HEAT TRANSFER MODEL FOR SUGAR BEET STORAGE PILE By Mona Shaaban Mahmoud Shaaban A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Horticulture Master of Science 2020 A BSTRACT HEAT TRANSFER MODEL FOR SUGAR BEET STORAGE PILE By Mona Shaaban Mahmoud Shaaban Harvested sugar beet ( Beta vulgaris L.) are stored in cold regions in large piles exposed to ambient weather conditions and fluctu ate temperatures during the winter storage period, which lasts for four months. To better understand the impact of air temperature on the pile temperature. A two - dimensional (2D) heat transfer steady - state model was designed to predict the temperature profile of the pile . To validate the model , temperatures obtained from the model were compared with the temperatures measured from onsite commercial piles during the storage seasons from November 2011 to January 2012 in the first season and fro m November 2012 to February 2013 in the second season in Reese, MI. T he model tended to underestimate the pile temperature ( °C ). The mean difference between . Daily rate of suga r loss (kg/metric ton/day) based on measured and modeled temperatures were calculated and compared for model accuracy. The mean of the daily sugar loss based on the modeled pile temperature was significantly . Additionally, three zones (upper, middle and lower) of the pile w ere stud ied for the model accuracy . There was a significant difference between the modeled and measured pile temperature between the three zones in the second season, whereas the first seaso 0.0 5). Moreover, a comparison of predicted sugar loss as a function of pile geometry was conducted under 2012 air temperature and a 3 ° C increase in air temperature relative to 2012 data . iii This thesis is de dicated to Dr. Sherine Awad and My Family . Thank you for being who you are. iv ACKNOWLEDGEMENT S All the praises and thanks be to Allah, the Lord of the 'Alamin (mankind and all that exists) . I would also like to express my deepest appreciation to my major advisor Dr. Rand olph Beaudry for his enthusiasm, continuous help and persistent guidance that have a great effect in my life either inside or outside the academic field . I would also like to sincerely thank Dr. Linda Hanson whose guidance towards many authentication contributions to my study has lasting effect. grateful to Dr. David Hodge for his valuable su pport and his welcoming attitude. I would greatly thank Michigan Sugar Company for its funding and technical support during my study. I would like to present my great thanks to the D epartment of Horticulture, the Graduate School, CVIP, Office of Internatio nal Students and Scholars and HOGS Club for providing financial help during my study. I would like to express my gratitude to my col l e a g u e s S afa Al zohairy , Shijian Zhuang, Pat Murad , Khaled Yousef and Daniel Wyrembelski for their contribution to technical support . I strongly express my thanks to my lab colleague Diep Tran for her lovely feelings and warm tho ughts. I would also like to thank my lab colleagues Dr. Sangeeta Dhingra, Dr. Sangram Dhumal, Dr. Nihad Smairat, Dr. Mah m ud Tengku Muda Mohame d , Patric k Abeli, Rossella Briano and George Henrique for their big influence during my research. The support before and during my master 's study from my husband, Dr. Ahmed Rady was crucial in completing my master 's program and his sustained ambition ins piring me to continue in my way. From the deep of my heart I want to thank my lovely children Yusuf, Jan a and Omar Rady for their patience for my absence from some of their important moments , but my wish that one day they will give me an excuse and be proud of m e v I also have so many feeling s but can no t find words to express my lo ve and appreciation to my mother Nadia, I can only say to her that my life, happiness and success ca nno t be possible without you. And to the spirit of my father Shaaban, I want to say to him that any time I write my name I write yours to remember to pray f or you even during my busy life, I will keep the last until the last day of my life whatever the responsibilities I have and the way is hard until we meet in your beautiful place that you told me in my dreams and you feel proud of me. To m y only brother and my first friend Mahmoud, I know that we have thousands of miles apart but you never stop thinking of me and I never do, please keep your praying and warm thoughts for me and my family becaus e we feel them in our life every day . Dr. Sherine Awad , although I was like a tree whose brown leaves were falling and its dry branches were fragile, you believed in me, stayed beside me, watered me and nourished me. Until new buds revived, green leaves gr ew and cheerful flowers bloomed. I cannot thank you enough. Last, and of big importance for me, I would like to give my unlimited thanks and appreciation to my extended family in Egypt for their continues praying and encouragement and to the Islamic commu nity in East Lansing for being family and friends who occupied a special place in my heart and for being a shelter and a shield for all my family in the hard days (Jazakom Allah Khairan). vi TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ................ vi LIST OF FIGURES ................................ ................................ ................................ ............... vi LIST OF ALGORITHMS ................................ ................................ ................................ ..... vi KEY TO SYMBOLS ................................ ................................ ................................ ............ vi INTRODUCTION ................................ ................................ ................................ .................. 1 ECONOMIC IMPORTANCE OF SUGAR ................................ ................................ ....... 1 SUGAR BEET PRODUCTION AND STORAGE ................................ ............................ 1 SUGAR BEET LOSS DURING STORAGE ................................ ................................ ..... 3 OPTIMIZING SUGAR BEET STORAGE CONDITIONS T HROUGH MATHEMATICAL MODELING ................................ ................................ ..................... 5 APPLICATION OF MODELING BIOLOGICAL SYSTEMS ................................ .......... 5 1. Models f or Individual a nd Packed Products ................................ ........................... 6 2. Models f or Bulk Stored Products ................................ ................................ ........... 6 3. Modeling Sugar Beet Storage ................................ ................................ ............... 10 MATERIALS AND METHODS ................................ ................................ ........................... 12 MODEL DESCRIPTION ................................ ................................ ................................ 13 MODEL PARAMETERS ................................ ................................ ................................ 14 1. Sugar Beet Thermal Properties ................................ ................................ ............. 14 2. Heat of Respiration of Sugar Beet ................................ ................................ ........ 1 4 3. Soil Thermal Properties ................................ ................................ ....................... 1 5 4. Air Thermal Properties ................................ ................................ ......................... 16 MODEL ASSUMPTIONS ................................ ................................ .............................. 1 6 GEOMETRY AND BOUNDARY CONDITIONS ................................ .......................... 1 7 MODEL EQUATIONS ................................ ................................ ................................ ... 18 1. Equations for Heat Transfer in the Sugar Beet Pile ................................ ............... 19 2. Equations for Heat Transfer in the Ground ................................ ........................... 2 0 3. Equations for Heat Transfer in the Air ................................ ................................ . 2 0 MONITORING PILE TEMPERATURE ................................ ................................ .......... 2 1 CALCULATING SUGAR LOSS ................................ ................................ ..................... 2 2 PILE ZONE COMPARISONS ................................ ................................ ........................ 2 3 PILE DESIGN EVALUATION ................................ ................................ ........................ 2 4 DATA HANDLING, STATISTICAL ANALYSIS AND EXPERIMENTAL DESIGN ... 2 6 R ESULTS ................................ ................................ ................................ ............................. 28 MODEL ACCURACY BASED ON PILE TEMPERATURE ................................ ........... 3 0 MODEL ACCURACY BASED ON SUGAR LOSS ................................ ........................ 3 2 vii 1. Daily Sugar Loss ................................ ................................ ................................ . 3 2 2. Cumulative Sugar Loss ................................ ................................ ........................ 3 5 PILE ZONE COMPARISON ................................ ................................ .......................... 38 1. Variation o f the Measured Temperature Inside t he Pile ................................ ........ 38 2. Daily Sugar Loss Comparison Between Pile Zones ................................ .............. 4 1 3. Cumulative Sugar Loss Comparison Between Pile Zones ................................ ...... 4 4 MODEL ACCURACY IN DIFFERENT ZONES ................................ ............................ 4 4 1. Temperature Comparison Between Pile Zones ................................ ..................... 4 4 EVALUATION OF PILE GEOMETRIES AND VENTILATION ON SUGAR LOSS ... 48 1. Daily S ugar Loss Comparison Between Pile Geometries a nd Ventilation ............. 48 2. Cumulative Sugar Loss Comparison Between Pile Geometries a nd Ventilation .... 5 1 DISCUSSION ................................ ................................ ................................ ....................... 5 3 SUMMARY AND CONCLUSIONS ................................ ................................ .................... 6 0 LITERATURE CITED ................................ ................................ ................................ .......... 6 3 viii LIST OF TABLES Table 1: Model parameters for sugar beet roots and soil used in developing the heat transfer simulation of stored sugar beet pile in Reese, MI (Ochsner et al., 2001; Tabil et al., 2003a) . ................................ ................................ ................................ ................................ .............. 14 Table 2: Air parameters used in developing the heat transfer simulation of stored sugar beet (Datta, 2002) . ................................ ................................ ................................ ......................... 1 6 Table 3 : Sugar beet pile geometry and boundary conditions used to develop the mathematical model ................................ ................................ ................................ .............. 18 Table 4: Dimensions for different pile designs used for developing heat transfer models in sugar beet storage piles ................................ ................................ ................................ .......... 2 4 Table 5: Measured and modeled beet pile temperatures ( ° C) for 2011 and 2012 averaged across the storage campaign ................................ ................................ ................................ .. 3 2 Table 6: Sugar loss estimates based on m easured and modeled beet pile temperatures for 2011 and 2012 ( ° C) averaged throughout the storage campaign ................................ .............. 3 5 Table 7: Calculated cumulative s ugar loss (kg/metric ton) in field - stored sugar beets based on measured and modeled pile temperature for 2011 and 2012 ................................ .............. 38 Table 8: Temperatures of the lower, middle and upper zones of the sugar beet pile in 2011 and 2012 averaged across the storage campaign. Means are the average of 37 d in 2011 and 97 d in 2012 ................................ ................................ ................................ ........................... 4 1 Table 9: Estimated rate of sugar loss (kg/metric ton/day) for field - stored sugar beets in the upper, middle and lower zones of the beet pile based on measured temperature ..................... 4 4 Table 10: Estimated cumulative sugar loss (kg/metric ton) in the 2011 season (37 days) and the 2012 season (95 days) calculated from the measured temperature of the lower, middle and upper zones of the beet pile ................................ ................................ ................. 4 4 Table 1 1: Significance level , resulting from ANOVA analysis , assessing whether predicted and measured pile temperatures differed in the upper, middle and lower zones over two storage seasons ................................ ................................ ................................ ...................... 48 Table 1 2: Modeled prediction of the d aily rate of sugar loss (kg/metric ton/day) for beet piles having different heights or ventilation in a sugar beet pile based on th e 2012 air temperature ................................ ................................ ................................ ........................... 5 0 ix Table 1 3: Daily rate of sugar loss (kg/metric ton/day) for beet piles having different heights or ventilation in a sugar beet pile based on 3°C increase in air temperature relative to 2012 data ................................ ................................ ................................ ................................ ....... 5 0 Table 1 4: Predicted total sugar loss (kg/metric ton) after 100 days of field storage based on the temperatures obtained from models varying pile height, ventilation and average temperature (+3 °C) for beet piles under the 2012 season ................................ ....................... 5 1 x LIST OF FIGURES Figure 1 : Photograph: Presents a cross section of the studied sugar beet pile showing the dimensions of the studied pile, which is located at Reese, Michigan for 2011 - 2012 and 2012 - 2013 seasons, to study the heat flux distribution inside the pile. ................................ ..... 13 Figu re 2: Schematic design: Illustrates the sugar beet pile and pile boundaries included in the model for the 2011 season . The data used to predict pile temperature included inlet air temperature, wind speed and ground temperature at 50 cm depth ( 5.5 ° C ( Schaetzl et al., 2005)) . The pile dimensions are 45.7 m, 4.9 m and 2 7 . 4 m for the base, height and top, respectively . The black dots are the positions of the thermocouples in the middle of the. The boundaries are assigned to be 9.75 m in height and 91.44 m in width . Arrows o n the left side show the direction of the inlet air ................................ ................................ .............. 1 8 Figure 3: Schematic design: Illustrates the sugar beet pile and pile boundaries included in the model for the 201 2 season . The data used to predict pile temperature included inlet air temperature, wind speed and ground temperature at 50 cm depth ( 5.5 ° C (Schaetzl et al., 2005)) . The pile dimensions are 45.7 m, 4.9 m and 2 7 . 4 m for the base, height and top, respectively . The black dots are the positions of the thermoco uples in the middle of the pile . The boundaries are assigned to be 9.75 m in height and 91.44 m in width . Arrows o n the left side show the direction of the inlet air. ................................ ................................ ........ 1 8 Figure 4: Schematic d esign: Illustrates the sugar beet pile divided into three zones, lower , middle ( ) and upper ( ) for the 2011 season . The colored symbols illustrate the thermocouple locations as distributed in each zone . ................................ ................................ 2 3 Figure 5: Schematic d esign: Illustrates the sugar beet pile divided into three zones, lower , middle ( ) and upper ( ) for the 2012 season . The colored symbols illustrate the thermocouple locations as distributed in each zone. ................................ ................................ 2 3 Figure 6: Schematic d esign: Illustrates The outer pile - shape describes the commercial pile, the insider pile - shape describes the ................................ .... 2 4 Figure 7: Schematic d esign: Illustrates th The outer pile - - shape describes the commercial pile. Arrows show the air direction for the model. ................................ ......... 2 5 Figure 8: Schematic d esign: Illustrates the s domain is surrounding the pile from all sides. Arrows show the air direction for the model. ................................ ................................ ................................ ................................ .............. 2 5 xi Figure 9: Heat transfer diagram: Illustrate the t emperature profile of the Gera Road beet pile on December 5 - 10, 2011. The average air temperature is given in the thermometer to the right of each panel. A, B, C, D and SP indicate five thermocouple harnesses; harness D was embedded approximately 5 cm into the soil. White circles indicate locations of in dividual thermocouples. ................................ ................................ ................................ ...... 29 Figure 10: Graph: Shows the m easured ( ) and modeled ( ) sugar beet whole pile temperatures for the 2011 season in relation to air temperature ( ). ................................ ........ 3 0 Figure 11: Graph: Shows the m easured ( ) and modeled ( ) sugar beet whole pile temperatures for the 201 2 season in relation to air temperature ( ). ................................ ........ 3 1 Figure 12: Boxplot : D isplay s the distribution of the values of measured and modeled temperatures for the beet pile temperature for the 2011 season. Diamonds represent the mean value s ; circles represent the outlier values. ................................ .............................. 3 1 Figure 13: Boxplot : D isplay s the distribution of the values of measured and modeled temperatures for the beet pile temperature for the 201 2 season. Diamonds represent the mean value s ; circles represent the outlier values. ................................ .............................. 3 2 Figure 14: Graph: Shows the d aily sugar loss (kg/metric ton /day) from field - stored sugar beets calculated from m easured ( ) and modeled ( ) pile temperatures for the 2011 season . ................................ ................................ ................................ ................................ ... 3 3 Figure 15: Graph: Shows the d aily sugar loss (kg/metric ton /day) from field - stored sugar beets calculated from m easured ( ) and modeled ( ) pile temperatures for the 201 2 season . ................................ ................................ ................................ ................................ ... 3 3 Figure 16: B oxplot : Displays the distribution of the daily rate of sugar loss (kg/metric ton/day) due to respiration as a function of measured or modeled sugar beet pile temperature for the 2011 season. D iamond represent the mean values ; circle s represent the outlier values. ................................ ................................ ................................ .... 3 4 Figure 17: B oxplot : Displays the distribution of the daily rate of sugar loss (kg/metric ton/day) due to respiration as a function of measured or modeled sugar beet pile temperature for the 201 2 season. D iamond represent the mean values ; circle s represent the outlier values. ................................ ................................ ................................ .... 3 4 Figure 18: Graph: Shows the cumulative sugar loss (kg/metric ton) from field - stored sugar beets calculated from m easured ( , Lm ) and modeled ( , Ld ) pile temperatures for the 2011 season . ................................ ................................ ................................ ........................... 3 6 Figure 19: Graph: Shows cumulative sugar loss (kg/metric ton) from field - stored sugar beets calculated from m easured ( , Lm ) and modeled ( , Ld ) pile temperatures for the 201 2 season . ................................ ................................ ................................ ........................... 3 6 xii Figure 20: Boxplot : Displays the distribution of measured or modeled Cumulative sugar loss (kg/metric ton) for the 2011 season. Diamonds represent the mean values; circles represent the outlier values. ................................ ................................ .............................. 3 7 Figure 21: Boxplot : Displays the distribution of measured or modeled C umulative sugar loss (kg/metric ton) for the 201 2 season. Diamonds represent the mean values; circles represent the outlier values. ................................ ................................ .............................. 37 Figure 22 : Graph: Shows the t emperatures (°C) of the lower ( ), middle ( ) and upper ( ) zones of the sugar for the 2011 season . ..... 39 Figure 23: Graph: Shows the t emperatures (°C) of the lower ( ), middle ( ) and upper ( ) for the 201 2 season . ..... 4 0 Figure 24 : Boxplot : Displays the average temperatures (°C) throughout the storage campaign of lower, middle and upper zones of the sugar beet pile for the 2011 season. Diamonds represent the mean values; circles represent the outlier values. ................... 4 0 Figure 25 : Boxplot : Displays the average temperatures (°C) throughout the storage campaign of lower, middle and upper zones of the sugar beet pile for the 201 2 season. Diamonds represent the mean values; circles represent the outlier values. ................... 4 1 Figure 2 6: Graph: Shows the c alculated daily rate of sugar loss (kg/ metric ton/day) based on measurements of pile temperature (°C) for the lower ( ), middle ( ) and upper ( ) zones of the pile for the 2011 season. The secondary vertical axis is the average daily air ................................ ................................ ................................ ..................... 4 2 Figure 27 : Graph: Shows the c alculated daily rate of sugar loss (kg/ metric ton/day) based on measurements of pile temperature (°C) for the lower ( ), middle ( ) and upper ( ) zones of the pile for the 201 2 season. The secondary vertical axis is the average daily air ................................ ................................ ................................ ..................... 4 2 Figure 28: Boxplot : Displays the average calculated daily sugar loss (kg/metric ton/day) based on measured temperatures (°C) of lower, middle and upper zones of a sugar beet pile for the 2011 season . Diamonds represent the mean value s; circles represent the outlier values. ................................ ................................ ................................ ......................... 4 3 Figure 29: Boxplot : Displays the average calculated daily sugar loss (kg/metric ton/day) based on measured temperatures (°C) of lower, middle and upper zones of a sugar beet pile for the 201 2 season . Diamonds represent the mean value s; circles represent the o utlier values. ................................ ................................ ................................ ......................... 4 3 Figure 30 : Graph: Shows the m easured ( ) and modeled ( ) temperatures (°C) of the upper zone of the sugar beet pile for the 2011 season in relation to .......... 4 5 xiii Figure 31 : Graph: Shows the m easured ( ) and modeled ( ) temperatures (°C) of the upper zone of the sugar beet pile for the 201 2 season in relation to .......... 46 Figure 32: Graph: Shows the m easured ( ) and modeled ( ) temperatures ( ° C) of the middle zone of the sugar beet pile for the 2011 season in relation to ........ 46 Figure 33 : Graph: Shows the m easured ( ) and modeled ( ) temperatures ( ° C) of the middle zone of the sugar beet pile for the 201 2 season in relation to ........ 47 Figure 34 : Graph: Shows the m easured ( ) and modeled ( ) temperatures (°C) of the lower zone of the sugar beet pile for the .......... 47 Figure 3 5: Graph: Shows the m easured ( ) and modeled ( ) temperatures (°C) of the lower zone of the sugar beet pile for the 201 2 .......... 48 Figure 36: Boxplot : D isplay s the distribution of the values for daily sugar loss rate (kg/metric ton/day) of different design s of sugar beet storage pile s based on 2012 air temperatures. Diamonds represent the mean values; circles represent the outlier values. ................................ ................................ ................................ ................................ .... 49 Figure 37: Boxplot : D isplay s the distribution of the values for daily sugar loss rate (kg/metric ton/day) of different design s of sugar beet storage pile s based on predicted 3°C increase in air temperatures relative to 2012 data . Diamonds represent the mean values; circles represent the outlier values. ................................ ................................ ................... 5 0 xiv LIST OF ALGORITHMS Equation 1: Respiration rate CO 2 (mL·kg - 1 ·hr - 1 ) = (outlet CO 2 (%) - inlet CO 2 (%)) x flow rate (mL·hr - 1 )/sample wt (kg) ................................ ................................ ................................ . 1 5 Equation 2: Respiration rate = 0.97 · T - 262.06 ................................ ................................ ....... 15 Equation 3: Q = exp ( - 6291 x (1/T) + 25.472)/182) ................................ ............................... 15 Equation 4: ................................ ................................ ....................... 19 Equation 5: ................................ ................................ ................................ ... 19 Equation 6: k eff p k p + (1 - p ) k + k disp ................................ ................................ ................ 19 Equation 7: ................................ ................................ ....................... 20 Equation 8: ................................ ................................ ................................ ....... 20 Equation 9: ................................ ................................ ....................... 20 Equation 10: ................................ ................................ ................................ ....... 2 0 Equation 11: L d = 0.7784 L m 0.021 ................................ ................................ ...................... 