1 ; A. Shaviv 2 ; H. Wang 1 1 Institute of Soi Science, Chinese Academy of Sciences, Beijing Street East, Nanjing, , China; e-mai of corresponding author: jmzhou@ns.issas.ac.cn 2 Facuty of Civi and Environmenta Engineering, IIT-Israe Institute of Technoogy, Haifa, 32000, Israe (Received 24 June 2003; received in revised form 2 March 2004; pubished onine 11 June 2004) An exact mathematica mode based on Fick s Second Diffusion Law was deveoped to predict the reease rate of poymer-coated fertiiser using a numerica soution and Fourier series expansion. From the expicit mathematica mode, an approximate soution for the nutrient reease was obtained. The mode showed that the nutrients reease was mainy controed by the diffusion coefficient, membrane thickness and granue radius. This mode was simper compared with the origina numerica soution, and different radii of poymercoated controed-reease fertiiser was used to verify the approximate reease mode. The nutrient was mainy reeased in the inear stage, and the cumuative percentage of nutrient reease decreased when the granue radius increased. The cumuative reease profie of potassium from the poymer-coated fertiiser into water agreed with the prediction of the mode on the whoe. # 2004 Sisoe Research Institute. A rights reserved Pubished by EsevierLtd 1. Introduction Poymer-coated controed-reease fertiisers are used to overcome and improve current ow nutrient use efficiency, and the potentia economic and environmenta benefits have been reported (Shaviv, 1999, 2000, 2001). The reease of nutrients shoud coincide with the requirement of pants and the objective for the modeing work is designed to ensure that products are manufactured that meet crop requirement. Baker(2000) deveoped a mode to predict the reease of drug from a sphere: ¼ 4 ¼ 1 8 p 2 Dt 05 p 2 for04 exp p2 Dt for where: D is the diffusion coefficient forthe poymerin mm 2 d 1 ; is the thickness of the poymermembrane in mm, is the mass diffusion in kg up to time t in d, and is tota mass in kg. The mode does not contain the parameter of sphere radius. A-Zahrani (1999) deveoped a mathematica mode for the nutrient reease from ð1þ ð2þ poymer-coated controed-reease fertiiser, and an approximate soution was deduced as foows: ¼ 6ð1 þ aþ td05 ðpbþ 05 where: D is the diffusion coefficient forthe poymerin mm 2 d 1 ; b is radius of the fertiiser granue in mm; and a is a constant. This mode is simpe, but the parameter a is difficut to obtain. Some othermodes were investigated (Abdekhodaie & Cheng, 1996; Abdekhodaie, 2002; Arnodus & Andries, 2002), but were found to be too compex for the poymer-coated fertiiser appication. Shaviv (2000) divided the reease course into three stage: (1) the initia stage during which amost no reease is observed (ag period), (2) the constant-reease stage, and (3) the stage where there is a gradua decay of reease rate. This three stage approach gives a good description of the reease course for poymer-coated fertiiser. Adopting this three stage of reease, the purpose of the paper is to deveop a mode that can adequatey represent K reease from poymer-coated fertiisers. ð3þ /$ # 2004 Sisoe Research Institute. A rights reserved Pubished by EsevierLtd

2 396 ARTICLE IN PRESS C. DU ET AL. Notation A 1, A 2, A 3 constants a radius of fertiiser granue, mm a 1, a 2 constants B 1, B 2, constants B 3, B 4 b radius of coated fertiiser granue, mm C diffusion concentration in the granue, c 1 nutrient concentration inside the granue, c 2 nutrient concentration outside the granue, c s saturated concentration of nutrients, c t nutrient concentration inside the granue (function of diffusion time), D diffusion coefficient, mm 2 d 1 g t cumuative percentage of nutrient reease up to time t, % g Y cumuative percentage of nutrient reease up to time Y, % J 0 diffusion rate, kg d 1 thickness of membrane, mm tota mass of nutrient, kg mass diffusion up to time t, kgd 1 n