CAPÍTULO III: RESULTADOS Y DISCUSIÓN
3.2 Resultados experimentales al ejecutar el percentrón multicapa en una sola computadora
The accuracy and robustness of the S-REA for heat treatment of wood under a constant heating rate is benchmarked by the experimental data of Younsi et al. (2007). The experimental details were reported in Kocaefe et al. (1990; 2007) and Younsi et al.
(2006b)and these are reviewed here for better understanding of the current approach.
A thermogravimetric analyser was used for the heat treatment of wood as shown in Kocaefe et al. (1990). Wood samples with dimensions of 0.035× 0.035 × 0.2 m were heat treated by suspending the samples on a balance with accuracy of 0.001 g. The heat treatment was conducted by exposing the samples to hot gas whose temperature was linearly increased according to the heating rate. The humidity of the gas was controlled by injection of steam into the second furnace placed under the main furnace (Younsi et al.,2006a,2007).
The samples were first heated to 120°C and held at this temperature for half an hour followed by heating under the preset heating rate (refer to Table 3.5) until the final temperature (also refer toTable 3.5) was achieved. During the heat treatment, the weight of the samples was recorded by the balance. In addition, the temperatures were measured by a T-type thermocouple placed inside the samples. The measurements indicated that the temperatures inside the samples were essentially uniform, perhaps due to the small size of the samples (Younsi et al.,2006b).
3.5.1 The mathematical modelling of wood heating using the S-REA
In the experiments reported by Younsi et al. (2006a; 2007), the subject of interest here, two-dimensional modelling with respect to x and y directions can be set up as the height of the sample (0.2 m) is much greater than the length (0.035 m) and width (0.035 m) of the sample. The mass balance of water in the liquid phase is written as (Chen,2007; Chong and Chen,1999; Putranto and Chen,2013; Zhang and Datta,2004): solids’ concentration (kg dry solids m−3) which can change if the structure is shrinking, I is the evaporation or wetting rate (kg H˙ 2O m−3s−1), I is>0 when evaporation occurs
Reaction engineering approach II: S-REA 149
locally, while the mass balance of water in the vapour phase is expressed as (Chen,2007;
Chong and Chen,1999; Putranto and Chen,2013; Zhang and Datta,2004):
∂Cv
The heat balance is represented as (Chen,2007; Chong and Chen,1999; Putranto and Chen,2013; Zhang and Datta,2004):
ρCp∂T where T is the sample temperature (K), k is thermal conductivity of sample (W m−1 K−1), ρ is the sample density (kg m−3) and Hv is the vaporisation heat of water (J kg−1).
The initial and boundary conditions for Equations (3.5.1) to (3.5.3) are:
t = 0, X = Xo, Cv = Cvo, T = To (initial condition, uniform initial concentrations
and temperature), (3.5.4)
x= 0, y= 0,d X
d x = 0, dCv
d x = 0,d T
d x = 0 (symmetry boundary), (3.5.5) x= L, y = L, −CsDwd X
d x = h(Tb− T ) (convective boundary for heat transfer). (3.5.8) Because the samples were dried uniformly from all directions and the lengths and widths were the same, the mass balance of water in liquid phase can be simplified into (Incropera and DeWitt,2002; Van der Sman,2003):
∂(CsX )
while the mass balance of water in vapour phase can be expressed as:
∂Cv
In addition, the heat balance can be represented as:
ρCp
Similarly to the convective and intermittent drying described in Sections 3.3 and 3.4, the internal evaporation rate (˙I), effective vapour diffusivity, tortuosity, solids concentration and porosity are evaluated using Equations 3.1.19, 3.2.1, 3.2.3, 3.2.4 and3.2.5, respectively.
The relative activation energy of heat treatment of wood is generated from the drying run in Case 2 (refer toTable 3.5) (Younsi et al.,2007). The activation energy during drying is evaluated using Equation (2.1.5) and divided with the equilibrium activation energy represented in Equation (2.1.7) to yield the relative activation energy as men-tioned in Equation (2.1.6). The relationship between the relative activation energy and average moisture content can be represented by simple mathematical equation obtained by the least-square method using Microsoft Excel (Microsoft Corp,2012). The relative activation energy can be represented as:
Ev For modelling using the S-REA here, the relative activation energy shown in Equation (3.5.12) is used, but the average moisture content X in Equation (3.5.12) is substituted by the local moisture content (X) as the REA represents the local evaporation rate here, instead of the global drying rate. In order to incorporate the effect of linearly increased gas temperature, the equilibrium activation energy shown in Equation (2.1.7) implements the corresponding gas temperature and humidity during heat treatment. In addition, the linearly increased gas temperature is used in Equation (3.5.8).
