International Journal of eISSN: 2475-5559 IPCSE

Petrochemical Science & Engineering
Research Article
Volume 3 Issue 4

An approach for determination of mass transfer parameters using finite integral transform method and experimental data for regular geometries

Mohammad Reza Talaghata
Department of Petroleum and gas engineering, Shiraz University of Technology, Iran
Received: March 26, 2018 | Published: October 16, 2018

Correspondence: Mohammad Reza Talaghata, Department of Petroleum and gas engineering, Shiraz University of Technology, Iran, Tel +989173170434, Email

Citation: Talaghata MR. An approach for determination of mass transfer parameters using finite integral transform method and experimental data for regular geometries. Int J Petrochem Sci Eng. 2018;3(4):152‒159. DOI: 10.15406/ipcse.2018.03.00089

Abstract

Mass transfer coefficient and diffusion coefficient are important for modeling of food processing operations. There have been many methods to determine mass diffusivity in the during mass transfer phenomena. Drying methods, simplified methods, simulation method, numerical methods, and regular regime method are procedures to determine this parameter. Experimental determination of these parameters would be valuable. This article offers an approach method for estimation of mass transfer parameters (diffusion coefficient (D) and mass transfer coefficient (kc)) using analytical solutions and experimental data for regular geometric shapes such as infinite slab, infinite cylinder, and sphere. Analytical solutions have a broad use in experimentally determining these parameters. Here, the method of Finite Integral Transform (FIT) was used for solutions of governing differential equations. The concentration ratio vs. time of regular shapes was recorded to determine both the mass transfer coefficient and diffusion coefficient. In this study, determination steps of diffusion coefficient during the mass transfer when this parameter is fixed or variable, also been described. The results showed that diffusion coefficients of the sodium tripolyphosphate solutions (2% w/v) in slab shaped before and after the barrier formation on the surface of meat samples was completed is and using the method presented in this paper. In this research, the mass transfer coefficient was also being determined using the analytical solutions when the diffusivity of substance is known.

Keywords: analytical, solutions, mass, transfer, parameters, diffusion, coefficient, finite integral transform

Introduction

Mass transfer is widely used in chemical engineering problems. It is used in reaction separations engineering, reaction, heat transfer, and many other sub-disciplines of chemical engineering. Usually, the difference in chemical potential is the driving force mass transfer. For single-phase systems, this is usually converted to uniform concentration throughout the phase, while for multiphase systems, chemical species often prefers a phase to others and reaches a uniform chemical potential. These mass transfer coefficients are usually published in terms of dimensionless numbers including Reynolds numbers, Sherwood numbers, and Schmidt numbers. The transport of mass in a phase depends directly on the gradient of the concentration of the transport species at that stage.

The mass may be transferred from one phase to another, and this process is called interphase mass transfer.1-3 Based on a theory of mass transfer, the mass transfer rate in the absence of the bulk flow directly proportional to the propulsion is expressed as a molar concentration difference.3 In the two-film theory, the kc is directly proportional to D and inversely proportional to the thickness of the film. In the theory of film propagation, the kc is a complex function of the D, the thickness of the film. In many cases, the kc value cannot be calculated from the basics, although how kc varies according to the operating conditions.1-6 Theoretical expressions for D in a mixture of low-density gases as a function of the molecular properties of the system by Sutherland,7 Jeans8 and Chapman & Cowling9 based on the kinetic theory of gases. Diffusivities of vapors are determined by Winkel Mann10 in which liquid is allowed to evaporate in a vertical glass tube. The diffusion D is not known to transmit another gas and the experimental determination is not practical. It is necessary to use one of the many predictable methods. A normally used method proposed by Gilliland11 and Fuller.12 Different experimental methods for determining the molecular diffusion coefficient. The measurement method is usually included microscopy,13,14 total internal reflection fluorescence (TIRF) spectroscopy 15,16 and Interferometry.17-19 Average balance interface model for equilibrium was developed by Hodgson20 and then improved by Deng.21The convective mass transfer coefficient in the equilibrium interface model introduced by Deng.21,22 To determine the mass transfer coefficient convection (hm), the empirical equations are widely used based on the analytical transfer of heat and mass.21 Mass transfer coefficient (kC) depending on the medium characteristics such as roughness of the surface, mass transfer equipment, shape, the surface temperature, size and profile of fluid flow.23 Different equations in literature have been used to predict mass transfer parameters such as various Sherwood numbers. The reflection of the process parameters experimental determination is important. A novel method for these parameters determination is to use the experimental time-mass concentration data. Analytical solutions of regular shapes such as an infinite cylinder, infinite slab, and sphere with initial and boundary conditions suitable to the experimental data obtained from food material itself may be used to experimentally determine these parameters.23 Therefore, the objective of this study is to explain the methodologies to use analytical solutions of regularly shaped geometries with the experimentally obtained data to determine the mass transfer parameters.

Mathematical modeling

Governing differential equations with initial and boundary condition and solutions for infinite geometries are given by the following equation:

1 x n x ( x n C x )= 1 D C t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaamiEaSWaaWbaaeqabaqcLbmacaWG UbaaaaaajuaGdaWcaaGcbaqcLbsacqGHciITaOqaaKqzGeGaeyOaIy BcLbmacaWG4baaaKqbaoaabmaakeaajugibiaadIhalmaaCaaabeqa aKqzadGaamOBaaaajuaGdaWcaaGcbaqcLbsacqGHciITcaWGdbaake aajugibiabgkGi2MqzadGaamiEaaaaaOGaayjkaiaawMcaaKqzGeGa eyypa0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWGebaaaK qbaoaalaaakeaajugibiabgkGi2kaadoeaaOqaaKqzGeGaeyOaIyBc LbmacaWG0baaaaaa@5C90@           (1)

In the above equation, C is mass concentration; x is the distance from the center of substance; n is a characteristic number (n = 0 for infinitely slab, n = 1 for infinitely cylinder, and n=2 for a sphere); and D is diffusivity coefficient (m2/s). Solutions for Eq. (1), for initial and boundary conditions of central symmetry and convection boundary of the surface, are given for these geometries as follows.24,25

Solution by the method of finite integral transform

A finite integer transform method (FIT) involves an operator that converts the main equation to a simpler domain. The solution in the new domain will be quite primitive. However, to be of practical value, it must be converted into the original space. The operation is formed for this reverse transformation, along with the main operator, which we call a pair of integral transforms.26 to demonstrate the development of the integral transform pair in a practical way, consider the Fick’s mass transfer problem with the regular geometry object. We see the use of the transfer technique as a temporary problem of mass transfer in a slab, cylinder or spherical as shown below:

L[ 1 xn x ( xn C x ) ]=L[ 1 D C t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaK qbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaa dIhajugWaiaad6gaaaqcfa4aaSaaaOqaaKqzGeGaeyOaIylakeaaju gibiabgkGi2kaadIhaaaqcfa4aaeWaaOqaaKqzGeGaamiEaKqzadGa amOBaKqbaoaalaaakeaajugibiabgkGi2kaadoeaaOqaaKqzGeGaey OaIyRaamiEaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaKqzGeGa eyypa0JaamitaKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacaaIXa aakeaajugibiaadseaaaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaam4q aaGcbaqcLbsacqGHciITcaWG0baaaaGccaGLBbGaayzxaaaaaa@5FFB@ 0<x<1 (2)

According to the initial and boundary conditions:

At t = 0     C= C i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4qai abg2da9iaadoealmaaBaaabaqcLbmacaWGPbaaleqaaaaa@3B63@ (3)

At x = 0   C x =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kaadoeaaOqaaKqzGeGaeyOaIyBcLbmacaWG4baa aKqzGeGaeyypa0JaaGimaaaa@3FC9@ (4)

At x = 1 C x =Bi( CC ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kaadoeaaOqaaKqzGeGaeyOaIyBcLbmacaWG4baa aKqzGeGaeyypa0JaamOqaiaadMgajuaGdaqadaqaaKqzGeGaam4qaK qzadGaeyOhIuAcLbsacqGHsislcaWGdbaajuaGcaGLOaGaayzkaaaa aa@49A3@ (5)

In the Eq. (2) n is a shape factor of the domain. It takes a value of 0, 1 or 2 for a slab, cylinder or spherical coordinates. Note that the boundary conditions (5) are heterogeneous.

To provide homogeneous boundary conditions, we need to solve the steady-state problem:

1 xn C x ( xn C x )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba qcLbsacaaIXaaajuaGbaqcLbsacaWG4bqcLbmacaWGUbaaaKqbaoaa laaakeaajugibiabgkGi2kaadoeaaOqaaKqzGeGaeyOaIyRaamiEaa aajuaGdaqadaqaaKqzGeGaamiEaKqzadGaamOBaKqbaoaalaaabaqc LbsacqGHciITcaWGdbaajuaGbaqcLbsacqGHciITcaWG4baaaaqcfa OaayjkaiaawMcaaKqzGeGaeyypa0JaaGimaaaa@51EB@ (6)

According to:

At x = 0   C x =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kaadoeaaOqaaKqzGeGaeyOaIyBcLbmacaWG4baa aKqzGeGaeyypa0JaaGimaaaa@3FC9@ (7)

At x = 1 C x =Bi( CC ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kaadoeaaOqaaKqzGeGaeyOaIyBcLbmacaWG4baa aKqzGeGaeyypa0JaamOqaiaadMgajuaGdaqadaqaaKqzGeGaam4qaK qzadGaeyOhIuAcLbsacqGHsislcaWGdbaajuaGcaGLOaGaayzkaaaa aa@49A3@ (8)

A solution of Eq. (6) is simply. The concentration is equal to:

C=C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4qai abg2da9iaadoeajugWaiabg6HiLcaa@3BAF@ (9)

Therefore, θ is a new dependent variable. In that sense,

θ=CC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde Naeyypa0Jaam4qaKqzadGaeyOhIuAcLbsacqGHsislcaWGdbaaaa@3EE1@ (10)

Substitution of Eq. (10) into Eq. (2), produces the following equations for θ with homogeneous boundary conditions:

1 x n x ( x n θ x )= θ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaamiEaKqbaoaaCaaaleqabaqcLbma caWGUbaaaaaajuaGdaWcaaGcbaqcLbsacqGHciITaOqaaKqzGeGaey OaIyRaamiEaaaajuaGdaqadaGcbaqcLbsacaWG4bqcfa4aaWbaaSqa beaajugWaiaad6gaaaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaeqiUde hakeaajugibiabgkGi2kaadIhaaaaakiaawIcacaGLPaaajugibiab g2da9KqbaoaalaaakeaajugibiabgkGi2kabeI7aXbGcbaqcLbsacq GHciITcaWG0baaaaaa@58AA@ (11)

At t = 0 θ=θi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde Naeyypa0JaeqiUdexcLbmacaWGPbaaaa@3D08@ (12)

At x = 0 < θ x =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kabeI7aXbGcbaqcLbsacqGHciITcaWG4baaaiab g2da9iaaicdaaaa@3EFA@ (13)

At x = 1 θ x =Bi( θθ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kabeI7aXbGcbaqcLbsacqGHciITcaWG4baaaiab g2da9iaadkeacaWGPbqcfa4aaeWaaeaajugibiabeI7aXLqzadGaey OhIuAcLbsacqGHsislcqaH4oqCaKqbakaawIcacaGLPaaaaaa@4AB0@ (14)

This new set of equations can now be easily solved by a finite integral transform method.

