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Analytical study of a moving boundary problem describing sublimation process of a humid porous body with convective heat and mass transfer

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Abstract

Sublimation of a humid porous body occurs commonly in food technology and thermal energy storage system . Especially, in accelerated freeze-drying, preservation of biological materials to be denatured is the prime interest. Despite the available literature on sublimation, there is a general lack of mathematical analysis of the effect of convection in the frozen and vapour regions, and rate of evaporation of water vapour in the vapour region. Therefore, it is essential to explore a mathematical model which accounts for these physical processes. This paper attempts to address these gaps in the modeling of sublimation of a humid porous body. For a specific form of the velocity profile, an exact solution of the current problem is obtained via similarity technique. Particularly, results from the current work are shown to be in strong agreement with the results of a previous work. The impact of various dimensionless problem parameters on the sublimation process is discussed extensively. Condition for sublimation limit is discussed. It is obtained that sublimation can take place only under limit of sublimation curve. It is found that, in the presence of convection, sublimation process becomes fast and the material requires less time than usual to sublimate. Furthermore, higher rate of evaporation of water vapour produces a lower temperature field and slower propagation rate of sublimation interface.

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Abbreviations

\(\alpha\) :

Thermal diffusivity (\({\text{m}^2} {\text{s}^{-1}}\))

\(c_{\mathrm{p}}\) :

Specific heat (\(\text{J} \text{kg}^{-1} \text{K}^{-1}\))

\(c_{\mathrm{pv}}\) :

Specific heat of water vapour (\(\text{J} \text{kg}^{-1} \text{K}^{-1}\))

k :

Thermal conductivity (\(\text{W} \text{m}^{-1}\text{K}^{-1}\))

\(\rho\) :

Material density (\(\text{kg m}^{-3}\))

L :

Latent heat of sublimation (\(\text{J}\ \text{kg}^{-1}\))

u :

Unidirectional molecular motion (\(\text{m}\ \text{s}^{-1}\))

\(C_{\mathrm{m}}\) :

Molar concentration of vapour moisture (\(\text{mol}\ \text{m}^{-3}\))

\(M_{\mathrm{m}}\) :

Molecular mass (\(\text{kg}\ \text{mol}^{-1}\))

\(p_{\mathrm{v}}\) :

Vapour pressure (\(\text{kg}\ \text{m}^{-1}\ s^{-2}\))

\(R_{\mathrm{0}}\) :

Universal gas constant (\(\text{kg}\ \text{m}^{2}\ \text{s}^{-2}\ \text{K}^{-1}\ \text{mol}^{-1}\))

t :

Time (s)

Exp(.):

Exponential function

Erf(.):

Error function

F :

Temperature (K)

\(F_{\mathrm{v}}\) :

Sublimation temperature (K)

\(F_{\mathrm{s}}\) :

Surface temperature (K)

\(T_{\mathrm{1}}\) :

Non-dimensional temperature profile \(\left( T_{\mathrm{1}}=1+\frac{F_{\mathrm{1}}-F_{\mathrm{v}}}{F_{\mathrm{s}}-F_{\mathrm{v}}}\right)\)

\(T_{\mathrm{2}}\) :

Non-dimensional temperature profile \(\left( T_{\mathrm{2}}=\frac{F_{\mathrm{2}}-F_{\mathrm{0}}}{F_{\mathrm{v}}-F_{\mathrm{0}}}\right)\)

:

Non-dimensional molar concentration

\(\theta\) :

Non-dimensional temperature/concentration profile

x :

Space coordinate (m)

\(\xi\) :

Non-dimensional space coordinate

s(t):

Moving sublimation front (m)

\(\alpha _{\mathrm{21}}\) :

Non-dimensional thermal diffusivity \(\left( \frac{\alpha _{\mathrm{2}}}{\alpha _{\mathrm{1}}}\right)\)

C :

Non-dimensional molar limit concentration \(\left( C=\frac{C_{\mathrm{m,0}}-C_{\mathrm{m,max}}}{C_{\mathrm{m,max}}-C_{\mathrm{m,s}}}\right)\)

\(\beta\) :

Non-dimensional quantity \(\left( \beta =\frac{ c_{\mathrm{pv}}M_{\mathrm{m}} C_{\mathrm{m,0}}}{c_{\mathrm{p}} \rho _{\mathrm{1}}}\right)\)

\(\gamma\) :

Non-dimensional heat flux \(\left( \gamma =\frac{k_{\mathrm{1}} {(F_{\mathrm{s}}-F_{\mathrm{v}})}}{k_{\mathrm{2}} {(F_{\mathrm{v}}-F_{\mathrm{0}})}}\right)\)

\(l_{\mathrm{0}}\) :

Non-dimensional latent heat of sublimation \(\left( l_{\mathrm{0}}=\frac{C_{\mathrm{m,0}}M_{\mathrm{m}} L}{k_{\mathrm{2}} {(F_{\mathrm{v}}-F_{\mathrm{0}})}}\alpha _{\mathrm{1}}\right)\)

\(\lambda\) :

Unknown constant \(\left( \lambda =\frac{s(t)}{2\sqrt{\alpha _{\mathrm{1}}t}}\right)\)

Pe :

Péclet number \(\left( Pe=\frac{u}{\sqrt{\frac{\alpha }{t}}}\right)\)

Lu :

Luikov number \(\left( Lu=\frac{\alpha _{\mathrm{m}}}{\alpha _{\mathrm{1}}}\right)\)

0:

Initial

1:

Vapour region

2:

Frozen region

m :

Moisture

s :

Surface \(x=0\)

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Acknowledgements

Vikas Chaurasiya, one of the authors is grateful to DST (INSPIRE)-New Delhi (India) for the Senior Research Fellowship vide Ref. No. DST/INSPIRE/03/2017/000184.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Authors and Affiliations

  1. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India

    Vikas Chaurasiya & Jitendra Singh

  2. Mechanical and Aerospace Engineering Department, The University of Texas at Arlington, Arlington , TX, USA

    Ankur Jain

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Contributions

JS: Supervision, formulation of model, reviewing, planning and writing article. VC: Formulation of model, analytical solution of mathematical model, figure plotting, writing, analysis and data curation and original draft preparation. AJ: Supervision, reviewing, planning and writing article.

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Correspondence to Jitendra Singh.

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Chaurasiya, V., Jain, A. & Singh, J. Analytical study of a moving boundary problem describing sublimation process of a humid porous body with convective heat and mass transfer. J Therm Anal Calorim 148, 2567–2584 (2023). https://doi.org/10.1007/s10973-022-11906-3

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