In order to yield the spatial profiles of moisture content, water vapour concentration and temperature in the heat treatment of wood, the mass and heat balances shown in Equations (3.5.9) to (3.5.11), in conjunction with the initial and boundary conditions represented in Equations (3.5.4) to (3.5.8) and the relative activation energy shown by Equation (3.5.12), are solved by method of lines (Chapra,2006; Constantinides,1999).
By this method, the partial differential equations are transformed into a set of ordinary differential equations with respect to time by firstly discretising the spatial derivatives.
The ordinary differential equations are then solved simultaneously by ode23s in Matlab (Mathworks Inc., 2012). The spatial derivative here is discretised into 10 increments;
application of 100 increments has been conducted and there is no difference in the profiles observed, as shown inFigure 3.18. No shrinkage is incorporated in the modelling, as Younsi et al. (2006b) indicated that the ratio between final and initial dimension is around 0.96. Similarly, the modelling implemented by Younsi et al. (2006a,b,2007) did not incorporate the shrinkage effect.
The average moisture content of wood during heat treatment is evaluated by:
X =
The profiles of average moisture content and temperature are then validated towards the experimental data of Younsi et al. (2007).
Reaction engineering approach II: S-REA 151
0.14
0.12
0.1
0.08
0.06
0.04
0.02
00 0.5 1 1.5
t(s)
Average moisture content (kg water/kg dry solids)
2 2.5
×104 3 S-REA 100 increments S-REA 10 increments Data
Figure 3.18 Profiles of average moisture content during heat treatment in Case 2 (refer toTable 3.5) solved by the method of lines using 10 and 100 increments.
3.5.2 The results of modelling wood heating using the S-REA
The S-REA is used to model the heat treatment of wood under a constant heating rate and the results of modelling are presented inFigures 3.19to3.27. The REA is implemented to model the local evaporation or condensation term and coupled with a system of equations in order to yield the spatial profiles of moisture content, water vapour concentration and temperature. It is noted that if locally there is no vacant pore space which is connected to other pores or channels, the internal mass transfer area is zero, hence the REA term is zero. In this study, the internal mass transfer coefficient (hm,in) shown in Equation (3.1.17) is chosen to be 0.001 m s−1. Application of hm,inhigher than 0.001 m s−1does not yield any noticeable differences in the profiles of moisture content and temperature profiles.
As mentioned before, no method has been presented anywhere in the literature to measure pore liquid diffusivity, and liquid diffusivity is obtained by numerical sen-sitivity to match the prediction with the experimental data of moisture content and temperature. Interestingly, in this study, for all cases the profiles of moisture content and temperature are independent of the liquid diffusivity value. Further explanation about this phenomenon is presented next.
For Case 1 (refer toTable 3.5), the results of modelling are presented inFigures 3.19 to3.22. The varied values of the liquid diffusivity in the range of 1 × 10−8 to 1 × 10−30m2s−1have been used and there are no noticeable differences in the profiles of moisture content and temperature as shown inFigures 3.19and3.20. This may indicate
0.12
Dw = 1e-8 m2/s Dw = 1e-12 m2/s Dw = 1e-18 m2/s Dw = 1e-24 m2/s Dw = 1e-30 m2/s Data
0.1
0.08
Average moisture content (kg water/kg dry solids)
0.06
0.04
0.02
00 1 2 3 4
t(s) × 104
5 6 7
Figure 3.19 Effect of liquid diffusivity on profiles of the moisture content during heat treatment in Case 1 (refer toTable 3.5).
500
450
400
350
300
2500 1 2 3 4
t(s)
5 6
× 104 7
Average moisture content (kg water/kg dry solids)
Dw = 1e-8 m2/s Dw = 1e-12 m2/s Dw = 1e-18 m2/s Dw = 1e-24 m2/s Dw = 1e-30 m2/s Data
Figure 3.20 Effect of liquid diffusivity on profiles of temperature during heat treatment in Case 1 (refer toTable 3.5).
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1 00
Average moisture content (kg water/kg dry solids)
0.12
0.1
0.08
0.06
0.04
0.02 0.14
2 3 4 5
t(s)
6 7
× 104 Model Data
Figure 3.21 Profiles of average moisture content during heat treatment in Case 1 (refer to Table 3.5).
1 2500
Temperature (K)
450
350 400
300 500
2 3 4 5
t(s)
6 7
× 104 Model Data
Figure 3.22 Profiles of temperature during heat treatment in Case 1 (refer toTable 3.5).
