1. bookVolume 29 (2020): Edition 3 (December 2020)
Détails du magazine
License
Format
Magazine
eISSN
2719-9509
Première parution
01 Jan 1992
Périodicité
4 fois par an
Langues
Anglais
Accès libre

Simulation of Heat and Mass Transfer of Cut Tobacco in a Batch Rotary Dryer by Multi-Objective Optimization

Publié en ligne: 31 Dec 2020
Volume & Edition: Volume 29 (2020) - Edition 3 (December 2020)
Pages: 145 - 155
Reçu: 09 Apr 2020
Accepté: 09 Nov 2020
Détails du magazine
License
Format
Magazine
eISSN
2719-9509
Première parution
01 Jan 1992
Périodicité
4 fois par an
Langues
Anglais
INTRODUCTION

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 (34), tea (5) and tobacco (6). Distinct dryers were developed and applied for biomass-drying based on a variety of treatment purposes. Among these dryers were pneumatic conveying dryers (7), tunnel dryers (8), conveyor belt dryers (9), fixed-bed dryers (10), spout-fluidized-bed 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 two-phase 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 semi-empirical models have been studied by several researchers such as models of Newton et al. (16), Page et al. (17), Henderson and Pabis (18), Menges and Ertekin (19), Midilli et al. (20) and Hii et al. (21). These models lack clear physical interpretation, so they could not be employed over a wide range of drying conditions. They also could not predict the surface temperature of the material to be dried because only the moisture content was considered. Many researchers sought mathematical models coupled with heat and mass transfer to describe the evolution of both moisture content and temperature during the drying process. Xu and Pang developed a mathematical model to simulate the drying of wooden chips in a rotary dryer (22). Gu et al. studied the influence of moisture content and humidity on flexible filamentous particles in a counter-current cascading rotary dryer (23). Zhu et al. simulated the evolution of cut tobacco temperature and moisture by a developed heat and mass transfer model (24). These models based on the classic theory of heat and mass transfer can be found in a number of articles and books on drying technology. However, these models were not suitable to solve the problem of heat and mass transfer in a rotary drying process due to their lack of adaptable parameters. In addition, the reaction engineering approach (REA) is seen as a new approach to simulate the drying process of single particles and thin-layer materials, which is proposed by Chen and his coworkers (25,26,27). This method considered the drying process as a chemical reaction, and the kinetic parameters of the drying process were obtained through experiments. Li et al. designed a batch rotary dryer to study the drying characteristic of cut tobacco (28). The REA model was well validated by experimental data at various wall temperatures, which illustrated that the REA model could well simulate the drying characteristics of cut tobacco. However, the calculation process of the REA model was complicated. It was difficult to evaluate whether the heat transfer coefficient or the mass transfer coefficient is optimal.

This study uses experimental data from Li et al. (28), it focuses on the development of a simpler and more easily comparable heat and mass transfer model to simulate the drying process of cut tobacco in a batch rotary dryer. Six different models of equilibrium moisture content were selected to calculate the driving force of mass transfer, and a mathematical model of heat and mass transfer was numerically solved by using multi-objective nonlinear optimization. The multi-objective nonlinear problem of heat and mass transfer coefficients was optimized by employing a weight factor. Moreover, two kinds of evaluation methods were used to investigate the fitting results of different equilibrium/classic models.

MODEL DEVELOPMENT

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.

Heat and mass transfer model

In order to simulate the mass transfer, the two-film theory was adopted (2930). 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]. dXdt=hmAm(ρbρe) {{dX} \over {dt}} = {h_m}{A_m}\left( {{\rho _b} - {\rho _e}} \right) where X is the moisture content of cut tobacco on a dry basis, hm Am = km is the product of the convective mass transfer coefficient hm and the mass transfer surface area Am, ρb is water vapor density in an equilibrium state of the gas film. Subscript b denotes the hot air and e denotes equilibrium zone.

According to the Antoine equation (31), the vapor pressure in hot air and in a film can be expressed as the product of the saturated vapor pressure and the relative humidity, which are given as equation [2]: pi=1.1939×1010exp(3826.36Ti45.47)RHi,i=b,e {p_i} = 1.1939 \times {10^{10}}\,\exp \left( { - {{3826.36} \over {{T_i} - 45.47}}} \right)R{H_i},\,i = b,e where ρb is the vapor pressure in hot air, RHb represents the humidity in hot gas, T denotes the temperature of cut tobacco, ρe is the vapor pressure in the gas film, and RHe represents the equilibrium humidity in the gas film.

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]: ρi=piMwRTi=1.1939×1010exp(3826.36Ti45.47)MwRHiRTi,i=b,e {\rho _i} = {{{p_i}{M_w}} \over {R{T_i}}} = {{1.1939 \times {{10}^{10}}\,\exp \left( { - {{3826.36} \over {{T_i} - 45.47}}} \right){M_w}R{H_i}} \over {R{T_i}}},\,i = b,e where R is the universal gas constant, and Mw is the molecular weight of water.

In above equations, the relative humidities of hot air RHb in each of the operation conditions are shown in Table 1.

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

In order to develop the mathematical model of heat and mass transfer, the equilibrium humidity of air (RHe) on the surface of cut tobacco (the same as the equilibrium humidity in the gas film) at different conditions are required, the model was used to calculate the driving force of mass transfer. According to Table 1, the equilibrium moisture content model (EMC) was introduced to establish a relationship between equilibrium temperature and moisture content of cut tobacco and equilibrium humidity of the air. In this study, six popular models were chosen to study the relative humidity in an equilibrium state. The models were Henderson (32), Modified Henderson (33), Modified Oswin (34), Chung and Pfost (35), Modified Chung-Pfost (36) and Halsey (37). Each model not only could be used for moisture content as a function of relative humidity and temperature, but also for relative humidity as a function of moisture content and temperature. The six models are summarized in Table 2 (see page 148).

Mathematical models of equilibrium moisture content.

