An Analysis of the Irreversible Thermodynaics Model for Coupled Heat and Moisture Transport Phenomena in Unsaturated Porous Media

Transcription

1 University of Arkansas, Fayetteville Technical Reports Arkansas Water Resources Center An Analysis of the Irreversible Thermodynaics Model for Coupled Heat and Moisture Transport Phenomena in Unsaturated Porous Media J. A. Havens University of Arkansas, Fayetteville Follow this and additional works at: Part of the Fresh Water Studies Commons, and the Water Resource Management Commons Recommended Citation Havens, J. A An Analysis of the Irreversible Thermodynaics Model for Coupled Heat and Moisture Transport Phenomena in Unsaturated Porous Media. Arkansas Water Resources Center, Fayetteville, AR. PUB This Technical Report is brought to you for free and open access by the Arkansas Water Resources Center at It has been accepted for inclusion in Technical Reports by an authorized administrator of For more information, please contact

2 A N A N A LY S IS O F THE IRREVERSIBLE T H E R M O D Y N A M IC S M O D E L FOR COUPLED HEAT A N D M O ISTU RE TR ANSPO R T P H E N O M E N A IN U N S ATU R ATED POROUS M E D IA by J.A. Havens Department of Chemical Engineering AW RRC Arkansas W ater Resources Research Center P ublication No. 67 In Cooperation W ith The ENGIN EERIN G EXPERIM ENT STATIO N P ublication No. 29 UNIVERSITY OF ARKANSAS Fayetteville 1980

3 PROJECT COMPLETION REPORT PROJECT NO. A-042-ARK AGREEMENT NO.: S ta rtin g Date: June, 1976 Ending Date: September, 1979 AN ANALYSIS OF THE IRREVERSIBLE THERMODYNAMICS MODEL FOR COUPLED HEAT AND MOISTURE TRANSPORT PHENOMENA IN UNSATURATED POROUS MEDIA by J. A. Havens Department o f Chemical Engineering Arkansas Mater Resources Research Center UNIVERSITY OF ARKANSAS in cooperation w ith Engineering Experiment S ta tio n UNIVERSITY OF ARKANSAS F a y e tte v ille, Arkansas January, 1980 (Contents o f th is p u b lic a tio n do not n e ce ssa rily r e fle c t the views and p o lic ie s o f the O ffic e o f Mater Research and Technology, U.S. Dept, o f the In te r io r, nor does mention o f trade names o r commercial products c o n s titu te th e ir endorsement or recommendation fo r use by the U.S. Government.) i

4 ABSTRACT The Irre v e r s ib le Thermodynamics-based model fo r the d e s c rip tio n o f coupled heat and m oisture tra n s fe r, a ttrib u te d to Cary and T a y lo r, was analyzed. The tra n s p o rt c o e ffic ie n ts appearing in the model equatio n s were independently determ ined, and the equations were num erica lly in te g ra te d to p re d ic t tem perature and m oisture content p r o file s fo r a closed system o f w a te r u n satu rate d glass beads. An experim ental in v e s tig a tio n o f the m oist glass beads medium provided measurements o f s te a d y-sta te p r o file s o f lo c a l temperatures and m oisture conte n t. These data, when compared w ith model p re d ic tio n s, in d ica te d the v a lid it y o f the Irre v e rs ib le Thermodynamics approach. The coupling c o e ffic ie n t re la tin g thermal gradie n ts to m oisture flu x was found to be s tro n g ly m oisture-dependent. The coupling c o e ffic ie n t which re la te s m oisture content g ra d ie n t to heat flu x was found to be extrem ely sm a ll, and the heat flu x associated w ith the m oisture content g ra d ie n t proved to be n e g lig ib le. i i

5 ACKNOWLEDGEMENTS The work upon which th is publication is based was supported in part by funds provided by the O ffice o f Water Research and Technology, U. S. Department o f the In te rio r, through the Water Resources Research Center at the U niversity o f Arkansas under Project A-042-ARK as authorized by the Water Research and Development Act of The experimental data and numerical analyses were presented as a doctoral dissertation in Chemical Engineering by Rajesh J. Parikh, presently Assistant Professor o f Chemical Engineering at the University o f Nebraska, Lincoln. J. Havens iii

6 TABLE OF CONTENTS Page ABSTRACT... ACKNOWLEDGEMENTS... LIST OF TABLES... LIST OF FIG U R E S... i i i i i v vi Chapter 1. INTRODUCTION HISTORICAL AND LITERATURE REVIEW THERMODYNAMICS OF IRREVERSIBLE PROCESSES DEVELOPMENT OF COUPLED HEAT AND MOISTURE TRANSFER MODEL DISCUSSION OF TRANSPORT EQUATIONS AND SOLUTION TECHNIQUE MEASUREMENT OF TRANSPORT COEFFICIENTS DESCRIPTION OF EXPERIMENT RESULTS AND DISCUSSION CONCLUSIONS AND RECOMMENDATIONS NOMENCLATURE LITERATURE CITED APPENDIX i v

7 LIST OF TABLES Table Page 1. Location s o f Thermocouples on the S urfaces o f the S p h e re s Coordinates o f Medium Thermocouples Coordinates o f M oisture Content Sampling Guides I n i t i a l M oisture Contents and Heating Rates fo r Experimental R u n s v

8 LIST OF FIGURES Figure Page 1. S pherical Porous Medium System Schematic o f Experim ental Apparatus f o r Thermal C o n d u c tiv ity D eterm ination Typical Heating Curves fo r Glass Beads Thermal C o n d u c tiv ity λ and Thermal D if f u s iv it y α vs. V olum etric M oisture Content θ H yd ra u lic C o n d u c tiv ity K vs. V olum etric M oisture Content θ Water P o te n tia l ψ vs. V olum etric M oisture Content θ D if f u s iv it y D vs. V olum etric M oisture Content Coupling C o e ffic ie n t β vs. V olum etric M oisture Content θ Coupling C o e ffic ie n t L vs. V olum etric M oisture Content θ Coupling C o e ffic ie n t β vs. V olum etric M oisture Content θ Schematic o f A p p a ra tu s Top and Bottom Access P l u g s Sampling Guide, Plug, and Bushing Cross-Section o f Inner Sphere Outer Sphere Support Assembly Thermocouple Support Schematic o f Thermocouple C ir c u it M oisture Content Sampling Device Block Diagram o f Temperature C ontrol System Local Tvs. Radial P osition--r un vi

9 Figure Page 21. Local 6 vs. Radial P osition-r un Local T vs. Radial Position Run Local θ vs. Radial Position--Run Local T vs. Radial Position--Run Local θ vs. Radial Position--Run Local T vs. Radial Position--Run Local θ vs. Radial Position--Run Local T vs. Radial Position--Run Local θ vs. Radial Position--Run Local T vs. Radial P osition-r un Local θ vs. Radial Position--Run Local T vs. Radial Position--Run Local θ vs. Radial Position--Run Local T vs. Radial Position--Run Local θ vs. Radial Position Run Local T vs. Radial Position--Run Local θ vs. Radial Position--Run Local T vs. Radial Position--Run Local 6 vs. Radial Position Run Local Tvs. Radial P osition Run Local θ vs. Radial Position Run Coupling C oefficient β* vs. Volumetric Moisture Content θ, fo r Runs 2, 4, 6, and v ii

10 CHAPTER 1 INTRODUCTION Simultaneous tra n s p o rt o f heat and mass in porous media occurs w id e ly in natu ra l and in d u s tria l processes. An understanding o f the associated phenomenology is a p p lic a b le, fo r example, to in d u s tria l processes such as recovery o f petroleum from underground sources, process dryin g o f g ra n u la r m a te ria ls, heat and mass tra n s fe r in packed beds, and in heat pipe technology. Simultaneous heat and m oisture tra n s p o rt in s o il occur as a r e s u lt o f d iu rn a l and seasonal tem perature flu c tu a tio n s. Engineered systems in v o lv in g s o il heating w ith power p la n t condenser e fflu e n t fo r enhancing a g ric u ltu ra l y ie ld s, and i r r i g a tio n, p a r tic u la r ly w ith warm w a te r, portend a need fo r p re d ic tiv e th e o rie s fo r such processes. Temperature g ra d ie n ts can induce w ater movement in porous media. For example, i f a sealed column f i l l e d w ith an unsaturated porous medium o f uniform m oisture content is subjected to a tem perature g ra d ie n t, m oisture movement occurs, and u ltim a te ly a ste a d y -s ta te m oisture content p r o file is e sta b lis h e d in the column. Such a s tru c tu re would not be a n tic ip a te d from c la s s ic a l phenomenological re la tio n s h ip s such as Darcy's Law. Thus, i t is "observed" th a t m oisture tra n s fe r in unsaturated porous media occurs in a s s o c ia tio n w ith tem perature gradie n ts in a d d itio n to m oisture content g ra d ie n ts. I t is commonly asserted th a t the associated heat and mass tra n s p o rt processes are "c o u p le d ". The Irre v e r s ib le Thermodynamics (IT ) theory which has been used to model coupled heat and m o is tu re tr a n s fe r processes was examined in 1

11 2 the present study. The coupled heat and moisture transfer equations attributed to Cary and Taylor [91] were numerically integrated to obtain temperature and moisture content profiles for a sealed, one dimensional spherical system o f water-unsaturated glass beads. A onedimensional (sph erical) water unsaturated glass bead medium was subjected to known heat flow rates, and the steady-state temperature and moisture content p ro file s were measured and compared with the Cary and Taylor model predictions.

