Model equations for determining the coupled heat, moisture and maturity changes within a concrete block are introduced and briefly examined. Preliminary results are obtained for the heat exchange between concrete slabs in contact driven by maturity differences

Ballim, Y

Fowkes, ND

Makinde, OD

Mambili Mamboundou, H

Patini, A

English

Mathematics in Industry Information Service Eprints Archive

MATURITY EFFECTS IN CONCRETE DAMS
N.D. Fowkes∗, H. Mambili Mamboundou †, O.D. Makinde ‡,
Y. Ballim and A. Patini §
Abstract
Model equations for determining the coupled heat, moisture and
maturity changes within a concrete block are introduced and briefly
examined. Preliminary results are obtained for the heat exchange be-
tween concrete slabs in contact driven by maturity differences.
1 Introduction
To allow shrinkage movement to take place, dam walls are constructed se-
quentially in discrete blocks with a second layer poured on top of the first
layer weeks after the first layer is set. A better understanding of this pro-
cess may lead to better construction practices. It has been observed that
hydration heat transferred across the interface from the upper block causes
enhanced hydration heat release within the first block, see Figure 1, from
Ballim and Graham (2003). The effect is primarily due to the strong depen-
dence of the hydration rate on temperature (an Arrhenius effect). To under-
stand the underlying processes Ballim and Graham (2003) have introduced
a maturity state variable to be evaluated at locations and time throughout
the concrete. The present study sets up a slightly more complete (but not
necessarily better) model to determine the maturity, temperature and mois-
ture changes within maturing concrete under a variety of conditions. It is
∗School of Mathematics and Statistics, University of Western Australia, Crawley WA
6009, Australia. e-mail: fowkes@maths.uwa.edu.au
†School of Computational and Applied Mathematics, University of the Witwatersrand,
Wits 2050, Johannesburg, S Africa. e-mail: mambili@cam.wits.ac.za
‡Department of Applied Mathematics, University of Limpopo, Sovenga 0727, South
Africa. e-mail: makindeo@unorth.ac.za
§School of Civil and Environmental Engineering, University of the Witwater-
srand, Wits 2050, Johannesburg, South Africa. e-mail: ballim@civil.wits.ac.za,
anthony@civil.wits.ac.za
59
60 N.D. Fowkes, H. Mambili Mamboundou, O.D. Makinde, Y. Ballim and A.Patini
This peak indicates
heat flow from the 
upper concrete layer 
Profile without 
sequential construction
T
Time
Figure 1: Temperature at a point in the lower layer of concrete.
hoped that the model will provide a better understanding of the underlying
processes.
2 The model
xL
Figure 2: The concrete slab.
We examine the thermal and moisture exchange within a one dimensional
concrete slab of thickness 2L subjected to environmental moisture and heat
exchanges, see Figure 2. In this report we mainly focus our attention on
processes within the concrete, leaving the detailed modelling of surface ex-
Maturity Effects in Concrete Dams 61
changes to another report. It is the liberation of heat within the concrete
due to the exothermic cement water (hydration) reaction that drives the
thermal exchanges of interest. The hydration rate is initially relatively large
and then continually decreases. The initial large hydration rate stage occurs
over periods of weeks, but significant hydration is still occurring even after
15 years.1 Following Ballim and Graham (2003) we introduce a maturity
(or degree of hydration) state function of position and time 0 ≤ m(r, t) ≤ 1.
This can for example be measured by chemical analysis or X-ray quantitative
analysis.
The relevant variables are the temperature T (x, t), moisture content 0 ≤
θ(x, t) ≤ θ0, where θ0 is the initial concentration of water in the concrete,
and the maturity m(x, t). The conservation and reaction equations are:
Heat conservation:
ρccc
∂T
∂t
= kT
∂2T
∂x2
+ γ
∂m
∂t
, (1)
where kt, ρc, ρc are the thermal heat parameters for concrete and γ
∂m
∂t is the
heat release rate due to hydration.
Hydration chemistry:
∂m
∂t
= µ(1−m)θe−E/RT , (2)
where the constant E is the apparent activation energy of the reaction,
R the gas constant, T the absolute temperature, and µ the reaction rate.
The reaction is assumed to occur at a rate proportional to the product
of concentrations of the reacting components; free water and ’unreacted’
concrete measured in terms of the level of maturity. Also the reaction is
strongly temperature dependent; an Arrhenius temperature dependence is
