THESIS we“. m\ “ti \ “‘\\ WW“ ‘ ii\\\i\\\i\\ii\\i\g\gl§l\3 ‘P—‘fiuaaaar : {222.122.2322 53am 112m “sitar ‘ _ A‘- n *7 ——— — This is to certify that the dissertation entitled Entropy Production By Biological Products in Storage presented by Robert Yemoh Ofoli has been accepted towards fulfillment of the requirements for Ph.D. degreein Agric. Engr. jor professor rge E. Merva Date MSU is an Affirmative Action/Equal Opportunity Inslilution 0-12771 MSU RETURNING MATERIALS: Place in book drop to LJBRARJES remove this checkout from “ your record. FINES will be charged if book is returned after the date stamped below. 113’: s M 3 lat: E228- ” EEB 0 5 '99? 513*1971 Ah/QL flt'ayLF/fe/L— . U“) ’97,? Al! ENTROPY PRODUCTION BY BIOLOGICAL PRODUCTS IN STORAGE BY Robert Yemoh Ofoli A DISSERTATION Submitted to Michigan State University In partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1984. C) Copyright by ROBERT YEMOH OFOL I 1984 ABSTRACT ENTROPY PRODUCTION BY BIOLOGICAL PRODUCTS IN STORAGE BY Robert Yemoh Ofoli Non-equilibrium thermodynamics was used to determine entropy production and energy dissipation by biological pro— ducts in storage subjected to cooling by forced convection. This approach resulted in a procedure that could be used for optimizing the ventilation rate. A model for the potato storage environment, incorporat— ing heat and mass transport and chemical reaction (respira- tion) was derived. The resulting equations were solved numerically using the finite difference method. The optimization procedure is based on an energy dissi- pation index (EDI) which is defined as the ratio of the total energy dissipated by the system to the input energy required to move the ventilation air. The parameter has an asymmetric profile which displays a minimum value when plot— ted against the ventilation rate. It predicts a zone of minimum entropy production that may be used for optimizing the ventilation rate. Robert Yemoh Ofoli The phenomenon of thermal diffusion has a negligible effect on the product mass loss and the temperature profile in the storage environment. Neglecting thermal diffusion is equivalent to setting the cross-phenomenological coefficient in the entropy production equation to zero. This reduces the rate of entropy production by 9%. However, even though the lower rate of entropy production reduces the magnitude of the EDI, the position of the minimum value of the EDI with respect to the ventilation rate is unaffected. As a result, thermal diffusion can be neglected without affecting the essential results of the study. TO My wife Sherry Our son Robert, Jr. (Bobby) and my friend always, Jonathan. ii ACKNOWLEDGEMENTS I would like to express my appreciation to Dr. G. E. Merva for serving as my major professor and for his willing— ness to assume that responsibility at such a late stage of my work. Very special thanks and gratitude are due Dr. Gary J. Burgess for his timely and critical review of thesis material, constant availability and service on the guidance committee. Many thanks to the members of the guidance committee -- Dr. Jim Steffe, Dr. Eric Grulke and Dr. Kris Berglund; and to Dr. Burt Cargill, the external examiner. I would like to thank the faculty, staff and students of the Department of Agricultural Engineering at Michigan State University for their continuous and sustained support throughout my studies. I especially appreciate the help and support of Dr. Donald Edwards, Beverly Anderson, Clara Kisch and Karen Dunn. Last, but certainly not the least, I am very grateful to my wife Sherry and our son Bobby for bearing with me dur- ing all those months when it seemed I would rather live in my office than in our home. iii TABLE 9g CONTENTS Page LIST OF TABLES O O O O O O O O O O O O O O 0 O O O O O O O O O O O O O O O O O O O O O O O Vii LIST OF FIGURES ...0............OOOOOOOOOOOOOOO...... Viii NOMENCLATURE O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O I O O O i x 1.6 INTRODUCTION 0.0.0.0000.........OOOOOOOOOOOOOOOOO 1 .l The potato in storage ...................... .2 ”Odeling approaCh ......OOOOOOOOOOO0.0.0.... .3 Data on potato storage ..................... l—‘HH \imm 2.“ OBJECTIVES ......0......O.......OOOOOOOOOOOOOOOOO 8 3.0 LITERATURE REVIEW 0.0.0.... ......OOOOOOCOOOOOOOO 9 3.1 Thermodynamics of irreversible processes ... 9 3.1.1 Comments and observations ........... 13 3.2 Porous media analyses ...................... 14 3.2.1 Comments and observations ........... 20 .3 Agricultural storage analyses .............. 21 .4 Entropy production in biological systems ... 25 4.0 MODEL DEVELOPMENT ......OOOOOOOOOOOO......OOOOOOO 27 4.1 Provisions of the model .................... 27 4.2 Assumptions ................................ 27 4.3 Geometric modeling of the potato ........... 28 4.4 The storage volume as a packed column ...... 30 4.5 Characterization of the transport processes .................................. 32 4.5.1 Transport processes in the fluid phase .................... 32 4.5.1.1 The Peclet Number ..... 32 4.5.2 Transport processes in the solid phase .................... 36 iv COMPUTER SIMULATION U1U'IU'I o o o WNH RESULTS AND ANALYSIS h .buawrdm Modeling of parameters 4.8.1 ##4## mmmm o o o o U'IIhUN Introduction Relevant equations matical relations System entropy production ........... Generalized fluxes and forces Chemical reaction Modification of the dissipation relationShip ...-......OOOOOOOOOOOOOO 4.6.4.1 The modified relations The thermal diffusion coefficient The resistance coefficient for mass transport The rate of local entropy PIOduction ......OOOOOOOOOO0.0.000... The dissipation function ............ thermodynamic relations First law analysis 4.7.1.1 Control volume 1 4.7.1.2 Control volume 2 Heat and mass transfer coefficients Moist air relations The heat capacities The rate of heat generation ......... Other input parameters .............. computer implementation ......COOOOCOOOCCOOO 5.3.1 Misener's experimental procedure Computer solution techniques Computer program listing and results The resistance coefficient for mass transport Entropy preduction 0.000..........OOOOOOOOOO Optimization by minimizing entropy production The effect of thermal diffusion 36 36 40 42 45 47 49 50 50 52 52 53 S3 68 68 68 69 70 71 72 74 74 74 81 82 85 85 88 88 90 99 107 7.0 CONCLUSIONS ....0.0.0.0.0.0...0.0000000000000000. 8.0 SUGGESTIONS FOR FUTURE RESEARCH ................. APPENDICES 0......00.0.0.........COOCOOOOOOOOOOOOOO... Appendix A Expression for the mass difoSiVity 00.00.000.000...0.0.0.... Appendix B Computer program listing ............ BIBLIOGRAPHY vi 1G8 110 111 111 114 119 LIST pg TABLES Title The heat generation expression versus experimental data, kJ/hr-kg ..................... Temperature-dependent input parameters .......... Parameters used in computer program ............. Comparison of simulated ang experimental air temperatures at 24 hours, K ......OOOOOOOOOOOOOO Comparison of simulated ang experimental air temperatures at 48 hours, K .................... Comparison of simulated and experimental air temperatures at 72 hours, K .................... Comparison of simulated ang experimental air temperatures at 92 hours, K ................... System entropy production ....................... System dissipation of energy .................... vii Page 73 79 80 91 92 93 94 98 101 LIST 9g FIGURES Title Page BaSiC potato unit ......OOOOOOOCCOOOOOOO00...... 29 The storage volume as a packed column: cross-sectional View ......OOOOOOOOOOOOO0.0.0.... 3]- Concentration of moisture in the potato and the air ......OOOOOOOOOOOOOO......OOOOOOOOOO 37 Control volume for the analysis of total entropy production ............................. 38 Control volume for analysis of the ventilation air ......OOOOOOOOOOOOOOCO00.0.0.0... 61 Misener's experimental set-up .................. 84 Program output: entropy production and energy diSSipatj-on O.......OOOOOOOOOOOOOOOO0.0... 86 Program output: heat and mass transfer ......... 87 The resistance coefficient, n = 447.9/RT ........ 89 Comparison of three heat and mass transfer models at 48 hours ............................. 95 Comparison of three heat and mass transfer mOdels at 92 hours ......OOOOOOO0.00.00.00.00... 96 Rate of entropy production ..................... 97 Rate of energy dissipation ...................... 100 Energy dissipation index (EDI) for various ventilation rates at four different inlet temperature (Bi) conditions ......OOOOOOOOOOOOCOO 105 viii NOMENCLATURE NOTE: The following list only includes symbols and nomenclature used from chapter 4 through the end of the thesis. Nomenclature in chapters 1 through 3 are defined locally. A affinity of chemical reaction, J/kg Aac cross-sectional area open to air flow, In2 AC cross-sectional area, m2 ACt potato-to-potato contact area, m2 AS surface area, m2 c concentration, kg/m3 ck concentration of species k, moles/m3 cs solute concentration, kg/m3 C heat capacity, .J/kg-OK Cp heat capacity at constant pressure, J/kg-OK CV heat capacity at constant volume, J/kg-OK Cva dry air heat capacity at constant volume, J/kg-°K Cvp potato heat capacity at constant volume, J/kg-OK va water vapor heat capacity at constant volume, J/kg-OK va heat capacity at constant volume of water, J/kg-OK ix EDI 0| 0 32:7 553‘ O u| 9| Lu <4 “I diameter of storage bin, m binary mass diffusivity, m2/hr mass diffusivity, mZ/hr diameter of product, m diameter of sphere, m modified thermal diffusion coefficient, kg/m—hr-OK total enerng J energy dissipation index, dimensionless friction factor, dimensionless gravitational acceleration, m/sec2 mass velocity, kg/mz-sec standard Gibbs free energy, J/mole heat transfer coefficient, J/mz—hr-OC vertical height, m mass transfer coefficient, m/hr heat of vaporization, J/kg molar enthalpy: J/mole Heavyside step function, dimensionless partial molar enthalpy of species k, J/mole standard enthalpy: J/mole 3 velocity of reaction, kg/hr-m diffusion current flux, kg/mz-hr heat current flux, J/mz-hr entropy current flux, J/mZ-hr-OK non-convective current flux, J/mz-hr X 9(0) Boltzman constant, erg/OK thermal conductivity, J/m-OC-hr conductivity of air, J/m-K-hr molar external force on species k, N/mol thermal diffusion ratio, OK-l characteristic length, m phenomenological coefficient of chemical reaction, kg-hr/m5 convective mass flux coefficient, kg-hr/m3 modified convective flux coefficient, kg-hr/m4 cross-kinetic coefficient, kg/m—hr modified cross-kinetic coefficient, kg/mZ-hr heat flux coefficient, J/m-hr modified heat flux coefficient, J/mz-hr mass of water in air, kg mass of water in potato, kg equilibrium moisture content, fraction molecular weight, 9 mass of dry air, kg molecular weight of species i, g/mole mass of potato solid matter, kg pressure, N/m2 (Pa) Peclet number, dimensionless absolute saturation pressure, N/m2 vapor pressure, N/m2 heat of respiration, J/kg-hr local production of internal energy: J/m3-hr xi <|>< x XI A\ ‘1 §< n. D. A 12 SS mass fraction of species k generalized force for mass diffusion, J/kg-OK generalized force for heat diffusion, (m—OK).1 Greek symbols thermal diffusivity, m2/hr surface area per linear meter of product, m2/m humidity ratio, kg-water/kg-dry air dissipation function, J/m3-hr porosity, dimensionless energy of molecular interactions, J skin resistance coefficient, dimensionless combined skin resistance coefficient (qc = qRT), mZ/hr2 mass per linear meter of water in the air, kg/m mass per linear meter of water in the potato, kg/m mass per linear meter of the dry air, kg/m mass per linear meter of the potato solids, kg/kg temperature, 0C absolute viscosity, kg/m-sec chemical potential of species i, J/kg chemical potential, J/kg concentration-dependent chemical potential, J/kg partial derivative of the chemical potential 2 with respect to the concentration, mS/kg-hr xiii 9—0“? W rfl rz 9‘8; El kinematic viscosity, mz/sec extent of reaction, mol density, kg/m3 local volumetric entropy production, stress tensor, kg/hr2 shear stress, kg/m-hr2 relative humidity, % total energy dissipation, J reference velocity, m/sec weight factor, dimensionless Subscripts and operators ventilation air stream relating to chemical reaction component identifier potato or product solute (water vapor) delta operator the Laplacian del or nabla operator partial differential operator integral operator xiv J/m3-hr-OK 1. 0 INTRODUCTION The subject of entropy has fascinated engineers and scientists for ages in spite of its very abstract nature (or, perhaps, because of it!) Engineering applications of entropy usually involve the determination of the efficien- cies or the degree of irreversibility of a process, espe- cially where the process involves steam and other idealized gases for which standard thermodynamic tables or charts are available. Given the abstractness of the concept, the analysis of entropy production in the storage environment of agricul- tural products would appear to be simply an academic exer- cise. This is, however, not so. As will be shown in this thesis, entropy production is very intimately linked, not only to energy transport, but to all mass transport and to chemical reactions that may be taking place in a given sys- tem. Thus, an examination of entropy generation can be used to gain further insight into these processes, as well as any accompanying energy transformations. It also provides a convenient scheme for optimization, as is done here. Perhaps, Adrian Bejan sums up the role of examining entropy generation best when he wrote that his engineering education led him to assume that "the importance of entropy generation through heat transfer was obvious and that the second law could sell itself. ... entropy generation should assume a central role in heat transfer, as central as the relation between temperature difference and heat transfer rate or the relation between pressure drop and flow through a channel" (Bejan, 1982). In general, biological products in storage constitute a porous medium in which several thermodynamic processes may take place simultaneously. Invariably, these processes are irreversible. In addition, in almost any practical situa- tion, the storage media are moist and subjected to tempera- ture, moisture concentration and pressure gradients, among other thermodynamic forces that may be present. As pointed out by Haase (1969), whenever a continuous multi-component system is subjected to gradients of concentration, pressure and temperature, six different transport processes could occur simultaneously. These processes are: 1. mass diffusion or mass convection: the transport of mass resulting from concentration gradients or concentration differences; 2. pressure diffusion: the transport of mass as a result of pressure gradients; 3. thermal diffusion (Soret effect): the transport of mass due to temperature gradients; 4. heat conduction or heat convection: the transport of thermal energy resulting from temperature gradients or temperature differences; 5. pressure thermal effects: the transport of thermal energy due to pressure gradients; and 6. Dufour effects: the transport of thermal energy as a result of concentration gradients. In addition to the possible occurrence of cross-kinetic effects, further complications are created by the fact that a porous medium is essentially a random structure, whose details are largely unknown (Slattery, 1975). A complete mathematical analysis, employing classical thermodynamics, is therefore quite difficult without the use of an excessive number of assumptions. Traditionally, Darcy's Law has been the foundation for the study of transport processes in porous media. It states that the rate of flow in a porous medium is directly propor- tional to the pressure gradient causing that flow (Carman, 1956). Mathematically, v = 1.2 (1-1) where v is the velocity, k the permeability coefficient, ‘VT is the pressure gradient and L is a characteristic length. This is the defining equation for the permeability coeffi- cient, k. However, Darcy's Law has been shown to have severe lim- itations at high and low velocities for gases, and at high 4 velocities for liquids (Scheidegger, 1960). Also, tempera- ture gradients can induce moisture movement in porous media -- the phenomenon of thermal diffusion. Therefore, mass transport in porous media does not necessarily occur only as a result of mass concentration or pressure gradients. This phenomenon is not anticipated by Darcy's Law (Havens, 1980). Classical thermodynamics is not very useful for describing processes in living organisms or biological pro- ducts since these normally constitute systems that are not only open, but also in non-equilibrium. Without the neces— sary postulates or constitutive relations required by clas- sical thermodynamics in the analysis of these systems, the mathematical description of the processes taking place require model-specific kinetic equations. Developing a proper kinetic relationship, however, often requires infor- mation that is more detailed than is readily available (Katchalsky & Curran, 1965). The thermodynamics of irreversible processes is an empirical tool that gives one the opportunity of studying heat and mass transport in their often inseparable associa- tion. Through the use of Onsager's principle, this metho- dology allows one to model systems in which cross-kinetic effects such as thermal diffusion take place. Since the general relationships can be reduced to the standard linear relations of classical thermodynamics (Fick's law, Kirchhoff's law, Fourier's law, Darcy's law, etc.), the irreversible thermodynamics procedure essentially provides 5 the researcher with additional useful information not readily available through the classical approach. 