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Analytical and numerical studies of heat and moisture transfer through porous insulation Zheng, Bi-Feng 1993

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)ANALYTICAL AND NUMERICAL STUDIES OF HEAT ANDMOISTURE TRANSFER THROUGH POROUS INSULATIONByBI-FENG ZHENGB. Sc. (Thermal Science and Engineering) Zhejiang University, ChinaM. Sc. (Thermal Science and Engineering) Zhejiang University, ChinaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1993© BI-FENG ZHENG, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of Mechanical EngineeringThe University of British Columbia2324 Main MallVancouver, CanadaV6T 1Z4Date:^0 cto bex^9)AbstractThis work contains both analytical and numerical studies of heat and moisture transportthrough a porous insulation in the presence of condensation, with impermeable, adiabaticvertical boundaries, and with one horizontal boundary facing a warm humid ambient andthe other facing a cold impermeable surface.The analytical model is developed for heat and water vapor transfer in flat-slab andround-pipe thermal insulations. The model is validated by comparing its predictionswith available experimental data. The effective thermal conductivity of the insulationin the presence of condensation depends on seven design and operating variables. Theeffect of these variables is determined by a parametric study. For practical operatingconditions, the effective thermal conductivity varies from about 1.5 to 15 times the dry-state value. The computed data are presented in the form of design curves which may beused to estimate the effective thermal conductivity for fiat-slab and round-pipe insulationsystems.The analysis in this work quantifies the process of energy and mass transport in aporous insulation. A rigorous and fundamental formulation of heat and mass transfer inthe insulation system is presented. The problem is modeled as one-dimensional, tran-sient, multiphase flow with variable properties. Four stages in the energy and moisturetransport process are identified, and they are formulated by a system of transient inter-coupled equations and several thermodynamic relations using a local volume averagingtechnique. The numerical results are compared with experimental data for five differentoperating conditions and for times up to 600 hours. The model predicts the tempera-ture distribution, heat transfer rate, the total moisture gain successfully. The predictedliquid distributions agree well with measured data for a period of up to 70 hours. Theinteresting effects of pertinent parameters on the energy and moisture transfer in theporous insulation are investigated. The present study, which for the first time presentsa full simulation of the problem considering the mobile condensate, can be applied toother classes of problems on heat and mass transfer with phase change through a porousmedium.iiiTable of ContentsAbstractList of Tables^ viiiList of Figures ixNomenclature^ xiiiAcknowledgement xvii1 INTRODUCTION 11.1 Background ^ 11.2 Physical Process Description ^ 21.3 Geometrical Characteristic of Fibrous Insulation ^ 51.4 Motivation of the Present Study ^ 52 A BRIEF REVIEW OF LITERATURE 82.1 Background ^ 82.2 Experimental Studies ^ 102.3 Theoretical Studies 122.4 Scope and Objectives of the Present Study ^ 143 MATHEMATICAL DESCRIPTION 163.1 Introduction ^ 163.2 Continuum Approach ^ 16iv3. Volume Average Technique ^Basic Assumptions ^Governing Equations Constitutive Correlations ^171819204 QUASI-STEADY ANALYTICAL MODEL 224.1 Background ^ 224.2 Analysis 224.3 Nondimensional Form^ 244.4 Solution Procedure 264.4.1^Dry Region ^ 264.4.2^Wet Region 274.4.3^Wet-Dry Interface^ 284.4.4^Trial and Error Procedure ^ 294.5 Heat Transfer ^ 304.5.1^Heat Flux at Cold Plate ^ 304.5.2^Effective Thermal Conductivity ^ 304.6 Round-Pipe Insulation ^ 314.7 Concluding Remarks 335 ANALYTICAL RESULTS AND DISCUSSIONS^ 345.1 Introduction ^  345.2 Comparison with Experimental Results ^  345.3 Comparison with Literature Reports  365.4 Effects of Condensation on Heat Transfer ^  385.5 Concluding Remarks ^  44v6 TRANSIENT NUMERICAL MODEL^ 466.1 Introduction ^  466.2 Analysis and Formulation ^  476.2.1 General Formulation  476.2.2 Stage 1: Initial Process  ^506.2.3 Stage 2: Immobile Liquid Accumulation ^  516.2.4 Stage 3: Mobile Liquid Diffusion ^  536.2.5 Stage 4: Long Term Liquid Accumulation and Diffusion ^ 566.3 Solution Methodology ^  566.3.1 Discretized Formulations ^  576.3.2 The Solution Algorithm  636.4 Numerical Considerations ^  677 NUMERICAL RESULTS AND DISCUSSIONS^ 697.1 Introduction ^  697.2 Physical Data  697.3 Diffusion and Condensation Processes ^  707.3.1 Initial Process  ^707.3.2 Quasi-steady State Period ^  737.3.3 Liquid Accumulation and Diffusion ^  737.4 Comparison with Experimental Results  757.4.1 Temperature and Heat Transfer ^  767.4.2 The Moisture Gain and Liquid Transport ^  767.5 Thermal and Transport Performances ^  857.5.1 The Variation of Heat Flux  857.5.2 Effect of Humidity Levels ^  87vi7.5.3 Effect of Convective Heat Transfer Boundary Conditions ^ 887.5.4 Effect of Slab Thickness ^  887.5.5 Effect of Porosity  ^937.5.6 Comparison of Different Thermal Conductivity Models ^ 938 CONCLUSIONS AND RECOMMENDATIONS^ 958.1 Analytical Study ^  958.2 Numerical Study  968.3 Recommendations^  97Bibliography^ 98Appendices 102A Water Vapor-Air Mixture Diffusion^ 102A.1 Vapor Diffusion Coefficient in Porous Media ^  102A.2 Mass Transfer Coefficient in Ambient ^  103A.3 Calculation of Vapor Concentration in Ambient ^  104B Thermal Conductivity of Moist Insulation 105viiList of Tables5.1 Comparison of Predicted and Measured [23] Heat Flux ^ 355.2 Comparison of Predicted and Measured [21,22] Effective Thermal Conduc-tivity  ^387.1 Physical Data. ^  70viiiList of Figures1.1 Schematic Diagram of Physical Problem ^31.2 Four Stages in Energy and Moisture Transport Process ^43.1 Macroscopic Average Volume Element ^ 184.1 Schematic Diagram of Round-Pipe Insulation.  325.1 Temperature distribution in slab; Comparison with experimental Data [23]. 355.2 Vapor Flux Distribution in slab, Comparison with experimental data [23]. 365.3 Comparison of analytical and numerical results, Conditions: L = 0.099 m,T. = 20°C, kdry = 0.037 W/m K, h* = 12 W/m2 K^  375.4 Variation of heat flux with temperature difference, T. = 40°C. ^ 395.5 Variation of heat flux with temperature difference, T. = 30°C. ^ 395.6 Variation of heat flux with temperature difference, Ta = 20°C. ^ 405.7 Variation of effective thermal conductivity ratio with temperature differ-ence, T. = 40°C ^  405.8 Variation of effective thermal conductivity ratio with temperature differ-ence, T. = 30°C ^  415.9 Variation of effective thermal conductivity ratio with temperature differ-ence, Ta = 20°C ^  425.10 Effect of h on the variation of the effective thermal conductivity ratio. . ^ 435.11 Effect of K on the variation of the effective thermal conductivity ratio. ^ 435.12 Effect of e on the variation of the effective thermal conductivity ratio. . ^ 44ix6.1 Grid-point Cluster for One Dimensional Transport Problem: (a). ControlVolume for the Internal Points, (b). Control Volume for the BoundaryPoints^  576.2 Grid-point Cluster for the Interface Location of Wet-dry Regions ^ 617.1 Temperature Distribution in the Initial Stage; p* = 53 kg/m3, L = 66mm, RH . 96%, h* = 12 W/m2 K^  717.2 The Distribution of Vapor Density in the Initial Stage; p* = 53 kg/m3,L . 66 mm, RH = 96%, h* = 12 W/m2 K^  717.3 The Time Variation of Condensation Rate in the Initial Stage; p* = 53kg/m', L = 66 mm, RH = 96%, h* = 12 W/m2 K. ^ 727.4 The Time Variation of Volumetric Liquid Fraction in the Initial Stage;p* = 53 kg/m3, L = 66 mm, RH = 96%, h* = 12 W/m2 K^ 727.5 Temperature Distribution in the Quasi-steady State Period, Comparisonof Numerical, Analytical and Experimental Results [22]; p* = 53 kg/m2,L = 66 mm, RH = 96%, h* = 12 W/m2 K^  747.6 The Time Variation of Liquid Fraction over a Long Term Period; p* = 53kg/m2, L = 66 mm, RH = 96%, 17,* = 12 W/m2 K, 71, = 33°C , T2 = 6.8°C. 747.7 The Temperature Distributions in Run 1, Comparison of Numerical Re-sults with Experiment Data [24]; p* = 53 kg/m2, L = 62.02 mm, RH =^97%, h* = 12 W/m2 K    777.8 The Temperature Distributions in Run 2, Comparison of Numerical Re-sults with Experiment Data [24]; p* = 53 kg/m', L = 68.65 mm, RH =^96.5%, h* = 12 W/m2 K    78x7.9 The Temperature Distributions in Run 3, Comparison of Numerical Re-sults with Experiment Data [24]; p* = 53 kg/m', L = 70 mm, RH = 96%,h* = 12 W/m2 K^  797.10 The Temperature Distributions in Run 5, Comparison of Numerical Re-sults with Experiment Data [24]; p* = 53 kg/m3, L = 76.13 mm, RH =96%, h* = 12 W/m2 K    807.11 The Heat Flux at the Cold Plate, Comparison of Numerical Results withExperiment Data for Four Runs [24]; p* = 53 kg/m3, h* = 12 W/m2 K. . 817.12 The Total Moisture Gain, Comparison of Numerical Results with Experi-ment Data for Four Runs [24]; p* = 53 kg/m3, h* = 12 W/m2 K ^ 837.13 The Liquid Distribution in Slab at Different Times, Comparison of Numer-ical Results with Experiment Data [24]; Run 1. p* = 53 kg/m3, h* = 12W/m2 K, L = 62.02 mm ^  847.14 The Variation of Heat Flux at Cold Boundary with Times; p* = 53 kg/m',h* = 12 W/m2 K, Ta = 33°C, T2 = 6.8°C, RH = 90%. ^ 867.15 The Variation of Equivalent Thermal Conductivity along the InsulationSlab; p* = 53 kg/m3, h* = 12 W/m2 K, T = 33°C, T2 = 6.8°C, RH = 90%. 867.16 The Variation of keff/kdry at Cold Boundary and Total Liquid Concentra-tion under Different Humidity Levels; p* = 53 kg/m3^ 897.17 The Variation of Iceffikh.v at Cold Plate and Total Liquid Concentrationunder Different Convective Heat Transfer Boundary Conditions; p* = 53kg/m', T. = 33°C, T2 = 6.8°C, L = 66 mm, RH = 90%. ^ 907.18 The Variation of Iceff/kdrv at Cold Boundary and Total Liquid Concen-tration under Different Slab Thickness; p* = 53 kg/m3, T. = 33°C,T2 = 6.8°C, h* = 12 W/m2 K, RH = 90%    91xi7.19 The Variation of Iceff/kdry at Cold Boundary and Total Liquid Concentra-tion under Different Porousities; p* = 53 kg/m', T. = 33°C, T2 = 6.8°C,L = 66 mm, h* = 12 W/m2 K, RH = 90%  927.20 The Variation of keff/kdry at Cold Surface Using Different k Models; p* =53 kg/m3, T. = 33°C, T2 = 6.8°C, L = 66 mm, h* = 12W/m2 K, RH = 90%. 94x i iNomenclaturea^constant in Clausius-Clapeyron relation, Eq. (4.26)a coefficient in linear discretized equationsb^constant in Clausius-Clapeyron relation, Eq. (4.26)bi constant in Equation (4.16)b2^constant in Equation (4.16)B Biot number, h*LIk*ffB,„^mass transfer Biot number, km* L/Dffcp dimensionless heat capacity at constant pressureC(z)^dimensionless concentration of vapourC. dimensionless concentration of vapour in ambientCc^dimensionless concentration of vapour at dry-wet interfaceDt, dimensionless vapor diffusion coefficientDi^dimensionless liquid diffusion coefficient defined in Eq. (6.26)Dfi dimensionless liquid diffusion coefficient defined in Eq. (3.4)D12^binary diffusion coefficient of Fick's law [m/s2]f(9) dimensionless function for concentrationl(0)^df/d8Fo t. Fourier number, L2/ao.,effg(T)^Clausius - Clapeyron relation [kg/kg(dry air)]G dimensionless condensation rate defined in analytical studyG*^dimensionless condensation rate defined in numerical studyh dimensionless heat transfer coefficient in ambienth19^ enthalpy of vaporization [kJ/kg]hm dimensionless mass transfer coefficient in ambientdimensionless thermal conductivitykdr y^ dry thermal conductivity [W/mke f f effective thermal conductivity defined in Eq. (4.29) [W/m K]keg^ equivalent thermal conductivity defined in Eq. (7.1) [W/m K]K(w) hydaulic conductivitylength of slab [m]Le^ Lewis number, a* ID:Leg equivalent slab length [m]m(x)^vapor concentration [kg/kg (dry air)]ma vapor concentration in ambient [kg/kg (dry air)]m2^saturation concentration at temperature T2 [kg/kg(dry air)]Pa dimensionless partial pressure of airPal^dimensionless total pressure of gas phasesPv dimensionless partial pressure of vapor— P5^dimensionless parameters defined in Eq. (6.10)Ra air gas constant [N m/kg K]vapor gas constant [N m/kg K]dimensionless heat flux at cold surface4^heat flux at cold surface [W/m2}dimensionless heat fluxQ'^dimensionless heat flux in Reference [38]radius of pipe insulation [m]To^outer radius of pipe insulation [m]3dvri^inner radius of pipe insulation [m]T(x) temperature distribution [K]To^reference temperature in Clausius-Clapeyron relation [K]Ta ambient temperature [K]T2^ temperature of cold surface [K]S liquid saturation^ volume of porous insulation [m3V„, main stream velocity [m/s]w^weight liquid contentx distance along slab [m]y^transformed variable [m]z dimensionless distancezd^dimensionless dry lengtha0,eff* effective thermal diffusivity, kcjil4C0P^dimensionless groupE volumetric fraction€o^porosity (void fraction)T vapour flux [kg/m2 s](1)^dimensionless vapour fluxPa density of air [kg/m3]I'^condensation rate per unit volume [kg/m3 s]T tortuosity factorB(z)^dimensionless temperature distribution19e dimensionless temperature at the dry-wet interfaceXVSubscripts0^initial value2 value at cold surfacea^ambienta air phasedryeast side control volume faceeff^effectiveeast side grid pointi-th phasecentral grid pointref^referencevapor phasewest east side control volume facewest side grid pointliquid phase7^gas phaseo- solid phaseSuperscriptsdimensional valuevalue in previous time stepxviAcknowledgementI wish to express sincere gratitude to Dr. M. Iqbal and Dr. E. G. Hauptmann for theirsupervision and continuous encouragement through all stages of this work. Financialassistance from the National Science and Engineering Research Council of Canada isgratefully appreciated.My thanks go to Dr. Wijeysundera from The National University of Singapore whogenerously offered the experimental data. His valuable suggestions, comments, and dis-cussions during his sabbatical leave at the Department of Mechanical Engineering, UBCwere also very helpful.This thesis is dedicated to my wife, Wei-Hung, her understanding and support madethe completion of this work possible.xviiChapter 1INTRODUCTION1.1 BackgroundHeat and mass transfer in porous media accompanied by phase change is a phenomenonwhich occurs frequently in nature and in many engineering applications. These applica-tions are in the area of chemical , environmental, mechanical and petroleum engineering,geology and others. In recent years, heat and moisture transfer through fibrous insulationhas drawn considerable attention among many investigators because of applications inthe energy management of buildings, and heated and refrigerated building envelopes.Fibrous insulation is used within building sections, such as partition walls, ceilings,roofs, floors etc, to separate the heated space from the unheated space. In air conditioningsystems, fibrous insulation is commonly used to insulate the chilled water pipes and coldair ducts. In these applications, vapor barriers are applied to prevent water vapor ingressfrom the ambient. However, because of factors such as poor installation, cracks developwith age, particularly near joints, clamps and supports. The water vapor will migrateinto the insulation. Water vapor condensation can take place anywhere in a porousinsulation when the vapor density is greater than the saturation vapor density whichcorresponds to the local temperature at that point. The condensation effect is especiallyprominent when the insulating material is exposed to large temperature differences andhigh humidity environments. As condensation occurs, the liquid phase resulting fromcondensation will cause a significant increase in heat transfer across the insulation and1Chapter 1. INTRODUCTION^ 2hence it affects the thermal performance of the insulation. The long term effects ofcondensation are corrosion of the metallic components of the insulated system and theeventual destruction of insulation.1.2 Physical Process DescriptionThe heat and moisture transfer in porous insulation is generally a complex multidimen-sional problem including vapor transport by diffusion and convection, flow of liquid dueto gravity and capillary action, and condensation or freezing accompanied by release orconsumption of latent heat. The solution of this entire problem is not attempted here.Instead, the problem is reduced to a one-dimensional configuration which still retainsmost of the important parameters of the original problem.The one-dimensional configuration investigated here is shown in Figure 1.1 An initialdry fibrous insulation has two boundaries which encounter two different environments:one of the boundary faces a cold, impermeable plate with temperature T2, the otherboundary is exposed to a warm humid ambient air, with temperature T., humidity RH,vapor concentration C., convective heat transfer coefficient h., and mass transfer coeffi-cient hm.The water vapor will migrate into the insulation slab due to the vapor concentrationgradient and the thermal gradient, and it is expected to exhibit four stages of transportprocesses as shown in Figure 1.2.The first stage is a relatively short initial transient stage in which the temperature andvapor concentration fields are developing within the insulation slab. During this process,a very small quantity of liquid water accumulates in the porous medium. Condensationis defined here as the accumulation of liquid beyond the adsorption process in which thevapor reaches the saturation concentration at a given temperature. The region in whichChapter 1. INTRODUCTION^ 3Warm, Humid Ambient AirTa, Ma, h, hmFigure 1.1: Schematic Diagram of Physical Problem.the air-vapor mixture is saturated is referred to as the wet region where condensationtakes place. The wet region is established , and the wet and dry region is separated bythe wet-dry interface after this stage.In the second stage, the heat and vapor transfer processes reach a quasi-steady state,and the temperature and vapor concentration fields are invariable with time. Liquid isaccumulated in the wet region, however the amount of liquid content is still low and doesnot have a significant effect on transport properties.When liquid accumulation exceeds a value high enough as to affect the propertiessignificantly, the liquid starts to move due to generated liquid pressure and will flowtowards the wet-dry interface and towards the drier, warmer surface by capillary action.The wet-dry interface is moving as the wet region expands due to the liquid outflow.In the last stage, the wet-dry interface eventually reaches the exposed surface, TheT2Dry RegionTa # # # # # # #Chapter 1. INTRODUCTION^ 44.1/ ./..