natura number p h waterpermeabiity of coated membrane, mm 2 Pa d 1 Q t quantity of diffusion up to time t, kg r radius of diffusion in coated granue, mm T function of diffusion time t time, d t 0 ag period, d t 1 ag period after diffusion of nutrient starts, d u function of diffusion distance and time, kg mm 1 d 1 V granue voume, m 3 X function of diffusion distance x diffusion distance in membrane, mm Y time to dissove a the soid nutrients in the granue, d a constant g tota granue porosity, % DP vapour pressure difference, Pa constant nutrient density, r s 2. Mathematica modeing The diffusiona reease of soute from a poymercoated fertiiser granue of spherica geometry into water with a certain externa voume is considered. The diffusion coefficient is assumed to be independent of concentration, and soute diffusion is assumed to be rate controing step rather than poymer sweing or nutrient dissoution. A fertiiser granue consists of a core containing fertiiser nutrients and a poymer coat, which is the rate-imiting eement in the reease process. For the granue of core radius a in mm and coated sphere radius b in mm, the thickness of the coating in mm is given as the difference (b a). A schematic diagram of cross-sectiona view of fertiiser granue is iustrated in Fig. 1. The first stage of reease according to Zaide (1996) and Shaviv (2000) is a ag period, in which water diffuses into the granue through the poymer membrane, and the ag period time t 0 in day is deduced as foowing: t 0 ¼ gr ð4þ 3p h DP where: g is tota granue porosity incuding aso voids between the nutrient core and the membrane, and has a vaue of between 5 and 10%; p h is waterpermeabiity of the membrane in mm 2 Pa 1 d 1, DP is the vapour Fig. 1. Schematic diagram of a cross-section view of a fertiiser granue: a, radius of granue; b, radius of coated granue;, thickness of membrane pressure difference between water and saturated nutrient soution in Pa; and r is radius of diffusion in mm in the coated granue. According Eqn (4) the nutrient reease is amost zero during the ag period. Therefore, the cumuative percentage of nutrient reease g t is g t ¼ Q t ¼ 0 for t4t 0 ð5þ where Q t is the quantity of diffusion in kg up to time t.

3 ARTICLE IN PRESS MATHEMATICAL MODEL FOR POTASSIUM RELEASE 397 After the ag period, the reease of nutrient begins. The nutrient concentration inside and outside the granue is maintained at constant concentrations c 1 and c 2, respectivey, in. The concentration in the poymeric membrane C in is a function of both time t in d and the position variabe r in mm, and is determined by transient diffusion according Fick s ¼ 2 þ Substitution u for Cr, Eqn ¼ u 2 ð7þ Eqn (7) defines inearfow in one dimension. The diffusion distance x in mm in the membrane is defined as x ¼ r a ð8þ In the region 05x5, the boundary conditions are uð0; tþ ¼c 1 a t > 0 ð9þ and the initia condition is uðx; 0Þ ¼0 ð6þ uð; tþ ¼c 2 b t > 0 ð10þ ð11þ The method of separation of variabes is appied to sove Eqn (7), subject to the above boundary and initia conditions. Assuming that uðx; tþ ¼X ðxþtðtþ ð12þ where: XðxÞ is function of x, andtðtþ is function of t, then by substitution into Eqn (7) and rearranging 1 dt DT dt ¼ 1 d 2 X X dx 2 ¼ 2 where is a constant. From Eqn (12) dt dt ¼ 2 DT ð13þ ð14þ d 2 X x 2 ¼ 2 X ð15þ Soving Eqns (14) and (15) gives ( X ¼ A 1x þ A 2 for ¼ 0 ð16þ B 1 sin x þ B 2 cos x for 6¼ 0 where A 1, A 2, B 1, B 2 are constants, and ( const for ¼ 0 T ¼ B 3 expð 2 ð17þ DtÞ for 6¼ 0 where B 3 is a constant. Therefore uðx; tþ ¼ ( a 1 x þ a 2 for ¼ 0 ða 3 sin x þ B 4 cos xþexp ð 2 DtÞ for 6¼ 0 ð18þ where a 1, a 2, A 3, B 4 are constants. Appying the boundary conditions, Eqns (9) and (10), to the soution for ¼ 0, and using the principe of superposition: uðx; tþ ¼c 1 a þ ðc 2b c 1 aþ x þða 3 sin x þ B 4 cos xþexp ð 2 DtÞ ð19þ Appying the boundary conditions to the above expression impied that the vaue for