Cortex Effect on Vacuum Drying Process of Porous Medium

Abstract

Corns, fruits, and vegetables are usually used as porous medium in drying process. But in fact, it must be considered as the cortex effect on mass transfer because the mass transfer of cortex is very difficult than inner medium. Based on the theory of heat and mass transfer, a coupled model for the porous medium vacuum drying process with cortex effect is constructed. The model is implemented and solved using COMSOL software. The water evaporation rate is determined using a nonequilibrium method with the rate constant parameter 𝐾_𝑟 that has been studied. The effects of different vapor pressures (1000, 5000, and 9000 Pa), initial moisture contents (0.3, 0.4, and 0.5 water saturation), drying temperatures (323, 333, and 343 K), and intrinsic permeability for cortex part (10⁻¹³, 10⁻¹⁴, 10⁻¹⁵ m²) on vacuum drying process were studied. The results facilitate a better understanding of the porous medium vacuum drying process that nearer to the reality.

1. Introduction

Scientists and engineers in China are currently studying vacuum drying equipment that could be used in corn drying [1–3]. However, the corn vacuum drying theory remains unclear. Hypothesized that corn is a porous medium, the vacuum drying of corn is a complicated heat and mass transfer process that has been the subject of intensive research [4–7]. All vacuum drying models have to address the water phase change during numerical solving. In one method, the vapor pressure is equal to its equilibrium value [8–11]. Another method is nonequilibrium method [12–16]. As the porous medium, the heat and mass transfer in vacuum drying process has been studied by nonequilibrium method [17]. The water evaporation rate is determined using a nonequilibrium method with the rate constant parameter 𝐾_𝑟. 𝐾_𝑟 values of 1, 10, 1000, and 10000 are simulated. The effects of different vapor pressures (1000, 5000, and 9000 Pa), initial moisture contents (0.4, 0.5, and 0.6 water saturation), drying temperatures (323, 333, and 343 K), and intrinsic permeability (10⁻¹³, 10⁻¹⁴, 10⁻¹⁵ m²) are studied. It was observed that the temperature increased quickly at the start of drying and then lowered gradually. As the drying process continued, the temperature increased slowly. In the absence of free water, temperature increased rapidly. As the drying process concluded, the temperature remained unchanged. The water evaporation rate could not be obtained during the porous medium vacuum drying process. The rate constant parameter is essential to the nonequilibrium method. When 𝐾_𝑟≥1000, the simulation of the drying process was not evidently affected. Vapor pressure and heat transfer affected the transfer of mass. A similar effect was found in the initial moisture and the heat temperature. Intrinsic permeability had a greater effect on the drying process. In the study above, it is as the uniform porous medium. But in fact, it must be considered as the cortex effect on mass transfer because the mass transfer of cortex is more difficult than inner medium. The study of about cortex effect is very few.

In this paper, heat and mass transfer of porous medium with cortex in the vacuum drying process is implemented by using a nonequilibrium method. The effects of vapor pressure, initial moisture, heat temperature, and cortex intrinsic permeability on the drying process were then examined.

2. Physical Model

A physical one-dimensional (1D) model that explains the drying process is shown in Figure 1. The heat and mass transfer is considered only in the 𝑦 direction. Because consider the cortex effect. The heat and mass transfer is through the surface of medium that is the cortex position. The total height of the porous medium is 1.05 cm, and the cortex height is 0.05 cm.

Figure 1 

1D model of porous medium with cortex.

3. Mathematical Model

The porous medium consists of a continuous rigid solid phase, an incompressible liquid phase (free water), and a continuous gas phase that is assumed to be a perfect mixture of vapor and dry air, considered as ideal gases. For a mathematical description of the transport phenomenon in a porous medium, we adopt a continuum approach, wherein macroscopic partial differential equations are achieved through the volume averaging of the microscopic conservation laws. The value of any physical quantity at a point in space is given by its average value on the averaging volume centered at this point.

The moisture movement of the inner porous medium is liquid water and vapor movement; that is, the liquid water could become vapor, and the vapor and liquid water are moved by the pressure gradient. The heat and mass transfer theory could be found in everywhere [8].