Name of the model Equation
Henderson (Hen) RHe=1exp(ATXeB) R{H_e} = 1 - \exp \left( { - ATX_e^B} \right)
Modified Henderson (M-Hen) RHe=1exp(A(T+B)Xe1/c) R{H_e} = 1 - \exp \left( { - A\left( {T + B} \right)X_e^{1/c}} \right)
Modified Oswin (M-Osw) RHe=11+(A+BTXe)1c R{H_e} = {1 \over {1 + {{\left( {{{A + BT} \over {{X_e}}}} \right)}^{{1 \over c}}}}}
Chung-Pfost (C-P) RHe=exp(ATexp(BXe)) R{H_e} = \exp \left( { - {A \over T}\exp \left( { - B{X_e}} \right)} \right)
Modified Chung-Pfost (M-C-P) RHe=exp(AT+Bexp(CXe)) R{H_e} = \exp \left( { - {A \over {T + B}}\exp \left( { - C{X_e}} \right)} \right)
Halsey (Hal) RHe=exp(ATXeB) R{H_e} = \exp \left( { - {A \over {TX_e^B}}} \right)

In Table 2 Xe is the equilibrium moisture content of cut tobacco corresponding to the equilibrium state, RHe is the equilibrium humidity of air (gas layer), T is the temperature of cut tobacco. A, B and C are parameters of each model. On the other hand, the heat transfer equation of the drying process is simple to describe since the temperature change of cut tobacco can be attributed to two main factors. The one is convective heat transfer from the hot air, the other is endothermic vaporization of water during the drying of cut tobacco. According to the energy conservation equation, the general heat transfer equation based on the experimental date could be written as: (XCp,w+Cp,t)dTdt=hAm(TbT)+ΔHwdXdt \left( {X{C_{p,w}} + {C_{p,t}}} \right){{dT} \over {dt}} = h{A_m}\left( {{T_b} - T} \right) + \Delta {H_w}{{dX} \over {dt}} where T is the temperature of cut tobacco, Tb is the bulk air temperature, hAm = kh denotes the convective heat transfer coefficient combining with inseparable heat transfer area, ΔHw is the latent heat of vaporization of water, Cp,w is the specific heat capacity of the water and Cp,t is the specific heat capacity of cut tobacco. In equation [4], the transfer of heat by conduction is neglected due to the fact that the particles of cut tobacco were fully subjected to the hot air in the rotary dryer, even in a packed state.

Since cut tobacco is a kind of irregular-shaped, 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: ΔHw=2891833(1T647.13)0.321 \Delta {H_w} = 2891833{\left( {1 - {T \over {647.13}}} \right)^{0.321}} Cp,w=1.459×106T41.971×103T3+1.005×T2228.7T+23750 {C_{p,w}} = 1.459 \times {10^{ - 6}}{T^4} - 1.971 \times {10^{ - 3}}{T^3} + 1.005 \times {T^2} - 228.7T + 23750

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 Cp,t=1.4286×103 {C_{p,t}} = 1.4286 \times {10^3}

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): mtdXdt=hmA[ ρv,sat(Tt)exp(ΔEvRTt)ρv,b] {m_t} {{dX} \over {dt}} = - {h_m}A\left[ {{\rho _{v,sat}}\left( {{T_t}} \right)\exp \left( { - {{\Delta {E_v}} \over {R{T_t}}}} \right) - {\rho _{v,b}}} \right]

Herein, mt (kg) is the mass of dried cut tobacco, t is the drying time (s), X (kg·kg−1) is the cut tobacco's moisture content on dry basis, hm (m·s−1) is the mass transfer coefficient, A (m2) is the surface area of cut tobacco for mass transfer, ρv,sat (kg·m−3) is the saturated vapor concentration corresponding to the cut tobacco temperature Tt (K), ρv,b (kg·m−3) is the vapor concentration of bulk drying air, and ΔEv (J·mol−1) is the apparent additional activation energy. More details of the REA model can be found in the literature (28). The simulation results of the REA model are also discussed in this study.

Calculation method

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 multi-objective nonlinear optimization problem, which can be defined as: minimize{f(hAm,hmAm)=j=1Mi=1N(Xcal,i,jXexp,i,jXexp,i,j)g(hAm,hmAm)=j=1Mi=1N(Tcal,i,jTexp,i,jTexp,i,j)22subjectto{(XcalCp,w+Cp,t)dTcaldt=hAm(TbTcal)+ΔHwdXcaldt,Tcal(0)=Texp(0)dXcaldt=hmAm(ρbρe),Xcal(0)=Xexp(0)Tb=338.15,358.15,378.15,398.15,418.15 \eqalign{ & {\rm{minimize}}{\left\{ {\matrix{ {f\left( {h{A_m},{h_m}{A_m}} \right) = \sum\limits_{j = 1}^M {\sum\limits_{i = 1}^N {\left( {{{{X_{{\bf{cal}},i,j}} - {X_{{\bf{exp}},i,j}}} \over {{X_{{\bf{exp}},i,j}}}}} \right)} } } \hfill \cr {g\left( {h{A_m},{h_m}{A_m}} \right) = \sum\limits_{j = 1}^M {\sum\limits_{i = 1}^N {{{\left( {{{{T_{{\bf{cal}},i,j}} - {T_{{\bf{exp}},i,j}}} \over {{T_{{\bf{exp}},i,j}}}}} \right)}^2}} } } \hfill \cr } } \right.^2} \cr & {\rm{subject}}\,{\rm{to}}\left\{ {\matrix{ {\left( {{X_{{\rm{cal}}}}{C_{p,w}} + {C_{p,t}}} \right){{d{T_{{\bf{cal}}}}} \over {dt}} = h{A_m}\left( {{T_b} - {T_{{\rm{cal}}}}} \right) + } \hfill \cr {\Delta {H_w}{{d{X_{{\bf{cal}}}}} \over {dt}},{T_{cal}}(0) = {T_{\exp }}(0)} \hfill \cr {{{d{X_{{\bf{cal}}}}} \over {dt}} = {h_m}{A_m}\left( {{\rho _b} - {\rho _e}} \right),{X_{{\rm{cal}}}}(0) = {X_{{\rm{exp}}}}(0)} \hfill \cr {{T_b} = 338.15,358.15,378.15,398.15,418.15} \hfill \cr } } \right. \cr} where M is the total number of experiments, N indicates the total amount of data in each experiment, Xcal,i,j is the predicted value of moisture content, Xexp,i,j is the experimental value of moisture content, Tcal,i,j is the predicted value of tobacco temperature, Xexp,i,j is the experimental value of tobacco temperature, the subscripts i and j are the number of data points on each trial and the number of test conditions, respectively.