B 4 is zero, and that Therefore ¼ n ¼ np n ¼ 1; 2; 3... uðx; tþ ¼c 1 a þ ðc 2b c 1 aþ x þ X1 A n sin n x exp ð 2 DtÞ The initia condition, Eqn (11), requires that c 1 a ðc 2b c 1 aþ x ¼ X1 A n sin n x Using the Fourier series expansion yieds A n ¼ 2 Z c 1 a ðc 2b c 1 aþ x sin npx dx 0 ¼ 2 np ½ð 1Þn c 2 b c 1 aš; n ¼ 1; 2; 3::: Thus the fina soution is uðx; tþ ¼c 1 a þ ðc 2b c 1 aþ x þ 2 X 1 1 p n ðð 1Þn c 2 b c 1 aþ sin npx From Eqns (8) and (24) cðr; tþ ¼ ac 1 r exp ½ n 2 pdt= 2 Š þ ðc 2b c 1 aþ ðr aþ r X 1 þ 2 pr exp ð n 2 pdt= 2 Þ 1 n ðð 1Þn c 2 b c 1 aþ sin npðr aþ ð20þ ð21þ ð22þ ð23þ ð24þ ð25þ

4 398 ARTICLE IN PRESS C. DU ET AL. From this expression, r¼a which is the current voume fux J 0 in kg d 1 (rate at which the diffusing substance emerges at the interface) is readiy cacuated and J 0 ¼ D r ðc 2b c 1 aþþ 2D pr exp ð n 2 pdt= 2 Þ From Eqns (26) and (27) Q t ¼ 4pab Dt r ðc 2b c 1 aþ þ 2Dt pr X 1 X 1 Q t ¼ J 0 4pabt 1 n ðð 1Þn c 2 b c 1 aþ 1 n ðð 1Þn c 2 b c 1 aþ exp ð n 2 pdt= 2 Þ ð26þ ð27þ # ð28þ As the exponentia coefficient appearing in the Eqn (28) is proportiona to n 2, the term in the series with a arge vaue for n orargervaue fordt= 2 decay very quicky with time. Thus, as a good approximation, ony the first and the second terms may be retained. The nutrient concentration c 1 is very sma when the water voume outside the granue is arge enough, and it can be regarded as c 1 ffi 0. As t!1, approaches the ine: Q t ¼ 4pab Dt r ðc 2b c 1 aþ 6 ¼ 4pDab2 c 2 r t 2 6D This has an intercept on the t axis given by ð29þ t 1 ¼ 2 ð30þ 6D where t 1 in d was defined as the ag period after diffusion of nutrient starts. From an observation of the intercept, D is deduced. When Dt 1 = 2 ffi 045 approximatey the steady state is achieved (Crank, 1967). However, in some cases is very sma, so t 1 is aso sma, and the tota ag period is sti decided by t 0. When is very sma, a ffi b ffi r, therefore and Q t ¼ 4pr 2 Dt ðc 2 c 1 Þ 6 g t ¼ Q t ð31þ ð32þ where is the tota mass of nutrient density r s in ¼ 4 3 pr3 r s ð33þ From Eqns (32) and (33) g t ¼ 3 Dðt t 0 Þ ðc 2 c 1 Þ ð34þ r s r 6 In case of poymer-coated fertiiser, assuming that: c 1! 0, and c 2!c s (Saturated concentration), then g t ¼ 3 r s r From Eqn (35) Dðt t 0 Þ c s 6 ð35þ dg t dt ¼ 3Dc s ð36þ r s r where c s is the saturated concentration of nutrients in. When there is no soid fertiiser in the granue (t5y), the concentration in the granue is no onger saturated: c t ¼ ð1 g tþ ¼ r V s ð1 g t Þ ð37þ where: c t is the nutrient concentration in the granue in ; Y is the time when a the soid nutrients dissoved in d; and V is the granue voume in m 3 : dg t dt ¼ 3D r s r r sð1 g t Þ¼ 3D r ð1 g tþ ð38þ When t ¼ Y, the boundary vaue for the cumuative percentage of nutrient reease g Y is g Y ¼ c s V From Eqns (38) and (39) g t ¼ 1 c s ¼ 1 c s r s ð39þ exp 3D ðt YÞ ð40þ r s r Therefore, the nutrient reease from poymer-coated granue is as foows: 8 0 t4t 0 3 Dðt t 0 Þ c g t ¼ s >< t 0 5t4Y r s r 6 ð41þ 1 C s exp 3D >: ðt YÞ t > Y r s r 3. Mode verification A poymer-coated fertiiser was provided by Haifa Chemica Ltd, Israe. The reease of potassium in distied water was determined with fame photometer. The K content was 1079%, and the thickness was

5 ARTICLE IN PRESS MATHEMATICAL MODEL FOR POTASSIUM RELEASE mm. Four uniform fertiiser granues were chosen, immersed in 10 m distied water for reease at 308C. When samping a the soution was obtained, and another10 m of distied wateradded, then kept reeasing at 308C. Different radii of fertiiser granues were carefuy chosen to verify the infuence of radius on reease. The diffusion coefficients were determined according to Zhang et a. (1994). Figure 2 shows that the experimenta resuts are cose to the mathematica prediction to a satisfactory extent. The sope of reease curve in Fig. 2(a) is sharper than that in Fig. 2(b), which means that the nutrient reease rate increases when the granue