12 CHAPTER 2 HISTORICAL AND LITERATURE REVIEW The study o f m oisture tra n s fe r in porous media was in itia t e d by Darcy [2 7 ] who observed th a t the flu x o f water through saturated s o il is p ro p o rtio n a l to the imposed pressure g ra d ie n t, an em pirical re la tio n now re fe rre d to as D arcy's Law. Buckingham [1 3 ] described the flo w in an unsaturated porous medium o ccu rrin g as a re s u lt o f below- atmospheric water pressure and termed i t c a p illa r y flo w. Darcy's Law was la te r "extended" to include m oisture flo w in unsaturated porous media as w e ll, and the c o e ffic ie n t o f d if f u s iv it y o f water in unsaturated media was defined [2 4 ]. Procedures fo r numerical treatm ent o f the unsaturated flo w Darcy equation w ith d iffe r e n t kinds o f boundary conditio n s have been proposed, and the lite r a tu r e is re p le te w ith examples o f numerical s o lu tio n s fo r various boundary c o n d itio n s. However, the work described above was re s tric te d to consideratio n o f m oisture tra n s fe r under isotherm al c o n d itio n s. I t was subsequently observed th a t tem perature g ra d ie n ts can cause s ig n ific a n t movement o f m oisture in s o il [10, 1 1 ], and considerable research e f f o r t has been focused on the "energy s ta tu s " o f m oisture in unsaturated porous media. Several authors have applied the theory o f E q u ilib riu m Thermodynamics to derive p ro p e rty re la tio n s among macroscopic v a ria b le s in w a te r-co n ta in in g porous media [7, 32, 36, 108]. S im ultaneously, e ffo r ts were made to e s ta b lis h tra n s p o rt equations fo r the tra n s fe r o f heat in porous media and to determine the thermal p ro p e rtie s o f porous media. A complete th e o re tic a l 3

13 4 description is d if f ic u lt, since a moist porous medium is a three-phase system (so lid phase, liq u id water, and a gas phase), and transfer of heat is invariably "coupled" to moisture movement. However, the special case of "pure" (Fourier) heat tra n sfe r has been considered, and the transport equations thus obtained have been solved fo r boundary conditions of practical in te re st [58, 59, 60, 95, 106]. Furthermore, experimental determination o f the "e ffe c tiv e " thermal conductivity of the porous medium, fo r use in these heat conduction equations, has been reported [46, 69, 70, 82]. In order to obtain a clearer understanding of the transport processes in a porous, three-phase system, i t is necessary to formulate equations fo r describing the simultaneous or coupled transport of heat and mass species present. Research in th is area has evolved along two paths: the classical Soil Physics approach and the phenomenological formalism o f the Thermodynamics of Irre ve rsib le Processes. The Soil Physics approach pioneered by P h ilip and devries [78, 79, 97, 98, 99, 100] suggests modified forms of Pick's Law and Darcy's Law fo r vapor and liq u id tra nsfe r of moisture in porous media. P h ilip and devries developed equations fo r describing the coupled heat and moisture tra n sfe r phenomena from the theories o f heat and mass tra nsfe r in so ils and from a knowledge of the soil structure. This model, however, does not provide guidance fo r determining the transport co e fficie n ts appearing in the model equations. Numerous reports of application o f the P h ilip and devries model have been made. These include extensions of the model to d iffe re n t types of s o ils, attempts to estimate the transport c o e ffic ie n ts, and solutions of the model equations fo r various boundary conditions [2, 23, 31, 33, 35, 40, 44,

14 5 45, 47, 51, 53, 55, 56, 61, 62, 64, 83, 84, 85, 88, 103, 105, 110]. However, these in ve stig a tio n s have not established a conclusive q u a n tita tive description in agreement with experimental evidence, prim arily because o f the d iffic u lt y of determining the transport c o e ffic ie n ts. The Irre v e rs ib le Thermodynamics approach, in contrast to the Soil Physics approach, does not require as detailed an understanding of the mechanism o f heat and mass tra n s fe r or o f the porous medium s tru c tu re. I t is a phenomenological theory, which postulates lin e a r re la tio n s between "flu x e s" o f heat and mass and "thermodynamic forces" ident if ie d by the entropy production equation. This application of the theory of Irre ve rsib le Thermodynamics was proposed by Cary and Taylor as a special application o f the general case involving transfers of heat and several species in a porous medium system [20, 21, 22, 90, 91, 92]. I t has since been extended by Cary to the consideration of the model's lim ita tio n s and it s application to systems o f practical in te re st [15, 16, 17, 18, 19]. One apparent advantage of the Cary and Taylor model over the P h ilip and devries model is th a t i t provides d e fin itio n s of the coupling c o e ffic ie n ts which fa c ilita te s th e ir determination and subsequent use in the model equations. Other investigators have discussed the determ ination o f the phenomenological c o e ffic ie n ts and the ap plicatio n o f the model to various porous media systems [1, 9, 14, 38, 41, 49, 55, 56, 81, 89]. To the author's knowledge, Gee [38] and Jury [55, 56] have made the only experimental attempts to q u a n tita tiv e ly determine the coupling e ffe cts. Gee reported a s ig n ific a n t increase in moisture transfer when a temperature gradient was applied to a soil column. He obtained values o f the coupling c o e ffic ie n t re la tin g the moisture flu x to the temperature gradient. Jury also reported a strong e ffe c t of

15 6 temperature gradient on moisture flu x. Jury's data indicate negligible heat flu x associated with moisture content gradient. He performed two experiments: a closed column one with zero net steady-state moisture flu x and an open column experiment with zero heat flu x. Jury solved the four simultaneous algebraic transport equations describing these two experiments. The so lu tio n provided the values o f the four phenomenological coefficients in these equations. Jury also reported relations between the c o e ffic ie n ts appearing in the Cary and Taylor model equation s and those in the P h ilip and devries transport equations. None of the work mentioned so fa r has, however, verifie d the Cary and Taylor model by solving a boundary value problem involving the phenomenological equations, conducting the corresponding laboratory experiment with a porous medium system, and comparing the experimental measurements w ith model predictions.

16 CHAPTER 3 THERMODYNAMICS OF IRREVERSIBLE PROCESSES Irre v e rs ib le Thermodynamics, the development o f which is f r e quently associated w ith Onsager [7 4 ], Meixner [6 6 ], and Prigogine [39, 80], is a comparatively recent and developing subject. Wh ile E q u ilibrium Thermodynamics or "therm ostatics", deals with the study of equilibrium states o f physical and chemical systems, Non-equilibrium or Irre versib le Thermodynamics attempts a phenomenological description of systems undergoing irre v e rs ib le processes. Such processes are characterized by the presence of fluxes (transport) of mass and/or d iffe re n t forms o f energy. Although Irre versib le Thermodynamics has recently been extended to the consideration of nonlinear system behavior [39, 63], the present discussion is lim ite d to the theory o f Linear Irre versib le Thermodynamics, which has been described in several standard works [28, 34, 39, 43, 63, 80]. In th is chapter, the basic principles o f Linear Irre versib le Thermodynamics are b r ie fly summarized. The entropy change of any system (defined by a closed surface in three-dimensional space) during an accounting period 6t may be w ritte n as ds = des + d i S, (3.1) where des is the to ta l entropy tran sfe rred from the surroundings to system during the accounting period. The qu a n tity des, in 7

17 8 general, may be negative, zero or positive depending on the nature of the interactions of the system with its surroundings, di S is the entropy production w ith in the system during the accounting period, which results from irre v e rs ib le processes. According to the Second Law o f Thermodynamics, di S > 0 f o r a ll real processes. (3.2) Invoking the so-called assumption of "local equilib riu m ", the local (specific) entropy is assumed related to the local values of the macroscopic state variables by the fundamental property re la tio n ship (Gibbs Equation) of Equilibrium Thermodynamics. The resulting equation, which relates the local entropy to internal energy, volume and composition, is du dt T ds dt P dv dt + Σ i μi dci d t (3.3) Application o f balance (conservation) equations fo r mass and energy and subsequent u t iliz a t io n o f Equation (3.3) leads to an entropy accountab ilit y relationship o f the form ds dt des dt + σ. (3.4) The local rate of entropy production ϭ is always nonnegative, and has the b ilin e a r form

18 Ϭ = ΣJi Xi i (3.5) 9 where is the i th "thermodynamic flu x ", Xi is the i th "thermodynamic fo rce". The fluxes appearing in the entropy production term are related to the fa m ilia r heat flu x, mass flu x, e le c tric a l current, e tc., and the thermodynamic forces Xi are the gradients o f intensive propertie s o f the system (or functions th e re o f). I t has been experimentally observed that the thermodynamic fluxes and forces appearing in the entropy balance equation can be related by empirical rate expressions. In general, the fluxes and forces may be described by the phenomenological expressions [43] Ji =ΣLikxk (3.6) where are known as the phenomenological c o e ffic ie n ts. Equation (3.6) expresses the "Irre v e rs ib le Thermodynamics" postulate th a t each thermodynamic flu x is lin e a rly related to a ll of the thermodynamic forces. The phenomenological coefficie n ts may be functions of the intensive properties o f the system such as temperature and pressure but not o f the forces. An important re la tio n between the phenomenological c o e ffic ie n ts, generally assumed v a lid w ith in the lin e a r region, is the Onsager re c ip rocal re la tio n [74] Lik= Lki. (3.7)

19 10 Equation (3.7) states that the matrix o f phenomenological coefficie n ts is symmetric, which reduces the number o f unknowns in the tra n s port equations. Equation (3.7 ), which was derived by Onsager fo r a rather re s tric te d class o f relaxation processes using the p rin cip le of "microscopic re v e rs ib ility " from s ta tis tic a l mechanics, has been v e rifie d experim entally fo r a large number o f processes [67].

20 CHAPTER 4 DEVELOPMENT OF THE COUPLED HEAT AND MOISTURE TRANSFER MODEL The f i r s t development of an Irre versib le Thermodynamics-based model fo r coupled heat and moisture transfe r in unsaturated porous media is commonly a ttrib u te d to Taylor and Cary [91]. In th is chapter, the development by Taylor and Cary w ill be described to illu s tr a te the Irre v e rs ib le Thermodynamics th e o re tic a l approach to th is problem. Concurre n tly, questions raised during the present work re la tin g to the formulation of the entropy balance equation fo r th is process and the id e n tific a tio n o f the associated thermodynamic "d rivin g " forces fo r heat and mass fluxes are discussed. The follow ing sim p lifyin g assumptions are made in Taylor and Cary's development o f the model equations fo r the coupled transfe r of matter and energy in a continuous porous medium system: 1. The concept o f "local equilibrium ", which holds that the Gibbs equation o f E quilibrium Thermodynamics (which provides an equation of state fo r entropy in equilibrium systems), applies lo c a lly in non-equilibrium or continuous systems. This is a fundamental, underlying assumptio n o f the theory o f Irre v e rs ib le Thermodynamics. 2. An id e a lize d, chemically in e r t, porous medium is considered, and e ffe c ts other than heat and moisture tra n s fe rs are neglected. 11

21 12 3. Moisture transfer is liq u id phase, and the moisture content o f the porous medium is s u ffic ie n tly high that a continuous liq u id phase exists. This assumption is im p lic it in Taylor and Cary's r e s tr ic tio n o f the "general" equations to the case with water being the only form o f flow ing m atter, and in p a rtic u la r to single component flow ( i.e., no d iffu s io n ). 4. The system is never "fa r" from equilibrium, so that the fluxes can be described by linear phenomenological equations. 5. The "water potential" is a unique function of the moisture content. (The id e n tific a tio n of the water potential with local thermodynamic properties w ill be discussed below.) 6. Hysteresis e ffe c ts are neglected. Phenomenological equations are f ir s t developed fo r the transport o f energy and several species in a general multicomponent system. These equations are then specialized to describe heat and liq u id moisture transfer in porous media. Considering an open, multicomponent thermodynamic system but excluding chemical reaction, the equation of change fo r component mass is [43] p DCi Dt - * Ji, (4.1) where p is the to ta l mass density o f the system, D Dt is the substantial d e riva tive operator, Ci = pi P is the mass fraction of component i,

22 13 ρi is the mass density of component i, Ji is the "d iffu s iv e " flu x of component i, or ρ i( vi - v), vi is the ve lo c ity o f component i, and v is the mass average v e lo c ity, or Σ C i vi, and V is the gradient operator. The equation o f change fo r to ta l internal energy, neglecting external forces and viscous dissipation e ffe c ts, is [43] DU P Dt = -v * JQ - P v * v, (4.2) where U is the specific internal energy, JQ is the thermodynamic heat flu x, and P is the pressure. * I t w ill be seen la te r that th is form o f the internal energy balance, when used in conjunction with Equations (4.1) and (4.3 ), leads to an entropy balance which requires a re in te rp re ta tio n of "heat" transfe r (defined in classical Equilibrium Thermodynamics as being s t r ic t ly separable from energy transfers associated with mass tra n sfe r). This problem has been discussed and proposals fo r its c la rific a tio n offered [111]. The application o f the equations derived herein to a closed (zero mass flu x ) system renders these d if f ic u lt ie s moot; however, v e rific a tio n o f the Taylor and Cary model fo r open systems in which the mass flu x is not zero requires a tte n tio n to th is problem.

