Building and Enrironment. Vol. 17. No. 4. pp. 257-262, 1982. 036~k-1323/82/t¼0257-1~$03.00/(t Printed in Great Britain. © 1982 Pergamon Press Lid.

Water Movement in Porous Building Materials--V. Absorptton and Shedding of Rain by Building Surfaces

CHRISTOPHER HALL* A. N. KALIMERIS*

We discuss in terms of unsaturated flow theory the absorption and shedding of rain falling on porous surfaces, and present a finite element solution of the one-dimensional nonlinear diffusion equation subject to a constant flux boundary condition. This shows how the water absorption depends on the rainfall rate and on the hydraulic diffusivity of the absorbing material. For a class of hydraulically similar porous building materials we establish the dependence of absorption on sorptivity and porosity, and give a general expression for the time to surface saturation. We briefly consider also the fluid mechanics of rainwater run-off .

1. INTRODUCTION

WHEN rain falls on porous building surfaces much of it is absorbed. The absorbed water is temporarily immobilised, to be released later by evaporation. On impermeable building surfaces all rain water is free to flow. This rather obvious distinction deserves the attention of building technologists interested in rain exclusion and penetration.

The absorbency of many types of brick and sedi- mentary building stone greatly reduces the quantity of free water which has to be shed during rain. Construction in absorbent materials thus requires less attention to joint sealing and to drainage details such as drips. On the other hand, impermeable facing materials such as glazed tile, plastics and painted timber cladding shed large amounts of water during rain, placing severe demands on the performance of jointing details. By way of example, we have recently seen a building in which severe leak- age of rainwater occurred through defective joints between glazed ceramic cladding tiles fixed to a vertical external wall and adjacent sills and soffits. The joints were grouted with a brittle cementitious mortar. Glazed tiles were no doubt chosen to provide a weathertight impermeable facing. But inevitably in rain, the wall surface was flooded with water which penetrated freely into any capillary channels opened up at joints by very minor differential movement in the wall structure. This design is, in fact, very in- tolerant of imperfections in construction. In this paper we present an analysis o f the absorption of rainwater by porous building surfaces using ideas

*Department of Building, UMIST, P.O. Box 88, Manchester M60 IQD, U.K.

257

drawn from earlier papers in this series [1, 2]. We link the absorbency of the surfaces to the hydraulic sorp- tivity and diffusivity of building materials, and est- ablish the time to reach surface saturation at different rainfall intensities. We also offer a brief discussion of run-off from impermeable or saturated inclined sur- faces.

2. ABSORPTION OF RAIN BY POROUS SURFACES 2.1 General remarks

Previously [I] we have looked in detail at the capil- lary absorption of water into initially dry materials, using the boundary condition 0 = O,(t t> 0), where 0m is a constant surface water content, normally taken as saturation.

In this paper we are concerned with rain falling on to building surfaces and it is more appropriate to use a constant flux boundary condition: - I/,, = K ( a ~ / a x ) at x = 0 and t ~> 0, where K is the hydraulic conduc- tivity and • is the hydraulic potential. However, this introduces new difficulties since solutions to the un- saturated flow equation subject to a flux boundary condition cannot be obtained analytically.

Finite difference solutions and quasi-analytical solutions to the constant flux infiltration and ab- sorption processes have been obtained in the past in the context of soil science [3, 4]. Here we describe the use of a finite element formulation of the one-dimen- sional flow equation with a constant flux boundary condition to model the absorption of rain by porous building surfaces.

2.2 Analysis o f absorption We consider absorption of water into the building

fabric from a surface exposed to rain falling at a

258 C. H a l l a n d A . N. K a l i m e r i s

constant rate. We assume that the material is homo- geneous and that the unsaturated flow equation [1] is applicable. The flow system is non-hysteretic and isothermal, and osmotic effects associated with solu- ble salts are absent.

The derivation of the equations describing un- saturated water flow is detailed elsewhere [5, 6]. It is based on the assumption that Darcy 's Law is valid for unsaturated flow through porous media. The ap- plicability of the extended Darcy equation to porous building materials has been suggested by Gummerson et al. [7].

The equation

ao, a (D(O~) ao~] O t--'~ = a--x O"ffx ] (1)

is solved subject to the following boundary and initial conditions:

0 r = 0 t , = 0 x~>0 (la)

D(0r)~x' = - Vo t,>~O X = 0 ( lb)

00____~ = 0 tr > 0 x = L, (lc) 8x

where 0r is the normalised water content 0r = (0 - 0o)/(0,- 0o), t, is the reduced time t, = tl(O, - 0o), x is the horizontal coordinate, D(O,) is the hydraulic diffusivity, V, is the applied constant flux and L is the length of the medium considered.