The following integral transform is derived as:

θ,Kn = 0 1 xnθ( x,t ) Kn( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaamaaba qcLbsacqaH4oqCcaGGSaGaam4saKqzadGaamOBaaqcfaOaayzkJiaa wQYiaKqzGeGaeyypa0tcfa4aa8qCaOqaaKqzGeGaamiEaKqzadGaam OBaKqzGeGaeqiUdexcfa4aaeWaaOqaaKqzGeGaamiEaiaacYcacaWG 0baakiaawIcacaGLPaaaaSqaaKqzGeGaaGimaaWcbaqcLbsacaaIXa aacqGHRiI8aiaadUeajugWaiaad6gajuaGdaqadaGcbaqcLbsacaWG 4baakiaawIcacaGLPaaajugibiaadsgacaWG4baaaa@5A53@ (15)

Where the kernel of the transform is obtained from the following associated Eigen problem: 

LKn( x )+ λ n 2 Kn( x )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitai aadUeajugWaiaad6gajuaGdaqadaGcbaqcLbsacaWG4baakiaawIca caGLPaaajugibiabgUcaRiabeU7aSLqbaoaaDaaaleaajugibiaad6 gaaSqaaKqzGeGaaGOmaaaacaWGlbqcLbmacaWGUbqcfa4aaeWaaOqa aKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcaaIWaaaaa@4DA2@ (16)

At x = 0   Kn( x ) x =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kaadUeajugWaiaad6gajuaGdaqadaGcbaqcLbsa caWG4baakiaawIcacaGLPaaaaeaajugibiabgkGi2kaadIhaaaGaey ypa0JaaGimaaaa@43E2@ (17)

at x = 1  K n x =Bi k n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kaadUeajuaGdaWgaaqaaKqzadGaamOBaaqcfaya baaakeaajugibiabgkGi2kaadIhaaaGaeyypa0JaamOqaiaadMgacq GHflY1caWGRbqcfa4aaSbaaeaajugWaiaad6gaaKqbagqaaaaa@4905@ (18)

The solution for θ is easily seen:

θ= n1 n θi 1,Kn exp( λ n 2 D L 2 t ) Kn Kn,Kn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde Naeyypa0tcfa4aaabCaOqaaKqzGeGaeqiUdeNaamyAaKqbaoaaamaa keaajugibiaaigdacaGGSaGaam4saKqzadGaamOBaaGccaGLPmIaay PkJaaaleaajugibiaad6gacqGHsislcaaIXaaaleaajugWaiaad6ga aKqzGeGaeyyeIuoaciGGLbGaaiiEaiaacchajuaGdaqadaGcbaqcLb sacqGHsislcqaH7oaBlmaaDaaabaqcLbmacaWGUbaaleaajugWaiaa ikdaaaqcfa4aaSaaaOqaaKqzGeGaamiraaGcbaqcLbsacaWGmbqcfa 4aaWbaaSqabeaajugibiaaikdaaaaaaiaadshaaOGaayjkaiaawMca aKqbaoaalaaakeaajugibiaadUeajugWaiaad6gaaOqaaKqbaoaaam aakeaajugibiaadUeajugWaiaad6gajugibiaacYcacaWGlbqcLbma caWGUbaakiaawMYicaGLQmcaaaaaaa@6DAE@  (19)

And therefore:

θ θi = n1 n θi 1,Kn exp( λ n 2 D L 2 t ) Kn Kn,Kn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba qcLbsacqaH4oqCaKqbagaajugibiabeI7aXLqzadGaamyAaaaajugi biabg2da9KqbaoaaqahakeaajugibiabeI7aXLqzadGaamyAaKqbao aaamaakeaajugibiaaigdacaGGSaGaam4saKqzadGaamOBaaGccaGL PmIaayPkJaaaleaajugWaiaad6gacqGHsislcaaIXaaaleaajugWai aad6gaaKqzGeGaeyyeIuoaciGGLbGaaiiEaiaacchajuaGdaqadaGc baqcLbsacqGHsislcqaH7oaBlmaaDaaabaqcLbmacaWGUbaaleaaju gWaiaaikdaaaqcfa4aaSaaaOqaaKqzGeGaamiraaGcbaqcLbsacaWG mbWcdaahaaqabeaajugWaiaaikdaaaaaaKqzGeGaamiDaaGccaGLOa Gaayzkaaqcfa4aaSaaaOqaaKqzGeGaam4saiaad6gaaOqaaKqbaoaa amaakeaajugibiaadUeacaWGUbGaaiilaiaadUeacaWGUbaakiaawM YicaGLQmcaaaaaaa@721E@  (20)

For three different shapes, the expressions for Kn (x), λn, and are:

Slab

Kn( x )=Cos( λ n x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4saK qzadGaamOBaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaeyypa0Jaam4qaiaad+gacaWGZbqcfa4aaeWaaOqaaKqzGe Gaeq4UdWwcfa4aaSbaaSqaaKqzadGaamOBaaWcbeaajugibiaadIha aOGaayjkaiaawMcaaaaa@4A4B@ (21)

λnSin( λn )=BiCos( λ n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW wcLbmacaWGUbqcLbsacaWGtbGaamyAaiaad6gajuaGdaqadaGcbaqc LbsacqaH7oaBjugWaiaad6gaaOGaayjkaiaawMcaaKqzGeGaeyypa0 JaamOqaiaadMgacqGHflY1caWGdbGaam4BaiaadohajuaGdaqadaGc baqcLbsacqaH7oaBjuaGdaWgaaWcbaqcLbsacaWGUbaaleqaaaGcca GLOaGaayzkaaaaaa@5323@ (22)

1, K n = Sin( λ n ) λ n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaamaake aajugibiaaigdacaGGSaGaam4saSWaaSbaaeaajugWaiaad6gaaSqa baaakiaawMYicaGLQmcajugibiabg2da9Kqbaoaalaaakeaajugibi aadofacaWGPbGaamOBaKqbaoaabmaakeaajugibiabeU7aSLqbaoaa BaaaleaajugWaiaad6gaaSqabaaakiaawIcacaGLPaaaaeaajugibi abeU7aSTWaaSbaaeaajugWaiaad6gaaSqabaaaaaaa@4EF3@ (23)

Kn,Kn = 1 2 [ 1+ Si n 2 ( λn ) λn ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaamaake aajugibiaadUeajugWaiaad6gajugibiaacYcacaWGlbqcLbmacaWG UbaakiaawMYicaGLQmcajugibiabg2da9KqbaoaalaaabaqcLbsaca aIXaaajuaGbaqcLbsacaaIYaaaaKqbaoaadmaabaqcLbsacaaIXaGa ey4kaSscfa4aaSaaaeaajugibiaadofacaWGPbGaamOBaKqbaoaaCa aabeqaaKqzadGaaGOmaaaajuaGdaqadaqaaKqzGeGaeq4UdWwcLbma caWGUbaajuaGcaGLOaGaayzkaaaabaqcLbsacqaH7oaBjugWaiaad6 gaaaaajuaGcaGLBbGaayzxaaaaaa@5C8A@ (24)

Cylinder

Kn( x )=Jo( λnx ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4saK qzadGaamOBaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaeyypa0JaamOsaKqzadGaam4BaKqbaoaabmaakeaajugibi abeU7aSLqzadGaamOBaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaa@49C3@ (25)

λnJ1( λn )=BiJo( λn ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW wcLbmacaWGUbqcLbsacaWGkbGaaGymaKqbaoaabmaakeaajugibiab eU7aSLqzadGaamOBaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcaWGcb GaamyAaiabgwSixlaadQeajugWaiaad+gajuaGdaqadaGcbaqcLbsa cqaH7oaBjugWaiaad6gaaOGaayjkaiaawMcaaaaa@520B@ (26)

1,Kn = J1( λn ) λn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaamaake aajugibiaaigdacaGGSaGaam4saKqzadGaamOBaaGccaGLPmIaayPk JaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGkbqcLbmacaaIXa qcfa4aaeWaaOqaaKqzGeGaeq4UdWwcLbmacaWGUbaakiaawIcacaGL PaaaaeaajugibiabeU7aSLqzadGaamOBaaaaaaa@4DBF@ (27)

Kn,Kn = 1 2 J 1 2 ( λn )[ 1+ ( λn Bi ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaamaake aajugibiaacUeajugWaiaac6gajugibiaacYcacaWGlbqcLbmacaWG UbaakiaawMYicaGLQmcajugibiabg2da9KqbaoaalaaabaqcLbsaca aIXaaajuaGbaqcLbsacaaIYaaaaiaadQealmaaDaaajuaGbaqcLbma caaIXaaajuaGbaqcLbmacaaIYaaaaKqbaoaabmaabaqcLbsacqaH7o aBjugWaiaad6gaaKqbakaawIcacaGLPaaadaWadaqaaKqzGeGaaGym aiabgUcaRKqbaoaabmaabaWaaSaaaeaajugibiabeU7aSLqzadGaam OBaaqcfayaaKqzGeGaamOqaiaadMgaaaaajuaGcaGLOaGaayzkaaWc daahaaadbeqaaiaaikdaaaaajuaGcaGLBbGaayzxaaaaaa@61E1@ (28)

Sphere

Kn( x )= Sin( λnx ) x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4saK qzadGaamOBaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaam4uaiaadMgacaWGUb qcfa4aaeWaaOqaaKqzGeGaeq4UdWwcLbmacaWGUbqcLbsacaWG4baa kiaawIcacaGLPaaaaeaajugibiaadIhaaaaaaa@4C4E@ (29)

λnCos( λn )=( 1Bi )Sin( λn ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW wcLbmacaWGUbqcLbsacaWGdbGaam4BaiaadohajuaGdaqadaGcbaqc LbsacqaH7oaBjugWaiaad6gaaOGaayjkaiaawMcaaKqzGeGaeyypa0 tcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiaadkeajugWaiaadMga aOGaayjkaiaawMcaaKqzGeGaam4uaiaadMgacaWGUbqcfa4aaeWaaO qaaKqzGeGaeq4UdWwcLbmacaWGUbaakiaawIcacaGLPaaaaaa@56D2@ (30)

1,Kn = [ Sin( λn )λnCos( λn ) ] λ n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaamaake aajugibiaaigdacaGGSaGaam4saKqzadGaamOBaaGccaGLPmIaayPk JaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcfa4aamWaaOqaaKqzGeGaam 4uaiaadMgacaWGUbqcfa4aaeWaaOqaaKqzGeGaeq4UdWwcLbmacaWG UbaakiaawIcacaGLPaaajugibiabgkHiTiabeU7aSLqzadGaamOBaK qzGeGaam4qaiaad+gacaWGZbqcfa4aaeWaaOqaaKqzGeGaeq4UdWwc LbmacaWGUbaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaeaajugibi abeU7aSTWaa0baaeaajugWaiaad6gaaSqaaKqzadGaaGOmaaaaaaaa aa@618F@ (31)

Kn,Kn = 1 2 [ 1+ Co s 2 ( λn ) Bi1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaamaake aajugibiaadUeajugWaiaad6gajugibiaacYcacaWGlbqcLbmacaWG UbaakiaawMYicaGLQmcajugibiabg2da9KqbaoaalaaabaqcLbsaca aIXaaajuaGbaqcLbsacaaIYaaaaKqbaoaadmaabaqcLbsacaaIXaGa ey4kaSscfa4aaSaaaeaajugibiaadoeacaWGVbGaam4CaSWaaWbaaW qabeaacaaIYaaaaKqbaoaabmaabaqcLbsacqaH7oaBjugWaiaad6ga aKqbakaawIcacaGLPaaaaeaajugibiaadkeajugWaiaadMgajugibi abgkHiTiaaigdaaaaajuaGcaGLBbGaayzxaaaaaa@5C25@ (32)

The dimensionless concentration ratio for non-finite regular geometries is limited to:

Infinitely slab

ϕ= CC CiC = 2Sin( λn ) λn+Sin( λn )Cos( λn ) exp( λ n 2 Fo ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqy1dy Maeyypa0tcfa4aaSaaaOqaaKqzGeGaam4qaiabgkHiTiaadoeajugW aiabg6HiLcGcbaqcLbsacaWGdbGaamyAaiabgkHiTiaadoeajugWai abg6HiLcaajugibiabg2da9KqbaoaaqaeakeaajuaGdaWcaaGcbaqc LbsacaaIYaGaeyyXICTaam4uaiaadMgacaWGUbqcfa4aaeWaaOqaaK qzGeGaeq4UdWwcLbmacaWGUbaakiaawIcacaGLPaaaaeaajugibiab eU7aSLqzadGaamOBaKqzGeGaey4kaSIaam4uaiaadMgacaWGUbqcfa 4aaeWaaOqaaKqzGeGaeq4UdWwcLbmacaWGUbaakiaawIcacaGLPaaa jugibiabgwSixlaadoeacaWGVbGaam4CaKqbaoaabmaakeaajugibi abeU7aSLqzadGaamOBaaGccaGLOaGaayzkaaaaaaWcbeqabKqzGeGa eyyeIuoacqGHflY1ciGGLbGaaiiEaiaacchajuaGdaqadaGcbaqcLb sacqGHsislcqaH7oaBlmaaDaaabaqcLbmacaWGUbaaleaadaahaaqc fayabeaajugWaiaaikdaaaaaaKqzGeGaamOraiaad+gaaOGaayjkai aawMcaaaaa@8580@  (33)

