Mathematical Modelling of Orange Slices during Microwave, Convection, Combined Microwave and Convection Drying

Yıl 2014, Cilt: 1 Sayı: 2, 143 - 149, 26.07.2014

Sevil Karaaslan Tunahan Erdem

Öz

In this work, the microwave (180, 360, 540, 720 and 900 W), convective (100, 150, 200 ºC), combined microwave (180, 360 and 540 W) and convective drying (100, 150, 200 ºC) behaviours on drying time, moisture ratio of orange slices were investigated. The drying data were applied to nine different mathematical models, namely Page, Henderson and Pabis, Logarithmic, Wang and Singh, Diffusion Approach, Verma, Two Term, Two Term Exponential, Midilli-Kucuk Equation Models. The performances of these models were compared according to the coefficient of determination (R2), standard error of estimate (SEE) and residual sum of squares (RSS), between the observed and predicted moisture ratios. Results showed that the Midilli-Kucuk equation gave the best prediction to the drying kinetics evidenced by coefficient of determination, R2 ranging from 0.9964 – 0.9999.

Anahtar Kelimeler

Orange, drying, microwave, convection

Mikrodalga, Sıcak Hava ve Mikrodalga-Sıcak Hava Kombinasyonu ile Kurutulan Portakal Dilimlerinin Matematiksel Modellemesi

Öz

Bu çalışmada, mikrodalga (180, 360, 540, 720 ve 900 W), sıcak hava (100, 150, 200 °C), mikrodalga (180, 360 ve 540 W) ve sıcak hava (100, 150, 200 °C) kombinasyon yöntemlerinin, portakal dilimlerinin kuruma süresi ve nem oranı üzerine etkileri incelenmiştir. Page, Henderson ve Pabis, Logaritmik, Wang ve Singh, Difüzyon Yaklaşımı, Verma, İki terimli, İki terimli Üssel, Midilli-Küçük olmak üzere dokuz farklı matematiksel modeller birbirleri ile karşılaştırılmıştır. Bu modellerin performansları gözlemlenen ve tahmini nem oranları arasında belirtme katsayısı değeri (R2), tahmini standart hatası (SEE) ve kalanların kareleri toplamına (RSS) göre karşılaştırılmıştır. Sonuçlar göstermiştir ki R2‘si 0.9964-0.9999 aralığında olan Midilli- Küçük modeli en iyi tahmini vermiştir.

