Heat and Mass Transfer

, Volume 54, Issue 4, pp 1121–1134 | Cite as

Convective drying of osmo-dehydrated apple slices: kinetics and spatial behavior of effective mass diffusivity and moisture content

  • Juarez Everton de Farias Aires
  • Wilton Pereira da Silva
  • Kalina Lígia Cavalcante de Almeida Farias Aires
  • Aluízio Freire da Silva Júnior
  • Cleide Maria Diniz Pereira da Silva e Silva


The main objective of this study is the presentation of a numerical model of liquid diffusion for the description of the convective drying of apple slices submitted to pretreatment of osmotic dehydration able of predicting the spatial distribution of effective mass diffusivity values in apple slabs. Two models that use numerical solutions of the two-dimensional diffusion equation in Cartesian coordinates with the boundary condition of third kind were proposed to describe drying. The first one does not consider the shrinkage of the product and assumes that the process parameters remain constant along the convective drying. The second one considers the shrinkage of the product and assumes that the effective mass diffusivity of water varies according to the local value of the water content in the apple samples. Process parameters were estimated from experimental data through an optimizer coupled to the numerical solutions. The osmotic pretreatment did not reduce the drying time in relation to the fresh fruits when the drying temperature was equal to 40 °C. The use of the temperature of 60 °C led to a reduction in the drying time. The model that considers the variations in the dimensions of the product and the variation in the effective mass diffusivity proved to be more adequate to describe the process.



Effective mass diffusivity of water (m2 s−1)


Value of D on the interface b of neighboring control volumes

Lx, Ly

Dimensions of the apple slabs (m)

Lx0, Ly0

Initial values of slab dimensions (m)

P, E, N, W and S

Nodal points

\( \overline{S} \)

Average value of sucrose gain (g of sucrose / 100 g of initial dry matter)


Time (s)


Drying temperature


Local value of the dimensionless moisture content

\( \overline{X^{\ast }} \)

Average value of dimensionless moisture content

x, y

Position variables (m)


Transport variable in the diffusion equation

\( \overline{\Phi} \)

Average value of the transport variable


Chi-square function


Transport coefficient

Δxe, Δxw,Δyn and Δys

Distances between nodal points (m)



The second author would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for the support given to this research and for his research grant (Processes Number 302480/2015-3 and 444053/2014-0).

Compliance with ethical standards

Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Juarez Everton de Farias Aires
    • 1
  • Wilton Pereira da Silva
    • 2
  • Kalina Lígia Cavalcante de Almeida Farias Aires
    • 1
  • Aluízio Freire da Silva Júnior
    • 2
  • Cleide Maria Diniz Pereira da Silva e Silva
    • 2
  1. 1.Federal Institute of Education, Science and Technology of ParaíbaJoão PessoaBrazil
  2. 2.Federal University of Campina GrandeCampina GrandeBrazil

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