Infinitely cylinder

ϕ ¯ v= Cv ¯ C CiC = 4 λ n 2 J 1 2 ( λn ) J 0 2 ( λn )+ J 1 2 ( λn ) exp( λ n 2 Fo ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaaba qcLbsacqaHvpGzaaqcLbmacaWG2bqcLbsacqGH9aqpjuaGdaWcaaGc baqcfa4aa0aaaOqaaKqzGeGaam4qaKqzadGaamODaaaajugibiabgk HiTiaadoeajugWaiabg6HiLcGcbaqcLbsacaWGdbqcLbmacaWGPbqc LbsacqGHsislcaWGdbqcLbmacqGHEisPaaqcLbsacqGH9aqpjuaGda aeabGcbaqcfa4aaSaaaOqaaKqzGeGaaGinaaGcbaqcLbsacqaH7oaB juaGdaqhaaWcbaqcLbsacaWGUbaaleaajugibiaaikdaaaaaaaWcbe qabKqzGeGaeyyeIuoacqGHflY1juaGdaWcaaqaaKqzGeGaamOsaSWa a0baaKqbagaajugWaiaaigdaaKqbagaajugWaiaaikdaaaqcfa4aae WaaeaajugibiabeU7aSLqzadGaamOBaaqcfaOaayjkaiaawMcaaaqa aKqzGeGaamOsaSWaa0baaKqbagaajugWaiaaicdaaKqbagaajugWai aaikdaaaqcfa4aaeWaaeaajugibiabeU7aSLqzadGaamOBaaqcfaOa ayjkaiaawMcaaKqzGeGaey4kaSIaamOsaSWaa0baaKqbagaajugWai aaigdaaKqbagaajugWaiaaikdaaaqcfa4aaeWaaeaajugibiabeU7a SLqzadGaamOBaaqcfaOaayjkaiaawMcaaaaajugibiGacwgacaGG4b GaaiiCaKqbaoaabmaakeaajugibiabgkHiTiabeU7aSTWaa0baaeaa jugWaiaad6gaaSqaamaaCaaajuaGbeqaaKqzadGaaGOmaaaaaaqcLb sacaWGgbGaam4BaaGccaGLOaGaayzkaaaaaa@97A3@ (34)

Sphere

ϕ ¯ v= Cv ¯ C CiC = 6 λ n 3 [ sin( λn )λncos( λn ) ] λnsin( λn )cos( λn ) 2 exp( λ n 2 Fo ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaaba qcLbsacqaHvpGzaaqcLbmacaWG2bqcLbsacqGH9aqpjuaGdaWcaaGc baqcfa4aa0aaaOqaaKqzGeGaam4qaKqzadGaamODaaaajugibiabgk HiTiaadoeajugWaiabg6HiLcGcbaqcLbsacaWGdbqcLbmacaWGPbqc LbsacqGHsislcaWGdbqcLbmacqGHEisPaaqcLbsacqGH9aqpjuaGda aeabGcbaqcfa4aaSaaaOqaaKqzGeGaaGOnaaGcbaqcLbsacqaH7oaB juaGdaqhaaWcbaqcLbsacaWGUbaaleaajugWaiaaiodaaaaaaaWcbe qabKqzGeGaeyyeIuoacqGHflY1juaGdaWcaaqaamaadmaabaqcLbsa ciGGZbGaaiyAaiaac6gajuaGdaqadaqaaKqzGeGaeq4UdWwcLbmaca WGUbaajuaGcaGLOaGaayzkaaqcLbsacqGHsislcqaH7oaBcaWGUbGa ci4yaiaac+gacaGGZbqcfa4aaeWaaeaajugibiabeU7aSLqzadGaam OBaaqcfaOaayjkaiaawMcaaaGaay5waiaaw2faaaqaaKqzGeGaeq4U dWwcLbmacaWGUbqcLbsacqGHsislciGGZbGaaiyAaiaac6gajuaGda qadaqaaKqzGeGaeq4UdWwcLbmacaWGUbaajuaGcaGLOaGaayzkaaqc LbsacqGHflY1ciGGJbGaai4BaiaacohajuaGdaqadaqaaKqzGeGaeq 4UdWwcLbmacaWGUbaajuaGcaGLOaGaayzkaaaaaSWaaWbaaKqbagqa baqcLbmacaaIYaaaaKqzGeGaeyyXICTaaiyzaiaacIhacaGGWbqcfa 4aaeWaaOqaaKqzGeGaeyOeI0Iaeq4UdW2cdaqhaaqaaKqzadGaamOB aaWcbaWaaWbaaKqbagqabaqcLbmacaaIYaaaaaaajugibiaadAeaca WGVbaakiaawIcacaGLPaaaaaa@AA9F@ 35)

Where Fo= Dt ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOrai aad+gacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGebGaamiDaaGcbaqc LbsacqaH+oaEjuaGdaahaaWcbeqaaKqbaoaaCaaabeqaaKqzadGaaG Omaaaaaaaaaaaa@41E9@ is Fourier number (ξ is L, thickness of half an infinite slab and R, the radius of an infinite cylinder or a sphere); D is the diffusivity coefficient, and the λn are the roots of the following equations (Eqs. (36)- (38)) for infinitely slab, infinitely cylinder and sphere, respectively:25

Bi= λ n tan( λ n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqai aadMgacqGH9aqpcqaH7oaBjuaGdaWgaaqaaKqzadGaamOBaaqcfaya baqcLbsacqGHflY1ciGG0bGaaiyyaiaac6gajuaGdaqadaGcbaqcLb sacqaH7oaBjuaGdaWgaaqaaKqzadGaamOBaaqcfayabaaakiaawIca caGLPaaaaaa@4BBD@ (36)

Bi= λ n J 1 ( λ n ) J o ( λ n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGcbGaamyAaiabg2da9iabeU7aSLqbaoaaBaaabaqcLbmacaWGUbaa juaGbeaajugibiabgwSixNqbaoaalaaakeaajugibiaadQeajuaGda WgaaqaaKqzadGaaGymaaqcfayabaWaaeWaaOqaaKqzGeGaeq4UdWwc fa4aaSbaaeaajugWaiaad6gaaKqbagqaaaGccaGLOaGaayzkaaaaba qcLbsacaWGkbWcdaWgaaadbaGaam4Baaqabaqcfa4aaeWaaOqaaKqz GeGaeq4UdWwcfa4aaSbaaeaajugWaiaad6gaaKqbagqaaaGccaGLOa Gaayzkaaaaaaaa@586D@ (37)

Bi=1 λ1 tan λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGcbGaamyAaiabg2da9iaaigdacqGHsisljuaGdaWcaaGcbaqcLbsa cqaH7oaBjugWaiaaigdaaOqaaKqzGeGaciiDaiaacggacaGGUbGaeq 4UdWwcfa4aaSbaaeaajugWaiaaigdaaKqbagqaaaaaaaa@4882@ (38)

Where Bi= K c ξ D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGcbGaamyAaiabg2da9KqbaoaalaaabaqcLbsacaWGlbqcfa4aaSba aeaajugWaiaadogaaKqbagqaaKqzGeGaeqOVdGhajuaGbaqcLbsaca WGebaaaaaa@434A@ is Biot number (Bi) and Jo and J1 are the 0th and 1st order of the first kind of Bessel functions, respectively. The mass transfer is obtained by integrating o Eqs. (33) - (35) for the total volume due to the experimental results of mass transfer for the total volume. The results of these integrations for the infinite slab, infinite cylinder, and sphere are obtained by the Eqs. (39) to (41), respectively.

Infinitely slab

ϕ= CC CiC = 2Sin( λn ) λn+Sin( λn )Cos( λn ) exp( λ n 2 Fo ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqy1dy Maeyypa0tcfa4aaSaaaOqaaKqzGeGaam4qaiabgkHiTiaadoeajugW aiabg6HiLcGcbaqcLbsacaWGdbGaamyAaiabgkHiTiaadoeajugWai abg6HiLcaajugibiabg2da9KqbaoaaqaeakeaajuaGdaWcaaGcbaqc LbsacaaIYaGaeyyXICTaam4uaiaadMgacaWGUbqcfa4aaeWaaOqaaK qzGeGaeq4UdWwcLbmacaWGUbaakiaawIcacaGLPaaaaeaajugibiab eU7aSLqzadGaamOBaKqzGeGaey4kaSIaam4uaiaadMgacaWGUbqcfa 4aaeWaaOqaaKqzGeGaeq4UdWwcLbmacaWGUbaakiaawIcacaGLPaaa jugibiabgwSixlaadoeacaWGVbGaam4CaKqbaoaabmaakeaajugibi abeU7aSLqzadGaamOBaaGccaGLOaGaayzkaaaaaaWcbeqabKqzGeGa eyyeIuoacqGHflY1ciGGLbGaaiiEaiaacchajuaGdaqadaGcbaqcLb sacqGHsislcqaH7oaBlmaaDaaabaqcLbmacaWGUbaaleaadaahaaqc fayabeaajugWaiaaikdaaaaaaKqzGeGaamOraiaad+gaaOGaayjkai aawMcaaaaa@8580@ (39)

Infinitely cylinder

ϕ ¯ v= Cv ¯ C CiC = 4 λ n 2 J 1 2 ( λn ) J 0 2 ( λn )+ J 1 2 ( λn ) exp( λ n 2 Fo ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaaba qcLbsacqaHvpGzaaqcLbmacaWG2bqcLbsacqGH9aqpjuaGdaWcaaGc baqcfa4aa0aaaOqaaKqzGeGaam4qaKqzadGaamODaaaajugibiabgk HiTiaadoeajugWaiabg6HiLcGcbaqcLbsacaWGdbqcLbmacaWGPbqc LbsacqGHsislcaWGdbqcLbmacqGHEisPaaqcLbsacqGH9aqpjuaGda aeabGcbaqcfa4aaSaaaOqaaKqzGeGaaGinaaGcbaqcLbsacqaH7oaB juaGdaqhaaWcbaqcLbsacaWGUbaaleaajugibiaaikdaaaaaaaWcbe qabKqzGeGaeyyeIuoacqGHflY1juaGdaWcaaqaaKqzGeGaamOsaSWa a0baaKqbagaajugWaiaaigdaaKqbagaajugWaiaaikdaaaqcfa4aae WaaeaajugibiabeU7aSLqzadGaamOBaaqcfaOaayjkaiaawMcaaaqa aKqzGeGaamOsaSWaa0baaKqbagaajugWaiaaicdaaKqbagaajugWai aaikdaaaqcfa4aaeWaaeaajugibiabeU7aSLqzadGaamOBaaqcfaOa ayjkaiaawMcaaKqzGeGaey4kaSIaamOsaSWaa0baaKqbagaajugWai aaigdaaKqbagaajugWaiaaikdaaaqcfa4aaeWaaeaajugibiabeU7a SLqzadGaamOBaaqcfaOaayjkaiaawMcaaaaajugibiGacwgacaGG4b GaaiiCaKqbaoaabmaakeaajugibiabgkHiTiabeU7aSTWaa0baaeaa jugWaiaad6gaaSqaamaaCaaajuaGbeqaaKqzadGaaGOmaaaaaaqcLb sacaWGgbGaam4BaaGccaGLOaGaayzkaaaaaa@97A3@ (40)

Sphere

ϕ ¯ v= Cv ¯ C CiC = 6 λ n 3 [ sin( λn )λncos( λn ) ] λnsin( λn )cos( λn ) 2 exp( λ n 2 Fo ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaaba qcLbsacqaHvpGzaaqcLbmacaWG2bqcLbsacqGH9aqpjuaGdaWcaaGc baqcfa4aa0aaaOqaaKqzGeGaam4qaKqzadGaamODaaaajugibiabgk HiTiaadoeajugWaiabg6HiLcGcbaqcLbsacaWGdbqcLbmacaWGPbqc LbsacqGHsislcaWGdbqcLbmacqGHEisPaaqcLbsacqGH9aqpjuaGda aeabGcbaqcfa4aaSaaaOqaaKqzGeGaaGOnaaGcbaqcLbsacqaH7oaB juaGdaqhaaWcbaqcLbsacaWGUbaaleaajugWaiaaiodaaaaaaaWcbe qabKqzGeGaeyyeIuoacqGHflY1juaGdaWcaaqaamaadmaabaqcLbsa ciGGZbGaaiyAaiaac6gajuaGdaqadaqaaKqzGeGaeq4UdWwcLbmaca WGUbaajuaGcaGLOaGaayzkaaqcLbsacqGHsislcqaH7oaBcaWGUbGa ci4yaiaac+gacaGGZbqcfa4aaeWaaeaajugibiabeU7aSLqzadGaam OBaaqcfaOaayjkaiaawMcaaaGaay5waiaaw2faaaqaaKqzGeGaeq4U dWwcLbmacaWGUbqcLbsacqGHsislciGGZbGaaiyAaiaac6gajuaGda qadaqaaKqzGeGaeq4UdWwcLbmacaWGUbaajuaGcaGLOaGaayzkaaqc LbsacqGHflY1ciGGJbGaai4BaiaacohajuaGdaqadaqaaKqzGeGaeq 4UdWwcLbmacaWGUbaajuaGcaGLOaGaayzkaaaaaSWaaWbaaKqbagqa baqcLbmacaaIYaaaaKqzGeGaeyyXICTaaiyzaiaacIhacaGGWbqcfa 4aaeWaaOqaaKqzGeGaeyOeI0Iaeq4UdW2cdaqhaaqaaKqzadGaamOB aaWcbaWaaWbaaKqbagqabaqcLbmacaaIYaaaaaaajugibiaadAeaca WGVbaakiaawIcacaGLPaaaaaa@AA9F@ (41)

As can be seen in Eqs. (39) – (41), it is important to know how many terms of infinite series solutions are necessary to obtain the correct solution. This is a general knowledge that the use of the first term is sufficient if the Fo number is more than 0.2. In this case, the concentration ratio is then linear after that time. This first approach may easily be used to determine this parameter with the value of the mass transfer coefficient (kc).25

Let's assume that the sodium tripolyphosphates concentration were determined in red meat in an experiment for the regular geometries such as infinitely slab, infinitely cylinder and sphere while meat samples were immersed in a solution for a given period of time. It is very easy to determine D as long as the available experimental data when the Fourier number is greater than 0.2. In the following experiment, we measuring the concentration of STP in the beef sample with flat plate shape for determination of constant diffusion coefficient using natural logarithm concentration ratio of STP in the beef vs. time.

