Claude Boulay
email: claude.boulay@lcpc.fr

Jean-Michel Michel Torrenti

Jean Luc

Richard Saintilan

Jean-Luc Andre

Quasi-adiabatic calorimetry for concretes : Influential factors

Quasi-adiabatic calorimetry for concretes: Influential factors AbstrAct ■ Quasi-adiabatic calorimetry performed on concrete specimens (known under the French acronym QAB) is a method first proposed at the LCPC Laboratory in the 1980's; it involves transposing to concrete the same procedure applied for the standardized determination of cement hydration heat . This input is essential for predicting, by numerical simulation, the thermomechanical behavior of structures. The present article offers a description of the calorimetry equipment and its performance, along with the various calculations that yield: the amount of heat released, the kinetics of this release (if the test had been adiabatic), heat flux, chemical affinity and degree of hydration. The principles behind QAB calorimeter calibration are reviewed and the magnitudes of influences assessed in order to estimate uncertainty in the heat release calculation. on the other hand, the level of dispersion inherent in the material still needs to be estimated.

Calorimétrie quasi adiabatique pour bétons : facteurs d'influence

■ résumé La calorimétrie quasi adiabatique sur éprouvettes de béton (QAB), méthode proposée au LCPC dans les années 1980, est une transposition au béton de la procédure appliquée pour la détermination normalisée de la chaleur d'hydratation des ciments . Cette donnée est essentielle pour la prédiction, par simulation numérique, du comportement thermomécanique des structures. Cet article regroupe une description du matériel et de ses performances et les différents calculs permettant d'obtenir la chaleur dégagée, la cinétique de ce dégagement de chaleur si l'essai avait été adiabatique, le flux de chaleur, l'affinité chimique et le degré d'hydratation. Les principes de l'étalonnage des calorimètres QAB est rappelé et l'examen des grandeurs d'influences permet une estimation de l'incertitude sur la chaleur dégagée. Une estimation de la dispersion inhérente au matériau serait encore à faire.

IntroduCtIon

Since the end of the 1980's, the development of powerful finite element computation codes has provided a glimpse into the prediction of temperature fields within massive concrete elements at an early age [2]. One of the data inputs, critical for this prediction, is the concrete heat of hydration. For this reason, the time was ripe for a joint LCPC -CECP-Angers project team [3] to develop a calorimeter labeled QAB (French acronym for "Quasi-Adiabatic for Concretes"), based on the Langavant-type semi-adiabatic calorimeters used for cement. This calorimeter and its associated testing method have since been refined given that attempts to deduce the heat release inside concrete from tests conducted on cement pastes or mortars were, 1 at the time, inconclusive.

The QAB test makes it possible to estimate the heat being released due to the hydration of cement contained in the concrete specimen. In knowing that all anhydrates will not be hydrated by the end of the hydration process, the heat considered herein does not allow determining the specific heat of cement hydration 2 [20]. On the other hand, this test incorporates all possible interactions between concrete components, including the effects of admixtures.

The QAB calorimeter is composed of a double-walled caisson (external wall made of PVC, internal wall of fiberglass-reinforced polyester) filled with an insulating material (polyurethane foam) approx. 14 cm thick (see Fig. 1). The central housing of this set-up accommodates a two-part steel cylindrical shell, in which the cylindrical concrete specimen (dimension: Ø16 × 32 cm) in its cardboard mold is placed following fabrication.

Two identical calorimeters are used and placed in a test room held at constant temperature (20°C for a standard calorimetric test). The fresh concrete specimen is set into the first calorimeter, while a control specimen (made of concrete aged at least 3 months) is positioned in the second. It is assumed that the heat capacity of the control specimen is close to that of the test specimen. Mounanga demonstrated that a variation in this parameter with respect to hydration did exist, yet remained rather weak [17] (see also [14, 15]). Each calorimeter has been calibrated and is required to display similar characteristics. The temperature measurement chain has also been calibrated. The test then consists of recording, at regular intervals (10-15 min), the internal temperatures of both calorimeters, as well as the external temperature measured between the two calorimeters, in order to ensure room conditions are being well controlled (see [6]).

The next section will provide a review of: the equations used to transform these temperature recordings into heat released by the concrete; instrumentation employed; operating protocols for calorimeter calibration and testing; and the procedure adopted to estimate test measurement uncertainty (without incorporating reproducibility).

determInatIon of heat released In a qab CalorImeter

Let's use the following notations:

C

concrete [in J/°C]: heat capacity of the concrete alone; this value is calculated from concrete composition and specimen mass m S measured after fabrication, 2 The specific heat of cement hydration is the quantity of heat per unit mass (in J/g) required for complete hydration. In the QAB calorimeter, part of the heat released due to cement hydration increases the specimen temperature while another part increases the calorimeter temperature, and the remainder gets discharged to the outside. For the QAB calorimeter, these effects are expressed as follows:

(1)

This equation is derived by first writing the heat balance for both the calorimeter containing the sample and the calorimeter containing the control specimen, and then by combining the two resulting equations. This procedure allows skirting the complex task of having to measure ambient temperature by assuming that the cooling speeds of the two calorimeters are equivalent, a condition that may be reflected by:

(2) are roughly equal since the control set-up contains a specimen composed of the same pre-hardened concrete and since the two calorimeters have been developed using the same technology (with nearly equal heat capacities when empty and very similar heat loss coefficients). Nonetheless, given that α depends on temperature, this hypothesis cannot hold throughout the test duration. The only way to overcome this obstacle would be to regulate ambient temperature and then ensure that the initial control temperature is close to this regulated temperature (see [10])

The expression for q(t) constitutes the test result. Figure 16 offers a glimpse of such a result for a conventional concrete whose mix design components are listed in Table 2.

Additional calculations become feasible once the quantity of heat released has been determined. Since the heat capacity of concrete is known, it is possible for example to calculate [3] the temperature rise in this specimen should it have been placed under adiabatic conditions, i.e.:

(3)

The association between T adia and t in this initial correction cannot be assigned any real meaning; had the test actually been adiabatic, then thermo-activation of the cement hydration reaction would have accelerated the heat release over time [3, 10]. In this case, a second correction would need to be applied on the time parameter, by applying Arrhenius' Law in order to introduce activation energy E a . This corrected time is then given by the following expression:

(4)

The determination of activation energy E a (in J/mol) has been discussed in the technical manual entitled "Concrete strength in structures: Maturity measurements" [13]. Note: R is the perfect gas constant (i.e. equal to 8.314 J mol -1 K -1 ) Calculation of the two previous relations produces (Fig. 2) the curve T adia , (t cor ), which ultimately yields the adiabatic evolution in concrete temperature. This analysis of semi-adiabatic test results on concrete, cement or mortar does not generate an evolution exactly identical to the results of an adiabatic test [10, 12]. For one thing, these corrections undoubtedly introduce deviations, though the authors emphasize that chemical species formed at one temperature may differ from those formed at other temperatures.

Heat flux proves straightforward to determine, except for the difficulties inherent when evaluating a slope based on experimental data, which always seem to contain noise. Data noise can be reduced by various conventional techniques (analog or digital filtering, etc.).

With respect to the same mix design presented in Table 2, this determination is depicted (Fig. 3) in the form of two calculations. The first slope evaluation of the q(t) curve focuses on two successive points, and this calculation is associated with a noise. The second calculation includes 6 points on both sides of the central point (± 30 min until reaching a time of 29 h and ± 6 h beyond). Under these conditions, smoothing of the second calculation turns out to be entirely satisfactory, as the peak height remains unchanged and no temporal offset is introduced. For concretes with fast kinetics, it would most certainly be necessary to alter the search interval in order to narrow the smoothing time window.

For this numerical processing step, it is preferable to utilize a regular measurement interval (e.g. 10-15 min), but this was not the case for the test presented herein. Another tool employed during these calculations to estimate changes in both the degree of hydration and rate of heat release vs. temperature history is chemical affinity.

The variation rate in degree of hydration, denoted ξ(t), can be expressed as a function of chemical affinity along with a term expressing thermo-activation [18], i.e.:

(5)

where ξ ∞ is the final degree of hydration and q ∞ the final amount of heat released.

Chemical affinity can then be written as follows:

(6) Time t eq is the equivalent time required to reach, at constant temperature T 0 , the degree of hydration obtained (in real time) after any temperature history profile.

The degree of hydration ξ(t) corresponds to the ratio of the quantity of cement that has entered into reaction to the initial quantity of anhydrates. This degree of hydration does not necessarily reach a value of 1 once hydration has stopped, but instead a value ξ ∞ . A number of models are available in the literature to estimate this value. As an example, Waller [20] proposed, based on a collection of results from the literature and his own research, an expression adapted to CEM I-type cements related to the water/cement (w/c) ratio as follows:

(7)
The degree of hydration is also expressed as a function of the degree of reaction progress, denoted here as γ(t), according to the equations:

with:

(8) and (9)

which leads to:

(10)

A good estimation of the final value q ∞ of q(t) can then be produced should the semi-adiabatic test be extended for a long enough period of time [6].

Figure 4 shows the chemical affinity deduced from a calorimetric test conducted on the concrete described in Table 2. testIng InstrumentatIon

Description of the calorimeter ■

The calorimeter is composed of a PVC box insulated using polyurethane foam. The specimen housing, located at the center of the caisson, has been designed for placement inside an intermediate steel cylinder 1.5 mm thick. An insulated cover provides access for the cable connecting the temperature probe, which has been set into a sealed copper tube placed at the base and filled with oil (Fig. 5).

