Volume 5, Number 1 (2020) pp 113 doi 10.20448/808.5.1.1.13  Research Articles
The need for storage is one of the crucial factors considered in food preservation, due to deterioration and spoilage of food materials. Drying as a means of preservation has been adopted in processing of food materials by exposing the material to heat to heat for the reduction of moisture content to the level unfavorable for mold growth. Monkey cola is a nutritional indigenous seed known with its pleasant taste and medicinal values. Moisture kinetics and drying time were used to examine the drying characteristics of Monkey cola seed. Monkey kola seed and a suitable mathematical model was developed by fitting the drying data gotten from the open sun drying and oven drying of monkey kola at different temperatures (40, 50, 65, 60 and 700C) to properly predict the drying characteristics of the seed using statistical tools. This gives the highest R₂ and lowest RMSE and provided the longterm performance of correlation for each degree of temperature. The best model fitted for each temperature at 40, 50, 65, 60 and 700C for oven drying were Diffusion approach, Verma, Hii, and Midili Kucuk models, respectively, and Midili Kucuk model, for open sun drying. Temperature dependence of the effective diffusivity (Deff) coefficient was expressed by an Arrhenius type relationship and the activation energy (Ea) was determined.
Keywords: Drying Kinetics, Monkey cola, Modelling thin layer models.
DOI: 10.20448/808.5.1.1.13
Citation  Olabinjo, Oyebola Odunayo; Adeniyan, Adedotun Timothy (2020). Modelling the Drying Kinetics of Monkey Cola (Cola Parchycarpa). Scientific Modelling and Research, 5(1): 113.
Copyright: This work is licensed under a Creative Commons Attribution 3.0 License
Funding : This study received no specific financial support.
Competing Interests: The authors declare that they have no competing interests.
History : Received: 29 October 2019 / Revised: 2 December 2019 / Accepted: 7 January 2020 / Published: 12 February 2020 .
Publisher: Online Science Publishing
Highlights of this paper