Experiments

Analysis of Sodium tripolyphosphate in meat sample with a flat plate shape Sodium tripolyphosphate solution and beef sample were used in this experiment. Therefore, Sodium tripolyphosphate solution must be made before testing begins. Tripolyphosphate solution was prepared using distilled water at 65-70°C. in the various concentrations. Different concentrations of sodium tripolyphosphate solution such as 4%, 6% and 8% (w/v) were used in this experiment. Also, the purchased meat is cut into 25×25×25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca aIYaGaaGynaiabgEna0kaaikdacaaI1aGaey41aqRaaGOmaiaaiwda aaa@3FA6@ mm pieces and used for different stages of testing. It should be noted that the frozen specimens should be melted before testing at room temperature (25-25°C). Beef samples were also immersed in distilled water as a control group. After the preparation of the raw material, the test began several times. To begin each experiment, after cooling the STP solution with desired concentration to ambient temperature, beef samples were immersed in it. The ratio of volume to weight of solutions to beef samples is approximately 5:1(v/w). Then, phosphate level changes in both beef samples and solutions versus time were determined using a modified spectrophotometer ammonium molybdate method. In this experiment, every 10 minutes after immersion, each sample is removed from the solution and the sodium tripolyphosphate concentration is measured. It should be noted that any sample taken out of the solution before measuring the amount of STP in it, first, after leaving their surfaces were washed and dried with a tissue and then they are crushed well in the blender. Five grams of crushed sample was homogeneous in 40 ml of water and allowed to be stored at 4°C for 30 minutes. The homogenous are filtered with a Watten filter paper and then filtered with distilled water to a volume of 50 ml. To do this, a vacuum pump is used. Then, to 5 ml of the filtered sample, 5 ml TAC 10% aqueous solution was added and then centrifuged at 40,000 rpm for 5 minutes. After the supernatant was completed to 25 ml, the pH of the solution was adjusted to 8 using concentrated ammonia solution. Then remove 2 ml of solution and 10 ml combined reagent was added. The volume of this solution reaches 50 ml by adding distilled water. Similarly, at each sampling time, solutions STP phosphate levels were also evaluated. After removing the beef samples, the solution is diluted. The pH of 4 ml of diluted solution is adjusted to 8 and then, 10 ml combined reagent was added, and the resulting solution volume is added to distilled water to 50 ml. The dilution process causes phosphate levels within a measurable range, and by hydrolysis of phosphates into orthophosphates. These mixtures are placed inside the incubator for 10 minutes at a temperature of 37oC. In this experiment, the absorbance was read at 690 nm using UV/VIS spectrophotometry. Also, in order to calculate the concentration of phosphate, the necessary curve was first prepared by standard STP solutions, and then phosphate concentration of samples in term of orthophosphates was calculated using this curve. All of these experiments were repeated 3 times.

Results and discussion

Determination of constant diffusion coefficient value
Suppose that the change in concentration was recorded in a specific location of regular geometries such as an infinite slab, infinite cylindrical shape and spherical object in a medium to determine the diffusion coefficient (D) and mass transfer coefficient (kc). Then, the roots of Eqs. (36)-(38) would be (π/2, 3π/2, 5π/2,….) for an infinite slab, (2.4048, 5.5200, 8.6537,….) for an infinite cylinder and (π, 2π, 3π,….) for a sphere, respectively, when the Bi is infinite. Due to the fact that the concentration ratio ( Cv ¯ C CiC ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaSaaaeaadaqdaaqaaiaadoeajugWaiaadAhaaaqcfaOaeyOeI0Ia am4qaKqzadGaeyOhIukajuaGbaGaam4qaKqzadGaamyAaKqbakabgk HiTiaadoeajugWaiabg6HiLcaaaKqbakaawIcacaGLPaaaaaa@48D8@ becomes linear when the Fourier number greater than 0.2, the first term of Eqs. (39) to (41), are then used to characterize the linear change in that regions.25 When the natural logarithm of both sides of Eqs. (39) to (41) are taken with the first term approximation (n =1), Eqs. (42) to (44) are obtained for infinite slab (λ1 = π/2), infinite cylinder (λ1 = 2.4048) and sphere (λ1 = π), respectively:

ln[ Cv ¯ C CiC ]= A 1 λ 1 2 D L2 t= A 1 ( π 2 ) 2 D L 2 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaciiBai aac6gajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqbaoaanaaakeaajugi biaadoeajugWaiaadAhaaaqcLbsacqGHsislcaWGdbqcLbmacqGHEi sPaOqaaKqzGeGaam4qaiaadMgacqGHsislcaWGdbqcLbmacqGHEisP aaaakiaawUfacaGLDbaajugibiabg2da9iaadgeajuaGdaWgaaqaaK qzadGaaGymaaqcfayabaqcLbsacqGHsisljuaGdaWcaaGcbaqcLbsa cqaH7oaBlmaaDaaabaqcLbmacaaIXaaaleaajugWaiaaikdaaaqcLb sacqGHflY1caWGebaakeaajugibiaadYeajugWaiaaikdaaaqcLbsa caWG0bGaeyypa0JaamyqaSWaaSbaaWqaaiaaigdaaeqaaKqzGeGaey OeI0scfa4aaSaaaOqaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsa cqaHapaCaOqaaKqzGeGaaGOmaaaaaOGaayjkaiaawMcaaKqbaoaaCa aabeqaaKqzadGaaGOmaaaajugibiaadseaaOqaaKqzGeGaamitaSWa aSbaaWqaaiaaikdaaeqaaaaajugibiaadshaaaa@74E9@ (42)

ln[ Cv ¯ C CiC ]= A 1 λ 1 2 D R 2 t= A 1 ( 2.4048 ) 2 D R 2 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaciiBai aac6gajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqbaoaanaaakeaajugi biaadoeajugWaiaadAhaaaqcLbsacqGHsislcaWGdbqcLbmacqGHEi sPaOqaaKqzGeGaam4qaKqzadGaamyAaKqzGeGaeyOeI0Iaam4qaKqz adGaeyOhIukaaaGccaGLBbGaayzxaaqcLbsacqGH9aqpcaWGbbqcfa 4aaSbaaeaalmaaBaaajuaGbaqcLbmacaaIXaaajuaGbeaaaeqaaKqz GeGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeq4UdW2cdaqhaaqaaKqzad GaaGymaaWcbaqcLbmacaaIYaaaaKqzGeGaeyyXICTaamiraaGcbaqc LbsacaWGsbWcdaWgaaadbaGaaGOmaaqabaaaaKqzGeGaamiDaiabg2 da9iaadgealmaaBaaameaadaWgaaqaaiaaigdaaeqaaaqabaqcLbsa cqGHsisljuaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaaGOmaiaac6 cacaaI0aGaaGimaiaaisdacaaI4aaakiaawIcacaGLPaaajuaGdaah aaqabeaadaahaaqabeaajugWaiaaikdaaaaaaKqzGeGaeyyXICTaam iraaGcbaqcLbsacaWGsbqcfa4aaSbaaWqaaSWaaSbaaWqaaSWaaSba aKqbagaajugWaiaaikdaaKqbagqaaaadbeaaaeqaaaaajugibiaads haaaa@7CEC@ (43)

Ln( CvC CiC )= A 1 λ 1 2 D R 2 t= A 1 π 2 D R 2 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitai aad6gajuaGdaqadaqaamaalaaabaqcLbsacaWGdbqcLbmacaWG2bqc LbsacqGHsislcaWGdbqcLbmacqGHEisPaKqbagaajugibiaadoeaju gWaiaadMgajugibiabgkHiTiaadoeajugWaiabg6HiLcaaaKqbakaa wIcacaGLPaaajugibiabg2da9iaadgeakmaaBaaaleaadaWgaaadba WcdaWgaaadbaqcLbmacaaIXaaameqaaaqabaaaleqaaKqzGeGaeyOe I0scfa4aaSaaaOqaaKqzGeGaeq4UdW2cdaqhaaqaaKqzadGaaGymaa WcbaqcLbmacaaIYaaaaKqzGeGaamiraaGcbaqcLbsacaWGsbGcdaah aaWcbeqaamaaBaaameaacaaIYaaabeaaaaaaaKqzGeGaamiDaiabg2 da9iaadgeajuaGdaWgaaqaaKqzadGaaGymaaqcfayabaqcLbsacqGH sisljuaGdaWcaaGcbaqcLbsacqaHapaCjuaGdaahaaqabeaajugWai aaikdaaaqcLbsacaWGebaakeaajugibiaadkfalmaaCaaajuaGbeqa aKqzadGaaGOmaaaaaaqcLbsacaWG0baaaa@7203@ (44)

Where A1 in these equations is calculated from the following expressions:

For infinite slab

A 1 =Ln[ 2sin( λ1 ) λ1+sin( λ1 ) cos( λ1x/L ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqaK qbaoaaBaaabaqcLbmacaaIXaaajuaGbeaajugibiabg2da9iaadYea caWGUbqcfa4aamWaaOqaaKqbaoaalaaakeaajugibiaaikdacqGHfl Y1ciGGZbGaaiyAaiaac6gajuaGdaqadaGcbaqcLbsacqaH7oaBjugW aiaaigdaaOGaayjkaiaawMcaaaqaaKqzGeGaeq4UdWwcLbmacaaIXa qcLbsacqGHRaWkciGGZbGaaiyAaiaac6gajuaGdaqadaGcbaqcLbsa cqaH7oaBjugWaiaaigdaaOGaayjkaiaawMcaaaaajugibiGacogaca GGVbGaai4CaKqbaoaabmaakeaajugibiabeU7aSLqzadGaaGymaKqz GeGaamiEaiaac+cacaWGmbaakiaawIcacaGLPaaaaiaawUfacaGLDb aaaaa@694A@ (45)

For infinite cylinder

A 1 =ln[ 2. J 1 ( λ 1 ) λ 1 [ J 0 2 ( λ 1 )+ J 1 2 ( λ 1 ) ] J 0 ( λ 1 r R ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqaK qbaoaaBaaabaqcLbsacaaIXaaajuaGbeaajugibiabg2da9iaadYga caWGUbqcfa4aamWaaeaadaWcaaqaaKqzGeGaaGOmaiaac6cacaWGkb qcfa4aaSbaaeaajugWaiaaigdaaKqbagqaamaabmaabaqcLbsacqaH 7oaBjuaGdaWgaaqaaKqzadGaaGymaaqcfayabaaacaGLOaGaayzkaa aabaqcLbsacqaH7oaBjuaGdaWgaaqaaKqzadGaaGymaaqcfayabaqc LbsacqGHflY1juaGdaWadaqaaKqzGeGaamOsaSWaa0baaKqbagaaju gWaiaaicdaaKqbagaajugWaiaaikdaaaqcfa4aaeWaaeaajugibiab eU7aSLqbaoaaBaaabaqcLbmacaaIXaaajuaGbeaaaiaawIcacaGLPa aajugibiabgUcaRiaadQealmaaDaaajuaGbaqcLbmacaaIXaaajuaG baqcLbmacaaIYaaaaKqbaoaabmaabaqcLbsacqaH7oaBlmaaBaaame aacaaIXaaabeaaaKqbakaawIcacaGLPaaaaiaawUfacaGLDbaaaaqc LbsacqGHflY1caWGkbqcfa4aaSbaaeaajugWaiaaicdaaKqbagqaam aabmaabaqcLbsacqaH7oaBjuaGdaWgaaqaaKqzadGaaGymaaqcfaya baWaaSaaaeaajugibiaadkhaaKqbagaajugibiaadkfaaaaajuaGca GLOaGaayzkaaaacaGLBbGaayzxaaaaaa@847D@ (46)