To ensure adequate test reproducibility, the thicknesses of polyurethane foam insulating layers must match as closely as possible. As an order of magnitude, this insulating thickness is sufficient to maintain, in the steady state, a temperature of 35°C (i.e. above a 20°C ambient temperature) for a 3.5 W internal heat source.

Rubber type joints provide the necessary seal. The calorimeter bases are essential for ventilation beneath the caisson, so that wall temperature of the device is actually being dictated by the ambient air. Without this precaution, the heat loss coefficient derived by calibration might have to be modified. It has even been recommended to install the caisson on a table at a height that allows for stirring on the under-surface yet low enough to easily place the specimen inside the caisson (i.e. at a height of approx. 40 cm).

The control calorimeter ■

To conduct a test, two calorimeters are introduced. A control specimen is positioned inside a QAB calorimeter (shown as no. 6 here) that matches the calorimeter receiving the fresh concrete specimen (no. 5). Control specimen temperature is close to ambient temperature and evolves with fluctuations in ambient temperature using nearly the same time constant as for the fresh concrete specimen temperature. The following experiment yielded an estimation of time constants for the two matching calorimeters. In order to test for the delayed effect in comparison with external temperature variations, two concrete cylinders aged at least 3 months were placed in the calorimeters and their temperature was recorded throughout the test duration. Room temperature was initially set at 16°C, and temperature stabilization was anticipated. A recommended temperature of 26°C was then applied; the rise in room temperature from 16° to 26°C took 2.5 hours.

It can be seen that in 2 days' time, the concrete specimens only increased in temperature to 21.5°C. These recordings were nearly identical in both calorimeters (Fig. 6).

In order to quantify the differences in thermal behavior between the two calorimeters subjected to this external temperature loading, a numerical simulation was implemented; this simulation consisted of calculating, according to a step-by-step procedure, internal temperature as a function of the previous time step (a numerical resolution of systems of differential equations using Euler's method). Just a single parameter, i.e. the time constant, underwent adjustment, as based on a set of recorded points (see Fig. 6, from 0 to 44 hours). The time constant τ 5 of Calorimeter 5 equals 55.7 h, while that of Calorimeter 6 (τ 6 ) is 53 h. At this stage, it can be affirmed by observation that the time constants corresponding to these two different calorimeters differ by no more than 5%. Let's recall that the curve can then be extrapolated out to two weeks .(Fig. 7). 273 hours are required for the internal temperature to reach 99% of the level imposed on the outside, equivalent to 5 times the time constant value What is the impact on the measurement of θ from such a deviation in time constants? The modeling approach adopted offers an answer to this question by replacing external temperature by dummy fluctuations (that are still realistic) of a test room over a two-week period. Fluctuations with a 20-min period and 1°C amplitude were applied and assumed to represent temperature variations due to test room regulation. These fluctuations were then superimposed by daily 1°C fluctuations, assumed to simulate the daytime/nighttime alternation often found in test rooms. This scenario was then repeated for two weeks (Fig. 8).

Figure 9 shows the trend in temperature deviation for the time constants previously determined (their ratio τ 5/ τ 6 equals 1.051). This set of points is considered to be a population of temperature deviations between the two calorimeters as ambient temperature fluctuates; this population can be characterized by its standard deviation.

The next step is intended to observe how this standard deviation evolves with respect to the ratio τ 5 / τ 6 by varying just τ 5 in a way that allows spanning the 0.8-1.2 interval. The tracking of 2 times this standard deviation (which is considered the standard uncertainty with an expansion factor of 2) is displayed in Figure 10.

It is observed that when the time constant of one of the calorimeters deviates by 20% in either direction from the other calorimeter constant, the expanded uncertainty (k = 2) remains less than 0.025°C. For the calorimeters studied herein (whose time constants differ by 5%), expanded uncertainty is estimated at 0.005°C. As will be discussed at greater length in the section devoted uncertainties, it can already be concluded, with inspiration from the findings drawn by Mounanga [17], that the influence of variations in ambient temperature on the accuracy of temperature deviation between the two calorimeters is negligible given that such variations remain relatively mild.

Temperature probes ■

The temperature probes to be used contain platinum resistors (100 Ω at 0°C); they need to be cabled according to the Kelvin method (4 connection wires leading to the signal conditioner, 2 for the constant current power supply and the other 2 for voltage readings at the resistance terminals). If probe responses between 0° and 100°C were assimilated to a straight line, then the error committed would be that shown in Fig. 11.

Even though this error is quite small, the level of accuracy can still be improved by applying a correction. The resistance between 0 and 100°C is, according to ITS-90 (i.e. International Temperature Standard -90), given by the following equation:

R T = R 0 (1 + A T + B T 2 ) (11)
where R T (Ω) is resistance of the platinum probe at temperature T (°C).

The coefficients A, B and R 0 are found by calibration, with nominal values of: A = 3.9083 10 -3 , B = -5.775 10 -7 , and R 0 = 100 Ω (which is the resistance value at 0°C).

The practical problem to be solved here is the inverse, i.e. to calculate temperature once resistance is known (after being deduced from voltage and current measurements). In this case, the practical function becomes:

T = -245.66 + 2.3556 R T + 0.0010115 R T 2 (12)
with R T expressed in Ω and T in °C.

The probes are placed at the specimen core in a copper tube Ø8 mm inside and 10 mm outside, with an opening towards the bottom and a lip (flared nipple) on top. This tube crosses the flexible cap (after cutting out a hole at the center) that is sold with the packaged molds. Once the specimen has been positioned inside the calorimeter, the probe is slid into the tube and paraffin oil is poured until the tube is entirely filled. Measurement resolution must be at least 0.01°C. It is possible to reach an expanded uncertainty of 0.17°C over the full calibration chain, in which case the standard uncertainty can be rounded to 0.1°C. Despite the strong recommendation to only supply these platinum probes during the actual measurement phase (so as to avoid self-heating, which could be detrimental to probe accuracy), the question can still be raised whether a continuously-supplied probe leads to a heat release that interferes with a precise reading of specimen heat release. The heat dissipated by means of the Joule effect during a test on concrete is not completely offset when using a set-up with both an active calorimeter and a control calorimeter. The excess heat in the active calorimeter stems from the phenomenon of its platinum resistance value not following the same trend as that of the control calorimeter. This excess is given by the expression:

(13)
where R TA (t) and R TT (t) are probe resistances of the active and control specimens, respectively. The currents circulating in the two probes are assumed to be equal. On an actual test (using the Figure 12 shows the evolution in heat release for the two extreme currents over the fixed interval.

Figure 13 indicates the relative error (for a total heat release in concrete of 725,600 Joules) committed on the heat release measurement vs. supply current. In conclusion, it is preferable (if possible) to select a 1 mA current, as this leads to a relative error for the concrete of just 0.03%, which depends on total heat released by the cement. For cements with a low heat of hydration, it is also preferable to select a supply current on the order of 1 mA. For CEM I (which applies to the present study), the supply current should not exceed 3 mA (with a dissipated power of 0.9 mW at 0°C and 1.25 mW at 100°C).

Another approach to this same problem consists of acknowledging the heat release and subtracting it from the result derived in Equation 1. Under these conditions, only the temperature accuracy criterion (self-heating of the probe by means of the Joule effect) is to be taken into account.

Figure 12 Excess heat released by the Joule effect in platinum

probes. The differential impact of measurements using two calorimeters does not fully compensate for this effect, since temperature histories for the two calorimeters differ.

Table 1 Excess heat released by the Joule effect, at 4 weeks into the test period in the active calorimeter and with the platinum probes being continuously supplied

CalorImeter CalIbratIon

The calibration procedure for QAB calorimeters involves determining both the heat loss coefficient α, which is a linear function of temperature θ (with b and a being the corresponding linear coefficients), and the heat capacity C cal of the calorimeter [1].

Determination of heat loss coefficient ■

Determining the heat loss coefficient entails positioning an aluminum cylinder fitted with a heating resistance inside the calorimeter to be calibrated, alongside a matching control calorimeter containing an identical cylinder that has not been equipped with a resistance. Temperatures are recorded at the core of both cylinders (θ denotes the difference in temperature between the two calorimeters). In the vicinity of the calorimeters, the average temperature must lie between 19 and 21°C (monitored required using an external probe), with fluctuations around this average not to surpass ±0.5°C. Air inside the test room must be regularly circulated. Voltage at the resistance terminals along with the circulating current are measured whenever the temperature reaches a plateau, at which point the energy dissipated by the Joule effect (P = U I = R I 2 ) has been completely channeled to the outside.

In analyzing calorimeter calibration data (4 temperature plateaus), the existence of an apparent linear relation is readily observed (Fig. 14) between the power emitted by the Joule effect and the temperature deviation between the reference cylinder and the control cylinder. For the calorimeter targeted herein, the slope of this linear relation equals 369.8 J/h/°C.

Figure 13

Percentage of error committed on the estimation of heat released by concrete vs.

continuously-applied platinum probe supply current (calculation conducted for the concrete described in Table 2).