Anahtar Kelimeler

Portakal, kurutma, mikrodalga, sıcak hava

Kaynakça

• 1 MR=exp(-kt n ) Page (Agrawal and Singh, 1977) 2 MR=a exp(-kt) Henderson and Pabis (Akpınar et al., 2006) 3 MR=a exp(-kt)+c Logarithmic (Yaldız et al., 2001) 4 MR=1+at+bt 2 Wang and Singh (Wang and Singh, 1978) 5 MR=a exp(-kt)+(1-a)exp(-kbt) Diffusion Approach (Toğrul and Pehlivan, 2003) 6 MR=a exp(-kt)+(1-a)exp(-gt) Verma (Verma et al., 1985) 7 MR=a exp(-kt)+bexp(-k 1 t) Two Term (Alibas, 2012) 8 MR=a exp(-kt)+(1-a)exp(-kat) Two Term Exponential (Sharaf et al., 1980) 9 MR=a exp(-k(t m )+bt Midilli and Kucuk (Sacılık and Elicin, 2006)
• Convective drying experiments were conducted at the temperature values of 100, 150, and 200 °C. The effect of changing the temperature in the microwave oven on the moisture ratio curve of orange samples is shown in Fig 3. The total drying times to reach the final moisture content of the orange slices at the temperatures of 100, 150 and 200 °C were 184, 98 and 68 min, respectively. It is clearly shown that the air temperature had a importantly effect on the drying time. From the range analysis of the experiments, it can be found that the drying time is longest at 100 °C, and shortest at 200 °C. Similar findings were reported by (Alıbas, 2006) for chard leaves, (Vega-Galvez et al., 2008) for apple samples.
• Mathematical modelling of drying curves The nine thin layer drying models were compared in terms of the statistical parameters R 2 (Coefficient of determination), SEE (Standard error of estimate), RSS (residual sum of square) Nine thin layer drying models were used as described by several researchers and were shown in Table 1. The statistical analyses results applied to 9 drying models at drying process at 180, 360, 540, 720 and 900 W microwave output powers; 100, 150, 200 ºC drying air temperatures; 100, 150 and 200 ºC drying air temperatures at constant microwave powers of 180 W, 360 W and 540 W are given in Tables 2, 3, 4 and 5 for orange slices. In this work, the thin layer drying model in which (R) value was closest 1.0000 and smallest SEE and RSS values were chosen to be the most optimum model. To take into account the effect of the drying variables on the Midilli–Kucuk model constants a, k, m and b were regressed against those of drying air temperatures using multiple regression analysis (Ertekin and Yaldız, 2004). Based on the multiple regression analysis, the accepted model was as follows: bt kt a M M M M b m k a MR m e e       ) exp( ) , , , ( (2)
• Figure 3. Variation of experimental and predicted moisture ratio by Midilli model with drying time at selected temperatures Table Non-linear regression analysis results for microwave drying of orange slices under microwave power Microwave power Statistics No 1 2 3 4 5 6 7 8 9 180W R 2 0.9969 0.9559 0.9971 0.9928 0.9237 0.9237 0.9559 0.9237 0.9996 SEE(±) 0.0180 0.0675 0.0176 0.0273 0.0899 0.0899 0.0691 0.0899 0.0064 RSS 0.0140 0.1960 0.0131 0.0320 0.3393 0.3393 0.1960 3393 0.0017 360W R 2 0.9984 0.9081 0.9886 0.9810 0.8640 0.8640 0.9081 0.8640 0.9996 SEE(±) 0.0144 0.1086 0.0408 0.0494 0.1358 0.1358 0.1148 0.1358 0.0074 RSS 0.0040 0.2241 0.0166 0.0465 0.3317 0.3317 0.2241 0.3317 0.0009 540W R 2 0.9997 0.9452 0.9886 0.9822 0.9162 0.9162 0.9452 0.9162 0.9998 SEE(±) 0.0064 0.0853 0.0408 0.0486 0.1106 0.1106 0.0943 0.1106 0.0059 RSS 0.0005 0.0800 0.0166 0.0260 0.1224 0.1224 0.0800 0.1224 0.0003 720W R 2 0.9983 0.9326 0.9874 0.9821 0.9066 0.9066 0.9326 0.9066 0.9994 SEE(±) 0.0159 0.1013 0.0465 0.0523 0.1265 0.1265 0.1149 0.1265 0.0110 RSS 0.0023 0.0924 0.0173 0.0246 0.1279 0.1279 0.0924 0.1279 0.0008 900W R 2 0.9991 0.9330 0.9856 0.9804 0.9098 0.9098 0.9330 0.9098 0.9995 SEE(±) 0.0121 0.1040 0.0521 0.0562 0.1303 0.1303 0.1230 0.1303 0.0106 RSS 0.0010 0.0757 0.0163 0.0221 0.1019 0.1019 0.0757 0.1019 0.0006
• SEE: Standard error of estimate; R 2 : Coefficient of determination; RSS: residual sum of square Drying time decreased substantially with increased microwave power and temperature. Different mathematical models, namely Page, Henderson and Pabis, Logarithmic, Wang and Singh, Diffusion Approach, Verma, Two Term, Two Term Exponential, Midilli-Kucuk Equation Models used to describe the drying kinetics of orange slices. The Midilli-Kucuk model gave excellent fit for all data points with higher R 2 values and lower SEE and RSS values. Conclusions In this study, an experiment of microwave and convective drying orange slices are presented. The effects of different microwave power and temperature levels on the drying of orange slices were considered based on the drying parameters such as the drying time and moisture ratio. Table Non-linear regression analysis results for microwave drying of orange slices under microwave powerair temperature No No Table Non-linear regression analysis results for microwave drying of orange slices under microwave powerair temperature 540 W 100 ºC 150 ºC 200 ºC No R 2 SEE (±) RSS R 2 SEE (±) RSS R 2 SEE (±) RSS 1 0.9995 0.0078 0.0007 0.9992 0.0104 0.0011 0.9973 0.0185 0.0034 2 0.9497 0.0816 0.0732 0.9207 0.1011 0.1023 0.9311 0.0943 0.0889 3 0.9898 0.0385 0.0148 0.9902 0.0375 0.0127 0.9940 0.0294 0.0078 4 0.9854 0.0439 0.0212 0.9846 0.0446 0.0199 0.9902 0.0356 0.0127 5 0.9256 0.1040 0.1082 0.8821 0.1300 0.1520 0.9009 0.1191 0.1277 6 0.9256 0.1040 0.1082 0.9906 0.0120 0.0385 0.8962 0.1219 0.1338 7 0.9497 0.0902 0.0732 0.9207 0.1131 0.1023 0.9311 0.1054 0.0889 8 0.9256 0.1040 0.1082 0.8821 0.1300 0.1520 0.9009 0.1191 0.1277 9 0.9997 0.0064 0.0004 0.9997 0.0064 0.0003 0.9996 0.0076 0.0005
• SEE: Standard error of estimate; R 2 : Coefficient of determination; RSS: residual sum of square Table Non-linear regression analysis results for microwave drying of orange slices under air temperature; SEE Standard error of estimate; R 2 , coefficient of determination; RSS, residual sum of square 100 ºC 150 ºC 200 ºC No R 2 SEE (±) RSS R 2 SEE (±) RSS R 2 SEE (±) RSS 1 0.9926 0.0232 0.0981 0.9991 0.0087 0.0073 0.9997 00052 0.0018 2 0.9727 0.0445 0.3620 0.9754 0.0451 0.1977 0.9616 0.0551 0.2035 3 0.9997 0.0046 0.0038 0.9981 0.0124 0.0148 0.9943 0.0213 0.0300 4 0.9997 0.0049 0.0043 0.9951 0.0202 0.0394 0.9878 0.0311 0.0647 5 0.9595 0.0544 0.5376 0.9504 0.0644 0.3982 0.9207 0.0798 0.4201 6 0.9595 0.0544 0.5376 0.9504 0.0644 0.3982 0.9207 0.0798 0.4201 7 0.9727 0.0447 0.3620 0.9754 0.0456 0.1977 0.9616 0.0559 0.2035 8 0.9595 0.0544 0.5376 0.9504 0.0644 0.3982 0.9207 0.0798 0.4201 9 0.9997 0.0044 0.0035 0.9999 0.0035 0.0012 0.9997 0.0051 0.0017 SEE: Standard error of estimate; R 2 : Coefficient of determination; RSS: residual sum of square References
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