And for a sphere

A1=Ln( 2[ sin( λ 1 ) λ 1 cos( λ 1 ) ] λ 1 sin( λ 1 )cos( λ 1 ) sin( λ 1 r/R ) ( λ 1 r/R ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqai aaigdacqGH9aqpcaWGmbGaamOBaKqbaoaabmaakeaajuaGdaWcaaGc baqcLbsacaaIYaqcfa4aamWaaOqaaKqzGeGaci4CaiaacMgacaGGUb qcfa4aaeWaaOqaaKqzGeGaeq4UdWwcfa4aaSbaaeaajugWaiaaigda aKqbagqaaaGccaGLOaGaayzkaaqcLbsacqGHsislcqaH7oaBjuaGda WgaaqaaKqzadGaaGymaaqcfayabaqcLbsacqGHflY1ciGGJbGaai4B aiaacohajuaGdaqadaGcbaqcLbsacqaH7oaBjuaGdaWgaaqaaKqzad GaaGymaaqcfayabaaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaeaa jugibiabeU7aSLqbaoaaBaaabaqcLbmacaaIXaaajuaGbeaajugibi abgkHiTiGacohacaGGPbGaaiOBaKqbaoaabmaakeaajugibiabeU7a STWaaSbaaWqaaiaaigdaaeqaaaGccaGLOaGaayzkaaqcLbsacqGHfl Y1ciGGJbGaai4BaiaacohajuaGdaqadaGcbaqcLbsacqaH7oaBlmaa BaaajuaGbaqcLbmacaaIXaaajuaGbeaaaOGaayjkaiaawMcaaaaaju gibiabgwSixNqbaoaalaaakeaajugibiGacohacaGGPbGaaiOBaKqb aoaabmaakeaajugibiabeU7aSLqbaoaaBaaabaqcLbmacaaIXaaaju aGbeaajugibiaadkhacaGGVaGaamOuaaGccaGLOaGaayzkaaaabaqc fa4aaeWaaOqaaKqzGeGaeq4UdWwcfa4aaSbaaeaajugWaiaaigdaaK qbagqaaKqzGeGaamOCaiaac+cacaWGsbaakiaawIcacaGLPaaaaaaa caGLOaGaayzkaaaaaa@988D@ (47)

As can be seen in Eqs. (42) to (44), slope m of the concentration ratio vs. time curve for the infinite slab, infinite cylinder, and sphere is shown in the following equations:

For infinite slab

m= λ 1 2 D L 2 =[ ( π/2 ) 2 D L 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBai abg2da9iabgkHiTKqbaoaalaaakeaajugibiabeU7aSTWaa0baaeaa jugWaiaaigdaaSqaaKqzadGaaGOmaaaajugibiaadseaaOqaaKqzGe GaamitaKqbaoaaCaaabeqaaKqzadGaaGOmaaaaaaqcLbsacqGH9aqp juaGdaWadaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiWdaNaai4laiaaik daaOGaayjkaiaawMcaaKqbaoaaCaaabeqaaKqzadGaaGOmaaaajuaG daWcaaGcbaqcLbsacaWGebaakeaajugibiaadYeajuaGdaahaaqabe aajugWaiaaikdaaaaaaaGccaGLBbGaayzxaaaaaa@585E@ (48)

For infinite cylinder

m= λ 1 2 D R 2 =[ ( 2.4048 ) 2 D R 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBai abg2da9iabgkHiTKqbaoaalaaakeaajugibiabeU7aSTWaa0baaeaa jugWaiaaigdaaSqaaKqzadGaaGOmaaaajugibiaadseaaOqaaKqbak aadkfadaahaaqabeaajugWaiaaikdaaaaaaKqzGeGaeyypa0tcfa4a amWaaOqaaKqbaoaabmaakeaacqGHsislcaaIYaGaaiOlaiaaisdaca aIWaGaaGinaiaaiIdaaiaawIcacaGLPaaajuaGdaahaaqabeaajugW aiaaikdaaaqcfa4aaSaaaOqaaKqzGeGaamiraaGcbaqcfaOaamOuam aaCaaabeqaaKqzadGaaGOmaaaaaaaakiaawUfacaGLDbaaaaa@58DA@ (49)

And for a sphere

m= λ 1 2 D R 2 =[ π 2 D R 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBai abg2da9iabgkHiTKqbaoaalaaakeaajugibiabeU7aSTWaa0baaeaa jugWaiaaigdaaSqaaKqzadGaaGOmaaaajugibiaadseaaOqaaKqbak aadkfadaahaaqabeaajugWaiaaikdaaaaaaKqzGeGaeyypa0tcfa4a amWaaOqaaKqbakabgkHiTiabec8aWnaaCaaabeqaaKqzadGaaGOmaa aajuaGdaWcaaGcbaqcLbsacaWGebaakeaajuaGcaWGsbWaaWbaaeqa baqcLbmacaaIYaaaaaaaaOGaay5waiaaw2faaaaa@5410@ (50)

Then, with the known slope and thickness of infinite slab or radius of the infinite cylindrical or sphere materials, the diffusion coefficient (D) value may be determined. As we know, this method does not need to know the location where test data is recorded, regardless of location.27

Diffusion coefficients for the STPs diffusing into the beef samples may easily be determined using these results in both stages before and after the diffusion. After the phosphate content change of the beef samples was experimentally determined, natural log of concentration ratio values ( ln CtC CiC ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaqcLb saciGGSbGaaiOBaKqbaoaalaaakeaajugibiaadoeajugWaiaadsha jugibiabgkHiTiaadoeajugWaiabg6HiLcGcbaqcLbsacaWGdbqcLb macaWGPbqcLbsacqGHsislcaWGdbqcLbmacqGHEisPaaaakiaawIca caGLPaaaaaa@4B5A@ was calculated and graphed versus immersing time. Figure 1 & Figure 2 show this change with respect to the differences in values ln CtC CiC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaciiBai aac6gajuaGdaWcaaGcbaqcLbsacaWGdbqcLbmacaWG0bqcLbsacqGH sislcaWGdbqcLbmacqGHEisPaOqaaKqzGeGaam4qaKqzadGaamyAaK qzGeGaeyOeI0Iaam4qaKqzadGaeyOhIukaaaaa@49C7@  in the meat samples before and after the barrier formation was completed. Starting from this point, assuming the natural log of concentration changes after a certain time would be linear, diffusion coefficients may be easily determined with this approach that was applied to the experimental data before and after the barrier formation was completed. As can be seen in Eq. (42), the slope of the concentration ratio versus time directly gives the diffusion coefficient value with the known (Eq. (36). The assumption of an infinite mass transfer coefficient (k) between the surface and the solution interphase was a general approach for this kind of problems. Then, the constant diffusion coefficient of STP in beef with slab shaped may be determined using the following equation:

D= slop 3 λ 1 2 L 2 = 4 3 slop π 2 L 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamirai abg2da9KqbaoaalaaakeaajugibiaadohacaWGSbGaam4Baiaadcha aOqaaKqzGeGaaG4maiabeU7aSTWaa0baaeaajugWaiaaigdaaSqaaK qzadGaaGOmaaaaaaqcLbsacqGHflY1caWGmbqcfa4aaWbaaeqabaqc LbmacaaIYaaaaKqzGeGaeyypa0JaeyOeI0scfa4aaSaaaOqaaKqzGe GaaGinaaGcbaqcLbsacaaIZaaaaKqbaoaalaaakeaajugibiaadoha caWGSbGaam4BaiaadchaaOqaaKqzGeGaeqiWda3cdaahaaadbeqaai aaikdaaaaaaKqzGeGaeyyXICTaamitaKqbaoaaCaaabeqaaKqzadGa aGOmaaaaaaa@5F10@

Figure1 The increases in the sodium tripolyphosphate (STP) concentration ratio of the meat samples dipped in STP solutions before the barrier formation was completed.

Figure 2 The increases in the sodium tripolyphosphate (STP) concentration ratio of the meat samples dipped in STP solutions after the barrier formation was completed.

Table 1 shows the values of the diffusion coefficient determined according to the concentration of STP solutions. As observed in these results, they increased with increasing concentration of STPs in the early stages of immersing, before the barrier formation was completed. Secondly, after completing the barrier, Diffusion coefficient values have not changed. Minor variations in numerical values may be due to experimental errors.

Concentration of 
STP solutions %

Diffusion coefficient (D) before barrier
formation was completed m2/s

Diffusion coefficient (D) after barrier
formation was completed m2/s

2

1.81×10-9

5.40×10-9

4

1.99×10-9

5.48×10-9

6

4.00×10-9

5.36×10-9

Table 1 Diffusion coefficients of the sodium tripolyphosphate solutions for slab shaped before and after the barrier formation on the surface of meat samples was completed (Fo > 0.2)

Determination of average diffusion coefficient       

As described above, determining the of diffusion coefficients was based on the assumption that Fourier number is more than 0.2 and the diffusion coefficient of the STPs to be constant through the full immersing process. When the determined diffusion coefficients were used to determine the Fourier number, it is clear that this assumption is not correct. For example, if a 20 mm infinite slab shaped beef sample is immersed in a 2% STP solution, it should take more than 3 hours for this assumption. Although it was thought that after a certain time of immersion, the phenomena of propagation changed. Clearly, the rate of release of STPs and orthophosphates changes through the entire process, leading to a variable diffusion coefficient. The diffusion coefficient values also seemed to be high. When this value was applied to the diffusion equations, the amount of penetration STPs in the meat sample was also determined to be high. According to these problems, another method should be used to determine the minimum mean diffusion coefficient for describing the STP diffusion process. In order to determine the mean diffusion coefficients, the least squares method was used for the experimental data to determine these mean values as the follows:

The results of integration for infinitely slab, infinitely cylinder, and sphere using of Eqs. (33) - (35) through the total volume would be obtained by Eqs. (39) to (41), respectively as the following:

For infinitely slab

θv ¯ = Cv ¯ C CiC = 2 λ 1 A 1 exp( λ 1 2 Dt ξ 2 )+ 2 λ 2 A 2 .exp( λ 2 2 Dt ξ 2 )+..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiabeI7aXjaadAhaaaGaeyypa0tcfa4aaSaaaOqaaKqbaoaa naaabaGaam4qaKqzadGaamODaaaajugibiabgkHiTiaadoeajugWai abg6HiLcGcbaqcLbsacaWGdbqcLbmacaWGPbqcLbsacqGHsislcaWG dbqcLbmacqGHEisPaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsaca aIYaaakeaajugibiabeU7aSLqbaoaaBaaabaqcLbmacaaIXaaajuaG beaaaaqcLbsacqGHflY1caWGbbqcfa4aaSbaaeaajugWaiaaigdaaK qbagqaaKqzGeGaeyyXICTaciyzaiaacIhacaGGWbqcfa4aaeWaaOqa aKqzGeGaeyOeI0Iaeq4UdW2cdaqhaaqaaKqzadGaaGymaaWcbaqcLb macaaIYaaaaKqbaoaalaaakeaajugibiaadseacaWG0baakeaajugi biabe67a4LqbaoaaCaaabeqaaKqzadGaaGOmaaaaaaaakiaawIcaca GLPaaajugibiabgUcaRKqbaoaalaaakeaajugibiaaikdaaOqaaKqz GeGaeq4UdWwcfa4aaSbaaeaajugWaiaaikdaaKqbagqaaaaajugibi abgwSixlaadgeajuaGdaWgaaqaaKqzadGaaGOmaaqcfayabaqcLbsa caGGUaGaciyzaiaacIhacaGGWbqcfa4aaeWaaOqaaKqzGeGaeyOeI0 Iaeq4UdW2cdaqhaaqaaKqzadGaaGOmaaWcbaqcLbmacaaIYaaaaKqb aoaalaaakeaajugibiaadseacaWG0baakeaajugibiabe67a4Lqbao aaCaaabeqaaKqzadGaaGOmaaaaaaaakiaawIcacaGLPaaajugibiab gUcaRiaac6cacaGGUaGaaiOlaiaac6cacaGGUaaaaa@9D19@ (51)