Figure 14

The heat (loss) flux through the QAB calorimeter sidewalls appears to be proportional to the temperature deviation between the reference cylinder and the control cylinder. The linear regression is forced at the point 0,0 (i.e. zero temperature deviation = zero flux).

As an initial approximation, these findings reveal that the heat loss coefficient (i.e. the slope of this functional representation) does not vary with respect to temperature. As previously mentioned by Wadsö [19], it would seem that heat transfer through calorimeters equipped with an insulated sidewall (such the QAB calorimeter) primarily takes place via conduction, whereas transfer with Langavant-type calorimeters mainly occurs, as indicated by Alègre [5], through radiation, thus making these latter calorimeters temperature-dependent.

In reality, flux deviations relative to the regression line, as magnified in Figure 15, show a nonlinearity whose shape (in this figure) has been smoothed by a second-degree polynomial. The effects of radiation on heat transfer are nonetheless visible at this scale. A similar nonlinearity has been found for a set of calibration curves from various QAB calorimeters.

For the sake of accuracy, when determining and using α, the calibration protocol for Langavanttype calorimeters is selected for the QAB calorimeters. The heat loss coefficient, α, is calculated for each of the four temperature plateaus (power divided by θ). Coefficients a and b are obtained by means of linear regression for α/θ.

Figure 16 indicates the influence of the associated heat conduction coefficient, which is considered to either vary with temperature or remain constant during heat release in a QAB calorimeter test. This deviation is observed to stay small (on the order of 3%). In summary, heat loss through the sidewalls of a QAB calorimeter mainly involves conduction. The little amount of radiation contributing to this heat transfer makes the heat conduction coefficient response slightly parabolic. The interpretation of heat release results from QAB calorimeter tests is not very sensitive to the choice of heat loss coefficient (i.e. constant vs. proportional to temperature). In contrast and out of concern for accuracy and homogeneity relative to Langavant calorimeter calibration procedures, it is preferable to introduce 4 temperature thresholds in order to determine a temperature-based heat loss coefficient, as proposed in .

Determination of calorimeter heat capacity ■

Determining the calorimeter heat capacity (C cal ) consists of measuring the temperature drop once the current has stopped circulating following the last threshold (in other words, when the Joule effect heat source falls to 0). The calculation proceeds by assuming that the heat loss coefficient (α) and heat capacity of the reference cylinder (C R ) are both known. C cal is the difference between total heat capacity (C tot ) and C R . The focus then turns to determining C tot .

Once the current has been shut off, heat loss in both the calorimeter and reference cylinder will equal, at every point in time, the losses incurred through the calorimeter sidewalls (as the product of flux α θ multiplied by time interval dt). This equation is written as follows:

-C tot dθ = (a + b θ) θ dt (14)
The solution to this differential equation yields: Temperature measurements are selected for 24, 26, 28 and 30 hours. At these times, the total capacity is calculated. An average of the 4 values obtained can then be calculated, with the heat capacity of the reference cylinder being subtracted out so as to leave the heat capacity of the empty calorimeter.

For the active calorimeter under study herein (no. 5), C tot equals 16,751 J/°C on average and C cal amounts to 3,266 J/°C (with C alu = 13,485 J/°C).

Note: In reality, the calorimeter heat capacity calculation is affected by a bias stemming from the fact that calorimeter temperature is not homogeneous; instead, it varies between the value θ to the outside temperature value according to a slightly nonlinear profile [8]. This nonlinear trend however does not affect the heat of hydration estimation since the temperature profile in the calorimeter during the QAB test resembles the calibration profile.

unCertaInty In determInIng heat release If q ∞ is to be found experimentally, then the test would need to be stopped once θ moves below its expanded uncertainty. Let u a and u c be the standard uncertainties of temperature measurements in the active and control calorimeters respectively, then the expanded uncertainty on θ (with an expansion factor of k = 2) is given by: (16)

As an example, let's take a standard uncertainty on the order of 0.1°C. Under these conditions, U θ equals 0.3°C. Uncertainty in the determination of q(t), according to Equation (1), depends on the elementary standard uncertainties of each term in this equation. To derive this result, a number of numerical calculations have been applied to a real-world case (using the conventional concrete described in During an initial step, the most comprehensive inventory possible (see Table 3) of elementary standard uncertainties, u xi , is built (with Equation (1) being detailed to a point of deriving each parameter involved in the determination). In a subsequent step, the sensitivity, λ i , of q(t) with small variations, Δx i , on each of these parameters is calculated numerically, i.e.

(17)

Lastly, the standard uncertainty on the heat release determination can be calculated from the following expression [20]:

(18)
where ρ ij is the correlation coefficient between variables x i and x j . This coefficient is considered to assume a value of 0, 1 or -1. Should these variables be independent, then ρ = 0 and the expression for uncertainty on the heat release can be simplified to: .

In this inventory of elementary uncertainties, all mass values are considered as independent. Mix components are weighed separately, and each weighing exerts no influence on the subsequent weighing.

The standard uncertainties of these weight recordings for a given concrete mix (10 to 30 liters) are then transposed into standard uncertainties on the quantities present in the specimen (6.4 liters) contained inside the calorimeter. The mass recording of concrete in the specimen is also independent of the other recordings. This mass of concrete poured into the mold allows retrieving the masses of components actually present in the specimen through mass proportions found in the concrete formulation.

The uncertainties (Table 3) on coefficients stemming from the calibration (C cal , α) step are provided in the calorimeter calibration report.

The concrete specimen heat capacity can be calculated as shown in Standard NF EN 196-9, using conventional terms, by means of the following expression: In definitive terms, given a lack of greater precision, the standard uncertainty adopted for these specific heat capacities is set equal to 5% of their nominal value (Table 3), or 10% for expanded uncertainty with an expansion factor of 2.

C concrete = 800 m s + 3800 m e (19)
For the purpose of visualizing the relative impact of each parameter on the heat release calculation, a comparison is drawn of the expressions, which are considered as the indicators of elementary uncertainties (Fig. 17).

It is seen that during the first week of the test, uncertainties relative to specific heat capacity values capture the greatest share, followed by uncertainties on temperature measurements and then uncertainties on heat loss coefficients. It might be possible to improve the accuracy of these specific heat capacities, but it would not be very realistic to assume that temperature measurement accuracy could be improved. The other variables account for just a slight, and negligible, contribution.

Accuracy on the order of 0.1°C is not entirely infeasible, although such a goal would justify the introduction of platinum probes, along with a Kelvin (4-wire) assembly and a highly meticulous calibration process. In addition to these precautions, sheathed probes would need to be used in a stainless steel sleeve of sufficient length (16 to 20 cm) so that no water entry could be suspected, especially at the time of calibration.

Accuracy on the heat loss coefficient is no simpler to improve. It is also noteworthy that the calorimeter heat capacity determination exerts a negligible impact on the assessment of uncertainties, which correlates well with the observation of a determination protocol taking place under nonhomogeneous conditions (see the section entitled "Determination of calorimeter heat capacity").

Since temperature measurement accuracy is the most critical parameter after one week of testing, it makes sense to examine how this accuracy is correlated with uncertainty on the heat release calculation; such an investigation requires successively applying various values for the standard uncertainty (0.05-0.3°C) on temperature measurements (Fig. 18), as an input to calculating the uncertainty on q(t). This standard uncertainty range corresponds to an interval from 0.1° to 0.6°C for expanded uncertainties (with the expansion coefficient 3 , k, equal to 2).

On this diagram, the uncertainty on heat released is seen not to behave like a constant; it begins at 5% and then declines for a week (inaccuracy on specific heat capacities), followed by a nearly linear increase thereafter as a result of inaccuracy on temperature measurements. By setting a 10% threshold for relative uncertainty on the heat released (expansion factor = 2), the expanded uncertainty for temperature measurements must be less than 0.2°C. This limit leads to an interval like the one illustrated in Figure 19, which also includes uncertainties on both specific heat capacities and 3 When k = 2, the probability that values lie outside the expanded uncertainty interval equals 0.05. the heat loss coefficient. For this example, extending measurements beyond two weeks is clearly of no utility4 . This approach has not been applied to the other quantities capable of being determined (i.e. heat flux, adiabatic temperature rise, chemical affinity).

ConClusIon

The test to determine the heat of hydration of a concrete specimen using the QAB calorimeter has been analyzed in detail herein. This analysis has served to validate the draft test method, which has been shaped along the lines of the standard on cement and mortar calorimetry.

It would appear that heat is transferred through this type of calorimeter primarily by conduction. A very small share of radiation leads to processing the calibration data, for the sake of accuracy, as is the case with Langavant bottles, whose heat is mainly lost due to radiation.

With the aim of improving accuracy, temperature measurements need to be carried out on an active specimen placed in a first calorimeter at the same time as an inert concrete specimen is held in a second, matching calorimeter. Platinum resistance probes are also recommended as a means of achieving the required accuracy levels.

Figure 18 Variation in relative uncertainties on the heat release measurement as standard uncertainty on the temperature measurements varies. In this context, relative

uncertainty is defined as the expanded uncertainty of q(t) divided by the value q(t).

Figure 19

Interval covering potentially 95% of the calculated values using the dataset in Table 3.

Calibration uncertainties on QAB calorimeters exert a secondary influence with respect to the accuracy required for temperature measurements performed on both the concrete test specimen and control specimen. Another way to improve uncertainty on the heat release measurement would be to develop greater knowledge of the specific heat capacity of solid components and water vs. progress in the hydration reaction.

