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

In order to simulate the mass transfer, the two-film 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]. [1] dXdt=hmAm(ρb−ρe) {{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, h_m A_m = k_m is the product of the convective mass transfer coefficient h_m and the mass transfer surface area A_m, ρ_b 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]: [2] pi=1.1939×1010 exp(−3826.36Ti−45.47)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 ρ_b is the vapor pressure in hot air, RH_b represents the humidity in hot gas, T denotes the temperature of cut tobacco, ρ_e is the vapor pressure in the gas film, and RH_e 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]: [3] ρi=piMwRTi=1.1939×1010 exp(−3826.36Ti−45.47)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 M_w is the molecular weight of water.

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

In order to develop the mathematical model of heat and mass transfer, the equilibrium humidity of air (RH_e) 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).

In Table 2 X_e is the equilibrium moisture content of cut tobacco corresponding to the equilibrium state, RH_e 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: [4] (XCp,w+Cp,t)dTdt=hAm(Tb−T)+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, T_b is the bulk air temperature, hA_m = k_h denotes the convective heat transfer coefficient combining with inseparable heat transfer area, ΔH_w is the latent heat of vaporization of water, C_p,w is the specific heat capacity of the water and C_p,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: [5] ΔHw=2891833(1−T647.13)0.321 \Delta {H_w} = 2891833{\left( {1 - {T \over {647.13}}} \right)^{0.321}} [6] Cp,w=1.459×10−6T4−1.971×10−3T3+1.005×T2−228.7T+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 [7] 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): [8] 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, m_t (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, h_m (m·s⁻¹) is the mass transfer coefficient, A (m²) is the surface area of cut tobacco for mass transfer, ρ_v,sat (kg·m⁻³) is the saturated vapor concentration corresponding to the cut tobacco temperature T_t (K), ρ_v,b (kg·m⁻³) is the vapor concentration of bulk drying air, and ΔE_v (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.

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: [9] minimize{f(hAm,hmAm)=∑j=1M∑i=1N(Xcal,i,j−Xexp,i,jXexp,i,j)g(hAm,hmAm)=∑j=1M∑i=1N(Tcal,i,j−Texp,i,jTexp,i,j)22subject to{(XcalCp,w+Cp,t)dTcaldt=hAm(Tb−Tcal)+ΔHwdXcaldt,Tcal(0)=Texp(0)dXcaldt=hmAm(ρb−ρe),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, X_cal,i,j is the predicted value of moisture content, X_exp,i,j is the experimental value of moisture content, T_cal,i,j is the predicted value of tobacco temperature, X_exp,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 (41–42).

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: [10] minimize∑j=1M∑i=1N(rXcal,i,j−Xexp,i,jXexp,i,j)2+∑j=1M∑i=1N((1−r)Tcal,i,j−Texp,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.

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: [11] RMSEX=∑i=1N(Xexp,i−Xcal,i)2N, RMSET=∑i=1N(Texp,i−Tcal,i)2NRMSEtotal=∑i=1N(Xexp,i−Xcal,j)2+∑i=1N(Texp,i−Tcal,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 } [12] MRDX=1N∑i=1N|Xexp,i−Xcal,i|Xexp,i,MRDT=1N∑i=1N|Texp,i−Tcal,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.

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 RH_e 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.

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 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

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

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

Figure 3

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

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

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

Figure 6

The value of k_h 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 k_m or between the values of RMSE and k_h.

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, 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 RMSE_X 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

Overall MRD_T 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 MRD_T value of the M-Hen/C model was significantly lower than in other models at 398.15 K. Comparing the overall MRD_X 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.