••^ //// It- CondensateDry InsulationT2 ^P.Wet-dry InterfaceTa + +^ Ta ++44+4+444(a) Initial Stage: Condensation^(b) Quasi-steady Stage:at Cold Plate.^ Liquid Accumulation inthe Wet Region.(c) Liquid Diffusion Stage:^(d) Long Term Stage:Liquid Diffusion Starting Liquid Accumulation andfrom Cold Surface. Diffusion in Entire Slab. Dry RegionLiquid Diffusion RegionWet RegionFigure 1.2: Four Stages in Energy and Moisture Transport Process.Chapter 1. INTRODUCTION^ 5effects of liquid accumulation and liquid flow become dominant. The accumulation ofliquid in the insulation is believed to have two important effects: (1). decrease in thelocal vapor diffusion coefficient, (2) increase in local thermal conductivity of the slab.These factors will both influence the heat transfer and condensation rate in the slab.1.3 Geometrical Characteristic of Fibrous InsulationThe Fiberglass batts used as thermal insulation consist of layers of Fiberglass filamentsthat are held together with a phenolic binder. The batts usually have a very high voidfraction (95-99 percent). The Fiberglass filaments have varying diameters with an averagevalue of 3.56(10)-6m. Each Fiberglass layer differs from its adjacent layers and has adifferent void fraction. The Fiberglass filaments are dispersed randomly in each layer.However, the number of fibers along the batt is slightly larger than that across the batt,and the number of fibers that are perpendicular to the batt and run from one layer tothe other is very small.Generally speaking, fibrous insulation is an inhomogeneous assemblage of Fiberglassfilaments and phenolic binder. The insulation is characterized by an anisotropic fiberdensity and a nonuniform void fraction.1.4 Motivation of the Present StudyA detailed study for the entire transport process in fibrous insulation is complicated bythe following factors:• Heat and mass transport in porous insulation is a problem with multiphase flow.In general a wet porous insulation consists of three phases: the solid matrix, theliquid water, and a binary gas phase composed of air and water vapor.Chapter 1. INTRODUCTION^ 6• The transport mechanisms involved in the process are quite complicated. Energytransport in such a medium occurs by conduction in all of the phases as well asby convection with those phases which are able to move. In addition, there isheat transfer caused by phase change at the interface between liquid and gas. Masstransfer occurs within the voids of the medium. In an unsaturated state, these voidsare partially filled with a liquid, whereas the rest of the voids contains air and watervapor. In the gas phase, there is vapor diffusion due to the vapor concentrationgradients, bulk convection due to the density variation induced by temperaturegradients, and air infiltration due to the small difference in gas pressure acrossthe insulation. The flow of liquid is caused by internal forces, such as capillary,intermolecular and osmotic forces, and external forces, such as imposed pressuredifference, and gravity.• The transport properties involved vary strongly with structure of the porous medium,moisture content and temperature.• It is almost impossible to study quantitatively the transport processes for the ir-regular void configurations which exist in general in porous insulation. It is stilldifficult even for a regularly shaped matrix.Thus there is indeed a need for a rigorous and extensive investigation of the heat andmass transport processes in porous insulation. This may lead to better estimation ofinsulation properties for building design purposes.In the present work, the focus is on the analysis of the process of moisture accumu-lation and transport in the fore-mentioned insulation system, and as well as the effect ofmoisture on thermal performance of insulation. The entire process of energy and masstransport in the porous insulation system is quantified and the significant transport mech-anisms are identified. A simplified formulation for four stages of the transport process isChapter 1. INTRODUCTION^ 7presented to simulate the transient and spatial variation of the pertinent variables. Ananalytical model is also presented to perform a parametric study on the thermal per-formance of porous insulation in the presence of condensation. Both the analytical andtransient numerical model are validated by comparing with experimental data. Finallythe interesting effects of variation of pertinent parameters on the energy and moisturetransfer are investigated.Chapter 2A BRIEF REVIEW OF LITERATURE2.1 BackgroundThe importance of heat and mass transport in a porous medium is well appreciated. Theearly treatments of liquid flow and heat transfer in a porous medium were more empiricalrather than rigorously theoretical. The initial macroscopic treatments on water flowthrough a porous medium were made in the 1880's [2]. The porous medium, which is aheterogeneous system made of a solid matrix with its void filled with gases and liquids, canbe treated as a continuum by properly accounting for the role of transport through eachphase in this system of phases. The attempt at microscopic (pore-level) transport studiesbegan in the early 1900's [2]. The pore-level analysis offered a better understanding ofheat and mass transport, however the description and solution of a transport problemat the microscopic level is impractical and, perhaps impossible because the geometry ofthe surface that bounds the phase is not observable and is too complex to be described.The macroscopic level approach, at which continuous and differentiable quantities maybe determined, is needed to model and predict the heat and mass transport in a porousmedium. Progress was achieved in the second half of this century with the more rigorousapproaches of local volume averaging techniques [3].Early studies on transport in porous insulation were mainly on the motion of liquidthrough unsaturated porous media. In early papers, heat transfer was not given sufficientattention until Krischer (1940) [4]. Probably one of the most significant contributors to8Chapter 2. A BRIEF REVIEW OF LITERATURE^ 9heat and mass transfer in porous media, Krischer first considered seriously the intimaterole that transport of energy may play in moisture transfer in a porous medium. Priorto Philip and DeVries (1957) [5], moisture movement was explained by a simple diffusiontheory, the flux being given by Fides law. Philip and DeVries (1957) [5] and DeVries(1958) [6] extented the theory and generalized the effects of capillary flow, vapor transportand energy transfer, and represented the transport due to capillary forces in terms ofgradients of moisture content and temperature. Their work consolidated most of theprevious knowledge on the influence of temperature gradients on moisture movementand laid the groundwork for a better understanding of simultaneous heat and watermovement in soil.Extensive research on heat and mass transfer in porous media has been done bythe Soviet scientist, Luikov (1966, 1975)[7, 8]. He described heat and mass transfer ascaused both by the temperature and the liquid concentration gradient, and developed amechanistic approach for the representation of simultaneous heat and moisture transferin dying processes. A set of linear mass and heat transport equations were derived.However, the effect of phase change was not considered and the dependence of transportcoefficients on temperature and concentration were neglected.Whitaker (1977) [3] presented general formulations of simultaneous heat, mass andmomentum transfer in porous media. The intrinsic phase averaged quantities were definedat the appropriate level and the continuum point equations which are averaged weredescribed.Although simultaneous heat and moisture transfer in a porous medium has beenstudied by many other researchers [9, 10, 11], most of the work has been directed towardsstudies of soils and sands. Very little attention has been given to fibrous insulatingmaterials until the last fifteen years.Chapter 2. A BRIEF REVIEW OF LITERATURE^ 102.2 Experimental StudiesThe experimental investigation of porous insulation has focused mainly on two aspects:1). the effect of moisture gain by porous insulation on thermal conductivity of and heattransfer through the insulation; 2). transport of liquid condensate within insulation.Jespersen (1953) [12] measured the thermal conductivity of a 62 kg/m3 glass fiber,using a steady heat flow method, but with a very small gradient of temperature acrossthe sample in order to get a uniform moisture distribution. Joy (1957) [13] worked on96 kg/m3 fibrous insulation, using a transient heat flow method. Their results show asignificant increase of thermal conductivity even for a very low moisture contents.Similar measurements to Jespersen and Joy's work have been performed by Langlais(1982) [14], using a classic steady-state method. The thermal conductivity of a highdensity mineral fiberboard was determined as a function of moisture content. However,their results show that thermal conductivity is far less affected for low moisture content.They explained that their differences with Joy's results are due to the non-uniform mois-ture distribution under the influence of the temperature gradient. Bomberg and Shirtliffe(1978)[15] also demonstrated the influence of this redistribution on thermal conductivity.The moisture redistribution process was also studied experimentally by Kumaran (1987,1988) [16, 17] in an insulation slab under the influence of a temperature gradient.Langlas et al. (1983) [18] and Langlais and Klarsfeld (1984) [19] further studied theeffects of moisture transfer through an insulation slab under two sets of boundary condi-tions and pointed out that the rapid variations in thermal conductivity values were dueto moisture phase change and diffusion. Thomas et al. (1983) [20] measured the moistureand temperature distribution , as well as the thermal conductivity versus time. Modiand Benner (1985, 1986)[21, 22] measured the moisture gain of spray-applied fiber-glassand cellulose insulation slabs from the surroundings and also the effect of the moistureChapter 2. A BRIEF REVIEW OF LITERATURE^ 11gain on the thermal conductivity.Wijeysundera et al. (1989) [23, 24] conducted two series of experiments to study thediffusion and condensation of water vapor in fiber-glass. The first set of experimentswere tests on fibrous insulation. In the second series of experiments, one of the facesof the insulation slab was exposed to a warm and humid ambient and the other onewas maintained at a low temperature. The temperature and liquid distribution, thetotal moisture gain, the heat flux were measured for a range of experimental conditions.Further experimental studies on effects of moisture gain by a fiber glass insulation slabin a second set experimental conditions were reported by Wijeysundera (1992) [25]. Themajor heat and moisture transfer parameters were measured over a period of up to 600hours.The transport of the liquid condensate within fibrous insulation has received lessattention. Motakef and El-Masri (1985) [26] presented a model for the isothermal liquiddiffusion in fibrous insulation in the absence of gravitational forces. The model relatedthe liquid diffusion to the characteristics of the insulation, such as void fraction, spatialdistribution, directionally index, tortuousity factor etc. Experimental data was alsoreported to verify the model. Timusk and Tenende (1988) [27] studied the capillary riseof water in fiber glass slabs. Cid and Crausse (1990) [28] measured the coefficient of liquiddiffusion which was correlated to moisture content and the nominal density of the solidphase, based on the principle of attenuation of two radioactive emissions. Although thethermal effects were not added to the measurements, so doing would makes it possibleto predict the behavior of the materials when they are submitted to moisture.Extensive field studies of heat transfer through wet insulation have been reported byHeldlin (1988) [29], during all seasons of the year.Chapter 2. A BRIEF REVIEW OF LITERATURE^ 122.3 Theoretical StudiesAs mentioned in the beginning of this Chapter, simultaneous heat and mass transferhas been extensively studied for various systems [3-11]. However, these studies havebeen only recently extended to heat and moisture transfer in insulation . Dinulescu andEckert (1980) [30] analyzed the moisture migration in a slab of an unsaturated porousmedium between two impermeable surfaces.The problem of condensation in insulation was first studied by Ogniewicz and Tien(1980) [31]. The condensation process in insulation is characterized in terms of threeregimes, and a quasi-steady model was presented to studied the condensation effectsin the second regime, where coupling between temperature and concentration of thecondensing vapor was taken into account.Motakef and El-Masri (1986) [32] presented a one-dimensional analytical model forheat and mass transport with phase change in a porous slab similar to Oganiewcz andTien [31], and the closed-form approximate solutions for two limiting regimes of con-densate diffusivity were reported. The same problem was also modeled by Vafai andSarkar(1986) [33] as a transient, multiphase flow, with variable properties. The formula-tion was based on the local volume-averaging technique for each phase to come up withthe governing equations for the condensation process in fibrous insulation. However, thesimplifying assumption that liquid accumulation is small was made in arriving at thesolution.Shapiro and Motekef (1989) [34] extended their analysis to unsteady transport pro-cesses and reduced the unsteady process to that of quasi-steady fields in time-dependentdomains corresponding to mobile and immobile condensates. The model underpredictsthe energy transfer in the wet zone and the range of validity of the solutions is limited,especially for the mobile condensate.Chapter 2. A BRIEF REVIEW OF LITERATURE^ 13Vafai and Whitaker (1986) [35] first reported a two-dimensional unsteady numericalwork with simplifying assumptions. Vafai and Tien (1988, 1989) [36, 37] performed amore thorough numerical simulation of the two-dimensional multiphase transport processin a porous medium. Information was presented on the variation and the intercouplingeffects of the important field variables, the effects of some parameters such as humid-ity levels, the porosity etc. on the condensation rate, liquid accumulation and energytransfer. However, No attempt was made to verify numerical solutions by comparisonwith experimental data. The computational results presented in the paper only showthe transport process within the first 500 seconds. In such a short initial period, theabsorption process of bounded liquid is still in the initial stage, the effects of tempera-ture gradient and condensation are not significant, and the contribution to the heat andmass transfer in insulation is small considering the long term impact of accumulation andtransport of the liquid condensate through the insulation. In addition to, the treatmentused in their scheme to determine when condensation should be taken into account lackseither experimental validity or theoretical support.Wijeysundera et al. [23, 25] used a semi-empirical model to interpret some of theirexperimental results. The analytical model developed for transport process during thequasi-steady phase has given good predictions of heat and vapor flow through insulationfor low level of moisture transfer.Tao et al. (1991) [38] analyzed the moisture and frost accumulation in a glass-fiberslab. An approach similar to the study of Vafai and Sarker [32], who used the localvolume average technique, was applied to the case with temperature below the triplepoint of water. A numerical simulation was performed for one-dimensional, transient,vapor diffusion with phase changes and variable properties.Chapter 2. A BRIEF REVIEW OF LITERATURE^ 142.4 Scope and Objectives of the Present StudyIt is clear from the literature survey that the transfer processes in insulation can bequantified by a). field and laboratory