Ambient temperature fluctuations over a ±2°C range only cause negligible temperature deviations between the active calorimeter and control calorimeter.

Despite being advised to only supply the platinum resistance temperature probes while measurements are being recorded, these probes can still be supplied on a continuous basis provided power supply current is on the order of 1 mA. Beyond this value, the quantity of heat applied to the specimen becomes significant.

This estimation of heat release measurement uncertainty in a QAB calorimeter does not take into account reproducibility of the operating procedure. Quantifying these uncertainties can only be fully accomplished upon completion of a cross-testing campaign.

  figure 1Quasi-adiabatic calorimeters used for concrete specimens (QAB).

Figure 2

 2 Figure 2Derivatioin of the temperature curve under adiabatic conditions.

Figure 3

 3 Figure 3 Heat flux obtained by numerical derivation of the heat release curve.

  Figure 4Chemical affinity smoothed using a mathematical model[16]

Figure 5

 5 Figure 5 Cross-section diagram of the QAB calorimeter (actual proportions are shown).

Figure 6

 6 Figure 6Internal temperature recordings of both calorimeters in response to temperature level inside the test room.

Figure 7

 7 Figure 7 Imposed temperature level outside the calorimeters, internal responses and model indicating long-term extrapolation.

Figure 8

 8 Figure 8Ambient temperature imposed at the start of the on a 24-hour mesh -this pattern is repeated 14 times.

  Figure 9Evolution in temperature deviation between the two calorimeters when their time constants are same as those of the studied calorimeters.

  time elapsed since shutting off the resistance power supply of the reference cylinder, θ 0 = heating at time 0, α 0 = heat loss coefficient at time 0 (for θ = θ 0 ), θ t = heating at time t d, α t = heat loss coefficient at time t d (for θ = θ t ). Note: The numerator in the current version of Standard NF EN 196-9 is erroneous, appearing as α t d instead of a t d .

  Figure 17Proportion of elementary uncertainties vs. concrete age.

Table 2

 2 mix design), the calculation (in Table1) is performed for a current varying from 1 to 10 mA. Let's note that this choice of current is not without consequences.

	Current (mA)	1	2	3	4	5	6	7	8	9	10
	Heat release (J)	248	991	2,229 3,963 6,192 8,916	12,136	15,851	20,061	24,767

Table 2 ).

 2

	Components	Mass (in kg/m 3 )
	CEM I 52.5 N PMES CP2	340
	Bernières 0/4 sand	739.45
	Bernières 8/22 gravel	1,072.1
	Total water	184.22

Table 2

 2 Composition of the concrete tested in the QAB calorimeter.

. L., torrenti J. M., «

  Essais calorimétriques sur béton : banque de données, Paris : LCPC, Bordeaux : CETE du sud-ouest, rapport interne, 1990, 63 pages. Projet de mode opératoire pour la détermination de la chaleur dégagée lors de l'hydratation du ciment d'un béton placé dans un calorimètre Quasi Adiabatique pour Bétons (QAB) », BLPC, n° 278, 2010, pp. 37-42.

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	8 9 10 11	BriFFaut torrenti J. M., BuFFo-LaCarrière M., BenBoudJeMa F., nahas g., L., Prévision et évaluation de la fissuration précoce des ouvrages en béton. Thèse de l'Institut National des Sciences Appliquées de Toulouse, 2007. d'aLoia L., de Larrard F., de sChutter 12 g., taerwe L., haMFLer 14 h., röhLing s., van BreugeL k., "Methods for calculating temperatures in hardening concrete structures", RILEM TC119TCE, Chapter 6, draft 4, Rilem, 1993. JoLiCoeur 15 C., Étude calorimétrique de l'hydratation du ciment : détermination de l'énergie d'activation, rapport interne, Université de Sherbrooke, 1994. LaCkner 16 r., piChLer C., Mang h. a., Thermo-chemomechanics of cement-based materials at finer scales of observation: Application to hybrid analyses of shotcrete tunnel linings, in: Engineering Structures under Extreme Conditions -Multi-Physics and Multi-Scale Computer Models in Non-Linear Analysis and Optimal Design, A. Ibrahimbegovic, B. Brank (Hrg.); IoS Press, 2005, ISBN: 1 58603 479 0, pp. 170-199. Mounanga 17 p., kheLidJ a., Bastian g., uLM 18 F., torrenti J. M., wadsö 19 L., An experimental comparison between isothermal calorimetry, semi-adiabatic calorimetry and solution calorimetry for the study of cement hydration, Nordtest report TR522, 2003, March, 42 p. waLLer 20 v.,

It can nonetheless be advised not to stop testing too early since direct analyses often lead to difficulties.

whereby 800 and 3800 are respectively the specific heat capacities of the solid components and water, in J/°C/kg. m s and m e are the masses of solid components and water, in kg. The solid mass equals the sum of sand, gravel and cement masses.

In this expression, the uncertainties on masses are known, while uncertainties on the specific heat capacities are not listed in the Standard and moreover their values are not explained.

A number of researchers have offered some indications on this topic [4, 11, 14, 15, 20]. When the concrete is fresh, specimen heat capacity equals the sum of heat capacities for each component. When specimen concrete has hardened, its heat capacity tends to increase with temperature and decrease with water content (as maturity advances). These two phenomena nearly offset one another, with on the whole a slight increase (0.5% to 2.5% depending on type of binder). It is readily recognized that the heat capacity calculation for fresh concrete accurately reflects the average value for concrete during setting and hardening. For semi-adiabatic tests, temperature increases are not as significant as those recorded during adiabatic tests; this approach is capable of overestimating the average heat capacity value of concrete (with, as a corollary, the temperature rise of concrete being underestimated). For this reason, AFNOR (AFN88) recommends introducing a lower specific heat capacity value for water (i.e. 3,800 instead of 4,180 J/°C/kg).

The specific heat capacity value for solid components has also been averaged, taking into account both aggregates and cement, given that the individual values are not straightforward to determine [14]: they vary between 710 and 890 J/°C/kg depending on the type of solid (Table 4). An average value of 800 J/°C/kg is considered acceptable.

Components

Specific heat capacity (in J/°C/kg) 20°C [4, 20].


Sandrine Rousseaux

Personal carbon card: prospects and challenges for ICT

Keywords:

The carbon card is a generic notion which includes several categories of programmes aiming at limiting the environmental impact of the individuals. Carbon compensation cards and green loyalty cards are offered by the private sector and municipalities. Issuing a carbon credit card to all the eligible inhabitants of a country (personal quota system) is also considered.

The purpose of this paper is to emphasize the major role of ICT for implementing individual carbon card schemes. The different functions to be fulfilled by ICT are underlined from a detailed analysis of the way each scheme is electronically managed. This analysis enables to identify potential prospects for the ICT sector, as well as the technological challenges to take up.

INTRODUCTION

Rapid and substantial reduction of greenhouse gas emissions is needed to contain the global warming of the Earth surface below the threshold of dangerous and irreversible disturbance of the climate system, estimated at approx. 2°C by the end of the century relative to pre-industrial values [START_REF]Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF]. The world emissions, expressed in carbon dioxide equivalent (CO 2 eq.), must be reduced by at least 50% in 2050 relative to their 1990 level, and those of the industrialised countries by at least 80% at that time. The necessity of this reduction is again emphasised by a recent state of the art of climate science, which indicates that climate change is accelerating more rapidly than the most pessimistic scenarios of the last report of the Intergovernmental Panel on Climate Change adopted in 2007 [START_REF]Climate Change Compendium[END_REF]. The quite radical proposal of personal carbon rationing is in keeping with this context of ecological emergency. Indeed the individuals contribute significantly to climate change, directly by the consumption of non-renewable energy for housing and transport, indirectly by purchasing products whose life cycle is energy-intensive.

Reducing the individuals' emissions requires far-reaching change in habits and behaviours. The necessity for this change was mentioned by the Intergovernmental Panel on Climate Change for the first time in its last report [START_REF]Summary for Policymakers, Climate Change 2007: Mitigation. Contribution of Working Group III to the Fourth Assessment Report[END_REF]. Two economic instruments, taxation and tradable quotas, may be used to encourage to adopt low-carbon behaviours by sending a price signal. The main difference between both these instruments lies in the choice of the uncertainty, which may be put either on the environmental result (taxation) or on the price of pollution (quotas). A tradable quotas system may be preferable when the certainty of the environmental result is needed [START_REF]Trade and climate change: WTO[END_REF], which is the case of climate change considering the 2°C threshold that should not be exceeded.

A cap and trade system is gradually being established by the most emitting countries in order to limit the emissions from the energy and industry sectors, and sometimes from other economic activities. Engaging the individuals with such a system proves increasingly interesting at international level, on the politics and research and development plane. Personal carbon trading was initially developed by English researchers at the end of the 1990s [START_REF] Fleming | Energy and the Common purpose[END_REF], before to be taken into consideration by political institutions in the United-Kingdom [1] [4]. Other countries and a few international organisations are also showing an interest in this instrument. It has never been established as yet, apart from certain pilot experiments in the United-Kingdom.