1.1 The Potato lg Storage The need for potato storage stems from the fact that while most of the nation's potato production occurs in the fall, potatoes must be made available to the consumer on a year-round basis (Smith, 1968). The goals of potato storage (Plissey, 1976) are to: a) retain water in the tuber; b) hold respiration to a minimum; c) hold reducing sugars to a minimum; and d) maintain external appearance. Ultimately, the goal of potato storage is to have the product come out of storage as close to its pre-storage con- dition as possible. The two primary storage environmental variables that can be manipulated to achieve this objective are the temperature and the relative humidity. These two factors are instrumental in determining the degree of weight loss of the product. Weight losses of less than 5% provide for a smooth- skinned, firm product; losses of 5 to 10% result in soft potatoes; while losses of greater than 10% lead to wrinkled products (Cargill, 1976). The limits of storage tempera- tures lie between l.7°C and 18.3°C (Cargill, 1976). The storage must aim for relative humidities of 90—95% (Smith, 1968); this promotes early suberization and reduces shrinkage. l.2_ Modeling Approach The local production of entropy in any biological storage system is the result of heat transport, mass tran- sport and chemical reactions in the system. However, it is nearly impossible to measure entropy production experimen- tally in a biological storage. The approach taken in this thesis, therefore, was to formulate a model for heat and mass transport that predicts convective air temperatures, and product mass losses in the storage environment of suber- ized potatoes. The heat and mass transport relations resulting from this model were then used, in addition to appropriate chemical reaction equations, to examine the pro- duction of entropy and the dissipation of energy in the sys- tem. Models for moisture loss in the environment of stored potatoes have usually been based on the vapor pressure defi- cit as the thermodynamic driving force (Misener, 1973; Hunter, 1976; Lerew, 1978; Brugger, 1979). This work devi- ates slightly from the approach of these and other research- ers and treats the mass transport relationship as an expli- cit function of moisture concentration differences rather than vapor pressure deficits. While vapor pressure deficits are implicit functions of concentration, it is the contention of the author that an explicit concentration-dependent expression provides a better casting for the mass transport equations and a more correct physical sense of the mass transfer process. The phenomenological coefficients for heat and mass transport are determined by curve-fitting to experimental data. 1.3 Data 93 Potato Storage Since neither the rate of entropy production nor the total entropy production could be measured experimentally, the data required for the testing of this model is that of air temperatures, product temperatures and mass losses in the potato storage environment, and thermodynamic tables of entropy of moist air as a function of temperature. There is already a tremendous amount of data available in the literature on convective air temperatures and product mass losses in the environment of agricultural products. Specifically, in the cooling or ventilation of potato storages, the work by Misener (1973), Hunter (1976), Grahs (1978), Lerew (1978) and Brugger (1979) provide sufficient data. It was felt that any additional experimental work would be a duplication of existing efforts. As a result, no new data was collected. 2.0 OBJECTIVES The primary objectives of this study are to apply the principles of the thermodynamics of irreversible processes to an open biological system subjected to forced convection to: l. derive a model for the quantitative analysis of entropy production resulting from heat transport, mass tran— sport and chemical reaction processes in the storage environment; and 2. examine the feasibility of using the model as a tool for the optimization of the ventilation rate. 3.0 LITERATURE REVIEW Each section of this chapter is arranged chronologi- cally to help give the reader a notion of the progress of research and analytical work in each area. 3.1 Thermodynamics 9f Irreversible Processes Miller (1960) tested the validity of the Onsager reciprocal relations and linear phenomenological equations for a variety of irreversible processes: thermoelectricity, electrokinetics, transference in electrolytic solutions, isothermal diffusion, conduction of heat and electricity in anisotropic solids, thermomagnetism and galvanomagnetism. Experimental results confirmed the validity of the Onsager relations for thermoelectricity, electrokinetics, isothermal diffusion and anisotropic heat conduction. Electrolytic transference showed good validity. In the other areas, experimental errors were too large to yield a significant test. Coleman and Truesdell (1960) showed that, given any linear relation between forces and fluxes in irreversible thermodynamics, a redefinition of forces and fluxes by 9 10 linear combination yields a relation with a symmetric matrix. That is, the rate of entropy production is invari- ant with respect to the redefinition. Cary and Taylor (1962a, 1962b) and Cary (1963) experi- mented with thermally-induced water vapor diffusion through air and through moist soil at temperatures ranging from 15°C to 45°C to test the theory of the thermodynamics of irrever- sible processes and the Onsager reciprocal relationships. The general rate equations for heat and mass transfer were transformed into simple relations of the flux as a function of temperature. In addition to verifying the rate equa- tions, it was found that the coefficient between heat and vapor flow and that between vapor and heat flow are identi- cal, essentially validating Onsager's reciprocal theory. In 1964, Cary considered a particular heat and water vapor transfer problem in a nonisothermal steady state sys- tem under conditions similar to what might develop during the normal drying of porous materials and again demonstrated that Onsager's reciprocal relations are valid. Cary and Taylor (1964) gave a theoretical treatment of some transfer processes in soils and showed that the general theoretical relations of irreversible thermodynamics encom- pass Darcy's Law and the mass diffusion equation. Wei (1966) recommended that the use of irreversible thermodynamics be restricted to situations where it improves the ability to describe, measure and predict outcomes, and when forces and fluxes are coupled; i.e., if the phenomeno- 11 logical coefficients Lij are not zero, whenever i and j are different. Sliepcevich and Hashemi (1968) analyzed a one- dimensional, one-component system in which the properties of temperature, pressure and chemical potential were assumed to be uniform throughout. Their results validated the Onsager reciprocal relations without using the theorem of macros- copic reversibility. I Fortes (1978) and Fortes and Okos (1978) developed a set of equations describing heat and mass transport in unsa— turated hygroscopic-porous media. The phenomenological coefficients were obtained by making use of the commonly accepted expressions for liquid and vapor fluxes in porous media. A model was developed to analyze heat and moisture transport in sand induced by a spherical heat source, using available data from the literature. While the approach was basically one of irreversible thermodynamics, the work also attempted to utilize concepts from mechanistic or classical thermodynamics. It was postulated that both liquid and vapor fluxes are driven by temperature gradients and rela- tive humidity gradients. Fortes and Okos (1981a), in an analysis of transport phenomena in porous media, postulated that the gradient of the moisture content is the driving force for both liquid and vapor movement in the media. However, it is possible for water, in a capillary-porous matrix, to move against the moisture content gradient. On the contrary, vapor always 12 moves in the direction of the equilibrium moisture content gradient. This finding was based on the principle of local equilibrium. It was proposed, further, that the driving force depends on (Vh)T and not on'Vh, where p is the chemi- cal potential. To evaluate the phenomenological coeffi- cients, the equations derived by non-equilibrium thermo— dynamics were compared to those obtained by using standard mechanistic principles. Parikh, Havens and Scott (1979) utilized a transient method to measure thermal diffusivity of low thermal conduc- tivity materials (specifically, glass beads and silt loam soil) over a range of volumetric moisture contents. The method was based on the solution of the transient heat con- duction equation in a cylinder 5 2 r where T is the temperature, t is the time and r is the radius. The transient method was used because steady state measurements were found to be difficult to obtain. This was due to the fact that moisture content gradients at steady state were also accompanied by steady state heat transfer. Havens (1980) analyzed the Cary and Taylor model for the description of coupled heat and moisture transport in water unsaturated glass beads. The transport coefficients appearing in the model equations were independently deter- mined, and the equations were numerically integrated to predict steady state temperature and moisture content 13 profiles. It was concluded that the coupling coefficient relating thermal gradients to moisture flux is strongly moisture—dependent; the coupling coefficient relating mois- ture content gradient to heat flux is extremely small; and the heat flux associated with the moisture content gradient was negligible. Kung and Steenhuis (1982) investigated the movement of water in soil under freezing conditions, using what was termed "irreversible thermodynamic equilibrium" and a modi- fied finite difference formulation to analyze the problem. Major conclusions were that the equations derived by the thermodynamics of irreversible processes provide correct values of heat, vapor, and water fluxes in a partially frozen soil. 3.1.1 Comments And Observations Several of the pieces of literature reviewed here represent milestones in‘ the development of irreversible thermodynamics. The findings helped open up new fields and/or applications to analysis by non-equilibrium princi- ples. Perhaps the most important principle of irreversible thermodynamics is the reciprocal relation of Onsager which relates the cross-coefficients. This principle has been experimentally validated by several researchers, including Miller (1960), Cary and Taylor (1962a), Cary (1963 and 1964), and Sliepcevich and Hashemi (1968), thus providing 14 analytical and experimental evidence of the soundness of the concept. Coleman and Truesdell (1960) made a significant contri- bution with the proof of the invariance of entropy produc- tion with respect to a redefinition of forces and fluxes. Just as important a contribution to researchers in porous media was made by Cary (1963, 1964 and 1966) when he showed that the theoretical development of the relations in irreversible thermodynamics includes Darcy's Law and the mass diffusion equation. Fortes and Okos (1978) [heat and moisture transport in sand], and Parikh, Havens and Scott (1979) [measurement of the diffusivities of low thermal conductivity materials] provided valuable lessons in application. The study by Fortes and Okos (1981a) also confirmed the validity of the standard practice of seeking coefficients in irreversible thermodynamics through a parallel analysis of equilibrium thermodynamics. Another result claimed by this study is that the driving force for mass transport depends on (Vb)T rather than on ‘Vh. This conclusion follows directly from a formal derivation of the mass transport expression, and is therefore not a direct result of that study. 3.2 Porous Media Analyses Furnas (1930) compiled extensive experimental data on heat transfer from gases (similar to air) to porous media of 15 iron ore, coke, limestone, bituminous coal, anthracite and a typical blast-furnace charge and provided many graphical correlations of parameters. In 1937, Muskat derived the generalized Darcy Law for gases .1. m (l+m)< u Ap(1+m)/m= k f0 %E (34) where1¢p is the pressure gradient, Po is the initial fluid density, k is the permeability coefficient, t is the time, m = 1 for isothermal expansion and m = Cv/Cp for adiabatic expansion. Muskat stipulated that the density of a gas flowing in a homogeneous porous medium must obey this dif- ferential equation for both steady state and transient con- ditions. Green and Duwez (1951) outlined a method of correlating experimental data obtained from studies of the flow of gases and liquids through porous metals. The correlation is based on expressing the pressure gradient accompanying liquid flow through a porous medium by a simple quadratic equation valid at both high and low Reynolds numbers _ g; = xpv + PPVZ (3-3) where k and P are viscous and inertial resistance coeffi- cients, respectively. This derivation was necessary because Darcy's Law, which defines the permeability coefficient, is valid only for low velocity liquid flows. 