studies, b). analytical/numerical studies. Generallyspeaking, the field and laboratory studies are time consuming due to the slow process ofmoisture transfer. Furthermore, it is difficult to cover the effects of such a wide range ofoperating and design parameters.The available analytical studies mainly dealt with the quasi-steady phase of the trans-fer processes. Although several investigators have reported good predictions with theirmodels, the performance of insulation in the presence of condensation has not been suf-ficiently analyzed. From a design stand-point, it is important to study the effects ofmoisture on thermal performance of insulation under practical operating conditions.Less work has been done to simulate numerically the full transport processes whichthe insulation may undergo for a long time period in the presence of condensation. Thelong term effects of condensation on the performance of insulation has not been studiedtheoretically.The present study focus on the analytical and numerical investigation of the transferprocesses through fibrous insulation in the presence of condensation. The objectives ofthis study are1). To perform a detailed and parametric study on effects of condensation on thethermal performance of insulation in the quasi-steady phase. The analytical modelsbased on the quasi-steady models by Wijeysundera et al. [22] will be extended to coverthe main design and operating parameters on which the effective thermal conductivitydepends, with the hope that the modified model will be able to to predict the mostimportant heat and mass transfer performance for engineering design and applicationpurposes.Chapter 2. A BRIEF REVIEW OF LITERATURE^ 152). To perform a numerical simulation study that covers all the phases mentionedin Chapter one, and also uses measured physical property variations available in theliterature. The numerical results will be compared with the experimental data by Wijey-sundera et al. {22,24 Accordingly, the analytical and numerical models will be derived,and the computer codes developed.Chapter 3MATHEMATICAL DESCRIPTION3.1 IntroductionThis chapter presents the background for the general mathematical models that describetransport of heat and mass in porous insulation at a macroscopic level. The generalformulation of this problem is based on the local volume-averaging technique [3] for mass,momentum, and energy equations for each phase which make up the governing equationsfor heat and mass transport process in insulation. The general governing equations forthe transport of energy, water vapor and liquid, together with constitutive relations aredescribed. The limitations and restrictions of the governing equations are stated and thevalidity of the assumptions are discussed.3.2 Continuum ApproachPorous insulation is a multiphase system, where the various phases are separated fromeach other by an abrupt interface. It is difficult to formulate the transport problemincluding the interface boundaries between each phase. The conventional treatment isto replace the real system, where the multiphases together occupy disjoint subdomainswithin a porous medium, with a model in which each phase is assumed to behave as acontinuum over the whole domain.With the continuum approach, the average values of phase variable are taken overelementary volumes, centered at the point. By assigning average values to every point,16Chapter 3. MATHEMATICAL DESCRIPTION^ 17the variables which are differentiable functions of the space coordinates can be obtained.The continuum approach is thought reasonable since it circumvents the needs tospecify the exact configuration of the interface boundaries, and it also is useful in solvingfield problems of practical interest due to its measurable quantities. The informationregarding interface boundaries and the actual variation of quantities within each phasestill remain in the form of coefficients. The numerical values of these coefficients must bedetermined experimentally for a specific porous media, in the laboratory, or in the field.3.3 Local Volume Average TechniqueThe local volume technique is applied by associating with every point in the porousmedium an averaging volume V that is bounded by a closed spatial surface A, as shownin Fig. 3.1. In general, the volume V is composed of three phases. These are: the solidphase V,, the liquid phase Vfi(t), and the gas phase V.,f(t). At any location in space z, aquantity y is said to be spatially averaged when it is defined as1< y > (z) = v(z) fv(z)ydV.A quantity in phase a is said to be intrinsic phase averaged when it is defined as(3.1)1<y >a= —v (t) iva(t)yadV,^ (3.2)where ya is zero in phases other than the a phase, and <y >a is the mean value of y inVa. In this study, the averaging symbols are omitted in order to simplify the notation.The derivation of the general governing equations for heat and mass transfer in aporous insulation is based on Whitaker's work [3]. Several assumptions are made toarrive at the governing equations.Liquid Phase (13)Chapter 3. MATHEMATICAL DESCRIPTION^ 18Figure 3.1: Macroscopic Average Volume Element.3.4 Basic Assumptions(1). The fibrous insulation is homogeneous and isotropic. This assumption is the com-mon simplying procedure to rationally tackle the problems for heat and mass transfer inporous materials. The structure of the porous media is difficult to characterize. Motakefet al. [26] has been able to correlate diffusion of liquid using the void fraction, pore sizedistribution, tortuosity and so on. It seems clear that describing the parameters char-acterizing the structure will be a most difficult task for porous insulation, therefore thepresent theoretical development is restricted to the case where the structure parametersare independent of the spatial coordinates.(2). The solid-liquid-gas system is in local thermal equilibrium. Since the heat andmass transport process investigated here, as well as most of drying processes, are char-acterized by no or relatively low convective transport rates, under these circumstancesone could therefore assume that conductive transport is sufficient to eliminate significanttemperature differences between the separated phases.Chapter 3. MATHEMATICAL DESCRIPTION^ 19(3). No convective gas phase flow occurs in the insulation matrix; any moistureaccumulation is caused by vapor diffusion only. In the present physical model, the effectsof air infiltration are not included. This restriction might be removed in further studies.(4). The total gas phase pressure in the insulation matrix is constant. This assump-tion is usually justified for drying process and building envelopes.5). The gravitational effect on the liquid motion is negligible. Under the assumptions(3) and (4), the gas and liquid pressure distribution can be considered to be hydrostatic.Thus the forces exerted on the liquid phase are of two kinds: surface tension and gravityforces. For porous insulation with very small pores, the effect of gravity can be neglecteddue to the large surface tension forces. The experimental observation by Wijeysunderaet al. [25] showed that after long term testing up to more than 300 hours, no liquid wasobserved to drip from a totally wet insulation slab.3.5 Governing EquationsThe derivation of governing equations requires considerable algebraic manipulation. Asa result, they are taken directly from Whitaker's work [3]. The governing equations forwater vapor, liquid, and energy transport are given as the following:Vapor diffusion equation: a^op:r _ a(c-r4) -[D:;(11)-a-;1 - - at*Liquid transport equation:a^*^act;*aeg,PA-ai[DA(w)Til+ r PA at**Energy equation:(3.3)(3.4)Chapter 3. MATHEMATICAL DESCRIPTION^ 20aT—a [e(w)—(371 j + hfgr = p`c* .Ox^ax^P at. (3.5)3.6 Constitutive CorrelationsVolumetric Constrain: The volume fractions for the three phases are defined ase„ = 11,7711,^Eg = vo (t)/ v,^el, = 14/V(t).^(3.6)Therefore the sum of these fraction is one,fa-Fe9+ey=1.^ (3.7)Thermodynamic Relations: Some thermodynamic relations are needed to connect the thermal energy equationwith the vapor diffusion equation. By treating the gas phase as ideal gas, the air andvapor density can be determined by^p: = Rep:T,^ (3.8)^P: = Rv P:71 .^ (3.9)The vapor pressure-temperature relation for the vaporizing gas can be represented bythe Clausius-Clapeyron equation:h191^1P: = P:oexP[—(— —^)i-Re T Tref (3.10)Chapter 3. MATHEMATICAL DESCRIPTION^ 21The limitation of the Clausius-Clapeyron relation is that the gas-liquid interface is as-sumed flat, that is surface tension effects are not significant. When the effects of curva-ture and surface tension are important, the Kelvin equation may be used to representthe vapor pressure-temperature relation [45].Assumption (3) gives the other relation for gas phase pressures:P: = Ptotal P:. (3.11)The above three governing equations and five correlations can be used to solve thefollowing eight unknown: pa, pa, egg, c, T, r, p„, pa. The effort required to solve thecoupled transport equation is significant, although the numerical methods to solve theseequations are well known. What appears to be extremely difficult is that special modelsneed to be constructed so that the various transport stages in the problem can be studiedproperly, and comparison of the theoretical model with experiments in order to deter-mine the parameters that appear in the transport equations. The general mathematicaldescription of transport process needs to be extended to a series of specified theories thatadequately describe the whole heat and mass transport process. Also we need some the-oretical and experimental basis for estimating the values of the parameters which appearin the transport equations.Chapter 4QUASI-STEADY ANALYTICAL MODEL4.1 BackgroundExperimental studies [23, 24] have shown that after a short initial transient period, theenergy and vapor transfer processes reach a quasi-steady state. The numerical simulationsby Tao et al. [38] have shown that quasi-steady conditions are reached within about onehour for practical situations. During this quasi-steady phase, heat flux and therefore theeffective thermal conductivity attain high values.In this section a detailed analysis is developed for energy transfer during the quasi-steady phase. Expressions for heat flux and effective thermal conductivity are derivedfor the flat-slab system. These are easily extended to a round-pipe system by a simpletransformation of coordinates.4.2 AnalysisA schematic diagram of the slab is shown in Fig. 1.1. The outer surface of the insulationis exposed to warm humid ambient air from which heat and water vapor flow towards thecold surface. These flows are caused by the temperature and vapor pressure gradientswithin the insulation. When the local vapor concentration at a point exceeds the satu-ration concentration corresponding to the local temperature, condensation occurs. Thewet region is established and separated from dry region by the wet-dry interface. Thevapor that reaches the impermeable surface condenses at this surface.22Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 23In addition to the basic assumptions made in Chapter Three, the following assump-tions are made in the analytical model.(1) The liquid phase is immobile and its effect on properties such as the vapor diffusioncoefficient and the thermal conductivity is negligible. Since the quasi-steady state isreached in a short time (about an hour) for practical situations [38], the quantity of liquidcondensed is relatively small, and it will therefore have little effect on the properties.(2) All properties are independent of temperature. This assumption is justified be-cause the temperature range of the present study is limited to about 0-40°C. The prop-erties of the slab are therefore, evaluated at the mean temperature.(3) The initial adsorption process is not taken into account because of its relativelysmall contribution to the total liquid fraction; only the liquid accumulation due to con-densation in the wet-region is considered.Subject to the above assumptions, the governing equations of energy and mass transferfor the quasi-steady state may be written as:d2Tle— + 'lig = 0,dx2andd2m r = O.dx2In the dry region the condensation rate r . 0 and Equations (4.1) and (4.2) areuncoupled. In the wet region the vapor concentration m is a function of the local tem-perature.The boundary conditions at the exposed surface, x = 0, of the slab are:d– k* T—dx = h* [T„ – T(0)],(4.1)(4.2)(4.3)Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 24and- p:D:,—dm = h:n[rna — m(0)].dx (4.4)At the impermeable surface, x = L, the boundary conditions are:T(L) = T2 and m2 = g(T2).^ (4.5)4.3 Nondimensional FormIn order to generalize the equations, the following nondimensional variables are intro-duced.=z XL9 ^T —T2 — Ta—T2Dimensionless distance,Dimensionless temperature,a _ 171- m 2 ma—m2B h* L= k*Bm = 11LAID:nr, — P:DZ (In a—m2)I L_ _AL_— lc* (Ta—T2)T.L- p:D.(rna—m2)Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 25Dimensionless concentration,Heat transfer Biot number,Mass transfer Biot number,Dimensionless condensation rate,Dimensionless heat flux,Dimensionless vapou flux,Ratio of latent heat to conductive heat, # p:D:h f kna—m2)— lc* (Ta—T2)The meaning of the symbols is defined in the Nomenclature.The governing equations in terms of the dimensionless variables are:d2B+ GO = 0,dz2and(4.7)(4.6)d2C+G=0.dz2(4.8)Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 26The boundary conditions become: at z = 0,d9— —dz = B[1— 0(0)j,^ (4.9)and— —dC = — (4.10)B„,[1^C(0)];dzat z = 1,0(1)^=^0,(4.11)1C(1)^=^0.4.4 Solution Procedure4.4.1 Dry RegionThe solution of Equations (4.7) and (4.8) in the dry-region are obtained by setting thecondensation rate G = 0 and using the following conditions at the wet-dry interface. AtZ = zd,O(zd) = C.^0(zd) = Oc, 1^(4.12)The expressions for the temperature and concentration in the dry region are as follows:—[1 — Oc]z + Pc/ B + zd] 0(z) —  ^ (4.13)[zd + 1/13]and—41 — Cclz + [Cc/ Bm + zd] C(z) —^ (4.14)^[zd + 1/13,7,]^•Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 274.4.2 Wet RegionEliminating the condensation rate G between Equations (4.7) and (4.8), the total energyequation can be obtained as:d20 ad2Cdz2 + ''' dz2 = °' (4.16)where i3 is a nondimensional group which may be interpreted as the ratio of latent heattransfer due to vapor flow to that of sensible heat transfer.Since in the wet region the vapor is saturated, the vapor concentration is a uniquefunction of local temperature, C = 1(0); the form of the function f(9) will be givenlater in this section. Therefore, Equation (4-15) can be directly integrated to obtain thetemperature distribution in the wet region as0 + fiC(0) = biz + b2,^ (4.16)where the concentration C is a function of the local temperature, bi and b2 are constantsof integration.The following boundary and interface conditions are used to obtain b1 and b2.1Z = zd, 0 = 0c, and C = f (0 c);z =1, 8 = 0, and C = 0.This gives the temperature distribution in the wet-region as:(4.17)1 — z0 + Of(0) = [ec + 0 f (9 c)1( 1 — Xcl )(1 < z < zd).^(4.18)Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 284.4.3 Wet-Dry InterfaceBoth temperature distributions given by Equation (4.13) for the dry region and Equation(4.18) for the wet region contain two unknowns, Oc and zd. This requires a careful studyof the wet-dry interface.At the dry-wet interface the concentration, C = AO), therefore we have:dC^dedz I.=^ dz I. • (4.19)Substituting in Equation (4.19) for^and ficz- from Equations (4.13) and (4.14), thefollowing expression is obtained for the dry region length zd[1— f(0c)]/B —[f(0c)(1-0c)]/B,„, (4.20)zd^[P(0,)(1 — 8) — (1 — 1(8))]^•By applying the following interface condition:dO^de ,6-1.; Itvet= (7E; Wry)^ (4.21)this gives the equation[9.+ fif(9.)1 ^(1 —Or) (1 — zd)[1 ger(Oc)]^zd + 1 / B^ (4.22)•Thereforezd +1/B — [1 + f(Oc)](1 — OciSubstituting for zd from Equation (4.20) in Equation (4.23), the following nonlinearequation is obtained for the interface temperature Oc.(1 +1/13)  [1 —f(Oc)]^1^1^Oc-f-Af(Oc) (1 --0c).= (1 +1/B,n) P(Oc) -F (B Brn )( 1+10P(9c) ].^ (4.24)1 — zd _ ^Oc fii(Oc) (4.23)Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 29The function f(0) in Equation (4.24) is the temperature dependence of the saturationconcentration. In dimensional form this is given by1^1g(T) = a exp[b( — )],^ (4.25)which is the Clausius - Clapeyron relation.The values of the constants a, b, and To for the temperature range 0 58°C are [24a = 0.0232,^b = 5271.2,^and To = 302K.In nondimensional terms, Equation (4.25) has the formaexp[b(A ^^e(Ta-T2 )4-T2 )1^M2ma — 7712 ma — 7712Note that at the impermeable surface, 8 = 0, and AO) = 0.An examination of Equation (4.24) shows that Bc depends on three nondimensionalnumbers B, Bfl, and fl. However, Equation (4.26) reveals that f(9c) depends on theoperating conditions Ta, T2 and ma. When these variables are specified, the solution ofEquation (4.24) for the unknown Oc is easily accomplished by a trial and error procedure.Once Oc is found, all the important heat transfer parameters can be obtained directly.4.4.4 Trial and Error ProcedureThe trial and error procedure for solving the interface temperature 9c is as following:1). Assuming an initial value for the dry-wet interface temperature 9, the localsaturated vapor concentration and its first derivation are obtained from the Clausius-Clapeyron relationship. These values are substituted into Equation (4.20) to obtain thedry region length zd.2). Applying the boundary condition 9 = 83 at the cold plate, one can obtained thewet region length from Equation (4.18).f(19) = (4.26)Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 303). The total slab length L is the sum of dry and wet region lengths. The computedslab length L is compared with the actual slab length.4). If the computed slab length does not match the actual slab length, the Oc value isadjusted and the procedure from step 1 to 3 is repeated until the computed slab lengthmatches the actual slab length.4.5 Heat Transfer4.5.1 Heat Flux at Cold PlateThe heat flux at the cold surface is given bydT^dmqL.