Session3-3.pdf

A MODEL FOR MOISTURE-CONTENT EVOLUTION IN POROUS BUILDING

ELEMENTS: TREATMENT OF IRREGULAR REGIONS

Regina Katsman ,

NRC/IRC, Bld. M-24, Montreal Rd., Ottawa

ON K1A 0R6, Canada

Rachel Becker

Department of Civil Engineering, Technion

Israel Institute of Technology, Haifa, 32000,

Israel

Becker@tx.technion.ac.il

Regina.Katsman@nrc.ca

ABSTRACT

The effects of moisture movement and residual

moisture content are becoming a recognized part of a

performance based design of buildings. Despite the

long-time existence of a basic 3-D formulation of heat

and mass transfer within porous materials, the

generalized hygrothermal behavior of construction

elements that contain hygrothermal bridges and/or air

voids has not been studied. Neither have the processes,

which occur at the interfaces between different

materials, been investigated. Relying on basic energy

and mass conservation principles, a general integral

form of presentation was applied to establish the

coupled heat and mass transfer field equations, which

were further transformed to a discrete set of algebraic

equations. This form of presentation is more suitable

for treatment of irregular regions, such as material

interfaces and internal air voids. To treat these regions,

specific mathematical procedures were implemented.

Two Software packages were written to simulate the

evolution of moisture content in non-homogeneous flat

planar walls, without and with air voids. Finally, the

results of modeling are presented for both types of

constructions, thus enabling one to analyse a

hygrothermal behavior of irregular wall's regions.

building material [4 to 9]. Despite the long-time

existence of a basic 3-D formulation of heat and mass

transfer within porous materials, the generalized

hygrothermal behavior of construction elements that

contain hygrothermal bridges and air voids has not been

studied yet. Neither have the processes, which occur at

the interfaces between different materials, been

investigated. Moreover, the effects of commonly

existing air-voids (in the form of large hollow-cores or

small deffects) within the materials composing the wall,

or at the interfaces between the materials (as usually

stems from the construction methods for the jointing),

have not been studied either.

This paper is devoted to the mathematical modeling and

numerical solution of the moisture-content evolution in

walls with hygrothermal bridges, including the case of a

construction with air voids, concentrating mainly on

methods of mathematical treatment of irregular wall's

regions.

MATHEMATICAL MODEL

The basic physical considerations and previous research

indicate that the mutual influence of the heat and mass

transfer is an inherent property of the explored system.

Moreover, the mutual influence of these phenomena on

the moisture-content evolution may be particularly

significant at the boundaries, where solid meets humid

air with variable humidity, and at interfacial zones

between different materials.

Generally, the above-mentioned Philip and De-Vries

differential equations are actually an outcome of the

energy and mass conservation principles. However, for

the present aims, the integral form of presenting these

basic priciples is the most suitable, as it enables easy

inclusion of interfaces between different materials,

internal air-voids and their internal boundaries.

INTRODUCTION

For most planar building elements (usually designated

by the general term walls), the main moisture

movement is in the direction perpendicular to the

element’s air-exposed surfaces. However, the non-

homogeneous structure of the element within its main

plane causes multidirectional moisture movement. The

general formulation of simultaneous heat and mass

transfer in a porous material has been performed by

Luikov [1] in order to analyze moisture transport

phenomena in soils. It has then been implemented for

building materials by Philip and De-Vries [2] and since

then by others [3,4]. The field equations, which

emerge from this formulation, have been used since

then for analyzing various cases. These works include

investigations of 1-D heat and mass transfer phenomena

in building elements composed of a homogenous

General Scheme

The heat and mass field equations at every material

point are derived from the basic energy and mass

conservation principles:

(1)

∂

⋅

(2)

∂

Where:

is the heat stored in an arbitrary control-

volume V (with a total surface area S) during

the time interval dt

(7)

∂

⋅

∂

is the total resultant in-flowing heat across S

caused by the temperature gradient

(8)

Conducting several transformations, the field equations

are thus given by:

n (at

the surface, with the normal n to S pointing

outwards at every point on S)

∂

is the total resultant heat produced within the

volume V by internal heat sources and sinks

(e.g. due to heat of hydration of cementitious

compounds)