The compressibility effects of the liquid phase are negligible, and the phase is homogeneous: 𝜌_𝑙=constant. (1) The solid phase is rigid and homogeneous: 𝜌_𝑠=constant. (2) The gaseous phase is considered an ideal gas. This phase ensures that 𝜌_𝑎=𝑚_𝑎𝑃_𝑎𝑅𝑇,𝜌_𝑣=𝑚_𝑣𝑃_𝑣𝑅𝑇,𝑃_𝑔=𝑃_𝑎+𝑃_𝑣,𝜌_𝑔=𝜌_𝑎+𝜌_𝑣. (3)

The assumption of the local thermal equilibrium between the solid, gas, and liquid phases involves 𝑇_𝑠=𝑇_𝑔=𝑇_𝑙=𝑇. (4)

Mass conservation equations are written for each component in each phase. Given that the solid phase is rigid, the following is given: 𝜕𝜌_𝑠𝜕𝑡=0. (5)

The averaged mass conservation of the dry air yields 𝜕𝜀⋅𝑆_𝑔𝜌_𝑎𝜕𝑡+∇⋅𝜌_𝑎𝑉_𝑎=0. (6)

For vapor, 𝜕𝜀⋅𝑆_𝑔𝜌_𝑣𝜕𝑡+∇⋅𝜌_𝑣𝑉_𝑣=𝐼.̇ (7)

For free water, 𝜕𝜀⋅𝑆_𝑤𝜌_𝑙𝜕𝑡+∇⋅𝜌_𝑙𝑉_𝑙=−𝐼.̇ (8)

For water, the general equation of mass conservation is obtained from the sum of the conservation equations of vapor (𝑣) and free water (𝑙). The general equation is written as follows: 𝜕𝑊1𝜕𝑡+∇⋅𝜌_𝑠𝜌_𝑙𝑉_𝑙+𝜌_𝑣𝑉_𝑣=0,𝑊=𝜀⋅𝑆_𝑤𝜌_𝑙+𝜀⋅𝑆_𝑔𝜌_𝑣(1−𝜀)𝜌_𝑠. (9)(10)

For the Darcy flow of vapor, 𝜌_𝑣𝑉_𝑣=𝜌_𝑣𝑉_𝑔−𝜌_𝑔𝐷_eff⋅∇𝜔. (11)

For the Darcy flow of air, 𝜌_𝑎𝑉_𝑎=𝜌_𝑎𝑉_𝑔−𝜌_𝑔𝐷_eff⋅∇𝜔, (12) where the gas and free water velocity is given by 𝑉_𝑔=𝑘⋅𝑘_𝑟𝑔𝜇_𝑔⋅∇𝑃_𝑔−𝜌_𝑔𝑔,𝑉_𝑙=𝑘⋅𝑘_𝑟𝑙𝜇_𝑙⋅∇𝑃_𝑙−𝜌_𝑙𝑔. (13) The effective diffusion coefficient [8] is given by 𝐷_eff=𝐷𝐵₌. (14)

The vapor fraction in mixed gas is given by 𝜌𝜔=^𝑣𝜌_𝑔. (15) The pressure moving the free water is given by 𝑃_𝑙=𝑃_𝑔−𝑃_𝑐. (16) For capillary pressure, 𝑃_𝑐=56.75×10³1−𝑆_𝑙exp1.062𝑆_𝑙. (17) The saturation of free water and gas is 𝑆_𝑔+𝑆_𝑙=1. (18) Free water relative permeability is given by 𝑘_𝑟𝑙=𝑆_𝑙−𝑆_𝑐𝑟1−𝑆_𝑐𝑟³𝑆_𝑤>𝑆_𝑐𝑟0𝑆_𝑤≤𝑆_𝑐𝑟. (19) Gas relative permeability is given by 𝑘_𝑟𝑔=𝑆_𝑔. (20) The water phase change rate is expressed as 𝐼̇=𝐾_𝑟𝑚_𝑣𝑎_𝜔𝑃_sat−𝑃_𝑣𝑆_𝑔𝜀.𝑅𝑇 (21) Water saturation vapor pressure is given by 𝑃_sat=101325760×10^{(8.07131−(1730.63/(233.426+(𝑇−273))))}. (22)

By considering the hypothesis of the local thermal equilibrium, the energy conservation is reduced to a unique equation: 𝜕𝜌ℎ𝜕𝑡+∇⋅𝜌_𝑎𝑉_𝑎ℎ_𝑎+𝜌_𝑣𝑉_𝑣ℎ_𝑣+𝜌_𝑙𝑉_𝑙ℎ_𝑙−𝜆_𝑒⋅∇𝐼̇𝜆𝑇−Δ𝐻⋅=0,_𝑒=(1−𝜀)𝜆_𝑠+𝜀𝑆_𝑙+𝑆_𝑔𝜔𝜆_𝑣+(1−𝜔)𝜆_𝑎,𝜌ℎ=𝜌_𝑠ℎ_𝑠+𝜀⋅𝑆_𝑔𝜌_𝑎ℎ_𝑎+𝜀⋅𝑆_𝑔𝜌_𝑣ℎ_𝑣+𝜀⋅𝑆_𝑙𝜌_𝑙ℎ_𝑙. (23)

4. Boundary Condition and Parameters

The air pressure on the external surface of the porous medium is fixed, and the boundary condition for air is given by 𝑃_𝑎=𝑃_𝑎𝑣. (24)

The boundary condition for vapor at the surface of the porous medium is given by 𝑃_𝑣=𝑃_𝑣𝑏. (25)

To simulate the vapor pressure of the vacuum drying chamber effect on the drying process, four different vapor pressure boundary values are used.