In the above equation, A values in the above equation are given by Eq. (52)

for n= 1,2,…….    An= sin 2 ( λn ) λn+sin( λn )cos( λn ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiyqaK qzadGaaiOBaiabg2da9SWaaSaaaKqbagaajugibiGacohacaGGPbGa aiOBaSWaaWbaaWqabeaajugWaiaaikdaaaWcdaqadaqcfayaaKqzGe Gaeq4UdWwcLbmacaWGUbaajuaGcaGLOaGaayzkaaaabaqcLbsacqaH 7oaBjugWaiaad6gacqGHRaWkjugibiGacohacaGGPbGaaiOBaSWaae WaaKqbagaajugibiabeU7aSLqzadGaamOBaaqcfaOaayjkaiaawMca aKqzadGaeyyXICDcLbsaciGGJbGaai4BaiaacohalmaabmaajuaGba qcLbsacqaH7oaBjugWaiaad6gaaKqbakaawIcacaGLPaaaaaaaaa@655B@ (52)

For infinitely cylindrical shape

θv ¯ = Cv ¯ C CiC = 4 λ 1 2 A 1 exp( λ 1 2 Dt ξ 2 )+ 4 λ 2 2 A 2 .exp( λ 2 2 Dt ξ 2 )+..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiabeI7aXjaadAhaaaGaeyypa0tcfa4aaSaaaOqaaKqbaoaa naaabaGaam4qaKqzadGaamODaaaajugibiabgkHiTiaadoeajugWai abg6HiLcGcbaqcLbsacaWGdbqcLbmacaWGPbqcLbsacqGHsislcaWG dbqcLbmacqGHEisPaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsaca aI0aaakeaajugibiabeU7aSTWaa0baaKqbagaajugWaiaaigdaaKqb agaajugWaiaaikdaaaaaaKqzGeGaeyyXICTaamyqaKqbaoaaBaaaba qcLbmacaaIXaaajuaGbeaajugibiGacwgacaGG4bGaaiiCaKqbaoaa bmaakeaajugibiabgkHiTiabeU7aSTWaa0baaeaajugWaiaaigdaaS qaaKqzadGaaGOmaaaajuaGdaWcaaGcbaqcLbsacaWGebGaamiDaaGc baqcLbsacqaH+oaEjuaGdaahaaqabeaajugWaiaaikdaaaaaaaGcca GLOaGaayzkaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaI0aaa keaajugibiabeU7aSTWaa0baaeaadaWgaaadbaqcLbmacaaIYaaame qaaaWcbaqcLbmacaaIYaaaaaaajugibiabgwSixlaadgeajuaGdaWg aaqaaKqzadGaaGOmaaqcfayabaqcLbsacaGGUaGaciyzaiaacIhaca GGWbqcfa4aaeWaaOqaaKqzGeGaeyOeI0Iaeq4UdW2cdaqhaaqaaKqz adGaaGOmaaWcbaqcLbmacaaIYaaaaKqbaoaalaaakeaajugibiaads eacaWG0baakeaajugibiabe67a4LqbaoaaCaaabeqaaKqzadGaaGOm aaaaaaaakiaawIcacaGLPaaajugibiabgUcaRiaac6cacaGGUaGaai Olaiaac6cacaGGUaaaaa@9DE7@ 53)

Where A values in the Eq. (51) are given by Eq. (54):

An= J 1 2 ( λn ) J 0 2 ( λn )+ J 1 2 ( λn ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqaK qzadGaamOBaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamOsaSWa a0baaeaajugWaiaaigdaaSqaaKqzadGaaGOmaaaajuaGdaqadaGcba qcLbsacqaH7oaBjugWaiaad6gaaOGaayjkaiaawMcaaaqaaKqzGeGa amOsaSWaa0baaeaajugWaiaaicdaaSqaaKqzadGaaGOmaaaajuaGda qadaGcbaqcLbsacqaH7oaBjugWaiaad6gaaOGaayjkaiaawMcaaKqz GeGaey4kaSIaamOsaSWaa0baaeaajugWaiaaigdaaSqaaKqzadGaaG OmaaaajuaGdaqadaGcbaqcLbsacqaH7oaBjugWaiaad6gaaOGaayjk aiaawMcaaaaaaaa@6067@ for n= 1,2,……. (54)

For spherical shape

θv ¯ = Cv ¯ C CiC = 6 λ 1 3 A 1 exp( λ 1 2 Dt ξ 2 )+ 6 λ 2 3 A 2 .exp( λ 2 2 Dt ξ 2 )+..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajugibiabeI7aXjaadAhaaaGaeyypa0tcfa4aaSaaaOqaaKqbaoaa naaabaGaam4qaKqzadGaamODaaaajugibiabgkHiTiaadoeajugWai abg6HiLcGcbaqcLbsacaWGdbqcLbmacaWGPbqcLbsacqGHsislcaWG dbqcLbmacqGHEisPaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsaca aI2aaakeaajugibiabeU7aSTWaa0baaKqbagaajugWaiaaigdaaKqb agaajugWaiaaiodaaaaaaKqzGeGaeyyXICTaamyqaKqbaoaaBaaaba qcLbmacaaIXaaajuaGbeaajugibiGacwgacaGG4bGaaiiCaKqbaoaa bmaakeaajugibiabgkHiTiabeU7aSTWaa0baaeaajugWaiaaigdaaS qaaKqzadGaaGOmaaaajuaGdaWcaaGcbaqcLbsacaWGebGaamiDaaGc baqcLbsacqaH+oaEjuaGdaahaaqabeaajugWaiaaikdaaaaaaaGcca GLOaGaayzkaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaI2aaa keaajugibiabeU7aSTWaa0baaeaadaWgaaadbaqcLbmacaaIYaaame qaaaWcbaqcLbmacaaIZaaaaaaajugibiabgwSixlaadgeajuaGdaWg aaqaaKqzadGaaGOmaaqcfayabaqcLbsacaGGUaGaciyzaiaacIhaca GGWbqcfa4aaeWaaOqaaKqzGeGaeyOeI0Iaeq4UdW2cdaqhaaqaaKqz adGaaGOmaaWcbaqcLbmacaaIYaaaaKqbaoaalaaakeaajugibiaads eacaWG0baakeaajugibiabe67a4LqbaoaaCaaabeqaaKqzadGaaGOm aaaaaaaakiaawIcacaGLPaaajugibiabgUcaRiaac6cacaGGUaGaai Olaiaac6cacaGGUaaaaa@9DED@ (55)

Where A values in the Eq. (55) are given by Eq. (56):

An= [ sin( λ n ) λ n cos( λ n ) ] λ n sin( λ n )cos( λ n ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqai aad6gacqGH9aqpjuaGdaWcaaGcbaqcfa4aamWaaOqaaKqzGeGaci4C aiaacMgacaGGUbqcfa4aaeWaaOqaaKqzGeGaeq4UdW2cdaWgaaadba GaamOBaaqabaaakiaawIcacaGLPaaajugibiabgkHiTiabeU7aSLqb aoaaBaaabaqcLbmacaWGUbaajuaGbeaajugibiGacogacaGGVbGaai 4CaKqbaoaabmaakeaajugibiabeU7aSLqbaoaaBaaabaqcLbmacaWG UbaajuaGbeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaKqzGe Gaeq4UdW2cdaWgaaadbaGaamOBaaqabaqcLbsacqGHsislciGGZbGa aiyAaiaac6gajuaGdaqadaGcbaqcLbsacqaH7oaBjuaGdaWgaaqaaK qzadGaamOBaaqcfayabaaakiaawIcacaGLPaaajugibiabgwSixlGa cogacaGGVbGaai4CaKqbaoaabmaakeaajugibiabeU7aSLqbaoaaBa aabaGaamOBaaqabaaakiaawIcacaGLPaaaaaWaaWbaaSqabeaadaah aaadbeqaaKqzadGaaGOmaaaaaaaaaa@74B6@  for n=1,2,…  (56)

In Eqs. (51), (53) and (55), the diffusion coefficient value (D) is only unknown assuming kc of the medium, hence the Bi and the λ values, is known. In this state, the diffusivity (D), the value is determined by solving the above equations. Every other numerical method such as Newton-Raphson may be used to solve this equation. when the Newton-Raphson technique is used, the iterative solution of the Eqs. (57)-(63) give the result of D value for regular geometries:

For infinitely slab

f( D )= n1 2 λ n A n .exp( λ n 2 Dt L 2 ) θ v ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaabmaakeaajugibiaadseaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaabCaOqaaKqbaoaalaaakeaajugibiaaikdaaOqaaKqzGe Gaeq4UdWwcfa4aaSbaaeaajugWaiaad6gaaKqbagqaaaaaaSqaaKqz adGaamOBaiabgkHiTiaaigdaaSqaaKqzadGaeyOhIukajugibiabgg HiLdGaeyyXICTaamyqaKqbaoaaBaaabaqcLbmacaWGUbaajuaGbeaa jugibiaac6caciGGLbGaaiiEaiaacchajuaGdaqadaGcbaqcLbsacq GHsisljuaGdaWcaaGcbaqcLbsacqaH7oaBlmaaDaaabaqcLbmacaWG UbaaleaajugWaiaaikdaaaqcLbsacaWGebGaamiDaaGcbaqcLbsaca WGmbqcfa4aaWbaaeqabaqcLbmacaaIYaaaaaaaaOGaayjkaiaawMca aKqzGeGaeyOeI0scfa4aa0aaaOqaaKqzGeGaeqiUdexcfa4aaSbaaS qaaKqzGeGaamODaaWcbeaaaaqcLbsacqGH9aqpcaaIWaaaaa@72A6@ (57)

f'( D )=[ n1 2 λ n t L 2 A n .exp( λ n 2 Dt L 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzai aacEcajuaGdaqadaGcbaqcLbsacaWGebaakiaawIcacaGLPaaajugi biabg2da9iabgkHiTKqbaoaadmaakeaajuaGdaaeWbGcbaqcfa4aaS aaaOqaaKqzGeGaaGOmaiabeU7aSLqbaoaaBaaabaqcLbmacaWGUbaa juaGbeaajugibiaadshaaOqaaKqzGeGaamitaKqbaoaaCaaabeqaaK qzadGaaGOmaaaaaaaaleaajugibiaad6gacqGHsislcaaIXaaaleaa jugibiabg6HiLcGaeyyeIuoacqGHflY1caWGbbqcfa4aaSbaaeaaju gibiaad6gaaKqbagqaaKqzGeGaaiOlaiGacwgacaGG4bGaaiiCaKqb aoaabmaakeaajugibiabgkHiTKqbaoaalaaakeaajugibiabeU7aSL qbaoaaDaaaleaajugibiaad6gaaSqaaKqzGeGaaGOmaaaacaWGebGa amiDaaGcbaqcLbsacaWGmbqcfa4aaWbaaeqabaWaaWbaaeqabaqcLb macaaIYaaaaaaaaaaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@6F2A@ (58)

For cylindrical shape

f( D )= n1 4 λ n 2 A n .exp( λ n 2 Dt R 2 ) θv ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGMbqcfa4aaeWaaOqaaKqzGeGaamiraaGccaGLOaGaayzkaaqcLbsa cqGH9aqpjuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqzGeGaaGinaaGcba qcLbsacqaH7oaBjuaGdaWgaaqaaSWaaSbaaKqbagaajugWaiaad6ga aKqbagqaaaqabaWcdaahaaadbeqaaiaaikdaaaaaaaWcbaqcLbmaca WGUbGaeyOeI0IaaGymaaWcbaqcLbmacqGHEisPaKqzGeGaeyyeIuoa cqGHflY1caWGbbGcdaWgaaWcbaqcLbmacaWGUbaaleqaaKqzGeGaai OlaiGacwgacaGG4bGaaiiCaKqbaoaabmaabaqcLbsacqGHsislcqaH 7oaBlmaaDaaajuaGbaqcLbmacaWGUbaajuaGbaqcLbmacaaIYaaaaK qbaoaalaaabaqcLbsacaWGebGaamiDaaqcfayaaKqzGeGaamOuaKqb aoaaCaaabeqaamaaCaaabeqaaKqzadGaaGOmaaaaaaaaaaqcfaOaay jkaiaawMcaaKqzGeGaeyOeI0scfa4aa0aaaeaajugibiabeI7aXLqz adGaamODaaaajugibiabg2da9iaaicdaaaa@7556@ (59)

f'( D )=[ n1 4t R 2 A n .exp( λ n 2 Dt R 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGMbGaai4jaKqbaoaabmaakeaajugibiaadseaaOGaayjkaiaawMca aKqzGeGaeyypa0JaeyOeI0scfa4aamWaaOqaaKqbaoaaqahakeaaju aGdaWcaaGcbaqcLbsacaaI0aGaamiDaaGcbaqcLbsacaWGsbqcfa4a aWbaaeqabaWaaWbaaeqabaqcLbmacaaIYaaaaaaaaaaaleaajugWai aad6gacqGHsislcaaIXaaaleaajugWaiabg6HiLcqcLbsacqGHris5 aiabgwSixlaadgeajuaGdaWgaaqaaKqzGeGaamOBaaqcfayabaqcLb sacaGGUaGaciyzaiaacIhacaGGWbqcfa4aaeWaaeaajugibiabgkHi TiabeU7aSTWaa0baaKqbagaajugWaiaad6gaaKqbagaajugWaiaaik daaaqcfa4aaSaaaeaajugibiaadseacaWG0baajuaGbaqcLbsacaWG sbqcfa4aaWbaaeqabaqcLbmacaaIYaaaaaaaaKqbakaawIcacaGLPa aaaOGaay5waiaaw2faaaaa@6EB0@ (60)