38 xv KEY TO SYMBOLS T ambient temperature (K). Q energy (W · m - 3 ) . density (kg · m - 3 ) . C p the specific heat at constant pressure (J · kg - 1 ·K - 1 ) . u the velocity (m · s - 1 ) . T the temperature gradient (K · m - 1 ) . q heat flux (W · m - 2 ) . k eff the effective thermal conductivity (W · m - 1 · K - 1 ) . p porosity (%) . k thermal conductivity (W · m - 1 · K - 1 ) . k disp Dispersive thermal conductivity (W · m - 1 · K - 1 ). L m = Estimated cumulative sugar loss (kg/metric ton) based on pile measured temperature. L d = Estimated cumulative sugar loss (kg/metric ton) based on pile modeled temperature. 1 INTRODUCTION ECONOMIC IMPORTANCE OF SUGAR Sugar (sucrose) is an important product around the world for home use and in numerous industries (Asadi, 2005; FAOSTAT, 2014) . Sugar is mainly produced from two sources, sugar cane ( Saccharum officinarum ) and sugar bee t ( Beta vulgaris L. ) (FAOSTAT, 1994) . In 201 4 , the wor l d production of sugar cane and sugar beet were 2010.4 and 2 77.7 million metric ton s of raw product , respe ctively (FAOSTAT, 2014). The total crop values of production in 20 1 4 were about $ 224.9 and $1 4.8 b illion for sugar cane and sugar beet , respectively (FAOSTAT, 2014). SUGAR BEET PRODUCTION AND STORAGE USA was ranked as the 1 1 th largest producer of sugarcane in 201 4 with 27. 6 million metric ton s . The production of sugar beet in 201 4 was 2 8 . 4 million metric ton s , making the USA the 4 th largest sugar beet producer (FAOSTAT, 2014). Sugar beet accounted for 57% of the USA sugar production in 2014, while sugarcane accounted for 43% (McConnell and Riche , 2015). This represents a substantial increase of the sugar beet share from 25% in 1920 ( Reference for Business , 2011) . Sugar beet was originally grown in temperate zones in winter and / or summer (Draycott, 1972) . However, it is now cultivated all over the world i n summer in cooler regions and with supplemental irrigation in arid and semi - arid regions (Draycott, 1972) . USA sugar beet production is distributed over several regions; the Red River Valley (Minnesota and North Dakota), the f ar w est area (California, Idaho, Oregon and Washington), t he C entral High p lain area (Colorado, 2 Montana, Nebraska and Wyoming) and t he Great Lakes area (Michigan) with a contribution of 47.45%, 24.13%, 14.41% and 14.01% of the total USA sugar beet production , respectiv ely (USDA, 2015) . In production areas with relatively cold temperatures, h arvested roots are stored in large piles and exposed to ambient environmental c onditions during winter (Bugbee, 1993; Huijbregts et al., 2013) . A single hectare of sugar beet produces approximately 45 metric ton s of roots, with a volume of about 80 m 3 (Campbell and Klotz, 2006; Draycott and Christenson, 2003) . According to McConnell (2015) , approximately 538,000 ha of sugar beet i s grown in the US A . T h e scale of production prevents imm ediate process ing , as the processing capacity of the beet extraction plant can reach up to 17,000 metric ton s per day (BMA, 2010) , thus providing storage facilities is required . Beet h arvest in Michigan begins at the end of September and continues until the middle of November (Ruhlman, 2018) . A fter harvest , t he roots are transferred to the processing plant or placed in specially built piling for storage grounds until process ing (Bugbee, 1993; Huijbregts et al., 2013) . The storage period last s for approximately 1 2 0 days (Van Eerd et al., 2012) and du ring this period, the beet piles are exposed to the surrounding environment. During sugar beet storage period in Michigan (October - April) , the air temper ature range s from - 2 8.7 ° C to 29.8 ° C (according to the data obtained by our team from MSU extension station, located at the MSU Saginaw Valley Beet and Bean Research Farm, Frankenmuth, MI. (Enviro - weather, 2011) for air temperature in the piling areas for 2009 - 2014 storage seasons). In 2005, e levated ambient temperatures during storage in the Great Lakes area contribute d to increas ed respiration and promote d many cases of microbial activity (Poindexter, 2012) . Both phenomena were responsible 3 for a considerable amount of sugar loss that reach ed $25 million after such storage season in Michigan alone (Beaudry and Loescher, 2008) . SUGAR BEET LOSS DURING STORAGE Elevated storage temperature enhances the rate of root metabolism and thereby raises the respiration rate (Vukov, 1977) . During respir ation, stored sugar is oxidized to CO 2 , converted to needed metabolites and yield s the energy for maintaining cell metabolic activities (Siedow and Day, 2017) . Over a storage campaign of 130 d , sugar losses are around 14 pounds per ton of roots , about 4% of the total sucrose (Wyse and Dexter, 1971) . Barr et al. (1940) estimated that 60% of this sugar loss was attribut able to respirat ory losses and Wyse and Dexter (1971) calculated that respiration accounted for roughly 80% of the sugar loss, with the remainder being lost due to the interconversion of sucrose to invert sugars and other impurities. Wyse and Dexter (1971) also noted that the respired CO 2 exceeded that a ccou nted for by sugar loss and that a significant proportion (~38%) of the respired CO 2 was derived from non - sucrose compounds . F luctuat ing temperature s during storage of bulk fresh crops can cause considerable loss (Ullah et al., 2014; Wyse, 1978) . F luctuat ing storage temperatures increase the respiration rate and encourage condensation (Hylmó et al., 1976) . Free water , in combination with elevated temperature , can promote virulence and root deterioration by some storage pathogens such as P homa betae (Cormack and Moffatt, 1961) . Temperature f luctuation additionally reduces the root quality when the temperature decreases sufficiently to cause freezing (Wyse, 1978) . Upon freezing , the root cell membranes rupture and cellular leakage occurs (Ullah et al., 2014; Wyse, 1978) . Upon thawing, t he leak ed sucrose - rich solution is available to bacteria, which form polysaccharide gums that interfere with sugar extraction during the process ing (Campbell and Klotz, 2006) . 4 Storage rot of beet roots is another reason for sugar loss and var ies in severity according to storage temperature (Gaskill and Seliskar, 1952; Liebe and Varrelmann, 2016) and is an important reason for understand ing the temperature patterns inside the storage pile. Phoma betae Frank, Botrytis cinerea L , Penicillium vulpinum (Cooke & Massee) Seifert & Samson (formerly Penicillium claviforme Bainier) and Rhizopus stolonifera are storage rot pathogens (Bugbee, 1982; Bugbee, 1986) . L ow storage temperature (<10 ° C ) slows the development of the rot by P . betae and R . s tolonifera (Cormack and Moffatt, 1961; Fugate and Campbell, 2009; Miles et al., 1977) , while B . cinerea and P . vulpinum can maintain their activity under a wide r range of temperature (Gaskill, 1952; Mumford and Wyse, 1976) . Respiratory sugar loss and fungal growth can be minimized by maintaining storage temperature between 1.5 and 5 ° C , relative humidity between 95 % to 98% and oxygen and carbon dioxide levels at 5% and 6%, respectively (Karnik et al., 1970) . These conditions avoid cell rupture and reduce dehydration and fungal growth (Bugbee, 1993; Campbell and Klotz, 2006; Wyse, 1978) . I n areas such as the Red River Valley region and parts of Canada where the winter temperature is cold enough to ' deep freeze' beet piles (i.e., w h ere weather temperature s remain below - 5 ° C during the storage period ) , piles are kept frozen until processing (Bugbee, 1993; Campbell and Klotz, 2006; Wyse, 1978) . Deep freezing can be used for large ' super ' piles with a base equal to 66 m wide (Bugbee, 1993) , but is also useful for smaller piles (e.g., 23 m wid th) if an increased s urface area for heat exchange is needed (Bugbee, 1993) . 5 OPTIMIZING SUGAR BEET STORAGE CONDITIONS THROUGH MATHEMATICAL MODELING Em pirical approaches have been employed to achieve optimal storage condition s . These include covering the pile with straw (Akeson et al., 1974) or woven polypropylene (Perry, 1989) , forced air cooling or ventilation (currently applied in some locations in Michigan) (Clark, 2012) , deep - freezing (Bugbee, 1993) and storing in small piles or covered clamps (List, 2015) . Although e mpirical approaches help solv e some of the storage problems to obtain uniformity and condition optimization (Kumar and Kalita, 2017) , they are not very useful for modifying storage techniques or developing ne w approaches. On the other hand, mathematical models can test possi ble solutions before practical implementation, with relatively low costs (Ambaw et al., 2013; Xie et al., 2006) . Therefore , mathematical models for optimizing post - harvest handling and storage conditions ha ve been of interest to researchers (Ambaw et al., 2013; Verboven et al., 2006) and n umerical models have been proposed to simulate and predict fluid flow and heat and mass transfer during transportation and stor age for agricultural commodities (Ambaw et al., 2013; Verboven et al., 2006) . APPLICATION OF MODELING BIOLOGICAL SYSTEMS Modeling approaches have been extensively applied to biological systems since the early 2000s (Rennie and Tavoularis, 2009a; Rennie and Tavoularis, 2009b; Verboven et al., 2006) . Many studies were recently performed to simulate and predict the cooling behavior and airflow for individual produc ts and package d and bulk food or have simulated the effect of cooling and/or airflow o n the change in storage conditions and vent ilation . 6 1. Models for Individual and Packed Products Thorpe G. (2006) formulated a transi ent numerical model u sing individual produc t s as discrete entities to determine the efficiency of a hydrocooler for cooling spherical elements . The model quantifies the time needed for a horticultural produce to reach a target temperature after exposure to specific water temperatures. T he study recommen ds the use of the mass - weighted average temperature of the produce to calculate the time needed for cooling instead of using the core temperature , because reaching the target temperature is significantly faster in the recommended method . A transient mathematical model f or forced air cooling of apple was created by Arêdes Martins et al. (2011) to describe the temperature of two apples as a function of the surrounding airflow. The model explains the cooling in the tandem arrangements on appl e trays. The results demonstrated a delay of cooling in the downstream apple relative to the upstream apple . Ferrua and Singh (2009a); Ferrua and Singh (2009b); Ferrua and Singh (2009c ) developed a m athematical model to predict the beh avior of the airflow and temperature of the strawberry and the clamshell packaging during the forced - air cooling of strawberry packages . A irflow calculations were validated us ing a particle image velocimetry (PIV) approach as well as temperature calculations. Later, Ferrua and Singh (2011) improved the commercial strawberry storage system by improving t he design of the clamshells and trays as well as the behavior of the airflow across the cooling chambers , using the early designed mode l . 2. Models for Bulk Stored Products To investigate the change of the airflow , temperature and moisture content in bulk stored potatoes, Xu and Burfoot (1999) developed a transient three - dimensional ( 3D ) c omput ational f luid d ynamics (CFD ) model f or forced air cooling of potato . T emperature and weight loss were used 7 for model validation . The difference between simulated temperature and experimental temperature of a potato bed with 2.4m height and 0.7m diameter reached 1.4 ° C lower in the model than the measured temperatures . T he re also were variation in weight loss between experimental and simulated results reached 5%, due to excessive water evaporation of potato tubers and the possible fluctuation of air speed and increase o f the humidity near the inlet air. As a further example, a t ransient 3D air flow heat and mass transfer model was designed for bulk storage of chicory roots to study the commercial storage systems , and validate the ability of the refrigerated store to maintain root quality (Hoang et al., 2003) , by predicting airflow , dehydration and temperature of chicory roots through a wind tunnel (a closed chamber that can measure and/or control the parameters of the air passing through) . Measured and predicted results of temperature and moisture content were compared to validate the model . T he simulated results underestimate d the actual root temperature . Also, the roots lost more moisture than predicted , which was assumed to be due to the low relative humidity (RH) of the ambient air in the storage , variation in the size of the voids between the roots , non - uniformity in the size of the roots and the large size of the roots compar ed with the size of the voids between the roots . Markarian et al. (2006) proposed a mathematical model that predict s potato temperature and the surrounding relative humidity ( RH ) throughout cold stor a g e. The model also described the res piration and transpiration rate of the tubers during storage . The experiment was conduct ed in a highly controlled condition and the findings were compared to measurements found in the literature with a considerable agreement . The mean of absolute differenc e in temperature was 0.01 ° C, the maximum absolute difference was 0.49 ° C . T o study the airflow and energy transfer characteristics in ventilated layered and bulk apple packages, Zou et al. (2006a) designed transient computational fluid dynamics CFD model based 8 on equations used for porous media . In a second study , Zou et al. (2006b) introduce d user - friendly software to solve the mathematical equations, then the model results were compared with experimental temperature measurements for validation. The model underpredict ed the product temperature . This was thought to be due to errors in the position s of thermocouples, model input values and /or model assumptions. For p otato, Chourasia and Goswami (2007a) exa mine d the eff ect of several parameters, such as rate of generated respiratory heat, void diameter in the medium, RH, tuber size and temperature, on heat and mass transfer patterns in potato bulk storage. The study also included steady - state and transient models and val idated the models using experimental measurements of temperature and moisture loss . In general, the simulation overestimated temperature and moisture loss with an average error of 1.2 ° C and 11.5%, respectively. Later, in another study by Chourasia and Goswami (2007b) , a 2D CFD - based model was developed to compute the change in airflow, heat transfer and moisture content in potato bulk storage and they again examine d the model accuracy by comparing the modeled data with experimental data obtained from commercial potato storage. T he model was able to predict the potato temperature with an average error of approximately to 0 .5 ° C , while the model over - predict ed moisture loss by 61% . T he difference between simulated and experimental measurement s was thought to be a result of chang ing storage condition s for stored tubers due to the unloading of stored potatoes throughout three months of storage. Thorpe G.R. (2008) measured the change in temperature and moisture content of stored grain and design ed a CFD model . T he model predict ed the variation of temperature and moisture content of the stored grains , but the results were not evaluated for accuracy by comparing the modeled data with experimental measurements . 