16 Scheidegger (1965) discussed four standard attempts at statistical mechanics models of transport phenomena in porous media: random walk of individual fluid particles, quasi-turbulent flow analogy, analogy with thermodynamics and general statistical mechanics. Scheidegger concluded that while nothing in the transport processes in porous media is really random, a stochastic process is employed due to the fact that current knowledge of the porous medium is of such an incomplete nature that the only reasonable way to deal with it is in terms of averages. Whitaker (1966) formulated the equations of continuity and motion in Cartesian coordinates for an anisotropic porous medium. In deriving the continuity equation, two assumptions were made: 1) that while the porosity may be a function of the spatial coordinates, the porous volume is large enough so that the porosity is independent of the volume; and 2) that the macroscopic velocity varies "slowly" with position. In 1969, Whitaker proposed three distinct (though not mutually exclusive) approaches to the problem of flow in porous media: statistical analysis, geometric model- ing and averaging. Each approach results in unspecified parameters that can be determined experimentally. Pinker and Herbert (1967) tested eight single gauzes placed perpendicular to the direction of air flow, and of such diameter and mesh to provide porosities between 0.3 and 0.7. Pressure losses were measured from Mach numbers of choking to 0.1. The data was used to provide a single rela- 17 tion between the incompressible loss factor and a form of Reynolds number. Emmanuel and Jones (1968) treated the case of gas flow in a straight frictionless, insulated duct of constant cross-section, with a porous plate of uniform thickness situated in it. A simple one-dimensional equation was given for steady, compressible, adiabatic flow of a perfect gas through the porous plate. Darcy's equation was modified after Shapiro (1953) to allow for compressible adiabatic flow and high mass flow rates. The authors concluded that compressibility should be important when the mass flow rate (and hence the pressure across the plate) is large. Curry (1970, 1974) investigated the flow of a compres— sible gas through a porous matrix externally heated at the fluid exit surface and formulated a numerical solution of the conservation equations of coupled heat transfer and fluid flow in a high temperature porous matrix. Analytical results were presented for one and two dimensions utilizing both non-isothermal and isothermal conditions between the solid and fluid phases. Curry concluded that the validity of the assumption of thermal equilibrium depends upon the type of porous material being studied. Siegel and Goldstein (1970, 1972) devised a technique for obtaining exact solutions for the heat transfer behavior of a two-dimensional porous medium subjected to cooling and used Darcy's Law to arrive at an analytical solution. The problem involved forcing an ideal gas from a reservoir at 18 constant temperature and pressure through a two-dimensional porous region, with the exit surface at a different uniform temperature and pressure. Both studies assumed isothermal conditions. Moore (1973) investigated the convective heat and mass transport in granular porous media saturated with a wetting liquid. Small glass beads and granular aluminum oxide were used to constitute the porous medium between two concentric cylinders, each surface maintained isothermally. Moore con- cluded that the thermal conductance of a porous medium (with particle size small enough to allow surface tension effects to dominate) depends on the quantity of liquid contained in the medium and may be significantly increased by convective transfer. Slattery (1975) postulated that characterizing mass transfer in a real porous medium requires empiricism, since with the unknown structural detail of the porous medium, mass transfer cannot be described totally on the basis of first principles. Yaron (1975) used the well-established conservation equations for separate phases of the porous media, solved them separately, matched them at the common interphase and applied an averaging procedure to establish a constitutive relationship. Yaron recognized and commented on the concep- tual problem of volume averaging: the need that the volume be small enough to be statistically homogeneous is often in serious conflict with the requirement that it be large 19 enough to enclose a sufficiently representative portion of the system. Duguid and Reeves (1976) developed a two-dimensional transient model for flow of a dissolved constituent through a porous medium. The model includes advective transport, hydrodynamic dispersion, chemical adsorption and radioactive decay. The model is expected to serve two purposes: allow the simulation of the transport of toxic materials through saturated-unsaturated porous media to predict future concen- trations in ground water and to provide the toxic-material concentration data necessary for human-dose calculations. Surkov and Skakum (1978) solved three versions of the heat and mass transfer equation in a porous medium (from linear to progressively non-linear) for a situation that may occur in the transpiration cooling of a rocket engine. The authors concluded that by allowing temperature dependence of the thermophysical properties of the porous wall (thus mak- ing the problem non-linear), a more exact description of the temperature field could be obtained. Steady state and isothermal conditions between the porous matrix and the coolant were assumed. Montakhab (1979) presented a closed-form solution to the initial value problem governing convective heat transfer between a fixed bed of granular solids and a steady flow of heating or cooling gas at constant mass velocity. The model was for a storage system initially at thermal equilibrium at a uniform temperature before a step change in the tempera— 2O ture of the convective fluid is imposed. Singh and Dybbs (1979) carried out an experimental and analytic study of non-isothermal flow in sintered metallic porous media. The experiments consisted of measuring the one- and two-dimensional temperature distributions and effective thermal conductivities of water-saturated copper and nickel sintered fiber metal wicks. The objective was to provide a self-consistent method for determining the heat transfer characteristics of porous metals. Volume-averaged forms of the conservation equations were used and local thermal equilibrium between the solid and liquid phases and the validity of Darcy's Law were assumed. The authors noted that for Reil0, inertia effects become important and Darcy's Law ceases to be applicable. To extend the range of vali- dity of Darcy's Law, an inertial term was added to the Darcy momentum equation. 3.3.1. Comments And Observations The standard assumption in studies on transport processes in porous media has been that Darcy's Law is valid. In many studies, this assumption was made without showing justification, even under conditions that appear to be unsuitable to analysis by Darcy's Law, e.g., high flow velocities. There have been several attempts at modifying Darcy's Law to enable the concept to be used in applications other than the restricted conditions under which Darcy proposed 21 his law. For example, Muskat (1937) modified Darcy's Law to make provision for both isothermal and adiabatic expansion. Another modification was effected by Green and Duwez (1951) to allow for processes with high flow rates to be analyzed. Emmanuel and Jones (1968) also modified the Darcy expression to make it suitable for application in a process with compressible adiabatic flow and large mass flow rates. Singh and Dybbs (1979) added an inertia term on discovering that for Re 2 l0, inertia effects become important and that this is not anticipated by Darcy's Law. The lack of a complete understanding of the behavior of porous media makes rigorous mathematical analysis rather difficult. Recognizing this, several researchers, espe- cially Scheidegger (1965) and Whitaker (1969) used stochas- tic or statistical models involving averages. 3.3. Agricultural Storage Analyses Schaper and Hudson (1971) found from small lot studies that the influence of temperature and relative humidity may be time-dependent, suggesting that this dependency may be utilized to modulate storage conditions according to elapsed time and thereby reduce insulation requirements. Schippers (1971) examined the effects of storage condi- tions on weight loss, change in specific gravity, suscepti- bility to damage, blackspot and color of chips and concluded that 1) storage conditions do not have much effect on sus- 22 ceptibility of blackspot and 2) whether temperature or rela— tive humidity is the more important storage factor depends on the quality characteristic under consideration. Schippers also suggested that since chemical sprout inhibi- tors are being used extensively, low storage temperatures (intended to limit sprouting) are no longer necessary. Thus, potatoes can be stored above 5°C. This would reduce fan hours, weight loss, susceptibility to damage and black- spot. Misener (1973) studied the drying characteristics of potatoes immediately following suberization and developed a model for simulating deep bed cooling of potatoes from known ambient conditions. The study included an optimization pro- cedure for the ventilation rate as measured against potato shrinkage and ventilation costs and concluded that the optimum air flow for cooling a 4.8 meter deep bed of pota- toes ranges from 5 to 7 m3/min-m2. Roa (1974) developed a model to simulate different sys- tems of the natural drying of layers of cassava. Experimen- tal and analytical studies were conducted on several varieties of cassava to evaluate conventional (natural) and newly-designed systems of drying under variable weather con- ditions. Hunter (1976) developed a steady-state simulation model for the white potato in storage. The model predicted air temperatures, relative humidity and weight losses and found the critical range of air velocities to be 0 to 3.05 meters 23 per minute. A major conclusion was that the vapor pressure deficit (rather than boundary layer effect) provides the thermodynamic driving force for weight loss in white pota- toes. Yaeger (1977) stored potatoes for six months in con- stant temperature chambers at three levels of humidity and pressure. The effects of various treatments on weight loss and pressure flattening were evaluated. It was concluded that relative humidity has more influence on weight loss and pressure flattening than potato pile pressure. Lerew and Bakker-Arkema (1977) gave a finite difference solution of the simultaneous heat and mass transfer equa— tions in bulk stored potatoes. Temperature and weight loss were simulated for various types of instrumentation and ven- tilation system management schemes commonly applied to com- mercial storages. Effects of fan control by temperature sensors and time clock to that of continuous fan operation were examined. Cloud and Morey (1977) presented an analysis of the effect of various parameters on the uniformity of air discharge from potato storage ventilation ducts. The param- eters used were the ratio of discharge area to duct cross— sectional area, static pressure, equivalent duct diameter, duct wall roughness, discharge dynamic loss coefficient and duct length. Charts were developed to predict uniformity of air discharge for three ratios of discharge area to duct cross-sectional area. 24 Dwelle and Stallknecht (1978) measured weekly carbon dioxide release by whole tubers of six potato varieties from November through May. The objective was to examine respira- tion and sugar content in an attempt to compare the respira- tion rates of whole tubers and tissue slices under storage temperatures ranging from 1.7 to 10°C and a relative humi- dity of 95%. It was found that generalizations could not be made, even within a given potato variety, because prior phy— siology in the field and at harvest influence subsequent physiological behavior in storage. Pratt and Buelow (1978) studied the relationship between vapor pressure deficit and time, on weight loss of potatoes in storage. It was found that stored potatoes ven- tilated with incoming air at 100% humidity still lost weight, leading to the conclusion that the potatoes must be slightly warmer than the ventilating air. Brugger (1979) developed a mathematical model for the two-dimensional air flow and heat and mass transfer within a potato pile and reported that temperature and humidity of the storage air are the most important factors influencing the quality of stored potatoes and that these affect weight loss, respiration rates and biochemical reactions, among other factors. Davis, 33 31. (1980) developed a mathematical model to predict air flow and pressures in perforated corrugated ducts. Even though the study was supposed to address air flow in agricultural storage (principally potato storage), 25 it appears that the theoretical development and experimental results were for empty cylindrical containers. Peterson, Wyse and Neuber (1981) compared respiration with the storage indicators of maturity, dry matter, invert sugars, harvest injury, bruise susceptibility and weight loss. The objective was to use respiration analysis as a guide to internal potato quality and a parameter for storage management. Respiration rates were found to be high immedi- ately after harvest (as have other investigators), reaching steady state at about 162 hours. Respiration rates were calculated by measuring the CO2 evolved per unit weight and time. 3.4 Entropy Production lg Biological Systems Very little previous work was found on the analysis of entropy production in biological systems. None of these has to do with agricultural storage. Briedis (1981) studied the link between engineering thermodynamics and experimental and theoretical biology. The work includes derivations of material and energy bal- ances for open, growing systems at thermodynamic states far from equilibrium. The derived relations were then used to study growth and development in the avian egg and microbial systems. The energy flows were linked to the relationship between oxygen consumption by and heat loss from the system. 26 Balmer (undated) developed a model for entropy rate balance for complex biological systems, using the growth equation of Bertalanffy and postulated that all living sys- tems experience a continuously decreasing total energy, reaching biological death (defined as a state of minimum total entropy) at the end of their lifetime. It is not clear what Balmer means by "decreasing total entropy", since it is the £E£E of entropy production rather than the total entropy production that reaches a minimum at biological death. Bornhorst and Minardi (1970) developed a phenomenologi- cal theory for contracting muscle, based on the sliding filament theory. The phenomenological equations were obtained for the whole muscle by assuming that each cross bridge is a linear energy converter with constant coeffi- cients. 4 . 