⋅

∂

⋅

∂

is the total resultant in-flowing latent heat

(associated with vaporization and

condensation processes) across S, caused by

the partial water-vapor pressure gradient

∂

⋅

∂

(at the surface, with the normal n as above)

(9)

is the total moisture stored in the same volume

V during the time interval dt

∂

⋅

∂

is the total resultant moisture produced within

the volume V by internal moisture sources and

sinks ( e.g. due to the hydration process of

cementitious materials)

∂

⋅

∂

is the total resultant in-flowing moisture across

S caused by the temperature gradient

(10)

The system of partial differential equations for the

simultaneous heat and mass transfer at a regular point

within the porous material, which is obtained from the

integral equations 9 and 10 when

is the total resultant in-flowing moisture across

S caused by the moisture-gradient

∂

n (at

V , corresponds

to the classical Philip and De-Vries formulations and to

those given by the IEA Annex 24. However, as

presented in the following sections, the integral

presentation is more suitable for the general

development of the equations in irregular regions, such

as material interfaces and internal voids. Moreover, it is

more amenable for establishing the numerical

integration procedures at irregularities and at all the

internal and external boundary points.

→

the surface, with the normal n as above).

For the case without internal sources or sinks,

delineation of each factor then yields the following

integral presentations:

∂

= V

⋅

∂

(3)

∂

⋅

∂

Analytical Treatment of Irregular Regions

The following continuity equations emerge at materials’

interfaces when the zero-limit is reached for two

infinitesimal control volumes that have a mutual

interface S:

(4)

∂

⋅

∂

(5)

−

∂

(11)

= V

⋅

∂

−

(12)

(6)

∂

Derivation of the equations for an element with air-

voids is now straightforward. Figure 1 depicts the

schemes for the relevant control volumes within the

regular porous material (Volume 1) and within the air-

void (Volume 2).

[

(

⋅

)

∂

⋅

]

∂

−

∂

[

(

⋅

)

∂

Volume 2

∂

⋅

]

∂

(13)

∂

Volume 1

[

⋅

∂

⋅

]

∂

−

Figure 1: Scheme of general control-volumes within

the element’s regular material region (Volume 1)

and an air-void (Volume 2).

∂

[

⋅

∂

As a first order approximation, it is assumed that within

the void the air temperature and partial water-vapor

pressure distributions are uniform, except for an

extremely thin boundary layer at the interface with the

surrounding solid. Thus, across the void the

temperature and partial water-vapor pressure are

represented by unique values,

∂

⋅

]

∂

(14)

The following boundary-condition equations emerge at

material-air boundaries when the zero-limit is reached

for an infinitesimal control volume with an air-exposed

surface S:

p . Wherever

part of the solid material continuum is replaced by an

air-void, equations 9 and 10 re-written for the total

entrained air volume

T and

V that is bounded by the

S (with a normal n into the porous medium).

After substitution of 15 and 16 this yields:

surface

∂

(

⋅

)

⋅

(15)

∂

⋅

∂

⋅

∂

⋅

∂

⋅

∂

⋅

−

∂

⋅

(16)

(17)

∂

⋅

∂

∂

i,j+1

⋅

∂

dy(j)

∂

⋅

∂

i-1,j

i+1,j

i, j

−

⋅

dy(j-1)

(18)

This model does not take into account gravitational

moisture movement that occurs in pores or cracks

outside the capillary range, or within vertical air-voids.

Neither does it address the convection inside the porous

material, effects of direct rainfall, solar radiation or air

transport.

dx(i-1)

i,j-1

dx(i)

Figure 2: Scheme of the mesh at a general point i,j.

The integration points (nodes i,j) are on an orthogonal

grid, with main grid line along all material boundaries.

The grid is composed between its main lines by a

variable-size mesh. The control volume around every

node extends until the mid-distance to the next mesh-

node in every direction (rectangle ABEFDCHG in

Figure 2). Each of the four parts of the control volume,

1 to 4, may be composed of a different material.

