The presented problem was to determine an appropriate heat transfer boundary condition at the surface of a concrete slab exposed to the environment. The condition obtained involves solar radiation and convective heat transfer, other terms were shown to be small compared to these. It is shown that this boundary condition leads to a temperature variation that has qualitative agreement with experiments carried out by the Cement and Concrete Institute

Ballim, Y.

Charpin, J.P.D.

Fitt, A.D.

Myers, T.G.

Patini, A.

English

Mathematics in Industry Information Service Eprints Archive

MODELLING SURFACE HEAT EXCHANGES
FROM A CONCRETE BLOCK INTO THE
ENVIRONMENT
J.P.D. Charpin, T.G. Myers∗, A.D. Fitt †,
Y. Ballim and A. Patini‡
Abstract
The presented problem was to determine an appropriate heat transfer
boundary condition at the surface of a concrete slab exposed to the
environment. The condition obtained involves solar radiation and con-
vective heat transfer, other terms were shown to be small compared to
these. It is shown that this boundary condition leads to a temperature
variation that has qualitative agreement with experiments carried out
by the Cement and Concrete Institute.
1 Introduction
The long term performance of concrete structures, e.g. durability and
strength, can be greatly influenced by the initial thermal field, see for ex-
ample Schindler et al (2004), Wojcik et al (2003), Yu et al (1998). Hence,
the removal of heat from concrete during the initial drying stage is essen-
tial to minimize undesirable effects. Temperature fluctuations can also be
a problem, leading to shrinkage and expansion which in turn may lead to
large stresses and cracking, see Liu et al (2002).
When concrete is prepared and placed under conditions of high ambient
temperature, low humidity, solar radiation or wind, an understanding of the
∗Department of Mathematics and Applied Mathematics, University of Cape Town,
Rondebosch 7701, South Africa. e-mail: jcharpin@maths.uct.ac.za,
myers@maths.uct.ac.za
†School of Mathematics, University of Southampton, Southampton, UK. e-mail:
adf@maths.soton.ac.uk
‡School of Civil and Environmental Engineering, University of the Witwatersrand, Wits
2050, Johannesburg, South Africa. e-mail: ballim@civil.wits.ac.za,
anthony@civil.wits.ac.za
51
52 J.P.D. Charpin, T.G. Myers, A.D. Fitt, Y. Ballim and A. Patini
effects these environmental factors have on concrete properties and construc-
tion operations is required. Once these factors are understood, measures can
be taken to eliminate or minimize undesirable effects (Alabidien 1991). To
describe the heat loss from a concrete slab into the environment the Cement
and Concrete Institute (CCI) fixes the concrete temperature at the surface
to the ambient temperature. Experiments have shown that in reality the
surface temperature differs significantly from ambient, see Schindler et al
(2004), Bentz (2000), Gilliland & Dilger (1997). Laboratory experiments
reported by the CCI also show that the surface temperature variation is out
of phase with the ambient variation. This is confirmed by the results of
Gilliland & Dilger (1997) obtained during the construction of a bridge in
Canada.
In this paper we attempt to improve on the current boundary condition
and explain both the temperature difference and phase lag.
2 Modified boundary condition
We will consider a one-dimensional model to simplify the analysis and allow
us to focus on the boundary condition. Physically this corresponds to a
block with a large top surface area (this will be discussed later) and we only
take temperature readings away from the edges of the block. However, this
one-dimensional analysis is illustrative. In reality a block losing heat from
all surfaces would require the same boundary condition applying to each
surface.
In one dimension the heat flow is modelled by
ρccc
∂T
∂t
= κc
∂2T
∂z2
, (2.1)
where ρc, cc and κc are the density, specific heat and conductivity (respec-
tively) of concrete. This may be solved subject to a fixed temperature at
the base T = Tg, where Tg is the ground temperature. This could be taken
equal to the ambient temperature. Initially, when the concrete has been
poured, we can assume the temperature is constant throughout the block
T (x, 0) = T0. At the top surface, which is in contact with the air, the current
boundary condition used by the CCI is
T = Ta(t)|z=L , (2.2)
where Ta is the ambient temperature. The ambient temperature may be
Modelling Surface Heat Exchange 53
approximated by
Ta =
(
Tmax + Tmin
2
)
−
(
Tmax − Tmin
2
)
sin
(
2pi(t+ tw)
24 × 3600
)
, (2.3)
where tw is the time at which the temperature is a minimum. This occurs
usually about an hour before sunrise. In the following we assume that
the temperature is a minimum at 5.30am which implies that tw ≈ 5.5 ×
3600sec. Obviously a more accurate expression could be obtained through
local meteorological data.
Equation (2.2) states that there is continuity of temperature across the
air-concrete interface. Anybody who has witnessed hot air rising from a