0 MODEL DEVELOPMENT 4.1 Provisions 9f The Model The model is for entropy production and energy dissipa- tion resulting from non-isothermal heat transport, mass transport and chemical reactions in a porous medium. It provides for: a) heat generation within the product, resulting from biological metabolism; b) mass transport from the product to the void space; c) heat transport from the product to the void space; d) evaporation of water from solid surfaces; and e) chemical reaction resulting from respiration. 4.2 Assumptions l. The medium is isotropic, i.e., the system has the same properties in all directions. 2. Magnetic and electric field effects are negligible. 3. The only significant chemical reaction taking place in the storage environment is that of respiration, as 27 28 governed by the relationship C6H1206 + 60 ----> 6C0 + 6H 0 (4-1) 2 2 2 4. The walls of the storage volume are adiabatic and impermeable. 4.3 Geometric Modeling 2; the Potato The experimental data of Misener (1973) was collected using potatoes that averaged 9.50 cm in length and 5.10 cm in diameter. These dimensions are used to first model the potato as an ellipsoid with minor axes 4.75, 2.55 and 2.55 cm (Figure 4.1) and then as an equivalent sphere. The volume represented by the ellipsoid is modeled as the basic potato unit. The volume of an ellipsoid of minor axis a, b and c is given by Ve11 = 4nabc/3 (4-2) A sphere of radius r has the volume relation v = 4nr3/3 (4-3) sph From equations (4-2) and (4-3), the radius of a sphere of a volume equivalent to the volume of the ellipsoid is given by r = (abc)0'333 (4-4) 29 Figure 4.1 Basic potato unit 30 Given the relationship in equation (4-4) and the dimen- sions given above for the basic unit, the diameter of the equivalent sphere is computed as 0.063 m. 4.4 The Storage Volume gg‘g Packed Column Given that the potatoes are viewed as equivalent spheres, the storage volume can be modeled as a packed column (Figure 4.2). The Ergun (1952) relations for fluid flow through packed columns are used in this study. The Reynolds number based on the particle is defined as DuSP Rep = n—(l - 4) (4-5) where D is the particle diameter, uS is the superficial velocity (the velocity the fluid would have in the storage volume if the storage volume were completely empty), P is the density, p is the absolute viscosity and 4 is the poros- ity of the medium. The pressure drop is given by (Ergun, 1952; Bennett and Myers, 1974) f Lu 2(1 - 4) P p 5 AP = 3 (4'6) D4 where fp is the friction factor and L is the length of the column. For intermediate Reynolds numbers -- between 1.0 and 103 -- (Bennett and Myers, 1974), the friction factor is _ 150 fp - -R—e—- + 1.75. (4.7) P 31 Figure 4.2 The storage volume as a packed column: cross-sectional view 32 4.5 Characterization g; the Transport Processes The storage volume comprises a solid phase (the pro- duct) and a fluid phase (the convective air). The primary transport processes involve heat and mass. Heat can be transported by radiation (considered negligible in this study), conduction and/or convection, while mass transport occurs by convection and/or diffusion. 4.§.l Transport processes lg the fluid phase In the fluid phase, the relative importance of mass diffusion and convection can be evaluated by an examination of the Peclet Number, Pe. The same approach can be used for the comparison of heat conduction and heat convection in the fluid phase. 4.§.4.l The Peclet Number Given an arbitrary volume of fluid with no sources or sinks and a mass diffusion coefficient independent of con- centration, a mass balance yields the equation 6c - _ KE- 4’ VVC - DG vzc (4.8) where c is the concentration of matter, V is the velocity vector, t is time, DG is the mass diffusivity, V is the del operator defined as V = 21 33(— (4-9) 33 and'V‘2 is the Laplacian operator. [The derivation of the expression for the mass diffusivity, as it is used for this study, is presented in Appendix A.] For the steady state situation and for flow in the x direction only, equation (4—8) becomes 5c _ 82c vx 5? - DG 3 2 (4-10) x where vx is the velocity in the x direction. Equation (4- 10) is made dimensionless by the use of the following param- eters: ; c = ES" x = (4-11) 0 where v0 and co are some reference velocity and concentra- tion respectively and L is a characteristic length. When the relations in equation (4-11) are substituted in equation (4-10) and the appropriate simplifications made, the following equation is obtained: (4-12) The coefficient on the right hand side of equation (4-12) is the reciprocal of the dimensionless Peclet number for mass transfer, defined as (Levich, 1962) VOL Pe = —D (4-13) 34 It is worth noting the similarity between this Peclet number for mass transfer and its heat transfer counterpart. The thermal diffusivity, used in the heat transfer Peclet number expression, has been replaced by the mass dif- fusivity. Otherwise, the two expressions are identical. The left side of equation (4-12) represents the convec— tive transport of matter in the fluid in the x-direction; the right side accounts for molecular diffusion. Therefore, just as the Peclet number in heat transfer is a measure of the ratio of energy transport by convection to that by con- duction in a given direction, in mass transfer, the parame- ter represents the relative importance of the convective and diffusive transport of matter (Levich, 1962) in the direc- tion of flow. When the Peclet number is much less than unity, molecu- lar diffusion predominates, mass transfer by convection is negligible and the concentration gradients are dependent only on the process of diffusion. On the other hand, when the Peclet number is much greater than unity, molecular dif- fusion becomes negligible and the concentration gradients are determined solely by convective mass transport. For the forced convection situation studied here, equa- tion (4-13) yields a mass transfer Peclet number of 780, which is greater than unity by approximately three orders of magnitude. It is postulated from this that mass transport by convection predominates and that the fluid is, for all practical purposes, at a single concentration. 35 While it is important to keep moisture losses to a minimum in the storage environment, the cooling of the pro- duct to an acceptable and safe storage temperature is the major focus of fruit and vegetable storages. Therefore, even though the Peclet number for mass transfer offers needed insight into the nature of the mass transfer process, the more important aspect of this analysis is the heat transfer process. As already indicated, the Peclet number for heat transfer is given by P=— - e d (4 14) where d is the thermal diffusivity of the fluid and has the relation k FE where k is the thermal conductivity of the moist air (90 d = (4'15) J/m-OC-hr -- Lerew, 1978), P is the fluid density (1.25 kg/m3) and C is the heat capacity of the air stream (1007 J/kg-OC). Given the approximate values in parentheses, the expression in equation (4-15) yields a value of 0.0715 mZ/hr for the thermal diffusivity. When this value is substituted in equation (4—14), with a velocity of 24.0 m/hr and the length of the storage bin (2.4 m) as the characteristic dimension, a Peclet number of 806 is obtained, which is approximately the same as what was obtained for the mass transfer situation. A Peclet number of this magnitude indi— cates that internal temperature gradients are practically 36 non-existent and that the fluid is at a uniform temperature. 4.§.3 Transport processes lg the solid phase The experimental work of Misener (1973) showed that gradients of moisture concentration and temperature across the potato are negligible. Thus in this study, mass diffu- sion and heat conduction within the potato will be con- sidered negligible. It is surmised from the discussions above that there is a step change in moisture concentration (Figure 4.3) and temperature across the boundary of the product and the con- vective air. The gradients of moisture and concentration at the boundary can be considered infinite. 4.6 Mathematical Relations The mathematical derivation revolves around obtaining an appropriate expression for the local entropy production and assumes at least an elementary knowledge of the thermo- dynamics of irreversible processes. Readers who are unfami- liar with irreversible thermodynamics are directed to DeGroot and Mazur (1963), Haase (1969), Katchalsky and Cur- ran (1965), Luikov (1966) or Prigogine (1967). 4.4.4 System Entropy Production Since entropy production is a global quantity in this study, the control volume comprises the entire storage volume (Figure 4.4). Being an open system, it exchanges 37 has one ccw Opmpoa one ca mpsumfloe mo cofiuwAucoocoo m.v mpswfim cOApmooA pcoficmum coHpaupcoocoo maficaaafi J 1|l'lllllllll N? A UV Q 3 oawpoa cw muzumfloe mo cofiuapucoocoo I A UV Ham cw manpmfioe mo coflpwnpcoocoo 63 A DC A UV elnisrom JO uoIieJIueouog Control volume Ti' ¢1' “1 Figure 4.4 Control volume for the analysis of total entropy production (actual calculations were based on twelve subdivisions of the above control volume.) 39 both heat and mass with its environment. However, for mathematical convenience, the points of these exchanges are restricted to the entry and exit points of the ventilation air. The rate of entropy production can be represented as bsv _ _ W = - V. (JS) 4' O’ (4’16) where _ '50 _ Js = ‘T’ + 2k Ckskvk ”’17) is the entropy current flux and represents flows of entropy into and out of the system, while “I -__2 -1 - E G — TZ‘VT T2 5k ckavk7VT + T‘V’P JChA T (4-18) +12 Eff 2 "v(£’i)+ T k k k ' k Ckvk T represents the local production of entropy due to the occurrence of irreversible processes in the system (Haase, 1969). The production of entropy (rather than entropy flows into and out of the control volume) is the quantity of interest in this study since it relates directly to the dis- sipation of energy. After several manipulations, the entropy production term reduces to 3 J A = _ Q 1 — o — Ch _ 40 In arriving at equation (4-19), the following relations have been used: “k ”k T = ’T_ ' Sk ‘4’”) and JR = ck (Gk - V) (4-21) in addition to the Gibbs-Duhem equation at constant-tempera- ture, given by -VP + 2k Ckmk)T = 9 (4-22) 4.4.; Generalized Fluxes And Forces The control volume comprises potatoes and moist air. An examination of equation (4-19) shows three contributions to the local production of entropy: heat transfer due to a temperature difference between the potatoes and the air» stream, mass transfer (almost entirely from the potatoes into the air stream) and chemical reaction due to metabolic processes inside the potatoes. Given this framework, the summation sign can be removed from equation (4-19) and the subscript k used to refer to the water component of the potato. One can define a dissipation function, r, such that r = T 0' (4'23) 41 Then 3 r - Jk [Kk .. mmT] - —T 'V T + JchA (4-24) Assuming linearity for the heat and mass fluxes and the chemical reaction, - - " SE - Jk ‘ LKK [Kk ' m’T] " LKQ T (4 25) for mass transport, _ — _ _ S711" _ for heat transport, and Jch = Lch A (4'27) for chemical reaction, with the Onsager reciprocal relations giving Lox = LKQ (4-23) Equations (4-25), (4-26) and (4-27) illustrate the application of the Curie-Prigogine principle which, for an isotropic system, forbids the coupling of vectorial quanti- ties that differ in order by an odd number. Thus, heat and mass transfer, first order vectorial quantities, can be cou- pled with each other, but not with chemical reaction, a zero order vectorial (scalar) quantity. The gradient terms of chemical potential and concentra- tion, (Vb C)T and‘VbS, are related (Katchalsky and Curran, s 1965) by 42 c 6"is (V115 )T ' Eaves = nss v'Cs (4'29) For ideal gases and ideal solutions, the concentration-dependent part of the chemical potential has the relationship (Katchalsky and Curran, 1965) u = R T ln c + p o S s S (T) (4-30) It follows from equation (4-30) that bu 53E = ‘155 = izT (4-31) S S With the above relations, the heat and mass transport equations (equations 4-25 and 4-26) then become ‘- - -' __§ - SE: - Jk - LKK [Kk - RT (35] LKQ T (4 32) and _ _ VcS JQ = LQK [Kk - RT 'E—s—l - LQQ T (4-33) 4.4.4 Chemical Reaction The following assumptions were made in the analysis of the contribution made by chemical reaction to the local entropy production. a) the combustion of glucose is the only significant chemical reaction in the system; 43 b) neither carbon dioxide nor oxygen is significantly accumulated in the product; and c) the respiration quotient for oxygen and carbon dioxide is approximately unity. The respiration quotient is the ratio of moles of oxygen consumed to moles of carbon dioxide produced. The entropy production induced by the reaction is represented by (4-34) where the term, Jch' is the rate or velocity of reaction per unit volume and A is the chemical affinity. If the storage volume is assumed to be a linear system, then (as was done for the case of heat and mass transfer), the reaction velo- city can be written as J = L ch A (4-35) ch as represented in equation (4-27). The phenomenological coefficient, Lch’ must be determined from experimental data. Since respiration data is usually available in terms of carbon dioxide evolved, J can be directly related to ch experimental data by expressing it as d as (4-36) - l Jch - V where Q is the extent of reaction, or the reaction coordi- nate. 44 Ideally, the reaction coordinate must be written in terms of mass of moisture released by the reaction per unit time. Since the rate of carbon dioxide evolution is charac- terized by rC h (with units of mg. CO evolved per kilogram 2 of product per hour), the extent of reaction is represented in this thesis as d _ lg 1 3% ' 4 196 rch Pp V (4-37) where pp is the potato density and V is the total volume of potatoes. The factor 18/44 accounts for the fact that for every 44 grams of carbon dioxide evolved, 18 grams of water is released; and 106 provides a conversion from milligrams to kilograms. Given the expression above, the relationship for J ch becomes 4 J = 4.44 x 10- r (4-38) ch ch The chemical affinity, A, for the reaction is a function of the chemical potential of each of the species involved in the reaction. In particular, for the respiration reaction, A = ”a + 6gb - 6pc - 6nd (4-39) where a, b, c and d represent glucose, oxygen, carbon diox- ide and water vapor, respectively. While the chemical affinity can be computed as in equa- tion (4-39) above, in many practical applications, it can be approximated by the standard heat of reaction, 'Afioch° By 45 the Gibbs-Helmholtz equation (Smith and Van Ness, 1975), 0 ch 0 d‘RT’_ AHch 44g —a'r " 2 (-) Ifqéfioch is assumed to be constant over the temperature range of interest (278-2880K, in this case), then it is apparent from equation (4-40) above that O .. 0 A5 ch " A“ ch (4-41) For the respiration reaction, the value of ‘égoch is -2870 kJ/mol of glucose combusted (Lehninger, 1970), or 26,600 kJ/kg of water released, assuming complete oxidation. g.