assumed.
Water conservation:
∂θ
∂t
=
∂
∂x
(Dm(θ)
∂θ
∂x
)− η∂m
∂t
, (3)
where Dm(θ) is the moisture diffusivity of concrete. The reaction removes
free water from the concrete; η is the Stoichiometric ratio for the hydration
reaction, assumed constant.
The water conservation equation is generally not invoked in such studies,
presumably because water is normally present in abundance and so does not
limit maturation. This is likely to be the case throughout most of the slab,
1but with relatively little heat release
62 N.D. Fowkes, H. Mambili Mamboundou, O.D. Makinde, Y. Ballim and A.Patini
because of the small moisture diffusivity, but is unlikely to be the case at
the surface under dry conditions; certainly if the surface quality of concrete
is important this is an important issue. The moisture diffusivity Dm is
strongly dependent on the moisture level2, so that moisture fronts are to be
expected. Such effects could significantly alter the global moisture balance.
Initially we assume uniform conditions prevail:
m(x, 0) = 0, θ(x, 0) = θ0, T (x, 0) = T0.
The surface temperature is assumed to be prescribed. Except for a constant
average temperature, daily and monthly fluctuations will dominate other
Fourier components, so
T (L, t) = T¯0 + Td sinωdt+ Tm sinωmt, (4)
is a useful model. A Robin condition
−Mm∂θ(1, t)
∂x
= ξ(θ¯ − θ(1, t)), (5)
with the exchange coefficient ξ constant, can be used to describe environmen-
tally driven moisture fluxes to and from the concrete surface, where θ¯(t) is
the effective moisture content of the air in contact with the surface. As noted
earlier the aim of the present work is not to understand surface exchanges,
so the above boundary conditions simply represent sensible conditions to
use for understanding aspects of the physics. Symmetry requirements at
the centre of the slab give
∂θ(0, t)
∂x
=
∂T (0, t)
∂x
= 0.
Scaling
There are a variety of time scales in the problem with associated length
scales:
• surface heating: daily temperature variations with a penetration depth
of about 10 cm,
• seasonal temperature variations with associated penetration depth 2m,
• the diffusion time associated with the slab thickness, typically 10m
with associated time scale 15 years.
2varying by a factor of 103 over the moisture range of interest
Maturity Effects in Concrete Dams 63
Of engineering concern is the stresses set up during construction3 and specif-
ically during, and soon after, the pouring of the second slab on top of the
first. With this in mind we choose a time scale t0 of 1 month; the typical
time span between the pouring of the two slabs. Based on this time scale
the sensible scaling for the problem is given by:
x = Lx′, T = T¯ + T T ′, with T = γm˙0t0
ρc
, θ = θ0θ
′, (6)
in terms of which the problem reduces to (after dropping primes)
∂T
∂t
= κL
∂2T
∂x2
+
∂m
∂t
, (7)
∂m
∂t
= µ(1−m)θeξ(T−1), (8)
∂θ
∂t
= κm
∂
∂x
(km(θ)
∂θ
∂x
)− Sm˙. (9)
The dimensionless groups are
κL =
kT t0
L2ρc
, µ = νt0θ0e
−E/RT¯ , ξ =
E
RT¯
, (10)
and the scaled Stochiometric ratio S. The surface boundary condition re-
duces to
T (1, t) = 1 + αd sinωdt+ αm sin t, (11)
with ωd ≈ 30 1. Symmetry and initial conditions complete the specifica-
tion of the problem.
Suggested parameter values
Over the time scales of interest the heat and moisture diffusion length scales
will be relatively small compared with the slab thickness so that κL, κM  1
(for computational purposes we’ll use κL = κM = 0.001). The reaction
parameter µ will be of unit order, reflecting the observation that significant
hydration heat is liberated over the adopted time scale of a month (µ = 1
is used). A significant modification of the reaction rate is observed due
to the second slab which indicates ξ is of order 1 (ξ = 2 is used). The
moisture diffusivity is known to vary rapidly with θ, κm(θ) = e
3θ used. The
Stochiometric number is expected to be of order 1 (we will use S = 2).
3Such stresses generate undesirable cracking
64 N.D. Fowkes, H. Mambili Mamboundou, O.D. Makinde, Y. Ballim and A.Patini
3 Simulations
A simple slab on slab model
An in-depth investigation of this system is well beyond the scope of the
MISG, however a few simulations have been performed to illustrate some
features of the physics. Of special interest is the effect of a second slab on the
thermal history on the first slab. We will model the effect of the introduced
second slab as a constant temperature boundary condition, T (1, t) = T1,
prescribed. Whilst crude this condition reflects the fact that the newly
placed slab is more ‘active’ in a heat production sense than the other slab.