23 14 The fundamental property re la tio n, or Gibbs equation, which applies lo c a lly, is [43] DS T Dt ' DU Dt + P DV Dt - n Σ i = l μi DC. i Dt (4.3) where is the sp e cific entropy, V is the sp e cific volume, 1/ρ, and is the chemical potential of component i. Combination of Equations (4.1 ), (4.2 ), and (4.3) gives, a fte r some algebraic manipulations, DS 1 n Σ i =1 T Δ Ji ρdt - τ Δ JQ (4.4) Noting that TΔ JQ = Δ J Q VT and μi τ J i = μi Ji Τ - J i μi Τ (4.5) and u tiliz in g the d e fin itio n o f the substantial d e riva tive, Equation (4.4) can be rew ritten as

24 15 Ϭ p S Ϭ t = -Δ ρ S^v + T n Σ i = l μ i T (4.6) ΔT + n Σ i = l J i V T Further, noting that μi = Hi - TSi (4.7) where the overbar denotes partial properties, i.e., Hi = Ϭ H Ϭmi J Equation (4.6) becomes Ϭ p S Ϭ t = -Δ ps v + n Σ 1= 1 J isi - n Σ i = l J ihi T + T ϬρS Ϭt ' ΔT + n Σ i= l J i V T (4.8) η _ n _ Since ρ Sv + Σ J isi = Σ piv isi, Equation (4.8) reduces to i=l i=l

25 16 ϬρS Ϭ t = - Δ * η Σ i =1 ρ iv is i - Σ τ JQ Τ JQ τ2 Δτ + η Σ i = 1 J i Δ i μi Τ (4.9 ) The f i r s t term in the brackets on the RHS o f Equation (4.9 ) is commonly in te rp re te d as the tra n s fe r o f entropy between the system and surroundings due to mass and heat tra n s fe rs, w h ile the second bracketed term represents the entropy production associated w ith irre v e rs ib le processes ta kin g place in the system. A re d e fin itio n o f thermodynamic heat flu x (see fo o tn o te, page 13) is then made as fo llo w s. Noting th a t μί τ ( mj τ τ Ηi i τ2 (4.10) where (Δ μ i ) T denotes eva luatio n a t a given tem perature, and d e fin in g the c a lo rim e tric heat flu x as Jq=JQ η Σ i = 1 (4.11) Equation (4.9 ) becomes ϬρS Ϭ t = - Δ η Σ i = l pi vi Si + J q τ J + η Σ i = 1 Τ * ( μi) Τ (4.12)

26 The local volumetric rate o f entropy production is therefore 17 Jq o' * - v1 - Σ Ji ( μi) T2 i= l T n and the dissipation function (rate of lo s t work) is T σ' = -Jq In T - n Σ i=l J, ( μ i ) 1 T (4.13) The fluxes in Equation (4.13) are then assumed related to the associated forces id e n tifie d in the equation by the lin e a r phenomenological expressions Ji - - n Σ k=l Li k ( μ i ) Liq VlnT (4.14) and Jq=- n Σ k=l Lqk( μi)t - Lqq lnt (4.15) where the phenomenological co e fficie n ts are related by the Onsager relations [74 ], Lik Lki and (4.16) L iq = L qi

27 18 Considering water the only species tra n s fe rre d, Equations (4.14) and (4.15) reduce to Jw = -LWW ( μw) - Lwq In T (4.17) and Jq = -L qw, ( μ w ) - Lq In T. (4.18) Follow ing T a ylor and Cary, the "w ater p o te n tia l" ψ is defined as the d iffe re n c e in chemical p o te n tia l between water in the porous medium and pure fre e water [9 1 ], i. e., ψ = μw - uw constant; hence where the chemical p o te n tia l o f pure fre e w ate r, μ, is a μw = ψ. (4.19)

28 19 S ubstitution of Equation (4.19) in Equations (4.17) and (4.18) gives Jw = -Lwνψ - Lwq ln T (4.20) and J = -Lqw ψ - Lq l n T, (4.21) which describe the coupled flows of heat and moisture in a porous medium. When a sealed column of unsaturated porous medium is subjected to a constant temperature gradient, the steady-state moisture flu x is zero, and Equation (4.20) reduces to 0 = -Lw ψ - Lwq lntt, (4.22) *The careful reader has noted that the isothermal gradient of the water p o ten tial, (νψ )γ, has been dropped in favor of νψ. Such an assumption appears ju s tifie d in practice since the water potential is a much stronger function of moisture content than of temperature. This is probably more apparent from consideration of the temperature dependence o f air-w ater surface tension which undoubtedly is a major component of the water potential as defined by Taylor and Cary.

29 20 which can be s im p lifie d to give v Ψ _ Lwq _. lnt Lww β (4.2 3 ) which is to be considered the d e fin itio n o f the c o e ffic ie n t β. Now, re w ritin g Equation (4.20, Jw = -Lww (νφ + β lnt) (4.24) and using the Onsager re la tio n [7 4 ], Equation (4.21) becomes Jq = - Lww βνψ - L lnt. (4.25) In order to use Equation (4.24) and (4.25) fo r systems of p ractical in te re s t, i t is necessary to re la te ψ to a p ra c tic a lly measurable variable. A single-valued fu n c tio n a lity between ψ and water content is assumed, so tha t Jw = - ρw D 0 - Lwq In T, (4.26) where 0 is the volumetric moisture content, ρ is the density o f water, and D =wdψ is the c o e ffic ie n t of d iffu s iv ity. p w d0 At steady-state, in a sealed column, Jw = 0, and Equation (4.26) reduces to

30 21 ν_θ = -Lwq = _β* ln Τ pw D (4.27) which is to be considered the d e fin itio n o f the c o e ffic ie n t 8*. Applying the above re la tio n to Equation (4.26), Jw = -PWD( 0 + β* lnt), (4.28) and s im ila rly, using Equation (4.25) and the d e fin itio n o f D given in (4.26) Jq = -pwdβ 0 - Lqq lnt (4.29) Equations (4.28) and (4.29) are known as the Cary and Taylor model.

31 CHAPTER 5 DISCUSSION OF TRANSPORT EQUATIONS AND SOLUTION TECHNIQUE The Cary and Taylor Irre ve rsib le Thermodynamics-based model fo r coupled heat and liq u id moisture tra n sfe r in a three dimensional unsaturated porous medium is : Jw = -pwd( θ + β* lnt) (4.28) Jq = -PwDβ θ- L qq ln T, (4.29) where Jw is the flu x o f liq u id water, Jq is the flu x o f heat, D is the c o e ffic ie n t o f d iffu s iv ity of water in the unsaturated porous medium, β* is the coupling c o e ffic ie n t, defined as - θ/ lnt a t steady-state and zero moisture flu x, $ is the coupling c o e ffic ie n t, defined as - ψ/vlnτ at steady-state and zero moisture flu x Lq is a phenomenological c o e ffic ie n t = λt, λ is the thermal c o n d u c tiv ity o f the porous medium, T is the absolute temperature, θ is the volumetric moisture content o f the porous medium, Pw is the density o f water, Ψ is the water potential o f liq u id moisture in the porous medium, and 22

32 23 V is the gradient operator. The transport coefficie n ts D, β*, β and have been reported to be strong functions o f the volumetric moisture content [38, 50, 55, 91] I t is in s tru c tiv e to consider the case where only one of the gradients 0 or lnt is present in the porous medium. For the case of isothermal equations ( lnt = 0), Equation (4.28) reduces to Jw = - ρw D 0, (5.1) which is a form of the classical Darcy's Law fo r movement of moisture in an isothermal unsaturated porous medium. Conversely, fo r heat conduction in a porous medium o f uniform moisture content, 0 is zero, and Equation (4.29) reduces to Jq = -Lqq InT = - λ T, (5.2) which is the well known Fourier's law of heat conduction. I t is thus seen that the theory o f Irre versib le Thermodynamics permits temperature gradients to be combined with moisture content gradients in the general flow equations of water and heat in unsaturated porous media in such a way that they reduce to the commonly used phenomenological equations under p a rtic u la r conditions. Further, the general phenomenological Equations (4.28) and (4.29) provide information regarding the in te ractio n between the coupled heat and moisture transfers and some in sig h t in to the nature of the phenomenological c o e ffic ie n ts. For example, fo r the isothermal case, Jq = -pw Dβ 0 (5.3)

