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A mathematical model of mould growth

on wooden material

A. Hukka, H. A. Viitanen

Summary A mathematical model for the simulation of mould fungi growth on

wooden material is presented, based on previous regression models for mould

growth on sapwood of pine and spruce. Quanti®cation of mould growth in the

model is based on the mould index used in the experiments for visual inspection.

The model consists of differential equations describing the growth rate of the

mould index in different ¯uctuating conditions including the effect of exposure

time, temperature, relative humidity and dry periods. Temperature and humidity

conditions favourable for mould growth are presented as a mathematical model.

The mould index has an upper limit which depends on temperature and relative

humidity. This limiting value can also be interpreted as the critical relative

humidity needed for mould growth depending also on the mould growth itself.

The model enables to calculate the development of mould growth on the surface

of small wooden samples exposed to arbitrary ¯uctuating temperature and

humidity conditions including dry periods. The numerical values of the

parameters included in the model are ®tted for pine and spruce sapwood, but

the functional form of the model can be reasoned to be valid also for other

wood-based materials.

List of symbols

M mould index [)]

T temperature [°C]

SQ surface quality (0 = resawn, 1 = original kiln-dried)

t time [d]

k

1

,k

2

correction coef®cients [)]

W wood species (0 = pine, 1 = spruce)

RH relative humidity [%]

Wood Science and Technology 33 (1999) 475±485 ÓSpringer-Verlag 1999

Received 18 May 1997

A. Hukka, H. A. Viitanen (&)

VTT Building Technology, Wood Technology,

P.O. Box 1806, FIN-02044 VTT, Finland

Present addresses:

A. Hukka

Valmet-Utec Oy,

P.O. Box 43, FIN-20251, Turku, Finland

H. A. Viitanen

FFRI Joensuu Research Station,

P.O. Box 68, FIN-80101 Joensuu, Finland

475

Introduction

Mould fungi is a heterogeneous and not a particularly well de®ned group of fungi.

Most of the mould fungi, as also the blue-stain and soft rot fungi, belong to Asco-

mycotina-(Ascomycetes) fungi. Mould and bluestain fungi are often both called

discolouring fungi. Discolouring fungi are often the initial microbial colonisers of

wood: logs can be infected in forests and in storage, sawn goods can be contami-

nated before and during drying, at storage or at the building site if the conditions for

fungal growth are favourable. In buildings, mould fungi cause problems in different

structures and materials: roofs, basements, ¯oors and walls. Except of wooden

substrates, surfaces of many other materials support growth of microbes and mould

problems are more common than decay damages. Typical mould fungi found in

moisture damaged wood are Alternaria alternata,Aspergillus species, Aur-

eobasidium pullullans,Cladosporium cladosporioides,Chaetomium globosum,

Paecilomyces variotii,Penicillium species and Trichoderma viride.

The growth of mould fungi on wooden material has been the subject of ex-

perimental research for a long time, but the knowledge thus gathered has been

mostly qualitative in nature (Henningsson 1980; Block 1953; Park 1982). The aim

has been to describe the material response and ®nd the critical conditions for

mould growth on surfaces of different materials with different treatments. Most of

this previous extensive research has been carried out in constant temperature and

humidity conditions but even such models are usually not applicable in arbi-

trarily varying conditions. Concerning ¯uctuating conditions, no mathematical

models seem to be available at all in the literature.

Especially dry periods have posed a problem in the published regression

models, as experimental knowledge concerning the effect of non-favourable

conditions for mould growth has been very limited. The few results published on

this subject are not applicable to humidity histories other than those used in the

experiments. From a practical point of view, it has not been possible to analyse

real building structures in actual varying climatic conditions.

Recently, Viitanen (1997a) has published comprehensive regression models for

mould growth in constant humidity and temperature conditions for pure pine

and spruce sapwood. Those regression models along with the experiments in

¯uctuating conditions are the basis for the present mathematical model. It is a

material model describing the response of pure wooden material to arbitrary

temperature and humidity conditions and will form an essential part of a more

complete structural model simulating the moisture behaviour of building struc-

tures in actual measured weather conditions.

