INFORMATION REQUIRED FOR ANALYSIS
CHAPTER 6--ADVANCED CHAPTER 6--ADVANCED NUMERICAL MODELS 93
1-D
I Hygrothermal Models
3 [ 2-D, 3-D ]1 I Stochastic
Fields: Pv, T, Constant Properties
l ....
Fields: Pv, T, Variable Properties
I :=
,i
J..J- !
Deterministic
Variable Properties Fields: Pv, T, Psuc,
Var ab e Properties I FieldSpsPVc,T, Pro=, Variable Properties
Stochastic Material i
Properties
Stochastic i Boundary Conditions
Fields: Pv, T, P=ot, Psuc, Building System
and Sub-System Effects, Variable Properties
FIG. 2--Classification including sub-system and system effects and durability.
the authors only correctly apply the model in applications.
In some instances, the preparation period for initiating even a simple 2-D simulation may requite several days or even months if additional material property data, laboratory sys- tem, or sub-system characterization of the envelope is needed. These models require not only a higher level of ex- pertise in preparing a simulation, but also significantly more sophisticated computational as well as experimental facili- ties. The computational time increases very non-linear as one goes from 1-D to 2-D, and from 2-D to 3-D simulations.
As of early 2001, the simulation of the 3-D transient moisture performance of a building envelope system does not exist; a model is not available. In some instances, the numerical re- quirements of an advanced moisture engineering model are difficult to acquire even with the more sophisticated com- mercial computational fluid dynamic (CFD) models.
The following section will give details on the current status of advanced hygrothermal numerical models, the essential attributes, the inputs, concerns, and outputs. Authors of some advanced hygrothermal models were contacted and in- vited to include relevant information on their models. Illus- trative examples are presented in Appendix 1 (Appendices A- K) at the end of the book.
Theoretical Background
Porous Medium
A porous construction material (medium) may contain a multi-phase assembly of solids, liquids, and gas. Heat and mass transfer essentially occurs at the microscopic level for each phase. To resolve the transport mechanisms at the mi- croscopic level, a detailed description of geometry and to- pology of the porous medium is required. The description of the porous medium and the associated transport processes at the microscopic level is difficult and complex. A contin- u u m approach is commonly employed to avoid analysis at the molecular level and to permit analysis at the macro- scopic level. The continuum approach resorts to phenome- nologically determining the mass/thermal transport coeffi- cients from mass/thermal fluxes imposed experimentally on a representative elementary volume (REV) (see Fig. 3).
Employing this basic assumption allows the development of differential balance equations for mass, momentum, and energy transfer. The porous medium may then be described by three distinct phases, the solid, s, the liquid water, l, and the gas phase, g. According to Whittaker [12], by employing the spatial average of a microscopic property P at a point m, the microscopic property P over an elementary volume can be defined as an average of:
V
>~
AV 1 A V
FIG. 3 - - A schematic of a representative elementary volume (REV) (left). The fluid volume is continuous. The density depends on the size of volume (AV) (right).
(p~ = ~ PdV
The phase average over a microscopic property Pi is given
by:
Ifv 1~
(P,) = ~ e i d V = ~, VflV i = s, t, g
This definition implies that for a constant microscopic prop- erty, Pi, the value of the phase average (Pi) is different from the local value of the microscopic property Pi. The macro- scopic value of the property must be independent of the size, the shape, and the orientation of the elementary volume V that is employed in the conceptualization [13]. The smallest elementary volume that satisfies this condition [14] is the representative elementary volume (REV).
(P) = limv_v~ v p Pdg
Whitaker [15] limits the validity of the averaging process as long as:
d ~ l ~ L
where d is a microscopic characteristic length over which significant variations in the mathematical point properties occur, l is a characteristic length of the representative ele- mentary volume VREV, and L is a macroscopic characteristic length. The use of the REV is convenient to describe the averaging procedure to obtain a field of macroscopic prop- erties for each phase. The macroscopic properties of many rigid and homogenous geometrical and topological porous materials are single-valued functions of time and space. The REV averaging concept is an important assumption in any modeling of mass transfer in porous media, as it provides upper and lower bounds of length scales in the modeling domain.
LOCAL T H E R M O D Y N A M I C E Q U I L I B R I U M In the representative elementary volume of Fig. 3, tempera- ture differences may exist between the fluids in the pores and the solid material. The assumption of local thermodynamic equilibrium imposes negligibly small temperature differ- ences between the various phases at any particular location,
relative to the applied temperature differences of the system.