§.g Modification 9E The Dissipation Relationship In view of the earlier discussion on the significance of the Peclet number and the subsequent assumption that there is no continuity in either concentration or tempera- ture across the product boundary, the relations in equations (4-32) and (4-33) which call for temperature and concentra- tion gradients are not very useful and do not lend them- selves to practical analysis of the situation here. As a result, the mass and heat flux equations in this study are modified as follows. With the nature of the application considered here, the external force term, Kk, is made up entirely of viscous forces in the system. Viscous forces are, generally, con- sidered internal to the system. However, a simple sign 46 change is all that is required in this case to change the frame of reference. Now, define a relationship between ER and the diver- gence of the stress tensor, V'E’, such that m K k Pk (4-42) where Pk is the mass density of component k. Given the one-dimensional treatment used here and the fact that the shear stresses are generated entirely as a result of the air flow, equation (4-42) can be reduced to 41 z Kk‘L (4-43) '13 where L is a characteristic dimension in the vertical direc- tion of the bin. In this thesis, L is taken as the length of the bin. From Ergun's analysis (Ergun, 1952), APAC< t' =—— rz St (4-44) where‘ép is the pressure drop, Ac is the cross-sectional area of the empty column, 4 is the porosity and St is the total surface area of the solids in the bed, given by Ergun (1952) as (4-45) where rh is the hydraulic radius, defined as (Bennett and Myers, 1974) 47 _ 4D In ‘ ETT'I'ZT (4-46) Combining equations (4-44), (4-45) and (4-46), D 4 trz = f 6(1 - 4) AP (4-47) Finally, combining equation (4-47) with equation (4-6) for the pressure drop 2 f u tiz = _E__§§_E (4-48) 64 where fp is defined as in equation (4-7) and us is the superficial velocity. g.§.g.l The Modified Relations Given the need to use temperature and concentration differences rather than gradients, the heat and mass transfer equations (equations 4-32 and 4-33) are modified to read L' trz _ _ _§-: AT .. k‘ KKIP'WCCS] LKQT (449) for mass transfer and 1' Ac _ . ._££ _ __§ _ u T _ J -LQK[P We Cs] 1.00% (4 50) '0 for heat transfer, with trz as given in equation (4-48). The factor qc in equations (4-49) and (4-50) is termed the combined skin resistance coefficient. It is used to account for the resistance to mass transport presented by 48 the skin membrane of the potato. Without the coefficient, the use ofqéps in equation (4-49) and (4-50) essentially implies a free water—to—air interface, which is not true for the situation considered in this thesis. Note also that the gas constant and the absolute temperature have been absorbed into the combined skin resistance coefficient, i.e., qc = qRT, where q is the skin resistance coefficient. The major difference between the equation pairs, (4-32, 4-33) and (4—49, 4-50), is that the temperature and concen— tration gradients in the former have been replaced by simple differences in the latter. This step was made necessary because of the discontinuities in the system. The modifica- tion of the equations makes it necessary to redefine the phenomenological coefficients, The mass, cross- L... 13 phenomenological and heat transport coefficients are, there- fore, respectively modified to take the forms h . - DP L KK - g D (4-51) DTT L KO = L QK = ——5—2 (4-52) L'QQ = h T (4-53) where hD is the mass transfer coefficient, P is the air den— sity, g is the gravitational constant used here to provide dimensional homogeneity, D is the diameter of the storage T volume, D is a modified Soret coefficient and h is the heat transfer coefficient. 49 Given the mass flux expression, 3*, the rate of mass transport out of the potatoes is obtained by integrating the scalar product of 3k and the directed differential area. Thus, a. a. "L? = - jg Jk°ai (4-54) In a one-dimensional formulation using equations (4- 49), (4-51), and (4-52), the mass transport expression becomes h dm .__E = _. S [ D dt g P (trz ACs T AT t D - q 7;—) - D -—J (4-55) F s D Even though this equation is derived from a global for- mulation, it can be used as a local equation based on the fact that the potatoes are the only materials inside the control volume producing moisture. 3.6.; The Thermal Diffusion Coefficient The thermal diffusion coefficient, in a binary system, is often related to the thermal diffusion ratio, K through T! the expression (Bird, et al., 1960; Bennett and Myers, 1974) where P is the density, DT the binary thermal diffusion coefficient, DAB the binary mass diffusivity, c the concen- tration, and the M1 are the molecular weights. 50 Using equation (4-56) as a defining equation for the thermal diffusion coefficient would require a knowledge of K a quantity that is not available for the application TI considered here. To circumvent this problem, the expression for DT is modified for this study. The defining equation used here is 0 0T = ————A2 Pa (4-57) 9 where Tp is the absolute temperature of the product and DAB is the binary diffusivity. 6.6.6 The Resistance Coefficient for Mass Transport An analytical expression could not be obtained a priori for this coefficient. It is therefore determined by curve— fitting to experimental data. It is expected that, once value(s) is (are) obtained for the coefficient(s), appropri- ate regression equations can be developed for use as a predicting tool. 6.6.1 The Rate 66 Local Entropy Production From equation (4-19), the local entropy production is 255 '3 _ _ J A "BYV'="="%OVT+%JI<'[KI<‘ M’T] +—%—(4‘58) where the subscript T on the chemical potential gradient term implies that the gradient is spatial isothermal. From 51 equations (4-32) and (4-33), equation (4-58) can be written as L' ‘Vb L' =-__Q§. — _ _§ JQE 0' [Kk RT Cs]'V'I'+ 2 T'VI‘ T2 T L' V0 VG KK _ S o — S + ——T [Kk — RT -—-—CS ] [Kk - RT —CS ] u - 1%? '[Ek - RT :35] + JC: A (4-59) An examination of equation (4-59) reveals immediately that it presents the same type of problem encountered with the mass transport equation and which necessitated the introduction of the skin resistance coefficient. Three of the terms in equation (4-59) contain the gradient of the temperature, a quantity that has physical meaning but no real mathematical form or significance owing to the discon- tinuity of the temperature across the potato skin membrane. It became necessary therefore to modify the first, second and fourth terms of the equation that contain the tempera- ture gradient. This was done by replacing the derivative of the temperature resulting from the scalar product operations by the quotient of the temperature and a characteristic dimension L. From the Onsager reciprocal relationship and the defin- ition of the dot or scalar product, the first and fourth terms on the right hand side of equation (4-59) are identi— cal. After combining these terms, performing the inner 52 product and using the modification described above, with equation (4-43) incorporated, equation (4-59) becomes TL P L' t’ J A KK rz 2 ch + TL ( P - RT) + T (4-60) 6.6.6 The Dissipation Function From equation (4-23), the dissipation function is r = T a (4-61) Given equation (4-60) above, the expression for the dissipation function is = QQ _ KQ rz _ r I. L l P RT] L' ‘f KK rz 2 + I, ( P - RT) + JCh A (4-62) 6.1 Other Thermodynamic Relations The principal objective of this thesis is to calculate the rate of entropy production and from that the dissipation of energy in the system. However, the relations for these two quantities contain coefficients that are dependent on the potato temperature, the moist air temperature and the concentration of moisture in the system. 53 To enable the calculation of the air and potato tem- peratures in the storage environment, an energy balance analysis must be done on the system. For the purpose of performing appropriate balances, the storage volume may be divided into two sub-systems: the potato and its water con- tent and the moist air and its moisture content. The entire storage volume is a combination of the two. 6.1.1 First Law Analysis The thermodynamic system under consideration is the storage bin, which contains potatoes and moist air, as shown schematically in Figure 4.4. The variables of interest are the potato and air temperatures, Tp and Ta, the mass of the water content of the potato, mp, and the mass of the water content of the moist air, ma. All four variables are func- tions of time and location inside the bin. Each of the two control volumes discussed above is con— tained in the boundaries of the bin. The first law is applied only to the collection of particles located inside the control volume at time t. 6.7.1.1 Control Volume 1 The fixed collection of particles contained in this control volume consists of the potatoes bounded by the planes x and x+dx and their water at time t. The formula— tion assumes that the potatoes and their water content are 54 in thermal equilibrium. The first law is given by do + dW = dE (4-63) where heat added to and work done on the fixed collection of particles are considered positive here. The total change in energy, dB, is made up of the change in energy of the potato solid mass and the change in energy of the water content of the potato over a time inter— val dt; note that some of this water may leave the potato during the time interval dt. Mathematically, E = M - - d pcvpdTp + mpCVWdTp + ( dmp) hfg H( dmp) + (-dmp) va(Ta - Tp) H(-dmp) (4-64) where Mp is the solid mass of the potato, assumed to be a constant, Cvp is the constant volume heat capacity per unit mass of the potato, va and va are the constant volume heat capacities per unit mass of the water vapor and liquid water, respectively, Tp is the potato temperature, Ta is the air temperature, hfg is the heat of vaporization per unit mass of water at the potato temperature, dmp is the change in moisture content of the potato and H is the Heavyside step function defined by if x>0 = 1‘ H(x) 0; if x60 (4'65) 55 The variables Ta and Tp are absolute temperatures and are functions of both time and location. Note that the energy associated with the change of phase of the moisture lost from the potato is treated in equation (4-64) as an internal energy term rather than a heat flow term. This is because during the time interval dt, the moisture lost from the potato remains a part of the control volume associated with the potato, even though part or all of it may be outside the boundary of the potato. Thus the conversion from a liquid to a vapor does not con- stitute an exchange of energy with the environment; it sim- ply constitutes a change in the internal energy of the con- trol volume. The heat flow to the system of particles is given by dT d0 = hAS(Ta - Tp) dt - (chct 33(9)): dt 6T __2 + “(Act dx)x+dx dt + q(Mp + mp)dt (4-66) where As is the surface area of the potato, A is the ct potato-to-potato contact area, h is the convective heat transfer coefficient, k is the thermal conductivity of the potato, q is the rate of heat release by metabolic processes and x and x+dx refer to spatial locations. The sum in parenthesis in the last term of equation (4-66) is the total mass of the potatoes, including their water content. This is because the expression for the meta— bolic term (derived later in this chapter) is based on the 56 total mass of the potato, even though only the carbohydrate portion is undergoing the chemical reaction that generates the heat. Assuming that the potatoes are "arranged" in the bin in such a manner that only point contacts occur between any set of adjacent pieces, A is very small; thus the thermal con- ct duction term is negligible. Therefore, equation (4-66) can be reduced to do = hAS(Ta - Tp)dt + q(Mp + mp)dt (4-67) As water is lost from the potato, it comes to the sur— face of the product as liquid water which must be evaporated into the air stream. Due to the volume expansion that occurs as the liquid becomes a gas, negative work is done by the environment on the gas. If it is assumed that the pota- toes experience a negligible change in volume, then de is nearly zero and the total work associated with the control volume can be represented as dW = - 9 WV (4-68) where dVv is the volume change of the water vapor. Since water vapor behaves as an ideal gas, equation (4-68) can be rewritten as dW = - vavTv -—— (4-69) where mv, Rv and Tv are the mass, gas constant for water vapor and temperature of the water vapor respectively. A 57 reasonable assumption is that the water vapor and the air reach thermal equilibrium instantaneously, so that Tv is essentially Ta. The volume of the liquid before evaporation is much smaller than its volume after being vaporized, therefore the change in volume is nearly equal to the final volume, so that V 1.0 (4-70) Equation (4-69) becomes dWon = - (-dmp) RvTa H(-dmp) (4-71) where mv has been replaced by (-dmp) H(-dmp). Substituting equations (4-64), (4-67) and (4-71) in equation (4-63), the first law yields hAs(Ta - Tp)dt + q(Mp + mp)dt - (-dmp) RvTa H(-dmp) =M .. .- pCvpdTp + mpvadTp + ( dmp)hfg H( dmp) + (-dmp) va (Ta - Tp) H(-dmp) (4-72) or, simplifying, hAs(Ta - Tp)dt + q(Mp + mp)dt + [- RvTa - hfg - va (Ta - Tp)] (-dmp) H(-dmp) 58 =M .. pCvpdTp + mpvadTp (4 73) Equation (4-73) can be transformed into a continuum formulation by using the following definitions: M =‘Ab dx (4-74) mp = Ap(x,t) dx (4—75) om b\ dmp = 1figdt = 1%? dx dt (4-76) A = d dx (4’77) where _ total potato mass in bin (4-78) ‘46 _ length of bin is a constant, \ _ mass of water in potato at time t 4 79 'p - (differential length dx of potato ( - ) is a function of time and location and _ total potato surface area C(p " length of bin (4'80) is a constant. Using equations (4-74) to (4-77), equation (4-73), divided by (dt dx) becomes 3% hdp(Ta - 'rp) + guy, + ,\p) - (- 37:9) RvTa H(- TE) DT 63 b; = LApCvp + |F\pcvw] Tfi? + (- tz) hfg H(- 7“?) 59 b\ b\ + (- fifmwwa - Tp) H(— {—3) (4-81) Solving for the substantial or material derivative of the potato temperature DT M M Tfi? -.1fi? + u 15? (4-82) where u = 0 for the potatoes, equation (4-81) becomes 251' 1 17:2 2¥Cvp+'\c [mpma ' Tp) *9 V\p+o\p) p vw -bx b) + (- RvTa ' hfg) (_5t2) H(— ) -b\ b\ o o - CW(Ta - Tp) (—5EE> n(-«1fi§)1 (4-83) Since the cross-sectional area is treated as constant, the continuity and momentum equations are respectively (Eck- ert and Drake, 1972) §§(Pu) + gg = a (4-84) and g%4p + Puz) + §%(Pu) + Pg = a (4-85) where P is the density, u the velocity and p the pressure, all relating to the potato, x is the vertical coordinate and g is the standard acceleration due to gravity. Since p is essentially constant and u = 0 for the potato, these equa- tions give 60 gg = ‘ P9 (4-86) or P = P 9 (L “ X) (4-87 which says that the potato pressure at x is due to the weight of the potatoes above location x. Only the moisture given up by the potatoes is in rela- tive motion (on exit from the potato). Since this moisture eventually appears as a moisture gain for the moving air mass, the continuity equation expressing this fact is deferred until the control volume comprising the moist air is analyzed. 6.1.1.1_ Control Volume 1 This control volume comprises the air and its water vapor located in the bin between the planes x and x+dx at time t. At time t + dt, this collection of particles would be located partially outside the original control volume due to the movement of air (Figure 4.5). Therefore, this ther- modynamic system is an open one. Again the first law statement of equation (4—63) is used. The total differential energy, dB, is given by dE = ManadTa + manvdTa (4-88) where Ma is the mass of the dry air, ma is the mass of mois- ture in the air and Cva and va are the specific heats of 61 ufia :ofiuwafip:o> on» no mamaaaca sow oesflo> Hosucoo m.v ousmfim a mafia pd oESHo> Houucoo 0’0 I. O .0 to u 00.0 H O ‘0 0 n O. O o l 00 § 0‘ O. 00'. o. 0'. 