Drying, also known as dehydration is an aged and a crucial unit operation involved in food processing. It is the application of hot air to a food material that leads to transfer of moisture within the material to its surface and water removal from the material to the atmosphere. Drying involve both heat and mass transfer operation simultaneously. Drying is the most affordable techniques for food preservation; it reduces spoilage and improves the quality of a product. The major essence of drying food materials is to remove the free water available in the food material. The removal of moisture can be due to concurrent heat and mass transfer from the heat source to the food material. Drying is usually the last step in the series of processing and handling operation. The major aim of drying agricultural produce is to maintain the quality of the product for storage, create an unfavorable environment for mold growth, to eliminate unnecessary water that contribute to the weight of agricultural products and also make packaging of product easier. The main characteristics of dried materials are reduced porosity, high apparent density and reduced sorption capacity that results in colour change. Achievement of drying aid better handling of dried materials, transportation minimization, effective storage, reduction in packing cost and time.
The study of the kinetic model is the rate of reactions that occurred during drying process of each particular product. Drying kinetics must be done in other to evaluate the drying behavior of fruits and vegetables. The drying kinetics helps to understand the process of moisture removal from a food material and it is crucial in determination of the drying conditions, which are significant parameters in equipment design and product quality improvement [1]. The drying kinetics is affected by the type of dryer and behavior of the material to be dried. The most acceptable method for the determination of the drying kinetics of fruit and vegetables is thinlayer drying.
The thinlayer modeling is useful in the determination of the drying kinetics behaviour gotten from the experimental data.
The thinlayer modeling is helpful in the estimation of the drying kinetics from the experimental data gotten from drying. It describes the drying properties, improve and auspicate the drying process and also optimize energy required for drying. The vital aspects of thinlayer drying technology are the modeling of drying process and the design equipment use in drying. Drying model is therefore used to predict the changes and the rate of bio chemical and physical reactions which occurs with drying kinetics. The aim of the research is to select appropriate model that can predict the drying kinetics of monkey cola.
The sample of Monkey Cola (Cola Parchycarpa) studied was obtained from its tree in a forest located in Ipetu Ile, Obokun local government, Osun state. The good and healthy fruits were sorted and treated from the contaminated ones and the pod of the fruits was separated from the seeds for further experiments. The seed gotten from the separation was naturally fermented for 23 days using banana leaves as the antioxidizing and fermentation agent. 100 g of the fermented seed and 50 g of the separated fruit pod were weighed into several separate cans and labelled for thin layer drying experiments.
Thin layer drying experiments were carried out at varying degrees of temperature ranging from 35, 40, 45, 50, 6070°C using electrical laboratory oven (TT9083; Gallenkamp Devices, UK) and at normal atmospheric temperature using the natural solar drying system (open sun drying). The weight of each sample were checked during drying using the weighing scale at each interval of 1 hour until they were dried to the temperature degree variation of ±1°C for 1g of weighed samples, where there was no moisture content present in the samples.
The moisture content of the seeds was measured by using the hot air (oven) method set at 103 ±2 °C for 72 hours. Four samples were heated in the oven until there was no change in weight using ASABE S352 method and applied by Okoro and Osunde [2]; Abodenyi, et al. [3]; Oloyede, et al. [4]; Oniya, et al. [5] for sour sop seeds. The moisture content was calculated using Equation 1.
where:
MC_{wb} is moisture content (% wet basis).
Mb is the weight of wet sample (g).
Ma is the weight of oven dried sample (g).
Mc is weight of empty can (g).
The experimental data gotten from the open sun and oven drying were stated in terms moisture ratio, and drying rate. The moisture ratio (MR) and the drying rate (DR) of the seeds were calculated using the Equations 2 and 3 [6]:
For these three drying methods, the equilibrium moisture content (M∞) was presumed to be zero, consequently the equation simplified then become [7, 8] .
Where Mt is the moisture content at any given time (g water/g dry matter), Mo is the initial moisture content (g water/g dry matter), M∞ is the equilibrium moisture content (g water/g dry matter), is the moisture content at t + dt (g water/g dry base) and t is drying time (min).
Effective moisture diffusivity can be evaluated using the analytical solution for spherical geometry, as followed Equation 4;
For the evaluation of the effective moisture diffusivity at different temperature conditions, the slope (ko) was determined by plotting ln(MR) versus time using Equation 5;
The activation energy for diffusion was evaluated by using Arrhenius equation [9].
Activation energy (Ea) was evaluated by plotting ln(Deff) against 1/T.
Where, Do is the constant equivalent to the diffusivity at infinitely high temperature (m2 min1),
Ea is the activation energy (kJ/mol), R the universal gas constant (8.314 x 103 kJ/mol).
T is the absolute temperature (K).
The drying behavior of Monkey cola seeds was evaluated by plotting the moisture ratio against the drying time. The experimental data were fitted to thirteen thinlayer mathematical models Table 1 to depict the drying process. The numerical calculations of the data were done utilizing the software package, Excel 2016 (Microsoft Inc.). The models' parameters were assessed with the nonlinear regression techniques of MarquardtLevenberg until insignificant error was achieved between experimental and calculated values. The coefficient of determination, R2; normalized Moisture content against time was changed to moisture ratio. The drying data were input into the thirteen selected thin layer drying models. The models were analysis utilizing statistical tools; coefficient of determination (R2), sum of estimated error (SEE), root mean square error (RMSE) and chisquare (X2) as reported by Chukwunonye, et al. [10] described by Equation 7 to 9. The reduced chi square (x2) and RMSE values and the higher R2 values were selected as the basis for goodness of fit for the model.
a, b, c and d are constants and coefficients in the drying models.
The average initial moisture content of Monkey Cola (Cola Parchycarpa) was determined using laboratory method to be around 59.8% (w.b) the seeds were oven dried in laboratory oven (TT9083; Gallenkamp Devices, UK) to the moisture content of about 5.77% at 70°C, 11.8% at 65°C, 11.4% at 60°C, 13.6% at 50°C and 26.99% at 40°C (w.b) until no further changes in their mass were observed. Sample dried at 40°C showed the highest moisture content of 26.99% while sample dried at 70°C showed the least moisture content value of 5.77%. Sample dried at 70°C had better tendency for longer shelf life due to its lower moisture content. Lower moisture content infers reduction of water and microbial activity on dried food samples as reported by Ajala, et al. [25].
The drying characteristics of Monkey Kola (Cola Parchycarpa) evaluated in this study comprised drying curves showing the relationship between the drying rate, moisture content and drying time. The drying kinetics of the monkey kola seed for the oven and open sun drying were determined and the effect of temperature on the moisture content, moisture ratio and drying rate for the two methods of drying.