This assumption is used by almost all advanced hygrother- mal models.
In the cases where large latent heat transfer occurs, such as during phase change of vapor to liquid or vice versa, the assumption of thermodynamic equilibrium does not hold ex- actly. Similarly, when the boundary conditions at the faces of the building envelope change significantly over a short time, or when the thermal properties of the solid and fluid phases are significantly different, serious deviations from thermodynamic equilibrium may be present. Adopting the thermodynamic equilibrium assumption in hygrothermal models allows the lumping of all phases with respect to ther- mal transport and reduces the number of governing differ- ential equations and interphase couplings. This limiting assumption of thermodynamic equilibrium is employed throughout this chapter and reflected in the presentation of the governing equations for heat and mass transfer.
T R A N S P O R T M E C H A N I S M S
The objective of this section is to provide a sample set, but not an exhaustive theoretical review of the definition of the transport mechanisms present in hygrothermal processes in porous media. Works by Kuenzel [16], Krus [17], Kohonen [18], Janssens [19], and Luikov [20] provide details on the theoretical development of various transport potentials. The choice of moisture transport potential was made based on what was familiar to the author. Transport mechanisms may also be defined for moisture flow using chemical potential t**, or even concentration gradient X, and a similar theoret- ical development is required.
In the past, many moisture transport models were devel- oped based on discontinuous potentials such as moisture content [20,21]. The moisture content at an interface be- tween two materials is discontinuous. This requires an elab- orate flux analysis developed at each inhomogeneous mate- rial intersection, and an iterative treatment is required to satisfy mass conservation. The main advantage in employing moisture content as a mathematical transport potential is that the liquid diffusivity is directly measured as a function of moisture content. LATENITE [22] and TRATMO2 [23]
employ this approach in their formulation. Recently, contin-
C H A P T E R 6 - - A D V A N C E D N U M E R I C A L M O D E L S 95 uous driving potentials have been employed successfully
[16,24,25]. Examples are vapor pressure, suction pressure, relative humidity, and chemical potential. They appear to be more efficient in both formulating the governing equations and execution time. The issue of flux splitting, the separation of the quantity of mass that is transported in vapor or liquid phase, becomes a focal point for differences among all ad- vanced hygrothermal moisture models. Apparently, a straightforward approach has not yet been recognized and each model employs some limiting assumptions.
Mass Transfer
An excellent description of the transport mechanism in po- rous media is described by Kerestecioglu et al. [26] and Ka- viany [27]. It is important to recognize that the transport coefficients may not only be strong functions of the indepen- dent variable, but may change as a function of time and ex- posure. Some of the more important modes by which mois- ture may be transported are:
Molecular vapor diffusion, by partial vapor pressure gradients.
Molecular liquid diffusion, movement of the liquid phase due to liquid filled pores.
Capillary liquid flow, movement of the liquid phase due to capillary suction pressures.
Knudsen vapor diffusion, movement of the vapor phase in small pores and at low pressures; the mean free path is greater than the pore diameter and collisions of molecules with the pore walls occur more frequently than collisions with other diffusing molecules.
Evaporation-condensation vapor flow, movement occurs in conjunction with heat transfer, moisture evaporates and recondenses in a similar fashion to a heat pipe.
Gravity-assisted diffusion liquid flow, movement occurs due to gravity and occurs mostly in macroporous materials.
Vapor Transport
The diffusion of water vapor under isothermal conditions may be described by Fick's first law for unimpeded flow in still air:
qv = - D ~ V X
where qv is the mass flux rate of vapor flow (kg/m 2 9 s), D v the diffusion coefficient of vapor in air m2/s, and X the vapor concentration (kg/m3).
M v q" = - D r R T Pv
where R is the universal gas constant (8.314 J/(mol 9 K)), Pv is the partial vapor pressure, T is temperature (K), and M is the molar weight of water (0.018 kg/mol).
In a porous material, diffusion is reduced in comparison to that in still air by a resistance that corresponds to the volume fraction of air-filled open pores a and a tortuosity factor a. This is expressed as:
qv = - ~ a Dv--~T V Pv
In most European countries a resistance factor is introduced
as ix = (1/e~ a) leading to the following flux equation for vapor flow:
D v M v p ~ qv - p. R T
In the measurement of the water vapor permeance using the ASTM E 96 standard or the EN 12086:1997, the factor 5v is
D~ M
used, and that is equivalent to -~- ~--~. This transport coeffi- cient is termed as the water vapor permeability and has units of kg/Pa 9 m 9 s.