9 Xie et al. (2006) applied a CFD model to obtain the optimal flow and temperat ure parameters for controll e d cold storage of apples . The proposed 2D model was used to show the behavior of airflow and temperature tra nsport during forced air cooling . T he model was also used to evaluate the effect of several stacking patterns on the airflow and temperature of stored foodstuffs. The model was reliable ; the error range in temperature was ± 2 ° C. Airflow through a random arrangement of horticultural products in package s was modeled by Delele et al. (2008) , t he model was developed based on measurements of low resistance according to change s in a confinement ratio, void to product ratio and box vent ratio in random stacking . Literature measurements were used for validation with good agreement. The influence of product position and package vent arrangement on airflow and t emperature characteristics was studied by Tutar et al. (2009) . A transient CFD model was used for the simulation. The study showed a significant effect of the inflow rate compar ed with the vent rate on airflow and product temperature s . However, model validation was n o t performed . To investigate the ch ange of spherical produce temperature according to a change of vent area, Dehghannya et al. (2011) used solid polymer balls for simulation and develop ing a transient 2D airflow and energy transfer simulation model. The model results confirm the hypothesis that im p roving ventilation by increasing the number of package vents from 1 to 5 ( with area equal to 2.4 % to 12.1% of the package area, respectively ) can improve temperature uniformity inside the package between the produce units as demonstrated by a reduced heterogeneity index from 61.5 % to 5.6% . Validation occurred by comparing the modeled core temperature by measured temperature of the balls . Deviation from the m odeled temperature was found d ue to some inaccurate parameter inputs such as the velocity and temperature of airflow, the thermal properties of the simulating balls and the fail ure of the numerical model to reach an it e rative solution . 10 Tanaka et al. (2012) used a CFD modeling system to study the behavior of the air flow in a semi - loaded truck and its influence on product temperature . T he model was validated with measurements of temperature and air velocity with a mean error of 1.4 ° C and 0.36 m s - 1 , respectively. Later the model was used to find the best loading configuration of the packages as the results suggest ed arranging the packages flat with considerable gaps is optimal for homogeneity of air cooling and temperature distribution. These findin gs were not validated with a real experiment. However, the results were consistent with expectations , because this way of arranging packages allow s the cold air to consistently reach the commodities and achieve temperature uniformity. 3. Modeling Sugar Beet Storage There have been several attempts to m odel sugar beet storage . Bakker - Arkema and Bickert (1966) design ed a model to simulate the airflow and energy tran sfer of a ventilated deep - bed of sugar beet . Significant d ifferences between measured and modeled cooling ra te were found and thought to be from excluding mass transfer between beets and surrounding fluid. La ter, Andales et al. (1979) used the finite difference approach to develop a 2D model for the temperature and weight loss of a v entilated s ugar beet pile . They compar ed measured and modeled values for model validation . De viations between measured and modeled values were mainly attributed to the change of porosity and heat loss at the pile walls . Holdredge and Wyse (1982) developed a relatively simpl e model , compar ing to that of Andales et al. (1979) , the model was reduced to one dimension as they found no significant variation of temperature in the horizontal plane. The model was verified and tested using an insulated box simulating a section of the commercial pile. The model validation show ed good agreement between experiment al and simulated values under the low and moderate air flow 5.2 and 10.4 m 3 /k s - metric ton (10 to 20 cfm/ton) , respectively . However, the fit 11 was poor under the high air flow rate applied ( 20.8 m 3 /k s - metric to n ; 40 cfm/ton ) , which the authors attributed to non - continuous fan operation hot air into the pile . The previous studies focused on modeling a ventilat ed pile. No models exist to describe the temperature profile insid e unventilated beet pile s. However, in Michigan about 50% of the piles are currently unventilated ( J . Stewart, Michigan Sugar Company , personal communication ) . Further , there is a 2 to 4 °C increase in the average global temperature is predicted within this century (New et al., 2011) . Thus, more effective pile architectures could be of increasing value. Therefore, the main objective of this project w as to develop a two - dimensional mathematical model using a finite - element approach t o simulate heat transfer in commercial beet pile s under non - ventilated condition s. Such a model can be used to describe the temperature profile of an unventilated sugar beet pile that dire ctly affect s sugar loss and microbial activ ity during storage. D esigning such a model can assist in decisions regarding pile management (size and duration) . The s econd objective was to study the effect of the spatial variation of the pile on tem perature distribution and sugar loss . Finally, we use d the model to better understand how pile shape and ventilation presence affects sugar loss during storage . 12 MATERIALS AND METHODS Modeling a involved several steps . T he first step contains developing and solving model equations for beets, air and ground thermal properties and input data (i.e. air temperature, air velocity and relative humidity ( RH ) ) as stated in the model description section . The second step was monitoring the pile temperature by measuring the temperature of predefin ed points in a storage pile then comparing the collected temperatures with the temperatures obtained from the model at the same points to test the accuracy of the model. The third step was calculating sugar loss using the measured temperature and comparing it with the sugar loss calculated using modeled temperature. The fourth step was to evaluate the pile as three horizontal zones to help understand the model accuracy for each zone. The fifth step was to use the model to predict how new systems of storage affect root storability then giving a recommendation for the best storage system according to decreased predicted sugar loss . The model design and simulations in this research were based on the pile structure and conditions measured at a commercial sugar beet pile of the Michigan Sugar Company at the Gera road piling ground, Reese, Michigan (43.409337 ° N, 83.739412 ° W). The dimen sions of the commercial pile were 4 5 . 7 m, 2 7 . 4 m and 4.9 m for the base, top and height, respectively as illustrated in Fig. 1. 13 Figure 1 : Photograph: Presents a cross section of the studied sugar beet pile showing the dimensions of the studied pile, which is located at Reese , Michigan for 2011 - 2012 and 2012 - 2013 season s, to study the heat flux distribution inside the pile . Important input parameters were required for the model includ ing : soil and sugar bee t thermal properties (described below) and heat of respiration generated by the sugar beet, in addition to air temperature, air velocity and relative humidity (RH) were obtained from the weather station located at the MSU Saginaw Valley Beet and Bean Research Farm, Frankenmuth, MI. , ( 43.3995° N, 83,6980° W ) (Enviro - weather, 2011) . MODEL DESCRIPTION Input parameters underwent in a finite element analysis through a mathematical model to calculate the rate of heat gain from the ground and respiratory activity and heat loss to the environment. The model was built and integrated using finite element software COMSOL (COMSOL Multiphysics ® 4 . 3 b , COMSOL AB, Stockholm, Sweden ). For simplification of the calculations , t he model was developed assuming a steady - state heat transfer condition in two dimensions (2D) o n a daily basis, which means the environmental conditions changed from day to day but wa s considered constant during the day. However, this assumption might not be accurate during the day especially with the fluctuating weather in the piling site based on observations. The 14 model included heat convection at the pile surface and heat conduction inside the pile between beets and between the pile and the ground. MODEL PARAMETERS 1. Sugar Beet Thermal Properties Tabil et al. (2003a) , measured the thermal proper ties of sugar beet roots including density and specific heat ( Table 1 ) which were used i n the model. Root thermal conductivity (k p ) was calculated as a function of temperature (K) and was taken to equal 0.6 W. m - 1 .K - 1 . T he thermal conductivity for frozen roots on the other hand was taken as 1.16 W.m - 1 .K - 1 ; freezing occurs at temperatures equal to or lower than - 5 to - 2 ° C (Campbell and Klotz, 2006) . Table 1 : M odel parameters for sugar beet roots and soil used in developing the heat transfer simulation of stored sugar beet pile in Rees e , MI (Ochsner et al., 2001; Tabil et al., 2003a) . Parameter Value Unit Soil density 1700 kg · m - 3 Soil thermal conductivity 0.525 W · m - 1 ·K - 1 Soil specific heat 1 . 615 k J · kg - 1 ·K - 1 Root specific heat 3.5464 k J · kg - 1 ·K - 1 Root density 1169.9 kg ·m - 3 2. Heat of Respiration of Sugar Beet The respiration rate of sugar beet roots was calculated on the average rate measured for 38 cultivars. Three beet samples from each cultivar were stored in three different temperatures ( 3, 10 and 20 ° C ) . Each sample was weighed then stored in 20 - L high - d ensity polyethylene pails. The respiration rate was measured using the closed system method (Guevara et al., 2006; Hagger et al., 15 1992; Lee, 1987; Song et al., 1992) . To obtain accurate readings, respiration rate for each sample was measured only after reaching system equilibrium. Each measurement took place by manually injecting gas samples that were derived from the desired pail into a CO 2 analyzer ( Model ADC 225 - MK3, Analytical Development Co., Hoddesdon, England) that uses N 2 as the carrier gas (N 2 flow rate = 100 mL · min - 1 ). Respiration rate was calculated using Eq.1 Equation 1: Respiration rate CO 2 (mL · kg - 1 · hr - 1 ) = (outlet CO 2 ( % ) - inlet CO 2 ( % ) ) x flow rate (mL · hr - 1 )/sample wt (kg) A simple linear regression relationship was performed to obtain the best - fit line to predict the respiration rate as a function of the ambient temperature . Eq. 2 shows the best - fit respiration rate for CO 2 production (mg · k g - 1 · h - 1 ) as a function of ambient temperature T ( K) Equation 2: Respiration rate = 0.97 · T - 262.06 where : T is ambient temperature (K) . To calculate the portion of heat released from the total energy of respiration, Siedow and Day (2017) estimated that plant cells retain approximately 33% of the respiration energy for metabolic processes. Therefore, 67% of the energy associated with respir ation was assumed to be released as heat , producing the exponential relationship between the energy of respiration (Q) per unit volume (W · m - 3 ) and the ambient temperature ( T ) as follows: Equation 3: Q = exp ( - 6291 x (1/T) + 25.472)/182) Where T is in units of K. E quation 3 can be applied in the case of unfrozen roots, while for frozen roots respiration ceases (Campbell and Klotz, 2006) . 3. Soil Thermal Properties Reese, MI is characterized by its clay and loam soil type (Boring, 2009; Meyer, 2009) for which thermal properties are known ( Table 1 ) (Ochsner et al., 2001) . 