For spherical shape

< f( D )= n1 6 λ n 3 A 1 .exp( λ n 2 Dt R 2 ) θv ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGMbqcfa4aaeWaaOqaaKqzGeGaamiraaGccaGLOaGaayzkaaqcLbsa cqGH9aqpjuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqzGeGaaGOnaaGcba qcLbsacqaH7oaBjuaGdaWgaaqaaSWaaSbaaKqbagaajugWaiaad6ga aKqbagqaaaqabaWcdaahaaadbeqaaiaaiodaaaaaaaWcbaqcLbmaca WGUbGaeyOeI0IaaGymaaWcbaqcLbmacqGHEisPaKqzGeGaeyyeIuoa cqGHflY1caWGbbqcfa4aaSbaaeaajugWaiaaigdaaKqbagqaaKqzGe GaaiOlaiGacwgacaGG4bGaaiiCaKqbaoaabmaabaqcLbsacqGHsisl cqaH7oaBlmaaDaaajuaGbaqcLbmacaWGUbaajuaGbaqcLbmacaaIYa aaaKqbaoaalaaabaqcLbsacaWGebGaamiDaaqcfayaaKqzGeGaamOu aKqbaoaaCaaabeqaamaaCaaabeqaaKqzadGaaGOmaaaaaaaaaaqcfa OaayjkaiaawMcaaKqzGeGaeyOeI0scfa4aa0aaaeaajugibiabeI7a XLqzadGaamODaaaajugibiabg2da9iaaicdaaaa@761D@ (61)

f'( D )=[ n1 6t λ n R 2 A n .exp( λ n 2 Dt R 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGMbGaai4jaKqbaoaabmaakeaajugibiaadseaaOGaayjkaiaawMca aKqzGeGaeyypa0JaeyOeI0scfa4aamWaaOqaaKqbaoaaqahakeaaju aGdaWcaaGcbaqcLbsacaaI2aGaamiDaaGcbaqcLbsacqaH7oaBjuaG daWgaaqaaKqzadGaamOBaaqcfayabaqcLbsacaWGsbqcfa4aaWbaae qabaqcLbsacaaIYaaaaaaaaSqaaKqzadGaamOBaiabgkHiTiaaigda aSqaaKqzadGaeyOhIukajugibiabggHiLdGaeyyXICTaamyqaKqbao aaBaaabaqcLbsacaWGUbaajuaGbeaajugibiaac6caciGGLbGaaiiE aiaacchajuaGdaqadaqaaKqzGeGaeyOeI0Iaeq4UdW2cdaqhaaqcfa yaaKqzadGaamOBaaqcfayaaKqzadGaaGOmaaaajuaGdaWcaaqaaKqz GeGaamiraiaadshaaKqbagaajugibiaadkfajuaGdaahaaqabeaaju gWaiaaikdaaaaaaaqcfaOaayjkaiaawMcaaaGccaGLBbGaayzxaaaa aa@7392@ (62)

Then, the new diffusion coefficient for any shape is calculated by the following equation:

D n+1 = D n f( D n ) f'( D n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGebqcfa4aaSbaaeaajugWaiaad6gacqGHRaWkcaaIXaaajuaGbeaa jugibiabg2da9iaadseajuaGdaWgaaqaaKqzadGaamOBaaqcfayaba qcLbsacqGHsisljuaGdaWcaaqaaKqzGeGaamOzaKqbaoaabmaabaqc LbsacaWGebqcfa4aaSbaaeaajugWaiaad6gaaKqbagqaaaGaayjkai aawMcaaaqaaKqzGeGaamOzaiaacEcajuaGdaqadaqaaKqzGeGaamir aKqbaoaaBaaabaqcLbmacaWGUbaajuaGbeaaaiaawIcacaGLPaaaaa aaaa@55DA@ (63)

In this way, only knowing the concentration ratio at any time will be enough to determine the diffusion coefficient value, instead of using a set of experimental data. Constant D value assumption is checked with this approach using testing data obtained at different times. Table 2 shows the average diffusion coefficients of the sodium tripolyphosphate solutions including 2%, 4% and 6%, (w/v) before and after the barrier formation on the surface of meat samples was completed. As can be seen, the average diffusion coefficient values were found again to increase with increasing STP concentration. Using these values, the diffused amount of STPs into the meat samples may be easily determined.

Concentration of 
STP solutions %

Diffusion coefficient (D) before barrier
formation was completed m2/s

Diffusion coefficient (D) after barrier
formation was completed m2/s

2

1.76×10-11

47.22×10-11

4

5.57×10-11

87.50×10-11

6

31.82×10-11

105.10×10-11

Table 2 Average diffusion coefficients of the sodium tripolyphosphate solutions for slab shaped before and after the barrier formation on the surface of meat samples was completed (Fo> 0.2)

Determination of variable diffusion coefficient value
Unal27 published variable diffusion coefficient values for diffusion of sodium tripolyphosphate in red meats. The variable diffusion coefficient may be explained using the above method using Eqs. (57) to (63).

If changes in the entire process continue, it may be simpler to determine an average value for describing the total process, or it should use the numerical finite difference to consider changes. Therefore, to minimize the sum of squares (S), the difference between experimental data and the results of analytical solutions can be used to minimize the equation Eq. (64) as follows:

S= i1 n ( θvE ) i 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGtbGaeyypa0tcfa4aaabCaOqaaKqbaoaabmaakeaajugibiabeI7a XLqzadGaamODaKqzGeGaeyOeI0IaamyraaGccaGLOaGaayzkaaaale aajugWaiaadMgacqGHsislcaaIXaaaleaajugWaiaad6gaaKqzGeGa eyyeIuoalmaaDaaajuaGbaqcLbmacaWGPbaajuaGbaqcLbmacaaIYa aaaaaa@505F@ (64)

In Eq. (64), n is the number of experimentally obtained data, and E is the testing data. The diffusivity value is determined by minimizing S. In the other words should be 0.

Deriving from Eq. (64) relative to D, then the following equations are obtained:

S D = i1 n [ 2( θvE ) θv D ] i=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcfa4aaS aaaOqaaKqzGeGaeyOaIyRaam4uaaGcbaqcLbsacqGHciITcaWGebaa aiabg2da9KqbaoaaqahakeaajuaGdaWadaGcbaqcLbsacaaIYaGaey yXICDcfa4aaeWaaOqaaKqzGeGaeqiUdexcLbmacaWG2bqcLbsacqGH sislcaWGfbaakiaawIcacaGLPaaajugibiabgwSixNqbaoaalaaake aajugibiabgkGi2kabeI7aXLqzadGaamODaaGcbaqcLbsacqGHciIT caWGebaaaaGccaGLBbGaayzxaaaaleaajugWaiaadMgacqGHsislca aIXaaaleaajugWaiaad6gaaKqzGeGaeyyeIuoacaWGPbGaeyypa0Ja aGimaaaa@64DA@ (65)

S D = i1 n 2{ ( [ 2 λ 1 A 1 exp( λ 1 2 Dt ξ 2 )+ 2 λ 2 A 2 exp( λ 2 2 Dt ξ 2 )+ ]E ) i × ( 2 λ 1 t ξ 2 A 1 exp( λ 1 2 Dt ξ 2 ) 2 λ 2 t ξ 2 A 2 .exp( λ 2 2 Dt ξ 2 ) ) i } =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcfa4aaS aaaOqaaKqzGeGaeyOaIyRaam4uaaGcbaqcLbsacqGHciITcaWGebaa aiabg2da9KqbaoaaqahabaqcLbsacaaIYaGaeyyXICDcfa4aaiWaae aadaqadaqaamaadmaabaWaaSaaaeaajugibiaaikdaaKqbagaajugi biabeU7aSLqbaoaaBaaabaqcLbmacaaIXaaajuaGbeaaaaqcLbsacq GHflY1caWGbbqcfa4aaSbaaeaajugWaiaaigdaaKqbagqaaKqzGeGa eyyXICTaciyzaiaacIhacaGGWbqcfa4aaeWaaeaajugibiabgkHiTK qbaoaalaaabaqcLbsacqaH7oaBlmaaDaaajuaGbaqcLbmacaaIXaaa juaGbaqcLbmacaaIYaaaaKqzGeGaamiraiaadshaaKqbagaajugibi abe67a4LqbaoaaCaaabeqaaKqzadGaaGOmaaaaaaaajuaGcaGLOaGa ayzkaaqcLbsacqGHRaWkjuaGdaWcaaqaaKqzGeGaaGOmaaqcfayaaK qzGeGaeq4UdWwcfa4aaSbaaeaajugWaiaaikdaaKqbagqaaaaajugi biabgwSixlaadgeajuaGdaWgaaqaaKqzadGaaGOmaaqcfayabaqcLb sacqGHflY1ciGGLbGaaiiEaiaacchajuaGdaqadaqaaKqzGeGaeyOe I0scfa4aaSaaaeaajugibiabeU7aSTWaa0baaKqbagaajugWaiaaik daaKqbagaajugWaiaaikdaaaqcLbsacaWGebGaamiDaaqcfayaaKqz GeGaeqOVdGxcfa4aaWbaaeqabaqcLbmacaaIYaaaaaaaaKqbakaawI cacaGLPaaajugibiabgUcaRiabgwSixlabgwSixlabgwSixlabgwSi xlabgwSixdqcfaOaay5waiaaw2faaKqzGeGaeyOeI0Iaamyraaqcfa OaayjkaiaawMcaamaaBaaabaWaaSbaaeaajugWaiaadMgaaKqbagqa aaqabaqcLbsacqGHxdaTjuaGdaqadaqaaKqzGeGaeyOeI0scfa4aaS aaaeaajugibiaaikdacqaH7oaBjuaGdaWgaaqaaKqzadGaaGymaaqc fayabaqcLbsacaWG0baajuaGbaqcLbsacqaH+oaEjuaGdaahaaqabe aajugWaiaaikdaaaaaaKqzGeGaeyyXICTaamyqaKqbaoaaBaaabaqc LbmacaaIXaaajuaGbeaajugibiabgwSixlGacwgacaGG4bGaaiiCaK qbaoaabmaabaqcLbsacqGHsisljuaGdaWcaaqaaKqzGeGaeq4UdW2c daqhaaqcfayaaKqzadGaaGymaaqcfayaaKqzadGaaGOmaaaajugibi aadseacaWG0baajuaGbaqcLbsacqaH+oaEjuaGdaahaaqabeaajugW aiaaikdaaaaaaaqcfaOaayjkaiaawMcaaKqzGeGaeyOeI0scfa4aaS aaaeaajugibiaaikdacqaH7oaBjuaGdaWgaaqaamaaBaaabaqcLbma caaIYaaajuaGbeaajugibiaadshaaKqbagqaaaqaaKqzGeGaeqOVdG xcfa4aaWbaaeqabaqcLbmacaaIYaaaaaaajugibiabgwSixlaadgea juaGdaWgaaqaaKqzadGaaGOmaaqcfayabaqcLbsacaGGUaGaciyzai aacIhacaGGWbqcfa4aaeWaaeaajugibiabgkHiTKqbaoaalaaabaqc LbsacqaH7oaBlmaaDaaajuaGbaqcLbmacaaIYaaajuaGbaqcLbmaca aIYaaaaKqzGeGaamiraiaadshaaKqbagaajugibiabe67a4Lqbaoaa CaaabeqaaKqzadGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaqcLbsacq GHsislcqGHflY1cqGHflY1cqGHflY1cqGHflY1cqGHflY1cqGHflY1 aKqbakaawIcacaGLPaaadaWgaaqaaSWaaSbaaKqbagaajugWaiaadM gaaKqbagqaaaqabaaacaGL7bGaayzFaaaabaqcLbmacaWGPbGaeyOe I0IaaGymaaqcfayaaKqzadGaamOBaaqcLbsacqGHris5aiabg2da9i aaicdaaaa@2D79@ (66)