The flux Vo may be identified with the rate at which rain is received per unit area of surface. Lacy [8] has suggested the expression

V. = 0.222 W R °8~,

where W is the wind speed and R is the rainfall rate. The geometry is defined in Fig. I.

3. F . E . M . F O R M U L A T I O N

The finite element method [9] provides a numerical solution to equation (1) in a finite solution domain fl of length L subject to boundary conditions (In, b, c).

Dividing fl (Fig. 2) into two-noded finite elements, the distribution of 0r within each element has the form:

O/"'(x, h)= ~, N~(x)O.(t,)= [NJ{O/"}. i = 1

A linear variation of the normalised water content within each element is assumed. The normalised water content can then be written in terms of its known nodal values and the coordinates of the nodes, a s

O/'~(x, t , )= x,-_ x O , , + x - x ~ X2 - - X~ X2 - - X i

: t s , N:j l 0.-J"

Or,_

m

I Fig. 1. The rainfall geometry. R is the rate of rainfall, W is

the wind speed and I'0 is the rate of driving rain.

v

0

-'~=0

m ~ I ,t I X. x X [

Fig. 2. The distribution of 8, within one element of the domain 1"1.

Galerkin's criterion uses the interpolation functions as the weighting functions and equates the integrals of the weighted residuals to zero:

( no , ~ ' a ~,, no) "~ ( , ,

Applying integration by parts,

atr / \ ax /In('~

I ( + D(O, (e~) d l l ~eJ = 0. ~(~ OX OX

Using our definition of O, ....,

= ( N O ) n,~,'

where

Q = D(8] ~') nO, ''~ ax "

For a typical line element

(NQ)[n . . . . N2Q2 - N~ Q,,

where Q~, Q2 and N,, N2 are the fluxes and the

Water Movement in Porous Building Mater ia l s - - V. 259

interpolation functions at the boundary nodes of the line element. But N, = N2 = 1 by definition.

Hence

[Kt](e}{Or}(e) + [K]"{O,} (~) = {Q},

where

scheme is employed

([SYSK]':~+,~ + [SYSM] ~t . ) " {O.~ ,}'k' "

+ ([SYSK]':+~, (1 - ~.) - [SYSM] ~t~)

K,,i = fn,e, (NINj)dlT"

The hydraulic diffusivity is assumed to be constant within each element; it is estimated at the centre of the element, i.e.

D( Or,,)) = D( O,,) + D( O,2) 2

This assumption is justifiable provided that the ele- ment length is kept small•

After assembly, properties of the system are given by the following set of nonlinear differential equa- tions:

[SYSM]{I~r} + [SYSK]{0,} = {SYSQ},

where SYSM is the system 'mass ' matrix, SYSK the system 'stiffness' matrix and

{SYSQ} =

An implicit method is used to link two time levels n and n + 1, where t,, = nat,, At, being the step size in time. The time derivatives are approximated by a forward difference at n and a backward difference at n + 1. Thus

([SYSK]v. + [SYSM] ~ t r ) " {0.+,}

+ ([SYSK)(I - p , ) - [SYSM] ~t~)

• {0.} = {SYSQ},

where ~. may be varied to produce any number of different schemes. The resulting equation is un- conditionally valid, i.e. stable and convergent for 0.5 ~< p, ~< 1.0.

Since the SYSK matrix is time-dependent--non- linearity is introduced via the dependence of diffusivity D on water content 0,--iteration within each time step is needed. The following iteration

• {0.} = {SYSQ},

where the superscript (k) is the iteration number (k ~< 8).

Cholesky's method is utilised for the solution of the set of system equations. It is particularly suited to the problem since the tridiagonal system matrices SYSK and SYSM are symmetric as well as banded and sparse.

Full use was made of the SERC Finite Element Library of subroutines FELIB [10] in the program- ming effort. The computing work was carried out on the PRIME 750 of the Interactive Computing Facility ICF, UMIST, which is part of the UK-wide SERC network.

4. RESULTS 4.1 Data

Water content profiles were computed (Fig. 3) using data for a typical solid clay common brick obtained previously, namely

00 = 0.002 cm3/cm 3

0~ = 0.262 cm3/cm 3

and

S = 1 mm/min "2.

The hydraulic diffusivity was calculated from the expression

2 -~ •

D(O,) = ~ 0.00224 exp (7.5 Or) cm-/mm,

which has been found experimentally to describe the diffusivity for a number of building materials [I 1].