16 4. Air Thermal Properties The thermal properties of moist air are built in the COMSOL software and were used to develop the model. Table 2 shows the air thermal properties at 0 ° C . Table 2 : A ir parameters used in developing the heat transfer simulation of stored sugar beet (Datta, 2002) . Parameter Value Units Air density 1.225 kg · m - 3 Air thermal conductivity 0.0243 W · m - 1 ·K - 1 Air specific heat 1.005 kJ · kg - 1 ·K - 1 MODEL ASSUMPTIONS Several a ssumptions w ere r equired to s implify m odel d evelopment ; t hese i nclude: Heat energy transfers through the shortest two dimensions of the pile (width and height), whereas the energy transfers through the length can be negligible due to the relatively long dimension. Root density and specific heat do not vary significantly within the tempe rature range for beet piles through the storage period. Ground density, specific heat and thermal conductivity do not vary significantly within the temperature range throughout the storage period. The pile was considered as a porous material. Enthalpy due to the water vapor diffusion from beets was negligible. 17 Inlet air conditions changed daily depending on the average temperature , RH and the average wind velocity. However, the air direction will be assumed constant from south to north based on prevailing w ind conditions . COMSOL Multiphysics predicts the flow pattern of the air, either laminar or turbulent according to Reynolds number as a function of the air velocity and pile dimensions, which is provided as an input in the model. T he ground temperature from November to April at 50 cm deep at Reese MI is 5.5 ° C (Schaetzl et al., 2005) . Therefore, ground temperature was considered constant at that temperature and depth. R oot moisture content was co nsidered constant at 70% to 80% (Tabil et al., 2003b) during the experiment. Porosity is assumed to be constant during the experiment as 41.37% (Tabil et al., 2003b) . GEOMETRY AND BOUNDARY CONDITIONS A geometric representation of the studied pile base, height and top, the dimensions of which are 45.7 m, 4.9 m and 2 7 . 4 m , respectively, is constructed ( Table 3 , Fig. 2 for 2011 and Fig. 3 for 2012 ). We assumed that there are defined boundaries for the active air that interact with the pile , beyond these limits there is no effect from the air on the pile . These boundaries are assigned to be 9.75 m in height and 91.44 m in width , roughly 2x pile dimensions (Table 3) . The assumed bounda ries were obtained based on our preliminary work and for simplicity . 18 Table 3 : Sugar beet pile geometry and boundary conditions used to develop the mathematical model. Type Base (m) Height (m) Top (m) Pile geometry 45.7 4.9 27.4 Boundary dimensions 91.44 9.75 91.44 Figure 2 : Schematic design: Illustrate s the s ugar beet pile and pile boundaries included in the model for the 2011 season . The data used to predict pile temperature included inlet air temperature, wind speed and ground temperature at 50 cm depth ( 5.5 ° C (Schaetzl et al., 2005)) . The pile dimensions are 45.7 m, 4.9 m and 2 7 . 4 m for the base, height and top, respectively . The black dots are the positions of the thermocouples in the middle of the. The boundaries are assigned to be 9.75 m in height and 91.44 m in width . Arrows o n the left side s how the direction of the inlet air. Figure 3 : Schematic design: Illustrate s the sugar beet pile and pile boundaries included in the model for the 201 2 season . The data used to predict pile temperature included inlet air temperature, wind speed and ground temperature at 50 cm depth ( 5.5 ° C (Schaetzl et al., 2005)) . The pile dimensions are 45.7 m, 4.9 m and 2 7 . 4 m for the base, height and top, respectively . The black dots are the positions of the thermocouples in the middle of the pile . The boundaries are assigned to be 9.75 m in height and 91.44 m in width . Arrows o n the left side show the direction of the inl et air. MODEL EQUATIONS For the numerical solution of the proposed problem, the heat transfer governing differential equations were built in and calculated using COMSOL software (COMSOL_Multiphyics, 2013a; COMSOL_Multiphyics, 2013b) . 19 1. Equations for Heat Transfer in the Sugar Beet Pile The beet pi le was considered a porous medium with sugar beet root s as the solid phase and moist air as the fluid phase as described by equations 4 , 5 and 6 (COMSOL_Multiphyics, 2013a; COMSOL_Multiphyics, 2013b) . Equation 4: where: Equation 5: and: Equation 6: k eff p k p + (1 - p ) k + k disp where: : is the density of the beet (kg · m - 3 ), C p : is the specific heat of the beet at constant pressure (J · kg - 1 ·K - 1 ), u: is the velocity of the moist air inside the pile (m · s - 1 ), T: is the temperature gradient (K · m - 1 ), q: is the conductive heat flux (W · m - 2 ), Q: is the respiration heat per unit volume (W · m - 3 ), k eff : is the effective thermal conductivity (W · m - 1 · K - 1 ), p : porosity (%) , k p : beet thermal conductivity (W · m - 1 · K - 1 ), k: moist air thermal conductivity (W · m - 1 · K - 1 ), k disp : Dispersive thermal conductivity (W · m - 1 · K - 1 ). 20 2. Equations for Heat Transfer in the Ground The ground was considered as a solid material and th e following equations were used to solve for the ground temperature (COMSOL_Multiphyics, 2013a; COMSOL_Multiphyics, 2013b) : Equation 7: Equation 8: where: k g = ground thermal conductivity (W · m - 1 · K - 1 ) , : is the density of the soil (kg · m - 3 ), C p : is the specific heat of the soil at constant pressure (J · kg - 1 ·K - 1 ), u: is the velocity field defined by the t ranslational m otion sub - node when parts of the model are moving in the material frame (m · s - 1 ), T: is the temperature gradient (K · m - 1 ), q : i s the conductive heat flux (W ·m - 2 ), Q: is the heat flux (W ·m - 3 ) . 3. Equations for Heat Transfer in the Air Pile cooling occurs by natural convection (Beukema, 1980) ; therefore , moist air was considered as a fluid material and relative humidity (RH) was u sed as an input quantity of moisture . The following equations solve for air temperature (COMSOL_Multiphyics, 2013a; COMSOL_Multiphyics, 2013b) : Equation 9: Equation 10: 21 Where : is the density of moist air (kg · m - 3 ), C p : is the specific heat of the moist air at constant pressure (J · kg - 1 ·K - 1 ), u: is the air velocity (m · s - 1 ), T: is the temperature gradient (K), q: is the conductive heat flux (W ·m - 2 ), Q: is the heat flux from the heat source (or sink) (W ·m - 3 ), k : moist air thermal conductivity (W/m. K). MONITORING PILE TEMPERATURE To monitor the pile temperature , wiring harnesses of T - type (copper/constantan) thermocouples ( OMEGA Engineering , INC., Norwalk , CT, USA) , encased in 6 - mm i.d. polypropylene tubing for protection, were installed in the middle of the beet piles in late October in 2011 and in early November in 2012 . Harnesses were placed on the sloping face of the pile when the pile was partially constructed. Following placement of the harnesses , pile construction was completed , burying the thermocouples within the pile at predefined locations (Fig. 2 for 2011 and Fig. 3 for 2012 ). Each harness had from 1 to 10 thermocouples, depending upon their p osition in the pile. One harness was placed vertically down the face of the pile at the midpoint and another harness was placed diagonally across the face of the pile from its outer shoulder to the base at its midpoint. A third harness ran horizontally al ong the base of the pile to its midpoint and a fourth harness (thermocouples only, no protective tubing used) was buried about 5 cm below the soil surface along the base of th e pile to its midpoint. O ne additional thermocouple was embedded in the pile 22 betw een the vertical harness and the diagonal harness and another two thermocouples were embedded in the pile between the diagonal harness and the horizontal harness. A total of 26 (in 2011 season) and 25 (in 2012 season) locations were monitored. There was 1 failed thermocouple in the 2012 season leading to a decrease in the number of locations in the later season. Temperature measurements were collected every minute using digital dataloggers ( CR - 10, Campbell Scientific, Inc., Logan, Utah, USA) and the average for each hour recorded. Temperature data that was used for model validation was collected from November to January in 2011, and from November to February in 2012. The 2011 storage season was dry and w a rm, limiting storage to two months . CALCULATIN G SUGAR LOSS Evaluating the sugar loss and decrease in quality in the roots during storage are the main way s to evaluate any storage system for sugar beet (Huijbregts et al., 2013) . Thus, Cumulative sugar loss (kg/metric ton) was estimated using measured and modeled temperatures to evaluate the model accuracy and to study the effect of various modeled geometries and ventilation designs of the pile which can be recommended sugar beet storage in the future . Cumulative sugar loss was calculated as a function of the daily average of either measured temperature or modeled temperature. To calculate t he daily sugar loss , E q. 2 was used to solve for the temperature to obtain the predicted CO 2 respiration rate (mg kg - 1 hr - 1 ) . Daily sucrose loss ( m g kg - 1 hr - 1 ) is equal to the daily CO 2 re s p i ration rate divided (mg kg - 1 hr - 1 ) by 1.55, as the mass of CO 2 is 1.55 times that of s ucrose ( Azcón - Bieto and Osmond, 1983; Siedow and Day, 2017) . To predict Cumulative sugar loss , calculated daily sugar loss was added for every day of the period tested. Sugar loss was also used to compare between different pile zones, measured 23 and modeled temperature s and to test three designs of sugar beet storage system s (described in the Pile des ign evaluation section later ) compared to the actual sugar loss . T hree differing pile designs were developed to determine the effect of pile architecture on beet root temperature, assuming a 3 ° C increase in air temperature relative to 2012 data . The Cumulative sugar loss was calculated for the modeled scenarios . PILE ZONE COMPARISONS To simplify the interpretation of the results, the pile was divided into t hree zones . T he lower zone; from the base of the pile up to 1.6 m high, the middle zone; from the lower z one to 3.2 m high and the up per zone; from the middle zone to the top of the pile ( 4.9 m ) . Each zone contain ed 6 to 14 collected temperature points (Fig. 4 for 2011 and Fig. 5 for 2012 ) . The average temperature of the points in each zone was used as the temperature of that zone. A c omparison between sugar loss w as obtained based on average zone temperature . Figure 4 : Schematic d esign: Illustrate s the sugar beet pile divided into three zones , lower , middle ( ) and up per ( ) for the 2011 season . The colored symbols illustrate the thermocouple locations as distributed in each zone . Figure 5 : Schematic d esign: Illustrate s the sugar beet pile divided into three zones, lower , middle ( ) and upper ( ) for the 2012 season . The colored symbols illustrate the thermocouple locations as distributed in each zone. 24 PILE DESIGN EVALUATION Three pile designs differing in height (50% decrease and 50% increase relative to a commercial pile) and ventilation (by creating a pass underneath the pile that allows cold air to flow under the pile to cool the base) , were designed using the built model with s ome modifications in each system to predict the effect of these design variables on temperature (Table 4 ) . Modeled t emperature obtained from each design w as used to estimate the daily and Cumulative sugar loss during storage , which was used to compare the three designs . Table 4 : Dimensions for different pile designs used for developing heat transfer models in sugar beet storage piles. Pile s hape Pile height (m) Pile width (m) Commercial 4.8 45.7 50% Decrease 2.4 45.7 50% Increase 7.3 45.7 Ventilated 4.8 45.7 In the d ecreased height beet pile system ( Fig. 6 ) , the main pile height was decreased by 50% , wh ile all other parameters were the same as the main pile. Figure 6 : Schematic d esign: Illustrate s the sugg es t e d decreased height sugar beet pile. The outer pile - shape describe s the commercial pile, the insider pile - shape describe s the decrease d height pile. Arrows show the air direction for the model . 25 In the i ncreased height beet pile system shown in Fig. 7 , the main pile height was increased ` by 50% , while all other parameters were the same as in the main pile. Figure 7 : Schematic d esign: Illustrate s the suggested increased height sugar beet pile. The outer pile - shape d escribe s the increased height pile , the inside r pile - shape describe s the commercial pile. Arrows show the air direction for the model . In the ventilated system (Fig. 8 ), t he area of that system was k ept the same as the commercial pile area and natural convection ventilation was added below the p ile to cool the base while all other parameters were the same as the main pile. Figure 8 : Schematic d esign: Illustrate s the suggested ventilated sugar beet pile. Th e a ir domain is surrounding the pile from all sides . Arrows show the air direction for the model . For validation , the calculated temperatures were compared with measured temperatures obtained during 132 days total for both years from an actual commercial storage pile . In the first year , the air temperature was higher than usual, leading to a short storage period , and we were able to obtain data for only 37 days. In the second year , the temperature was at the normal range, so we were able to obtain data for 95 days. The pile temperature was used to calculate the respiration rate of stored beets and thereby estimate sugar loss during the storage campaign. 