Liquid Transport
Liquid flow is defined in two ways since it is transported differently within the two regions of interest in building ma- terials. The first region is defined as the capillary water re- gion. It follows the hygroscopic sorption region and extends until free water saturation. This region can be characterized by states of equilibrium. Liquid transport occurs under the influence of a suction pressure or force in the capillary re- gime, and moisture liquid transport occurs mainly in this region. The second region, the supersaturated capillary re- gion, follows the capillary water region. Normal suction pro- cesses are not physically plausible in it. Liquid flow in this region occurs through diffusion under a temperature gradi- ent or by external pressure under suction. In this region, there are no states of equilibrium.
In 1856, Darcy [28] developed the theory of laminar trans- port in capillary tubes, which applies directly to suction flow through building materials. Extending the original formu- lation, to account for gravity forces, the transport of liquid in the capillary regime is given by
qw : - D , V 4) + OP~--h- Ou
where q~ is the mass flux of liquid (kg/m 2. s), D+ is the liquid coefficient (kg/m 9 s), g is the gravitational acceleration (m/s2), and p,~ is the density of water (kg/m3). The suction pressure is usually described by employing a cylinder capillary model and can be presented as:
2 (r cos 0
P h - - - -
r
where a is the surface tension of water (72.75 10 3 N/m at 20~ r the capillary radius (m), and 0 the contact angle or wetting angle (~
Using thermodynamic equilibrium conditions for a cylin- der capillary model, the relationship between relative hu- midity qb over a concavely curved water surface and the cap- illary pressure P~ is defined by Kelvin's equation:
- G
where Pw is the density of water (kg/m3), Ph is the capillary pressure (Pa), R~ is the gas constant for water vapor (J/kg K), temperature T is in Kelvin (K) and d) is relative humidity (-).
Heat Transfer
W i t h i n c o n s t r u c t i o n envelopes, h e a t t r a n s f e r m a y o c c u r b y c o n d u c t i o n , convection, a n d r a d i a t i o n transfer. Table 1 lists the g o v e r n i n g e q u a t i o n s of state for t h e s e m o d e s of h e a t transfer. The t h e r m a l c o n d u c t i v i t y k is a f u n c t i o n of t h e con- t e n t o f ice a n d l i q u i d p r e s e n t in t h e p o r o u s m a t e r i a l a n d m a y be s t r o n g l y i n f l u e n c e d b y t e m p e r a t u r e . I n a d d i t i o n , t h e ther- m a l c o n d u c t i v i t y m a y also b e d i r e c t i o n a l l y d e p e n d e n t .
P h a s e Change
The p h a s e c h a n g e c o n v e r s i o n e n t h a l p i e s c o n t r i b u t e a local s o u r c e of h e a t t h a t is s t o r e d o r r e l e a s e d in a p o r o u s m a t e r i a l w h e n m o i s t u r e a c c u m u l a t i o n o r d r y i n g is present. If one c o n s i d e r s t h a t the a m o u n t of m o i s t u r e I~i o f p h a s e i is con- v e r t e d to j t h e c o n t r o l v o l u m e receives t h e following a m o u n t of heat:
qt = Ahii " Iii
w h e r e Ah~i is the c h a n g e in enthalpy, 3.34 9 105 J / k g for con- v e r s i o n of ice to w a t e r a n d 2.45 9 106 J / k g at 20~ for w a t e r to vapor.
This q u a n t i t y of h e a t m a y be significant w h e n d r y i n g o r a c c u m u l a t i o n is p r e s e n t in p o r o u s m e d i a . A m o d e l i n g chal- lenge exists to p r o p e r l y a c c o m m o d a t e this l a t e n t h e a t t e r m in t h e g o v e r n i n g e q u a t i o n of energy.
Air Transport
Airflow is d r i v e n b y a d i f f e r e n c e of a i r p r e s s u r e . The m a s s flux of a i r t h r o u g h a p o r o u s m a t e r i a l m a y he e x p r e s s e d as:
m a = - k a V P ~
w h e r e m~ is the a i r m a s s flux (kg/m2s), k, is t h e a i r p e r m e - a b i l i t y ( k g / m 9 s Pa), a n d Pa is the a i r p r e s s u r e (Pa).