4.2 Reduced variables in terms of Vo If the reduced variables

and

X l -~- Vox

Tm= Vo:t,

are employed, equation (1) is reduced to

( 00,) oo, = a D( O,) ~ , , OTI OXl

(2)

260 C. Hall and A. N. Kal imeris

i

0 I

0

I

, i 0

1

X c m

Fig. 3. F.E.M. predicted water content profiles for a typical solid clay common brick subject to constant flux boundary conditions. (a) 75 ram/h; (b) 50mmth; (c) 25 mm/h. l =0.25 cm;

At, = 0.05 rain; ~ = 0.5 (Crank-Nicolson method).

1

0 . 5

subject to the following conditions: the applicability to a number of building materials of the equation

0r = 0 T, = 0 X, > 0 (2a) D(Or) = D, f (0,) ao,

D(O,)-ZV- = - 1 T, >!0 X, = 0 (2b) S -~ u z x l

- 0 1 - Oo f(O') 00, oX, = 0 T~ > 0 X, = L Vo. (2c) S 2

- 01 - 0o as exp (/30,).

Therefore, if the reduced variables T, and X, are employed, the process of one-dimensional constant flux absorption into a porous material with a uniform initial water content produces the same time-history of water content profiles regardless of the value of Vo. This is demonstrated in Fig. 4, which shows the F.E.M. results for one-dimensional constant flux ab- sorption into a solid clay common brick using the data given in 4.1 above.

Here only D, (= S"I(O,- 0,1)) varies from material to material. That is, these materials form a class having diffusivity functions of the same shape f(O,). Their diffusivities at all water contents 0, differ only by the scaling factor D,, which may be expressed in terms of the sorptivity S. We call such materials hydraulically similar.

Consequently, if the dimensionless quantities

4.3 Reduced variables in terms of Vo and material properties

Experimental evidence provides strong support for

1

0

0.5

( m e n / h )

o 2 5

[] o 5 0

[2 7 5 O 0

o @

T : 0 . 0 2 4 cm2 rnin -~

0 t oo ~ o

0 . 0 5 0 X I ' c m 2 ra in -1

Fig, 4. Normalised water content Or versus reduced distance XI( = Vox) master curve for one-dimensional constant flux absorption into solid clay common brick. The profiles were obtained from the F.E.M. results (Fig. 3) at the value of reduced time TI(= V0Zt,) shown. The symbols refer to fluxes

used, as shown.

and

X~ = Vox= Vox( Os~ 0o) - O t

1". = V°2 tr V°2 t " D , = S ~

are employed, equation (I) reduces to

00, aT~

subject to conditions

Or=O

ao, f ( O , ) ~ - ~ 2 = - 1

ao, = o ax~

a ( oo, 0)(2 f ( O , ) a X 2 ] (3)

7",_ = 0 )(2 > 0 (3a)

T2 >~ 0 X2 = 0 (3b)

LVo T2 > 0 )(2 = D, " (3c)

The practical implication is that for any group of hydraulically similar porous materials, the time-his-

Water M o v e m e n t in Porous Building M a t e r i a l s - - V . 261

tory of the water content profile for any value of flux V,, and sorptivity S is obtainable from a single set of experimental (or computed) profiles determined for one particular value of Vo and of S.

Water content profiles obtained by solving equation (3) with the aid of the F.E.M. are shown in Fig. 5.

5. RUN-OFF FROM INCLINED SURFACES

On striking inclined or vertical surfaces, water which is not absorbed forms a downward flowing film. To complete our analysis, we consider run-off from a surface inclined at an angle B to the vertical. We assume that the surface is wetted by rainfall forming a uniform film, rather than trickles. Taking the z-axis downwards parallel to the surface, then by mass balance

d6 V o - u dz f '

where 6 is the film thickness, u is the absorption rate, I,'o is the rainfall rate (normal to the surface) and f is the mean film velocity parallel to the surface in the direction z.

But f = p,,g6"- cos B/3~ [12], where p. is the density and rl the viscosity of water. Therefore

6 = (9( Vo- u)vlz/p,g cos B ) " .

On saturated surfaces, maximum run-off occurs when u <~ I,'o. For water at 10°C on a vertical surface

8/mm = 0.069((Vo/mmlh)(zlm)) '/3.

It follows that run-off films are normally 0.1-0.3 mm thick, and the mean velocity is 25-250 mm/s.