From equation (66), this can appear to ∂S/∂D to be 0:

i1 n 2{ ( 2 λ 1 t ξ 2 A 1 exp( λ 1 2 Dt ξ 2 ) 2 λ 2 t ξ 2 A 2 .exp( λ 2 2 Dt ξ 2 ) ) i } =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcfa4aaa bCaeaajugibiaaikdacqGHflY1juaGdaGadaqaamaabmaabaqcLbsa cqGHsisljuaGdaWcaaqaaKqzGeGaaGOmaiabeU7aSLqbaoaaBaaaba qcLbmacaaIXaaajuaGbeaajugibiaadshaaKqbagaajugibiabe67a 4LqbaoaaCaaabeqaaKqzadGaaGOmaaaaaaqcLbsacqGHflY1caWGbb qcfa4aaSbaaeaajugWaiaaigdaaKqbagqaaKqzGeGaeyyXICTaciyz aiaacIhacaGGWbqcfa4aaeWaaeaajugibiabgkHiTKqbaoaalaaaba qcLbsacqaH7oaBlmaaDaaajuaGbaqcLbmacaaIXaaajuaGbaqcLbma caaIYaaaaKqzGeGaamiraiaadshaaKqbagaajugibiabe67a4Lqbao aaCaaabeqaaKqzadGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaqcLbsa cqGHsisljuaGdaWcaaqaaKqzGeGaaGOmaiabeU7aSLqbaoaaBaaaba WaaSbaaeaajugWaiaaikdaaKqbagqaaKqzGeGaamiDaaqcfayabaaa baqcLbsacqaH+oaEjuaGdaahaaqabeaajugWaiaaikdaaaaaaKqzGe GaeyyXICTaamyqaKqbaoaaBaaabaqcLbmacaaIYaaajuaGbeaajugi biaac6caciGGLbGaaiiEaiaacchajuaGdaqadaqaaKqzGeGaeyOeI0 scfa4aaSaaaeaajugibiabeU7aSTWaa0baaKqbagaajugWaiaaikda aKqbagaajugWaiaaikdaaaqcLbsacaWGebGaamiDaaqcfayaaKqzGe GaeqOVdGxcfa4aaWbaaeqabaqcLbmacaaIYaaaaaaaaKqbakaawIca caGLPaaajugibiabgkHiTiabgwSixlabgwSixlabgwSixlabgwSixl abgwSixlabgwSixdqcfaOaayjkaiaawMcaamaaBaaabaWcdaWgaaqc fayaaKqzadGaamyAaaqcfayabaaabeaaaiaawUhacaGL9baaaeaaju gWaiaadMgacqGHsislcaaIXaaajuaGbaqcLbmacaWGUbaajugibiab ggHiLdGaeyypa0JaaGimaaaa@BAEC@ (67)

And the Eq. (67) can be solved numerically for the value of the diffusion coefficient using the Newton-Raphson method or any other solving method.

Constant mass transfer coefficient determination

Mass transfer coefficient can also be determined using the analytical solutions. Therefore, the diffusivity of substance must be known. Then, λ1 is determined λ 1 =ξ ( m D ) 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsacq aH7oaBjuaGdaWgaaqaaKqzadGaaGymaaqcfayabaqcLbsacqGH9aqp cqaH+oaEcqGHflY1juaGdaqadaGcbaqcLbsacqGHsisljuaGdaWcaa GcbaqcLbsacaWGTbaakeaajugibiaadseaaaaakiaawIcacaGLPaaa juaGdaahaaqabeaalmaaliaajuaGbaqcLbmacaaIXaaajuaGbaqcLb macaaIYaaaaaaaaaa@4E71@ using the slope (m) of the concentration ratio vs time curve. And it is then used to determine the Bi number and thus kc. However, this technique is easy to use. The general approach is to assume in the literature infinity mass transfer coefficient (kc) and then to determine the diffusivity value (D). Nevertheless, can be used the analogy of Chilton-Colburn for determination of the kc and D using heat transfer coefficient when a simultaneous heat and mass transfer is occurring such as drying process:

K c =( h ρ C p ) ( Pr Sc ) 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGlbqcfa4aaSbaaeaajugWaiaadogaaKqbagqaaKqzGeGaeyypa0tc fa4aaeWaaOqaaKqbaoaalaaakeaajugibiaadIgaaOqaaKqzGeGaeq yWdiNaam4qaKqbaoaaBaaabaqcLbmacaWGWbaajuaGbeaaaaaakiaa wIcacaGLPaaajugibiabgwSixNqbaoaabmaakeaajuaGdaWcaaGcba qcLbsaciGGqbGaaiOCaaGcbaqcLbsacaWGtbGaam4yaaaaaOGaayjk aiaawMcaaKqbaoaaCaaaleqabaWaaSqaaWqaaKqzadGaaGOmaaadba qcLbmacaaIZaaaaaaaaaa@56A8@ (68)

Where kc is mass transfer coefficient, h is the heat transfer coefficient, ρ and Cp are density and heat capacity of the heating medium, respectively, Pr is the Prandtl and Sc are the Schmidt number. Essentially, Eqs. (69) to (74) is used to determination of diffusivity value by knowing kc and experimental data. Then, determination of λ value (Eqs. (69), (71) and (73)) and therefore the D value through the slope of the concentration ratio vs. time curve (Eqs. (70), (72) and (74)).

Bi= λ n tan( λ n )= K c R m l 2 λ 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGcbGaamyAaiabg2da9iabeU7aSLqbaoaaBaaabaqcLbmacaWGUbaa juaGbeaajugibiabgwSixlGacshacaGGHbGaaiOBaKqbaoaabmaake aajugibiabeU7aSLqbaoaaBaaabaqcLbmacaWGUbaajuaGbeaaaOGa ayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaam4saK qbaoaaBaaabaqcLbmacaWGJbaajuaGbeaajugibiaadkfaaOqaaKqb aoaalaaakeaajugibiabgkHiTiaad2gacaWGSbqcfa4aaWbaaeqaba qcLbmacaaIYaaaaaGcbaqcLbsacqaH7oaBlmaaDaaabaqcLbmacaaI YaaaleaajugWaiaaigdaaaaaaaaaaaa@61A2@ (69)

D= m l 2 λ 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGebGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeyOeI0IaamyBaiaadYga juaGdaahaaqabeaajugWaiaaikdaaaaakeaajugibiabeU7aSTWaa0 baaeaajugWaiaaikdaaSqaaKqzadGaaGymaaaaaaaaaa@45CF@ (70)

Bi= λ n J 1 ( λ n ) J o ( λ n ) = K c R m R 2 λ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGcbGaamyAaiabg2da9iabeU7aSLqbaoaaBaaabaqcLbmacaWGUbaa juaGbeaajugibiabgwSixNqbaoaalaaakeaajugibiaadQeajuaGda WgaaqaaKqzadGaaGymaaqcfayabaWaaeWaaOqaaKqzGeGaeq4UdWwc fa4aaSbaaeaajugWaiaad6gaaKqbagqaaaGccaGLOaGaayzkaaaaba qcLbsacaWGkbWcdaWgaaadbaGaam4Baaqabaqcfa4aaeWaaOqaaKqz GeGaeq4UdWwcfa4aaSbaaeaajugWaiaad6gaaKqbagqaaaGccaGLOa GaayzkaaaaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaam4saKqb aoaaBaaabaqcLbmacaWGJbaajuaGbeaajugibiaadkfaaOqaaKqbao aalaaakeaajugibiabgkHiTiaad2gacaWGsbqcfa4aaWbaaeqabaqc LbmacaaIYaaaaaGcbaqcLbsacqaH7oaBlmaaDaaabaqcLbmacaaIXa aaleaajugWaiaaikdaaaaaaaaaaaa@6DAB@ (71)

D= m R 2 λ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikGaamirai abg2da9KqbaoaalaaakeaajugibiabgkHiTiaad2gacaWGsbqcfa4a aWbaaeqabaqcLbmacaaIYaaaaaGcbaqcLbsacqaH7oaBlmaaDaaaba qcLbmacaaIXaaaleaajugWaiaaikdaaaaaaaaa@4526@ (72)

Bi=( 1 λ1 tan λ 1 )= k c R m R 2 λ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikqcLbsaca WGcbGaamyAaiabg2da9KqbaoaabmaakeaajugibiaaigdacqGHsisl juaGdaWcaaGcbaqcLbsacqaH7oaBjugWaiaaigdaaOqaaKqzGeGaci iDaiaacggacaGGUbGaeq4UdWwcfa4aaSbaaeaajugWaiaaigdaaKqb agqaaaaaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaK qzGeGaam4AaKqbaoaaBaaabaqcLbmacaWGJbaajuaGbeaajugibiaa dkfaaOqaaKqbaoaalaaakeaajugibiabgkHiTiaad2gacaWGsbqcfa 4aaWbaaeqabaqcLbmacaaIYaaaaaGcbaqcLbsacqaH7oaBlmaaDaaa baqcLbmacaaIXaaaleaajugWaiaaikdaaaaaaaaaaaa@609A@ (73)

D= m R 2 λ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaikGaamirai abg2da9KqbaoaalaaakeaajugibiabgkHiTiaad2gacaWGsbqcfa4a aWbaaeqabaqcLbmacaaIYaaaaaGcbaqcLbsacqaH7oaBlmaaDaaaba qcLbmacaaIXaaaleaajugWaiaaikdaaaaaaaaa@4526@ (74)

Determining the mass transfer coefficient using a diffusion coefficient is not a simple solution. Therefore, it is assumed that the mass transfer coefficient is known, and then the diffusion coefficient is calculated using the following steps for each of regular geometries such as infinitely slab, infinitely cylinder, and sphere: Determination mass transfer coefficient value from Eq. (68), Determination λ1 value from Eqs. (69), (71) and (73) for infinitely slab, infinitely cylinder, and sphere, respectively, Determination D value from Eqs. (70), (72) and (74) for infinitely slab, infinitely cylinder, and sphere, respectively.

Conclusion

Various methods in literature are described in detail for experimental determining of D and kc. because these approaches require experimental data from interesting materials with analytical solutions. These methods are more advantageous than using preferred methods such as the lumped system approach or use of empirical equations to determine these mass transfer parameters. As seen in these approaches, it is necessary that one of the parameters is known so that another parameter can be determined. Therefore, it is still important to develop a procedure to determine both parameters simultaneously.

In this study, D and kc of regular shapes were estimated using the method of Finite Integral Transform and the experimental data. Diffusion coefficients for the STPs diffusing into the slab-shaped of beef samples may easily be determined using these results before and after the diffusion. Determination of the diffusion coefficient for STPs with the mass transfer coefficient at different stages of the immersion process will be useful for studies on further diffusion and optimization. On the other hand, knowing the degree of penetration of STPs in samples under different conditions leads to useful results in STP in meat. In addition to these, the diffusion coefficient was known to be strongly affected by temperature. In this study, all experiments were accomplished at room temperature (20oC). A small increase in temperature may affect the intensity of the STP release. Therefore, research on the effects of temperature on the release of STP may also be necessary for further studies on this subject.

Highlights

  1. New approach for determination of mass transfer parameters.
  2. Using Finite Integral Transform for solutions of governing equations.
  3. Prediction of constant and variable diffusion coefficient (D).
  4. Using concentration ratio vs. time for determination mass transfer coefficient and D.
  5. Determination of mass transfer coefficient, when the diffusivity of substance is known and vice versa

Nomenclature

A: Constant

C: Mass concentration

Ci: Initial mass concentration

C: Medium mass concentration

Cp: Specific heat, J/kg-K

D: Diffusion coefficient, m2/s

E: Experimental data

h: Heat transfer coefficient, W/m2K

Jo, J1: The first kind 0th and 1st order Bessel functions

k: Thermal conductivity, W/m K

kc: Mass transfer coefficient, m/s

L: Half-thickness for an infinite slab, m

m: Slopes of concentration ratio vs. time curves, l/s

µ: Dynamic viscosity, kg/ms

θ: Dimensionless mass concentration ratio

θv: Volume average dimensionless mass concentration ratio

r, x: Distance from the center, m

R: Radius of an infinite cylinder or a sphere, m

ρ: Density, kg/m3

ξ: Characteristic length, m

λ: Root of Eqs. (36)-(38)

ν: kinematic viscosity, m2/s

Acknowledgements

The author is grateful to Fars Technological and Environmental Research Center and the Shiraz University of Technology for supporting this work.

Conflict of interest

The author declares that there is no conflict of interest.

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