26 DATA HANDLING, STATISTICAL ANALYSIS AND EXPERIMENTAL DESIGN In the current study , the model yielded one temperature value per day for each po int among the 26 (in the 2011 season) and 25 (in the 2012 season) predefined points in the pile. However , each data logger recorded 24 temperature measurement s (i.e. one value per hour) for each thermocouple of the 2 6 or 25 thermocouples used in each pile . To achieve consistency between the measured and modeled temperatures, the daily average of the 24 measurement s from each thermocouple was obtained . The averag e temperature of all the predefined points was calculated to obtain the pile temperature for each day for the primary analysis. Analysis of variance (ANOVA) was used to test the relationship between the measured and modeled temperatures and estimated cumulative sugar loss . In this stage of analysis, only the average daily temperature of the whole pile was calculated from the 26 daily temperature values in 2011 and the 25 daily temperature values in 2012 . Thus, the total number of analyzed poi nts was 37 and 95 for the 2011 and 2012 seasons, respectively. Additionally, an other ANOVA was conducted to study the effect of pile zones (lower, middle and upper) on the difference between the measured and modeled temperature s . In this stage, the average daily temperature was considered for each zone and each season , calculated from 14, 6 and 6 daily temperature values for the lower, middle and upper zones, respecti vely for 2011, and 12, 7 and 6 daily temperature values for the lower, middle and upper zones, respectively for 2012 . Thus, the total number of analyzed point s for the 2011 season were 37 points for each zone . Whereas, the total number of analyzed points f or the 2012 season were 95 points for each zone . In each of the abo ve tests , th e significance level was chosen to be 0.05. If a significant effect of any of the studied independent variables was found a ccording to the ANOVA test, a mean comparison was carried out using the least significant difference ( LSD ) 27 T - t est with a significance level of 0.05 . Besides , a c orrelation analysis was conducted to study the relationship between cumulative sugar loss for measured and modeled temperatures with a level of significance as ( 0.05 ) and the correlation coefficient (r) w as used to estimate the strength of this relationship . To predict Lm ( the Cumulative sugar loss based on measured temperature ) as a function of Ld ( the Cumulative sugar loss based on modeled temperature ) , a simple linear regression analysis was conducted and the coefficient of determination ( R 2 ) was calculated. The statistical analysis was performed using the PROC GLM procedure in SAS 9.4 (2014, SAS Institute Inc., Cary, NC, USA). 28 RESULTS T he pile temperature distribution changed throughout the day in response to changing air temperature (Fig. 9 ) . T he temperature of the pile typically declined from the i nside to the outside of the pile . The center of the pile was usually the warmest region of the pile and of the three zones chosen for analysis, the lower zone of the pile, nea r the ground surface, was usually warm compared to the upper zone of the pile. The base of the pile t ypically ranged from 2. 8 to 11 ° C warmer than the surface of the pile. During a warm period , when the air temperature was around 2 ° C , large portions of the pile (>70%) had root temperatures above 7 ° C . During a cool period , wh en the air temperature was - 7 ° C , about 30% of the pile still had temperatures in the 4.5 ° C range, and almost half of the pile had temperatures below freezing. 29 Figure 9 : Heat transfer diagram : Illustrate the t emperature profile of the Gera Road beet pile on December 5 - 10, 2 011. The average a ir temperature is given in the thermometer to the right of each panel . A, B, C, D and SP indicate five thermocouple harnesses; harness D was embedded approximately 5 cm into the soil. White circles indicate locations of individual thermocouples. 30 MODEL ACCURACY BASED ON PILE TEMPERATURE The modeled daily average temperatures for the whole pile were compared with measured daily average temperatures for both storage seasons. T he model tended to underestimate temperature ( Fig. 10 for 2011 and Fig. 11 for 2012 ) . The m ean difference between measured and modeled temperature values was significant (P ( Table 5 ), and a boxplot show s the mean of the model ed temperatures was lower than the mean of measured temperatures for both seasons ( Fig. 12 for 2011 and Fig. 13 for 2012 ) . Due to some technical problems with dataloggers, some measuring data was lost, resulting in gaps in temperature measure ments in Fig. 10 and similar subsequent figures . Figure 10 : Graph: Show s the m easured ( ) and modeled ( ) sugar beet whole pile temperatures for the 2011 season in relation to air temperature ( ). 31 Fig ure 11 : Graph: Show s the m easured ( ) and modeled ( ) sugar beet whole pile temperature s for the 201 2 season in relation to air temperature ( ) . Figure 12 : Boxplot : D isplay s the distribution of the values of measured and modeled temperatures for the beet pile temperature for the 2011 season. Diamonds represent the mean value s ; circles represent the outlier values. 32 Figure 1 3 : Boxplot : D isplay s the distribution of the values of measured and modeled temperatures for the beet pile temperature for the 201 2 season. Diamonds represent the mean value s ; circles represent the outlier values. Table 5 : M easured and modeled beet pile temperatures ( ° C) for 2011 and 2012 averaged across the storage campaign. Pile temperature (°C) Data source 2011 a 2012 b Measured 6.21 A 5.68 A Modeled 4.3 B 3.64 B * Values followed by different letters within a column differ based on LSD test ( 0.05) . a LSD = 0.7714 for the 2011 season . b LSD = 0.6222 for the 2012 season . MODEL ACCURACY BASED ON SUGAR LOSS 1. Daily Sugar Loss D aily rate of sugar loss (kg/ metric ton/day) based on measured and modeled temper atures were calculated ( Fig. 14 for 2011 and Fig. 15 for 2012) and boxplot ( Fig. 16 for 2011 and Fig. 17 for 2012 ) and mean dif ferent analysis (Table 6 ) were obtained. The mean of the daily sugar loss 33 based on the modeled pile temperature was significantly lower than the mean of the daily sugar loss based on the measured pile temp erature, which means that the calculations based on the model ed pile temperature values underpredict the amount of sugar loss . Figure 14 : Graph: Show s the d aily sugar loss (kg/metric ton /day) from field - stored sugar beets calculated from m easured ( ) and modeled ( ) pile temperatures for the 2011 season . Figure 1 5 : Graph: Show s the d aily sugar loss (kg/ metric ton /day) from field - stored sugar beets calculated from m easured ( ) and modeled ( ) pile temperatures for the 201 2 season . 34 Figure 16 : B oxplot : Displays the distribution of the daily rate of sugar loss (kg/metric ton/day) due to respiration as a function of measured or modeled sugar beet pile temperature for the 2011 season. D iamond represent the mean values ; circle s represent the outlier values. Figure 1 7 : B oxplot : Displays the distribution of the daily rate of sugar loss (kg/metric ton/day) due to respiration as a function of measured or modeled sugar beet pile temperature for the 201 2 season. D iamond represent the mean values ; circle s represent the outlier values. 35 Table 6 : Sugar loss estimates based on m easured and modeled beet pile temperatures for 2011 and 2012 ( ° C) averaged throughout the storage campaign . 2011 a 2012 b Rate of sugar loss (kg/metric ton/day) Type Rate of sugar loss (kg/metric ton/day) Type 0.14 A Measured 0.13 A Measured 0.11 B Modeled 0.10 B Modeled * Values followed by different letters within a column differ based on LSD test ( 0.05 ) . a LSD = 0.0116 for the 2011 season. b LSD= 0.0094 for the 2012 season. 2. Cumulative Sugar Loss There was a high correlation between Cumulative sugar loss (kg/ metric to n ) based on daily average values of measured temperatures (Lm) and modeled temperatures (Ld) with a coefficient of correlation (r) of 0.99 7 and 0.999 for the 2011 and 2012 seasons, respectively. The total number of days in storage was 37 days in 2011, and 95 days in 2012 ( Fig. 18 for 2011 and Fig. 19 for 2012 ) . A mean comparison (Table 7 ) and a boxplot ( Fig. 20 for 2011 and Fig. 21 for 2012 ) show that there was a significant difference T he mean values of Ld w ere significantly lower tha n the mean values of Lm by 1.06 ( kg/ metric ton ) for the 2011 season and 2.91 (kg/ metric ton) for the 2012 season . 36 Figure 18 : Graph: Shows the c umulative sugar loss (kg/metric ton) from field - stored sugar beets calculated from m easured ( , Lm ) and modeled ( , Ld ) pile temperatures for the 2011 season . Figure 19 : Graph: Shows c umulative sugar loss (kg/metric ton) from field - stored sugar beets calculated from m easured ( , Lm ) and modeled ( , Ld ) pile temperatures for the 201 2 season . 37 Figure 20 : B oxplot : Displays the distribution of measured or modeled Cumulative sugar loss (kg/ metric ton ) for the 2011 season. D iamonds represent the mean values ; circles represent the outlier values. Figure 21 : Boxplot : Displays the distribution of measured or modeled Cumulative sugar loss (kg/metric ton) for the 201 2 season. Diamonds represent the mean values; circles represent the outlier values. 38 Table 7 : Calculated cumulative s ugar loss (kg/metric ton) in field - stored sugar b e ets based on measured and modeled pile temperature for 2011 and 2012 . 2011 a 2012 b Cumulative sugar loss (kg/metric ton) Type Cumulative sugar loss (kg/metric ton) Type 2.91 A Measured 6.97 A Measured 2.18 B Modeled 5.43 B Modeled * Values followed by different letters within a column differ based on LSD test ( 0.05) . a LSD = 0.6142 for the 2011 season. b LSD= 0.943 for the 2012 season. The cumulative sugar loss (kg/metric ton) for the 2011 and 2012 seasons were pooled together to conduct s imple linear regression analysis between sugar loss estimated from modeled and actual temperatures . Cumulative sugar loss was estimated by the pile temperature based on the model was used as an independent variable to predict Cumulative sugar loss based on the actual temperature measured in the beet pile . T he coefficient of determination (R 2 ) was equal to 0.999 . T he equation ( Eq. 11 ) can be used to correct the modeled data using the cumulative sugar loss based on measured temperature s . The equation can be used to obtain an estimate of the cumulative sugar loss for models of different designs of the storage piles . Equation 11: L d = 0.7784 L m 0.021 L m = Estimated cumulative sugar loss (kg/ metric ton ) based on pile measured temperature. L d = Estimated cumulative sugar loss (kg/ metric ton ) based on pile modeled temperature. PILE ZONE COMPARISON 1. Variation of the Measured Temperature Inside the Pile To study the spatial variation of temperatures ins ide the pile, the pile was virtually divided into upper, middle and lower zones . T he temperature in the upper zone was the lowes t followed 39 by the temperature of the middle zone and finally the lower zone ( Fig. 22 for 2011 and Fig. 23 for 2012 ) . Boxplot analysis ( Fig. 24 for 2011 and Fig. 25 for 2012 ) and m ean comparison t - test between measured temperatures of the different zones were conducted ( Table 8 ). The average temperature for the storage season of the three zones differed ( p 0.05 ). T he temperature of the upper zone was significantly lower than the temperature of the middle and lower zones, and the temperature of the middle zone was significantly higher than the temperature of the lower zone in both seasons . Figure 22 : Graph: Shows the t emperature s (°C) of the lower ( ) , middle ( ) and upper ( ) zones of the sugar beet pile and the average daily air temperature ( ) for the 2011 season . 40 Figure 23 : Graph: Shows the t emperatures (°C) of the lower ( ), middle ( ) and upper ( ) zones for the 201 2 season . Figure 24 : B oxplot : Displays the average temperatures ( °C ) throughout the storage campaign of lower , middle and upper zones of the sugar beet pile for the 2011 season. D iamonds represent the mean values ; circles represent the outlier values. 41 Figure 25 : Boxplot : Displays the average temperatures (°C) throughout the storage campaign of lower, middle and upper zones of the sugar beet pile for the 201 2 season. Diamonds represent the mean values ; circles represent the outlier values. Table 8 : Temperatures of the lower , middle and upper zones of the sugar beet pile in 2011 and 2012 averaged across the storage campaign. Means are the average of 37 d in 2011 and 97 d in 2012 . 2011 a 2012 b Zone Temperature (°C) Temperature (°C) Lower 7.65 A 6.45 A Middle 5.05 B 5.05 B Upper 2.75 C 3.85 C *Values followed by different letters within a column differ based on LSD test ( 0.05) . a LSD = 0.9678 for the 2011 season. b LSD= 0.6171 for the 2012 season . 