In general, an optimal solution for a multi-objective 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 Pareto optimal solution (4142).

In this method, the utilization of weight factor r is to combine multi-objectives into a single objective. So, the final objective function of our optimization problem can be written as: minimizej=1Mi=1N(rXcal,i,jXexp,i,jXexp,i,j)2+j=1Mi=1N((1r)Tcal,i,jTexp,i,jTexp,i,j)2 {\rm{minimize}}\sum\limits_{j = 1}^M {\sum\limits_{i = 1}^N {{{\left( {r{{{X_{{\rm{cal}},i,j}} - {X_{\exp ,i,j}}} \over {{X_{\exp ,i,j}}}}} \right)}^{2}}} } + \sum\limits_{j = 1}^M {\sum\limits_{i = 1}^N {{{\left( {({1} - r){{{T_{{\rm{cal}},i,j}} - {T_{\exp ,i,j}}} \over {{T_{\exp ,i,j}}}}} \right)}^2}} } where r is the weight factor varying from zero to one.

In MATLAB a fourth-order Runge-Kutta method was used to integrate the system of differential equations [1] and [4] and that the non-linear optimization problem was solved by an iterative algorithm. The multi-objective nonlinear problem of heat and mass transfer coefficients was optimized by employing the weight factor r.

Model evaluation

This is a multi-objective 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: RMSEX=i=1N(Xexp,iXcal,i)2N,RMSET=i=1N(Texp,iTcal,i)2NRMSEtotal=i=1N(Xexp,iXcal,j)2+i=1N(Texp,iTcal,i)22N \matrix{ {RMS{E_X} = \sqrt {{{{{\sum\limits_{i = 1}^N {\left( {{X_{\exp ,i}} - {X_{{\rm{cal}},i}}} \right)} }^2}} \over N},\,} RMS{E_T} = \sqrt {{{{{\sum\limits_{i = 1}^N {\left( {{T_{\exp ,i}} - {T_{{\rm{cal}},i}}} \right)} }^2}} \over N}} } \hfill \cr {RMS{E_{total}} = \sqrt {{{{{\sum\limits_{i = 1}^N {\left( {{X_{\exp ,i}} - {X_{{\rm{cal}},j}}} \right)} }^2} + {{\sum\limits_{i = 1}^N {\left( {{T_{\exp ,i}} - {T_{{\rm{cal}},i}}} \right)} }^2}} \over {2N}}} } \hfill \cr } MRDX=1Ni=1N|Xexp,iXcal,i|Xexp,i,MRDT=1Ni=1N|Texp,iTcal,i|Texp,iMRDtotal=MRDX+MRDT2 \matrix{ {MR{D_X} = {1 \over N}\sum\limits_{i = 1}^N {{{\left| {{X_{\exp ,i}} - {X_{{\rm{cal}},i}}} \right|} \over {{X_{\exp ,i}}}}} ,MR{D_T} = {1 \over N}\sum\limits_{i = 1}^N {{{\left| {{T_{\exp ,i}} - {T_{{\rm{cal}},i}}} \right|} \over {{T_{\exp ,i}}}}} } \hfill \cr {MR{D_{total}} = {{{MRD_X} + MR{D_T}} \over 2}} \hfill \cr } where the subscript X represents moisture content, subscript T is temperature and subscript total indicates moisture content and temperature together.

RESULTS AND DISCUSSION
Fitting of EMC models

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 adsorption-desorption process but was also employed to characterize the non-isothermal 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 RHe not only affected the predicted moisture content of cut tobacco during the drying process but also influenced the predicted temperature on the tobacco surface.

Table 3 shows the estimated coefficients and effects of fitting six EMC models. When comparing the RMSE value, the models of Modified Henderson and Modified Oswin shared the best value, those of Henderson model and Modified Chung-Pfost model were in the middle, and Chung-Pfost model had the least best fit. When comparing the MRD value, Modified Oswin model slightly outperformed the Modified Henderson model. By combining the two evaluation methods, the results indicated that the Modified Oswin model was the best, and the Chung-Pfost model was the worst.

Estimated coefficients and criteria for comparing EMC models for cut tobacco.

Parameter Henderson Modified Henderson Modified Oswin Chung-Pfost Modified Chung-Pfost Halsey
A 3.5239 −0.9313 −0.0914 3397.00 1.63·105 22.6107
B 3.0239 −516.0073 0.0006 32.1078 2.09·104 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.

The influence of factor r on the fitting result

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 Modified Henderson (M-Hen) model gave the best result on fitting among the six EMC models, this model was selected. Comparing the calculated data with the experimental data, it can be seen that the weight factor r had an obvious influence on the fitting results.

Figure 1

Comparison between experimental data and simulations given by M-Hen/Classic model at varying weight factor r values.

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 S-shaped 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.

Figure 2

Comparison between RMSE and MRD at various weight factor r values ranging from 0.9 to 0.01. Model: M-Hen/Classic model.

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 km and heat transfer coefficient kh with respect to weight factor r. With the decrease of r from 0.9 to 0.01, the value of kh dropped slowly at first, then rose, and finally declined slowly. However, the value of km steadily increased as the coefficient r decreased from 0.9 to 0.1. When the weight factor r was at 0.1, km reached its maximum at 0.07 m3·kg−1·s−1, and kh is 31.4 W·kg−1·K−1. These heat and mass transfer coefficients could also be considered as the optimal values.

Figure 3

Coefficients of km and kh at different weight factor r values in the range of 0.9 to 0.01. Model: M-Hen/Classic model.