Experimental material

The experimental material consisted of small samples (7 ´15 ´50 mm) of pure

kiln dried pine and spruce sapwood. Two different surface qualities differing in

nutrient content were studied: original kiln-dried and resawn. The experimental

results have been presented in detail by Viitanen and Ritschkoff (1991), Viitanen

and Bjurman (1995) and Viitanen (1997a).

Certain preconditions must be assumed concerning these experiments:

The small samples used do describe the mould growth on wooden material

reaching the equilibrium moisture content without delay. The ®nite size of the

samples may thus be neglected as well as the delay in the wood cell wall

attaining the equilibrium moisture content prescribed by the surrounding

relative humidity.

476

The growth of mould fungi takes place only on the material surface and it may

thus be modelled using only surface temperature and moisture content as input

data.

The existence of mould fungi on the material surface does not in¯uence the

moisture behaviour of the material, e.g. sorption properties.

The mould was measured applying an existing standard index based on the visual

appearance of the surface under study. Some re®nement has been made concerning

the scale and as a result this mould index assumes following integer values:

0 no growth

1 some growth detected only with microscopy

2 moderate growth detected with microscopy (coverage more than 10%)

3 some growth detected visually

4 visually detected coverage more than 10%

5 visually detected coverage more than 50%

6 visually detected coverage 100%

The same scale is applied in the present mathematical model, but the index is not

limited to integer values.

Model development

Conditions favourable to initiation of mould growth

Moisture content of wood depends on ambient humidity, temperature, exposure

time, dimensions and moisture absorption capacity of wood; water can also exist

in wood as free water in cavities or bound water within cell walls (Siau 1981;

Cloutier and Fortin 1991; Hartley et al. 1992). Being a hygroscopic material, the

equilibrium moisture content of wood is easily affected by the ambient humidity.

At low moisture content, the most direct measure of water availability is water

potential (w) de®ned as the free energy of water in a system relative to that of a

reference pool of pure water (Schniewind 1980). Water activity (a

w

) is, like water

potential, related to actual availability of water and it is determined by both matric

and osmotic components. Relative humidity (RH) is a percentage relation of

actual vapour pressure (p) and saturated vapour pressure (p

0

). The a

w

can also be

de®ned as the relative humidity at equilibrium (ERH) divided by 100, i.e. rela-

tive vapour pressure (p/p

0

) of the atmosphere in equilibrium with the substratum.

The water activity in the ambient air or in the cavities of the material is critical

for active stages of mould growth. Growth of mould fungi and time period needed

for the initiation of mould growth is mainly regulated by water activity, tem-

perature, exposure time and surface quality of the substrate. The experiments

suggest that the possible temperature and relative humidity conditions favouring

initiation of mould growth on wooden material can be described as a mathe-

matical diagram in Fig. 1. The favourable temperature range is 0±50 °C, and the

critical relative humidity required for initiation of mould growth is a function of

temperature. Based on experiments covering the temperature range 5±40 °C this

boundary curve can be described using a polynomial function

RHcrit ÿ0:00267T30:160T2ÿ3:13T 100:0 when T 20

80% when T >20

1

477

The behaviour of RH

crit

in the vicinity of the upper end of the temperature range

is only an approximation, but it is of only very little importance in practical

applications.

The largest possible mould growth

As known from experience, mould growth once initiated does not necessarily lead

to visually detectable mould (Viitanen and Bjurman 1995). Also, the ®nal coverage

of mould fungi on a surface is dependent on the temperature and humidity con-

ditions suggesting that a certain limiting value exists above which the mould index

does not rise irrespective of time available in basically favourable conditions. To

construct this limit, it is natural to assume that in conditions critical for the ini-

tiation of mould growth, Eq. (1), this upper limit for growth is 1, i.e. just some

growth can be detected microscopically no matter how much time passes. In the

other extreme it may be concluded that at 100% relative humidity the mould will

eventually cover the whole surface regardless of temperature (in the range 0±50 °C)

and M thus reaches a value of 6. In between these two ®xed points the experiments

suggest that the largest possible value of the mould index assumes a parabolic form:

Mmax 17RHcrit ÿRH

RHcrit ÿ100 ÿ2RHcrit ÿRH

RHcrit ÿ100

2

2

The contents of Eq. (2) may also be interpreted by stating that the critical RH

needed for mould growth does not only depend on temperature but also on the

Fig. 1. Conditions favourable for

initiation of mould growth on wooden

material as a mathematical model

Fig. 2. Temperature-

dependent critical relative

humidity needed for mould

growth at different values of

mould index

478

stage of mould development, i.e. the mould index itself. This result is arrived at by

solving Eq. (2) for RH, which now represents the temperature-dependent RH

needed for mould index reaching a value of M

max

. This result is depicted in Fig. 2.

Growth rate in favourable conditions

The present model is based on mathematical relations for the growth rate of

mould index in different conditions. The model is purely mathematical in nature

and as mould growth is only investigated by visual inspection, so it does not have

any connection to biology in the form of modelling the number of live cells. Also,

the mould index resulting from computation with the model does not re¯ect the

visual appearance of the surface under study, because traces of mould growth

remain on wood surfaces for a long time. The correct way to interpret the results

is that the mould index represents the possible activity of the mould fungi on the

wood surface.

As a basis for the growth model, Viitanen (1997a) presents a regression

equation for the response time (weeks) needed for the initiation of mould growth

on wooden material in constant temperature and humidity conditions:

tmexpÿ0:68 ln T ÿ13:9 ln RH 0:14W ÿ0:33SQ 66:02 3

If the mould index M is presumed to increase linearly in time and time is mea-

sured in days, Eq. (3) may be interpreted as a differential relation

dM

dt 1

7 expÿ0:68 ln T ÿ13:9 ln RH 0:14W ÿ0:33SQ 66:02;M<1

4

This conversion extends the applicability of Eq. (3) into variable conditions such

that the relative humidity is constantly above the critical value de®ned by Eq. (1)

and the temperature is in the range 0±50 °C. Linear growth in the range M < 1 is

only a mathematical description and in principle any other growth model could

be utilised. When interpreting the results of the model all values of M below 1

indicate no growth.

As the growth proceeds above the initial stage (M = 1), Eq. (4) is no longer

valid. For a larger growth Viitanen (1997a) presents another regression model

describing the response time needed for the ®rst visual appearance of mould

growth (M = 3):

tvexpÿ0:74 ln T ÿ12:72 ln RH 0:06W 61:50:5

If growth of the mould index is presumed to proceed from M = 1 to M = 3 on

a constant rate in constant conditions, Eqs. (3) and (5) can be combined to

give the growth rate on that range. The result is a correction coef®cient if

Eq. (4) is used as a basis:

k11 when M<1

2

tv=tmÿ1when M >1

6

Although based on constant conditions, the experiments suggest that Eq. (6) is

valid also for mould growth in ¯uctuating conditions as long as the conditions are

continually favourable to growth. Based on data from growth after the visual

479

appearance of mould fungi it may be concluded that the same correction for

growth rate applies for the entire range M > 1.

Taking into account the upper limit for mould growth de®ned by Eq. (2) may

also be accomplished by using a correction coef®cient. Assuming the delay to

affect the growth rate by 10% at 1 unit below the maximum value of the index

gives this coef®cient to the following form

k21ÿexp2:3MÿMmax 7

The complete model in conditions favourable for mould growth consists of

Eqs. (4), (6) and (7):

dM

dt 1

7 expÿ0:68 ln T ÿ13:9 ln RH 0:14W ÿ0:33SQ 66:02k1k28

As an initial condition M must be a known value, usually equal to zero imme-

diately after arti®cial drying of wood in a kiln with wood temperature exceeding

50 °C.