—[k*--Fhigp:D:dx (4.27)It should be noted that the heat flux includes both sensible heat transfer and the latentheat released at the impermeable surface due to condensation.Using Equations (4.18) and (4.20), the dimensionless heat flux is obtained as:[19c + Pf(9.)][(1 — f(9c))— f'(9)(1 — 9c)] 41, = [(1 + 1/B,„),P(0,)(1 — Oc)— (1 + 1/B)(1 — f(9c))).4.5.2 Effective Thermal Conductivity(4.28)The equivalent thermal conductivity of the slab in the presence of condensation is definedby the same expression that is used for a dry slab. This is given byIn the nondimensional form_ 1^L41,(1-17 +^f.) = (Ta — T2). (4.29)Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 3191{1+ (kekdry )1 = 1. (4.30)B^ffSubstituting for qL in Equation (4.30) gives the following expression for (keff/kdry) interms of Oc:Iceff^ [9c + flf(ec)][1— f(e9c)— f(8c)(1— 9c)} k^[Oc + 13 f (9c) — (1 + 11 Bm)jfi(e9c)(1 — Oc) — [Oc + 13 f (I) — (1 + 1/B)(1 — f(Oc))]•(4.31)4.6 Round-Pipe InsulationFor the round-pipe insulation system shown in Fig. 4.1 , moisture diffusion is assumedto occur uniformly across the circumference of the cross-section. Making the same as-sumptions as for the flat-slab, the governing equations of heat and vapor transfer can bewritten as:—d[rk*—dT]-1- higrr = 0,dr^dr(4.32)and—d [rp* D* —dm ] — rr = 0.dr " dr(4.33)Consider the transformation:y = ro inPi,^ (4.34)rwhere 7.0 is the outer radius of the insulation. Applying the transformation (4.34), Equa-tions (4.32) and (4.33) become(4.35)Chapter 4. QUASI-STEADY ANALYTICAL MODEL^ 32Wet-dry InterfaceTa, ma, h, hmFigure 4.1: Schematic Diagram of Round-Pipe Insulation.andc-1-[*D* —dm]- r(-7-7 . o.dpy " dy^roThe boundary conditions at the outer surface in the transformed system are:at r = 7-0, i.e y=0k* —dT = /eV'. — T(0)],dyand(4.36)(4.37)p:D:dm = h:i[mc, — m(0)].^ (4.38)dyIt can be seen that in the dry region where r . o, Equations (4.35) and (4.36) are thesame as for a flat-slab as are the boundary conditions (4.37) and (4.38). In the wet-regionthe combined energy equation obtained by eliminating II(,'; )2 between Equations (4.35)and (4.36) is the same as Equation (4.15). Therefore it is clear that the final expressionsChapter 4. QUASI-STEADY ANALYTICAL MODEL^ 33derived for the heat flux and effective thermal conductivity for flat slab can be appliedto round-pipe insulations by taking the equivalent flat-slab thickness asroLeg = ro ln( —)where ri is the inner radius of insulation.(4.39)4.7 Concluding RemarksThe analytical expressions for temperature distribution, heat flux and effective thermalconductivity in the presence of moisture gain have been derived. The main design andoperating parameters on which the effective thermal conductivity depends are identified.The results of the parametric study using the above expressions and the verification ofthe model will be presented in the next Chapter.Chapter 5ANALYTICAL RESULTS AND DISCUSSIONS5.1 IntroductionThe quasi-steady model presented in the previous chapter was used to perform a detailedparametric study. It is useful to examine carefully the variables on which the heat fluxand the effective thermal conductivity depend. The main dimensionless groups involvedare B, B,, and 0. The Lewis number for heat and mass transfer at outer surface may beassumed to be approximately unity [39]. With this assumption the dimensionless groupsdepend on three basic characteristics of the insulation viz, the thermal conductivity, theporosity and the thickness. The other variable is the outside heat transfer coefficient.The parameter (3 and the function 1(8), depend on the ambient temperature, the ambientrelative humidity and the cold surface temperature.In view of the above seven independent design and operating variables involved, it isdifficult to represent the results in the most general form. Therefore for ease of applicationthe results are presented in dimensional form, covering ranges of parameters that are ofpractical interest.5.2 Comparison with Experimental ResultsThe predictions made by the analytical model derived in previous chapter for the heatflux and temperature distribution were compared with the experimental data presented inRef. [23] for a flat-slab. The comparison of the measured [23] and computed temperature34Wet Region Line^tAnalYlleal resultsSymbols :Experimental data DryoWet RegionDryChapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 35^ RH: 96%- RH: 80%RH: 70%Wet Region 'N•,0.25^0.50^0.75^1.00Nondimensional Distance along the Slab Z/LFigure 5.1: Temperature distribution in slab; Comparison with experimental Data [23].T.(°C)Tcold(°C)RelativeHumidity (%)Heat FluxMeasured (W/m2)Heat FluxPredicted (W /m2)33.0 6.8 96 35.0 38.4632.0 6.8 90 32.0 3433.1 6.8 80 30.2 32.9533.2 6.8 70 27.4 29.56Table 5.1: Comparison of Predicted and Measured [23} Heat Fluxdistribution is shown in Fig. 5.1. The conditions are as following: T. = 30°C, T2 = 6.8°C,dry density of insulation slab p. 53 kg/m2, L = 66 mm, and the relative humiditiesare 96%, 80% and 70% respectively. The computed temperature distributions agree wellwith the measurements. The computed wet-dry interface is also indicated in the figure.The comparison of the computed and measured heat flux at the cold plate is shownin Table 5.1. There is satisfactory agreement between the calculated and measured heat0.001.000.750.500.250.00flux.Chapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 361.00e 0.90/". 0.80.2 0.700.600.500.00^0.25^0.50^0.75^1.00Nondimensional Distance along the Slab ZILFigure 5.2: Vapor Flux Distribution in slab, Comparison with experimental data [23].The comparison of predicted and measured vapor flux in the slab is shown in Fig.5.2. The conditions are the same as in Fig. Comparison with Literature ReportsFigure 5.3 shows a comparison of the predicted heat flux using the present analyticalmodel with the numerical results of Tao et al. [38]. The nondimensional heat flux wasdefined by Tao et al. [38] as the ratio of sensible heat flux with condensation in the slabto the dry-state heat flux with the same temperatures across the slab. This heat fluxis shown as Q' in Fig 5.4. The slight discrepancy between the results can be attributedto the physical properties used. The present analytical model assumes constant meanphysical properties while the numerical model [38] uses variable properties that dependon the liquid content.In the present study the heat flux at the cold plate was defined by Equation (4.29) .2.5Present analytical ModelNumerical Model [38]5^10^15Cold Plate Temperature T2 (C)1= 0.50.00 20Chapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 37Figure 5.3: Comparison of analytical and numerical results, Conditions: L = 0.099 m,T. = 20°C, kdry = 0.037 W/m K, h* = 12 W/m2 K.This includes the sensible heat flux and the latent heat released by the vapor condensingat the cold plate. The variation of the nondimensional total heat flux is indicated as Q inFig. 5.3. It is seen that the magnitude of Q is more than twice that of Q'. The variationof the total heat flux with cold surface temperature will be discussed later in this section.Modi and Benner [21, 22] measured the effective thermal conductivity of flat-slab in-sulations for different relative humidities and two values of the cold surface temperature.There is considerable scatter in their experimental data. However, there is an identifi-able quasi-steady regime in the graphs. The quasi-steady values of the measured, andcomputed values of (Iceff/kdry) are given in Table 5.2. There is satisfactory agreementwhen the cold surface temperature is 3°C, but for a cold surface temperature of 10°Cthere is some disagreement.Chapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 38T.(°C)Td(°C)RelativeHumidity (%)ke f f I kdryMeasuredIce f f I kdryPredicted30 3 70 2.22 2.3730 3 50 1.75 1.80730 3 30 1.375 1.28530 10 70 1.72 2.5030 10 50 1.4 1.7230 10 30 1.0 1.02Table 5.2: Comparison of Predicted and Measured [21,22] Effective Thermal Conductivity5.4 Effects of Condensation on Heat TransferThe variation of the heat flux with the temperature difference is shown in Figs. 5.4-5.6,for different values of the ambient relative humidity and slab thickness.The variation of the (icaffikdry) with the temperature difference is shown in Figs. 5.7- 5.9, for different values of the ambient relative humidity and slab thickness.It is instructive to explain in physical terms the shape of these curves. Consider apractical situation where the ambient temperature and the ambient humidity are main-tained constant and the cold surface temperature is decreased progressively starting froma value close to the ambient temperature. When the cold surface temperature T2 is suchthat the saturation concentration, m(T2) > ma, the vapor concentration in the ambient,no vapour diffusion will occur. The value of (ken /kdry) is then equal to unity and theheat flux is only due to sensible heat transfer. When m(T2) < ma, vapour will diffusethrough the dry slab and condense at the cold surface. This small quantity of condensedliquid may be assumed to have negligible influence on the transport processes in theporous slab. The heat flux leaving the impermeable cold surface will include both thesensible heat flux and the latent heat flux due to condensation. The heat flux at thecold surface increases with the temperature difference due to the increased sensible heat150Ta ■ 40°Ck ■ 0.035 W/m Xh ■ 8 W/m2XE ■ 98%00 5 10 15 20 25 30 35 40Temperature Difference Between Warm And Cold SidesTa ■ 30°Ck ■ 0.035 W/m Xh ■ 8 W/m2ICE ■■ 98%RH■90%L■25 mm•, 40%,7500 5 10 15 20 25 30Temperature Difference Between Warm And Cold SidesChapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 39Figure 5.4: Variation of heat flux with temperature difference, T. = 40°C.Figure 5.5: Variation of heat flux with temperature difference, T. = 30°C.rig 30g 204010500 0^5^10^15^20Temperature Difference Between Warm And Cold SidesTa ■ 20°Ck ■ 0.035 W/m Kh 8 W/m2 KE ■ 98% L■25 mm-%,- • •'^.....-^„...• 50%,...".., .--^..^ .. .... .......-• ,^..^L■75 mm 7.. ,^.. ...^. RH-90%...^..' ^..^. ...•^,..-^.....,..^,-^...-. . ...-Chapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 40Figure 5.6: Variation of heat flux with temperature difference, Ta = 20°C.0^5^10^15^20^25^30^35^40Temperature Difference Between Warm And Cold SidesFigure 5.7: Variation of effective thermal conductivity ratio with temperature difference,= 40°C.Chapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 415.0-r1V0 0.0oa .14$4.22.0Ei1 .000.0 0^5^10^15^20^25^30Temperature Difference Between Warm And Cold SidesFigure 5.8: Variation of effective thermal conductivity ratio with temperature difference,Ta = 30°C.transfer and the increased vapour flux. The ratio (Iceffilcdry) also increases due to thesame reason.At the turning point P in Fig. 5.7 and the corresponding point Q in Fig. 5.4, thevapour condensation is just beginning to occur within the insulation slab. At this pointthe wet-dry interface is just at the cold surface and the dry length zd given by Equation(4.20) is equal to one. When the temperature difference is increased further, the wet-dryinterface moves towards the exposed surface i.e. zd decreases progressively. Figures. 5.4-5.6 show that the heat flux increases continuously with increasing temperature difference.However, the rate of increase of the total heat flux is reduced because of the decreasein the function I'M with temperature. Due to the different rates of variation of and(T. — T2), the value of (ke f f kdry) increases up to P and then decreases slowly.The definition of ke f f is based on Equation (4.29), which is the design energy equationR11■50%Chapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 422500V M2.0a0•.-10 g 1•5g 2. Eig tooOZ0.0L■50 mm ___/^------ _ _ _L■75^_______________Ta • 20°Ck • 0.035 W/m Xh 8 W/siXE ■ 98%0^5^10^15^20Temperature Difference Between Warm And Cold SidesFigure 5.9: Variation of effective thermal conductivity ratio with temperature difference,Ta = 20°C.for dry insulation systems. This definition facilitates the inclusion of the effects of con-densation in more broader energy conservation studies by the use of a modified thermalconductivity value. This definition, however, leads to the somewhat unusual variation ofkeff with temperature difference as discussed earlier.The heat flux and the effective thermal conductivity increase with decreasing slabthickness. As expected higher ambient temperatures and humidities lead to higher effec-tive thermal conductivities.The sensitivity of the effective thermal conductivity to the heat transfer coefficient,the thermal conductivity and porosity of the insulation is shown in Figs. 5.10, 5.11and 5.12 respectively. Its magnitude increases with decreasing ha and k and increasingporosity.Design data for round-pipe insulations may be easily deduced from Figs. 5.7-5.9 using1,/ ,.._ •^---^"------ —_. _ --___14/„P- 1r14I^1^I^I^1^I^I^I^I^I^,Ta ■ 30°CL ■ 67 mmh ■ 8 W/m2 X6 ■ 98%RH ■ 96%^ K=0.038 W/M K K=0.035 W/M K K=0.033 W/M— — - K=0.030 W /m K5^10 15^20^25^300.00Chapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 43>44.0'CI• ko 3.0"▪ •11^2.0014 0O -443*-14,1) g 1.00.0 0^5^10^15^20^25^30Temperature Difference Between Warm and Cold SidesFigure 5.10: Effect of h on the variation of the effective thermal conductivity ratio.Temperature Difference Between Warm and Cold SidesFigure 5.11: Effect of K on the variation of the effective thermal conductivity ratio.Ta ■ 30°Ch ■ 8 W/m2 XL ■ 67 mmk ■ 0.035 W/mRE ■ 96%^ EPSIW98% EPSIV=96%EPSIV=94%0.0 0^5^10^15^20^25^30Temperature Difference Between Warm and Cold SidesChapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 44Figure 5.12: Effect of c on the variation of the effective thermal conductivity ratio.the equivalent flat-slab length given by Equation (4.39). For example, a 35 mm thickpipe insulation applied on a pipe of diameter 50 mm has an equivalent fiat-slab thicknessof about 52.5mm. The curves in Figs. 5.7-5.9 may now be used to find the effectivethermal conductivity for L = 52.5 mm.5.5 Concluding RemarksAn analytical model is developed for the quasi-steady phase of the heat and water vaportransfer through flat and pipe insulations. The computational effort involves the solutionof the nonlinear equation (4.18), by a trial and error method, to determine the dimen-sionless temperature 8, at the wet-dry interface. With a knowledge of Oc, the heat fluxand the effective thermal conductivity are obtained by directed substitution in Equation(4.28) and (4.31) respectively. The same procedure is appliable for pipe insulations withChapter 5. ANALYTICAL RESULTS AND DISCUSSIONS^ 45the use of the equivalent flat slab thickness given by Equation (4.39).The effective thermal conductivity and the heat flux in the presence of condensationdepend on seven independent design and operating variables of system.The effective thermal conductivity increases with increasing temperature differenceacross the insulation and reaches a maximum when condensation begins to occur in theslab. Thereafter, it decreases slightly as the temperature difference is increased further.For practical operating conditions the effective thermal conductivity varies from about1.5 to 15 times the dry-state value.The results of the parametric study are presented as curves which may be used toestimate the effective thermal conductivity over practical conditions for design purposes.The same curves can be used for pipe insulations with the use of equivalent flat-slabthickness.Chapter 6TRANSIENT NUMERICAL MODEL6.1 IntroductionThe moisture transport processes in this physical model have four main stages whichhave been stated in Chapter One. There is an initial transient period which may last upto about an hour for most practical situations before a quasi-steady phase is attained.The quasi-steady state may last anywhere up to several days before the accumulation andtransport of liquid water begins to significantly affect the heat transfer due to reevapo-ration and changes in physical properties of the insulation. The quasi-steady analyticalmodel presented in Chapter Four has given reasonable prediction of the heat and masstransfer parameters for the quasi-steady phase. However, many simplifying assumptionshave been made in order to obtain the quasi-steady solutions. For numerical studies ontransient moisture transfer through porous insulation, only work on the effect of con-densation during the early time period has been reported in the literature [35-37]. Inthis Chapter, the long-term transient moisture transport processes are analyzed, and thesignificant transport mechanisms are identified. Accordingly, the rigorous and detailedformulation of heat and moisture transfer for all the stages is presented. The long termeffects of condensation are investigated numerically, and the mobile condensate is takeninto account.46Chapter 6. TRANSIENT NUMERICAL MODEL^ 476.2 Analysis and Formulation6.2.1 General FormulationThe assumptions made in the transient formulation are the same as that stated in ChapterThree. The formulation is based on the local volume-averaged technique. In orderto generalize the mathematical formulation, the following new set of nondimensionalvariables have been introduced:Dimensionless distance,Dimensionless temperature,Dimensionless density,Dimensionless time,Dimensionless condensation rate,Dimensionless heat capacity,Dimensionless thermal conductivity,Dimensionless vapor diffusivity,Dimensionless liquid diffusivity,0PiG*cp1)(0)DAM====(6.1)Ta-T2P?Pijt.L2/a,I01,•Poao,ff/L2cp„—)4P421a0,ef0,effThe symbols are also defined in the Nomenclature.Chapter 6. TRANSIENT NUMERICAL MODEL^ 48The governing equations (3.3)-(3.5) in terms of the dimensionless variables are:Vapor diffusion equation; any^a(p) az_a EDv(0)---] — G* = atax Liquid transport equation; a^ae,^c* aft,Pt3—az[Di3(w)—az1+ ^= at'Energy equation; a^a()^UT)- 1—[k(w^+ P2G* = PCP at •az^az The corresponding constitutive correlations in terms of the dimensionless variablesare:Volumetric Constraint; ear + eo 4- E.7 ^1,Thermodynamic Relations; Pv = P3Pve,Pa = P4P09 )^1 ^1Pv = ExP[P5(w — eref)1,and(6.2)(6.3)(6.4)(6.5)(6.6)(6.7)(6.8)Matta = Pa + Pv•^ (6.9)Chapter 6. TRANSIENT NUMERICAL MODEL^ 49The dimensionless parameters are defined as the following:-F2 — cp0• ATD =^7_2,? *.3 Pt, , 0AT p: .0h f g P5^R,, AT(6.10)The spatial average density of porous insulation is defined asP =^efiP)9 67P7.