The drying curves for thin layer drying of Monkey Cola (Cola Parchycarpa) under various temperature conditions in oven and open are shown in Figure 1. The monkey kola seed were dried at varying interval at different temperature of 40, 50, 60, 65 and 700C respectively to reach constant mass showing moisture ratio decreased continuously with increasing drying time. The rate of drying was very high at the commencement of the drying process at each temperature. with the 650C having the highest initial value of 10.1783 and the lowest drying rate at 400C. While, at temperature 650C the rate of drying and the water content reduction was higher due to the fact that the drying temperature had a significant effect on the drying kinetics of the samples Monkey Cola. According to Kumar, et al. [26] which conforms to the information provided earlier, at higher water content, the increase in temperature has more considerable effect on the drying rates than at lower water content, which is almost negligible at the end. The increase in drying rate as temperature of drying air increases as shown in Figures 2 is due to increased heat transfer gradient between the air and the seeds which favor water evaporation from the seed, as agreed with the experiment carried out on fever leaves by Doymaz [27]. Figures 2. Shows that the drying rate decreases continuously with the decreasing moisture content or increasing drying time. In the drying rate/moisture content graph, at 650C the drying rate rises as the moisture content reduces to an equilibrium point 0f 10 (%/h) but decreases with the decreasing moisture content. These results are in agreement with the observations of earlier researchers based on thin layer drying of amaranth grains [28]. Also constant drying rate at each temperature were reached at different time interval. The open sun drying had the least drying rate as showed in Figures 2 and 3. The water content reduction was slow due to the significant effect of the unstable atmospheric condition on the drying temperature which also affect the drying kinetics of the samples Monkey Cola. These results are in agreement with the earlier observations [29, 30]. According to Figures 2 and 3 the rate of drying and moisture ratio reduces with increasing drying time.
Fick’s diffusion equation and Slab geometry were used for calculation of effective diffusivity by method of slopes expressed in equation and is shown in Table 2. The value of moisture diffusivity determined were 0.955, 2.379, 3.212, 3.435, and 6.945 for 40, 50, 60, 65 and 700C temperatures respectively, while the effective moisture diffusivity for open sun drying is . The effective diffusivity value of monkey cola seeds for oven drying ranged from 0.955 to 6.945. This was used to determine the value of the predicted moisture diffusivity for oven drying method also which ranged from 0.936 to 6.822 And the open sun having the value of 1.410 it can be seen that the effective diffusivity values increased greatly with increasing temperature, which in turns increased the vapor pressure. Similar trends were found in products such as grapefruit seeds [31] and grape seeds [32]. This indicates that as the moisture content decreases, the permeability to vapor increased, provided the pore structure remains open. Sharma and Prasad [33] also reported a similar trend in the variation in the moisture diffusivity with moisture content, as a result, it leads to fast drying.
The value of the activation energy (Ea) was determined by exploiting the Arhenius equation. The relationship between the effective diffusivities and temperature is assumed in the Arrhenius form of the type using Equation 10,
were 51.224, 51.223, 51.237, 51.237 and 51.240 respectively, while the activation energy value for open sun drying is 51.747. The activation energy values for the oven drying were found to be within the range of 51.22451.240kj/mol, this shows that values also increased with increasing temperature. According to Zogzas, et al. [34] the activation energy for farm products ranges from 12.7 to 110KJ mol. It is highlighted that in the drying processes, the lesser the activation energy, the greater the water diffusion within the product [35, 36].
Temperature 
70 °C 
65 °C 
60 °C 
50 °C 
40 °C 
Open sun 
Deff (1010) 
6.945 
3.435 
3.212 
2.379 
0.955 
1.439 
Do (1010) 
6.822 
3.372 
3.153 
2.334 
0.936 
1.410 
Ea (kj/mol) 
51.240 
51.237 
51.237 
51.233 
51.224 
51.747 
The drying curves obtained from experimental drying methods were fitted with moisture ratio equations into the thirteen (13) thin layer drying models. The best drying model was selected based on the maximum coefficient of determination (R²), minimum values of reduced chisquare (X²), and the percentage of root mean square error (RMSE).
The drying data gotten from the oven drying of monkey cola at different temperatures (35, 40, 50, 60 and 700C) were fitted into the thirteen (13) thin layer drying models, this gives the highest R₂ and lowest RMSE and provided the longterm performance of correlation for each degree of temperature , the best drying model fitted for the temperature at 400C is Diffusion approach having the highest R² of 0.8709,lowest chi square of 0.0088 and RMSE of 0.0521, Verma et al for 500C with the highest R² of 0.9927,lowest chi square of 0.0003 and RMSE of 0.0163, Hii et al for 60, 650C with the highest R² of 0.9945 and 0.9965,lowest chi square of 0.0003 and 0.0002 and RMSE of 0.0167 and 0.0118 and Midili et al for 700C having the highest R² of 0.9946,lowest chi square of 0.0005 and RMSE of 0.0206. From the analysis, the model fitted at an average temperature is Hii et al. Modified Henderson and Pabis is the most appropriate model having the highest R² of 0.8512, lowest chi square of 0.0029 and RMSE of 0.0521 for open sun drying Table 3.
Figure 4 and 5 shows the comparison of the predicted and experimental values for the oven drying and open sun drying at different temperatures. The model used is validate to check the suitability of the model in predicting the drying of the samples, it is validated by plotting a graph of predicted moisture ratio against experimented moisture ratio, if the coefficient of determination (R²) determined from the graph is ≥ 0.75 the model is valid and it can be used to predict the drying curve. The graphs of predicted moisture ratio and experimented moisture ratios for the monkey cola seed under the varying temperature conditions of oven drying had an average maximum coefficient of determination (R²) of 0.9965 and open sun drying had 0.9997. This indicated that all the models derived for drying of monkey cola seed under the two methods can correctly predict the drying characteristics of seed and are valid.
Drying is one of the aged methods used to preserve the quality of agricultural produce for availability throughout the year, as well as reduce postharvest loses. In this study, the various drying techniques (open and oven) used were capable to preserve the quality of the seed. Thirteen thinlayer model equations were used in testing the drying kinetics carried out on the thinlayer drying behavior of monkey cola seeds. It was observed that Two term model is the best and most appropriate in predicting drying kinetics of soursop seeds under the varying drying methods of study. It had highest R2 of 0.9993, lowest Chi square of 0.000, and RMSE of 0.0047, highest R2 of 0.991, Chi square of 0.0003, and RMSE of 0.0157 and highest R2 of 0.9961, Chi square of 0.0085 and lowest RMSE of 0.0093 for open, oven and oven drying methods respectively. The validation results established good concert between the experimental and predicted drying variables, however, two term model equation could be used satisfactorily to predict thin layer drying of monkey cola seeds for open sun and oven drying method respectively. It is highlighted that in the drying processes, the lesser the activation energy, the greater will be water diffusion within the product.
Temperature 
Model 
Model constant 
R² 
RMSE 
X² 