road or other such surface will know that this is not the case in reality. A
more accurate condition requires continuity of energy flux. To use such a
condition we must first determine the appropriate energy source and sink
terms. The standard terms are
Qsun = γabsQinc , Qr = σ(T
4 − T 4a ) , Qh = H(T − Ta) ,
where Qsun represents the contribution from the sun shining on a surface,
Qr represents radiation from the slab to the sky and Qh is convective heat
transfer. The constant γabs = 0.65 is the solar absorptivity of concrete,
Qinc ∼ 700W/m
2 is the solar radiation normal to the surface andQsun is zero
at night time. In the radiation expression,  = 0.9, σ = 5.7×10−8 W/m2 ◦C4.
If Ta > T radiation contributes to the heating, if Ta < T then heat is being
lost by radiation to the sky. The convective heat transfer coefficient H needs
to be determined experimentally, however simple formulae are reported in
Bentz 2000.
With these energy terms the appropriate boundary condition at the ex-
posed surface z = 0 is
−κc
∂T
∂z
= Qsunf(t) +Qr +Qh = Qs + σ(T
4 − T 4a ) +H(T − Ta) , (2.4)
where f(t) is simply a switch function to turn off the solar radiation term
at night. For example f(t) = 1, t ∈ [6, 18]hrs, f(t) = 0, t ∈ [18, 6]hrs.
If the ambient temperature is Ta = 20
◦C then we require T ∼ 100◦C for
Qr to be of the same magnitude as the other terms. When T takes a more
typical value, T ∼ 60◦C then Qr/Qh ∼ 10
−2 and so Qr may therefore be
neglected in all realistic calculations. A wet concrete slab would also lose
energy due to evaporation. This was estimated during the study week to
be negligible compared to other terms. However, it is retained in certain
published models, see Schindler (2004) for example.
54 J.P.D. Charpin, T.G. Myers, A.D. Fitt, Y. Ballim and A. Patini
A realistic boundary condition to replace T = Ta at the surface is there-
fore
−κc
∂T
∂z
= Qs +H(T − Ta) . (2.5)
This is a standard cooling condition. When the heat transfer between the
air and the concrete is large, H →∞, it reduces to the condition currently
used by the CCI. However, in general this condition can lead to significantly
different results to those obtained by setting T = Ta.
3 Verification of the new boundary condition
Now consider the laboratory experiment reported by the CCI. The results
showed a temperature variation on the concrete surface less than that of the
ambient temperature and with a time lag.
To describe the experiment we first scale the governing equation, equa-
tion (2.1). With length-scale L and a time scale of one day, τ = 24× 3600s,
we obtain
ρcccL
2
κcτ
∂T
∂t′
=
∂2T
∂z′2
, (3.1)
where z′ = z/L, t′ = t/τ . We note that the diffusion length-scale
L =
√
κcτ/ρccc ≈ 0.24m ,
that is, over the period of one day we expect the energy loss from the surface
to have an effect over distances of the order 20cm. Gilliland & Dilger (1997)
give data for temperature profiles at depths 7.5, 27.5 and 157.5cm inside a
concrete block which is exposed to the atmosphere. The ambient variation
is closely mimicked by the temperature profile at 7.5cm (with a time lag).
At 27.5cm the variation is barely noticeable and at 157.5cm the ambient
variation has no effect. Hence the length-scale of 20cm appears entirely
reasonable. Effectively this means that the bottom of a standard block is
sufficiently far from the surface for it to notice the daily variation. Further,
to carry out experiments using a one-dimensional model we would require
a block with a top surface significantly larger than 20x20cm2. We therefore
solve the system
∂T
∂t′
=
∂2T
∂z′2
, (3.2)
Modelling Surface Heat Exchange 55
−
∂T
∂z′
= Λ(T − Ta)
∣∣∣∣
z′=0
,
∂T
∂z′
= 0
∣∣∣∣
z′→−∞
, T (z′, 0) = 20 , (3.3)
where
Λ = LH/κc = H
√
τ/ρcκc. (3.4)
Note solar radiation is neglected since the experiment is indoors. We also
replace T = Tg with a zero flux condition, since, as mentioned, the base in
this scaling is a long way from the area of interest.
Numerical solutions for this system are shown in Figures 1–3 for a 10m
deep block initially at 20◦C up to time t = 72 hours. With a length-scale
L = 0.24m this means the domain extends for z ∈ [0, 41.7]. The ambient
temperature varies according to equation (2.3), with Tmax = 30, Tmin =
10◦C.
0 10 20 30 40 50 60 70 80
10
12
14
16
18
20
22
24
26
28
30
time (hours) 
Temp (C) (a) 
(b) 
A 
0 10 20 30 40 50 60 70 80
10
12
14
16
18
20
22
24
26
28
30
(a) 
(b) 
time (hours) 
Temp (C) 
B 
Figure 1: Temperature variation for 3 days, at the top surface of the concrete
with A) Λ = 0.1, B) Λ = 1 and boundary conditions a) ambient temperature,
b) cooling condition.
It is clear from Figures 1 A) and B) that the new boundary condition
exhibits the required behaviour. On figure 1 A) the numerically calculated
temperature depicted by curve (b) varies between 16.5 and 23.5◦C and lags
behind the ambient temperature. The form of the numerical curve depends
on the single parameter Λ. Increasing Λ increases the correspondence be-
tween the curves, both the amplitude increases and the time lag decreases.
This may be observed on Figure 1 B) which has Λ = 1. The condition