0 ~ 0 O O O O .0 t Q‘Olooooo\OoOooooO 0 no. 00. h o .0 ...-soo- \ O O O A}... .. ........ a. as: ......A no 0 o 0.09.! lo oo.‘\ 8 \C s o O o o o o co .....l.. ....u.. or. ......1 as... ... ..J ‘0 o no to. to“ ’00. u”. M. 00... DIG D... _ xv '— 62 the dry air and the water vapor, respectively. Now, define M = flhdx (4-89) and m = hadx (4-90) with = total dry air mass in bin at time t (4_91) ”E length ofgbin a function of Ta and _ mass of water in the air at time t 4a — length of bin (4-92) a function of the temperature and relative humidity of the air, and the history of moisture transmission from the pota- toes into the air stream. Substituting terms, the change in the internal energy is given by dE =‘AhcvadTa dx + AacvvdTa dx (4-93) The heat flow into the system of particles is dT _ _ a 50 - hAS(Tp - Ta) dt (kaAaC?EF)X dt dTa + (kaAac757)x+dx dt (4'94) where Aac is the cross-sectional area open to air flow, a function of the porosity and ka is the thermal conductivity 63 of the air. The thermal conductivity of the air, ka, is a small quantity; therefore convection predominates. As a result, the last two terms of equation (4-94) may be dropped. With equations (4-74) to (4-77), (4-89) and (4-90), equation (4-94) without the conduction terms reduces to = h _ - dQ dp(Tp Ta)dt dx (4 95) The total work done consists of four parts: air pres- sure work, potato pressure work, work against gravity and work done by the expanding dmp of the first control volume. Mathematically, dW = [(pAa u) dt — (pAaCu)x+dx dt] + pa dV C X P - (Ma + ma) 9 u dt + (-dmp) RVTa H(-dmp) (4-96) or, again making the assumption that de is negligible, _ b dW - -Aac 3§(pu) dx dt - (flfi + ha) 9 u dx dt + (-1%?) RVTa H(-1%?) dx dt (4-97) where u is the speed of the convective air. The above equation introduces two new variables, p and u, that must be transformed into previously defined vari- ables. This can be done by using the continuity and momentum equations (equations 4-84 and 4—85) written with respect to 64 the air stream, in addition to the perfect gas law, given by P = RaTa (4-98) The Mach number for this system is the ratio of the velocity of air flow to the speed of sound in air, approxi- mately 340 m/s for the temperatures of interest in this study (3°C 6 T 6 17°C). When the Mach number for a system is low, the flow can be considered incompressible (Cambel and Jennings, 1958; Benedict, 1983); that is gg = a. (4-99) where the term on the left hand side of equation (4-99) is the substantial derivative of the air density. For the system under consideration, the Mach number is 5 approximately 2.0 x 10. , thus incompressibility is assumed. The continuity equation can be rewritten as b b b D 3§(Pu) + SE = g = P8% + 5% (4-100) from which it follows, given the assumption of incompressi— bility, that g; = 9 (4-101) Thus the velocity is constant along the bin, even though it may still be a function of time. The momentum equation can be rewritten as g; + 61x (Pu'u) + 2?? (P'u) + Pg = 9 (4-102) 65 or gg + u g; (Pu) + Pu §%-+ u gg + P %% + Pg 0 (4-103) or g; + u P§% (Pu) + gE] + Pu %% + P %% + Pg 0 (4-104) The terms in square brackets in the above equation con- stitute the continuity equation and thus sum to zero. When equation (4-101) is incorporated, equation (4-104) reduces to gg = - P %% - Pg (4-105) For the ventilation system under consideration, the inlet velocity is kept constant. Therefore, at x = 0, %% = 0 for all t. (4-105) In addition, from equation (4-101), the velocity is not a function of x; therefore, it follows that b“ = a for all x and t. (4-107) 6? from which g5 = - Pg (4-108) Therefore equation (4-97) becomes dW = [Aachu - (Ah + 3a) 9 u b\ b\ + (-7E?) RvTa H(-1%?)] dx dt (4-109) 66 However, by the definition of P' AacP dx = (”E + ha) dx (4—110) Therefore the air pressure work cancels the work against gravity and b\ b\ dw = (-1%§) RVTa H(-1%?) dx dt (4-111) Combining dE (equation 4-93), dQ (equation 4-95) and dW (equation 4-111), flhcvadTadx + AanvdTadx = hdp(Tp - Ta) dt dx b\ b\ + ("67?) RvTa H(-%t2) dx at (4-112) Dividing through by dt-dx and solving for the material of substantial derivative of the air temperature, DTa _ 1 Dt /\ana + '\aC [hd (T - T ) vv P P a b\ b\ + RvTa (-7gg) H(-1%?)] (4-113) where DT 6T M a a a .51— = W + u W (4-114) Finally, b'r a _ 1 _ Ti? ",Racva + A C [hdp(Tp Ta) avv 67 b\ 6\ 6T +1.. .2 .2 -.— It must be true that since the air receives moisture from the potatoes, dka = - dkp H(-d)p) (4-116) and since these are both material derivatives, b\ b\ b\ B\ .— + u .2- <- .2 There are four unknowns in equations (4-83), (4-115) '\p together with the mass loss equation developed from irrever- and (4-117): Tp, T and Ta' These three equations, a! sible thermodynamics (4-55) constitute a sufficient number of field equations to enable solutions to be obtained for the listed unknowns. Equation (4-55) can be cast in the four unknown vari- ables by using the following definitions. 25m 6\ b\ 1E? = (— 1%?) H(- 1%?) dx (4-118) 3 = d dx (4-119) ll *3 I *3 Ag (4—120) The concentration of moisture in the potato and in the air stream are given respectively by (c ) = 9 (4-121) 68 (C ) = ——— (4-122) where AC is the total cross-sectional area of the bin. From these relations, ACs Pp Pa Act ' Aac ( - -—-)'( Cs Act ' Aac ac A ) (4-123) P which, on simplification and noting that PP >> Pa' gives ——— = 1.0 (4-124) With the above equations, equation (4-55) can be writ- h t' _ DP rz “'15? _ - dP [§_5'(_F_ - qc) (Tp - Ta)] (4-125) 6.6 Modeling 66 Parameters Several parameters have been employed in the develop- ment of this model. These are discussed below. 6.6.1. Heat And Mass Transfer Coefficients Following Misener (1973), the mass transfer coefficient is defined as hD = 8.96 x 19'4 h (4-126) where h is the heat transfer coefficient which is modeled 69 after the McAdams relationship (Holman, 1976) for heat transfer from spheres to a flowing gas, primarily because it is quite simple, yet accurate. The McAdams equation is 0 6 h = 0.37 Re ° ka / D (4-127) SP where Re is the Reynolds number, ka is the conductivity of the convective air and Ds is the diameter of the sphere. P The values obtained by the use of equation (4-127) are in very good agreement with generally-accepted values of the heat transfer coefficient for suberized potatoes in storage. 6.6.1 Moist Air Relations Based on the ideal gas law, the humidity ratio, Y5, is given by sat yh = 0.622 89 sat (4-128) P - ¢P where 0 is the relative humidity, Psat is the saturation pressure and P is the atmospheric pressure. Assuming that Dalton's Law of partial pressures is valid, the relative humidity 0 is defined as the ratio of the vapor pressure to the saturation pressure, i.e., (4-129) where Pv is the vapor pressure. 70 For vaporization processes at low pressures (about one atmosphere), the Clausius—Clapeyron equation is valid and gives the following relationship for the saturation pressure sat h dp = _£2 93 (4-130) sat R P 'v T where hfg is the heat of vaporization, T is the absolute temperature and Rv is the gas constant for water vapor. Using data available from standard steam tables, hfg was derived for this study as h = (3147.4 - 2.365T) x 103 fg (4-131) This expression is valid for temperatures between 273.150K and 313.150K. Using equation (4-131), equation (4-130) then becomes sat dPsat = 314754 dT _ 2.R365 quT (4_132) P RvT v After integrating from a lower limit of 273.15°K to T, the expression for the saturation pressure becomes 6814.03 psat = exp (60.08 - T - 5.12 In T) (4-133) 6.6.6 The Heat Capacities The heat capacities for liquid water, water vapor and moist air are all defined as having acceptable values of 4184, 1880 and 1007 J/kg-K, respectively (8012 & Tuve, 1976). The heat capacity for potatoes results from an 71 expression, derived for this study, relating the heat capa- city of the product to its moisture content. This expres- sion is Cv = 1430 + 2765 m.c. (4-134) where m.c. is the moisture content of the potato. This expression provides excellent agreement with published data (Mohsenin, 1980), with absolute errors of less than 4%. 6.6.6 The Rate 66 Heat Generation Misener (1973) derived the following expression for the heat generation rate, for 6p > 4.0OC Qres = 1.45 6p - 3.3 (4-135) where 6p is the product temperature, 0C. Given the units reported for Q (J/kg-hr), the above expression would res yield low and inaccurate values for the heat generation rate. Expressions by several other researchers (Brugger, 1979; Hunter, 1976; Lerew, 1978) were found to be similarly unsatisfactory. Based on the experimental data of Listov and Kalugina, 1964 (as reported by Mohsenin, 1980), the expression below was derived for the heat generation rate for the temperature range 4.00C < 6p < 16.0OC: ores = 5.6 6p + 21.4 (4-136) 72 Except at 10°C where a reliable experimental value for the heat generation rate could not be obtained, this expres- sion compares quite well with available data (Table 4.1). 6.6.6 Other Input Parameters The value of the bulk volume of potatoes used in this study is 1085 kg/m3. The initial storage temperature, the incoming ventilation air temperature and the incoming rela- tive humidity, as reported by Misener (1973) are, respec- tively, 15.50 C, 6.70 C, and 60%. The ventilation rate is 24 m3/hr/m2. The porosity of the Misener experimental sample was calculated to be 0.641. Porosity, in this study, is defined as the ratio of the pore volume to the total volume. 73 Table 4.1 The heat generation expression vs. experimental data, kJ/hr—kg. 2222' Exp. Data Eq. 5-73 % diff. 4 43.7 43.8 0.2 6 54.8 55.0 0.4 15 105.0 105.4 0.4 Source: Listove & Kalugina, 1964. (from Mohsenin, 1980) 5.0 COMPUTER SIMULATION 5.1 Introduction The determination of energy dissipation is accomplished three steps. The first involves the determination of qi, the skin resistance coefficients, by a trial and error pro- cedure based on a comparison between actual data and predicted values for heat and mass transfer in the potato storage environment. The second involves the use of regres- sion formulas, if necessary, for the coefficients. The last step uses the regression results together with the energy dissipation equation (equation 5-2) below to calculate the rate of entropy production and the dissipation of energy in, the system. 6.1 Relevant Equations The entropy production and energy dissipation terms are given by LI I o = 00 _ 2L KQ trz —TL‘ ‘ 2 I ' RT] TL P L' t’ J A KK rz h + TL ( P - RT)2 + CT (5_1) 75 and r = o T (5-2) The four relevant equations (with the Heavyside func- tion suppressed) that must be solved before solutions can be obtained for o and r are M = 1 - T122 /\3Cvp +W'pcvw [hdpfl‘a Tp) + q % + 3p) 6» + (RvTa + hfg + Cw”. - Tpn (TEN (5-3) for the potatoes, bTa 6T8 1 15? + u 15? ”,Kacva + Pacvv [hdp(Tp ' Ta) 5% - RvTa ( t)1 (5-4) for the ventilation air stream, 6. _ 61. PA. " TEE ' ‘6? + u “6.? (5'5) for continuity and _ :62 _ _ c1phot‘rz + dphD P t - g D g D qc T d D + 430— ("Pp - Ta) (5-6) for the mass transfer. 76 These equations can be simplified with the use of the following definitions: h dp A = (5-7) '\p va Apcvp q 4' \ ) B = YCLAP+ '5 (5'8) '9 vw ‘AP vp hf a = (5'9) \Cp vw *g/‘pcvp R + Cv b = V (5.13) “R vw +’Apvcvp c VV c = (5-11) Ppcvw +”Pcvp th c = (5-12) \a CVV ...BAacva RV d = \a CVV +/¥CV3 (5-13) T a D e = LI) (5-14) and d h d h f = .5431: ,1 - _Pl:_g_ (5-15, Note that in general, all of the above parameters may be functions of Ta' Tp, Pa and Pp and that the term f embo- 77 dies the skin resistance coefficient. The mass transfer equation can then be written as 5% - 1“? = f + e (Tp - Ta) (5-16) Substituting this into the two temperature derivative equations and the continuity equation and simplifying with the use of the above definitions, 6T t = (A - b f + a e) Ta - (A - c f + a e) Tp 2 + (c e - b e) Tan + b e Ta 2 - c e Tp + B - a f (5-17) bTa 6T3 2 I't_+u'5_x_=-deTa +(df-C)Ta + d T + _ e an C Tp (5 13) and 5A3 5A3 - f T Ed'U-K- +e(p-Ta) (5'19) Several boundary and initial conditions are required to solve equations (5-16) through (5-19). The necessary condi- tions are the initial potato and air temperature profiles in the bin, the initial air and potato moisture profiles in the bin, and the moisture content and temperature of the venti- lation air at the entrance to the bin. 78 Equations (5-16) through (5-19) are non—linear. Note that if the potato moisture loss is negligible, then by equations (5-17) and (5—20), f ---—> 0; e —---> 0 (5-20) which reduces equations (5-18) and (5-19) to 6T TtE = A (Ta - Tp) + B (5-21) and bTa bTa T? + u '67 = c (Tp - Ta), (5-22) both of which are linear equations. It immediately becomes apparent that the non-linearity of equations (5-18) and (5-19) is associated with the loss of moisture from the potato and its subsequent evaporation from the system. This result is as would be expected. Many of the parameters in the above equations are temperature-dependent but since the range of temperatures over which the system operates is rather small (277.0 < Ta < 290.0°K), it is not crucial to adhere to the temperature- dependency. As a result, these parameters are all evaluated at a temperature of 10.5°C and treated as constants for all temperatures between 4°C and 17°C (Table 5.1). Other param- eters used in the analysis and their numerical values are tabulated in Table 5.2. 79 Table 5.1 Temperature-dependent input parameters Parameter Values and units at 4°C at 17°C at 10.50C Max. error, % ka, J/m-sec-OC 0.024 0.025 0.025 0.04 pa, kg/m3 1.274 1.218 1.246 2.25 pa, kg/m-sec x 105 1.736 1.799 1.767 1.81 . va, m2/sec x 105 1.362 1.477 1.420 4.01 h hfg, J/kg x 10'6 2.49 2.46 2.47 0.81 DG, mz/sec x 105 1.97 2.15 2.06 4.37 80 Table 5.2 Parameters used in computer program. Parameter Value used Initial storage temperature 15.50C Heat capacities: Liquid water 4184 J/kg-OK Potato (at 0 % m.c.) 1430 J/kg-OK Water vapor 1880 J/kg-OK Air 1007 J/kg-OK Porosity 0.641 Bulk density of potatoes 1085 kg/m3 Material densities: Air 1.25 kg/m3 Potato 1800 kg/m3 Liquid water 1000 kg/m3 Initial potato moisture content 80% Outside air temperature 6.7°C Outside air relative humidity 60% Air velocity 37.4 m/hr Characteristic length 0.2 m Time step 0.00534 hr. 81 5.6 Computer Implementation The rate of entropy production and the system energy dissipation are given by equations (5-1) and (5-2), respec- tively. These equations are functions of the air and pro- duct temperatures, which in turn depend on the product mass losses and the degree of saturation of the convective air. As pointed out previously, the solution is a trial and error process, since the skin resistance coefficient, q, is not known a priori. The process begins with guessing a value for q to obtain a solution for the rate of mass loss. This result then becomes an input to equations (5—17) and (5-18), which then give the parameter f, along with values for the time rates of change of the convective air tempera- tures. These rates of change are then used to compute the temperature profiles via finite differences for the next time step. This process is repeated until the potato and air temperature profiles are known over the length of the bin for a period of time covering 24 hours. These profiles are then compared to the experimental results of Misener (1973). If the simulated results are in reasonable agree- ment with the experimental values, the value of the skin resistance coefficient is considered valid. If not, the process is repeated until convergence is obtained. Once convergence is achieved, equations (5-1) and (5-2) are solved to find the rate of entropy production and the dissipation of energy in the system, respectively. 