This condition allows heat to enter the first slab across the boundary located
at x = 1. Also to greatly simplify matters we will assume both slabs are
equally moist (so θ = 1); near the interface between the slabs this is likely
to be the case, unless the interface has dried out before the second slab is
poured. (A more accurate model would include both slabs with continuity
conditions across the interface and different initial maturity levels in the two
slabs. The present model simply represents the simplest sensible model.)
Under the above circumstances the equations reduce to
∂T
∂t
= κL
∂2T
∂x2
+
∂m
∂t
, (12)
∂m
∂t
= µ(1−m)eξ(T−1) , (13)
with
∂T (0, t)
∂x
= 0, T (1, t) = T1, T (x, 0) = 0, m(x, 0) = 0. (14)
Strictly speaking we should specify the boundary condition at the other face,
x = −1, of the slab but we will retain the symmetry condition ∂T (0,t)∂x = 0;
the results of interest are not sensitive to this condition.
The ξ = 0 case
Firstly note that in the ξ = 0 case there is no coupling between the matu-
ration equation and the heat equation, so that m(x, t) ≡ m(t) is uniform in
x and approaches m = 1 in a time t of order 1. The temperature increases
uniformly in t due to the hydration heat release, except close to the interface
x = 1, Additional heat diffuses from the second slab across the interface at
x = 1 effecting a change in temperature over a distance of order
√
κLt; i.e.
initially slowly but then slowing down rapidly, see Figure 3.
Maturity Effects in Concrete Dams 65
0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
0.25
0.5
0.75
1
1.25
1.5
1.75
2
t = 0.2
t = 0.5
t = 1
t = 2
T (x, t)m(x, t)
0 t x0
Figure 3: Maturity m(x, t) and temperature T (x, t) profiles in the ξ = 0
case with κL = 0.001 and µ = 1.
The ξ > 0 case
In the ξ > 0 case, see Figure 4, the heat inflow from the second slab en-
hances the maturation in the first slab so that m increases more rapidly to
1 close to the interface; Figures 3 and 4 are very different. Also there is
additional hydration heat (due to the Arrhenius factor) released near the
surface, so that the temperature rise is larger than would be expected if
Arrhenius effects were ignored. Such near surface effects will only be felt
within the diffusion distance from the interface. Note that m reaches 1 im-
mediately at the interface because the higher temperature at the surface
enhances maturation. Note, however, that the temperature throughout the
slab is increasing due to hydration so that enhanced maturation will occur
throughout the slab; the reaction in fact completes by time t = 1 compared
with t ≈ 2 in the ξ = 0 case, see Figure 4.
The difference between the cases is most easily seen in Figure 5 which
displays spatial variations in maturity; compare with Figure 3.
66 N.D. Fowkes, H. Mambili Mamboundou, O.D. Makinde, Y. Ballim and A.Patini
0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
0.25
0.5
0.75
1
1.25
1.5
1.75
2
t = 0.5
t = 1T (x, t)m(x, t)
x = 1 x = 0
t0 x0
Figure 4: Maturity m(x, t) and temperature T (x, t) profiles in the ξ = 2
case with κL = 0.001 and µ = 1.
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
t = 0.3
t = 0.2
t = 0.5
m(x, t)
x
Figure 5: Maturity m(x, t) profiles for fixed values of t,in the ξ = 2 case
(κL = 0.001, µ = 1).
4 Surface effects
Because of the low conductivity of concrete most of the interesting behaviour
occurs on or close to interfaces; the surface is particularly important. For
the same reason moisture transport effects are confined to the surface or
interfaces. This suggests that a semi-infinite slab model is appropriate for
Maturity Effects in Concrete Dams 67
investigations, and similarity solutions could be useful for understanding the
fronts moving in from the interfaces.
5 Recommendations and conclusions
Evidently the above just scrapes the surface of this problem, but at least the
simulations performed illustrate the phenomenon under investigation. The
indicated work requires the careful investigation and time input that would
be best achieved through postgraduate work.
References
Ballim, Y. and Graham, P. C. (2003). A maturity approach to the rate of
heat evolution in concrete. Magazine of Concrete Research, Thomas Telford,
55, 249-256.
Carslaw, H. S. and Jaeger, J. C. (1959). Conduction of Heat in Solids,
Oxford University Press, Oxford.


A maturity approach to the rate of heat evolution in concrete.

Conduction of Heat in Solids,

Maturity effects in concrete dams

Maturity effects in concrete dams

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