The personal carbon trading system consists in capping the global quantity of CO 2 eq. emissions which may be generated by the households, then in apportioning among them the authorised emissions by individual allowances. These allowances -or carbon budgets -would be fractioned into carbon credits 1 . The credits would be stored on a carbon account, and surrendered by bank card or direct debit according to the carbon content of the purchases covered by the system. These emission rights would be traded on an open and organised market between those who have exceeded their carbon budget, and those who have not used it up. The carbon budget -or ration -is thus flexible. It would be gradually reduced so as to cut the emissions to the level required. The participation of the eligible individuals in the system would be compulsory. Their direct CO 2 emissions, associated with home energy use and (surface and air) transport, would be covered first. Extending the coverage of the system to indirect emissions, associated with the consumption of products such as foodstuffs, depends in particular on the capabilities for measuring the emissions released during their life cycle.

Pending the possible introduction of personal carbon trading systems by public authorities, numerous carbon card programmes have been launched since the beginning of the 2000s by the private sector and local authorities. Banks are offering their clients to compensate for the carbon footprint of the purchases made with their bank card, by financing emission reduction projects. Loyalty cards aiming at supporting sustainable consumption and behaviours are also offered by private bodies or municipalities, in partnership with retailers. The carbon card is therefore a generic notion which includes several schemes that may obey different logics.

The three kinds of carbon card programmes (quotas, compensation, and loyalty) are not as environmentally efficient. The voluntary compensation and loyalty programmes contribute to making individuals sensitive to their contribution to climate change. But their purpose is not an absolute cut in emissions, contrary to a mandatory quota system.

The different carbon card schemes share however a common feature: they are ICT-based. Their implementation involves electronic monitoring of the consumption of the goods and services covered. This monitoring consists in counting the quantity of emissions generated, or the amount of loyalty points earned, by the individuals. The data are sent to an electronic database that holds the accounts of the participants in the programmes. The accounts are debited or credited on the basis of these data, in conventional or in parallel currency (carbon credits, points).

The major role of ICT can be highlighted by explaining the design and the details of implementation of the different carbon card programmes 2 . The analysis of the precise functions devolved to ICT is based on a state-of-the-art of the programmes established or contemplated in Europe and in the United-States [START_REF] Rousseaux | Etat des lieux international des programmes de 'carte carbone' pour les particuliers (Europe et Etats-Unis) : Report to ADEME[END_REF]. It helps to identify the technological needs, as well as the technical solutions tested or considered. The methods of electronic monitoring of the individual consumption are set out in the first instance. Then automated data processing in a database is examined. The conclusion stresses the main challenges that the ICTs should take up.

ELECTRONIC MONITORING OF THE INDIVIDUAL CONSUMPTION

The aim of individual consumption monitoring is to count the amount of CO 2 eq. emissions generated, or of loyalty points earned, by the participants in the carbon card programmes. Data are electronically collected and transmitted to a database.

Such a monitoring has financial consequences: payment for the emissions generated (purchasing offset credits, surrendering carbon credits), financial remuneration for the emissions avoided (selling or saving unused carbon credits, redeeming the loyalty points earned for gifts or discounts).

Measuring the individual ecological footprint

Measuring the individual ecological footprint consists in recording the environmental value assigned to each good and service covered by the carbon card programmes. This value may correspond to the carbon content of the products, or to a lump number of points. The products are consequently given a double value, monetary and environmental.

The environmental price is shown by means of labelling, with the aim of informing the consumers. It must appear on a medium enabling its electronic recording, for instance on a barcode. Implementing a quota system implies adding a carbon functionality to price calculation software packages for some products, such as the fuel purchased in petrol stations or the domestic energy use (heat and electricity), in order to determine their total carbon value.

Issues raised by measuring

Whereas counting the loyalty points earned is rather widespread and does not present any difficulty, measuring the individuals' emissions raises some technological issues.

The existing carbon compensation card programmes provide for a flat rate for emissions. This method for calculating lies in the fact that banks, which either manage or are in partnership with the programmes, are solely aware of the amount of the expenses made by electronic payment instruments (card, direct debit). Average emission values are therefore given to the expenses classed according to a list of economic activities.

Establishing a personal carbon trading system requires to accurately monitoring the individuals' emissions, by recording the carbon content of each of their purchases. The carbon content corresponds to the emissions released during the life cycle of the products. Though it can be quite easily determined as regards energy used for housing and transport (all fuels would be given carbon ratings), the process is more complex with regard to manufactured products. Methods for calculating their carbon footprint are being developed in several countries. They could forebode a future supranational standardisation.

The environmental effectiveness of a personal carbon trading system depends on a reliable and accurate measuring of all of the emissions covered.

Measuring tools

The various carbon card programmes involve using a card. It is most often a bank card. A loyalty card and an electronic identity card are also used.

The carbon compensation card only records the price of the purchases, on the basis of which emissions are calculated.

Conversely the loyalty card only records the environmental value of the purchases. An ecological accounting functionality is however added to the electronic identity card used by the participants in the 'De e-portemonnee' loyalty programme, implemented in Belgium (Flanders region) 3 . The loyalty card used in the trial of real-time calculation of personal carbon emissions, previously mentioned, is also fitted with carbon functionality.

A multifunction card is more practical for the individuals, since it saves them keeping an additional card in their wallet. The same chip thus acts as a medium for processing various data, among which are carbon data.

Smart meters might also be used for monitoring the emissions associated with the home energy use. To this end, a carbon accounting functionality should be added to those tools in full expansion 4 .

The purchases may be recorded by mobile phones as well [START_REF]Climate Bonus Final Report[END_REF].

Like the other technical solutions, mobile phones must present guarantees in terms of accurate and reliable monitoring of emissions associated with the individuals' way of life so as to be used in a personal carbon trading system. The personal carbon accounts are indeed debited on the basis of the emission data collected.

Report

Once collected by the electronic monitoring tools, environmental data must be reported to a database holding the individuals' accounts. Issuing loyalty points, purchasing offset credits on the carbon market and surrendering carbon credits are based on this report.

The environmental integrity of a cap and trade system depends on the reliability of the emission data reported. The existing systems, applying to States and companies, provide for the emission data to be verified before their report. The emissions are monitored by the participants themselves, and then verified by third parties accredited by the regulation authorities. The auditors certify the compliance with the monitoring rules and the correctness of the data. As the personal carbon trading system would be electronically administered, the certification of the emissions monitoring tools might prove to be necessary. This certification would attest to the accuracy and the reliability of the tools used for implementing a policy instrument.

Data could be reported by a tool that simultaneously recorded the environmental and monetary values of the purchases, such as a bank card with carbon functionality. In such a case, this tool should be able to communicate at the same time with two accounts, one provisioned with conventional currency, the other with carbon currency. These accounts may be held by the same database, or by two different ones (see below).

When the emission data are reported on the points of sale at the time of payment for the purchases, a specific carbon card or electronic payment instruments fitted with carbon functionality must be used to communicate with the individuals' carbon accounts and debit them with an amount of carbon credits equivalent to the emissions associated with the purchases. If the need arises, the retailers would be responsible for the electronic report of their customers' emissions. They would have to comply with the obligation of surrendering carbon credits to cover these emissions. The retailers should purchase the necessary number of credits on the market and integrate their price in that of the products sold. This procedure is called 'pay as you go'. It might also apply in case of omission, loss or nondelivery of the carbon card (in particular for temporary residents who would not be subjected to a similar scheme in the State to which they belong).

AUTOMATED PROCESSING OF PERSONAL DATA IN A DATABASE

The data collected by the consumption measuring tools are sent to a database that holds the individuals' accounts. Recording these data entails movements into or out of accounts.

Movements on accounts

Reporting the data relating to individual consumption leads to credit or debit one or two accounts, provisioned with conventional or parallel currency.

The procedure is conventional within the framework of the compensation programmes, inasmuch as the price of the purchases is solely recorded. The individuals' account is debited for that price. An account provisioned with conventional currency is also debited following the purchase of offset credits on the carbon market. This account is held by banks, since they are bearing the cost of the carbon compensation service they are offering to their clients.

The loyalty programmes do not raise any particular difficulty either. Recording the number of loyalty points associated with the purchases involves crediting an account provisioned with this parallel currency.

The procedure would be more sophisticated with personal quotas programmes, as they would state a legally binding obligation of surrendering carbon credits to cover the emissions associated with the purchases of goods and services that fall within their field. The carbon credits would have to be surrendered to the authority in charge of their allocation (in the event of a system applying to individuals only), or to the emission producers (in the case of an economy-wide system). Surrendering carbon credits would take the form of a transfer between carbon accounts.

Communication between the emissions measuring tools and the individuals' carbon accounts would consist in specifying the number of carbon credits to be surrendered. The accounts would be debited when the purchases are paid by card (on the points of sale, online or over the phone) or by direct debit. If the emissions data were reported on the point of sale, while the purchases are not paid for by electronic instruments, the seller's carbon account would be debited (see above section 2.2). Paying for emissions with carbon credits implies changes to be made to existing billing and payment infrastructures, so as to handle a new unit and to calculate the number of carbon credits to be surrendered [START_REF] Lane | An analysis of the technical feasibility and potential cost of a personal carbon trading scheme[END_REF].