NUMERICAL SOLUTION

General Scheme

The solution procedure adopted here is based on the

presentation of the integral equations in a discrete form,

including the boundaries and air-voids. An implicit

scheme was chosen for the numerical integration

process. The ensuing discrete set of algebraic equations

with non-constant coefficients is solved by means of an

iterative convergence process. In addition, at this stage

the solution was limited to the case of an orthogonal x-y

coordinate system, as most building walls have only

orthogonal internal and external surfaces.

Using an implicit scheme for the linearized set of

equations provides the stability of the numerical

solution without dependence on time and space steps.

In particular, for the iterative process used to solve the

equations with variable coefficients this scheme is

preferred because of its monotonous behavior. The

examples presented in this paper were solved using the

“optimal” time steps that stem from the spatial mesh,

which was chosen in accordance with the geometric

features of the analyzed construction, where the term

“optimal” refers in this case to the time step that would

be recommended as maximal in order to render an

explicit

Numerical Treatment of Irregular Regions

In establishing the field and boundary condition

equations, the temperature and moisture content within

each part are presented by their values at the node i,j. If

at a certain node nm different materials meet (nm

≤

4),

moisture-content in these materials at the node,

W ,

(

k=1,…,nm), is different, while the temperature,

T ,

and the partial water-vapor pressure,

p ,

, are identical

, are also identical).

The different moisture-contents at the node, generally

presented by the four points 1 to 4, are inter-dependent.

Their dependence is obtained via the sorption isotherms

of the different materials. The following procedure was

used for instantaneous linearization of the discrete set

of equations: At every iteration-step, m, undertaken

within a given time step,

i ,

scheme

stable

(i.e.

dt , from n to n+1 (with n=0

t , where dn designates a

mesh step in any direction and

∆

min[(

)

⋅

)]

at t=0):

max

∂

any of the relevant

diffusivity coefficients in the equations).

The scheme for the numerical model is shown in Figure

(

)

(

)

⋅

(

)

∂

∂ϕ

∂

(

⋅

)

⋅

(

)

∂ϕ

∂

(i.e. Relative-Humidity values,

∂

EXAMPLES

Two examples are given to demonstrate the effects

occurring in the irregular regions that are highlighted by

the suggested 2-D analysis for a drying process of a

block-wall with joints and surface-renderings. These

examples are restricted to the case of a symmetrical

AAC wall with cementitous-mortar joints and

renderings, without and with air-voids, drying

symmetrically from a nearly saturated initial state.

(

)

⋅

(

)

(19)

for k that runs from 2 to 4.

Or, re-written in the discrete form:

(

)

(

)

(

)

⋅

((

)

−

(

)

Construction data

The element without air-voids is shown in Figure 3,

(20)

For the sake of brevity,

(

W )

and

(

)

are

Joint

Rendering

AAC Block

, +

W .

For equations at internal and external boundaries, the

values of

W ,

and

p are unknowns. They depend, however,

non-linearly on the unknowns

, +

100

W .

Disceretization and linearization are thus performed by

the following:

T and

, +

19 20

240

(

)

(

)

14 15

⋅

(

−

)

⋅

(

−

)

(21)

1 2 3

Rendering

(24)

(32)

Where:

(

)

(

)

⋅

(

)

245

sat

(22)

(

∂

)

Figure 3: The element without air-voids.

while the one with air-voids is depicted in Figure 4.

∂

(23)

Joint

Rendering

AAC Block

∂

(

)

sat

∂

Void 1

Void 2

100

(24)

The local instantaneous values of all the terms on the

RHS of equations 23 and 24, at the intermediate time

step m, are thus derived from the local instantaneous

values

19 20

240

127

14 15

W , . These parameters are further

calculated by means of additional inherent series of

sub-iterations applied for each iterative convergence

stage, from time step n to n+1.

Two Software packages, MA2DW and MA2DWV,

were written to solve numerically the evolution of

moisture content in flat planar walls and verified by

specific examples. The first one is designed for the

solution of walls without air voids and the second for

those with.

T ,

and

1 2 3

Rendering

245

255

Figure 4: The element with air-voids.

Air-void No. 1 represents a cavity in the joint-mortar in

accordance with the manner of construction frequently

adopted for block-walls. Air-void No. 2 represents a

denoted in the sequel by

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