Drying as a process of removing moisture has been widely used in different kinds of biomass treatments, such as with biofuels (1), fruits and vegetables (2), milk (3–4), tea (5) and tobacco (6). Distinct dryers were developed and applied for biomassdrying based on a variety of treatment purposes. Among these dryers were pneumatic conveying dryers (7), tunnel dryers (8), conveyor belt dryers (9), fixedbed dryers (10), spoutfluidizedbed dryers (11) and rotary dryers. Among them, the rotary dryer is an effective equipment used in the tobacco industry for drying cut tobacco. In a rotary dryer, cut tobacco particles are moved, exposing them to hot air and the hot walls of the dryer, thus achieving a better drying efficiency. The physical and chemical properties of cut tobacco could be improved by optimizing the treatment conditions, such as wall temperature, air temperature, air flow rate, air humidity, and initial water content. However, the study of the detailed drying mechanisms of cut tobacco is difficult because of the complex heat and mass transfer between the twophase flow occurring in a rotary dryer (13,14,15).
To investigate the drying process in a rotary dryer, it is common to build mathematical models on the basis of experiments. Many empirical and semiempirical models have been studied by several researchers such as models of N
This study uses experimental data from L
Cut tobacco from the top leaves with a moisture content of 22.5 ± 0.1% (wet basis) was used in this study (28). The drying experiments were carried out in a batch rotary dryer (Xiamen Yingde Industry and Trade Co. Ltd., Fujian, China). The temperature of the inlet air was tested at five levels, 338.15 K, 358.15 K, 378.15 K, 398.15 K and 418.15 K respectively, while the average velocity of the air flow was kept at 0.067 ms^{−1}. During the drying process, the sampling interval was set from 30 s to 300 s as the drying process proceeded.
In order to simulate the mass transfer, the twofilm theory was adopted (29–30). We assumed that on the surface of cut tobacco particles, there existed a film of wet air, which kept a balance between water vapor density in the gas layer and moisture content in the cut tobacco particles. We also assumed that the convective mass transfer occurs as the primary process during drying, and the diffusion effect among cut tobacco particles was insignificant. Therefore, the mass transfer flux could be written as the product of the convective mass transfer coefficient and the driving force of mass transfer, which is expressed as the difference of water vapor density between the hot air and the gas film on the surface of the cut tobacco. The drying process of cut tobacco can be described as the following differential equation [1].
According to the A
In order to get the vapor density, the hot air in the rotary dryer is assumed to satisfy the ideal gas equation, and its temperature is equal to the wall temperature of the rotary dryer. The vapor density in hot air and the gas film are given as equation [3]:
In above equations, the relative humidities of hot air
The drying conditions and their corresponding equilibrium moisture content in tobacco on dry basis
Drying conditions  Tobacco equilibrium moisture content (%)  

Temperature (K)  Relative humidity (%)  
338.15  0.115  0.048 
358.15  0.050  0.035 
378.15  0.024  0.027 
398.15  0.012  0.022 
418.15  0.007  0.018 
In order to develop the mathematical model of heat and mass transfer, the equilibrium humidity of air (
Mathematical models of equilibrium moisture content.
Name of the model  Equation  

H 
(H 

M 
(MH 

M 
(MO 

C 
(CP) 

M 
(MCP) 