^ (6.11)The mass fraction weighted average quantity of heat capacity is defined ascp = EcrPacna €13P0cA15 c7P7cw (6.12)The problem is modeled as a porous insulation with impermeable and adiabatic ver-tical boundaries. The upper horizontal boundary is impermeable and subjected to acold temperature, and the lower one is exposed to moist ambient at a warm and humidcondition. The insulation slab is assumed to be fully dry and initially has a uniformtemperature. It encounters a temperature drop at upper surface and is suddenly exposedto the ambient air at the lower surface.Therefore, the initial condition for the model isChapter 6. TRANSIENT NUMERICAL MODEL^ 50pv(z,0) = p„,,ei3(z,0) = 011(z,0) = Ga.The boundary conditions to complete the formulation areae5; 10,0= B[90 — 19(0,t)],aPvOz ko,o= Bm[pv, - pi,(0,t)b(6.13)(6.14)(6.15)0(1,0 = 02,^ (6.16)pu(1,t)==!7(02),^ (6.17)where g(0) is saturated vapor density which can be represented by the Clausius-Clapeyronrelation1^1g(0) = exp[P5(—n — n )j/P382.U2^Uref6.2.2 Stage 1: Initial Process(6.18)Initially, it is assumed that there is no water vapor condensate inside the insulation slab.The temperature and vapor density fields are governed by:Vapor diffusion equation; Chapter 6. TRANSIENT NUMERICAL MODEL^ 51Energy equation; apt?,^apvaz1[1),,(61)1 -= 67^,a r,^ a19,^ae—azilgAu)-a–;1= PCP-a-t-,(6.19)(6.20)where the dry thermal conductivity is the only thermal property which is a function oftemperature. Although there might be some initial adsorbed water inside the slab inpractical situations, it is assumed that the insulation slab is totally dry initially, and theliquid fraction is zero,- 0.^ (6.21)6.2.3 Stage 2: Immobile Liquid AccumulationStage 2 is identified as that when condensation occurs within the insulation slab. Theair-vapor mixture is saturated when the actual vapor density in a region is larger than orequal to the local saturated vapor density which can be represented by equation (6.18).The new two-phase regions (dry-wet) are then determined and the wet-dry boundary islocated where the actual vapor density is just equal to the local saturation vapor density.It should be noted physically, due to the inhomogeneities of actual fibrous insulation, thewet-dry boundary is not a single line but has a finite volume. However according to thelocal average technique, this fuzzy boundary volume can be averaged and represented bya line.• Dry RegionIn the dry region, the condensation rate G* is set to zero, as well as the liquidfraction for the above mentioned reasons. Equations (6.20)-(6.19) can be still usedChapter 6. TRANSIENT NUMERICAL MODEL^ 52for solving the temperature and vapor density fields.• Wet RegionDuring this stage, the liquid which is accumulated due to condensation is smalland has an insignificant effect on the properties. Therefore, the liquid is practicallyimmobile, and is accumulated at a constant rate.The temperature and vapor density fields are governed by Equations (6.2) and(6.3). Since the vapor is saturated in the wet region, vapor density is the onlyproperty which is a function of the local temperature, and it can be represented bythe Clausius- Clapeyronrelation (6.18).Differentiating of Equation (6.18) against t and z respectively, one obtains:apt,^aog'(e)—az'LIZ (6.22)andapt,^aewet g1(8)—at at. (6.23)Eliminating the condensation rate G* between Equations (6.2) and (6.3), and mak-ing use of relations (6.22) and (6.23), the vapor diffusion equation (6.2) and energyequation (6.3) can be combined as:ao aE a ae[pc., + P2E-d(o)]—at + P2g(0)—a 1 = -[(k(W) P2D„(0)V(6))—az]. (6.24)t azTogether with boundary conditions on the cold side (6.16) and on the wet-dryinterface, which will be stated later, Equation (6.24) can be used to solve for thetemperature field in the wet region without the need to deal with an unknownsource term. Once the temperature distribution in the wet region is known, theChapter 6. TRANSIENT NUMERICAL MODEL^ 53saturated vapor density profile in the wet region can be obtained easily from Eq.(6.18), and the condensation rate can be solved either from the vapor diffusionequation (6.2) or the energy equation (6.3).Since the liquid is in a pendulous state, the liquid diffusion equation (6.4) is reducedtoaEa G*at = pi • (6.25)6.2.4 Stage 3: Mobile Liquid DiffusionWhen the amount of liquid has been collected to some extent, the liquid can no longer betrapped in the pores. The liquid will diffuse in the direction of liquid content gradient.Since all the vapor reaching the impermeable cold plate is considered to have condensedat the cold plate, the liquid fraction is highest near the cold plate at the beginning, anddiffuses towards the warmer, and drier direction.The critical liquid fraction eo denotes the liquid fraction below which the liquidis immobile. It is a complex function of the structure of fibers, liquid pressure, surfacetension etc.. According to the experimental results of Cid and Crausse [28] for fibrousinsulation with dry state density of 4 . 53 kg/m3, the liquid diffusivity abruptly becomessignificant (about 10 m2/s) when the liquid fraction is between 0.035-0.075.After the liquid starts to flow, there are two situations to be distinguished. In thefirst case, the liquid diffuses inside the wet region. There is no apparent liquid front, theliquid diffusion equation is applied where the liquid fraction exceeds the critical value.In the second case, the liquid flows out the wet region. It enlarges the wet region andmakes the wet-dry interface move due to the mobile liquid front.The temperature and vapor density fields in the wet region are formulated in the sameChapter 6. TRANSIENT NUMERICAL MODEL^ 54way as that in stage 2. However, treatment of the moving wet-dry interface is needed forthis stage, and will be stated in the section of solution methodology.The Equation (6.4) was developed to govern the liquid diffusion. However, the liquiddiffusivity is a phenomenological quantity. Without any available experimental data,the liquid diffusion coefficient is not a known quantity. Cid and Crausse [28] measuredthe liquid diffusion coefficient for several commonly used fibrous insulation. Their liquiddiffusion coefficient Di(w) is defined as in the following liquid transport equation:aa * Oww —[.D1(w)-e; + e^11=^l--34 K (w )J + —.^(6.26)at* Ox POwhere w is weight liquid content, K(w) is hydraulic conductivity and is the function ofthe liquid saturation S. The relations between w, S and liquid fraction ep are(6.27)= 43E0^ (6.28)= COS = (1 — ca)S,^ (6.29)where 60 is the void fraction of porous insulation.Note that some important relations exist between the differentiations of w, S, K(w)and liquid fraction co:OK (w) 0.1C(w) OwOx — Ow ax' (6.30)OK(w) OK(w) as _ 1 p;,`  OK(w)— as aweo 4 as (6.31)Chapter 6. TRANSIENT NUMERICAL MODEL^ 55ow4 ô€13_, —ox^(4; ax' (6.32)aw _.= 4 ae,,at*^pc*, atSubstituting the above relations into the Equation (6.26), the liquid diffusion equationwith experimentally determined liquid diffusion coefficient, in terms of the liquid fraction,is obtained:ae,3^a^ae„,^1 OK(w) act,^r(6.34)at* = a—x[Dr(Ei9)—axi+ Eo as ax + --7pf3.Applying the same dimensionless variables as in Eq. (6.1), the liquid diffusion equa-tion in nondimensional form isae,^a^ae,3^(9E, G*at = -a-i[D1(")-ai]+ F(S)-; + ---P1'where Pi is defined in (6.10), and F(S) can be viewed as a coefficient due to hydraulicconductivity, and is defined as:1 OK(w) L F(S)=  ^. (6.36)€0 as cq; ,ef fThe liquid diffusion coefficient and hydraulic conductivity in terms of the liquid sat-uration are regressed from Cid and Crausse's experimental data [28]. The followingpolynomials have less than 5% discrepancy from the original experimental data:Dr(S) = 1.29(10)-8 + 9.913(10)-7S + 1.6018(10)-6S2 — 7.8408(10)-6V+1.053(10)-5S4;^0.45 > S > 0.05,(6.37) (6.33)(6.35)Chapter 6. TRANSIENT NUMERICAL MODEL^ 56K(S) = 4.502(10)-i — 4.658(10)-6S + 1.05486(10)-3V — 2.56279(10)-3S-3—1.7555(10)'S4 + 3.956(10)-3S5;^0.45 > S > 0.05.(6.38)6.2.5 Stage 4: Long Term Liquid Accumulation and DiffusionThe liquid front eventually reaches the exposed surface, the entire insulation slab becomeswet. The liquid is accumulated continuously because the water vapor from the ambientair keeps condensing inside the slab and on the cold plate. The liquid will move underthe liquid fraction gradient, and will be redistributed continuously.The formulation for this stages is quite the same as that for the wet region in previousstage except that convective boundary conditions are introduced into the formulation forthe wet region. The treatment of these boundary conditions on the exposed surface willbe stated in detail in the next section.6.3 Solution MethodologyThe above introduced formulation can not be solved analytically. The numerical scheme isbased on the finite difference form of the above-stated formulation. The solution domainis divided into a finite number of control volumes or cells. A grid-point is placed at thegeometric center of each control volume. This arrangement has the following advantages:• The value of the general variable which is available at the center of the controlvolume represents the volume averaged value over the control volume.• The physical properties and the source terms can be calculated at the center of thecontrol volume.Chapter 6. TRANSIENT NUMERICAL MODEL^ 57kis •^^ZA z^______.interface^interface(a)k^(6 z) e ..1E..,^ZA Zinterface^interface(b)ww^P^Eell^Figure 6.1: Grid-point Cluster for One Dimensional Transport Problem: (a). ControlVolume for the Internal Points, (b). Control Volume for the Boundary Points.• Discontinuities at the boundaries can be conveniently handled by locating boundarycells where the discontinuities occur.6.3.1 Discretized FormulationsThe discretized system of the governing equations is formed with a grid-point clustershown in Fig. 6.1. The dashed line show the faces of the control volume. For the one-dimensional problem under consideration, we assume an unit thickness in the directionsother than z. Modification of these control volumes near the boundaries is straightfor-ward.For the numerical purpose, the governing equations described in the previous sectioncan be represented by the general transport equationa^a^as:k-&(4)) = a-i[B(Tz.)] + S4^(6.39)where (I) can be replaced for different equations, B is a general diffusion coefficient, and StChapter 6. TRANSIENT NUMERICAL MODEL^ 58a general source term. The discretized form of Equation (6.39) is obtained by integratingover the respective control volumes, i.e.,ft+At ft+at al,^ti-At it+At a^al,1 it —at dtdz = ft^4^[—az (B(-5---z ) + Ndzdt.^(6.40)For detailed derivation method of the discretized equation, one could refer to Patankar'swork [42]. The fully implicit scheme is chosen for the requirements of stability, simplityand physically satisfactory behavior. In general, the central difference form is used forinternal nodes and the backward or forward difference used for the boundary nodes. Thebackward difference form is used for the time derivative. The discretized forms of theformulations for this problem are given as the following.Stage 1: Initial ProcessThe discretized equation for energy transfer is:at the internal nodes,apOp = aEOE + awew + b,wherekeaE = ^(6x)e'kwaw = ^(64,',,o^Pcpllz-'1' - At 'b= 4,4,ap = aE + aw + 4.(6.41)(6.42)(6.43)(6.44)(6.45)(6.46)For the dry slab, the conductivity variation only depends on temperature distribution.The interface conductivities Ice, kw in terms of these grid-point values areChapter 6. TRANSIENT NUMERICAL MODEL^ 597._^2kpkENe =^kp + kE'2kpkwkw — kp + kw •At z = 0, half cell near boundary is introduced, the boundary condition iskepcP Az(9 p - OCIDI ) = (6z)e (9E — Op) — h(OP — O.),At 2(6.47)(6.48)(6.49)where h is dimensionless convective heat transfer coefficient in the ambient, and € 7 isconstant for a dry slab.The vapor diffusion equation has a similar discretized form to that of the energyequation. The equation system (6.41)-(6.46) can be used for vapor diffusion, if thecorresponding variable and coefficients are replaced. The boundary condition for thevapor diffusion equation at z = 0 is677 ^,^Dye— kPv P — P°v' p) = ^Pv,p) — hni(pv,p — p,,,a),At 2^' 6z^' (6.50)where kr, is dimensionless mass transfer coefficient in the ambient.At z = 0, the temperature at the cold plate is given; no problems are introducedin the discretized energy equation. The vapor density at this boundary is equal to thesaturated vapor density at T2.Stage 2: Immobile Liquid AccumulationIn this stage, the discretization equations for the dry region are the same as that in stageone. For the wet region, the combined energy equation (6.24) can not be solved directlybecause the gas phase volume fraction Gy is not known. An iterative procedure is neededto solve for the temperature field in the wet region.[PCP + P2e-ygi (0)]13 11(9 p — Op) = P(w)-1-P2 ap:(9),91 PA. (0 E _ O) 1—[11(OP — Ow) + (P2f:)w (Pp — Pw)].Chapter 6. TRANSIENT NUMERICAL MODEL^ 60The first derivative term of c,, is omitted for the first iteration, the combined energyequation becomesae a^ae[pcp + P2e.yi(0)]-5--t- = --a—z-[(k(w) + P2D„(0)i(0))--a-z],which has the following discretized form for the internal grids,1[pcp + p2e_ygt( 9)]p rt ( op 07,) = i( k(tV)+ P2aD:(9)g ' (9)1,  (I 9 E .. 0 p)PM+ P2 f: (8 )g '(9)118  (Op ■.■ 0 W) .(6.51)(6.52)In the dry-wet interface, the boundary condition for the combined energy equationcan be derived from the energy balance at the interface. Assuming that the grid sizecan be taken fine enough so that the dry-wet interface is approximately located at theinterface of control volumes, as shown in Fig. 6.2, the energy balance of the grid wherethe dry-wet interface locates can be written as(6.53)In the above discretized equation, the vapor densities in grid P and W are unknown atthe present time step. As a numerical approximation, the values at the previous timestep are used instead. Thus the vapor density term is treated as a source term, and aniterative procedure is performed to minimize the error.The vapor density is calculated directly from the Clausius- Clapeyron equation (6.18)in the wet region, and is computed from the vapor diffusion equation in the dry region.The condensation rate in the wet region can be computed explicitly from the vapordiffusion equation or from energy equation. The vapor diffusion equation was used tocalculate the condensation rate,Chapter 6. TRANSIENT NUMERICAL MODEL^ 61Dry Region^wet Region; P;Wet-dry InterfaceFigure 6.2: Grid-point Cluster for the Interface Location of Wet-dry Regionsf^ I^Di,^ lev(P.,P P.N) (67Pv)P — ■EyPv tGp^-a-7—z 2- le lPv,E Pv,P) (Az)2 (6.54)The following boundary conditions are used for solving the condensation rate:at the wet-dry interface, z = zd,= 0,^ (6.55)at the cold plate, z 1,2D„ e,,= Az2 lPv,P P.,w),which indicates that all vapor which reaches the cold plate will condense there.The liquid fraction is computed fromGaP 0EI3'P = + E".(6.56)(6.57)Once the gas phase volumetric fraction El. is obtained from Eq.(6.5), a source term0P2g(0)61'PZ"-P is added to the right hand side of Equation (6.52), and the discretizedenergy equation for the second iteration becomesChapter 6. TRANSIENT NUMERICAL MODEL^ 62[pcp + p2e7g1(95111(0 p _ 90p) :____ j(k(w)-FP2AD;(8)9i(eyie (9E _ 9 p)[(k(w)+P2D„(8)91 (Ow (9 PAz_p2g (0 ) Cry ,P -9,67)-€137,P  . 