Model during non-favourable conditions

In ¯uctuating humidity conditions Viitanen (1997a) states that the cumulative

time in high-humidity conditions can be used to a limited extent to quantify the

response time needed for the initiation of mould growth. This simpli®cation,

however, eventually always leads to a large mould activity as humidity cycles are

repeated. Instead of remaining on a constant level the activity of mould can be

thus regarded as decreasing during dry periods. Of course, the visual appearance

of the surface does not necessarily change during the dry period, but a ®nite delay

in growth after the dry period can be clearly observed. This delay does exist as

soon as after 6 h in dry conditions, but extending the dry period to 24 h does not

seem to signi®cantly affect the delay, if growth will initiate at all. After that the

delay is again prolonged. A mathematical description of the delay can be written

by using the time passed from the beginning of the dry period (t )t

1

):

dM

dt

ÿ0:032 when t ÿt16 h

0 when 6 h tÿt124 h

ÿ0:016 when t ÿt1>24 h

(9

The experimental data behind this equation cover dry periods between 6 h and 14

days, but the functional form of expression (9) is based only on a small number of

experiments and thus must not be seen as the best possible one. Knowledge of the

in¯uence of longer dry periods on mould growth is very limited as is the data

concerning the effect of temperatures below 0 °C. In lack of better information

Eq. (9) may be applied also for such situations, although the validity of such an

application must be questioned.

Comparing the model results against experimental data

The largest possible value of the mould index, Eq. (2), is based on 12-week ex-

periments in constant conditions. Figure 3 presents the values calculated using

Eq. (2) versus the experimental observations. Only points showing a clear limit of

mould index have been taken into account. It can be seen that the largest error in

480

the maximum value of mould index is 1.2 and that only one point includes an

error larger than 0.5 in value.

To compare the model results to the original experimental data, Figs. 4 and 5

present the experimental and simulated response times needed for the initiation

and the ®rst visual appearance of mould growth on the surface of resawn pine

sapwood. Similar results for spruce are presented in Figs. 6 and 7. The experi-

mental results represent the average of 6 parallel samples. It can be seen that there

are no major systematic errors in the model and that in all cases most of the

errors in response time are smaller than 25% in numerical value. However, also

some points with very large errors can be detected, indicating that a model with

only a very few numerical parameters may not be suf®cient for describing the

phenomenon of mould growth in the whole temperature and RH range, especially

in higher temperatures.

Figure 8 presents the results obtained when using the model to predict the

response time needed for the initiation of mould growth on pine sapwood in

Fig. 3. Comparison of the largest

possible value of mould index

produced by Eq. (2) against the

experimenta

Fig. 4. Comparison of

simulated and experimental

response times needed for

initiation of mould growth on

pine sapwood in constant

conditions

481

¯uctuating humidity conditions. The relative humidity has been kept at two

constant values, 75% and 95%, the period in each condition varying between

6 and 196 h. The temperature has been constant at 20 °C. It can be seen that

most of the simulated points (80%) are such that the error in response time is less

than 25%. The average error in simulated response time is only 1%, indicating

that there is essentially no systematic error in the model.

Discussion and conclusion

The mathematical model presented is throughout formulated in a differential

form. As such it allows to calculate the development of mould growth on a

wooden surface exposed to arbitrary temperature and humidity histories in-

cluding also dry periods. The numerical values of the parameters in the model

apply only for pure pine and spruce sapwood and aim at describing the average

response of the material based on a small number of parallel samples. The

Fig. 5. Comparison of

simulated and experimental

response times needed for

visual appearance of mould

growth on pine sapwood in

constant conditions

Fig. 6. Comparison of

simulated and experimental

response times needed for

initiation of mould growth on

spruce sapwood in constant

conditions

482

response does, however, exhibit a very large variation between samples origi-

nating from different stems of the same wood species. This variation is of the

same order in magnitude as is the variation in results between samples repre-

senting different species of wood. Based on this it may be reasoned that the same

functional form of the model could be utilised also for prediction of mould

growth on other wood-based materials, only the numerical values of the coef®-

cients must be re-evaluated.