The influence of heat and mass transfer models

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 Hen/REA and equilibrium /C models were acceptable, where “/C” following a model name indicates the use of the classical heat and mass transfer model. There were some differences between Hen/REA and classic models. The fitting results of the Hen/REA model were better in the lower temperature range from 338.15 K to 378.15 K, but became worse when the drying temperature increased to 398.15 K and higher. The fitting of moisture content and temperature for the classic models were best at 378.15 K. The larger the temperature difference from 378.15 K, the least good fitting results of classic models were obtained. Comparing the temperature curves of Hen/REA and the classic models, the fitting of temperature of classic models was better than that of the Hen/REA model when the moisture content of cut tobacco was in the range of 9 – 15%, but the Hen/REA model was better suited for a moisture content less than 5%.

Figure 4

Comparison between experimental data and simulations given by different models.

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 M-Osw/C is the smallest, followed by the M-Hen/C model, and those of Hen/REA model are the largest. Since the absolute deviation of temperature was significantly higher than that of moisture, the total RMSE value was mainly affected by temperature. As mentioned above, the temperature fitting results of the Hen/REA model were inferior to those of classic models. Therefore, the total RMSE values of the classic models were relatively lower. To eliminate the influence of magnitude, the MRD value displayed better comparability for estimating the fit of each model. It was found that the total MRD value of the M-Hen/C model was the lowest, while those of M-Osw/C and C-P/C models were higher with a poorer fitting quality. Moreover, the MRD value of the HEN/REA model was intermediate among all models, unlike the comparison of RMSE results.

Figure 5

Comparison between RMSE and MRD values under different heat and mass transfer models.

As discussed above, the fit of the modified Oswin model was the best with the lowest RMSE and MRD values. However, the MRD value of the M-Osw/C model was relatively high among the models in spite of its lowest RMSE value. For models of M-Hen/C and C-P/C, the fitting results of the EMC model were consistent with that of the heat-mass transfer models.

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 km and kh values that were optimized for numerous models. The value of km calculated by Hen/REA model was significantly lower than the one for the classic models, but the value of kh was significantly higher than the one for the classic models. Compared with the classic models, the HAL/C model obtained the largest km value which was 0.113 m3·kg−1·s−1.

Figure 6

Coefficients of km and kh for different heat and mass transfer models.

The value of kh in each model was similar in the range of 31.4 W·kg−1·K−1 – 31.7 W·kg−1·K−1.

Moreover, there was no significant correlation either between the values of RMSE and km or between the values of RMSE and kh.

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 RMSET 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, C-P/C and M-C-P/C model. The RMSE of M-Hen/C model was significantly lower than that of other models at 398.15 K. The HEN/REA model had higher RMSEX values than the classic models. Among the classic models, the RMSE value of M-Hen/C was the lowest. The RMSE values of the HEN/REA models were significantly lower than those of the classic models at 338.15 K and 358.15 K.

Figure 7

Comparison of RMSE and MRD values of models for various drying conditions.

Overall MRDT values (Figure 7) of Hen/REA model were higher than the ones in each classic model. Among the six classic models, the values of M-Osw/C and Hal/C models were relatively lower. The MRDT value of the M-Hen/C model was significantly lower than in other models at 398.15 K. Comparing the overall MRDX values, those of the Hen/REA model were only higher than those of the M-Hen/C model. Specifically, the MRD values of Hen/REA models were obviously lower than the values of the classic models at 338.15 K and 358.15 K, but higher at other temperatures. The M-Hen/C model had the lowest MRD value at all temperatures among the classic models. The results demonstrate that the M-Hen/C model had the best fitting on moisture content.

CONCLUSIONS

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 multi-objective 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 M-Osw/C and M-Hen/C fitted better than the models of Hen/C, Hal/C, C-P/C and M-C-P/C, but the REA model was inferior to each equilibrium/classic model. When MRD was used as a measure of deviation, the M-Hen/C model was obviously superior to other equilibrium/classic models and the REA model. For all the six equilibrium/classic models the fit was better for moisture content than for temperature. This study demonstrates an understanding of heat and mass transfer in the tobacco drying process and it provides a theoretical framework to predict temperature and moisture for various drying conditions.

Figure 1

Comparison between experimental data and simulations given by M-Hen/Classic model at varying weight factor r values.
Comparison between experimental data and simulations given by M-Hen/Classic model at varying weight factor r values.

Figure 2

Comparison between RMSE and MRD at various weight factor r values ranging from 0.9 to 0.01. Model: M-Hen/Classic model.
Comparison between RMSE and MRD at various weight factor r values ranging from 0.9 to 0.01. Model: M-Hen/Classic model.

Figure 3

Coefficients of km and kh at different weight factor r values in the range of 0.9 to 0.01. Model: M-Hen/Classic model.
Coefficients of km and kh at different weight factor r values in the range of 0.9 to 0.01. Model: M-Hen/Classic model.

Figure 4

Comparison between experimental data and simulations given by different models.
Comparison between experimental data and simulations given by different models.

Figure 5

Comparison between RMSE and MRD values under different heat and mass transfer models.
Comparison between RMSE and MRD values under different heat and mass transfer models.

Figure 6

Coefficients of km and kh for different heat and mass transfer models.
Coefficients of km and kh for different heat and mass transfer models.

Figure 7

Comparison of RMSE and MRD values of models for various drying conditions.
Comparison of RMSE and MRD values of models for various drying conditions.

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
Henderson (Hen) RHe=1exp(ATXeB) R{H_e} = 1 - \exp \left( { - ATX_e^B} \right)
Modified Henderson (M-Hen) RHe=1exp(A(T+B)Xe1/c) R{H_e} = 1 - \exp \left( { - A\left( {T + B} \right)X_e^{1/c}} \right)
Modified Oswin (M-Osw) RHe=11+(A+BTXe)1c R{H_e} = {1 \over {1 + {{\left( {{{A + BT} \over {{X_e}}}} \right)}^{{1 \over c}}}}}
Chung-Pfost (C-P) RHe=exp(ATexp(BXe)) R{H_e} = \exp \left( { - {A \over T}\exp \left( { - B{X_e}} \right)} \right)
Modified Chung-Pfost (M-C-P) RHe=exp(AT+Bexp(CXe)) R{H_e} = \exp \left( { - {A \over {T + B}}\exp \left( { - C{X_e}} \right)} \right)
Halsey (Hal) RHe=exp(ATXeB) R{H_e} = \exp \left( { - {A \over {TX_e^B}}} \right)

Estimated coefficients and criteria for comparing EMC models for cut tobacco.