1 (6.58)The other variables can be obtained from the corresponding formula for the firstiteration. Satisfactory results can be obtained after 4-5 time iteration.Stage 3: Mobile Liquid DiffusionThe discretized formulations and solution procedure for solving the temperature aresimilar to that of stage 2. The discretized form of the liquid transport equation isAz^DI ler^, Di ,^ N G* AAt (€0,P — e°0,P) =^k€0,E 60,P) A^z keis'p— €13,w)-1- F(S) IP (6.13,E — etLp)+Az Pi(6.59)The liquid starts to diffuse from the grid close to the cold boundary. Before the liquidfront reaches the quasi-steady wet-dry boundary, no special boundary formula is neededfor the liquid front.The liquid front is traced where the liquid volumetric fraction is equal to, or largerthan the critical liquid fraction cox,. = 0.5€0 (taken from Cid and Crausse's experimentaldata). When the liquid flows out the quasi-steady wet-dry interface, the new wet-dryinterface is located at the interface of two grids where the liquid fraction in one gridexceeds the critical value in the last time step. that is assuming at the i time step, thedry-wet interface is located at the interface between grids j-1 and j, and the value ofliquid fraction in grid j reaches the critical value, for the i+1 time step, the new wet-dryinterface will move to interface of grid j-1, and j-2. This approximation for tracing themoving boundary is reasonable as long as the grid size is taken fine enough.Chapter 6. TRANSIENT NUMERICAL MODEL^ 63At z = 1, the boundary condition isAz ,^0 \^I,„^ G* AzLut kfo,P — Ef3,13) = Az (EON Efi,P)+ F(S) Ip (Ei3,P EON) +At thet  moving wet-dry boundary,(6.60)Az ,^DI le,0 \kEl3 P E ) ^ kefi E E"f3 P) F(S) IP (€13,p — Es,w) —G* AzAz.^(6.61)Pi 2Stage 4: Long Term Liquid Accumulation and DiffusionIn this stage, the wet-dry interface reaches the exposed surface. The entire insulationslab is wet. The solution procedure for temperature is the same as mentioned for the wetregion. The discretized equations (6.54) and (6.58) for the wet region are applicable forthe internal grids. The boundary condition on the exposed surface is introduced as1{PCP + p2e7g/(8)} ip tist.(ep _ eop) ,_ t(k(w)-FP2AD,;(91g1(9)11. (0E _ op)(6.62)—11.(8p — 0.) — P2h,„(pp — pa)].Similar boundary conditions on the exposed surface are also introduced for computingthe condensation rate and liquid fraction.6.3.2 The Solution AlgorithmA Fortran 77 computer code has been developed to solve the above-stated system ofthe coupled nonlinear equations. Starting with the initial values of the temperature andvapor density, the solutions of the above model for the four stages involve several formats,as well as several iteration procedures. These formats are:Chapter 6. TRANSIENT NUMERICAL MODEL^ 64• Format 1. This format was followed for any time and location along slab for whichno condensation occurs in the insulation slab.1. Initial values of 0 and pt, are given from Equation (6.13), (step 1 is applied onlyat t = 0).2. The condensation rate is set to zero, 0 is obtained from Equation (6.20) and A,is obtained from Equation(6.19).3. The saturated vapor density p„,, is obtained from Eq. (6.18) based on thecomputed temperature profile.4. If, at any location, the pi, obtained in step 2 is greater than or equal to pi,,,obtained from step 3, then Format 2 is adopted for the next time step. Otherwisethe procedure starting from step 2 is repeated for the next time step.• Format 2. This format was followed for any time and location for which the wetregion is established and co < co,,..1. The temperature in the wet region is solved from Equation (6.52) with boundaryconditions (6.53) and (6.16) as the first iteration.2. The vapor density in the wet region is obtained from Equation (6.18).3. The temperature and vapor density in the dry region are obtained from Equation(6.20) and (6.19).4. The condensation rate is obtained from Equations (6.54) -(6.56).5. The liquid phase fraction co is obtained from Equation (6.57), and the gas phasefraction ey is obtained from the volumetric constraint Equation (6.5).6. The second iteration value of the temperature in the wet region is obtained fromEquation (6.58).Chapter 6. TRANSIENT NUMERICAL MODEL^ 657. The same procedures are followed from step 2 to step 5 to obtain the seconditeration values of vapor density, condensation rate and the liquid fraction.8. If, at any location, the liquid fraction obtained from step 7 is greater than orequal to the critical liquid fraction, Format 3 is adopted for the next time step.Otherwise, the procedure starting from step 1 is repeated for the next time step.• Format 3. This format was followed for any time and location for which the liquidvolumetric fraction exceeds the critical liquid volumetric fraction.1. The temperature field in the wet region is solved from Equation (6.52) withboundary conditions (6.53) and (6.16) as the first iteration.2. The vapor density in the wet region is obtained from Equation (6.18).3. The temperature and vapor density in the dry region are obtained from Equation(6.20) and (6.19).4. The condensation rate is obtained from Equations (6.54) -(6.56).5. The liquid phase fraction co is obtained from Equation (6.59), with the boundaryconditions (6.60), (6.61). The gas phase fraction e-y is obtained from the volumetricconstraint Equation (6.5).6. The second iteration value of the temperature in the wet region is obtained fromEquation (6.58).7. The same procedures are followed from step 2 to step 5 to obtained the seconditeration values of vapor density, condensation rate and the liquid fraction.8. If zd=0, Format 4 is adopted for the next time step. Otherwise, the procedurestarting from step 1 is repeated for the next time step.Chapter 6. TRANSIENT NUMERICAL MODEL^ 66• Format 4 This format was followed for any time and location for which all thelocations are wet.1. The first iteration value of temperature is obtained from Equation (6.52) withboundary conditions (6.16) and (6.6.62).2. The vapor density in the wet region is obtained from Equation (6.18).4. The condensation rate is obtained from Equations (6.54),(6.56) and (6.63).5. The liquid phase fraction efi is obtained from Equation (6.59), with the boundaryconditions (6.60), (6.61). The gas phase fraction e.,,, is obtained from the volumetricconstraint Equation (6.5).6. The second iteration value of the temperature in the wet region is obtained fromEquation (6.58).7. The same procedures are followed from step 2 to step 5 to obtained the seconditeration values of vapor density, condensation rate and the liquid fraction.The discretization equations for the above one-dimensional formulation can be solvedby a delightly convenient algorithm, called TriDiagonal -Matrix Algorithm (TDMA). Inprinciple, all the above-mentioned formulations have a form similar to the equation sys-tem (6.41)-(6.46). The nonzero coefficient of the set of equations defined by Equation(6.41) form a tri-diagonal matrix. The convergence of TDMA is fast because the bound-ary condition information is transmitted at once to the nodal points lying inside thesolution domain.It should be pointed out that although there are two groups of formulation for thewet and dry regions respectively in the stage 2 and 3, these two groups of discretizationequations should be solved in one matrix. Numerical errors may occur when the solutionsof field variables are attempted separately from the dry and wet regions. There are twoChapter 6. TRANSIENT NUMERICAL MODEL^ 67reasons for this. First, the dry region is being reduced due to the moving interface, andthe TDMA will lose effectiveness when the dimension of the matrix is less than 3. Thesecond reason is that nodal points inside the solution domain are blind to the informationof another domain.6.4 Numerical Considerations• Grid System and Time StepTwo checks were made in order to investigate the accuracy of the scheme. First,the three different grid systems, given by 50, 100 and 150 grids, were used to testthe effects of the grid size on heat transfer rate, the wet-dry interface location, andthe distributions of the field variables. It was found that an excellent agreementexists between the results from 100 and 150 grid systems. Next, the time step sizewas checked by fixing the grid size with 100 grids while varying the time step sizeas 30s, 60s, 120s, 180s, 240s, 360s etc.. As the results show, decreasing the timestep size beyond 180 seconds did not have an effect on the numerical results.The accuracy of the wet-dry interface position depends on the selection of the gridsize. For this reason, the grid size should be chosen as fine as possible. However, therequired time step size becomes much smaller as the grid is refined. Considering thebalance between accuracy and economy of computing time, a grid system with 100grids and a time step of 120 seconds were chosen for the numerical computation.• Iterative SchemeIn computing the field variables such as temperature 0 in the wet region, severaliteration schemes were needed. In order to handle nonlinearities in the iterationsolutions of the discretization equations, it was necessary to underrelax the iterationChapter 6. TRANSIENT NUMERICAL MODEL^ 68process. The solution was considered to be converged when the deviation of anyvariable from the last iterated value was within 0.01%.• Numerical Errors and InstabilitiesNumerical errors and instabilities can be caused by inappropriate numerical pa-rameters such as coarse grids, too large a time step and underrelaxation factors, aswell as unrealistic initial values. The correct evaluation of numerical parameterscan be achieved from the numerical experiments.Chapter 7NUMERICAL RESULTS AND DISCUSSIONS7.1 IntroductionThe numerical model presented in the previous Chapter was used to investigate transientheat and mass transport accounting for phase change in a fibrous insulation slab. Oneof the main objectives of this work was to simulate the thermal behavior of porousinsulation including dynamic response during the diffusion and condensation process.The numerical formulation and its solution methodology are validated by the comparisonof numerical results with the experimental data of Wijeysundera et al.. The variationand intercoupling effects of important field variables such as temperature, vapor density,condensation rate and liquid fraction are presented. Heat transfer rate through theinsulation is quantified, and the moisture accumulation and distribution are numericallysimulated. The long term effects of the moisture transport on thermal performance of theinsulation were studied, and the interesting effects of variation of physical characteristicson the moisture and energy transfer are discussed.7.2 Physical DataIn order to examine the validity of the numerical model, the physical data used in thenumerical computation are based on the fibrous insulation which was tested as an ex-perimental sample, and also based on the real experimental conditions [23]-[25]. Thephysical data are summarized in Table 7.1.69Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 70p(*)cp*,01c;53 kg/m3840 J/kg K0.037 W/m K4c;`k;999.87 kg/m'4200 J/kg K0.57 W/m Kc4,eff 8.84(10)-7 m2/s p: 1.15 kg/m3p: 2600 kg/m3 c„ 1005 J/kg Kc;. 836.8 J/kg K k: 0.57 W/m Kk:,EQ.0.762 W/m K0.02c;p,01882 J/kg K2337 bar14 461.89 J/kg K hfg 2450000 J/kgTable 7.1: Physical Data.7.3 Diffusion and Condensation Processes7.3.1 Initial ProcessThe dynamic response of porous insulation subjected to the specified boundary conditionshas been investigated by case studies. Figures 7.1-7.4 illustrate the distribution of thetemperature, vapor density, condensation rate, and liquid fraction in the initial stage.As can be seen in Figure 7.1, when a porous insulation slab is suddenly exposed to theambient air on one side, and is subjected to a temperature drop on the other side, thistemperature drop propagates from the cold side into the insulation slab with time. Underthe given boundary conditions, the temperature field becomes quasi-steady after abouthalf an hour (2040 seconds). The propagating behavior is also observed for vapor densityas shown in Fig. 7.2, and the vapor density field reaches the quasi-steady state afterabout the same period. It is apparent that the drop of vapor density propagates fasterthan that of temperature. It can be expected as the Lewis number which is a measureof the relative importance of heat transport to the vapor transport, is less than one.The distribution of condensation rate with time is shown in Fig. 7.3. The condensa-tion occurs first at the cold plate, and then takes place within the insulation slab after30gE-1 25e35i 20tt.,E 15E-(105Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 710.00^0.25^0.50^0.75^1.00Dimensionless Distance ZFigure 7.1: Temperature Distribution in the Initial Stage; p* = 53 kg/m', L = 66 mm,RH = 96%, h* = 12 W/m2 K.0.0350Aok 0' 0300-.V ca: 0.0250 -0.0200 Ta = 33°C^...^---^.... T2 = 6.8°C0.0150 -Ag. 0' 0100 -ett ›.0.00500.00.^I^-^I 0.25 0.50^0.75Dimensionless Distance Z1.00Figure 7.2: The Distribution of Vapor Density in the Initial Stage; p* = 53 kg/m3, L = 66mm, RH = 96%, h* = 12 W/m2 K.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 72e'E 0.002000.001500.001002 0.00050(.5 0.00000I^,^, I 1^- 0.00 0.25^0.50^0.75^1.00Dimensionless Distance ZFigure 7.3: The Time Variation of Condensation Rate in the Initial Stage; p* 53 kg/m3,L 66 mm, RH .96%, h* = 12 W/m2 K.`c" 0.00300 —CA)^Ta = 33°C^= 0.00250 —^T2 = 6.8°C0trc0.00200 —froPS 0.00150 —cr.41 0.00100 —c.•^3600s^2040s^960s^480sN 0.00050 -5"O °Loom ^^( ^,^,^1^,0.00^0.25^0.50^0.75Dimensionless Distance Z1.00Figure 7.4: The Time Variation of Volumetric Liquid Fraction in the Initial Stage; p* . 53kg/m3, L 66 mm, RH = 96%, h* = 12 W/m2 K.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 73about 480 seconds. The condensation occurs within the slab when the local saturationdensity is decreased below the actual vapor density. With the propagation of tempera-ture drop in the slab, the location of the condensation front moves towards the warmerside, and stops at some place when the temperature field becomes quasi-steady. Thecondensation rate is highest at the cold plate because all the vapor flux is condensedthere. The liquid fraction is accumulated with time due to condensation as shown in Fig.7.4. In the initial stage, it attains its highest value at the cold plate.7.3.2 Quasi-steady State PeriodFigures 7.1 and 7.2 show that after a period of time the distributions of temperature andvapor density remain constant. Fig. 7.3 shows that after about the same period, thelocation of the condensation front is fixed at some place which indicates the quasi-steadywet-dry interface. The quasi-steady behavior of temperature and vapor density withrespect to time indicates reduction of transient thermal, mass and diffusion processes.The liquid fraction et, still increases with time due to a constant condensation rate.For the illustrative case shown in Figs. 7.1-7.4, the quasi-steady state is reached in2040 seconds. Fig. 7.5 shows the temperature distributions along the insulation slab aftera quasi-steady state has been established; the numerical results are compared with theanalytical results in Chapter Five. The experimental results under the same conditionsare also plotted in Fig. 7.5, which shows good agreement.'7.3.3 Liquid Accumulation and DiffusionThe profiles of liquid volumetric fraction over a period of 120 hours are shown in Figure7.6. Initially, vapor condensation occurs mainly on the cold surface. The liquid generatedis accumulated at the impermeable cold plate and the nearby insulation layer. With theestablishment of a quasi-steady state, liquid is accumulated in the wet region at a constantNum .-96%RHAna.-96%.RHExp.-96%RHNum.-70%RHAna.-70%RHExp. 70%RH1.000.000.00^0.25^0.50^0.751.00 ^0.750.50E 0.25120 h100 h0.)0.075.C$▪^0.050*C▪^0.0250.0000.00^0.25^0.50^0.75Dimensionless Distance Z1.00Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 74Dimensionless Distance ZFigure 7.5: Temperature Distribution in the Quasi-steady State Period, Comparison ofNumerical, Analytical and Experimental Results [22]; p* = 53 kg/m3, L = 66 mm,RH = 96%, h* = 12 W/m2 K.Figure 7.6: The Time Variation of Liquid Fraction over a Long Term Period; p* = 53kg/m3, L = 66 mm, RH = 96%, h* = 12 W/m2 K, T. = 33°C , T2 = 6.8°C.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 75rate. The curve at 1 hour indicates that the liquid fraction at the impermeable coldboundary is higher than the critical value, and the liquid diffuses from the cold surfaceinto the insulation slab. From the curves from 5 hours to 60 hours, we can see clearly thatthe liquid diffusion front moves towards the dry side. Before the liquid diffusion frontarrives, the liquid fraction accumulates slowly with time. It increases sharply when theliquid diffusion front arrives. The liquid front can flow out the quasi-steady boundaryas indicated by the curve at 70 hours, and eventually reaches the exposed surface, asshown by the curves from 80 hours to 120 hours. In the later stage, the liquid fractionis higher than the critical value in the entire insulation slab. The liquid diffuses fromone layer into the adjoining layer with the resultant increase in liquid content and acorresponding leveling of the liquid content in the upstream layers. In addition to thistransport mechanism, the local liquid content still increases due to the local condensation.Since the liquid amount produced by condensation is still highest on the impermeable coldplate, the liquid will diffuse under the liquid fraction gradient, i.e. from the impermeablesurface towards the exposed surface.7.4 Comparison with Experimental ResultsWijeysundera et al. [23, 24, 25]] had measured the temperature distribution, heat flux,total moisture gain, and the liquid distribution for a range of experimental conditionsthrough five runs. The testing times ranged form 300 to 600 hours. In the presentwork, case studies have been performed by comparing the computational results withthe experimental data. The comparisons of the temperature distributions, heat transferrates at the impermeable plate, the total moisture gain and liquid distribution with theexperimental data are presented respectively in the following sections.