T = Ta is therefore reasonable (in the absence of sunlight) for high Λ. The
amplitude difference and time lag therefore decrease with an increase in H
or τ and a decrease in ρc, κc. Note, there is a square root dependence on
56 J.P.D. Charpin, T.G. Myers, A.D. Fitt, Y. Ballim and A. Patini
all parameters except H. The variation with time-scale τ indicates that a
long period variation, such as seasonal changes, do not have a great effect on
results obtained via the two conditions. However, short period variations,
of the order of days, lead to a significant difference in the results.
−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0
18.5
19
19.5
20
20.5
21
21.5
22
z 
Temp (C) 
(a) 
(b) 
Figure 2: Temperature throughout the concrete at a) t = 72 hours, b) t = 63
hours.
Figure 2 shows the temperature from the surface to a distance 20L inside
the block at times t = 63, 72 hours. When t = 72hours the block is coolest at
the surface and the temperature increases as z decreases up to a maximum,
it then starts to decrease again. This is because the cooling from the outside
has not reached all the way in and parts of the block are still feeling the effect
of an earlier heating period. For t = 63 the opposite occurs, i.e. the block
is hottest at the outside, cools inwards to a minimum and then increases
again.
Figure 2 also gives us a further indication about the length-scale over
which temperature variations occur. The calculated length-scale L indicates
the order of magnitude over which significant temperature changes are ex-
pected in the stated time-scale of 1 day. This does not mean that at a
distance L from the surface a numerical calculation should stop. In fact,
from Figure 2 it is clear that as far back as 10L ∼ 2.4m (corresponding to
z = −10 on the figure) differences can be observed in the two solutions.
However, these differences are a result of heating and cooling over 3 days
and so the temperature variation has penetrated much further than the
length-scale implies.
Finally in Figure 3 we show the full temperature profile variation through-
Modelling Surface Heat Exchange 57
out the block for 3 days.
−20
−15
−10
−5
0
0
20
40
60
80
18
18.5
19
19.5
20
20.5
21
21.5
22
time z 
Temp (C) 
Figure 3: Temperature throughout the concrete for t ∈ [0, 72] hours.
4 Conclusions
The cooling condition (2.5) appears to be a sensible replacement for the
current condition. Results show that the smaller than ambient temperature
variation and time lag observed in experiments are reproduced.
Of course there remains a great deal to be done with this work. In
particular, the heat transfer coefficient H is always problematic to pin down.
It’s value depends on a number of parameters, including wind speed, surface
roughness and orientation of the slab (dam walls are usually almost vertical
and have a lower heat loss rate). Tables and formulae for H do exist for
various surfaces and wind speeds, however, in practice, it may be sensible
to carry out an experimental study to determine H for specific conditions.
The example shown was aimed at modelling a one-dimensional labora-
tory experiment. In this case the solar radiation was neglected and so the
model depended on a single parameter Λ, see (3.4). The numerical results
showed that increasing Λ produced results closer to those obtained by set-
ting T = Ta. This is to be expected since in the limit H → ∞ the new
boundary condition reduces to T = Ta. The addition of solar radiation to
the model will significantly change results since the solar radiation term is
of the same order of magnitude as the heat transfer term.
Finally, whilst the model has been shown to exhibit the correct qualita-
tive behaviour, it must be tested much more thoroughly. To properly verify
58 J.P.D. Charpin, T.G. Myers, A.D. Fitt, Y. Ballim and A. Patini
this model it should be:
a) compared against data from the laboratory experiments
b) compared against one-dimensional data made outside the laboratory (so
including the solar radiation term).
A study of the heat transfer coefficient must also be carried out for various
realistic situations.
Acknowledgements
T.G. Myers acknowledges support of this work under the National Research
Foundation of South Africa grant number 2053289. J.P.F. Charpin acknowl-
edges the support of the Claude Leon Harris Foundation.
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Analysis of concrete pavement responses to temperature and wheel loads measured from instrumented slabs.

Concrete pavement temperature prediction FHWA HIPERPAV and case studies with the models.

Heat removal from mass concrete footing, Concrete

Hot weather concreting,

Modelling the interaction between the atmosphere and curing concrete bridge decks with the SLABS model.

Monitoring concrete temperature during construction of the Confederation Bridge,

Modelling surface heat exchanges from a concrete block to the environment

Modelling surface heat exchanges from a concrete block to the environment

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