2. Daily Sugar Loss Comparison Between Pile Zones The calculated daily sugar loss es (kg/ton/day) , based on measured temperatures from each of the three zone s , were higher at the beginning of the storage season tha n the end of the season ( Fig. 26 for 2011 and Fig. 27 for 2012 ). T he upper zone was predicted to have the lowest sugar 42 loss of the three zones and the lower zone had the highest ( Fig. 28 for 2011 and Fig. 29 for 2012 ) and ( Table 9 ). Figure 2 6 : Graph: Shows the c alculated daily rate of sugar loss (kg/ metric ton/day) based on measurements of pile temperature (°C) for the lower ( ) , middle ( ) and upper ( ) zones of the pile for the 2011 season . The secondary vertical axis is the average daily air temperature . Figure 27 : Graph: Shows the c alculated daily rate of sugar loss (kg/ metric ton/day) based on measurements of pile temperature (°C) for the lower ( ), middle ( ) and upper ( ) zones of the pile for the 201 2 43 Figure 28 : B oxplot : Displays the average calculated daily sugar loss (kg/ metric ton/day) based on measured temperatures ( ° C ) of lower, middle and upper zones of a sugar beet pile for the 2011 season . Diamonds represent the mean value s; circles represent the outlier values. Figure 29 : Boxplot : Displays the average calculated daily sugar loss (kg/metric ton/day) based on measured temperatures (°C) of lower, middle and upper zones of a sugar beet pile for the 201 2 season . Diamonds represent the mean value s; circles represent the outlier values. 44 Table 9 : Estimated rate of sugar loss (kg/metric ton/day) for field - stored sugar beets in the upper, middle and lower zones of the beet pile based on measured temperature . 2011 a 2012 b Zone Rate of sugar loss (kg/metric ton/day) Rate of sugar loss (kg/metric ton/day) Lower 0.161 A 0.143 A Middle 0.121 B 0.122 B Upper 0.087 C 0.103 C *Values followed by different letters within a column differ based on LSD test ( 0.05) . a LSD = 0.0145 for the 2011season. b LSD= 0.0093 for the 2012 season. 3. Cumulative Sugar Loss Comparison Between Pile Zones The amount of sugar loss was estimated to be highest in the lower zone and lowest in the u pper zone in both seasons (Table 10 ) . Table 10 : Estimated cumulative sugar loss (kg/ metric ton ) in the 2011 season (37 days) and the 2012 season (95 days) calculated from the measured temperature of the lower, middle and upper zones of the beet pile . Zone 2011 (37 days) 2012 (95 days) Upper 3.21 9.78 Middle 4.48 11.60 Lower 5.94 13.59 MODEL ACCURACY IN DIFFERENT ZONES 1. Temperature Comparison Between Pile Zones Based on the mea sured and modeled data , the tempe rature values obtained from the model were generally low er than the measured temperature s . However, in some days the modeled temperatures in the upper ( Fig. 30 for 2011 and Fig. 31 for 2012) and middle zones ( Fig. 32 for 45 2011 and Fig. 33 for 2012) overestimated the measured temperatures , possibly due to the rapid fluctuation in air temperatures. In the case of the lower zone, there was consistent temperature underestimation by the model ( Fig. 34 for 2011 and Fig. 35 for 2012 ) . According to ANOVA analysis (Table 1 1 ) , there was no significant difference between the measured and modeled temperatures in the upper and middle zones in the 2011 season . This illustrate s that the model has accuracy for predicting the pile temperatures in these zones . This finding contrasts with the results for the lower zone. In the 2012 season, there was a significan t 0.05). Figure 30 : Graph: Shows the m easured ( ) and modeled ( ) temperatures ( ° C ) of the upper zone of the sugar beet pile for the 2011 season in relation to . 46 Figure 31 : Graph: Shows the m easured ( ) and modeled ( ) temperatures (°C) of the upper zone of the sugar beet pile for the 201 2 season in relation to Figure 32 : Graph: Shows the m easured ( ) and modeled ( ) temperatures ( ° C ) of the middle zone of the sugar beet pile for the 2011 season in relation to air temperature 47 Figure 33 : Graph: Shows the m easured ( ) and modeled ( ) temperatures ( ° C) of the middle zone of the sugar beet pile for the 201 2 season in relation to Figure 34 : Graph: Shows the m easured ( ) and modeled ( ) temperatures ( ° C ) of the lower zone of the sugar beet pile for the 2011 season in relation to 48 Figure 3 5 : Graph: Shows the m easured ( ) and modeled ( ) temperatures (°C) of the lower zone of the sugar beet pile for the 201 2 Table 11 : Significance level , resulting from ANOVA analysis , assessing whether predicted and measured pile temperatures differed in the upper, middle and lower zones over two storage seasons . Pile zone 2011 2012 Upper 0.34 < 0.0001 Middle 0.06 < 0.0001 Lower < 0.0001 < 0.0001 EVALUATION OF PILE GEOMETRIES AND VENTILATION ON SUGAR LOSS 1. Daily Sugar Loss Comparison Between Pile Geometries and Ventilation F urther analysis was conducted to study the effect of pile height (decreased , increased and commercial) and t he effect of ventilation using the model developed above . P ile shape was predicted to affect temperature gradients and subsequently sugar loss . These models were developed one time under actual air temperature according to the air temperature records from 1 November 2012 to 8 Februar y 2013 , and the second time with 3 ° C increase in the air temperature 49 relative to 2012 data every day to evaluate different designs in the case of warm er winters . T he ventilated pile yielded lower daily sugar loss (kg/ metric ton /day) while the other designs did not vary significantly according to me an difference analysis under 2012 or increased air temperatures (Table 1 2 for 2012 air temperature and Table 1 3 for 3°C increase in the air temperature relative to 2012 data ) and ( Fig . 36 for 2012 air temperature and Fig. 37 for 3°C increase in the air temperature relative to 2012 data ) . Figure 36 : Boxplot : D isplay s the distribution of the values for daily sugar loss rate (kg/metric ton/day) of different design s of sugar beet storage pile s based on 2012 air temperatures. Diamonds represent the mean values; circles represent the outlier values. 50 Fig ure 3 7 : Boxplot : D isplay s the distribution of the values f or daily sugar loss rate (kg/ metric ton /day) of different design s of sugar beet storage pile s based on predicted 3 ° C increase in air temperatures relative to 2012 data . Di amonds represent the mean value s; circles represent the outlier values. Table 12 : Modeled prediction of the d aily rate of sugar loss (kg / metric ton /day) for beet piles having different heights or ventilation in a sugar beet pile base d on the 2012 air temperature . Treatment Pile height (m) Rate of sugar loss (kg/metric ton/day) a Decreased height pile 2.4 0.083 A Commercial pile 4.9 0.093 A Increased height pile 7.3 0.096 A Ventilated pile 4.9 0.04 B * Values followed by different letters within a column differ based on the LSD test ( 0.05) . a LSD value = 0.0171 Table 13 : D aily rate of sugar loss (kg/ metric ton /day) for beet piles having different heights or ventilation in a sugar beet pile based on 3 ° C increase in air temperature relative to 2012 data . Treatment Pile height (m) Rate of sugar loss (kg/metric ton/day) a Decreased height pile 2.4 0.105 A 51 Commercial pile 4.9 0.113 A Increased height pile 7.3 0.105 A Ventilated pile 4.9 0.085 B *Values followed by different letters within a column differ based on the LSD test ( 0.05) . a LSD value = 0.0167 2. Cumulative Sugar Loss Comparison Between Pile Geometries and Ventilation Table 14 represent s the predicted amount of Cumulative sugar loss (kg/ metric ton ) following 100 days of field storage in piles modeled with varying height or ventilated pile designs under the 2012 actual air temperatures and 3 ° C higher air temperatures relative to 2012 data . The amount of sugar loss was low er in the ventilated pile design than the other designs under both the 2012 and the increased pile temperatures according to the model . Table 14 : Predicted total sugar loss (kg/ metric ton ) after 100 days of field storage based on the temperatures obtained from models varying pile height , ventilat ion and average temperature (+3 °C) for beet piles under the 2012 season . Pile modification Pile height (m) Cumulative sugar loss over 100 days (kg/metric ton) Decreased height pile 2.4 8.37 Commercial pile 4.9 9.62 Increased height pile 7.3 9.33 Ventilated pile 4.9 6.03 Decreased height pile with 3°C increase in air temperature relative to 2012 data 2.4 10.49 Commercial pile with 3°C increase in air temperature relative to 2012 data . 4.9 11.29 52 Increased height pile with 3°C increase in air temperature relative to 2012 data . 7.3 12.09 Ventilated pile with 3°C increase in air temperature relative to 2012 data . 4.9 9.49 53 DISCUSSION T he model developed in the current study generally underestimate d the measured temperature as well as sugar loss in sugar beet pile during storage . The la ck of fit between the measured temperatures and modeled temperatures was possibly due to the steady - state heat transfer condition of the model that was assumed for simplicity for model development instead of accurately reflecting dynamic c ondition s . T he steady - state condition refer s to the situation when the change of the internal energy does not vary with time (Datta, 2002) , which we decided to be a day . A s teady - state condition may occur in a controlled system where the inputs (i.e. , air temperature and wind speed) do not change during the time . This was not the case in the studied beet pile as the environmental conditions change d frequently . On the other hand, developing a dynamic model for outdoor sugar beet pile storage would be highly challenging, because in the most available transient models of storage systems, they have controlled environment with limited variables, whereas in the case of this study we have the air tem perature, wind speed, relative humidity, precipitation, solar radiation, root temperature, respiration, desiccation, soil temperature, soil moisture content, water evaporation, and condensation as variables that give the model so much complexity . So that w e found that considering the most important factors in a simple model as a primary step , then modify that simple model in future studies working toward some complexity. Also employ ing a day to be the time for a steady - state was a relatively long time as we found that the pile temperature was responsive to the air temperature that change s as much as 1 5 ° C in one day . Also , neglecting the effect of moisture transfer and the change in the water content inside the pile may have contributed to an inaccur ate estimate of the thermal conductivity of the 54 pile material (beet, water and a ir ). Results obtained by Xu and Burfoot (1999) confirm the importance of accurate ly measu ring or calculating the moisture content to develop heat transfer models for biological materials. Al though their model ha d a good fit in most parts of the pile, a l ack of fit occur red when the moisture content was incorrectly calculated . Hoang et al. (2003) , also attributed the under estimation of the ir model to desiccation on the surface of chicory roots during storage , because the water heat transfer coefficient is 50 to 100 times higher than the air heat transfer coefficient (Datta, 2002) . S omething similar could be a possible reason in the current study, especially for the error found at the surface of the pile . Altho ugh lack of fit due to moisture content miscalculations was expected, we adapted the idea of a simple , more easily applied and less intensive model . For instance, m oisture sensors to cover every part in a huge body like the beet pile is very cost ly and time - consuming. Also, understanding the limitations of the current model can help future researchers decide the important aspects to monitor or model to develop a better model. N on - uniform root size of the beet pile during storage , leading to vari ed size s of voids between roots , can also lead to inexact estimates of in thermal conductivity, water content and air movement calculations (Hoang et al., 2003) . In addition to the above , using the average of the daily temperatures resulted in losing some model sensitivity to the change of temperature during the day . A s mentioned before , we used the daily average air temperature and wind speed as inputs when we developed the model . A t any time, these inputs change significantly during the day, using the average of such inputs sensitivity to chang e s in the environmental condition . This likely affected accuracy , as the pile temperature was sensitive to the change s in ambient conditi ons . 