82 The Fortran 5 computer programming language was used, and the solution was carried out on the MSU 750 Cyber com- puter. The ninety-two hour simulation required 25.1 seconds of Central Processing Unit (CPU) time. The output consisted of the air temperatures, product temperatures percentage mass loss of the product for each level of the storage bin. The cumulative entropy production by the products and the amount of energy dissipated were also calculated for the entire storage bin as a function of time. Printouts were initiated at intervals of four hours for the entire 92—hour simulation period. The simulations were based on the experimental data of Misener (1973). The Nordreco data of 1975 as reported by Lerew (1976) could have provided additional reliable material for testing the model. However, the geometric dimensions of the storage bin were not documented well enough to enable the simulation of the data by this model. To get around the problem of the lack of reliable data to test the heat and mass transport portion of the model, the first 24 hours of the Misener (1973) data was used to deter- mine qc, with the remaining 68 hours of data being used to test the model. The Misener experiment is described below. .6.6.1 Misener'g Experimental Procedure Misener (1973) created a vertical column by welding three 208 liter metal drums together. The column, 2.4 83 meters high and 0.7 meters in diameter, was wrapped with four layers of fiberglass insulation. A plenum was con- structed at the base. Iron-copper thermocouples were installed along the centerline of the column, and then relayed to a multipoint Honeywell recorder. Temperatures were taken on an hourly basis. The set-up is shown in Fig- ure 5.1. Air was fed through the base of the column by a centri- fugal fan, and the air flow rate was measured by a Flowtronic air velocity meter. Three hundred and sixty-four kilograms of freshly harvested Kennebec potatoes were loaded into the column and suberized at 15.50C for eleven days before beginning of the experimental procedure. The entire experiment took place in a constant temperature room. For the experiment, the potatoes were subjected to a constant volumetric air flow rate of 24 m3/hr-m2, with the temperature and relative humidity of the convective air held constant at 6.7°C, and 60%, respectively. To determine the moisture lost by the product, the potatoes were weighed before and after the cooling test. A second experiment was conducted to determine whether significant temperature gradients existed in the potatoes themselves. The results of this phase of the experiment led to the conclusion that there were negligible gradients. 84 "' A" 'A'A'A I | Four layers of 2" fiberglass batt insulation \\‘\Air outlet Honeywell multipoint recorder \ Potato surface Perforated floor I I '1 ' A'A‘ L'A“°A‘A'A'A'J'A' A‘A'A'A‘ L‘A ‘ s‘ A Centrifugal fan Figure 5.1 Misener's experimental set-up 85 6.3.1 Computer Solution Techniques The procedure used to solve the governing field equa- tions for the potato and air temperatures is similar to the thin layer approach often used in the solution of grain dry- ing problems. The 2.4 meter deep storage bin was divided into twelve "thin" layers, each 0.2 m deep. The ventilation air was brought in at the entry level at 6.7°C and 60% rela- tive humidity, and allowed to interact with the existing contents of the volume. After that, the mass lost by the product is computed, and the new temperatures for the air stream and the products in that layer are calculated. The body of ventilation air, now at a new temperature and new relative humidity, is then moved on to the next layer, where the procedure is repeated. Although the spacing, Ag, and the time step, ‘Afi, are independent,,¢¢ was chosen such that At = 41‘— (5-23) u where u is the interstitial velocity of the ventilation air. This was done so that whatever moisture is lost from the potatoes at location x and time t appears as a moisture gain for the air at the next assigned location x +‘AW after a time,¢¢. All of this makes the accounting much easier. 6.6.6' Computer Program Listing And Results A complete listing of the program is provided in Appen— dix B. Figures 5.2 and 5.3 show typical program outputs. DEPTH RAT! INT. PROD. 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Figure 5.2 Program output: 86 S I M U L A T I O N 5328. 5475. 5606. 5724. 5829. 5924. 6008. 6084. 6153. 6214. 6268. 6318. TIME: TOTAL BNT. 21588. 22131. 22613. 23039. 23417. 23751. 24046. 24306. 24534. 24735. 24910. 25063. energy dissipation (output R B S U L T S 4.0 HOURS PROD. INST. 1504404. 1549315. 1589686. 1625974. 1658593. 1687912. 1714267. 1737955. 1759247. 1778385. 1795586. 1811046. DISS. TOTAL DISS. 6101896. 6268855. 6417669. 6550139. 6667902. 6772440. 6865093. 6947071. 7019464. 7083255. 7139330. 7188482. entropy production and from Xerox 9700) 87 TIME : 4.0 HOURS DEPTH AIR TEMP POT. TEMP PCT. MASS L055 .2 9.2 15.1 .0330 .4 9.8 15.1 .0326 .6 10.4 15.2 .0322 .8 10.9 15.2 .0319 1.0 11.4 15.2 .0316 1.2 11.8 15.2 .0313 1.4 12.2 15.3 .0311 1.6 12.5 15.3 .0308 1.8 12.8 15.3 .0306 2.0 13.1 15.3 .0304 2.2 13.3 15.3 .0303 2.4 13.5 15.3 .0301 EDI I 12.78 Figure 5.3 Program output: heat and mass transfer (output from Xerox 9700) 6.0 RESULTS AND ANALYSIS 6.1 The Resistance Coefficient for Mass Transport From the results of the heat and mass transfer simula- tion, the skin resistance coefficient turns out to be a function of the reciprocal of the absolute temperature, ‘1: R T (6‘1) where qc is the combined coefficient introduced in equations (4-49) and (4-50), Rv is the gas constant for water vapor and T is the absolute temperature. From the experimental data of Misener (1973), the combined coefficient, qc was determined to be 447.0. The coefficient q is plotted in Figure 6.1. The result represented by equation (6-1) is a very important one for two reasons. First, the fact that qc is a constant means that q is a function only of the temperature. This is typical of many physical coefficients which occur in nature -- the heat transfer coefficient is a prime example. Second and just as important, it enables one to transfer this model to analysis of other agricultural products sub- jected to forced convection. To use the model, all one has 88 89 em\o.sv. Dom c .pcwfiowmmooo wocwuwfimon one H.o ohsmfih mo .THSDaquEoB 0mm owN - — Ohm ssetuotsuamtp ‘801 x u 90 to do is determine qc from reliable experimental data. Equation (6-1) represents the value of the coefficient that provides the best fit to Misener's experimental data. Tables 6.1 through 6.4 present the Misener (1973) experimen- tal air temperature data from which q was determined, along with the simulation results. The deviation of the simulated temperatures from the experimental values range from 0% to 0.4% In addition, as was explained in chapter 5, the first 24 hours of Misener's data was used to determine qc. The model was then tested using the remainder of the data and compared to those of Lerew (1978) and Misener (1973) at 48 hours and 92 hours. The comparisons, depicted in Figures 6.2 and 6.3, show that this model performs as well as the other two models. 6.1 Entropy Production The rate of entropy production is a smooth nearly linear profile (Figure 6.4) that reaches a steady state after 72 hours of ventilation. The rate of production ranges from 5.9 kJ/m3-oK-hr at the initiation of ventilation to a steady state value of 5.0 kJ/m3-OK-hr, with an average production rate of 5.4 kJ/mB-OK-hr for the 92-hour simula- tion. The results of the simulation are presented in Table 6.5. The total entropy produced in the system is 494 Table 6.1 air temperatures at 24 hours, Experimental 91 279.0 280.0 281.0 282.0 283.0 285.0 286.0 286.0 287.0 288.0 288.0 Simulated 279.0 280.0 281.0 282.0 283.0 284.0 285.0 285.0 285.0 286.0 286.0 287.0 °K Comparison of simulated & experimental Table 6.2 air temperatures at 48 hours, Experimental 92 279.0 280.0 280.0 280.0 280.0 281.0 281.0 283.0 284.0 285.0 286.0 287.0 Simulated 279.0 280.0 280.0 280.0 281.0 281.0 282.0 282.0 284.0 285.0 285.0 286.0 °K Comparison of simulated & experimental Table 6.3 air temperatures at 72 hours, Experimental 93 280.0 280.0 279.0 279.0 279.0 280.0 280.0 281.0 282.0 282.0 283.0 284.0 Simulated 280.0 280.0 280.0 279.0 279.0 280.0 281.0 282.0 282.0 283.0 283.0 283.0 °K Comparison of simulated & experimental Ln. Table 6.4 air temperatures at 92 hours, Experimental 94 280.0 280.0 279.0 279.0 279.0 279.0 280.0 280.0 280.0 281.0 282.0 283.0 Simulated 279.0 279.0 2790.0 280.0 280.0 280.0 280.0 281.0 281.0 282.0 283.0 283.0 °K Comparison of simulated & experimental 95 muco: wv um wamcoe uwmmampu mmaE can Hum: 009:» no acmfihameoo m.® musmfim E .cfin 2H :ofiudooq O.N O.H d - \\. soapafissfim Amsmfic A6660“: aofipaszsam Amsmflc amnmq aofipafiasam Avmmfic “Home «pap Hapgoefinoaxo AmhmHv Leann“: '0'.‘ l In 0 H ID H 30 ‘saxniazedmei 11v 96 mpson mm as mampoE hmwwcdup mwaE can paws mouse «0 comwamasoo m.® whamHm E .cfln cw cofipdooq o.m o.H cofipafissfiw Amsmfic 66:60.2 :oapafissam Amsmfic akoa cospafisefim Avwmfic Adamo «pap kuszHanxw Amhmfiv spawn“: I!) O H ID H ‘selniazadmaq 11V 30 97 :ofipospopa >Q0hpco yo ovum .An .6829 ow om ow v.6 mtsmam ON O u d I N {w Jq-yO-Em/fx ‘u01100p0Jd 10 3183 LG 98 Table 6.5 System entropy production 2122' 61 Production rate Cumulative production kJ/m3-OK-hr kJ/m3-OK 4 5.9 24.0 8 5.8 47.0 12 5.7 69.0 16 _ 5.6 93.0 20 5.5 115.0 24 5.5 137.0 28 5.5 160.0 32 5.4 183.0 36 5.4 205.0 40 5.4 227.0 44 5.3 248.0 48 5.2 269.0 52 5.1 290.0 56 5.1 311.0 60 5.1 331.0 64 5.1 352.0 68 5.1 372.0 72 5.0 393.0 76 5.0 413.0 80 5.0 433.0 84 5.0 453.0 88 5.0 474.0 92 5.0 493.0 99 kJ/m3-OK. To help put this result in perspective, standard ther- modynamic tables for moist air were used to calculate the average rate of entropy production associated with heating a given volume of air from 6.7°C (the temperature of the ven- tilation air on entry) to 15.0°C (the temperature of the exit air after 24 hours). The average rate of production was calculated to be 4.3 kJ/m3—oK-hr. In contrast, the average rate of entropy production over the first 24 hours of the simulation is 5.7 kJ/m3-OK-hr, which is 32% greater. This value is reasonable since the process involves a chemi- cal reaction not represented in the thermodynamic tables. The dissipation of energy in the system follows the same pattern as that of entropy production, as would be expected from equation (4-61). The rate of energy dissipa— tion, however, reaches steady state earlier than the rate of entropy production. The steady state occurs after 48 hours of ventilation, compared to 72 hours (Figure 6.5 and Table 6.6). 6.6 thimization 6y Minimizing Entropy Production The rate of entropy production provides a key parameter for system optimization. If a given system or process can be designed to operate in a zone of minimum entropy produc- tion, the system energy dissipation would be at a minimum, leading to higher efficiencies. The variables that can be 100 cofiuaafimmwu mwhmcm yo Team m.m waswfim .An .6829 ow om o8 on O d u - l-l N Jq-Ew/rx ‘UOInedISSIP :0 8128 Time, 61 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 101 Table 6.6 System dissipation of energy Dissipation rate kJ/m3-hr Cumulative dissipation kJ/m3 7.0 13.0 20.0 26.0 33.0 39.0 45.0 52.0 58.0 64.0 70.0 76.0 82.0 88.0 94.0 100.0 105.0 111.0 117.0 123.0 128.0 134.0 139.0 102 manipulated to provide the minimum entropy production would vary from system to system. The rate of entropy production for the system under analysis in this thesis is given by equation (5-1) and can be represented as 2 2 4 o = 9.- DAB p fp uS + 2 DAB R P +. hD p fp us L 2 DL 4 3 D L 4 T 36 g D L 4 T h R f u2 h R2T J A D P p s D P ch ' 2*7'5T+ T (6'2) 3gDL4 It is apparent from equation (6-2) that o is a function of both the velocity and the temperature. The dependence on temperature is, however, not as pronounced as the dependence on velocity. Under normal modes of cooling, most agricul- tural storages operate in a limited range of temperatures. In addition, equation (6-2) contains several terms involving powers of the velocity. Given these circumstances, it would appear that the minimization of the rate of entropy produc- tion can be done most effectively through control of the ventilation rates. One might be tempted to approach the optimization prob- lem by taking the partial derivative of equation (6-2) with respect to the velocity and, using the standard practice of calculus, determine the velocity at which the derivative is zero. While this approach is mathematically appealing, it would be physically meaningless because it would imply that the temperatures in the storage volume remain constant over 103 a range of ventilation rates, a situation that does not occur in practice. The global entropy production is Ag = ]Q j; O'dV dt (6-3) and the total dissipation of energy is g = At ft [x r dx dt (6-4) where‘éfi is the total change in entropy of the system, Q is the dissipation function and At is the total cross-sectional area of the system, a constant. Since the expression for o in equation (6-2) multiplied by the appropriate absolute temperature can be substituted in equa- tion (6-4) and the equation can be integrated over the length of the bin and over a specified length of time to obtain the total system energy dissipation. This approach enables one to define a dimensionless energy dissipation index (EDI) such that EDI = p A 9n. t (6-6) ac 1 where p is the internal pressure, Aac is the cross-sectional area available for air flow, 111 is the interstitial velocity and t is the elapsed time. 104 From a force balance, the internal pressure is related to the known external atmospheric pressure, pa, by p =‘__ (6-7) where 4 is the porosity of the porous medium. Equation (6-6) thus measures the energy dissipation through entropy production against the external energy input required to move the ventilation air. This approach normal- izes the result for all ventilation rates and thus allows direct comparisons to be made. The solution of equation (6-6) for various ventilation rates and for four different inlet temperature conditions is plotted in Figure 6.6. The four conditions used are inlet ventilation air temperatures of 3°C, 6.7°C (the temperature of the ventilation air for Misener‘s data), 10°C and a sinusoidal inlet temperature that satisfies the equation _ . n t _ e — 6.0 + 6.0 SID-73? (6 8) where t is the elapsed time in hours. Equation (6-8) represents an inlet temperature that oscillates between 0°C and 12°C, with a 24-hour period. The optimum ventilation rate for the inlet condition under which the data was collected is 22 m3/m2-hr. Given the size of the experimental storage volume and the high porosity, this rate is quite reasonable. The zone of optimum ventilation rates ranges from about 20 to 25 m3/m2- hr. The optimum ventilation rates for the 3°C, 10°C and 105 .msofiuficcoo Afiov musuwumaeop «Tana pcmammmfip a20w as wouau :ofipaaaa:m> msofiua> how Awamv xocafi