33 24 represents the transport of heat associated w ith the gradient in moisture content. The magnitude of β w ill, in such a case, be a measure of the e ffe c t of coupling o f the moisture content gradient on the heat flu x. An evaluation o f the c o e ffic ie n t ρwdβ*, or Lwq, which is the e ffe c t o f the In temperature gradient on the liq u id moisture flow, w ill be considered la te r in th is report. As described in the chapter on the development o f the model equations, β* is measured as the ra tio - 0 / ln T in sealed steady-state experiments with zero net moisture flu x. A large value of β* indicates a strong influence of temperature gradient on moisture tra n s fe r in open systems. Irre versib le Thermodynamics is a phenomenological theory which does not explain the mechanism of coupling or the reasons fo r i t, but several explanations have been proposed to account fo r the e ffects of the thermal gradient on moisture tra n s fe r [6, 18]. I t has been suggested th a t since surface tension o f an air-w ater interface decreases w ith increasing temperature,moisture in an unsaturated porous medium could flow from a warm region to a cooler region due to a surface tension gradient. Another proposed reason fo r coupled moisture flu x is the e ffe c t o f temperature on the adsorbed layer o f moisture on the so lid p a rtic le s. The observation of coupled moisture tra n sfe r in a saturated medium has led to the proposal that transfe r results from a net motion generated by random k in e tic energy changes associated with the hydrogen bond d is trib u tio n which develops under a thermal gradient [18]. I t is possible that one or more o f the above effects acts to cause coupling effects between heat and moisture. However, as Beatty [3 ] points o u t, i t seems constructive to th in k o f coupled heat and

34 25 mass transfers as "separable e ffe c t" processes. Beatty proposed that coupled processes are two or more macroscopically separable effects re s u ltin g from a single microscopic process in which the c a rrie r molecules with higher k in e tic energies than the average p a rticle s are transported as a re s u lt o f two or more independently variab le macroscopic potentials. The thermal conductivity c o e ffic ie n t is a measure of the e ffe c t of the temperature gradient on the heat flu x fo r the uncoupled case in the absence of moisture content gradients in the unsaturated porous medium. Porous medium thermal conductivity is a function o f the moisture content, but its measurement as such is not straightforw ard since a moisture content gradient invariably accompanies steady state heat transfer. The c o e ffic ie n t o f d iffu s iv ity of liq u id moisture in an unsaturated porous medium, D, is also a strong function o f the moisture content. The c o e ffic ie n t o f d iffu s iv ity is defined as Kdψ, where K is the hydraulic conductivity o f the porous medium and dψ / d0 is the slope of the water p o te n tia l versus moisture content (the so-called water retentio n curve), which is also a fun ction o f the moisture content. A v e rific a tio n of the model Equations Jw = -Pw D( 0 + β* ln T ) Jq q = -ρwdβ θ - L qq In T (4.28) (4.29) can be carried out in the follow ing manner. The transport coefficie n ts D, β*, β and L can be determined independently from separate exp eriments. With proper i n i t i a l and boundary co n d itio n s, Equations (4.28)

35 26 and (4.29) can be integrated, fo r steady-state conditions, yie ld in g p ro file s o f temperature and moisture contents. These p ro file s, when compared w ith experimentally observed temperatures and moisture content values fo r the system, provide a check o f the v a lid it y o f the model. For the steady-state condition in a closed s p h e ric a lly symmetrical system of unsaturated porous medium indicated schematically in Figure 1, Equations (4.28) and (4.29) reduce to d0 Jw = 0 = -pwd( dr + β* d In T dr ) (5.4) Jq Q 4πr = -pwdβ d0 dr - Lqq d In T dr (5.5) where Q is the to ta l steady state heat flow ra te. Equations (5.4) and (5.5) represent a set of nonlinear (due to the fu n c tio n a lity o f the c o e ffic ie n ts ) ordinary d iffe re n tia l equations which, with two boundary conditions, can be integrated to give T and 0 p ro file s in the annular region. A numerical in te gra tion technique u tiliz in g the boundary value problem "shooting method" was used, with the temperature T1 at radius as one of the boundary conditions and conservation of the to ta l mass of water in the system as an additional equation o f constraint. An in it ia l value of moisture content θ1, at radius R1, was in it ia lly guessed to s ta rt the in te g ra tio n. A fte r integration to the outside sphere radius R2, the to ta l mass of water in the system was calculated by integration o f the predicted moisture p ro file. I f the calculated mass o f water was d iffe re n t from the in it ia l mass of water in the system, the in it ia l value θ1 was ite ra te d and the in te g ra tio n repeated. When the calculated and o rig in a l masses of

36 27 Figure 1. Spherical Porous Medium System

37 28 water agreed, the temperature and moisture content p ro file s thus obtained were the predicted steady-state d is trib u tio n s in these variables fo r the given system. The Continuous System Modeling Program (CSMP)1 was used fo r the numerical in te g ra tio n. A sample program is given in the Appendix. System 360 Continuous System Modeling Program (360-A-CS-16X), User's Manual, International Business Machines Corporation, Data Processing D iv is io n, White P lains, New York.

38 CHAPTER 6 MEASUREMENT OF TRANSPORT COEFFICIENTS The transport coefficie n ts thermal conductivity, d iffu s iv ity, and the two coupling coefficie n ts were obtained independently from separate experiments, as functions o f moisture content. Thermal C o nductivity, λ The thermal conductivity o f an unsaturated porous medium is an "o vera ll" value fo r the three-phase so lid -w a te r-a ir system. A steady-state determination of the thermal conductivity c o e ffic ie n t as a function o f the volum etric moisture content of the porous medium, from heat flu x and temperature gradient measurements,is not straightforw ard, since moisture tra n sfe r resu lts from nonisothermal conditions in such a system. However, a tra n s ie n t method allows measurement o f thermal d iffu s iv ity, from which thermal conductivity can be determined, before moisture m igration occurs. An unsteady-state method fo r measuring thermal d iffu s iv ity o f low thermal conductivity m aterials [25, 75] was used to determine the thermal d iffu s iv ity o f moist porous media over a range o f volum etric moisture contents. The method is based on the solution o f the d iffe re n tia l equation fo r unsteady-state heat conduction in a c y lin d e r, 3T Ϭt = α Ϭ2T Ϭr ϬT r Ϭr (6. 1) with the in it ia l and boundary conditions 29

39 30 T = T0 at t = 0 ϬT Ϭr = 0 at r = 0 fo r t > 0 and l im ϬT _ h (Ta - T) fo r t > 0 r - rm Ϭr λ where T is the temperature, T0 is the in it ia l uniform temperature of sample, r is the radial distance from center of cylin d e r, rm is the radius o f cylin d e r, h is the surface heat tra n sfe r c o e ffic ie n t, λ is the thermal conductivity, Ta is the constant ambient temperature, a and t is the time. The series solution to Equation (6.1) converges ra p id ly, and a fte r a time of about one minute is well represented by the simple re la tio n Y = A 10- bt, (6.2) where A = 2 J1(x1) X1[J02(xi ) + J12 (x1)] J0(x1 rm ) -x1α (6.3) and b = r2m where Y is the ra tio of the temperature differences, (Ta - T ) / ( T a - T0).

40 31 A is a constant, -b is an exponent; gra p h ica lly, the slope of the log Y - t curve, J0( x) is a Bessel function o f f i r s t kind and zero order o f x J1(x) is a Bessel function o f f i r s t kind and f i r s t order o f x, and a is the thermal d if f u s iv it y o f the porous medium. Equation (6.2) implies a lin e a r re la tio n between log Y and t with slope -b. For the experimental conditions used in th is work, i t can be shown th a t a lim itin g value o f x1 = applies [2 5]. A diagram o f the experimental apparatus is shown in Figure 2. Samples o f the μ glass beads media were encased in th in-w a lle d aluminum cylind ers 15.2 cm long and 1.91 cm in diameter. A cm diameter two-bore ceramic thermocouple in sula to r supported a Chromel- Alumel thermocouple junction near the center of the sample. A fte r allow ing the sample temperature to e q u ilib ra te w ith the room temperature, T0 (20 C), the cylinder was immersed in the constant temperature (Ta) water bath, and the thermocouple temperature, T, recorded as a function of time. The recorded temperature was used in Equation (6.2) to graphically determine the slope, -b, from which the thermal d if f u s iv it y was calculated from Equation (6.3 ). Thermal conductivity was determined from the re la tio n λ = a p c (6.4)

41 Figure 2. Schematic of Experimental Apparatus fo r Thermal Conductivity Determination 32

42 33 where the volumetric heat capacity, pc, was calculated as the volume weighted average of the heat capacities of the sample constituents, glass, water and a ir. Measurements were made with bath temperatures of 30 C, 35 C and 40 C. Thermal d iffu s iv itie s calculated therefrom d iffe re d by less than 3%, which is w ithin the expected experimental error. A typical p lo t o f log Y versus t fo r the moist glass beads medium is shown in Figure 3. The measured temperature approaches the fin a l equilibrium value Y < 0.05 in about six minutes fo r the dry sample and three minutes fo r the moist sample. The transient period fo r the moist medium is long enough to s a tis fy the requirements fo r the model approximation (series convergence) but short enough to preclude s ig n ific a n t moisture re d is trib u tio n in the sample. Thermal d iffu s iv itie s and conductivities fo r the glass bead medium are plotted as functions o f volum etric moisture content in Figure 4. The thermal 2 d if f u s iv it y o f the beads was found to increase from cm /sec fo r dry 2 beads to cm /sec at 5% volumetric moisture content, above which i t remains e sse n tia lly constant with increasing moisture content, while the thermal conductivity o f the porous medium increases from a value of 0.53 mcal/ C cm sec fo r dry beads to 1.97 mcal/ C cm sec fo r saturated beads. D iffu s iv ity, D The d iffu s iv ity D o f water in an unsaturated porous system is a strong function o f the moisture content and is defined by the re la tio n ship D = K (6.5) dψ d0

43 34 Figure 3. Typical Heating Curves fo r Glass Beads

44 35 Figure 4. Thermal C onductivity λ and Thermal D iffu s iv ity α vs. Volumetric Moisture Content 0

45 36 where K is the unsaturated hydraulic conductivity, ψ is the water p o te n tia l, and 0 is the volum etric moisture content. Independent determinations were made o f K and d0 over a wide range of moisture contents and were then substituted in Equation (6.5) to obtain values o f D as a fun ction o f volum etric moisture content. A predictive method which has been used by s o il physicists to determine the hydraulic conductivitie s o f glass beads and s o ils [42] was employed to obtain K values fo r the μ glass beads - water system o f the present study. The method involved experimental de termination o f the hydraulic conductivity value at one moisture content (the saturated hydraulic conductivity) and substitu tio n in to the follow ing equation to calculate K values fo r other values o f moisture content: m Ks 30 Ϭ2 εp Κ(θ)1 = Σ [(2 j + 1-2i)hj-2)], Ksc pw9 n j = l k = l, 2,..., m (6. 6) where Κ(Θ)i is the hydraulic conductivity fo r a specified water content o r pressure, 0 is the moisture content, i denotes the class o f water content, Ks K is the matching fa cto r (measured standard conductivity/ Ksc calculated saturated conductivi t y ), Ϭ is the surface tension o f water,