55 L ack of fit between measured and predicted temperatures often is found in models of biological materials . For example, Zou et al. (2006b) explained the inaccuracy in the ir model by errors in the position of thermocouples, model in put values and /or model assumptions. Similarly , Hoang et al. (2003) found underestimation of chicory temperature due to water loss and condensation on the surface of the roots from evaporation due to low relative humidity in the storage room as well as non - uniform porous, variation in the product sizes and the relatively small voids compared to the size of the roots . On the other hand , Markarian et al. (2006) developed a mathematical model with high accuracy, due to the controlled storage system, the model included mass and heat t ransfer in three dimensions and for a limited storage time ( 1 hour ). Their success is likely a function of their control of conditions and highlights the difficulties in a complex, uncontrolled system that contribute to la ck of fit as in the case of our model. The c umulative sugar loss depending on measured and modeled temperatures were highly correlated. T h is means that the Cumulative sugar loss based on the modeled temperatures can be helpful to roughly predict the c umulative sugar loss based on the measured temperatures. T h is can be a helpful technique for storage managers to evaluate new pile designs. According to the pile zone comparison, d ifferences of measured temperatures between the three pile zone s described the existence of temperature variation s inside the pile and show that the up per zone wa s always colder than the middle and lower zones and the lower zone is always warmer than the middle and the up per zone . The results for sugar loss are in g ood agreement with the results f or temperature . The results are expected because the upper zone is expos ed to the relatively cold air and remove s the heat of the upper zone by convection. On the other hand , the lowe st zone gain ed heat from the ground by conduction , thereby increasing the temperature in that zone. 56 Estimated daily sugar loss es were higher early in the storage campaign . These results were similar to the findings of Fox (1973) and Wyse (1975) that 50% of sugar loss in sugar beets occur during the first two weeks of storage, and the majority of the sugar loss occur s during the first 40 days of storage. Cumulative sugar loss calculation s in different zones highlight an elevated sugar loss in the lower zone compared with the m iddle and the upper zones. This indicate s the importance of reducing the t emperature of this zone (Yang and Rao, 2006) . In the case of the effect of pile zones on modeled temperature accuracy, the modeled and measured temperatures showed good agreement in the upper and middle zones during the 2011 season for these zones . This may be a function of the model making better predictions at higher temperatures, this means that the model i s accurate for predicting the pile tem peratures in these zones. However, the model underestimated the temperature of the lower zone for both storage seasons. I t should be noted that the lower zone is mainly aff ected by the ground temperature and in the model , as we us ed the semiannual aver age temperature of the ground obtained from Reese, MI (Schaetzl et al., 2005) which resulted in losing part of the model sensitivity to the change s of temperature in that zone . Moreover, in the pile base , the beet weight is relatively high and that possibly cause d damage to the roots and consequently increase d respiration rate and thereby increase temperature (Cole, 1977) . T he compressed damage d roots may have also caused block ing of the voids which happened to potatoes (Prin gle et al., 2009) , which may have chang ed the the rmal conductivity of the pile material . Furthermore, beet roots in a field pile are often mixed with topsoil and stones . (Flegenheimer, 2015) noted there w ere an estimated 136,000 metric tons of topsoil and other debris are added to beet storage piles each year. The non - beet materials are 57 mainly accumulate d at the base of the pile , fill ing the voids and altering the physical and thermal properties of the pile material . In the m iddle zone , differences between modeled and measured temperatures are possibly from condensation in that zone . An increase in moisture content can subsequently affect porosity, thermal conductivity and air movement . The middle zone is also likely to be warmer than anticipated because of the heat gained from the lower zone. The w armer air strea m that mov es from the lower zone to the middle zone holds moisture and can cause a condensation layer as happened in a potato storage study by (Pringle et al., 2009) . Consequently, an increase of the moisture in the middle zone could lead to partially block ing t he voids with water that change the the rmal conductivity of that zone and cause a decline in respiration (Lafta and Fugate, 2009) . That could be an explanation of the model agreement with measured temperature in 2011 with the increase in air temperature and possibly increase in evaporation from the lower zone that cause s the formation of a condensation zone and reduce s the respiration rate of the beets in that zone. The up per pile zone is the most susceptible part of any change s in environmental conditions , as it has the largest surface interact s with ambient conditions (e.g., sunlight, wind, rain, snow, temperature, humidity) . Fluctuations in air temperature s lead to an increas e in respiration rate and sucrose loss even if occur of storage temperature s of - 1 ° C or below (Wyse, 1978) , which was the case in the commercial pile e specially in the upper zone . Therefore, evaluating the respiration rate under a constant storage temperature and using that value in the model a s a source of heat possibly led to underestimation in temperature i n this zone . Fluctuation can significantly increase sucrose loss due to the accumulation of reduc ing sugar s after respiration. A ddition ally , the possibility of root dehydration in this zone is much higher than the other zones (Campbell and Klotz, 2006; List, 2015) . D ehydration damages the cells and can result in los s of permeability 58 control and electrolyte leakage , which significantly increase respiration (Lafta and Fugate, 2009; Yang and Rao, 2006) . Dehydration also occur s due to cell freezing, ce lls start to freeze (or damage) at - 2 ° C and they are completely frozen at - 5 ° C (Campbell and Klotz, 2006; Wyse, 1978) . F reezing followed by thawing cause s the cells to lose metabolic control and permeability, which cause s increase d cell respiration, desiccation and roo t deterioration (Lafta and Fugate, 2009; Wyse, 1978). Lafta and Fugate (2009); Wyse (1978) noted a temperature fluctuation and dehydration within 60 cm from the pile surface , which likely i ncrease s respiration , as dehydration cause s more than 82% increase in the respiration rate compared with the initial respiration rate In the 2011 season , the air temperature was relatively higher than in the 2012 season and that contributed to less freezing damage . The temperatures in that year were closer to those used to calculate the respiration rate in the laboratory experiment , and may account for the increase in the model accuracy (i.e., lack of significant difference between model and measured temperatures) in the upper zone found that season. The model predicted that pile height and ventilation at the base of the pile could make a significant contribution to sugar loss . Through ventilation, the incre ase in the surface exposed to cold air convection lead to a significant decrease in the rate of daily sugar loss compared to other pile designs. T he ventilated pile yielded lower daily and Cumulative sugar loss comparing with the commercial, increased height and decreased height de s igns applying the 2012 air temperature and 3 ° C increase in 2012 air temperature . The result is consistent with the results of the zone comparison, which emphasize d the significant inc rease in sugar loss found in the lower zone , where heat is gain ed directly from the ground . R educing the heat in the base of the pile, the sugar loss significantly decrease d from 9.3 kg/ metric ton in the commercial pile to 3.95 kg/ metric ton in the ventilated pile , which is 68% decrease in sugar loss compar ed with the commercial pile . Thus, 59 after 100 days in the storage , a higher rate of sugar loss was found for the commercial pile ( 11.29 kg per metric to n per 100 days) compared to the ventilated pile ( 8.55 kg per metric ton per 100 days ) ; a 3 °C increase in air temperature would yield a 24% increase in sugar loss . On the other hand, the change in the pile height (either increase or decrease) does not affect the sugar loss significantly in comparison with the commercial beet pile based on the air temperature of 2012 or 3 °C increase in 2012 air temperature. The results are similar to th ose for a study by Michigan S ugar Company (List, 2015) in which they found that they need ed to reduce the pile width by 83.9% and the pile height by 38.7% to reach 1% of the sugar loss from harvesting to December . In addition to these findings , we recommend insertion of natural ventilation at the base of the pile during the storage period. 60 SUMMARY A ND CONCLUSIONS The enormous production of sugar beet prevents the immediate process ing of harvested roots and is responsible for the need for vast storage facilities until processin g. In Michigan, beets are stored in huge piles exposed to the fluctuat ion s in the surr ounding environmental condition. Approximately half of the storage piles are unventilated , t hus , n a tural convection is the only cooling technique for the top and sides of each of such pile s . Fluctuating air temperatures enhanc e respiration rate and microbial activity in the exposed piles , which result s in a reduction in root quality and increase d sugar loss . In the current study , ma thematical simulation methodology was applied to solve for pile temperature s . S uch simulation is intended to help storage managers not only for predicting temperature s but also to improve management decisions regarding pile structure (dimensions and shape), handling stored beet and installation of appropriate ventilation systems. A two - dimensional mathematical model was developed as a function of environmental parameters ( air temperature, air velocity, RH and an average ground temperature ) to simulate the heat transfer process during the storage period and to predict the temperature profile of an u nventilated sugar beet pile. The model was developed based on the finite element approach. Model validation was obtained by comparing the modeled temperatures with measured temperatures collected from a commercial beet pile using embedded thermocouples for two seasons , 2011 and 2012 , for 37 and 95 days, respectively . The model that was developed generally underpredict ed pile temperatures . This underprediction is attributed to several factors including the assumption by the model of steady - state condition s, mean ing that pile temperature is assumed to be in equilibrium with its 61 environment , which is likely rarely true . A dditionally , the model excluding moisture mass transfer taking place in the pile , w h ich can affect the heat loss and thermal condu ctivity and subseque ntly alter heat transfer calculations . The model accuracy was evaluated by comparing the modeled and measured temperatures for three spatial zones in the pil e up per , middle and lower zone s . T he model ed and measured temperatures showed fair agreement in the up per and middle zone s . However , the model underestimate d the temperature of the lower zone during the two storage season s . The measured and modeled temperatures were used to calcula te the Cumulative sugar loss ( kg/ metric ton). A s trong relationship was obtained between modeled and calculated values with a correlation coefficient of higher than 99 % . As expected , based on differences in model and actual temperatures, the Cumulative sugar loss calculated based on model temperature was lower than the Cumulative sugar loss calculated based on the measured temperature . T he effect of pile height or use of ventilation compared to commercial piles was also tested by assessing two different pile geometries, a 50% reduc tion of the height of the commercial pile or a 50% increase of the height of the commercial pile , in addition to taking into account the ventilation at the base of the commercial pile . T he three models were evaluated relative to a commercial pile design under normal and increased air temperature to examine the effects of the designs in the case of warmer winters . The results showed that the ventilated pile yielded significantly lower su gar loss compa red with the commercial pile under actual (2012) and increased temperature scenario s. T he increased and decreased height - pile s did not vary significantly from the commercial pile for sugar loss . In future work , we recommend developing a model for a transient heat transfer condition, including moisture mass transfer states and using hourly data instead of daily average values as t he air temperature for instant can change 15°C in one day during the storage period. The change 62 also occurs continuously in air velocity , RH and precipitation . A dynamic model (unsteady - state conditions) will take in to consideration the fluctuation in the environmental conditions which will increase the model accuracy as a consequence. T here is a need to include beet respiration rate values for a wide r range of temperatures and use various root sizes as it significantly affect s respiration (Wyse, 1978) . Further, non - uniform root size can lead to varied size s of voids between roots which causes an inexact result in thermal conductivity, water content and air movement calculations (Hoang et al., 2003) . We c an improve our model if we provide more precise parameters for the designed model such as ground temperature measurements, porosity and airflow inside the pile . 63 LITERATURE CITED 64 LITERATURE CITED Akeson, W., S. Fox, and E. Stout. 1974. Effect of topping procedure on beet quality and storage losses . J. Amer. Soc. Sugar Beet Tech nol . 18:125 - 135. Ambaw, A., M. Delele, T. Defraeye, Q.T. Ho, L. Opara, B. Nicolaï, and P. Verboven. 2013. The use of CFD to characterize and design post - harvest storage facilities: P ast, present and future . Comput. Electron. Agr . 93:184 - 194. Andales , S., C. 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