6. DISCUSSION

6.1 Influence of gravity on absorption The rain absorption analysis given in Sections 2, 3

and 4 above is strictly applicable only to horizontal

1.o

0 r

0.5

o o 0.5 1 .o

x 2

Fig. 5. Normalised water content Or versus dimensionless quantity X,(= VoxlDO curves for one-dimensional constant flux absorption into solid clay common brick. The profiles were determined by the F.E.M. at the values of dimension- less 72(= Vo2tJDO with f(0,)=0.00224 exp (7.5 0,). I= 0.25 cm; At, = 0.005 min; p. = 0.5 (Crank-Nicolson method).

(gravity-free) flow, since equation (1) contains no gravitational term. Such a term may be included [ l , 3], 4, 6], as it must be in soil infiltration analysis. If this is done, it is possible to distinguish the following two conditions: (a) 'do > KSAT; all rain falling on the sur- face is absorbed until the finite time of surface saturation t, has been reached; (b) Vo~ < KSAT, flOW takes place continuously whilst the surface volu- metric water content approaches a limiting value 0* such that

O* ~< O, and Vo = K(O*),

where 0~ is the volumetric water content at satura- tion.

However, porous building materials are generally fine-pored and have low permeabilities. A typical value of KSAT for a clay brick ceramic is 0.2 cm/day = 0.08 mm/h. The values of Vo used in the analysis (25-75mmlh) thus correspond to = 3 - 9 x l02 KSAT. In such cases the effect of gravity on rain absorption is negligible.

6.2 Time to attain surface saturation We noted in Section 4.2 that surface saturation

should occur at the same reduced time T~ = T, for all members of a class of hydraulically similar materials. When the particular diffusivity function defined in Section 4.1 is used, computed results show that T, -- 0,64. This is a value derived from an F.E.M. solution of equation (3) with f(O,) = 0.00224 exp (7.5 0,). Therefore the time to saturation

L = 0.64 V~- (4)

Values of t, at different rainfall intensities calculated from this formula are shown in Table 1. The values compare well with the individual values of t~ obtained directly from the F.E.M. solutions of the absorption equation (1), as expected. Discrepancies at small ts we attribute to the element length and time step used in the F.E.M. calculations.

6.3 Absorption after surface saturation is reached The analysis of rain absorption based on the con-

stant flux boundary condition is valid only for times t <~ t , Within this period the rate of absorption u is constant and equals Vo, while the cumulative ab- sorption i = Vot.

At longer times t > t, the rate of absorption u falls below Vo. A part ( V o - u) of the incident flux appears as run-off, as described in Section 5. The rate of absorption u diminishes with time. After saturation at the surface has been reached, the absorption process is modelled effectively by the unsaturated flow equa- tion subject to a constant concentration condition 0~ at the surface, with O(x) at t = t~ as the new initial condition.

7. CONCLUSIONS

We have shown that the time for an absorbent surface to become saturated when exposed to rain

262 C. Hal l and A. N. K a l i m e r i s

Table I. Surface saturation times for different rainfall intensity values

t~ from equation (I) t, from equation (4) (rain) Imin)

S Vo V,~ V~, V,, v~, V,~ ram/rain I/2 25 mm/h 50mm/h 75 mm/h 25 mm/h 50mm/h 75 mm/h

0.50 1.57 0.52 0.22 0.92 0.23 0.11 1.00 3.64 1.05 0.58 3.67 0.92 0.41 1.50 2.07 1.01 8.29 2.07 0.92 2.00 1.57 14.75 3.69 1.64

depends on $21 Vo 2. Since the S values of commercial bricks and of building stones range from as high as 2ram/rain ~2 to less than 0.5 mm/min "2, it is quite clear that the choice of material greatly effects the absorption/run-off characteristics of masonry walls. Under heavy driving rain (say 25 mm/h) low sorp- tivity materials will saturate in less than 1 min, whilst high sorptivity materials may absorb all incident rain for as long as 15 min.

The analysis also shows the power of the finite element method in obtaining numerical solutions to unsaturated flow problems in building science. Analytical solutions are very limited and a numerical

approach has great value. The F.E.M. is particularly suited to handling complicated geometries and boun- dary conditions. In the present case, the boundary condition is quite simple, but extension to a time- dependent flux presents no difficulties.

The use of reduced variables establishes clearly the functional dependence of the main parameters Vo and S. Once again, the central role of the sorptivity S in controlling the rate of water movement is apparent.

Acknowledgements--We thank the Science and Engineering Research Council for provision of computing facilities and the ICF staff at the SERC Rutherford Laboratory and UMIST for support.

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Unsaturated water flow within porous materials observed by NMR imaging. Nature, Lond. 281, 56-57 (1979).

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