Parameter Henderson Modified Henderson Modified Oswin Chung-Pfost Modified Chung-Pfost Halsey
A 3.5239 −0.9313 −0.0914 3397.00 1.63·105 22.6107
B 3.0239 −516.0073 0.0006 32.1078 2.09·104 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.cttr-2020-0013.tab.004

Nomenclature
A, B, C Parameter in equilibrium model
Cp Specific heat capacity (J·kg−1·K−1)
hAm Heat transfer coefficient (W·kg−1·K−1)
hm Am Mass transfer coefficient (m3·kg−1·s−1)
M Molecular weight (kg·mol−1)
MRD Mean relative deviation
p Vapor pressure (Pa)
R Universal gas constant (J·mol−1·K−1)
RH Relative humidity (1)
ρ Vapor density (kg−1·m−3)
Subscripts
b Hot gas
cal Calculated value
e Equilibrium zone
exp Experimental value
t Cut tobacco
w water

Stenström, S.: Drying of Biofuels from the Forest – A Review; Dry. Technol. 35 (2017) 1167–1181. DOI: 10.1080/07373937.2016.1258571 StenströmS. Drying of Biofuels from the Forest – A Review Dry. Technol. 35 2017 1167 1181 10.1080/07373937.2016.1258571 Ouvrir le DOISearch in Google Scholar

Yang, F., M. Zhang, A.S. Mujumdar, Q. Zhong, and Z. Wang: Enhancing Drying Efficiency and Product Quality Using Advanced Pretreatments and Analytical Tools – An Overview; Dry. Technol. 36 (2018) 1824–1838. DOI: 10.1080/07373937.2018.1431658 YangF. ZhangM. MujumdarA.S. ZhongQ. WangZ. Enhancing Drying Efficiency and Product Quality Using Advanced Pretreatments and Analytical Tools – An Overview Dry. Technol. 36 2018 1824 1838 10.1080/07373937.2018.1431658 Ouvrir le DOISearch in Google Scholar

Lin, S.X.Q. and X.D. Chen: The Reaction Engineering Approach to Modelling the Cream and Whey Protein Concentrate Droplet Drying; Chem. Eng. Process. 46 (2007) 437–443. DOI: 10.1016/j.cep.2006.05.021 LinS.X.Q. ChenX.D. The Reaction Engineering Approach to Modelling the Cream and Whey Protein Concentrate Droplet Drying Chem. Eng. Process. 46 2007 437 443 10.1016/j.cep.2006.05.021 Ouvrir le DOISearch in Google Scholar

Patel, K.C. and X.D. Chen: Sensitivity Analysis of the Reaction Engineering Approach to Modeling Spray Drying of Whey Proteins Concentrate; Dry. Technol. 26 (2008) 1334–1343. DOI: 10.1080/07373930802331019 PatelK.C. ChenX.D. Sensitivity Analysis of the Reaction Engineering Approach to Modeling Spray Drying of Whey Proteins Concentrate Dry. Technol. 26 2008 1334 1343 10.1080/07373930802331019 Ouvrir le DOISearch in Google Scholar

Temple, S.J. and A.J.B. van Boxtel: Equilibrium Moisture Content of Tea; J. Agric. Eng. Res. 74 (1999) 83–89. DOI: 10.1006/jaer.1999.0439 TempleS.J. van BoxtelA.J.B. Equilibrium Moisture Content of Tea J. Agric. Eng. Res. 74 1999 83 89 10.1006/jaer.1999.0439 Ouvrir le DOISearch in Google Scholar

Li, B., W. Zhu, P. Wang, D. Lu, L. Wang, and B. Wang: Fast Drying of Cut Tobacco in Drop Tube Reactor and its Effect on Petroleum Ether Tobacco Extracts; Dry. Technol. 36 (2018) 1304–1312. DOI: 10.1080/07373937.2017.1402022 LiB. ZhuW. WangP. LuD. WangL. WangB. Fast Drying of Cut Tobacco in Drop Tube Reactor and its Effect on Petroleum Ether Tobacco Extracts Dry. Technol. 36 2018 1304 1312 10.1080/07373937.2017.1402022 Ouvrir le DOISearch in Google Scholar

Rajan, K.S., K. Dhasandhan, S.N. Srivastava, and B. Pitchumani: Studies on Gas-Solid Heat Transfer During Pneumatic Conveying; Int. J. Heat Mass Transfer 51 (2008) 2801–2813. DOI: 10.1016/j.ijheatmasstransfer.2007.09.042 RajanK.S. DhasandhanK. SrivastavaS.N. PitchumaniB. Studies on Gas-Solid Heat Transfer During Pneumatic Conveying Int. J. Heat Mass Transfer 51 2008 2801 2813 10.1016/j.ijheatmasstransfer.2007.09.042 Ouvrir le DOISearch in Google Scholar

Mabrouk, S.B., B. Khiari, and M. Sassi: Modelling of Heat and Mass Transfer in a Tunnel Dryer; Appl. Therm. Eng. 26 (2006) 2110–2118. DOI: 10.1016/j.applthermaleng.2006.04.007 MabroukS.B. KhiariB. SassiM. Modelling of Heat and Mass Transfer in a Tunnel Dryer Appl. Therm. Eng. 26 2006 2110 2118 10.1016/j.applthermaleng.2006.04.007 Ouvrir le DOISearch in Google Scholar

El-Mesery, H.S. and G. Mwithiga: Performance of a Convective, Infrared and Combined Infrared-Convective Heated Conveyor-Belt Dryer; J. Food Sci. Technol. 52 (2015) 2721–2730. DOI: 10.1007/s13197-014-1347-1 El-MeseryH.S. MwithigaG. Performance of a Convective, Infrared and Combined Infrared-Convective Heated Conveyor-Belt Dryer J. Food Sci. Technol. 52 2015 2721 2730 10.1007/s13197-014-1347-1439731825892769 Ouvrir le DOISearch in Google Scholar