The boundary condition for free water at the top of the porous medium is 𝑛⋅−𝐷∇𝑆_𝑤=0. (26)

The boundary condition at the surface of the porous medium is 𝑇=𝑇_ℎ. (27)

Three different 𝑇_ℎ values are used in the simulation.

The initial moisture of the porous medium is represented by the liquid water saturation; different initial water saturation values are used. To compare the effects, drying base moisture content (d.b.) was also used, as shown in (9). The water phase change rate is used as 1000 that has been studied before [17]. Intrinsic permeability of inner medium is 10⁻¹³ m². Intrinsic permeability of cortex is 10⁻¹⁴, 10⁻¹⁵ m². In order to compare the results, the intrinsic permeability of cortex 10⁻¹³, that is no cortex effect is simulated. In order to easily converge in, the parameter changer is used a smooth method. The modeling parameters are shown in Table 1.

Table 1 

Parameters used in the simulation process.

5. Numerical Solution

COMSOL Multiphysics 3.5a was used to solve the set of equations. COMSOL is an advanced software used for modeling and simulating any physical process described by partial derivative equations. The set of equations introduced above was solved using the relative initial and boundary conditions of each. COMSOL offers three possibilities for writing the equations: (1) using a template (the Fick law and the Fourier law), (2) using the coefficient form (for mildly nonlinear problems), and (3) using the general form (for most nonlinear problems). Differential equations in the coefficient form were written using an unsymmetric-pattern multifrontal method. We used a direct solver for sparse matrices (UMFPACK), which involves significantly more complicated algorithms than solvers used for dense matrices. The main complication is the need to handle the fill-in in factors 𝐿 and 𝑈 efficiently.

A two-dimensional (2D) grid was used to solve the equations using COMSOL Multiphysics 3.5a. Given the symmetry condition setting at the left and the right sides, The 2D is applied to the the 1D model shown in Figure 1. The mesh consists of 2×200 elements (2D), and time stepping is 1 (from 0 s to 100 s of solution), 5 (from 100 s to 200 s of solution), 20 (from 200 s to 1000 s of solution), 30 (from 1000 s to 2000 s of solution), 40 (from 2000 s to 4000 s of solution), 50 (from 4000 s to 20000 s of solution), and 100 (from 20000 s to 50000 s of solution). Several grid sensitivity tests were conducted to determine the sufficiency of the mesh scheme and to ensure that the results are grid independent. The maximum element size was established as 1𝑒⁻⁴. A backward differentiation formula was used to solve time-dependent variables. Relative tolerance was set to 1𝑒⁻³, whereas absolute tolerance was set to 1𝑒⁻⁴. The simulations were performed using a Tongfang PC with Intel Core 2 Duo processor with 3.0 GHz processing speed, and 4096 MB of RAM running Windows 7.

6. Results and Discussion

6.1. Effect of Boundary Condition

In the study before [17], the heat and mass transfer direction is the same. That is, the heat boundary is on the bottom of the medium. The heat transfer is from the bottom to top. The mass transfer boundary is on the top of bottom. The mass transfer is from the bottom to top by the pressure driving. But in this study, the cortex is considered. And the heat transfer and mass transfer are on the reversed direction as Figure 1. The heat and mass transfer with dryer is on the surface of the medium. The both condition was simulated to be compared as Figure 2. The simulation parameters are 𝑆_𝑙0=0.5, 𝐾_𝑟=1000, 𝑇_ℎ=323 K, 𝑃_𝑣𝑏=1000 Pa, 𝑘_in=10⁻¹³ m², and no cortex. The moisture curve is just little different. The drying rate is little larger at the drying initial stage for same direction heat and mass transfer, and then it is little lower. The other condition is just reversed. The drying rate is little lower at the initial stage for reversed direction heat and mass transfer, and then it is little larger. The drying time for both is 6 hours.

Figure 2 

The effect of boundary condition, heat, and mass transfer is same and is reverse direction.