radius decreases. However, there sti are some differences between (a) Cumuative percentage of nutrient reease,% Day modeing and the experimenta resuts due to two factors: one is granue shape which is not exacty spherica. Since the surface area of spherica granue is the smaest under a certain mass, so the surface area (diffusion area) of granue used in the experiment wi be arger than that for modeing and, therefore, the modeing cumuative percentage of nutrient reease is owerthan experimenta one. The otherfactoris the granue thickness which is not competey uniform, and nutrient wi easiy reease from thinner part of the membrane, which aso eads to faster nutrient reease than predicted. In addition, this mode aso can evauate ag period through the intercept of the reease equation in the inearstage, this evauation method is much easierthan the theoretica cacuation, athough there remains some error which is of acceptabe magnitude. Further sensitivity anaysis shoud be done to check the mode. 4. Concusions A mathematica mode was estabished to predict the nutrient reease from poymer-coated fertiiser. The modeing resut agreed with the experimenta reease on the whoe. There sti existed a few differences, because some granue conditions do not competey agree with the mode assumptions. Further sensitivity anaysis shoud aso be done to check orimprove the mode. However, as a theoretica mode, it was satisfactory to assist in improving the production of poymer-coated fertiiser. (b) Cumuative percentage of nutrient reease,% Day Fig. 2. Comparison of experimentay measured (}&}) and modeing (}n}) reease from poymer-coated fertiiser (with membrane thickness of 0065 mm, nutrient density r s, of 309, diffusion coefficient D of 126mm 2 d 1, saturated concentration c s of 804kgm 3, ag period t 0 of 2 d, and boundary cumuative percentage of nutrient reease vaue g Y of 73%) for two vaues of granue radius a: (a) a ¼ 11 mm; (b) a ¼ 14mm Acknowedgements We give thanks forthe financia support given by Innovationa Project in Environment and Resources Fieds from Chinese Academy of Sciences (No. KZCX2-402) and Nationa Natura Science Foundation (No ), and Nationa Deveoping Project for High and New Technoogy (No. 2001AA246021); we are aso very gratefu to Prof. Brian Witney for his critica and kind correction. References Abdekhodaie M J (2002). Diffusiona reease of a soute from spherica reservoir into a finite externa voume. Journa of Pharmaceutica Science, 91, Abdekhodaie M J; Cheng YL (1996). Diffusiona reease of dispersed soute from a spherica poymer matrix. Journa of Membrane Science, 115,

6 400 ARTICLE IN PRESS C. DU ET AL. A-Zahrani S M (1999). Controed-reease of fertiisers: modeing and simuation. Internationa Journa of Engineering Science, 37(10), Arnodus J K; Andries T T (2002). Prediction of the reease characteristics of acohos from EVA using mode based on Fick s Second Law of Diffusion. Journa of Appied PoymerScience, 84, Baker R W (2000). Membrane Technoogy and Appications, pp Membrane Technoogy and Research Inc., Meno, Park, CA Crank J (1967). The Mathematics of Diffusion, 3rd Edn, pp Oxford University Press, London Shaviv A (1999). Preparation methods and reease mechanism of controed reease fertiisers: agronomic efficiency and environment significances. Proceeding, No Internationa Fertiiser Society, York, UK Shaviv A (2000). Advances in controed-reease fertiiser. Advances in Agronomy, 71, 1 49 Shaviv A (2001). Fertiisers and resource management for food security, quaity and the environment. Paper presented to the Internationa Fertiiser Society at Dahia Greidinger Symposium, pp Lisbon, Portuga Zaide E (1996). Modes of controed reease of fertiisers. Doctora Thesis, Israe Institute of Technoogy, Haifa, Israe Zhang M; Nyborg M; Ryan J T (1994). Determining permeabiity of coatings of poymer-coated Urea. Fertiiser Research, 38, 47 51

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