The carbon accounts holders would not be allowed to go overdrawn [14] [8]. The electronic payment instruments would therefore check whether the accounts are sufficiently provisioned. Transactions ought otherwise to be made on other accounts. Surrendering the missing credits by the retailer is a first possibility. The credits would be purchased by the retailer on the carbon market, and then recorded on his carbon account before to be transferred onto the account intended for surrendering. The second possibility consists for the individuals who exceeded the carbon budget they were allocated in putting additional carbon credits into their account. They would have to trade carbon credits on the market. Trading would entail transfers between carbon accounts. Exchanges of emission rights could be made by mutual agreement between two people, for instance at the bank, online or over the phone. They could also be performed by market intermediaries, such as banks, who would be commissioned to acquire credits for their clients. Cash flows would mirror the transfers between carbon accounts when credits are exchanged against payment.

The storage of credits on the chip of the carbon card is sometimes mentioned. It has however to be taken into account that the credits would also be deduced from the carbon account when purchases are made by direct debit [START_REF]Trade and climate change: WTO[END_REF].

Database

Implementing loyalty and quota programmes requires an electronic database to be built and maintained, wherein accounts provisioned with parallel currency (points, carbon credits) are open.

The database fulfils a bank function. This information system must be secured in order to prevent any risk of intrusion and fraudulent operation on the accounts. The risk is limited with loyalty programmes, since redeeming points illegally earned would involve a rather small financial loss. It is higher concerning the carbon credits allocated to the participants in a quota system, because their price should gradually rise as their scarcity is strengthened.

A major component of a personal carbon trading system is a central database, capable of holding a carbon account for all eligible participants and managing these accounts when the credits are issued, surrendered and exchanged. Setting up a central database should help the regulation authorities to oversee the implementation of the system, contrary to dispersed databases [START_REF]Technical requirements for Personal Carbon Trading: RSA Working Paper[END_REF]. Building and maintaining such a wide and complex database would however entail significant costs. Two information systems could therefore be required for implementing a personal carbon trading system: one for enrolment, identity verification and allocation, the other for accounting and transactions [START_REF] Lane | An analysis of the technical feasibility and potential cost of a personal carbon trading scheme[END_REF]. The enrolment process must ensure that eligible individuals cannot fraudulently open more than one carbon account, which requires appropriate identity verification [START_REF] Starkey | Domestic Tradable Quotas: a policy instrument for reducing greenhouse gas emissions from energy use: Technical report 39[END_REF], whilst the allocation process must ensure that the credits are issued to the entitled individuals and stored on the right accounts. Besides, all participants in the system will need a carbon bank account for the receipt, payment and recording of carbon credits, and for their exchange.

The central database handling enrolment in the system and allocation of credits could be the responsibility of a government organisation [START_REF] Lane | An analysis of the technical feasibility and potential cost of a personal carbon trading scheme[END_REF]. It might be a new database, or an existing one that would also be used for that purpose.

Two possibilities are considered with regard to the database that would hold the carbon accounts: building and maintaining a government carbon bank or opening the accounts in banks. The most cost-effective way is to use the existing banking infrastructures [10] [8]. Banks could be commissioned by the governments to manage the carbon accounts in their own databases. Opening the individuals' carbon accounts in the national registries set up in accordance with the emission trading mechanism rules, such as those provided for under the Kyoto Protocol, is another possibility. State carbon assets should be issued to the individuals for this possibility to be conceivable. This would require changing the rules relating to the existing domestic tradable quotas systems, so that the consumers instead of the producers would be responsible for the emissions released during the lifecycle of the products purchased.

The carbon accounts could be managed by authorized bodies such as banks, irrespective of their location. Managing these accounts consists in providing current services: credit and debit transactions, inter-account transfer, monthly statements, access to the account balance and customer help services [START_REF] Lane | An analysis of the technical feasibility and potential cost of a personal carbon trading scheme[END_REF].

CONCLUSION

The individual carbon card opens up a vast area of applications for ICT. Many functions are devolved to them, as regards both software and hardware, for implementing new schemes aiming at controlling the CO 2 eq. personal emissions.

Among the three kinds of carbon card programmes analysed, the quota system should be considered more particularly. Indeed this policy instrument has the potential to gradually and substantially reduce the individuals' emissions.

Implementing a personal carbon quota system raises several issues for the ICTs. Reliable and accurate emission monitoring is the first challenge to be taken up. It must be guaranteed by the electronic tools used for measuring the individuals' consumption. The second challenge is about handling two different currencies. It implies an adaptation of the existing billing and payment infrastructures. It also promotes the development of multifunction cards, such as bank cards with carbon functionality. The ability for these cards to not only debit but also credit several accounts at the same time might be useful in terms of practicability for individuals. In this case they could record loyalty points, should loyalty programmes supplement the implementation of quota systems to make access to low-carbon products easier for the individuals. The third challenge deals with computer security. Carbon credit cards need to be highly resilient to fraud, in order to prevent lost or stolen cards from being used, or their counterfeiting. Otherwise there is a risk for carbon accounts to be unduly debited, or for illicit emissions to be generated (without charging on carbon accounts). All carbon credits transactions could be protected through chip and pin, even if the purchases are made online or over the phone [START_REF] Starkey | Domestic Tradable Quotas: a policy instrument for reducing greenhouse gas emissions from energy use: Technical report 39[END_REF]. The carbon databases should be secured too. Intrusion into these information systems might involve the fraudulent debit with carbon credits or the creation of illicit carbon money. The last challenge relates to counting the individuals' emissions, providing they are within a jurisdiction where a personal quota system is introduced. Universally recognized communication technologies would make easier their acceptance and their connection to point-of-sale terminals [START_REF] Kampers Van Hilten | assignment of Belfast City Council and the Points Foundation[END_REF], and allow their holders' accounts to be updated even if they are open in another jurisdiction. The issue of standards relating to interoperability and security of the technologies used for implementing domestic quota systems applying to the individuals might be raised as these instruments are introduced in different countries.

  The emissions covered by the trials launched in the United-Kingdom are measured by the participants themselves. Expense statements, and carbon accounting computer software programs that give emission factors for the various budget items, are used for that purpose.

Manual calculation is however tedious. Moreover it does not guarantee any effective monitoring of emissions associated with many daily life acts, contrary to electronic measuring. A first trial of electronic measuring was run during the second half of 2008 within the CarbonLimited programme, led in the United-Kingdom by the Royal Society for the encouragement of Arts, Manufactures & Commerce (RSA). The aim of this experiment carried out in partnership with Atos Origin was to capture by card the emission data associated with purchases of fuel at BP filling stations

[START_REF]Practical solutions for personal carbon trading: RSA, Atos Origin[END_REF]

.

The carbon credits allocated to the individuals could be representative of the emission of one CO2 eq. kilogram, while the credits currently in circulation on the carbon market are representative of the right to emit one ton of CO2 equivalent.

Some schemes, which may be qualified as "greenwashing', are not taken into account. They consist in offsetting emissions or delivering points on an inclusive basis, according to the amount of the expenses paid for by credit card.

http://www.bondbeterleefmilieu.be/milieuopdekaart (accessed 2 Dec. 2009).

The Low Carbon Transition Plan of the United-Kingdom for instance, published in July 2009, foresees their installation in all homes by 2020.


Emmanuel Witrant
email: emmanuel.witrant@gipsa-lab.inpg.fr

Patricia Martinerie

A Variational Approach for Optimal Diffusivity Identification in Firns

Trace gas measurements in interstitial air from polar firn allow to reconstruct their atmospheric concentration time trends over the last 50 to 100 years. This provides a unique way to reconstruct the recent anthropogenic impact on atmospheric composition. Converting depth-concentration profiles in firn into atmospheric concentration histories requires models of trace gas transport in firn. A fundamental parameter of these models is firn diffusivity. Here we propose a new method to evaluate the diffusivity of polar firns using automatic control analysis techniques. More precisely, the diffusivity identification is formulated as an optimization problem in terms of partial differential equations (PDE). The proposed theorems generally apply to transport phenomena in non-homogeneous media (space-dependent coefficients), a variational approach is proposed and the optimization problem is solved with an adjoint-based gradient-descent algorithm.

Introduction

Polar ice cores provide a unique archive of atmospheric composition at time scales covering glacialinterglacial cycles (the last 800 000 years) to the rise of anthropogenic pollution (since about 1850 A.D.), as described in [START_REF] Solomon | The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF][START_REF] Wmo | Scientific Assessment of Ozone Depletion[END_REF] and references therein. Firn air pumping campaigns multiplied over the last decade for to major reasons: (1) large amounts of air can be retrieved, providing the opportunity to measure numerous trace gases [START_REF] Martinerie | Longlived halocarbon trends and budgets from atmospheric chemistry modelling constrained with measurements in polar firn[END_REF], (2) direct observation of air trapping in ice provide information about the age difference between the air bubbles and the surounding ice, which should be well constrained in order to finely interpret ice core signals [START_REF] Schwander | The age of the air in the firn and ice at summit, greenland[END_REF].