40 °C 
Newton 
K = 0.0163 
0.8786 
0.0891 
0.0081 

Henderson and Pabis 
k = 0.0093, a = 0.8051 
0.8597 
0.0381 
0.0015 

Page 
k = 0.0823, n = 0.5418 
0.9657 
0.0295 
0.0009 

Logarithmic 
k = 0.0107, a = 0.7522, c = 0.0626 
0.8637 
0.0376 
0.0015 

Two term 
k = 0.0226, g = 0.0315, a = 2.6029, c = 1.816 
0.7700 
0.0520 
0.0029 

Vermal 
k = 0.7386, g = 0.0114, a = 0.1592 
0.9549 
0.0328 
0.0011 

Diffusion approach 
k = 0.0219, g = 1.0107, a = 23.9471 
0.8769 
0.0912 
0.0088 

Midili Kucuk 
k = 0.0314, b = 0.0007, a = 0.862, n = 0.7472 
0.9505 
0.0247 
0.0006 

Wang and smith 
a = 0.0205, b = 0.0002 
0.8599 
0.0661 
0.0045 

Hii 
k = 0.0375, g = 0.147, a = 0.8714, c = 0, n = 0.675 
0.9547 
0.0236 
0.0006 

Modified Henderson and Pabis 
k = 5.3985, a = 0.0466, g = 5.4491, b = 0.0465, h = 0.7854, c = 0.0175 
0.9389 
0.0276 
0.0008 

