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Numerical Study to Predict Temperature and Moisture Profiles in Unventilated Grain Silos at Prolonged Time Periods

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Abstract

The experimental acquisition of data for monitoring cereal grain storage in silos for long periods is not practical in most cases; in such situations, mathematical modeling is an excellent alternative to predict temperature and mass transfer behavior. The model was developed in two dimensions using the equations of heat, mass, and momentum transport. The governing equations were solved using discretization of the spatial coordinates by orthogonal collocation with Jacobi polynomials and the resulting algebraic system was solved by employing nonlinear relaxation method that uses less computational resources. For the ambient boundary conditions, it was used as a more accurate equation to represent the diurnal–nocturnal temperature dynamic. This study analyzes the effects of moisture migration and temperature interactions between the interstitial air and the sorghum cereal grain bed defined as a porous medium in a cylindrical cavity, including the variations affected by meteorological conditions over 1 year. The mass–thermal gradients were predicted, and the effects of environmental temperature were analyzed for flow patterns, isotherms and moisture distribution changes, indicating how the environmental fluctuations influence the heat and moisture of the ambient spots in the grain storage. The maximum temperature inside the silo was 42 °C, considering the respiration heat of the grain, and the maximum temperature of the grain without the climatic boundary conditions was reported to be approximately 35 °C in previous studies. The proposed model can be easily extrapolated to be used for other cereal grains or to incorporate additional effects such as solar radiation, shadow sunlight path, wind velocity, equations to predict more precise ambient temperatures, etc. as boundary conditions.

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Abbreviations

A:

Height/radius ratio, L/R

\(a_{v}\) :

Grain–air interfacial area, m2·m−3

\(a_{w}\) :

Water activity, dimensionless

Bi:

Biot number, hcR/keff

C :

Concentration of component, kg·m−3

C A :

Concentration of component A (water vapor or grain moisture), kg·m−3

C P :

Specific heat, J·kg−1·°C−1

D :

Scalar diffusivity, m2·s−1

Da:

Darcy’s number, K/R2

Fo:

Fourier number or dimensionless time, αt/R2

g:

Acceleration of gravity, m·s−2

hc :

Heat transfer coefficient, W·m−1·°C−1

K :

Permeability of the porous media, m2

MA :

Molecular mass of water

MB :

Molecular mass of air

\(\bar{M}\) :

Average molecular mass

k eff :

Thermal conductivity of the porous media, W·m−1·°C−1

k y :

Mass transfer coefficient, m·s−1

L:

Height of the cavity, m

Le:

Lewis number, α/Deff, α/Da

mv :

Mass flow of evaporated water, kg·m−3·s−1

N:

Buoyancy ratio, βCρo(Y1− Y0)/β(T1 − T0)

p :

Hour when the minimum temperature is achieved

P :

Air pressure, mmHg

P o :

Volumetric generation of water by respiration, kg·m−3·s−1

P 0v :

Vapor pressure, mmHg

Pr:

Prandtl number, Cpμ/keff

Q O :

Volumetric heat of respiration of cereal grain, J·m−3·s−1

R:

Radius of the cavity, m

Raf :

Rayleigh number for homogeneous fluid, ρog β (T1 − T0) R3/μ α

Ra:

Rayleigh number for porous medium, ρog β K (T1 − T0) R/μ α

t1 :

Time, days

t2 :

Time, hours

T:

Fluid temperature, °C

u r :

Dimensionless radial velocity

u z :

Dimensionless axial velocity

vβ :

Volume of the continuous phase, m3

W :

Dimensionless humidity in the grain, (X − X0)/(Xi − X0)

X :

Humidity of the grain on a dry basis, kg H2O/kg dry grain

x :

Humidity of the grain on a wet basis, kg H2O/kg dry grain

X 0 :

Initial moisture content, kg H2O/kg dry grain

Y :

Absolute humidity of the air, kg H2O/kg dry air

Y 0 :

Initial absolute moisture, kg H2O/kg dry air

Y i :

Absolute humidity of air in the grain–air interface, kg H2O/kg dry air

α:

Thermal diffusivity of the porous medium, keff/ρCp

β :

Volumetric coefficient of thermal expansion, K−1

β c :

Volumetric coefficient of mass expansion, m3·kg−1

ζ:

Dimensionless axial coordinate, z/L

θ:

Dimensionless temperature, (T − T0)/(T1 − T0)

λ v :

Latent heat of vaporization of water, J·kg−1

μ :

Fluid viscosity, kg·m−1·s−1

ξ:

Dimensionless radial coordinate, r/R

ρ a :

Density of dry air, kg·m−3

ρ B :

Density of the continuous phase, kg·m−3

φ:

Dimensionless absolute humidity in air, (Y − Y0)/(Yi − Y0)

ω :

Dimensionless vorticity

\(\psi\) :

Dimensionless stream function

Β :

Fluid phase

eff :

Property for effective media

max:

Maximum value

min:

Minimum value

∞:

Environment property

0:

Reference state or property evaluated at time 0

1:

Reference property evaluated in the side or top wall of the silo

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Acknowledgments

We gratefully acknowledge the financial support of CONCYTEG (Grant 09-09-k662-065), Tecnológico Nacional de México, Grant 6655.18-P, and SEP-CONACYT, Grant CB-2011/167095 from Mexico.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: CONCYTEG (Grant 09-09-k662-065), Tecnológico Nacional de México, Grant 6655.18-P, and CONACYT, Grant CB2011/167095.

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Química-Bioquímica, Tecnológico Nacional de México en Celaya, Antonio García Cubas # 600, esq Av. Tecnológico, C.P. 38010, Celaya, Gto., México

    L. I. Quemada-Villagómez, F. I. Molina-Herrera, G. M. Martínez-González, J. L. Navarrete-Bolaños & H. Jiménez-Islas

  2. Unidad Profesional Interdisciplinaria de Ingeniería Campus Guanajuato, Instituto Politécnico Nacional, Avenida Mineral de Valenciana No. 200, Frac. Industrial Puerto Interior, C.P. 36275, Silao, Gto., México

    M. Carrera-Rodríguez

  3. Tecnológico Nacional de México, CRODE Celaya, Diego Arenas Guzmán 901, Zona de Oro I, C.P. 38023, Celaya, Gto., México

    M. Calderón-Ramírez

Authors
  1. L. I. Quemada-Villagómez
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  2. F. I. Molina-Herrera
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  3. M. Carrera-Rodríguez
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  4. M. Calderón-Ramírez
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  5. G. M. Martínez-González
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  6. J. L. Navarrete-Bolaños
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  7. H. Jiménez-Islas
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Corresponding author

Correspondence to H. Jiménez-Islas.

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Quemada-Villagómez, L.I., Molina-Herrera, F.I., Carrera-Rodríguez, M. et al. Numerical Study to Predict Temperature and Moisture Profiles in Unventilated Grain Silos at Prolonged Time Periods. Int J Thermophys 41, 52 (2020). https://doi.org/10.1007/s10765-020-02636-5

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