Governing E q u a t i o n s for Mass, M o m e n t u m , a n d Energy Transfer in a P o r o u s M e d i u m
A d v a n c e d m o d e l s r e q u i r e t h e s i m u l t a n e o u s s o l u t i o n of the heat, mass, a n d m o m e n t u m t r a n s f e r e q u a t i o n s . The govern- ing e q u a t i o n s a r e w r i t t e n for a h y g r o s c o p i c - c a p i l l a r y p o r o u s m e d i u m . The c o n s e r v a t i o n e q u a t i o n s for t h e p o r o u s m e d i u m will p r o v i d e m o i s t u r e c o n t e n t a n d t e m p e r a t u r e as a f u n c t i o n of s p a c e a n d t i m e of the m a t e r i a l . The m a c r o s c o p i c o r vol- u m e a v e r a g e e q u a t i o n s are d e r i v e d w i t h r e s p e c t to a v e r a g i n g v o l u m e s large e n o u g h for a c o n t i n u u m a p p r o a c h .
A s s u m p t i o n s
Several a s s u m p t i o n s a r e n e c e s s a r y for this d e v e l o p m e n t a n d m u s t be a c k n o w l e d g e d as t h e l i m i t a t i o n s of existing ad- v a n c e d models:
1. The m a t e r i a l is m a c r o s c o p i c a l l y h o m o g e n e o u s .
2. The solid p h a s e is a r i g i d m a t r i x , a n d t h e r m o p h y s i c a l p r o p e r t i e s a r e c o n s t a n t s w i t h space.
3. E n t h a l p y o f e a c h p h a s e is a f u n c t i o n of t e m p e r a t u r e a n d m o i s t u r e .
4. C o m p r e s s i o n a l w o r k a n d viscous d i s s i p a t i o n a r e negli- gible for e a c h p h a s e .
5. Diffusional b o d y force w o r k a n d k i n e t i c e n e r g y a r e small.
6. The gas p h a s e is a b i n a r y m i x t u r e of ideal gases.
7. The t h r e e - p h a s e s y s t e m is in local t h e r m o d y n a m i c equi- l i b r i u m ( s o l i & v a p o r - l i q u i d ) .
8. G r a v i t y t e r m s a r e i m p o r t a n t for the l i q u i d p h a s e m a s s t r a n s f e r b u t n o t the gas p h a s e m a s s transfer.
9. F l u i d s a r e N e w t o n i a n a n d i n e r t i a l effects a r e small.
10. The t r a n s p o r t p r o c e s s e s a r e m o d e l e d in a p h e n o m e n o - logical way.
A c c o r d i n g to the m a n n e r t h e a x i o m s of c o n s e r v a t i o n for t r a n s p o r t process, the r a t e of s t o r a g e of a n y e n t i t y w i t h i n a c o n t r o l v o l u m e at a n y given t i m e equals t h e r a t e of this e n t i t y e n t e r i n g the c o n t r o l v o l u m e t h r o u g h the s u r r o u n d i n g sur- faces p l u s the r a t e of g e n e r a t i o n of the e n t i t y w i t h i n the vol- u m e .
M a s s C o n s e r v a t i o n
C o n s e r v a t i o n o f m a s s of a i r a n d m o i s t u r e c a n b e e x p r e s s e d
a s :
A i r
opa
- - _ _ ~ m a
ot M o i s t u r e
0(upo)
- - 0t = -Vqv - Vqw - V(pv u) E n e r g y C o n s e r v a t i o n
C o n s e r v a t i o n of e n e r g y c a n be e x p r e s s e d as:
0(oCT)
- - - -Xrqc - Vqa - h~ Vq~ + qs at
w h e r e qs is the r a t e of h e a t g e n e r a t i o n p e r v o l u m e W / m 3 a n d i n c l u d e s the l a t e n t h e a t d u e to freezing, a n d o t h e r h e a t sinks o r sources.
T A B L E 1 - - E n e r g y t r a n s p o r t .
C o n d u c t i o n C o n v e c t i o n R a d i a t i o n
q~ -- - k V T w h e r e
k - t h e r m a l c o n d u c t i v i t y ( W / m K) T = T e m p e r a t u r e (~
qo - v povC, f
w h e r e
Pa - D e n s i t y o f a i r (kg/m3), v = v e l o c i t y (m/s),
C a = v o l u m e t r i c h e a t c a p a c i t y (J/m 3 K).
% = E ~r F(T~ T 4) w h e r e
e = e m i s s i v i t y o f g r a y s u r f a c e ( - ) ,
tr - S t e f a n B o l t z m a n n c o n s t a n t - (5.67 • 10 -s W / m 2 K4), F - v i e w f a c t o r ( - ) ,
T~, = s u r f a c e t e m p e r a t u r e (K), a n d T~ = s u r r o u n d i n g t e m p e r a t u r e (K).