In polar regions where no melting occurs, snow transforms into ice through the effect of its own weight. The transition between an open-porosity ma-terial (snow/firn) into an airtight material (ice) then occurs at about 50-100 meters depth, as depicted in Figure 1. Within snow and firn (i.e. compacted snow), atmospheric trace gases are mostly transported by diffusion through air channels. Air trapping in bubbles also generates an advection flux which is most important at sites undergoing high snow accumulation rates. Finally gravitational fractionation occurs, transporting heavy molecules downward more efficiently. This slight fractionation needs to be taken into account at least for isotopic ratios, measured at the per-mil precision level. Such phenomena lead to the modeling of firns as nonhomogeneous media where the trace gases move according to transport equations with space-dependent coefficients. The non-homogeneous transport of trace gases in firns can be modeled using the dynamic equations pro-posed in [START_REF] Rommelaere | Reconstructing recent atmospheric trace gas concentrations from polar firn and bubbly ice data by inverse methods[END_REF][START_REF] Rommelaere | Trois problèmes inverses en glaciologie[END_REF], where a first approach for model validation and diffusion profiles identification based on firn air measurements is described. This approach was successfully used to describe the firn diffusion property for specific geographical locations, where the diffusivity profiles are sufficiently smooth, but cannot be generalized due to numerical instabilities in the identification process. Our aim is then to revise the modeling and identification algorithms using modern automatic control methodologies.

The transport model assumes a perfect gas, hydrostatic (pressure computation), isothermal firn, stationary flow (constant accumulation rate, temperature etc.), no thermal diffusion, no interaction (chemical, particles collision, gas transfer) between the gas and the pore walls. The following PDE model is then established (e.g. for a single gas) to describe the trace gas quantity in open pores q(z,t) [START_REF] Rommelaere | Reconstructing recent atmospheric trace gas concentrations from polar firn and bubbly ice data by inverse methods[END_REF]:

       ∂ t q = D(z)∂ zz q + α(z)∂ z q + β (z)q q(0,t) = f (0)c s (t) . = q 0 (t) k 1 ∂ z q(z f ,t) + k 2 q(z f ,t) = 0 q(z, 0) = q I (z) (1)
with the compact notation ∂ x y . = ∂ y/∂ x and:

α(z) . = ∂ z D(z) -D(z)χ(z) -w(x) β (z) . = -D(z)∂ z χ(z) -∂ z D(z)χ(z) -λ + v(x)ε(z) f (z) ∂ z f (z) ε(z) -∂ z w(z)
where χ(z) . = Mg/RT + f z / f , q(z,t) is the trace gas quantity in open porosity, D(z) is the diffusivity, λ is the radioactive decrease, M is the molar mass, ε(z) the total porosity, f (z) the open porosity, v(z) the sinking speed of the layers, w(z) the airflow speed in open porosity. The depth in the firn is determined by 0 ≤ z ≤ z f and c s (t) is the atmospheric concentration of the trace gas. The mentioned physical variables are summarized in Table 1.

The model [START_REF] Solomon | The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF] writes in the compact form:

∂ t q = ∂ z [D ∂ z q -(Dχ + w)q] + -λ + v ∂ z f f - ∂ z ε ε q = ∂ z [D(z)∂ z q -V(z)q] + S(z)q
which generally describes non-homogeneous transport in a variable-porosity medium, characterized by its diffusivity D(z), an advection term V (z) and a source term S(z). It is related to (1) as α(z

) = D z (z) -V (z) and β (z) = -V z (z) + S(z).
In order to formulate the modeling and identification problems properly, we can distinguish between the air quantity in open pores R perfect gas constant (J

• mol -1 • K -1 ) T temperature (K) v(z)
firn sinking speed due to snow accumulation w(z) advection speed compensating the air trapping in bubbles (m/yr) z f total depth of the firn (m) z co depth level at which there is no remaining open porosity (m)

Table 1. Main physical variables and notations

chemical and physical parameters. The chemical variables (M, c s , λ ) depend on the gas considered while the physical parameters ( f , w, v, ε, D) depend on the geographical locations and are measured or calculated for each firn. The atmospheric time-trend history c s (t) is reasonably well known for specific gas such as CO 2 , CH 4 , SF 6 and some halocarbons such as CH 3 CCl 3 . Such gases can then be used to determine the diffusive property of a firn D provided that some distributed (depthdependent) measurement q(z,t f ) is available. The simulations illustrating the different theoretical results correspond to the transport of CH 4 in Dome C (Dome Concordia or Dome Charlie, located in Antarctica at an altitude of 3,233 m above sea level), and presented in [START_REF] Martinerie | Longlived halocarbon trends and budgets from atmospheric chemistry modelling constrained with measurements in polar firn[END_REF].

Optimal diffusivity identification

The diffusivity identification objective can be formulated as an optimization problem, where the diffusivity profile has to minimize the squared difference between the final concentration provided by the model and the measured one. The proposed approach is based on variational analysis applied to PDE optimization as described in [START_REF] Lions | Optimal control of systems governed by partial differential equations[END_REF]. More precisely, we extend some results presented in [START_REF] Jacquet | Macroscopic freeway modelling and control[END_REF] for conservation laws to transport in a variable-porosity medium, following the proposed formalism. A detailed bibliography, convective conservation laws, shock waves and state nonlinearities are considered in [START_REF] Jacquet | Macroscopic freeway modelling and control[END_REF].

Problem formulation

Our aim is to find the diffusivity profile D * that minimizes the difference between the model output and the measurements. This corresponds to a final-cost optimization problem with dynamics (transport equations) and inequality constraints that writes as:

min D J (D) = J meas + J reg under the constraints C (q, D) = 0 I (D) < 0
where J meas denotes the cost associated with the measurements, J reg is a regularization function, C (q, D) sets the dynamical constraint associated with the nonhomogeneous transport model and I (D) corresponds to the inequality constraints determined from the physics on the optimized variable D. More specifically, the measurements cost is chosen as a weighted leastsquares criterion and the equality constraints are included into the cost function J (D) by introducing the Lagrange or Kühn and Türker parameters λ i . The in- equality constraints can be introduced thanks to Valentine's method or the barrier function proposed in [START_REF] Boyd | Convex Optimization[END_REF] (chosen here). In the problem considered, the constraint J (D) is set on the diffusivity gradient, which necessitates a specific care that will be detailed later on. Considering the measurements of N gas, the optimization problem is:

min D J (D) . = N ∑ i=1 [J meas (q i , q meas ) +J trans (C (q i , D))] + J ineq (D) + J reg (D) (2)
with:

               J meas = 1 2 z f 0 r i (q meas -q i | t=t f ) 2 δ z dz J trans = t f 0 z f 0 λ i C (q i , D) dzdt J reg = 1 2 z f 0 s(z)D 2 dz
where z f and t f are the final depth and time, respectively, r i (z) ≥ 0 and s(z) > 0 are user-defined tuning functions, δ z (z) denotes the measurements location and J ineq describes the inequality constraint. Our aim is to find the set of parameters {D, q i , λ i } that minimizes the cost function J . Note that the dependence of the cost in q i could be removed from the fact that q i depends on D through C .

Linearized dynamics

In order to solve the previous optimization problem, the first step is to linearize the distributed dynam-ics and define an appropriate transport model. Consider the general transport equation

   ∂ t y + f 1 (z,t)y + f 2 (z,t)∂ z y = ∂ z [g(y, ∂ z y, u)] y(0,t) = y 0 (t), k 1 ∂ z y(L,t) + k 2 y(L,t) = 0 y(z, 0) = y I (z) (3)
Its linearized dynamics along the reference trajectory ( ȳ, ū, ȳI ) with perturbations ( ỹ, ũ, ỹI ) is given by

       ∂ t ỹ + f 1 (z,t) ỹ + f 2 (z,t)∂ z ỹ = ∂ z ∂ y ḡ ỹ + ∂ ∂ z y ḡ ∂ z ỹ + ∂ u ḡ ũ ỹ(0,t) = 0, k 1 ∂ z ỹ(L,t) + k 2 ỹ(L,t) = 0 ỹ(z, 0) = ỹI (z) (4)
where ḡ .

= g( ȳ, ∂ z ȳ, ū). Note that the particular choice of ȳ(z, 0) = y I (z) directly implies that ỹI (z) = 0, which may be a convenient condition in some applications. We can directly verify that ũ = 0 implies that ỹ = 0.

The previous theorem applies directly to the general porosity model:

     ∂ t q = ∂ z [D(z)(∂ z q -χ(z)q) -w(z)q] + S(z)q q(0,t) = q x=0 (t), k 1 ∂ z q(z f ,t) + k 2 q(z f ,t) = 0 q(x, 0) = q I (x) (5)
by considering the reference concentration q(x,t) satisfying [START_REF] Sowers | δ 15 N of N 2 in air trapped in polar ice: a tracer of gas transport in the firn and a possible constraint on ice agegas age differences[END_REF]. The linearized state q resulting from a variation in the diffusivity profile D then writes as:

Σ Lin :          ∂ t q = ∂ z [ D(∂ z q -χ q) -w q] +∂ z (∂ z q -χ q) D + S q q(0,t) = 0, k 1 ∂ z q(z f ,t) + k 2 q(z f ,t) = 0 q(x, 0) = qI (x) (6)
The linearized system is compared with the coupled dynamics on the test case as follows. A 5 % variation on y I induces a variation of the order of 4.5 % on the normalized concentration profile but a negligible error (of the order 10 -13 %, which corresponds to numerical noise) with the linearized model. This is related to the fact that the linear model obeys the same conservation law as the initial one. A 5 % variation on D varies the normalized concentration profile by 1 % (terminal value) while the normalized error associated with the linear model is less than 0.04 %, as depicted in Figure 2. It can be noticed that the concentration error tends to accumulate at the bottom of the firn over time.