Silva, D.I.S., G.F.M.V. Souza, and M.A.S. Barrozo: Heat and Mass Transfer of Fruit Residues in a Fixed Bed Dryer: Modeling and Product Quality; Dry. Technol. 37 (2019) 1321–1327. DOI: 10.1080/07373937.2018.1498509 SilvaD.I.S. SouzaG.F.M.V. BarrozoM.A.S. Heat and Mass Transfer of Fruit Residues in a Fixed Bed Dryer: Modeling and Product Quality Dry. Technol. 37 2019 1321 1327 10.1080/07373937.2018.1498509 Ouvrir le DOISearch in Google Scholar

Białobrzewski, I., M. Zielińska, A.S. Mujumdar, and M. Markowski: Heat and Mass Transfer During Drying of a Bed of Shrinking Particles – Simulation for Carrot Cubes Dried in a Spout-Fluidized-Bed Drier; Int. J. Heat Mass Transfer 51 (2008) 4704–4716. DOI: 10.1016/j.ijheatmasstransfer.2008.02.031 BiałobrzewskiI. ZielińskaM. MujumdarA.S. MarkowskiM. Heat and Mass Transfer During Drying of a Bed of Shrinking Particles – Simulation for Carrot Cubes Dried in a Spout-Fluidized-Bed Drier Int. J. Heat Mass Transfer 51 2008 4704 4716 10.1016/j.ijheatmasstransfer.2008.02.031 Ouvrir le DOISearch in Google Scholar

Silva, P.B., C.R. Duarte, and M.A.S. Barrozo: Dehydration of Acerola (Malpighia emarginata D.C.) Residue in a New Designed Rotary Dryer: Effect of Process Variables on Main Bioactive Compounds; Food Bioprod. Process. 98 (2016) 62–70. DOI: 10.1016/j.fbp.2015.12.008 SilvaP.B. DuarteC.R. BarrozoM.A.S. Dehydration of Acerola (Malpighia emarginata D.C.) Residue in a New Designed Rotary Dryer: Effect of Process Variables on Main Bioactive Compounds Food Bioprod. Process. 98 2016 62 70 10.1016/j.fbp.2015.12.008 Ouvrir le DOISearch in Google Scholar

Geng, F., Z. Yuan, Y. Yan, D. Luo, H. Wang, B. Li, and D. Xu: Numerical Simulation on Mixing Kinetics of Slender Particles in a Rotary Dryer; Powder Technol. 193 (2009) 50–58. DOI: 10.1016/j.powtec.2009.02.005 GengF. YuanZ. YanY. LuoD. WangH. LiB. XuD. Numerical Simulation on Mixing Kinetics of Slender Particles in a Rotary Dryer Powder Technol. 193 2009 50 58 10.1016/j.powtec.2009.02.005 Ouvrir le DOISearch in Google Scholar

Geng, F., Y. Li, X. Wang, Z. Yuan, Y. Yan, and D. Luo: Simulation of Dynamic Processes on Flexible Filamentous Particles in the Transverse Section of a Rotary Dryer and its Comparison with Ideo-Imaging Experiments; Powder Technol. 207 (2011) 175–182. DOI: 10.1016/j.powtec.2010.10.027 GengF. LiY. WangX. YuanZ. YanY. LuoD. Simulation of Dynamic Processes on Flexible Filamentous Particles in the Transverse Section of a Rotary Dryer and its Comparison with Ideo-Imaging Experiments Powder Technol. 207 2011 175 182 10.1016/j.powtec.2010.10.027 Ouvrir le DOISearch in Google Scholar

Geng, F., Y. Li, L. Yuan, M. Liu, X. Wang, Z. Yuan, Y. Yan, and D. Luo: Experimental Study on the Space Time of Flexible Filamentous Particles in a Rotary Dryer; Exp. Therm. Fluid Sci. 44 (2013) 708–715. DOI: 10.1016/j.expthermflusci.2012.09.011 GengF. LiY. YuanL. LiuM. WangX. YuanZ. YanY. LuoD. Experimental Study on the Space Time of Flexible Filamentous Particles in a Rotary Dryer Exp. Therm. Fluid Sci. 44 2013 708 715 10.1016/j.expthermflusci.2012.09.011 Ouvrir le DOISearch in Google Scholar

Jayas, D.S., S. Cenkowski, S. Pabis, and W.E Muir: Review of Thin-Layer Drying and Wetting Equations; Dry. Technol. 9 (1991) 551–588. DOI: 10.1080/07373939108916697 JayasD.S. CenkowskiS. PabisS. MuirW.E Review of Thin-Layer Drying and Wetting Equations Dry. Technol. 9 1991 551 588 10.1080/07373939108916697 Ouvrir le DOISearch in Google Scholar

Doymaz, İ. and O. İsmail: Drying Characteristics of Sweet Cherry; Food Bioprod. Process. 89 (2011) 31–38. DOI: 10.1016/j.fbp.2010.03.006 Doymazİ. İsmailO. Drying Characteristics of Sweet Cherry Food Bioprod. Process. 89 2011 31 38 10.1016/j.fbp.2010.03.006 Ouvrir le DOISearch in Google Scholar

Henderson, S. M. and S. Pabis: Grain Drying Theory: 1. Temperature Affection Drying Coefficient; J. Agric. Eng. Res. 6 (1961) 169–170. HendersonS. M. PabisS. Grain Drying Theory: 1. Temperature Affection Drying Coefficient J. Agric. Eng. Res. 6 1961 169 170 Search in Google Scholar

Menges, H.O. and C. Ertekin: Thin Layer Drying Model for Treated and Untreated Stanley Plums; Energy Convers. Manage. 47 (2006) 2337–2348. DOI: 10.1016/j.enconman.2005.11.016 MengesH.O. ErtekinC. Thin Layer Drying Model for Treated and Untreated Stanley Plums Energy Convers. Manage. 47 2006 2337 2348 10.1016/j.enconman.2005.11.016 Ouvrir le DOISearch in Google Scholar

Midilli, A., H. Kucuk, and Z. Yapar: A New Model for Single-Layer Drying; Dry. Technol. 20 (2002) 1503–1513. DOI: 10.1081/DRT-120005864 MidilliA. KucukH. YaparZ. A New Model for Single-Layer Drying Dry. Technol. 20 2002 1503 1513 10.1081/DRT-120005864 Ouvrir le DOISearch in Google Scholar