6.2. Effect of Cortex Resistance

Intrinsic permeability of porous medium is an inherent property and cannot be changed, and measuring it is difficult. Intrinsic permeability has a greater effect because the transfer of free water and vapor is affected by (13). Usually, the cortex intrinsic permeability is less than inner medium. Figure 3 is the moisture curves of no cortex (𝑘_in and 𝑘_cor=10⁻¹³, 10⁻¹⁴, and 10⁻¹⁵ m², resp.), with cortex (𝑘_in=10⁻¹³ and 𝑘_cor=10⁻¹⁴ and 10⁻¹⁵ m² resp.). The other simulation parameters are 𝑆_𝑙0=0.5, 𝐾_𝑟=1000, 𝑇_ℎ=323 K, and 𝑃_𝑣𝑏=1000 Pa. The drying time became evidently longer as the intrinsic permeability was reduced because the moisture movement velocity was lowered at the same pressure gradient. The different is not so obviously as the 𝑘_cor=10⁻¹⁴ m², that is lower 10-fold than inner medium. But when 𝑘_cor=10⁻¹⁵ m², the drying time is increased to 2-fold nearly. In order to compare, the no cortex, 𝑘_cor=10⁻¹⁴ m², and 10⁻¹⁵ m² was shown. Because the numerical calculation is not convergence, the cortex 𝑘_cor=10⁻¹⁶ m², 𝑘_in=10⁻¹³ m² result not gotten. The reason is that the material parameter changer is too big.

Figure 3 

Moisture curves of no cortex (𝑘in=10−13, 𝑘cor=10−13, 10−14, and 10−15 m2), (𝑘cor=10−14, 10−15, 𝑘in=10−13 m2).

6.3. Effect of Vapor Pressure in Vacuum Drying Chamber

The pressure of a vacuum drying chamber, especially vapor pressure, plays an important role in the vacuum drying process and is also linked to the drying cost. The moisture curves of 𝑃_𝑣𝑏=1000, 5000, and 9000 Pa are shown in Figures 4, 5, and 6 with different cortex intrinsic permeability. The other simulation parameters are 𝑆_𝑙0=0.5, 𝐾_𝑟=1000, and 𝑇_ℎ=323 K. The vapor pressure has a greater effect on the drying process; a lower vapor pressure results in greater pressure degradation. The movements of free water and vapor as well as the free water evaporation rate are quicker, as given by (13) and (21), respectively. But the pressure effect is lower by the cortex intrinsic permeability reduced.

Figure 4 

Moisture curves of different 𝑃𝑣𝑏=1000, 5000, 9000 Pa, 𝑘cor=10−15 m2, 𝑘in=10−13 m2.

Figure 5 

Moisture curves of different 𝑃𝑣𝑏=1000, 5000, 9000 Pa, 𝑘cor=10−14 m2, 𝑘in=10−13 m2.

Figure 6 

Moisture curves of different 𝑃𝑣𝑏=1000, 5000, 9000 Pa, no cortex, 𝑘in=10−13 m2.

6.4. Effect of Initial Moisture Content

The effect of initial moisture content on the moisture curve is shown in Figures 7, 8, and 9 for 𝑆_𝑙0=0.5, 0.4, 0.3 with different cortex intrinsic permeability. To compare the results, moisture is represented by the moisture ratio 𝑀/𝑀₀. The other parameters are 𝐾_𝑟=1000, 𝑃_𝑣𝑏=1000 Pa, and 𝑇_ℎ=323 K. The drying time is about 10 hours, 6.5 hours, and 6 hours for 𝑆_𝑙0=0.5, 0.4, 0.3, respectively. In the same intrinsic permeability, the drying time is almost the same. The initial moisture has a less effect when initial moisture is 0.4 and 0.3, especially cortex intrinsic permeability 𝑘_cor=10⁻¹⁵ m². The drying rate is little larger for 𝑆_𝑙0=0.5, and the drying rate is almost the same for 𝑆_𝑙0=0.4, 0.3.

Figure 7 

Moisture curves at different 𝑆𝑙0=0.5, 0.4, 0.3, 𝑘cor=10−15 m2, 𝑘in=10−13 m2.

Figure 8 

Moisture curves at different 𝑆𝑙0=0.5, 0.4, 0.3, 𝑘cor=10−14 m2, 𝑘in=10−13 m2.

Figure 9 

Moisture curves at different 𝑆𝑙0=0.5, 0.4, 0.3, no cortex, 𝑘in=10−13 m2.