The experimental data behind the model covers temperatures between 5 and

40 °C and relative humidities between 75 and 100%. The exposure time in con-

stant conditions has been at least 12 weeks and the time in ¯uctuating conditions

has varied between 6 and 24 weeks. This temporal scale is clearly shorter than will

be the application area of the model. This requires an extrapolation of experi-

mental results and causes an uncertainty in the results produced by the model in

such situations. Also the nature and range of the experiments conducted in

¯uctuating conditions will certainly need some revision as the numerical result

proposed by Eq. (9) is not totally satisfactory.

Fig. 7. Comparison of

simulated and experimental

response times needed for

visual appearance of mould

growth on spruce sapwood in

constant conditions

Fig. 8. Comparison of

simulated and experimental

response times needed for

initiation of mould growth on

pine sapwood in ¯uctuating

humidity conditions. Values of

surface quality SQ correspond

to original kiln-dried (1) and

resawn (0) surfaces

483

The maximum value of the mould index used in the model, Eq. (2), has been

deduced from experiments conducted in constant humidity conditions. It is

known that this parameter is lower in ¯uctuating humidity conditions, but the

numerical value of the maximum growth in such situations on a general level is

not comprehensively known. In this sense the present model obviously needs

further development. The same applies for ¯uctuating temperature conditions

with constant RH.

The present model describes mould growth on wooden material surfaces. It is a

pure material model trying to quantify the experimental results of mould expo-

sures conducted on small samples. The next natural step in the evolution of the

model is to insert the mathematical equations into a simulation program for

calculation of mould growth on an actual wooden structure. This would create a

new tool for prediction of the service life of a certain structural design in critical

temperature and humidity conditions. Another possible direction of further de-

velopment is to apply the model for quanti®cation of mould growth in materials

other than pure wood, which combined to a moisture simulation program would

give a powerful tool for analysing the moisture behaviour of a very wide range of

structures.

It has been found in laboratory studies and in practice that mould fungi are

more rapid invaders than decay fungi in wooden structures subjected to excess

moisture stresses (Viitanen 1997a, b). Also, the minimum humidity and tem-

perature requirements are in general lower for mould fungi than for brown rot

fungi. Growth of mould fungi and time period needed for the initiation of mould

growth are mainly regulated by water activity, temperature, exposure time and

surface quality of the substrate. Especially at low water activity and low tem-

perature, the latent period preceding the initial stages of mould growth and the

appearance of mould fungi on wood have been found to be several weeks al-

though the equilibrium moisture content of the small samples used was reached

in two weeks (Viitanen 1997a). However, Viitanen and Paajanen (1986) found no

mould growth at RH 75% and according to recent studies, the lowest humidity

condition allowing mould growth is RH 80±85% (i.e. water activity 0.80±0.85)

providing that the temperature is above 5 °C (Bjurman 1989; Park 1982; Wang

1992; Adan 1994; Hocking et al. 1994; Viitanen 1997a). Modeling of the critical

humidity and temperature conditions, and especially of the critical time required

for spore germination and the growth of mould fungi mycelium on pine and

spruce sapwood was presented by Viitanen (1997a). Temperature conditions

have a minor effect on the water activity and the main effect is concerned with

the growth and metabolic activity of the fungi. Often the selection of mould

species at high temperatures (above +30 to +35 °C) consists mainly in thermo-

tolerant or thermophilic mould fungi (Henningsson 1980). At ¯uctuating hu-

midity conditions, the growth of mould fungi is clearly retarded and latent period

is even longer than at constant favourable conditions (Adan 1994; Viitanen

1997a).

In general, the higher the temperature and the more favourable the nutrition at

a given relative humidity, the less time is required for spore germination. After

kiln drying, the concentration of nitrogen and low-molecular hydrocarbon

compounds on the surface layers of sawn timber are often higher than inside the

wood (Boutleje 1990; Theander et al. 1993). Terziev et al. (1994) and Viitanen and

Bjurman (1995) showed, that the growth of mould is clearly more rapid and

vigorous on the original kiln dried wood surface than on a resawn surface.

484

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