Hii, C.L., C.L. Law, and M. Cloke: Modeling Using a New Thin Layer Drying Model and Product Quality of Cocoa; J. Food Eng. 90 (2009) 191–198. DOI: 10.1016/j.jfoodeng.2008.06.022 HiiC.L. LawC.L. ClokeM. Modeling Using a New Thin Layer Drying Model and Product Quality of Cocoa J. Food Eng. 90 2009 191 198 10.1016/j.jfoodeng.2008.06.022 Ouvrir le DOISearch in Google Scholar

Xu, Q. and S. Pang: Mathematical Modeling of Rotary Drying of Woody Biomass; Dry. Technol. 26 (2008) 1344–1350. DOI: 10.1080/07373930802331050 XuQ. PangS. Mathematical Modeling of Rotary Drying of Woody Biomass Dry. Technol. 26 2008 1344 1350 10.1080/07373930802331050 Ouvrir le DOISearch in Google Scholar

Gu, C., X. Zhang, B. Li, and Z. Yuan: Study on Heat and Mass Transfer of Flexible Filamentous Particles in a Rotary Dryer; Powder Technol. 267 (2014) 234–239. DOI: 10.1016/j.powtec.2014.06.059 GuC. ZhangX. LiB. YuanZ. Study on Heat and Mass Transfer of Flexible Filamentous Particles in a Rotary Dryer Powder Technol. 267 2014 234 239 10.1016/j.powtec.2014.06.059 Ouvrir le DOISearch in Google Scholar

Zhu, W.K., L. Wang, K. Duan, L.Y. Chen, and B. Li: Experimental and Numerical Investigation of the Heat and Mass Transfer for Cut Tobacco During Two-Stage Convective Drying; Dry. Technol. 33 (2015) 907–914. DOI: 10.1080/07373937.2014.997882 ZhuW.K. WangL. DuanK. ChenL.Y. LiB. Experimental and Numerical Investigation of the Heat and Mass Transfer for Cut Tobacco During Two-Stage Convective Drying Dry. Technol. 33 2015 907 914 10.1080/07373937.2014.997882 Ouvrir le DOISearch in Google Scholar

Chen, X.D., W. Pirini, and M. Ozilgen: The Reaction Engineering Approach to Modelling Drying of Thin Layer of Pulped Kiwifruit Flesh under Conditions of Small Biot Numbers; Chem. Eng. Process. 40 (2001) 311–320. DOI: 10.1016/s0255-2701(01)00108-8 ChenX.D. PiriniW. OzilgenM. The Reaction Engineering Approach to Modelling Drying of Thin Layer of Pulped Kiwifruit Flesh under Conditions of Small Biot Numbers Chem. Eng. Process. 40 2001 311 320 10.1016/s0255-2701(01)00108-8 Ouvrir le DOISearch in Google Scholar

Putranto, A., X.D. Chen, Z. Xiao, and P.A. Webley: Mathematical Modeling of Intermittent and Convective Drying of Rice and Coffee Using the Reaction Engineering Approach (REA); J. Food Eng. 105 (2011) 638–646. DOI: 10.1016/j.jfoodeng.2011.03.036 PutrantoA. ChenX.D. XiaoZ. WebleyP.A. Mathematical Modeling of Intermittent and Convective Drying of Rice and Coffee Using the Reaction Engineering Approach (REA) J. Food Eng. 105 2011 638 646 10.1016/j.jfoodeng.2011.03.036 Ouvrir le DOISearch in Google Scholar

Putranto, A., X.D. Chen, and W. Zhou: Modeling of Baking of Thin Layer of Cake Using the Lumped Reaction Engineering Approach (L-REA); J. Food Eng. 105 (2011) 306–311. DOI: 10.1016/j.jfoodeng.2011.02.039 PutrantoA. ChenX.D. ZhouW. Modeling of Baking of Thin Layer of Cake Using the Lumped Reaction Engineering Approach (L-REA) J. Food Eng. 105 2011 306 311 10.1016/j.jfoodeng.2011.02.039 Ouvrir le DOISearch in Google Scholar

Li, Q., Y.F. Li, Y. Zhang, Q. Chen, H. Huang, H. Chen, Y. Lin, H. Xiao, Z. Liao, L. Che, W. Xie, and X.D. Chen: Drying Kinetics Study of Irregular Fibril Materials in a “Differential” Laboratory Rotary Dryer: Case Study for Cut Tobacco; Dry. Technol. 36 (2017) 523–536. DOI: 10.1080/07373937.2017.1341920 LiQ. LiY.F. ZhangY. ChenQ. HuangH. ChenH. LinY. XiaoH. LiaoZ. CheL. XieW. ChenX.D. Drying Kinetics Study of Irregular Fibril Materials in a “Differential” Laboratory Rotary Dryer: Case Study for Cut Tobacco Dry. Technol. 36 2017 523 536 10.1080/07373937.2017.1341920 Ouvrir le DOISearch in Google Scholar

Whitman, W.G.: The Two Film Theory of Gas Absorption; Int. J. Heat Mass Transfer 5 (1962) 429–433. DOI: 10.1016/0017-9310(62)90032-7. WhitmanW.G. The Two Film Theory of Gas Absorption Int. J. Heat Mass Transfer 5 1962 429 433 10.1016/0017-9310(62)90032-7 Ouvrir le DOISearch in Google Scholar

Hills, B.P. and M. Harrison: Two-Film Theory of Flavour Release from Solids; Int. J. Food Sci. Technol. 30 (1995) 425–436. DOI: 10.1111/j.1365-2621.1995.tb01390.x HillsB.P. HarrisonM. Two-Film Theory of Flavour Release from Solids Int. J. Food Sci. Technol. 30 1995 425 436 10.1111/j.1365-2621.1995.tb01390.x Ouvrir le DOISearch in Google Scholar

Smith, J.M., H.C. Van Ness, and M.M. Abbott: Introduction to Chemical Engineering Thermodynamics; 7th Edition, McGraw-Hill Education, NY, USA, 2004. SmithJ.M. Van NessH.C. AbbottM.M. Introduction to Chemical Engineering Thermodynamics 7th Edition McGraw-Hill Education NY, USA 2004 Search in Google Scholar