Modified Page I 
k = 0.01, n = 0.5136 
0.9785 
0.0251 
0.0006 

Modified Page II 
k = 0.8392, a = 0.0003, n = 0.7881, L = 0.0043 
0.9359 
0.0281 
0.0008 

50 °C 
Newton 
k = 0.0313 
0.9817 
0.0534 
0.0029 

Henderson and pabis 
k = 0.0259, a = 0.8732 
0.9826 
0.0252 
0.0007 

Page 
k = 0.0823, n = 0.7119 
0.9773 
0.0290 
0.0009 

Logarithmic 
k = 0.0295, a = 0.8242, c = 0.0589 
0.9823 
0.0254 
0.0007 

Two term 
k = 0.0292, g = 0.0309, a = 2.6877, c = 1.8162 
0.9825 
0.0252 
0.0007 

Vermal 
k = 0.7364, g = 0.0243, a = 0.1633 
0.9927 
0.0163 
0.0003 

Diffusion approach 
k = 0.0312, g = 0.9999, a = 24.6214 
0.9817 
0.0534 
0.0030 

Midili Kucuk 
k = 0.0326, b = 0.0007, a = 0.8944, n = 0.9633 
0.9822 
0.0254 
0.0007 

Wang and smith 
a = 0.0303, b = 0.0003 
0.9647 
0.0539 
0.0030 

Hii 
k = 0.0394, g = 0.1466, a = 0.9058, c = 0, n = 0.8927 
0.9838 
0.0243 
0.0007 

Modified Henderson and Pabis 
k = 5.3743, a = 0.0343, g = 5.2658, b = 0.0346, h = 0.9673, c = 0.0177 
0.9812 
0.0261 
0.0008 

Modified Page I 
k = 0.03, n = 0.7222 
0.9778 
0.0290 
0.0009 

Modified Page II 
k = 0.9076, a = 0.0003, n = 0.8906, L = 0.0043 
0.9839 
0.0242 
0.0006 

60 °C 
Newton 
k = 0.0365 
0.9935 
0.0277 
0.0008 

Henderson and pabis 
k = 0.0338, a = 0.9417 
0.9933 
0.0182 
0.0003 

Page 
k = 0.0542, n = 0.8786 
0.9924 
0.0198 
0.0004 

Logarithmic 
k = 0.0386, a = 0.8934, c = 0.0598 
0.9935 
0.0180 
0.0003 

Two term 
k = 0.0319, g = 0.0312, a = 2.7569, c = 1.8188 
0.9929 
0.0189 
0.0004 

Vermal 
k = 0.5178, g = 0.0323, a = 0.0886 
0.9943 
0.0169 
0.0003 

Diffusion approach 
k = 0.0362, g = 1, a = 25.2481 
0.9935 
0.0286 
0.0009 

Midili Kucuk 
k = 0.0349, b = 0.0008, a = 0.9486, n = 1.0132 
0.9937 
0.0178 
0.0003 

Wang and smith 
a = 0.033, b = 0.0003 
0.9917 
0.0290 
0.0009 

Hii 
k = 0.0386, g = 0.1573, a = 0.958, c = 0, n = 0.9704 
0.9945 
0.0167 
0.0003 

Modified Henderson and Pabis 
k = 5.244, a = 0.0252, g = 4.9145, b = 0.0262, h = 1.2522, c = 0.0181 
0.9944 
0.0168 
0.0003 

Modified Page I 
k = 0.0361, n = 0.8758 
0.9927 
0.0196 
0.0004 

Modified Page II 
k = 0.9695, a = 0.0003, n = 0.9153, L = 0.0042 
0.9934 
0.0181 
0.0004 

65 °C 
Newton 
k = 0.0569 
0.9883 
0.0495 
0.0025 

Henderson and pabis 
k = 0.0391, a = 0.8294 
0.9520 
0.0451 
0.0021 

Page 
k = 0.15, n = 0.641 
0.9891 
0.0208 
0.0005 

Logarithmic 
k = 0.0772, a = 0.7498, c = 0.1779 
0.9946 
0.0147 
0.0002 

Two term 
k = 0.0391, g = 0.0391, a = 8.7107, c = 7.8813 
0.9520 
0.0451 
0.0022 

Vermal 
k = 0.1154, g = 0.0134, a = 0.6439 
0.9907 
0.0215 
0.0005 

Diffusion approach 
k = 0.0495, g = 1, a = 27.5739 
0.9731 
0.0709 
0.0053 

Midili Kucuk 
k = 0.091, b = 0.0025, a = 0.9565, n = 0.8572 
0.9960 
0.0126 
0.0002 