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 767.4.1 Temperature and Heat TransferSixty six cases of measured temperature distribution tn different times for five operatingconditions have been simulated. For brevity, only the comparisons with the experimentaldata for 16 cases are presented in Figs. 7.7 - 7.10. There is very good agreement betweenthe measured and computed temperature distributions. It should be pointed out thatthe uncertainty of the thermocouples location in the insulation slab may be responsiblefor some fluctuations in the measured data.The comparisons of computed and measured heat flux at the impermeable cold platefor four runs are shown in Fig. 7.11. The computed heat flux at the cold plate issummed up from two components. The first component is conductive heat flux whichdepends on the thermal conductivity of the moisture laden insulation; so called 'solidthermal conductivity'. The second is latent heat due to the condensation of the vaporreached the cold plate. Fig. 7.11 shows that the numerical model underpredicts theheat flux at the cold plate. The reason for the discrepancy may be due to the accuracyof moisture laden thermal conductivity models. Several possible thermal conductivitymodels are listed in Appendix B. Among these, the bead arrangement model is chosenfor representative computation after a comparison of these models which will be discussedlater. The fluctuations in the heat flux curves are due to the nonuniform temperaturedifference between the ambient air and the cold plate in the experiments.7.4.2 The Moisture Gain and Liquid TransportThe total moisture gain per unit volume of the slab is obtained by integrating numericallythe liquid content over all the finite control volumes. The comparison of computed totalmoisture gain with measured data under four different operating conditions is shown inFig. 7.12. The numerical results show good agreement with measurements for the firstDimensionless Distance Z150.000^0.025^0.050Dimensionless Distance Z454U....e 35E.E4s3°4K25Ecl)E*200.000 0.025 0.050Ta=40.5°CT2=20.8°CTime: 24 hours45Ta=39.9°CT2=21.1°CTime: 311 hours150.000^0.025^0.050Dimensionless Distance Z0.000 0.025 0.050Ta=39.9°CT2=22.2°CTime: 214.3 hoursDimensionless Distance ZChapter 7. NUMERICAL RESULTS AND DISCUSSIONS^77Figure 7.7: The Temperature Distributions in Run 1, Comparison of Numerical Resultswith Experiment Data [24]; p* = 53 kg/m3, L = 62.02 mm, RH = 97%, h* = 12 W/rn3K.Ta=41.4°CT2=11.9°CTime: 48 hours12525et.5 2015100.000^0.025^0.050Dimensionless Distance Z4540P 35300.000 0.025^0.050Ta=40.7°CT2=14.4°CTime: 168.8 hours =Dimensionless Distance Z0.050100.000^0.025Dimensionless Distance ZTa=41.9°CT2=1 6.3°CTime: 337 hoursE.+ 30I4 25eu520a)15E-1454;3.35Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^78E-1j 25eloE 20a)15100.000^0.025^0.050Dimensionless Distance ZFigure 7.8: The Temperature Distributions in Run 2, Comparison of Numerical Resultswith Experiment Data [24]; p* = 53 kg/m3, L = 68.65 mm, RH = 96.5%, h* = 12 W/m2K.0.025 0.0500.000Dimensionless Distance Z Dimensionless Distance Z40355...) 3°E.., 2520a415E4 105Ta=36.5°CT2= 8.1°CTime: 94.8 hours40Ta=35.9°CT2= 9.9°CTime: 190.8 hours 12520aQ 15510500.000^0.025^0.050Dimensionless Distance Z35°c.) 30Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 79,••••••■2 5[0E 15CLoE 105Dimensionless Distance ZFigure 7.9: The Temperature Distributions in Run 3, Comparison of Numerical Resultswith Experiment Data [24]; p* = 53 kg/m', L = 70 mm, RH = 96%, h* = 12 W/m2 K.45Ta=40.0°CT2=20.2°CTime: 139.8 hours -_a0.... 35E-4to2554,)&..4 20Ta=39.9°CT2=20.4°CTime: 455.5 hours0.000 0.025 0.050 0.075Ta=39.9°CT2=20.5°CTime: 551.2 hours0.000 0.025 0.050 0.075Ta=39.8°CT2=20.5°CTime: 620.8 hoursChapter 7. NUMERICAL RESULTS AND DISCUSSIONS45435eaeti 25E„CD208015 ^150.000^0.025^0.050^0.075 0.000^0.025^0.050^0.075Dimensionless Distance Z Dimensionless Distance ZDimensionless Distance Z Dimensionless Distance ZFigure 7.10: The Temperature Distributions in Run 5, Comparison of Numerical Resultswith Experiment Data [24]; p* = 53 kg/m3, L = 76.13 mm, RH = 96%, h* = 12 Win?K.300 35050^100^150 200^250Time (Hams)(a) Run 160 -00 0 0 0 0 0 0 0 0 0 0:50 - _,......'w 40 -• • U •^ • -• • • • • •, M^:.530- L = 68.65 mm- - RH = 96.5%CT420• NumericalO Experimental [24]I^.^.50^100^150 200 250Time (Hams)(b) Run 2100 300 35030 -- L=70 mmRH = 96%.525 -- 13 Experimental [24]1550^100^150 200^250Time (Hums)(c) Run 310040 - _I 130/......_ • • • m m m mm •00 00 0 0 0 0 0 0 0 0^0 .• --• • •.5 20Cr10- L = 76.13 mmRH = 96%- • Numerical0 Experimental [24]250^500Time (Hums)(d) Run 5300 350504540 - a . 0 0 0 0 0 --35,.....- • . " •^o 0 0. • •• IICri 20. NumericalChapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 81Figure 7.11: The Heat Flux at the Cold Plate, Comparison of Numerical Results withExperiment Data for Four Runs [24]; p 53 kg/m3, h* = 12 W/m2 K.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 82three runs, especially for run 3 shown in Fig. 7.12(c). Fig. 7.12(d) shows that in thelonger times, the predicted total moisture gain is higher than the measured data. Thismay be due to the decrease in vapor diffusion coefficient with liquid concentration whichhas not been taken into account in this model.The measured spatial distribution of the average liquid concentration was obtainedby measuring the increase in mass of the different insulation layers. The comparison ofthe computed and measured spatial distribution of the average liquid concentration isshown in Fig. 7.13 for run 1. Fig. 13 (a) shows good agreement between the computedresults with the measured data for a period up to 70 hours. The liquid distributionsin this period display similar trends. The liquid produced by condensation is depositedmostly in the insulation layer adjacent to the cold plate, and is forced towards the nextlayer with time. Fig. 13(b) shows the comparison of computed results with experimentaldata at 120 and 144.5 hours respectively. Two apparent discrepancies can be observedfrom this plot. First the predicted liquid movement is faster than the measured. In themeasurement, the liquid is trapped in the layer adjacent to the cold plate untill the layerattains a very high liquid content. This indicates that the liquid diffusivity becomessignificant only when liquid fraction reaches a very high value (approximately co > 0.2for this case). This phenomenon did not appear in Cid and Crausse's experimentalwork [28]. The second discrepancy is on the average liquid concentration of the layerwith the exposed surface. The experimental data shows a rapid rise of average liquidconcentration of that layer after a certain period while the numerical model predicts afar slower growth of average liquid concentration in the layer. The present model cannot explain why the layer with the exposed surface has higher liquid content than inthe inner layers. A physical explanation for this behavior given by Wijeysundera etal. [25] is the tendency for the layer with the exposed surface to accumulate liquid dueto the surface tension forces between water and the last layer of fibers on the exposedoo oo50^100^150 200 250Time (Horns)(a) Run 1300 350 50^100^150 200 250Time (Hums)(b) Run 2300 350200^350^oo^0 50^100^150^200^250^300^250^500Time (Hours) Time (Hums)(c) Run 3 (d) Run 5- 0 •0 •0 0_0....- o -• Numerical0 Experimental [241-U0-300L= 62.02 mm250 o^-- RH = 97% •0•200 -o0 ••0150:_li_.GI100 -M • Numerical50 -^o ° Experimental [241-D.......V)2'—"0.a0.1•cA.103oE-1350o.8 2000,S 150tfl.0" 1003 500F-11 L = 68.65 mmRH = 96.5%O^o^••o -L= 76.13 mmE RH = 96%.o• -os0- _s• 00-o50 _-- o o. Numerical0 Experimental [24]-S • Numerical0^° Experimental [24]in-oto-.30002 250...."o• 2000i 150.4o 10030E-1LW) 1504 oL=70 mm- RH = 96%lee- _•O.o•02Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 83Figure 7.12: The Total Moisture Gain, Comparison of Numerical Results with Experi-ment Data for Four Runs [24]; p* = 53 kg/m3, li* = 12 W/m2 K.• Predicted, 24ho Measured [25], 24h--IV— Predicted, 48h0^Measured [25], 48h---A-- Predicted, 69.5h• Measured [25], 69.5hPredicted, 120h- 0^Measured [25], 120h"--•^Predicted, 144.5h- 0 Measured [25], 144.5h1^1^1^1^1^1^1^1^-300E 250o• 2001501001-4 5000.000 0.010 0.020^0.030 0.040 0.050 0.060Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 84250200I 150a 100• 5000.000 0.010^0.020^0.030^0.040^0.050^0.060Location, X (mm)(a)Location, X (mm)(b)Figure 7.13: The Liquid Distribution in Slab at Different Times, Comparison of Nu-merical Results with Experiment Data [24]; Run 1. p* = 53 kg/m3, h* = 12 W/m2 K,L = 62.02 mm.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 85surface. However, the reason for liquid penetrating the inner layers without becomingtrapped there is unknown. The effect of gravity may be an explanation for this. Sincethe mechanism of moisture movement and the liquid path through the fibers in fibrousinsulation are not known, theoretical simulation of liquid transport process in fibrousmaterials requires a more rigorous model of liquid transport coefficients.7.5 Thermal and Transport Performances7.5.1 The Variation of Heat FluxThe variation of heat flux at cold boundary with time under fixed temperature differencebetween the ambient and cold plate, as well as the variation of the two components whichcontribute to the total heat flux, are shown in Fig. 7.14. The conductive componentshows an increasing trend with times because of the increasing 'solid' thermal conductivity(which is a result of increased liquid content). The latent heat component decreases withtime due to the reduction of vapor flux from ambient. However, the total heat flux showsan increasing trend with time, which suggests that the apparent thermal conductivitydue to laden moisture has been increased adequately to bring this about.A dramatic increase in heat flux occurs when the liquid flows out the wet-dry interface.It attains its peak value when liquid front reaches the exposed surface, then decreasesto normal values and rates. This phenomenon is complicated by the combination ofevaporation and condensation which occur when liquid enters an unsaturated region.Liquid flows into the dry region forced by capillary action. When liquid encounters thewarmer, unsaturated water vapor-air mixture, evaporation will occur at the liquid front.Some vapor is produced in the local space. Since the liquid amount which is forcedfrom high liquid content region to the dry region is large, the result of liquid mixedwith unsaturated water vapor is a saturated liquid-gas mixture which reaches a new-N,-I^111111111114The moment when the liquid _-reaches the exposed surface -...-----= CI cond+q latent^ ,-----\-The moment when the liquid :flows out the wet-dry interface i-,q condqlatent ^■ 140100---ii Location of wet-dry interface3.01.0Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 860^25^50^75^100Fourier Number FoFigure 7.14: The Variation of Heat Flux at Cold Boundary with Times; p* = 53 kg/m',h* = 12 W/m2 K. T- = 33°C. T. = 6.8°C. Rif = 90%.0.00.25^0.50^0.75^1.00Dimensionless Distance ZFigure 7.15: The Variation of Equivalent Thermal Conductivity along the InsulationSlab; p* = 53 kg/m3, h* = 12 W/m2 K, T. = 33°C, T2 = 6.8°C, RH = 90%.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 87equilibrium state. The region which the liquid front occupies becomes saturated. Theoutflowing liquid empties its space in the wet region, which enables an increasing vaporflux into the insulation slab. The condensation rate and the condensation amount increasein the wet region. Subsequently, both the moisture laden conductive heat and latent heatincrease. After the liquid front reaches the exposed surface, the entire insulation slab issaturated. The temperature and vapor density fields regain their quasi-steady behavior,and the condensation rate returns to a constant rate.The equivalent thermal conductivity at any cross section is defined by the followingexpression:dTqi = —Iceg—dx 1x, . (7.1)The ratio of equivalent thermal conductivity to the dry state thermal conductivityrepresents the effect of condensation on the thermal performance of insulation. Thevariation of equivalent thermal conductivity ratio along the slab at selected moments isshown in Fig. 7.15. As can be expected, the equivalent thermal conductivity ratio is unityin the dry region. It increases significantly in the wet region and attains a maximumvalue near the wet-dry interface. The curves of equivalent thermal conductivity declinegradually in the direction of the cold side due to the reduction of vapor flux in thatdirection.7.5.2 Effect of Humidity LevelsThe effective thermal conductivity ratio and the total moisture gain under different hu-midity levels in the ambient are presented in Fig. 7.16. The term Iceffilcary in Fig. 7.16(a)represents the ratio of conventional effective thermal conductivity at cold boundary tothe dry state thermal conductivity of insulation (for brevity, all thermal conductivityChapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 88ratios which appeare in the following refer to the ratio at the cold boundary). Humiditylevels on the exterior boundary have a significant influence on the heat flux and the totalliquid concentration. This is because increasing the humidity level enhances the vaportransport, and the enhancement in vapor transfer will in turn cause an increase in heattransfer. It is also noted that a higher humidity level on the exterior boundary reducesthe time to reach a quasi-steady state. As can be observed in Fig. 7.16(a), decreasingthe humidity level reduces the quasi-steady wet region, thus reducing the time for liquidto flow out of the wet region.7.5.3 Effect of Convective Heat Transfer Boundary ConditionsInteresting results are obtained through the examination of the effect of the ambientconvective heat transfer coefficient. Fig. 7.17 shows the variations of keffikdry andthe moisture gain with time. It is noted that keff/kdry decreases with increasing heattransfer coefficient. Since increasing convective heat transfer at the exterior boundaryenhances the temperature penetration inside the slab, and the saturation vapor density ismainly dependent on the temperature distribution, the wet region in the slab is reducedas a result. The reduction of the wet region leads to decreases of condensation rate.Subsequently, the moisture laden thermal conductivity is decreased, as well as the latentheat flux. Therefore the total heat flux through the insulation slab decreases with anincreasing of ambient convective heat transfer coefficient. The above-mentioned resultsindicate the complex intercoupled nature of heat and mass transport in these types ofproblems.7.5.4 Effect of Slab ThicknessThe effects of insulation slab thickness on thermal performance and liquid accumulationin an insulation slab are shown in Fig. 7.18. As can be expected, with a decrease of slab100(4—'s,...5°JD 75-V0'M0F0w.-,o25030=1Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 890^25^50^75^100Fourier Number Fo(b)Figure 7.16: The Variation of Ice f f /1cdry at Cold Boundary and Total Liquid Concentrationunder Different Humidity Levels; p* = 53 kg/m3.1000^ h= 8 Wm-2K^ h=12 Wrii2 K- h=16 Wm"2KChapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 900^25^50^75^100Fourier Number Fo(b)Figure 7.17: The Variation of Iceff/kdry at Cold Plate and Total Liquid Concentration un-der Different Convective Heat Transfer Boundary Conditions; p* = 53 kg/m3, Ta = 33°C,T2 = 6.8°C, L = 66 mm, RH = 90%.00^25^50^75^100^125Fourier Number Fo(b)150Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 91Figure 7.18: The Variation of keff/kdry at Cold Boundary and Total Liquid Concentrationunder Different Slab Thickness; p* = 53 kg/m2, T. = 33°C, 7'2 = 6.8°C, h* = 12 W/m2K, RH = 90%.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 92g 2.75 4.1::=^2.50e2.25C.7stt2.0025^50^75^100Fourier Number Fo(a)0^25^50^75^100Fourier Number Fo(b)Figure 7.19: The Variation of ke f f kdry at Cold Boundary and Total Liquid Concentrationunder Different Porousities; p* = 53 kg/m', T. = 33°C, T2 = 6.8°C, L = 66 mm, h* = 12W/m2 K, RH = 90%.Chapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 93thickness, both keffilcdry and the average liquid concentration increase. This is becausetemperature diffusion is fast in a thinner slab, and vapor saturation will occur earlierwith a thinner slab. Within the same period, the liquid accumulated in a slab increaseswith decreasing slab thickness; thus the moisture laden conductive heat and latent heatalso increase.7.5.5 Effect of PorosityFig. 