46 37 pw is the density of water, g is the g ra vita tio n a l constant, η is the visco sity o f water, ε is the porosity o f the medium, p is the parameter that accounts fo r in te ra ctio n of pore classes, n is the to ta l number of pore classes, and h. is the pressure fo r a given class o f pores. J The saturated hydraulic co n d u ctivity Ks was measured in a laboratory experiment w ith a column of saturated glass beads medium. Ks was obtained from the Darcy's Law fo r flow in saturated media [2 7]: Δ H Qw = Ks A L (6.7) where Qw is the volume o f flow in time t, A is the cross-sectional area o f the column, ΔΗ is the hydraulic head, and L is the length o f the column. The value o f saturated hydraulic conductivity thus obtained, along with Equation (6.6 ), gave the hydraulic conductivity, K, which is p lo tte d as a fun ction o f volum etric moisture content in Figure 5. The potential ψ of water in the porous medium is related to the water content 0, and the graphical relatio nsh ip is usually referred to in Soil Physics lite ra tu re as the water retention curve. The water retention curve fo r the glass beads - water system was measured using the pressure p late method [5 0]. Measurements were made over a wide

47 38 Figure 5. Hydraulic Conductivity K vs. Volumetric Moisture Content Θ

48 39 range o f applied water p o te n tia ls, and the re la tio n sh ip obtained between ψ and 0 is plotted in Figure 6. As indicated in Figure 6, water potential decreases w ith increasing water content. I n it ia lly ψ decreases sharply fo r a s lig h t change in 0 near the dry region, followed by an almost lin e a r re la tio n sh ip fo r a wide range of moisture content. Near saturation range, ψ again ra p id ly decreases with increasing moisture content. The slope dψ o f the water re te n tio n curve, a t d i f fe re n t values o f moisture content, was obtained g ra p h ic a lly by a method1 which employs a French m irro r to draw a normal to the curve at the po in t in question. The values o f obtained from the water retention curve and the hydraulic conductivitie s were substituted in Equation (6.5) to get the d iffu s iv ity D, which is shown in Figure 7 as a function of moisture content. The d iffu s iv ity is seen to be a strong function of moisture content; i t increases w ith increasing moisture content at a ll moisture contents except a narrow range from 0 = 0.18 to 0 = 0.21, where D decreases w ith increasing 0. C o e fficie n t, β* The coupling c o e ffic ie n t β* is a fu n ctio n o f the moisture conte n t, and is defined fo r the spherically symmetric one-dimensional system by the re la tio n β* = - d0 / dr d In T / dr Jw = 0 (6.8) 1Kraus, Μ. N., "Drawing a Tangent to a Curve," Chemical Engineering, March 12, 1979.

49 40 Figure 6. Water P otential ψ vs. Volumetric Moisture Content Θ

50 Figure 7. D iffu s iv ity D vs. Volumetric Moisture Content 0 41

51 42 An evaluation o f β* from the above equation involves the experimental determ ination o f the steady-state d is trib u tio n s o f moisture content, 0, and temperature, T, in a sealed system o f unsaturated porous medium (the spherical one-dimensional system in the present study). Slopes, or f i r s t derivatives o f these d is trib u tio n s, and d ln T d r, are taken, and the negative o f th e ir ra tio gives β*. In order to obtain 3* as a fu n ctio n o f 0 over the e n tire range o f moisture contents fo r which the model applies, i t is necessary to carry out a closed experiment w ith a s u ffic ie n tly large value of heat rate, Q, that resu lts in a steady-state d is trib u tio n of 0 over the model range. As suggested by previous investigations [55, 91], a value of 0 = 0.01 was chosen as the lower lim it fo r the range over which the model requirement o f a continuous liq u id phase applies. Experimental Run 2, described in Chapter 8 on re su lts and discussion, exhibited such a wide range o f moisture contents, and β* calculations were made from data on T and 0 d is trib u tio n s obtained from th a t run. The slopes d0 and were graphically obtained w ith the (French m irro r) method described e a rlie r. The values o f β* as function of moisture content, calculated from these data, are p lo tte d in Figure 8. β* increases 0 w ith increasing moisture content u n til i t reaches a maximum value at 0 = 0.28, a fte r which i t decreases as 0 increases. A p lo t o f the phenomenological c o e ffic ie n t Lwq = Dβ* is also shown as a fu n ctio n o f 0 in Figure 9. In order to ascertain th a t 3* is a single-valued function o f 0, i t is necessary to prove th a t β* exhibits a unique re la tio n w ith θ fo r several experiments with d iffe re n t values of the gradients lnt and 0. This te s t w ill be examined in Chapter 8, where β* values w ill be shown fo r some sample experimental runs.

52 43 Figure 8. Coupling C o e ffic ie n t β* vs. Volumetric Moisture Content 0

53 Figure 9. Coupling C o e fficie n t Lwq vs. Volumetric Moisture Content 0 44

54 45 C o e ffic ie n t, β The coupling c o e ffic ie n t, β, which denotes the e ffe c t o f moistu re content on the heat f lu x, is defined by the re la tio n β = - dψ / dr Ϭψ ϬΘ d InΤ / dr Ϭ0 Ϭ InΤ Jw = 0 = β* Ϭψ Ϭ0 (6.9) and is a function of the moisture content, 0, since both β* and Ϭ 0 are fun ction s o f 0. A graph of β is shown in Figure 10. I t is to be remembered th a t β and β* are related through the Onsager reciprocal re la tio n, and calcula tio n o f β from Equation (6.9) re fle c ts th is re la tio n s h ip. I t has been suggested th a t β is too small to be calculated w ith s u ffic ie n t accuracy from experimental data (or even zero) [12, 55]. This w ill be examined in Chapter 8.

55 46 Figure 10. Coupling Coefficient β vs. Volumetric Moisture Content 0

56 CHAPTER 7 DESCRIPTION OF EXPERIMENT An open system experiment, in which heat and moisture fluxes occur simultaneously as a re s u lt of coupling between the temperature and moisture content gradients, could provide a te s t o f the v a lid ity of the Irre v e rs ib le Thermodynamics model Equations (4.28) and (4.29). However, the d if f ic u lt ie s associated w ith the measurement of the small values o f moisture fluxes and the s p e c ific a tio n o f exact boundary cond itio n s make such an open system experiment extremely d i f f ic u lt to perform. In lig h t o f these d if f ic u lt ie s, i t was decided to analyze in th is study a sim pler, closed system th a t would provide a v a lid te s t of the Cary and T aylor model. The low values o f thermal c o n d u c tiv ity o f porous media accentuate errors due to heat losses. A spherical system in which the porous medium is enclosed in the annular region between two concentric spheres precludes heat losses. Using a high thermal conductivity m aterial such as copper as the sphere m aterial provides an e ffe c tiv e means fo r applying constant and uniform temperature and heat ra te boundary cond itio n s. The choice o f μ glass beads and d is tille d water as the unsaturated porous medium was made to minimize e ffe c ts other than coupled heat and moisture tra n s fe r, and to provide a system in which measurements o f the required properties and variables could be made accurately enough to obtain a v e rific a tio n o f the IT model. 47

57 48 The system th a t was designed to meet these objectives is indicated diagrammatically in Figure 11. The primary components of the assembly are two concentric spheres which are mounted on a support stand. A heater encased in the inner sphere provided the heat input to the system, and the system was enclosed in a constant temperature a ir-b a th which provided the ou ter surface boundary cond itio n. Guides located on the outer sphere were part o f a moisture content sampling system fo r obtaining local measurements o f moisture content. Local temperatures in the medium were monitored w ith thermocouples and a potentiom etric recorder. The sphere assembly, the thermocouple arrangement and the moisture content sampling system were used by Moore [70] and are described by him in thorough d e ta il; however fo r the sake of c o n tin u ity, and to make th is study self-co nta ined, the experimental set-up w ill be described in th is chapter by considering each component assembly separately. Outer Sphere The outer sphere consisted o f two copper hemispheres, cm in diameter, spun from 0.47 cm th ic k copper sheet. The hemispheres were jo in e d together a t the bases w ith flanges extending 2.54 cm o u t ward. The top o f each hemisphere had a hole, 9.86 cm in diameter, machined in i t, and plugs, shown in Figure 12, were used to seal the openings. While the upper hemisphere opening had the plug permanently s ilv e r soldered to i t, the lower hemisphere opening was threaded to

58 49 Figure 11. Schematic o f Apparatus

59 50 Figure 12. Top and Bottom Access Plugs

60 51 accept the threaded plug, which could be e ith e r screwed in to positio n or removed fo r f i l l i n g and emptying the porous medium. Guides, through which the moisture content samples were extracted, are shown in Figure 13. These were aligned w ith 1.27 cm diameter holes in the outer sphere and bonded to the sphere surface which was fla tte n e d around the holes. The guides were then sealed by in se rtin g 1.27 cm diameter s ta in le s s steel rods w ith copper plugs, having the same surface curvature as the outer sphere, s ilv e r soldered at the ends. A 1.27 cm diamter hole was also d r ille d in the center o f the bottom plug and closed w ith a plug. This p o rt served as another sampling lc o a tio n. The thermocouple supports, described la te r in th is chapter, were positioned on th e ir outer ends by nylon pins which were inserted in holes d r ille d in threaded copper bushings. These bushings, which screwed in to 2.54 cm diameter holes in the outer sphere, are also shown in Figure 13. Thermocouples made from 30 gage, glass-wrapped Chromel -Alumel wires were positioned at three locations on the inner surface o f the outer sphere, to determine the temperature v a ria tio n on the outer sphere. The thermocouple wires were bonded to the sphere surface w ith epoxy glue, and the ju n c tio n s were l e f t exposed. Inner Sphere and Heater The inner sphere and the heating element are shown schem atically in Figure 14. The inner sphere consisted o f two copper hemispheres, 10.0 cm in diameter, joined by an in te rn a l annular rin g. The heater was potted in a 2.54 inch diameter c a v ity in the center o f the sphere. A

61 52 Figure 13. Sampling Guide, Plug, and Bushing

62 53 Figure 14. C ro ss-sectio n o f Inn er Sphere

63 hole was d r ille d in the center o f the top hemisphere to accept a 0.79 cm O.D., 0.47 cm I.D. steel rod which extended out from the ou ter sphere 54 surface through a hole in the top plug. An 0 -rin g was used to seal the clearance between the tube and the plug hole, and a set screw fix e d the position o f the rod in the plug, thus p o sitio n in g the inner sphere in the center o f the ou te r sphere cm deep holes were d r ille d on the ou ter surface o f the inner sphere and small nylon pins fixed in them fo r positio n in g the in ne r ends o f the thermocouple support tubes. The resistance heating element consisted o f a 1.27 cm diam eter, 2.22 cm long, Firerod C artridge heater1 w ith a cold resistance o f 3.67 ohms and a maximum rated power o f 100 w a tts. The heater was potted in the in n e r sphere c a v ity w ith RTV encapsulant rubber. Thermocouples used to determine the temperature v a ria tio n s on the ou te r surface o f the in ne r sphere were made from cm Chromel- Alumel, T eflo n-in sulated w ires, and the wires were bonded along th e ir e n tire length on the sphere surface w ith epoxy. These thermocouple wires entered the in ne r sphere p o s itio n in g tube through a small hole and extended out o f the ou te r sphere, along w ith the heater leads, from the top end o f the p o sitio n in g tube. Power fo r the heater was provided by a DC power supply and a constant voltage transform er connected to the house supply. The Lambda Model LP-531-FM2 regulated DC power supply had a v a ria b le voltage range 1Manufactured by Watlow E le c tric Manufacturing Company, St. Louis, M issouri. 2 Manufactured by Lambda E le ctro n ics, M e lv ille, Long Island, New York.