Henderson, S.M.: A Basic Concept of Equilibrium Moisture Content; Agric. Eng. 33 (1952) 29–32. HendersonS.M. A Basic Concept of Equilibrium Moisture Content Agric. Eng. 33 1952 29 32 Search in Google Scholar

Thompson, T.L., R.M. Peart, and G.H. Foster: Mathematical Simulation of Corn Drying - A New Model; Trans. ASAE 24 (1968) 582–586. Available at: https://www.ars.usda.gov/ARSUserFiles/30200525/34MathematicalSimulationofCornDrying.pdf (accessed November 2020) ThompsonT.L. PeartR.M. FosterG.H. Mathematical Simulation of Corn Drying - A New Model Trans. ASAE 24 1968 582 586 Available at: https://www.ars.usda.gov/ARSUserFiles/30200525/34MathematicalSimulationofCornDrying.pdf (accessed November 2020) 10.13031/2013.39473 Search in Google Scholar

Chen, C.C. and R.V. Morey: Comparison of Four EMC/ERH Equations; Trans. ASAE 32 (1989) 983–990. Available at: https://elibrary.asabe.org (accessed November 2020) ChenC.C. MoreyR.V. Comparison of Four EMC/ERH Equations Trans. ASAE 32 1989 983 990 Available at: https://elibrary.asabe.org (accessed November 2020) 10.13031/2013.31103 Search in Google Scholar

Chung, D.S. and H.B. Pfost: Adsorption and Desorption of Water Vapor by Cereal Grains and Their Products. Part II: Development of the General Isotherm Equation; Trans. ASAE 10 (1067) 552–555. Available at: https://elibrary.asabe.org (accessed November 2020) ChungD.S. PfostH.B. Adsorption and Desorption of Water Vapor by Cereal Grains and Their Products. Part II: Development of the General Isotherm Equation Trans. ASAE 10 1067 552 555 Available at: https://elibrary.asabe.org (accessed November 2020) Search in Google Scholar

Pfost, H., S.G. Maurer, D. Chung, and G. Milliken: Summarizing and Reporting Equilibrium Moisture Data for Grains; ASAE Paper No. 76–3520, St. Joseph, MI, USA, 1976. PfostH. MaurerS.G. ChungD. MillikenG. Summarizing and Reporting Equilibrium Moisture Data for Grains ASAE Paper No. 76–3520 St. Joseph MI, USA 1976 Search in Google Scholar

Halsey, G.: Physical Adsorption on Non-Uniform Surfaces; J. Chem. Phys 16 (1948) 931–937. DOI: 10.1063/1.1746689 HalseyG. Physical Adsorption on Non-Uniform Surfaces J. Chem. Phys 16 1948 931 937 10.1063/1.1746689 Ouvrir le DOISearch in Google Scholar

Yaws, C.L.: Chemical Properties Handbook; McGraw-Hill Education, NY, USA, 1999. YawsC.L. Chemical Properties Handbook McGraw-Hill Education NY, USA 1999 Search in Google Scholar

Linstrom, P.J. and W.G. Mallard: Nist Chemistry WebBook, Nist Standard Reference Database Number 69, National Institute of Standards and Technology, Gaithersburg, MD, USA, 2005. Available at: https://webbook.nist.gov/chemistry/ LinstromP.J. MallardW.G. Nist Chemistry WebBook, Nist Standard Reference Database Number 69 National Institute of Standards and Technology Gaithersburg, MD, USA 2005 Available at: https://webbook.nist.gov/chemistry/ Search in Google Scholar

Zhu, L., X. Qin, Z. Yuan, Y. Yan, D. Luo, and B. Li: Heat and Mass Transfer Characteristics of Filamentous Particles in Transverse Section of Rotary Dryer; Journal of Southeast University, Natural Science Edition, 44 (2014) 756–763. DOI: 10.3969/j.issn.1001-0505.2014.04.014 ZhuL. QinX. YuanZ. YanY. LuoD. LiB. Heat and Mass Transfer Characteristics of Filamentous Particles in Transverse Section of Rotary Dryer Journal of Southeast University, Natural Science Edition 44 2014 756 763 10.3969/j.issn.1001-0505.2014.04.014 Ouvrir le DOISearch in Google Scholar

Atashkari, K., N. Nariman-Zadeh, A. Pilechi, A. Jamali, and X. Yao: Thermodynamic Pareto Optimization of Turbojet Engines Using Multi-Objective Genetic Algorithms: Int. J. Thermal Sci. 44 (2005) 1061–1071. DOI: 10.1016/j.ijthermalsci.2005.03.016. AtashkariK. Nariman-ZadehN. PilechiA. JamaliA. YaoX. Thermodynamic Pareto Optimization of Turbojet Engines Using Multi-Objective Genetic Algorithms Int. J. Thermal Sci. 44 2005 1061 1071 10.1016/j.ijthermalsci.2005.03.016 Ouvrir le DOISearch in Google Scholar

Sarkar, D. and J.M. Modak: Pareto-Optimal Solutions for Multi-Objective Optimization of Fed-Batch Bioreactors Using Nondominated Sorting Genetic Algorithm; Chem. Eng. Sci. 60 (2005) 481–492. DOI: 10.1016/j.ces.2004.07.130. SarkarD. ModakJ.M. Pareto-Optimal Solutions for Multi-Objective Optimization of Fed-Batch Bioreactors Using Nondominated Sorting Genetic Algorithm Chem. Eng. Sci. 60 2005 481 492 10.1016/j.ces.2004.07.130 Ouvrir le DOISearch in Google Scholar

Sun, D.W.: Comparison and Selection of EMC/ERH Isotherm Equations for Rice; J. Stored Prod. Res. 35 (1999) 249–264. DOI: 10.1016/s0022-474x(99)00009-0. SunD.W. Comparison and Selection of EMC/ERH Isotherm Equations for Rice J. Stored Prod. Res. 35 1999 249 264 10.1016/s0022-474x(99)00009-0 Ouvrir le DOISearch in Google Scholar

Articles recommandés par Trend MD

Planifiez votre conférence à distance avec Sciendo