7.19 shows that the variation in porosity does not have significant effect on thethermal performance and liquid accumulation in a porous insulation. For practical use,the porosities for most of fibrous insulation fall into a narrow range due to their highlyporous characteristics. However, examining the effect of porosity can still give us thetrend of energy and mass transfer under the influence of porosity. The ken ficdry andaverage liquid concentration increase with increasing porosity.7.5.6 Comparison of Different Thermal Conductivity ModelsThe apparent thermal conductivity of porous insulation in the presence of liquid is greatlydependent on the liquid distribution in an insulation. Appendix B gives several possiblemodels for k available in literature [2], [43]. These models include bead arrangement,series arrangement, parallel arrangement and form arrangement, based on the manner ofliquid location in the pores of insulation.The variation of keff/kdri, with time using different k models is shown in Fig. 7.20.The parallel arrangement model gives maximum rates of heat transfer. The bead ar-rangement model gives better trend-wise agreement with the measured heat flux at coldplate for all runs; it is recommended for computational use. The actual distribution ofliquid in the insulation is not completely known. However, the above models give an in-dication of the sensitivity of the predicted quantities to the variation in these importantChapter 7. NUMERICAL RESULTS AND DISCUSSIONS^ 9425^50^75^100Fourier Number FoFigure 7.20: The Variation of keff/kary at Cold Surface Using Different k Models; p* = 53kg/m', T. = 33°C, T2 = 6.8°C, L = 66 mm, h* = 12W/m2 K, RH = 90%.physical properties.Chapter 8CONCLUSIONS AND RECOMMENDATIONSThe following conclusions can be drawn from the analytical and numerical studies onheat and moisture transfer through a porous insulation given in the previous chapters.8.1 Analytical Study• The analytical model developed for the heat and water vapor transfer through flat-slab and round pipe insulation is valid for describing the problem of condensationin porous insulation due to thermal and vapor diffusion.• The effective thermal conductivity and other heat transfer parameters are stronglyaffected by condensation. The effective thermal conductivity is a maximum whencondensation first occurs in the slab. For practical operating conditions its magni-tude varies from about 1.5 to 15 times the dry-state value.• The effective thermal conductivity and other heat transfer parameters depend onseven independent design and operating variables. A parametric study shows thatthe effective thermal conductivity increases with the increasing ambient humiditylevel and porosity, and decrease with increasing ambient heat transfer coefficient,slab thickness and dry-state thermal conductivity of insulation.• The design curves are presented for flat-slabs. They may be used for round-pipeinsulation by computing the equivalent flat-slab thickness of the insulation.95Chapter 8. CONCLUSIONS AND RECOMMENDATIONS^ 968.2 Numerical Study• The analysis in the present work has identified the four stages of heat and moisturetransfer in a porous insulation. A wave-like propagation phenomenon has beenobserved for the important field variables such as the temperature and vapor densityin the initial stage. The dynamic response and transient behavior with condensationand liquid diffusion processes are clearly shown. The numerical prediction of quasi-steady behavior has verified the analytical results, and the long term behavior ofenergy and moisture transfer through a porous insulation has been presented.• The numerical model simulates the heat and moisture transport process througha porous insulation successfully. The predicted temperature distributions, heattransfer rates and total moisture gains have good agreements with the experimentalresults over various operating conditions for a time period up to 600 hours.• The present model gives reasonable prediction of liquid distribution for the situa-tions in which liquid concentration in an insulation is low. For a complete inter-pretation of liquid distribution, a more accurate liquid diffusion coefficient modelbased on experimental work is required.• The effects of important parameters on heat and mass transfer have been investi-gated. The heat transfer is strongly affected by moisture distribution. The transienteffective thermal conductivity increases with increasing exterior humidity level andporosity, and also increases with decreasing convective heat transfer coefficient andslab thickness.Chapter 8. CONCLUSIONS AND RECOMMENDATIONS^ 978.3 RecommendationsIt is clear from previous comparisons and discussions that a complete simulation of heatand moisture transfer through a porous insulation will require more rigorous theoret-ical work, as well as more thorough and accurate experimental work. The followingrecommendations are suggested for a future work.• Due to lack of information on liquid diffusivity and hydraulic conductivity of fibrousinsulations, more accurate experimental work on physical properties and transportperformance of fibrous insulation is greatly needed.• A two-dimensional numerical model is needed in order to interpret a tendency ofliquid movement along the fibers. This may be important for the cases where thepressure gradient in an insulation is low.• The present numerical model can be improved by removing some simplifications.The effects of air infiltration, gravitation, and varied pressure in an insulation wouldbe taken into account for practical applications.• It would be interesting to extend the subject of the present study to a similarproblem with two permeable boundaries which represents the common situation inbuilding envelopes.Bibliography[1] Lotz, W. A., "Moisture problems in buildings in hot humid climates", ASHRAE J.,April 1989.[2] Kaviany, M., "Principles of heat transfer in porous media", Springer-Verlag, NewYork, 1991.[3] Whitaker, S., "Simultaneous heat, mass and momentum transfer in porous mediva:A theory of drying", Advance in Heat Transfer, Vol. 13, 119-203, 1977.[4] Krischer, 0., " The heat, moisture, and vapor movement during drying porousmaterials", VDIZ., Beih. 1, 17-24, 1990.[5] Philip, J.R., and De Vries, D.A, "Moisture movement in porous materials undertemperature gradients", Trans. Am. Geo. Union, 38:222-232, 1957.[6] De Vries, D.A., "Simultaneous transfer of heat and moisture in porous media",Trans. Am. Geo. Union, 39: 909-916, 1958.[7] Luikov, A.V., "Heat and mass transfer in capillary-porous bodies", Pergamon, Ox-ford, 1966.[8] Luikov, A.V., "System of differential equations of heat and mass transfer in capillaryporous bodies (review)", Int. J. Heat Mass Transfer 18: 1-14, 1975.[9] Cary, J.W., and Taylor, S.A., "The interaction of the simultaneous diffusion of heatand water vapor", Soil Sci. Soc. Am. Proc. 26:413:416, 1962.[10] Huang., C.L.D., "Multi-phase moisture transfer in porous media subjected to tem-perature gradient", Int. J. Heat Mass Transfer, 22:1295-1307, 1979.[11] Eckert, E.R.G., and Faghri, M., "A general analysis of moisture migration causedby temperature differences in an unsaturated porous medium", Int. J. Heat MassTransfer, 23: 1613-1623, 1980.[12] Jespersen, H.B., "Thermal conductivity of moisture materials and its measure-ments", J. Inst. Heat Vent. Engrs 1, pp. 216-222, 1953.[13] Joy, F.A., "Symposium on thermal conductivity measurements and applications ofthermal insulations", ASTM STP 217 65-80, 1957.98Bibliography^ 99[14] Langlais, C., Hyrien,M., and Klarsfeld,S., "Moisture migration in buildings", ASTMSTP 779 192-206, 1982.[15] Bomberg, M. and Shirtliffe, C. J., "Influence of moisture gradients on heat transferthrough porous building materials, thermal transmission measurement of insula-tion", ASTM STP 660, R.P. Tye, ed., American Society for Testing and Materials,211-233, 1978.[16] Kumaran, M. K.," Moisture transport through glass-fibre insulation in the presenceof a thermal gradient", J. of Thermal Insulation, 10:243-255, 1987.[17] Kumaran, M. K., "Comparison of simultaneous heat and moisture transport throughglass-fiber and spray-cellulose insulations", J. of Thermal Insulation, 12 :6-16, 1988.[18] Langlais, L. C., Hydrien, M. and Klarsfeld, S.," Influence of moisture on heat trans-fer through fibrous insulating materials, thermal insulation, materials and systemsfor energy conservation in the 80's." ASTM STP 789, Govan, F. A., Greason, D. M.and McAllister, J. D., eds., American Society for testing and materials, 563-581,1983.[19] Langlais, C. and Klarsfeld, S., " Heat and mass transfer in fibrous insulation", J.Thermal Insulation, 8:49-80, 1984.[20] Thomas, W. C., Bal, G. P. and Onega, R. J.," Heat and mass transfer in glass fibreroof insulating materials", ASTM STP American Society for Testing and Materials,pp. 582-601, 1984.[21] Modi, D. K., and Benner, S. M.," Moisture gain of spray applied insulations and itseffect on effective thermal conductivity - Part I", J. of Thermal Insulation, 8:259-277,1985.[22] Benner, S. M. and Modi, D. K., "Moisture gain of spray applied insulations and itseffect on effective thermal conductivity - Part II", J. of Thermal Insulation, 9:211-223, 1986.[23] Wijeysundera, N. E., Hawlader, M. N. A. and Tan, Y. T., "Water vapour diffusionand condensation in fibrous insulations", Int. J. Heat Mass Transfer, 32(10): 1865-1878, 1989.[24] Wijeysundera, N. E., Hawlader, M. N. A. and Lian, S. C.," An experimental studyof condensation in fiberglass insulations", Paper FE89-32 ASHRAE Far East Con-ference on Air Condition in Hot Climates, Kuala Lumpur, Malaysia, October 25-28,1989.Bibliography^ 100[25] Wijeysundera, N.E., and Hawlader, M.N.A., "Effect of condensation and liquidtransport on the thermal performance of fibrous insulations", Int. J. Heat MassTransfer, 35: 2605-2616, 1992.[26] Motakef, S. and El-Masri, M. A., " Liquid diffusion in fibrous insulation", ASME,J. Heat Transfer, 107:229-306, 1985.[27] Timusk, J., and Tenender, L.M., "Mechanism of drainage and capillary rise in glassfibre insulation", J. Thermal Insulation, 11: 231-241, 1988.[28] Cid, J., and Crausse, P., "Influence of the structural characteristic of fibrous heatinsulators upon their properties of moisture transfer", J. Thermal Insulation, 14:123-134, 1990.[29] Hedlin, C. P., "Heat transfer in a wet porous thermal insulation in a flat roof', J.of Thermal Insulation, 11:165-188, 1988.[30] Dinulescu, H.A., and Eckert, E.R.G., "Analysis of the one-dimensional moisturemigration caused by temperature gradients in porous medium", Int. J. Heat MassTransfer, 23: 1069-1078, 1980.[31] Ogniewiez, Y. and Tien, C. L., "Analysis of consensation in porous insulation" , Int.J. Heat Mass Transfer, 24:421-429, 1981.[32] Motakef, S. and El-Masri, M. A., "Simultaneous heat and mass transfer with phasechange in a porous slab", Int. J. Heat Mass transfer, 29(10): 1503- 1512, 1986.[33] Vafai, K., and Sarkar, S., "Condensation effects in a fibrous insulation slab", ASMEJ. Heat Transfer, 108:667-675, 1986.[34] Shapiro, A. P. and Motakef. S., " Unsteady heat and mass transfer with phasechange in porous slab: analytical solutions and experimental results", Int. J. HeatMass Transfer, 33(1): 163-173, 1990.[35] Vafai, K and Witaker, S., " Simultaneous heat and mass transfer accompanied byphase change in porous insulation", Trans. ASME, J. Heat Transfer, 108:132-140,1986.[36] Vafai, K and Tien, H. C., " A numerical investigation of phase change effects inporous materials", Int. J. Heat and Mass Transfer, 23(7): 1261-1277, 1989.[37] Tien, H.C., and Vafai, K., "A synthesis of infiltration effects on an insulation ma-trix", Int. J. Heat Mass Transfer, 33:1263-1280, 1990.Bibliography^ 101[38] Tao, Y. -X., Besant, R. W. and Rezkallah, "Unsteady heat and mass transfer withphase changes in an insulation slab: frosting effects", Int. J. Heat Mass Transfer,34(7): 1593-1603, 1991.[39] Edwards D. K., Denny, D. E. and Mills, A. F., "Transfer process: a introduction todiffusion, convection and radiation", New York, McGraw-Hill, 1979.[40] Woodside, W., "Probe for thermal conductivity measurements of dry and moistmaterials", Heat pipe. Air Cond. 30, pp. 163-170, 1958.[41] Crausse, P., Bacon, C. and Langlais C., " Experimental and theoretical study ofsimultaneous heat and moisture transfer in a fibrous insulation", J. of ThermalInsulation, 9:46-67, 1985.[42] Patankar, S.V., "Numerical heat transfer and fluid flow", New York, McGraw-Hill,1980.[43] Batty, W.J., O'Callaghan P.W. and Probert, S.D, "Apparent thermal conductivityof glass-fiber insulant: effects of compression and moisture content", Applied Energy,9:55:76, 1981.[44] Pallady, P.II. and Handley, P.J., "Evaluating moist air properties", Chemical Engi-neering, 10:, 1984.[45] Defay, R., Prigogine, I., and Bellemans, A., "Surface tension and adsorption", Wiley,New York, 1966Appendix AWater Vapor-Air Mixture DiffusionA.1 Vapor Diffusion Coefficient in Porous MediaThe mass transfer of a water vapor-air mixture within the pores of a porous mediumoccurs by two mechanisms: ordinary diffusion and Knudsen diffusion [39]. The Knudsendiffusion refers to the case that the pores are small or the gas density is very low, thereforethe molecules collide with the pore walls more frequently, and diffusion of molecules alongthe pore wall is described by the equations for free molecule or Knudsen flow.Porous insulation usually has a very high porosity. The vapor-air mixture is relativelydense. The vapor diffusion in porous insulation is dominated by ordinary diffusion whichcan be described by Fick's law.The vapor-air mixture diffusion coefficient in porous insulation, therefore, can betaken to be [39]:Dv* = 5-9-D12)T(A.1)where D12 is the binary diffusion coefficient of Pick's law; eo is the porosity; T is thetortuousity, accounting for the increased diffusion length due to tortuous paths of realpores, and for the effects of constrictions.Diffusion coefficients of gases at low pressure are almost composition independent, andincrease with temperature. According to reference [23], the binary diffusion coefficientfor water vapor-air mixture is given by:102Appendix A. Water Vapor-Air Mixture Diffusion^ 103D12 = 1.97(10)-5( 7'255.2 ')1.685^ (A.2)A.2 Mass Transfer Coefficient in AmbientTo determine the mass transfer coefficient between the ambient air and the insulationslab, an analysis of a slug flow over a slab surface is conducted. According to reference[39], the solution for the average heat transfer coefficient over a length of the flat surfaceis:2* Vh* =k^Y/270/2 atL (A.3)A similar solution can be derived for the average mass transfer coefficient over a lengthL:2p* D* Vh* = ^a 12  (  m  )1/2m^71/2 '1/4 DI2Lwhere V,n is the main stream velocity.Comparing the above two expressions, one could obtain:(A.4)14,^P:1)12( a* )1/2.^ (A.5)h*^k D12Note that (a*/D:) is the Lewis number, which characterizes the ratio of thermal andmass diffusion. For quasi-steady phase, the Lewis number may be assumed to be unity.Wijeysundera et al. [23] gave a modified expression for mass transfer coefficient:*^h*^p:D12m1047) k* ).(A.6)Appendix A. Water Vapor-Air Mixture Diffusion^ 104A.3 Calculation of Vapor Concentration in AmbientThe vapor concentration (or vapor density) of the ambient air needs to be calculated.This is done based on the saturated vapor pressure and relative humidity in the section.The equations for the determination of the saturated vapor pressure were obtained byusing a mathematical fit of the pyschometric chart by Pallady and Hanley [44].KD = —8.83(10)-13TR3 + 3.07(10)-6TR2 — 3.47(10)-3TR+ 4.4,^(A.7)F = KD(1 — 1165.7/TR) + 5,^ (A.8)PD = 1.668(10)F,^ (A.9)where, PD is the saturated vapor partial pressure in mm of mercury, and TR is temper-ature in °R.The vapor partial pressure in bar is:p: = RH PD760 .The vapor density in ambient can be obtained from gas state equation:P: = P:RvT.(A.10)(A.11)The concentration of vapor is:P: mv =^ (A.12)The air density can be obtained from gas state equation by assuming the total pressureis 1 atmospheres.Appendix BThermal Conductivity of Moist InsulationThe presence of moisture has a significant effect on apparent thermal conductivity ofporous insulation. It is believed that the apparent thermal conductivity of moist insula-tion depends on the manner that the liquid distributes inside the insulation [43].Four different physical models for liquid distribution in a porous medium are consid-ered in the present study.(a) Bead arrangement.In this model liquid is located as small beads throughout the insulant. The apparentthermal conductivity can be represent by:wherek* (€13) = Ic:1(1 eon(1 — esY) (Pij —y^(leh 21c:1)(B.13)(B.14)(b). Series arrangement.In this model the liquid is distributed in layers perpendicular to the direction of heatflow. The liquid would then have the maximum effect in inhibiting the heat flow.For this arrangement,41q; k* (€0) =IC 13 A(B.15)105Appendix B. Thermal Conductivity of Moist Insulation^ 106whereCe *^aX= —€7(ko — kd). (B.16)(c). Parallel arrangement.In this model the liquid is located in a continuous shape parallel to the direction ofheat flow.For this arrangement,wherek*(ep) = lc: + X,X = —" (Ivo* — kd*).el.(d) Form arrangement.In this model a liquid film surrounds each insulant fiber or particle; the films therebyforming a honeycomb structure.The same formula as that for bead arrangement can be used for the form arrangement.


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