64 55 o f 0-20 volts and a current range o f 45 ma A at ambient temperature s, and i t regulated the output voltages to w ith in 0.01% + 1 mv fo r lin e varia tio n s from VAC or fo r load changes from no load to f u ll load. Metered panels provided readings fo r both the voltage and the c u rre n t. The AC in p u t to the power supply was provided by a Sola 3 CVS constant voltage transform er connected to the house 125 VAC supply. The transform er had an output voltage o f 118 VAC. The power to the heater was also measured independently by 4 current and voltage readouts on a Fluke Model 8010A4 d ig ita l m ultim eter. The m ultim eter, when connected in series in the heater c ir c u it, measured the DC current with an accuracy o f + (0.3% of reading + 1 d ig it) and a resolution o f 1 ma, and when connected in p a ra lle l across the power supply output term inals, measured the DC voltage w ith an accuracy of + (0.1% o f reading + 1 d ig it) and a resolution o f 10 mv on the 20 v range th a t was used. Power supplied to the heater was varied fo r d iffe r e n t e x p e rimental runs by a d ju stin g the voltage u n til the desired power re q u ire ment was in d ica te d by the voltage and cu rre n t readouts on the meters. Sphere Support Assembly As described e a rlie r, the inner sphere was supported at the center o f the outer sphere by the positioning rod which extended out from a closely f it t in g hole in the outer sphere top plug. 3 Manufactured by Sola E le c tric, Elk Grove V illa g e, I llin o is. 4 Manufactured by John Fluke Manufacturing Company, In c., Mountlake Terrace, Washington.

65 The outer sphere support assembly is shown in Figure 15. The 56 hemispheres were joined at the flanges and positioned re la tiv e to each other by a 0.64 cm aluminum rin g f it t i n g closely around the outside surface o f both flanges. The flange o f the lower hemisphere rested on an aluminum supporting rin g, which was clamped to the positio ning ring above i t w ith 0.64 cm b o lts. The bolts were tightened by nuts on the positioning ring w ith the washers overlapping the surface of the upper hemisphere flange. A th in rubber gasket rin g was placed between the two flanges in order to provide a vapor tig h t seal. Occlusive seals were also ensured at the clearances around the guide plug ends by bonding th in rubber wiper rings around the holes on the inside surface o f the ou ter sphere. The sphere assembly was supported on two rota ta ble brackets on opposite sides to which the positio n in g and supporting rings were secured w ith bolts and nuts. The support bracket rods rested in holes in two hexagonal support stands fabricated from angle irons and iron bars in the Engineering Instrument Shop. The stands were clamped to a bench top. Rotation o f the sphere assembly along w ith the support brackets provided a c c e s s ib ility o f the required sampling location on the outer sphere. Temperature Measurement The temperatures on the inner and outer spheres were measured by thermocouples positioned on t h e ir surfaces, as discussed e a r lie r. The locations o f these thermocouple junctions were as shown in Table 1. (See Figure 11 fo r d e ta ils o f angular coordinates γ and φ.)

66 57 Figure 15. Outer Sphere Support Assembly

67 58 TABLE 1 LOCATIONS OF THERMOCOUPLES ON THE INNER SURFACE OF THE OUTER SPHERE AND THE OUTER SURFACE OF THE INNER SPHERE Angular coordinates o f thermocouple lo ca tio n s Inner Sphere Thermocouple Locations φ = 54 φ = 174 φ = 294 γ Outer Sphere Thermocouple Locations φ = 21 φ = 141 φ = 261 γ Local temperatures in the medium were measured w ith thermocouples fabricated from cm T eflon-insulated Chromel-Alumel wires. The thermocouple junctions lay at 1.27 cm in te rv a ls along radia l d ire c tio n s and were supported by cm diam eter, two bore, ceramic insula tin g tubes which were bonded to cm O.D. ceramic insulators extending from the inner sphere to the outer sphere. A thermocouple support assembly is shown in Figure 16. The thermocouple w ires, a fte r e x itin g the small diameter in sula to rs, were bonded along the surface o f the positio n in g in s u la to r from which they passed through holes in the nylon pins and came out of the outer sphere. The thermocouple supports, each w ith seven thermocouple ju n ctio n s, were positioned at four d iffe re n t locations in the medium,

68 59 Figure 16. Thermocouple Support

69 60 and the angular coordinates o f these measurement junctions are shown in Table 2. TABLE 2 COORDINATES OF MEDIUM THERMOCOUPLES Location γ (degrees) Φ (degrees) The accuracy o f temperature measurement w ith the Chromel- Alumel thermocouples was v e r ifie d by comparing the measured temperatures w ith those indicated by precision thermometers, and the uncertaintie s associated w ith the thermocouple measurements were found to be w ith in the range o f e rro r introduced by lin e a rly in te rp o la tin g between EMF versus temperature values given in the thermocouple conversion ta b le s. Therefore, no c o rre c tio n was applied to the temperatures in d i cated by the thermocouple EMF values. D is to rtio n o f the temperature fie ld caused by the presence of the thermocouple supports in the medium was found to be n e g lig ib le by Moore [7 0 ], and was, th e re fo re, not measured in the present study. Moore [70] also in s ta lle d thermocouples in the w all o f the in ne r sphere

70 61 positio n in g tube in order to measure the heat loss through the tube, and his re su lts indicated a minimal heat loss through the tube fo r values o f heat in p u t g re a tly higher than those used in these e x p e riments; therefore the p o sitio n in g rod heat loss was assumed n e g lig ib le. The thermocouple c ir c u it used to measure the thermocouple EMF's is shown schem atically in Figure 17. The thermocouple w ire s, a fte r coming out o f the ou ter sphere, were connected to term inals on two aluminum panels which were clamped w ith L - jo in ts to the top o f the ro ta ta b le brackets. 20 gage, PVC-insulated, stranded copper extension wires leading from the term inal panels were connected to the term inals 5 o f a 4 0 -p o sitio n, ro ta ry thermocouple selector switch5, which permitted sequential connections o f the thermocouples to a Laboratory S trip-c hart Potentiom etric Recorder6. 34 thermocouples were connected to the recorder through the switch in th is manner. The single-pen s trip -c h a rt recorder, w ith a 250 mm chart w idth, had switch-selected input spans o f 1 mv, 2 mv, 5 mv, 10 mv, 20 mv, 50 mv, 100 mv, 200 mv, 500 mv, 1 v, 2v, and 5v, and had chart speeds of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 cm/min and cm/hr. The zero was continuously ad ju sta ble from -100% to +100%. Most o f the thermocouple EMF's were measured w ith the 1 mv span w ith the zero set at 1 mv, and w ith the 2 mv span w ith the zero set at 1 mv to obtain the proper range o f voltages over the e n tire span. 5Model 0SW5-40, manufactured by Omega Engineering, In c., Stamford, Connecticut. 6Model , manufactured by Cole-Parmer Instrument Company, Chicago, I llin o is.

71 62 Figure 17. Schematic o f Thermocouple C irc u it

72 The o n ly lik e ly source o f e r r o r in the tem perature measurement could have been due to a net EMF in tro d u ce d in the c i r c u i t because o f 63 connections made between d is s im ila r w ire s. However,since connections between d is s im ila r w ire s were a t the same te m p e ra tu re, and the re fe re n ce ju n c tio n was d ir e c t ly connected to the te rm in a l p a n e l, the p o s s ib ilit y o f such an e r ro r was m inim ized. M o is tu re C ontent Measurement Local m o isture co n te n t v a lu e s, along th re e ra d ia l d ire c tio n s in the medium, were determ ined from a g ra v im e tric a n a ly s is o f samples e x tra c te d from the th re e lo c a tio n s by a sam pling d e vice. TABLE 3 COORDINATES OF MOISTURE CONTENT SAMPLING GUIDES L o ca tio n γ (degrees) φ (d egrees) Bottom Plug A fte r s te a d y -s ta te was reached in an experim ental ru n, a 1-27 cm core o f the porous medium was removed along a ra d ia l lin e w ith a s ta in le s s s te e l tube in s e rte d in the a n n u la r medium through the guide on the o u te r sphere. Then, the sam pling device shown in F ig ure 18 was in tro d u ce d through the guide to p e rm it e x tra c tio n o f e ig h t lo c a l samples o f the medium a t 1.27 cm in te r v a ls along the ra d ia l d ir e c tio n from the medium surrounding the core opening.

73 64 Figure 18. Moisture Content Sampling Device

74 65 The sampling device consisted o f a cm long, 1.27 cm O.D., and cm wall thickness, h a lf-c y lin d ric a l stain le ss steel tube, along the length o f which c irc u la r divid e rs were welded at 1.27 cm in te rv a ls to permit e xtra ctio n o f eight local samples from the medium by the e ig h t h a lf-c y lin d ric a l buckets so formed. The h a lf- c y lin d ric a l sampling tube was positioned by a cm steel rod soldered along its inside surface and extending through a p la s tic bushing inside a section of stain le ss steel tubing. f i t the guide on the outer hemisphere. The stainless steel tubing closely The outer end o f th is section extended beyond the end o f the guide to allow the h a lf-c y lin d ric a l buckets to be located and to provide fo r ro ta tin g the steel rod. The samples were weighed w ith a S a rto riu s e le c tro n ic balance.7 The balance has a capacity o f 160 g, re a d a b ility to 0.1 mg, and a sensitivity of 0.5 mg. The re la tiv e humidity in the laboratory was maintained close to saturation fo r a few hours p rio r to placing the medium in the sphere annul us at the beginning o f a run or e xtra ctin g the samples o f the medium at the completion o f a run, w ith a Sears Kenmore h u m id ifie r, in order to minimize loss o f moisture by evaporation from the moist porous medium. Temperature Control System I t was necessary to maintain a constant temperature on the outer sphere surface in order to implement the boundary conditions used fo r solving the model d iff e r e n tia l equations. 7 Model No. 2842, manufactured by Brinkmann Instruments, In c., Westbury, New York.