0.5 00
Average moisture content (kg water/kg dry solids)
0.12
0.1
0.08
0.06
0.04
0.02 0.14
1 1.5 2 2.5
t(s)
3
× 104 Model Data
Figure 3.23 Profiles of average moisture content during heat treatment in Case 2 (refer to Table 3.5).
that the modelling of heat treatment in Case 1 (refer toTable 3.5) is independent of the liquid diffusivity so this can be neglected and the modelling will only involve the vapour diffusion and evaporation/condensation. The phenomenon could be due to the initial moisture content being relatively low and the heating of wood implemented at a relatively high temperature.
By ignoring the liquid diffusion term on the mass balance of liquid water (refer to Equation3.5.9), a good agreement between the predicted and experimental data of moisture content and temperature is shown inFigures 3.23and3.24as well as confirmed by R2 and RMSE indicated inTable 3.6. Benchmarks against modelling implemented by Younsi et al. (2007) indicate that the S-REA yields comparable or even better results.
Therefore, it can be said that the S-REA models the heat treatment of wood in Case 1 well (refer toTable 3.5).
Figures 3.23to3.27show the results of modelling of heat treatment of wood in Case 2 (refer toTable 3.5). Similarly to Case 1 (refer toTable 3.5), the varied values of the liquid diffusivity in the range of 1× 10−8to 1× 10−30m2s−1have been used and there is no noticeable effect on the profiles of moisture content and temperature. By ignoring the liquid diffusion term in the mass balance of liquid water (refer to Equation3.5.9), a good agreement between the predicted and experimental data of moisture content and temperature is shown inFigures 3.23and3.24and confirmed by R2of 0.995 and 0.997 for moisture content and temperature, respectively and RMSE of lower than 0.005
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Table 3.6 R2and RMSE of modelling of heat treatment of wood under a constant heating rate using the S-REA.
Case R2for X R2for T RMSE for X RMSE for T
1 0.988 0.992 0.004 4.765
2 0.995 0.997 0.003 3.287
0.5 2500
Temperature (K)
450
350 400
300 500
1 1.5 2 2.5
t(s)
3 3.5
× 104 Model Data
Figure 3.24 Profiles of temperature during heat treatment in Case 2 (refer toTable 3.5).
and 3.287 for moisture content and temperature, respectively. The S-REA describes the profiles of moisture content and temperature very well. Benchmarks against modelling implemented by Younsi et al. (2007) reveal that the S-REA yields better agreement with the experimental data from moisture content.
The spatial profiles of moisture content, water vapour concentration and temperature are presented inFigures 3.25–3.27. The distribution of moisture content is not determined by liquid diffusivity. This may suggest that liquid diffusivity can be neglected, so the drying process is not governed by liquid diffusion as the initial moisture content is relatively low and temperature is relatively high.Figure 3.25indicates that the moisture content of the inner part of the samples is higher than that of the outer part, which indicates the moisture migrates outwards during drying.
Similarly, as shown inFigure 3.26, the water vapour concentration of the inner part of the samples is higher than that of the outer part. This could be because of the relatively
0.14
Moisture content (kg water/kg dry solids) 0.02
00
Figure 3.25 Profiles of spatial moisture content during heat treatment in Case 2 (refer to Table 3.5).
Figure 3.26 Profiles of spatial water vapour concentration during heat treatment in Case 2 (refer toTable 3.5).
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Figure 3.27 Profiles of spatial temperature during heat treatment in Case 2 (refer toTable 3.5).
high initial porosity of the samples, which allows evaporation at the core of the samples.
The water vapour seems to migrate outwards and at the surface it is removed by the gas. The water vapour concentration increases until a heating time of 15 500 s, followed by a decrease until the end of drying. The initial increase could be because the initial moisture content is still relatively high at the beginning of drying but as the heating pro-gresses, moisture content decreases and lower water vapour is generated. As shown in Figure 3.27, the temperature distribution of the samples is essentially uniform, in agree-ment with the observations of Younsi et al. (2007).
It can be observed that the S-REA can model the heat treatment of wood under constant heating rate very well for all cases investigated. The liquid diffusion term can be neglected so that the model for heat treatment of wood under a constant heating rate may be simplified as follows:
∂(CsX )
The S-REA has also the advantages of yielding profiles of water vapour concentration during the process. This enables better understanding of the transport phenomena during
the process. It can be said that the S-REA is an effective multiphase approach to modelling the heat treatment of wood.