6.5. Effect of Heat Temperature

The effect of heat temperature on moisture is shown in Figures 10, 11, and 12 for the heat temperature 𝑇_ℎ=323, 333, and 343 K. The other parameters are 𝑆_𝑙0=0.5, 𝐾_𝑟=1000, and 𝑃_𝑣=1000 Pa. The effect of heat temperature is obviously in all simulation. The drying time is more than 14 hours for 𝑇_ℎ=323 K, 𝑘_cor=10⁻¹⁵ m², and 𝑘_in=10⁻¹³ m². And it is about 10 hours and 8 hours for 𝑇_ℎ=333, 343 K, 𝑘_cor=10⁻¹⁵ m², and 𝑘_in=10⁻¹³ m². The drying time is more than 9 hours for 𝑇_ℎ=333 K, 𝑘_cor=10⁻¹⁵ m², and 𝑘_in=10⁻¹³ m². It is about 7 hours and 6 hours for 𝑇_ℎ=333, 343 K, 𝑘_cor=10⁻¹⁵ m², and 𝑘_in=10⁻¹³ m². The drying time is more than 7 hours for 𝑇_ℎ=333 K, 𝑘_cor=10⁻¹⁵ m², and 𝑘_in=10⁻¹³ m². It is about 6 hours and 5 hours for 𝑇_ℎ=333, 343 K, 𝑘_cor=10⁻¹⁵ m², and 𝑘_in=10⁻¹³ m².

Figure 10 

Moisture curves at different 𝑇ℎ=323, 333, 343 K, 𝑘cor=10−15 m2, 𝑘in=10−13 m2.

Figure 11 

Moisture curves at different 𝑇ℎ=323, 333, 343 K, 𝑘cor=10−14 m2, 𝑘in=10−13 m2.

Figure 12 

Moisture curves at different 𝑇ℎ=323, 333, 343 K, no cortex, 𝑘in=10−13 m2.

7. Conclusion

A coupled model of porous medium vacuum drying with cortex effect based on the theory of heat and mass transfer was implemented in this paper. The drying rate is little higher at the drying initial stage for the same direction of heat and mass transfer, and then it is little lower. The drying time became evidently longer as the intrinsic permeability was reduced because the moisture movement velocity was lowered at the same pressure gradient. The difference is not so obvious as the 𝑘_cor=10⁻¹⁴ m², that is lower 10-fold than inner medium. But when 𝑘_cor=10⁻¹⁵ m², the drying time is increased to 200% nearly. The vapor pressure has a greater effect on the drying process; a lower vapor pressure results in greater pressure degradation. But the pressure effect is lower by the reduction of the cortex intrinsic permeability. For initial moisture effect, in the same intrinsic permeability, the drying time is almost the same. The initial moisture is a less effect when initial moisture is 0.4 and 0.3, especially cortex intrinsic permeability 𝑘_cor=10⁻¹⁵ m². The drying rate is little larger for 𝑆_𝑙0=0.5, and the drying rate is almost the same for 𝑆_𝑙0=0.4, 0.3. The effect of heat temperature is obviously in all simulation.

Nomenclature

𝐵:	Diagonal tensor
𝐷:	Diffusivity (m2 s−1)
𝐷eff:	Diffusion tensor (m2 s−1)
𝑔:	Gravity vector (m s−2)
ℎ:	Intrinsic averaged enthalpy (Jkg−1)
𝐼:	Water phase rate (kg s−1 m−3)
𝑘:	Intrinsic permeability (m2)
𝑘𝑟:	Relative permeability
𝑚:	Mass (kg)
𝑛:	Outer unit normal to the product
𝑃:	Pressure (Pa)
𝑃𝑐:	Capillary pressure (Pa)
𝑅:	Universal Gas constant (J kmol−1 K−1)
𝑆:	Saturation
𝑡:	Time (s)
𝑇:	Temperature (K)
𝑊:	Moisture content (in dry basis).

Greek Letters

Δ𝐻:	Latent of phase change (Jkg−1)
𝜆ef:	Effective thermal conductivity tensor (Wm−1k−1)
𝜇:	Viscosity (kgm−1s−1)
𝜌:	Density (kgm−3)
𝜔:	Vapor fraction.

Subscripts

𝑎:	Dry air
𝑔:	Gas
𝑙:	Liquid
𝑠:	Solid
𝑣:	Vapor
sat:	Vapor saturation
in:	Inner medium
cor:	Cortex.

Mathematical Operators

Δ:	Gradient operator
∇⋅:	Divergence operator.

Acknowledgment

This research was supported by the National Natural Science Foundation of China (Grant nos. 31000665 and nos. 51176027).