Wang and smith 
a = 0.0454, b = 0.0006 
0.9714 
0.0592 
0.0037 

Hii 
k = 0.1035, g = 0.1648, a = 0.9644, c = 0.0029, n = 0.7828 
0.9965 
0.0118 
0.0002 

Modified Henderson Pabis 
k = 3.8577, a = 0.0273, g = 2.757, b = 0.0196, h = 2.0228, c = 0.0525 
0.9946 
0.0147 
0.0002 

Modified Page I 
k = 0.0518, n = 0.641 
0.9891 
0.0208 
0.0005 

Modified Page II 
k = 1.0075, a = 0.0028, n = 0.6346, L = 0.0018 
0.9892 
0.0208 
0.0005 

70 °C 
Newton 
k = 0.0516 
0.9559 
0.0721 
0.0054 

Henderson and pabis 
k = 0.0567, a = 1.0799 
0.9483 
0.0653 
0.0046 

Page 
k = 0.0122, n = 1.5041 
0.9831 
0.0383 
0.0016 

Logarithmic 
k = 0.0154, a = 2.5084, c = 1.4897 
0.9914 
0.0262 
0.0008 

Two term 
k = 0.1084, g = 0.1183, a = 11.9867, c = 11.044 
0.9811 
0.0393 
0.0018 

Vermal 
k = 0.0045, g = 0.0005, a = 6.8512 
0.9944 
0.0211 
0.0005 

Diffusion approach 
k = 0.1079, g = 1.0077, a = 114.8372 
0.9783 
0.0420 
0.0019 

Midilli Kucuk 
k = 0.0074, b = 0.0305, a = 0.9928, n = 0.3756 
0.9946 
0.0206 
0.0005 

Wang and smith 
a = 0.0336, b = 0.0001 
0.9944 
0.0211 
0.0005 

Hii 
k = 0.0063, g = 0.0006, a = 4.0365, c = 3.0565, n = 1.0683 
0.9946 
0.0206 
0.0005 

Modified Henderson and Pabis 
k = 11.3347, a = 0.1183, g = 5.7902, b = 0.1081, h = 6.4863, c = 0.1091 
0.9811 
0.0393 
0.0019 

Modified Page I 
k = 0.0535, n = 1.5041 
0.9831 
0.0383 
0.0016 

Modified Page II 
k = 0.9243, a = 0.0119, n = 1.8134, L = 1.7207 
0.9888 
0.0298 
0.0010 

Sun drying 
Newton 
k = 0.0255 
0.8741 
0.0789 
0.0064 

Henderson and pabis 
k = 0.0182, a = 0.8566 
0.8512 
0.0521 
0.0029 

Page 
k = 0.1166, n = 0.5187 
0.9623 
0.0265 
0.0007 

Logarithmic 
k = 0.1096, a = 0.5058, c = 0.4941 
0.9933 
0.0110 
0.0001 

Two term 
k = 0.018, g = 0.018, a = 9.2632, c = 8.407 
0.8511 
0.0521 
0.0030 

Vermal 
k = 0.1096, g = 0.0002, a = 0.5082 
0.9932 
0.0111 
0.0001 

Diffusion approach 
k = 0.0255, g = 1, a = 97.6266 
0.8741 
0.0789 
0.0067 

Midili Kucuk 
k = 0.09, b = 0.0072, a = 1.0184, n = 0.7732 
0.9896 
0.0137 
0.0002 

Wang and smith 
a = 0.0355, b = 0.0006 
0.9525 
0.0370 
0.0014 

Hii 
k = 0.0275, g = 0.0007, a = 5.6962, c = 4.6508, n = 0.3707 
0.9530 
0.0292 
0.0010 

Modified Henderson and Pabis 
k = 14.4079, a = 0.0465, g = 7.6862, b = 0.0616, h = 7.7277, c = 0.0329 
0.9959 
0.0086 
0.0001 

Modified Page I 
k = 0.0159, n = 0.5186 
0.9623 
0.0265 
0.0007 

Modified Page II 
k = 1.0388, a = 0.0022, n = 0.4821, L = 0.0002 
0.9649 
0.0252 
0.0007 
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