75 66 The constant temperature on the surface o f the outer sphere was maintained by enclosing the equipment assembly during an experimental run in a 76 cm x 76 cm x 76 cm wooden cabinet in which the a ir was contro lle d at a uniform temperature with the help o f a temperature control system. A block diagram o f the control system used is shown in Figure 19. I t consisted of a s o lid -s ta te time proportioning temperature c o n tro lle r8, with a 10-speed hot a ir blower9 as the control element. An Iron-Constantan thermocouple provided measurement of the controlled a ir temperature to the c o n tro lle r, and four 100 CFM box fans mounted on the inside w alls of the wooden cabinet maintained constant mixing of the a ir bath and uniform ity o f the controlled a ir temperature. A 43 CFM box fan mounted in the wall o f the wooden cabinet, when used w ith a variable autotransform er, provided a constant a ir throughput o f about 6 CFM, to remove the excess heat generated in the system. A room temperature control system maintained the a ir temperature va ria tio n in the laboratory w ith in a 3 C range, and i t was found tha t a temperature constancy of C fo r the a ir bath was achieved fo r the temperature con trolle d. However, the temperature of the outer sphere surface was held constant with a b e tte r uniform ity because o f its large thermal mass. The a ir bath temperature was continuously recorded fo r the in it ia l runs by a Chromel-Alumel thermocouple connected to a laboratory s trip -c h a rt potentiom etric recorder6 described e a rlie r. A mercury g Model No , manufactured by Cole-Parmer Instrument Company, Chicago, Illin o is. 9 Manufactured by Montgomery Ward Company.

76 Figure 19. Block Diagram o f Temperature Control System 67

77 thermometer in tro d u ced in the to p o f th e a ir bath provided independent measurement o f th e a ir tem perature in the c a b in e t. 68 E xperim ental Procedure The in n e r sphere was p o s itio n e d a t the c e n te r o f the o u te r sphere, which was then mounted on the support stand assembly as described e a r lie r. Thermocouple supports were in s ta lle d in the a n nula r space, and the leads extending o u t o f the o u te r sphere, as w e ll as the in n e r sphere therm ocouple leads emerging from the p o s itio n in g ro d, were connected to the copper hook-up w ire s on the te rm in a l panels. Plugs were in tro d u ce d in the two o u te r sphere guides to seal the openings. Porous medium was prepared by m ixing re q u ire d amounts o f glass beads and d i s t i l l e d w ater in e ig h t 1 g a llo n glass ja r s which were then allow ed to stand fo r a p e rio d o f 48 hours in o rd e r f o r the m oisture conte n t to a tta in u n ifo rm ity in each j a r. The sphere assembly was ro ta te d u n t il the p o rt in the low er hemisphere was a t the top fo r f i l l i n g. Small q u a n titie s o f the m o ist glass beads were in tro d u ce d in the annulus, w ith constant tapping on the o u te r sphere w ith a rubber hammer and evenly spreading out the f i l l e d medium w ith a s p a tu la, in o rd e r to pack the medium u n ifo rm ly. The f i l l i n g procedure re q u ire d about fo u r hours, and samples o f the medium were taken p e r io d ic a lly d u rin g the f i l l i n g procedure to a c c u ra te ly determ ine the i n i t i a l uniform m oisture conte n t o f the medium. The s ta rt-u p o f the experim ent was e ffe c te d by f i r s t p la c in g the wooden c a b in e t on the equipment assembly and a c tiv a tin g the a ir

78 69 bath tem perature c o n tro l system. A fte r the a ir tem perature had been c o n tro lle d a t a u n ifo rm value fo r a p e rio d o f 2 to 3 hours, the heater power supply was turned on. The c u rre n t through the h e ate r was increased s lo w ly to avoid breakage due to therm al shock. The tem perature a t a lo c a tio n in the medium was recorded c o n tin u o u s ly by the p o te n tio m e tric re c o rd e r, w ith a c h a rt speed o f 1 cm /hr. S te a d y -s ta te was reached in hours, depending on experim ental c o n d itio n s. A fte r th is temperatu re had remained co n sta n t f o r several hours, tem peratures a t a ll the lo c a tio n s in the medium and the in n e r and o u te r sphere surface temperatu re s were recorded by s w itc h in g w ith the therm ocouple s e le c to r s w itc h, a process th a t to o k about 15 m inutes to com plete. The h e a te r power supply and the tem perature c o n tro l system were then d e a c tiv a te d and the wooden c a b in e t l i f t e d from over the equipment to begin the m o istu re co n te n t sam pling o f the medium. The sam pling technique consiste d o f removing a core o f the medium w ith a s ta in le s s s te e l tube and then r o ta tin g the sphere u n t il the guide was in a h o riz o n ta l p o s itio n. The sam pling device was then in tro d u ced in to the opening from which th e core had been removed, and the buckets ro ta te d in a c o u n te r-c lo c k w is e d ir e c tio n f o r c o lle c tin g lo c a l samples o f th e medium around the opening. The sam pler was then p u lle d o u t w ith i t s open face v e r t ic a lly upward and the samples tra n s fe rre d w ith a s p a tu la to 10 ml s e a la b le v ia ls. This procedure was repeated f o r a ll the th re e sam pling lo c a tio n s. A fte r a ll the samples were c o lle c te d, the v ia ls were wiped clean from the o u ts id e and weighed. They were then p la ced, a f t e r removing the caps, in a d ry in g oven10 a t 105 C, 10Model S ta b il -Therm, m anufactured by Blue M E le c tr ic Company, Blue Is la n d, I l l i n o i s.

79 70 fo r about one hour. The v ia ls were then removed from the oven, allow ed to a tta in room tem perature, and weighed a fte r re p la c in g the caps. The porous medium samples were then removed from the v ia ls, which were again weighed in the empty s ta te. This procedure allow ed a g ra v im e tric d e te rm in a tio n o f the lo c a l m o isture conte n t values in the medium, which were then converted to v o lu m e tric measurements from a knowledge o f the medium d e n s ity. The tim e p e rio d, from the moment the h e ating and tem perature c o n tro l systems were d e a c tivate d to the tim e when sam pling o f the medium from the th re e lo c a tio n s was com pleted, was m inutes, an in te rv a l which was s h o rt enough to preclude more than a n e g lig ib le m o isture r e d is tr ib u tio n in the a n nula r medium, a fte r the hours re q u ire d fo r the system to reach steady s ta te. Thus, p o s s ib le e rro rs in the s te a d y -s ta te m o isture conte n t values determ ined f o r the samples must be a ttr ib u te d to c o lle c tio n and h a ndlin g o f the samples o n ly. The procedure was repeated f o r o th e r runs w ith the same i n i t i a l m oisture conte n t but d iffe r e n t heate r o u tp u ts. A fte r a s e rie s o f such runs was com pleted, the bottom plug was removed and the porous medium emptied from the a n n u la r space. The medium was placed in la rg e tra y s and d rie d in the oven. Broken thermocouples were re p la ce d, the sphere surfaces were th o ro u g h ly cleaned and d rie d, and the equipment was prepared f o r another s e rie s o f runs w ith a d iffe r e n t value o f i n i t i a l m oisture co n te n t.

80 CHAPTER 8 RESULTS AND DISCUSSION The re s u lts o f eleven steady-state experimental runs are presented and analyzed. Porous media components fo r the runs were nominally 250 μ ( μ) spherical glass beads and d is t ille d water. The i n i t i a l moisture contents o f the media and the applied heating rates are shown in Table 4. Excepting Run 1, three or more runs w ith d iffe re n t heating rates were made fo r each o f three i n i t i a l moisture contents. Steady-state temperature p ro file s measured on four d iffe re n t ra d ii along w ith inner and outer sphere temperatures, and moisture content p ro file s measured along three d iffe re n t ra d ii are shown fo r the eleven runs in Figures 20 through 41. TABLE 4 INITIAL MOISTURE CONTENTS AND HEATING RATES FOR EXPERIMENTAL RUNS Run No. I n it ia l Moisture Content, cm3/cm3 Heating Rate, Q, cal/day x x x x x x x x x x x

81 Figure 20. Local T vs. Radial P o s itio n --Run 1 72

82 Figure 21. Local 0 vs. Radial P osition--r un 1 73

83 Figure 22. Local T vs. Radial Position Run 2 74

84 Figure 23. Local 0 vs. Radial Position Run 2 75

85 Figure 24. Local T vs. Radial Position--Run 3 76

86 Figure 25. Local 0 vs. Radial P o s itio n Run 3 77

87 Figure 26. Local T vs. Radial P o s itio n Run 4 78

88 Figure 27. Local θ vs. Radial P o s itio n Run 4 79

89 Figure 28. Local T vs. Radial P osition--r un 5 80

90 Figure 29. Local 0 vs. Radial Position--R un 5 81

91 Figure 30. Local T vs. Radial Position--Run 6 82

92 Figure 31. Local 0 vs. Radial P osition--r un 6 83

93 Figure 32. Local T vs. Radial P o s itio n Run 7 84

94 Figure 33. Local θ vs. Radial P o s itio n Run 7 85

95 Figure 34. Local T vs. Radial Position--Run 8 86

96 Figure 35. Local 0 vs. Radial Position--R un 8 87

97 Figure 36. Local T vs. Radial P o s itio n Run 9 88

98 Figure 37. Local 0 vs. Radial Position--R un 9 89

99 Figure 38. Local T vs. Radial Position--R un 10 90

100 Figure 39. Local 0 vs. Radial Position--R un 10 91

101 Figure 40. Local T vs. Radial Position--R un 11 92

102 Figure 41. Local 0 vs. Radial P osition Run 11 93

103 94 Run 2 had a d is tr ib u tio n o f m oisture content values over the e n tire wetness range o f in te re s t and was th e re fo re selected fo r ca lcu la tio n o f the coupling c o e ffic ie n ts, β*, L wqand β using the methods de scribed in Chapter 6. The dashed curves in Figures 22 and 23, re p re senting "averaged" ra d ia l d is tr ib u tio n s o f temperatures and m oisture contents fo r Run 2, were obtained by least-square s f i t o f the mean lo ca l values by second order polynom ials. The c o e ffic ie n ts ca lcu la te d from these data were used to p re d ic t tem perature and m oisture content d is tr ib u tio n s fo r the o th e r runs. The s o lid curves in Figures 20 and 21 and in Figures 24 through 41 are the s te a d y-sta te p r o file s o f tem perature and m oisture content p re d icte d by the CSMP sim u la tio n program o f the model equations. The p re d icte d d is tr ib u tio n s showed good agreement w ith the e xp e rim e n ta lly observed tem perature and m oisture content p r o file s. Independent c a lc u la tio n s o f 3* values were made w ith data from Runs 4, 6 and 8, and p lo tte d as fu n c tio n s o f m oisture content in Figure 42. β* c a lc u la te d from Run 2 (see Figure 8), used to p re d ic t ste a d y-sta te p r o file s, are also shown in Figure 42 fo r comparison purposes. The d iffe r e n t curves fo r the β* vs. 0 re la tio n s h ip fo r the fo u r runs show reasonably close agreement, in d ic a tin g th a t 3* is a s in g le valued fu n c tio n o f 0 fo r the range o f flu x e s in v e s tig a te d. The β* vs. 0 curves are s im ila r in appearance to those observed by Gee [3 8 ] fo r s o il samples, although Gee's β* values fo r the s o il media are about 5 to 10 times those in Figure 42. Numerical e rro rs in the e va lu a tio n o f the temperature and m oisture content gradie n ts fo r subsequent c a lc u la tio n o f β* are probably responsible fo r the "unsmooth" appearance o f these curves.