cofiuwafimmfip mwuozm 0.0 Tasman unnne\ma .ouau :Owuaafipnm> on 04 on om 0H 0 u q d . q n TE mm cam m + m u flo H: 400 068.6 u a 106 sinusoidal conditions are 17, 28 and 22 m3/m2-hr, respec- tively. The fact that the EDI curve for the sinusoidal con- dition is essentially the same as the one for the 6.7°C entry condition is to be expected, since the mean tempera- ture for the sinusoidal condition is 6°C. These results suggest strongly that, from at least a heat transfer point of view, the EDI can be effectively used to optimize the ventilation rate. The optimum ventilation rates predicted by the EDI are very logical: a lower volumetric flow rate of air is required for cooling the potatoes when outside air is available at 3°C than when it is available at 10°C. The shapes of the curves also suggest that the need for optimization is more critical at high ven- tilation rates than at lower rates. The steepness of the EDI curves indicate that at high ventilation rates, the effect of the ventilation rate on the EDI coefficient is more pronounced. It must be pointed out, however, that these results are viewed only from the point of view of energy dissipation. Another important factor that must be considered is the effect of the ventilation rates on product quality. The quality factor was not considered in this thesis because of the inherent difficulties in characterizing "quality" mathematically. The information represented by Figure 6.6 would seem to indicate that the EDI does account in some measure for the cooling rates of the potatoes, although it may not totally account for the rate of mass loss. 107 6.6 The Effect 66 Thermal Diffusion The last term of the equation for the rate of mass transfer (equation 5-6) includes the Soret coefficient. The Soret coefficient is a measure of the importance of the phenomenon of thermal diffusion. The second term on the right hand side of the equation for the rate of local entropy production (equation 5-1) incorporates the cross- phenomenological coefficient which is a function of the Soret coefficient (equation 4-52). It is of interest to assess the effect of the thermal diffusion phenomenon on the analysis done in this study. Simulation results indicate that setting the Soret coefficient to zero has a negligible effect on both the pro- duct mass loss and the temperature profile in the storage environment. It does reduce the rate of entropy production by 9%, thus reducing the magnitude of the EDI. However, the location of the minimum value of the EDI with respect to the ventilation rate does not change. Thus the essential results of this study are unaffected by the Soret coeffi- cient. Since the EDI is designed to be a measure of the rela- tive (rather than the absolute) rate of entropy production, these results would suggest that the effect of thermal dif- fusion is negligible for the type of system analyzed in this thesis. 7.0 CONCLUSIONS A model for entropy production and energy dissipation in the storage environment of agricultural products has been derived using the principles of the thermodynamics of irreversible processes. The model includes a scheme for optimizing the rate of ventilation, based on the principle of minimum entropy production. To compute the entropy production within the storage environment, equations for heat transfer, mass transfer and chemical reaction (respiration) were derived and solved. The results of the heat and mass transfer part of the model compare favorably with those obtained by Misener (1973) and Lerew (1978). It has been shown that non-equilibrium thermodynamics can be used for the analysis of heat and mass transport in the storage environment of agricultural products. The heat and mass transport part of the problem requires a single temperature-dependent coefficient -- the skin resistance coefficient. This method of analysis is applicable to other forced-air ventilation systems if reliable data is available to determine the skin resistance coefficient. 108 109 The optimization scheme is based on a single parameter -- the energy dissipation index (EDI). The EDI is defined as the ratio of the total energy dissipated by the system to the input energy required to move the ventilation air. Under this definition, minimum entropy production and energy dissipation occur at the minimum value of the EDI. Simula- tion results indicate that this parameter may be a useful tool for system optimization. Setting the Soret coefficient (DT) to zero (thus neglecting the contribution of thermal diffusion to mass transfer) has little effect on the mass loss and the tem- perature profile in the storage environment, but it reduces the rate of entropy production by 9%. Although the lower rate of entropy production reduces the magnitude of the EDI, the position of the minimum value of the EDI with respect to the ventilation rate does not change. Thus the phenomenon of thermal diffusion can be neglected without changing the essential results of this study. 8.0 SUGGESTIONS FOR FUTURE RESEARCH Possible applications of irreversible thermodynamics in agricultural engineering are many and varied. This thesis has explored only one such application. Further applica- tions lie in the areas of food processing and in the con- trolled environmental storage of agricultural products. It is suggested that the methodology also be used in analyzing physiological responses in animals to various stimuli through a study of muscle movement as has been done with human muscle in the fields of bio-engineering. The area of biotechnology is becoming important for agricultural engineering. As Dr. Norman Scott of Cornell University remarked at the 1984 American Society of Agricul- tural Engineers (ASAE) summer meeting in Knoxville, Tennes- see, the role of agricultural enginners in this relatively new field would not be genetic improvements, but rather the modeling of biotechnological systems. Since the area is fairly new and all the required kinetic equations have not yet been developed, the field appears to be ripe for model- ing and analysis by irreversible thermodynamics. It is an area of endeavor that would be worth pursuing. 110 APPENDIX 6 APPENDIX A EXPRESSION FOR THE MASS DIFFUSIVITY In a binary system, the mass diffusivity of component A in relation to component B, DAB' has the relationship DAB = DBA = DG (A-l) where DG is the diffusivity of the gas. DG can be calculated for the diffusion of water vapor into moist air by using the kinetic theory of gases (Perry & Chilton, 1973). Wilke and Lee‘s 1955 modification of the equation by Hirschfelder, Bird and Spotz (derived in 1949) is used in this study. This modification, which gives a fairly accurate estimate of the diffusion coefficient, is B 93/2 _1.. + i \ M1 M2 DG = .2 (A-z) p r 1 12 n where B = (10.7 - 2.46 ML + Mi x 10'4) (A—3) \ 1 2 112 and the M1 are the molecular weights of the two gas com- ponents involved. P represents the absolute pressure in atmospheres, and r is the collision diameter in angstroms, 12 computed as = (A-4) where the ri are the collision diameters of the respective gases. From Perry and Chilton (1973), r = 1.18 V 1/3 0 b (A-S) where Vb is the molar volume of the liquid at its normal boiling point. For water vapor, Vb = 18.9 cc/g-mole, and for air, V = 29.9 cc/g-mole (Perry & Chilton, 1973). b I in eq. (A-2) is the collision integral for diffu- DI sion, and is a function of ke/412, where 412, the energy of molecular interactions, is given by (Perry & Chilton, 1973) ‘12 = \ <1 ‘2 ' (A's) 6 and k is the Boltzman constant, 1.38 x 10- erg/K. Using tabulated values from Perry and Chilton (1973), the following temperature-dependent relationship was derived for ID, the collision integral, = T-0.09007 4 I - 6.3 x 10- D 'T + 0.1764 (A-7) 113 Using eq. (A-7) to calculate values of the collision integral in the temperature range 0°C 6 t 6 40°C results in an absolute error of less than 0.4%. The use of eq. (A-Z) provides accuracies within 7—8% for most systems (Perry & Chilton, 1973). APPENDIX E APPENDIX B PROGRAM OFOLI (INPUT,OUTPUT,TAPE4-INPUT,TAPE6-OUTPUT) C DEFINITION OF VARIABLES A AC AK ALP AM AS CAPA CAPP CL CMP CUMDIS CVA CVP RHOA RV C C C C C C C C C C C C C C C C C C C C C C C C C C C (I ETA C C C C C C C C C C C C C C C C C C C C C C C C C C C S CHEMICAL AFFINITY TOTAL CROSS-SECTIONAL AREA AIR CONDUCTIVITY POTATO SURFACE AREA PER LINEAR METER AIR MASS AT ENTRY POINT SURFACE AREA OF POTATOES IN ELEMENT MASS OF AIR IN BIN PER LINEAR METER MASS OF POTATO PER LINEAR METER CHARACTERISTIC DIMENSION CURRENT MASS OF HATER IN POTATO CUMULATIVE ENERGY DISSIPATION HEAT CAPACITY OF AIR HEAT CAPACITY OF POTATO . HEAT CAPACITY OF WATER VAPOR HEAT CAPACITY OF LIQUID HATER DIAMETER OF BIN MASS DIFFUSIVITY TEMPERATURE DIFFERENCE BIN DEPTH INSTANTANEOUS ENERGY DISSIPATION DIAMETER OF SPHERICAL POTATO SORET COEFFICIENT TIME STEP SPATIAL DERIVATIVE OF AIR TEMPERATURE POROSITY OFOLI COEFFICIENT FRICTION FACTOR PERCENTAGE MASS LOSS GRAVITATIONAL ACCELERATION HEAT TRANSFER COEFFICIENT MASS TRANSFER COEFFICIENT HEAT OF VAPORIZATION HUMIDITY RATIO REACTION VELOCITY MASS TRANSFER PHENOMENOLOGICAL COEFF. CROSS-KINETIC PHENOMENOLOGICAL COEFF. HEAT TRANSFER PHENOMENOLOGICAL COEFF. MOISTURE CONTENT OF POTATO TOTAL MASS LOSS NUMBER OF THIN LAYERS MOLES OF AIR ON ENTRY ATMOSPHERIC PRESSURE RELATIVE HUMIDITY POTATO DENSITY SATURATION PRESSURE INITIAL TOTAL MASS OF POTATOES VAPOR PRESSURE RATE OF HEAT GENERATION UNIVERSAL GAS CONSTANT REYNOLDS NUMBER AIR DENSITY GAS CONSTANT FOR WATER VAPOR RATE OF TOTAL MASS L055 114 nnnnnnnnnnnnnnnnnnnnnnnnnnnnn SAPA SDIFF 115 MASS OF WATER IN AIR PER LINEAR METER MASS OF WATER IN POTATO PER LINEAR METER RATE OF CONVECTIVE MASS LOSS SOLID MASS OF POTATO TOTAL SURFACE AREA RATE OF MASS LOSS BY THERMAL DIFFUSION SUPERFICIAL VELOCITY TIME AIR TEMPERATURE POTATO TEMPERATURE SHEAR STRESS TOTAL DISSIPATION OF ENERGY TOTAL ENTROPY PRODUCTION ABSOLUTE AIR TEMPERATURE ABSOLUTE POTATO TEMPERATURE CUMULATIVE MASS LOSS INITIAL POTATO TEMPERATURE VOLUME OF AIR VENTILATION AIR TEMPERATURE AIR FLOW VELOCITY VENTILATION RATE VOLUME OF ELEMENT VOLUME OF STORAGE ENVIRONMENT VOLUME OF POTATO ' WATER MASS IN AIR WATER MASS IN POTATO TIME RATE OF CHANGE OF POTATO TEMPERATURE RATE OF ENTROPY PRODUCTION TIME RATE OF CHANGE OF AIR TEMPERATURE DIMENSION DELT (12), DISS (12), TDISS (12). s (12), TA (12) DIMENSION TP (12), TML (12), XI (12), TENTP (12), FR (12) REAL MC, NMA, LKK, LQK, LQQ, MT (12), JCH CALCULATE OR INPUT ALL PROGRAM PARAMETERS N I 12 VAT I 6.7 PHI I 0.6 MC I 0.8 PI I 3.1416 CL I 0.2 DT I 0.00534 TPI I 15.5 R I 8.314 EP I 0.641 RHOA I 1.246 AK I 90.0 D I 0.7 OS I 0.063 G I 9.81 DAB I 0.0742 AC I PI * D * D / 4.0 VE I 0.077 VA I 0.049 VP I 0.028 A5 I 0.0987 ALP I AS / CL ST I 1.184 CVA I 1007.0 CVV I 1880.0 CVW I 4184.0 CVP I 1430.0 GOOD 0' 1516 HFG I 2470000.0 PD I 394.0 P I 101353.0 JCH I 0.025 A I 26600000.0 INITIAL TOTAL MASS OF POTATOES PTM I PD * VE INITIAL MASS OF WATER IN POTATO WMP I PTM * MC SOLID MASS OF POTATOES SMP I PTM * (1.0 ' MC) INITIAL MASS OF WATER IN THE AIR NMA I P * VA / (R * (VAT + 273.15)) PSAT I EXP (60.08 - 6814.03 / (VAT I 273.15) ' +5.12 * LOG (VAT + 273.15)) HR I 0.622 * PHI ' PSAT / (P - PHI * PSAT) AM I 29.0 * NMA / 1000. WMA I 29.0 * NMA * HR / 1000. VENT I 24.0 BEGIN VELOCITY LOOP SVEL - VENT VEL - VENT / EP WRITE (6 5) VENT FORMAT ('1'. //////////////. T49, 'SIMULATION AT ', P4.1, + 1x, 'NII3/NII2- HR') CALCULATE PROGRAM VARIABLES RE - 13.755 I SVEL / (1 - EP) PP - 150.0/RE + 1.75 TAU - (PP I SVEL I SVEL I RR0A)/(6.0IEPIEP) H - 0.165 I (RE II0.6) I AK / 05 HD - 0.000896 I H INITIALIZE TEMPERATURES no 15 I - 1, N no 15 J -1,N TA (1) - 15.5 TP (1) - TPI START TIME LOOP T - 0.0 CALCULATE PROGRAM PARAMETERS CAPP - SMP / CL SAPP - WMP / CL CAPA - AM / CL SAPA - WMA / CL DIVl - SAPP I cvw + CAPP I CVP DIVZ - SAPA I CVV + CAPA I CVA CA - H I ALP / DIVl SA HFG / DIVl 53 (RV + CVV) / DIVl SC CVV / DIVl CC H I ALP / nzvz 50 RV / DIV2 501 - ALP I HD I RHOA / (G I 0) SF2 - ALP I HD I TAU / (G I 0) VOL - 0.924 ETA - 447.0 no 100 J - 1, 750 1137 T I T I DT PAC I 0.385 ' P * VEL ‘ T L I L I 1 CALCULATE DRIVING FORCES CONCENTRATION x - N IF (J .LE. N) M - J IF (L .EQ. 750) THEN CALCULATE THE ENERGY DISSIPATION INnEx CUMDIS - 0.0 no 1 NT - 1, N CUMDIS - CUMDIS + TDISS (NT) 1 CONTINUE EnI - CUMDIS I VOL / PAC WRITE (6.25) T 25 FORMAT ('1'. /////. T52, 'S I M U L A T I O N R E s U L T s +',/////, T62, 'TIME: ', F4.l, ' HOURS', /) END IF DEPTH - 0.0‘ no 30 I - 1, M DEPTH - DEPTH + 0.2 Q - 5.6 I T? (I) + 21.4 CMP - WMP - TML (I) 000 n C CALCULATE MASS LOSS THETA I (TA (I) I 273.15) THETAP I (T? (I) I 273.15) DST I DAB * RHOA / THETAP LKK I HD * RHOA / (3600. * 3600. * G ' D) LQK I DST * THETAP / D LQQ I H * TA (I) DELT (I) I TP (1) - TA (I) SF I SFl * ETA - SF2 SE I ALP * DST / D SDIFF I SF * CL STHERM I SE * CL * DELT (I) S (I) I SDIFF I STHERM XI (1) I LQQ/(CL * THETA) - (2.0*LQK / (CL ' THETA)) I * (TAU / (3600. * 3600. * RHOA) - RV * THETA) I I (LKK / (CL * THETA)) * ((TAU / (3600. * 3600. * RHOA) I - RV * THETA) ** 2) I JCH ‘ A / THETA TENTP (I) I TENTP (I) I II (I) ' DT DISS (I) I XI (I) * (TA (I) I 273.15) TDISS (I) I TDISS (I) I DISS (I) * DT TML (I) I TML (I) I S (I) * DT CALCULATE TEMPERATURE GRADIENTS AND NEW TEMPERATURES AIR TEMPERATURES IF (I .EQ. 1) THEN DTDE I 0.7 / CL IF (J .GT. 4496) DTDX I 0.4 / CL ELSE DTDX I (TA (I) - TA (I-l)) / CL END IF CB I Q * (CAPP I SAPP) / DIV1 C AIR TEMPERATURES Y I “SE * SD * TA (I) * TA (I) I (SD 5 SF - CC) ’ TA (I) I I SE ' SD * TA (I) ‘ T? (I) I CC * TP (I) I - VEL * DTDX C POTATO TEMPERATURES E I (CA - SB * SF I SA * SE) * TA (I) an none» 37 38 39 40 50 60 70 100 ILLS (CA - SC * SF I SA * SE) ' TP (I) (SE * SC - SE * SB) * TA (I) ‘ TP (I) SE * SB * TA (I) * TA (1) - SE * SC * TP (I) * TP (I) I CB - SA * SF TA (I) I TA (1) I Y * DT T? (I) I TP (I) I X * DT CONTINUE WRITE RESULTS PRINT VALUES EVERY FOUR HOURS 4444 44' I? (L .EQ. 750) THEN L - 0 WRITE (6. 37) FORMAT (/, T30, 'DEPTH', T41, 'RATE ENT. PROD.', T62, + 'TOTAL ENT. PROD.', T84, 'INST. DISS.', T101, + 'TOTAL DIss.'. //) DEPTH - 0.0 no 39 K - 1, M DEPTH - DEPTH + 0.2 WRITE (6,38) DEPTH, x: (K), TENTP (x), DISS (x), TDISS (K) FORMAT (//, T31, F3.l, T44, P7.0, T66, F7.0, T84, P10.0, + T99, P12.0) CONTINUE WRITE (6.40) T PORMAT ('1', /////, T44, 'TIME : ', F4.l, 2:, 'HOURS') WRITE (6.50) ‘ FORMAT (//, T27, 'DEPTH', 5:, 'AIR TEMP', 53, 'POT. TEMP', + 5x, 'PCT. MASS LOSS',/) DEPTH - 0.0 no 70 x - 1, M DEPTH - DEPTH + 0.2 FR (K) - TML (K) I 100.0 / PTM WRITE (6. 60) DEPTH, TA (x), TP (N), FR (K) FORMAT (//, T28, F3.1, T39, P4.1, T53, F4.l, T68, F6.4) CONTINUE WRITE (6,2) EDI FORMAT (//////. T47, 'EDI -', F6.2) END IP CONTINUE STOP END B IBL I OGRAPHY BIBLIOGRAPHY Bear, J. (1972). Dynamics of Fluids in Porous Media. Amer- ican Elsevier Co., N.Y., N.Y. Bejan, A. (1982). Entropy generation through heat and fluid flow. John Wiley, N.Y., N.Y. Balmer, R.T. (undated). Entropy and aging in biological systems. Univ. of Wisconsin, Mech. Engr. Dept., Milwaukee, Wis. Balmer, R.T. (undated). An entropy model for biological systems. Univ. of Wisconsin, Mech. Engr. Dept., Milwaukee, Wis. Benedict, R.P. (1983). Fundamentals of gas dynamics. John Wiley and Sons, N.Y., N.Y. Bennett, C.O., & J.E. Myers (1974). Momentum, heat, and mass transfer. McGraw-Hill, N.Y., N.Y. Bird, R.B., W.E. Stewart, E.N. Lightfoot (1960). Transport Phenomena. 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