H 
(H 

In Table 2
Since cut tobacco is a kind of irregularshaped, elastic and compressed material and its particles contain a certain amount of water, the mass heat capacity of cut tobacco is complicated to express accurately. For simplification, the mass heat capacity of cut tobacco is assumed to be a linear combination of mass heat capacity of water and mass heat capacity of dry cut tobacco.
For the integrity of the heat transfer equation, the latent heat of vaporization [J·kg^{−1}] (38) and the specific heat capacity [J·kg^{−1}·K^{−1}] of water (39) are assumed as functions of cut tobacco temperature (K), and they can be expressed as:
For simplification of the mathematical model the mass heat capacity (J·kg^{−1}·K^{−1}) (40) of cut tobacco remains unchanged with temperature, and is set to
In summary, the drying process of cut tobacco in a rotary dryer, is governed by both the mass transfer equation and the heat transfer process. The convective mass and heat transfer coefficients are the critical factors of the mathematical model, which are determined by the operating conditions such as inlet flow rate of hot air, the volume of cut tobacco and others. It is difficult to directly obtain accurate values of the two coefficients by independent experiments, because the heat transfer and mass transfer are coupled. However, they could be fitted by the temperature and moisture curves of cut tobacco simultaneously by an optimization method.
Besides the classic heat and mass transfer model, the reaction engineering approach (REA) is an application of chemical reaction engineering principles to simulate the drying process. The only difference between the classic model and REA model is the mass transfer equation, and the heat transfer equations of both models are the same. The mass transfer equation of the REA model is written as equation [8] (28):
Herein,
Our target is to find the optimal value of heat and mass transfer coefficients by fitting moisture content and temperature of cut tobacco to experimental data for all temperature conditions simultaneously. From the standpoint of mathematics, it is a multiobjective nonlinear optimization problem, which can be defined as:
In general, an optimal solution for a multiobjective optimization problem cannot be found because all the objectives usually cannot be achieved at the same time. However, the weighted sum is a popular approach to obtain a P
In this method, the utilization of weight factor r is to combine multiobjectives into a single objective. So, the final objective function of our optimization problem can be written as:
In MATLAB a fourthorder R
This is a multiobjective nonlinear optimization problem, which means that often there is no optimal solution. However, it is important for the evaluation of each objective for guiding practical work. The quality of the fitted curves was evaluated by using the Root Mean Square Error (RMSE) and the Mean Relative Deviation (MRD) (21, 43). The equation for RMSE and MRD can be written as:
The equilibrium moisture content model (EMC) provided the relationship between equilibrium moisture content, the temperature of the cut tobacco, and the ambient humidity. It was not only used to describe the isothermal adsorptiondesorption process but was also employed to characterize the nonisothermal process when the temperature difference between air and cut tobacco was not significant enough. Since the heat and mass transfer equations were coupled, the accurate prediction of the equilibrium humidity
Table 3 shows the estimated coefficients and effects of fitting six EMC models. When comparing the RMSE value, the models of M
Estimated coefficients and criteria for comparing EMC models for cut tobacco.
Parameter  H 
M 
M 
C 
M 
H 