104 95 Figure 42. Coupling C oefficient β* vs. Volumetric Moisture Content 0, fo r Runs 2, 4, 6, and 8

105 The extrem ely small values o f the coupling c o e ffic ie n t 3 (see Figure 10) in d ic a te n e g lig ib le in flu e n c e o f m oisture content gradients 96 on the heat flu x. As a numerical check, the model equations were also solved w ith a zero value f o r 3. and no change was observed in the pred ic te d tem perature and m oisture content p r o file s. This c o n s titu te s an in d ir e c t v e r ific a tio n o f the Onsager re la tio n s h ip since β* was ca lcu la te d from β* using the Onsager re la tio n. Surface tem perature u n ifo rm ity o f the two spheres and close agreement o f temperatures measured a t equal r a d ii fo r d iffe r e n t lo ca tio n s appear to support the assumption o f one-dimensional heat tra n s fe r. Local m oisture content determ inatio n by the g ra v im e tric method is subje c t to g re a te r e rro r than the tem perature measurement. Such e rro rs include the s p a tia l averaged nature o f the lo c a l m oisture content values, loss o f m oisture during h a ndlin g, and determ ination o f very small d iffe re n c e s in mass. The measured values o f m oisture co n tent, w h ile e x h ib itin g s c a tte r, are in reasonable agreement w ith p re d ic tio n s u t iliz in g the Cary and T a ylo r model. The agreement o f measured and p re d icte d tem perature p r o file s is b e tte r, but th is m ight be a ttrib u te d to the r e la tiv e unimportance o f coupling o f m oisture content g ra d ie n t to heat flu x and the g re a te r c e rta in ty associated w ith thermal conduct i v i t y values. The m oisture content data in d ic a te, however, th a t onedimensional tra n s fe r is not s t r i c t l y s a tis fie d since a general tre nd toward higher m oisture contents is obvious toward the bottom o f the annular medium. This observation m ight be q u a lita tiv e ly associated w ith g ra v ita tio n a l e ffe c ts.

106 CHAPTER 9 CONCLUSIONS AND RECOMMENDATIONS I. Methods have been developed fo r o b ta in in g the tra n s p o rt c o e ffic ie n ts in the Cary and Taylor equations, and a s o lu tio n technique was developed to o b ta in p re d ic tio n s o f the tem perature and m oisture content fo r a one-dimensional s te a d y-sta te system w ith zero mass flu x. A. A tra n s ie n t method has been used to determine the thermal c o n d u c tiv ity fo r the porous medium as a fu n c tio n o f m oisture content. The measurements were made in a time frame which precluded s ig n ific a n t m oisture tra n s fe r, thus o b v ia tin g the d i f f i c u l t y o f assigning thermal c o n d u c tiv ity values which do not r e fle c t the coupling between heat and m oisture tra n s fe r. B. The coupling c o e ffic ie n t β *, which measures the independent e ffe c t o f tem perature g ra d ie n t on liq u id m oisture f lu x, has been determined f o r a nom inally 250 μ glass bead- water unsaturated system, β *, although s tro n g ly dependent on the m oisture co n te n t, appears not to be a fu n c tio n o f the gradie nts o f m oisture o r tem perature fo r the range o f flu xe s stu d ie d. This conclusion is based on comparison o f c a lc u la te d values o f β* from runs w ith heating ra te s d iffe r in g by a fa c to r o f about two, and on the success o f p re d ic tio n o f ste a d y-sta te tem perature and m oisture p r o file s fo r d iffe r e n t boundary conditio n s using β* determined fo r one heating ra te. 97

107 98 C. The coupling c o e ffic ie n t β, which measures the independent e ffe c t o f m oisture content g ra d ie n t on heat f lu x, appears to be e s s e n tia lly zero fo r th is system, based on a p p lic a tio n o f Onsager's re la tio n s h ip and on num erical e va lu a tio n o f the e ffe c t o f β on p re d icte d ste a d y -s ta te tem perature and m oisture content p r o file s. D. The IBM Continuous System Modeling Program has been used to p re d ic t the ste a d y -s ta te tem perature and m oisture p r o file s in a one-dim ensional, s p h e ric a lly sym m etric, closed porous medium system. The thermal c o n d u c tiv ity, d if f u s iv it y, and coupling c o e ffic ie n t β* are tre a te d as fu n c tio n s o f m oisture content. I I. Comparison o f s te a d y-sta te experim ental data w ith model pred ic tio n s showed reasonable agreement, in d ic a tin g v a lid it y o f the Irre v e r s ib le Thermodynamics model over the range o f parameters studied. I I I. Precise g u id e lin e s regarding the a p p lic a b ility o f the theory o f lin e a r Irre v e r s ib le Thermodynamics to o th e r systems o f th is nature fo r both s te a d y-sta te and tra n s ie n t c o n d itio n s are not y e t known and need to be e s ta b lis h e d. A. The importance o f h yste re sis is not c le a rly understood and needs in v e s tig a tio n. B. A th e o re tic a l basis is needed fo r e s ta b lis h in g the ranges o f m oisture contents and temperatures in which liq u id - dominated flo w, which is assumed in the present fo rm u la tio n o f Cary and Taylor model, a p p lie s.

108 99 C. For high liq u id m oisture c o n te n ts, there is an in d ic a tio n th a t g ra v ity e ffe c ts may be im p o rta n t. Extension o f the p re d ic tiv e model to include such e ffe c ts should be in s tr u c tiv e. D. V a lid a tio n o f the Cary and T a ylo r model fo r processes in u n satu rate d porous media f o r open systems should be attem pted. E. F in a lly, the Irr e v e r s ib le Thermodynamics basis fo r the Cary and T a ylo r model needs c a re fu l s c ru tin y. I t appears th a t a more conceptually c o rre c t d e riv a tio n should id e n tify a p o te n tia l which includes surface e ffe c ts in the Gibbs Equation and th a t the id e n tific a tio n o f chemical p o te n tia l in the thermodynamic fo rce conjugate to mass flu x in th is system may not be rig o ro u s.

109 NOMENCLATURE A b constant exponent; g raph ically, the negative of the slope of the log Y -t curve, sec-1 C. mass concentration o f component i, g / g c sp e cific heat, cal/g C D c o e ffic ie n t o f d iffu s iv ity of liq u id water in an unsaturated porous medium, cm2 / day g g ravitatio nal constant, cm/sec2 H height, cm h surface heat tra nsfe r c o e ffic ie n t, cal / sec cm2 C Hi p a rtia l enthalpy o f component i, cal / g thermodynamic flu x of i th species Jq thermodynamic heat flu x, cal / cm day Jq heat flu x (= JQ - Σ J i Hi ), cal / cm2 day Jw flu x of liq u id water, g / cm day J0(x) Bessel function of f i r s t kind and zero order of x J1(x) Bessel function of f i r s t kind and f i r s t order of x K Ks hydraulic con ductivity, cm / day saturated hydraulic conductivity, cm / day 100

110 101 Ksc Κ (θ)i c a lc u la te d satu rated c o n d u c tiv ity, cm / day c a lc u la te d h y d ra u lic c o n d u c tiv ity fo r a s p e c ifie d water content o r pressure L le n g th, cm Li k Liq Lq i Lqq Lqw phenomenological c o e ffic ie n ts L, L wq ww M Mi n P p Q Qw mass o f system, g mass o f component i, g to ta l number o f pore classes 2 pressure, dynes / cm2 parameter th a t accounts fo r in te ra c tio n o f pore classes heat r a te, cal / day 3 volu m e tric flo w ra te, cm /d a y R1, R2 ra d ia l d ista n ces, cm r r m ra d iu s, cm radius o f c y lin d e r, cm S entropy, cal / C S s p e c ific entro p y, cal / g C T tem perature, K To i n i t i a l uniform tem perature o f sample, C T1 tem perature a t ra d iu s R1, K

111 102 T constant ambient water bath temperature, C a Trm t U V v Vi temperature of cylinder wall time variable sp e cific internal energy, cal / g spe cific volume, cm3 / g mass average v e lo c ity, cm / sec ve lo city o f component i, cm/sec x1 a constant (= 2.405) i th thermodynamic force Y a ra tio o f temperature difference, dimensionless thermal d iffu s iv ity, cm2 / sec 3 coupling c o e ffic ie n t (= d ψ/ d In T), cal / g β* coupling c o e ffic ie n t (= do/dlnt), dimensionless γ colatitude angular coordinate, degrees Δ notation fo r change Ϭ surface tension of water, dynes / cm ε porosity, cm3 /cm3 η visco sity o f water, g / cm sec 0 volumetric moisture content, cm3 water / cm3 medium 01 moisture content at radius R1, cm3 water / cm3 medium λ thermal conductivity, cal / cm sec C P chemical p ote te n tial, l, cal/q cal/g

112 103 μw chemical potential o f water, cal / g chemical potential o f free water, cal / g p density, g / cm3 density o f component i, g / cm3 ρw σ σ' Φ ψ density o f water, g / cm3 rate o f entropy production, cal / g C day volum etric rate o f entropy production, cal / cm3 C day la titu d e angular coordinate, degrees water p o te n tia l, dynes / cm2 gradient operator, cm-1

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122 APPEND IX CSMP PROGRAM FOR NUMERICAL INTEG RATIO N OF IR R E V E R S IB LE THERMODYNAMICS MODEL EQUATIONS