A  3.5239  −0.9313  −0.0914  3397.00  1.63·10^{5}  22.6107 
B  3.0239  −516.0073  0.0006  32.1078  2.09·10^{4}  1.1462 
C  0.4210  0.4553  26.4425  
RMSE  0.0010  0.0004  0.0004  0.0029  0.0015  0.0018 
MRD  0.0554  0.0255  0.0172  0.1751  0.0940  0.1052 
Interestingly, these EMC models were previously only used below a temperature of 373.15 K (34), but all of the models still showed good effects on the simulation of the equilibrium drying state of cut tobacco in a more wider range of temperature conditions (338.15 K – 418.15 K).
The result showed that they still had good applicability in high temperatures and low humidity drying conditions. Moreover, the effect of the EMC model on the simulation of the heat and mass transfer model needs to be confirmed further.
Figure 1 shows the influence of weight factor r on the simulation results of temperature and moisture content during cut tobacco drying. Due to the fact that the M
With the decrease of weight factor r in all drying situations, the fitting curves of water content were improving at first, then becoming worse, but the fitting curves of temperature were improving. When the weight factor r was 0.1, the simulated temperature was close to the experimental values in most conditions, especially at 378.15 K and at higher temperatures. While the weight factor r was further reduced to 0.01, the fitting quality for the temperature rose but slowly, but the fitting quality for the moisture content declined.
Figure 2 shows the total RMSE and MRD over the weight factor r. As it drops from 0.9 to 0.01, the total RMSE gradually decreases at first, then accelerates and finally decreases slowly. The curve of RMSE shows an Sshaped trend as a whole. The reason that the RMSE mainly reflects the fitting results of the temperature data might be due to the magnitude of the temperature data being much higher than that of moisture content.
It has turned out that the smaller the weight factor was, the better the fitting results of the temperature curves have proved. In dependency of r, the transformation of MRD values was different from that of the respective RMSE values. When the weight factor r was 0.1, the value of MRD was approaching a minimum. As the value of r decreased from 0.1 to 0.01, the MRD value began to increase significantly. Combining the two evaluation indices, the best fitting result was obtained when the weight factor r was chosen to be 0.1. Figure 3 shows the variation of mass transfer coefficient
Various models of heat and mass transfer were set up based on different EMC models as a driving force of mass transfer. Figure 4 shows the fitting curves of moisture content and temperature for all mathematical models in various drying conditions, where the weight factor r was set to 0.1. The fitting results of H
Figure 5 shows the total RMSE and MRD values of different models. The total RMSE contains the sum of the absolute fitting deviation of temperature and moisture. The results show that the deviation of MO
As discussed above, the fit of the modified O
The results indicate that the fitting results of the heat and mass transfer coefficients not only depend on the fitting quality of EMC model but are also related to those of applicability and prediction.
Figure 6 shows the comparison between
The value of
Moreover, there was no significant correlation either between the values of RMSE and
In order to further compare the fitting characteristics of each model, the RMSE and MRD values for various temperature conditions were analyzed. The Hen/REA model had the highest overall RMSE_{T} value (Figure 7) compared to the six classic models. The values of the REA model were relatively small at 358.15 K and 378.15 K, which was lower than some classic models such as HEN/C, CP/C and MCP/C model. The RMSE of MH
Overall MRD_{T} values (Figure 7) of H
Six different EMC models were introduced to calculate the driving force of mass transfer, and a mathematical model of heat and mass transfer was developed to investigate the drying of cut tobacco in a batch rotary dryer. The heat and mass transfer coefficients of the model were determined by solving a multiobjective nonlinear optimization problem using a weight factor. The simulation results show that the weight factor r is an important parameter for fitting moisture content and temperature. When setting the r value to 0.1, both RMSE and MRD for the whole data of moisture content and temperature almost reached their lowest values at the same time. However, the value of the mass transfer coefficient reached its maximum and the value of heat transfer coefficient almost reached its minimum. When RMSE was used as a measure of deviation, the models of MO
The drying conditions and their corresponding equilibrium moisture content in tobacco on dry basis Xe (28).
Drying conditions  Tobacco equilibrium moisture content (%)  

Temperature (K)  Relative humidity (%)  
338.15  0.115  0.048 
358.15  0.050  0.035 
378.15  0.024  0.027 
398.15  0.012  0.022 
418.15  0.007  0.018 
Mathematical models of equilibrium moisture content.
Name of the model  Equation  

H 
(H 

M 
(MH 

M 
(MO 

C 
(CP) 

M 
(MCP) 

H 
(H 

Estimated coefficients and criteria for comparing EMC models for cut tobacco.
Parameter  H 
M 
M 
C 
M 
H 

A  3.5239  −0.9313  −0.0914  3397.00  1.63·10^{5}  22.6107 
B  3.0239  −516.0073  0.0006  32.1078  2.09·10^{4}  1.1462 
C  0.4210  0.4553  26.4425  
RMSE  0.0010  0.0004  0.0004  0.0029  0.0015  0.0018 
MRD  0.0554  0.0255  0.0172  0.1751  0.0940  0.1052 
j.cttr20200013.tab.004
Nomenclature  

Parameter in equilibrium model  
Specific heat capacity (J·kg^{−1}·K^{−1})  
Heat transfer coefficient (W·kg^{−1}·K^{−1})  
Mass transfer coefficient (m^{3}·kg^{−1}·s^{−1})  
Molecular weight (kg·mol^{−1})  
Mean relative deviation  
Vapor pressure (Pa)  
Universal gas constant (J·mol^{−1}·K^{−1})  
Relative humidity (1)  
Vapor density (kg^{−1}·m^{−3})  
Subscripts  
b  Hot gas 
cal  Calculated value 
e  Equilibrium zone 
exp  Experimental value 
t  Cut tobacco 
w  water 
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