Adjoint state and gradient computation

Adjoint state

The proposed gradient computation implies to compute the adjoint state when the diffusivity variation is zero, e.g. D = 0. Such adjoint is given by the following theorem.

Theorem 3.1 Consider the linearized transport equation without input:

   ∂ t ỹ = ∂ z [ f 1 (z,t)∂ z ỹ + f 2 (z,t) ỹ] + f 3 (z,t) ỹ ỹ(0,t) = 0, k 1 ∂ z ỹ(L,t) + k 2 ỹ(L,t) = 0 ỹ(z, 0) = 0
The corresponding adjoint state is given as:

       ∂ t λ = -f 3 λ + ( f 2 -∂ z f 1 ) ∂ z λ -f 1 ∂ zz λ λ (0,t) = 0, f 1 ∂ z λ + [ f 1 k 2 /k 1 -f 2 ]λ | z=L = 0 λ (z, T ) = 0 (7)

Proof 1 The adjoint state equation is obtained as (integration by parts):

T 0 L 0 λ (∂ t ỹ -∂ z [ f 1 ∂ z ỹ + f 2 ỹ] -f 3 ỹ) dzdt = T 0 L 0 ỹ [-∂ t λ -f 3 λ + ( f 2 -∂ z f 1 ) ∂ z λ -f 1 ∂ zz λ ] dzdt + L 0 λ ỹ| T 0 dz - T 0 ([ f 1 ∂ z ỹ + f 2 ỹ] λ -f 1 ỹ ∂ z λ ) | L 0 dt
All the components in the previous integrals are directly set to zero from [START_REF] Rommelaere | Reconstructing recent atmospheric trace gas concentrations from polar firn and bubbly ice data by inverse methods[END_REF] and by noticing that at z = L:

[ f 1 ∂ z ỹ + f 2 ỹ]λ -f 1 ỹ ∂ z λ = [(-f 1 k 2 /k 1 + f 2 )λ -f 1 ∂ z λ ] ỹ
The adjoint state corresponding to the linearized system (6) without diffusivity variation is obtained thanks to the previous theorem by setting:

f 1 = D, f 2 = -( Dχ + w), f 3 = S,
and the adjoint state is: The time and space evolution of the adjoint state for a unitary terminal condition (λ (z,t f ) = 1) is presented in Figure 3. We can note the impact of the depth on the step dispersion in the adjoint model. This is directly related to the depth dependency of the gas diffusion speed.

       ∂ t λ = -Sλ + (-w -Dχ -∂ z D) ∂ z λ -D∂ zz λ λ (0,t) = 0, D ∂ z λ + [ Dk 2 /k 1 + ( Dχ + w)]λ | z=z f = 0 λ (z,t f ) = 0 (8)

Inequality constraint

The fact that the inequality constraint applies on the diffusivity gradient as ∂ z D < 0 motivates the change of variables ∂ z y IC = u and D = y IC with y IC (z f ) = 0 and where u is the new variable used for the optimization process. Introducing the associated Lagrange parameters λ IC (z,t) and a barrier function R(u), the cost is set with:

J ineq = L 0 λ IC (∂ z y IC -u) + R(u) dz = L 0 -y IC ∂ z λ IC -uλ IC + R(u) dz + λ IC y IC | z f 0
The fact that there is no dynamics involved is directly related to the algebraic nature of the constraint. The barrier function can be set as a log involving upper and lower bounds [START_REF] Boyd | Convex Optimization[END_REF], or with the augmented Lagrangian [START_REF] Cohen | Decomposition coordination methods in large scale optimization problems[END_REF].

Adjoint-based gradient

The last step to set the optimal diffusivity profile identification algorithm is to propose an adjoint-based gradient computation, based on the variational analysis. This is done thanks to the following theorem.

+ L 0 λ IC C IC (y IC , u) + R(u)dz (9)
where C i (y i , y IC , u) can be linearized as:

       ∂ t ỹi = ∂ z [ f 1 (z,t)∂ z ỹi + f 2 (z,t) ỹi ] + f 3 (z,t) ỹi + ∂ z [ f 4 (z,t) ỹIC ] ỹi (0,t) = 0, k 1 ∂ z ỹi (L,t) + k 2 ỹi (L,t) = 0 ỹi (z, 0) = ỹIi (z) and C IC (y IC , u) = ∂ y IC -u with y IC (z f ) = 0.
The gradients of J with respect to the decision variables u and y Ii along the reference trajectory ( ū, y( ū)) are given by:

∇ u J = R ( ū) - T 0 λ IC dt (10)
∇ y Ii J = -λ i (z, 0) (11)
where λ i are the solutions of:

       ∂ t λ i = -f 3 λ i + ( f 2 -∂ z f 1 ) ∂ z λ i -f 1 ∂ zz λ i λ i (0,t) = 0, k 1 f 1 ∂ z λ i + [k 2 f 1 -k 1 f 2 ]λ i | z=L = 0 λ i (z, T ) = -P ( ȳi (T )) (12)
and λ IC is obtained from:

∂ z λ IC = ∑ N i=1 f 4 ∂ z λ i λ IC (0,t) = 0, (13)
Proof 2 As we are interested in the gradient computation, achieved with a sufficiently small step, the analysis is established on the first order variation of the cost function around the reference trajectory ( ȳ, ū) (defining y as the set {y i , y IC }), denoted as J. Considering the perturbation ( ỹ, ũ), we have:

J ( ỹ, ũ) . = N ∑ i=1 L 0 P ( ȳi (z, T )) ỹi (z, T ) dz + T 0 L 0 λ [∂ y C ( ȳ, ū) ỹ + ∂ u C ( ȳ, ū) ũ] dzdt + L 0 R ( ū) ũ dz where λ . = {λ i , λ IC }, C .
= {C i , C IC } and ∂ y involves the partial derivatives with respect to the derivatives of y (omitted to keep simple notations). This expression implies the linearized version of C (•), obtained thanks to Section 2.2 as in [START_REF] Lions | Optimal control of systems governed by partial differential equations[END_REF]. Applying integration by parts as in the proof of Theorem 3.1, the double-integral term involving y i writes as:

T 0 L 0 λ i [∂ t ỹi -∂ z [ f 1 ∂ z ỹi + f 2 ỹi ] -f 3 ỹi -∂ z [ f 4 ỹIC ]] dzdt = L 0 λ i ỹi | T 0 dz - T 0 L 0 λ i ∂ z [ f 4 ỹIC ] dzdt
where the last equality is obtained with [START_REF] Cohen | Decomposition coordination methods in large scale optimization problems[END_REF]. Introducing J ineq as defined in Section 3.2, we have:

T 0 L 0 λ [∂ y C ( ȳ, ū) ỹ + ∂ u C ( ȳ, ū) ũ] dzdt - L 0 λ i ỹi | T 0 dz = - T 0 L 0 ũλ IC dzdt
where the successive simplifications are a direct consequence of ( 12)-( 13) and the definition of y.

Introducing the previous equality in the expression of J and considering the solutions defined as in [START_REF] Cohen | Decomposition coordination methods in large scale optimization problems[END_REF], it follows that:

J = L 0 - N ∑ i=1 λ i (z, 0) ỹIi (z) + R ( ū) - T 0 λ IC dt ũ dz
where the fact that ũ does not depend on time was used in the last equality. The gradients are finally obtained as in ( 10)- [START_REF] Boyd | Convex Optimization[END_REF].

The optimization problem (2) fits within the previous formalism by noticing that:

P = 1 2 r i (q meas -q i | t=t f ) 2 δ z R = - 1 M log(-∂ z D) + 1 2 s(z)∂ z D 2
and considering the linearized dynamics [START_REF] Lourantou | Contraindre l'augmentation en dioxyde de carbone (CO 2 ) lors des déglaciations basés sur son rapport isotopique stable du carbone (δ 13 CO 2 )[END_REF]. The gradients are then obtained thanks to Theorem 3.2 as (setting

I (D) = ∂ z D): ∇ ∂ z D J = 1 M∂ z D + s∂ z D - T 0 λ IC dt (14) ∇ q Ii J = -λ i (z, 0) (15)
and [START_REF] Cohen | Decomposition coordination methods in large scale optimization problems[END_REF] with: f 1 = D, f 2 = -( Dχ + w), f 3 = S, f 4 = ∂ z q -χ q, P = -r i (q measq i | t=t f )δ z

Considering the CH 4 experimental data, the gradient with respect to the input ∇ ∂ z D J is obtained from the previous results and presented in Figure 4. It can be noticed that the algorithm converges smoothly within about 500 steps, which is very satisfying for this problem. Nevertheless, it is highly sensitive to the design weights and to the constraint, which will be the topic of future work.

Conclusions

In this work, a new identification problem related to a challenging environmental application with high social impact has been presented: the diffusivity profile determination in firns based on trace gases measurements. This problem was formulated as an optimization problem involving distributed dynamics, a terminal cost constraint and an inequality constraint on the diffusivity gradient. Several new analytical results that apply generally to transport phenomena in variable-porosity media were derived, which allowed for a variational approach in the PDE framework. The optimal diffusivity profile was obtained thanks to an adjoint-based gradient-descent algorithm. The different results were illustrated and discussed on the basis of experimental measurements of CH 4 in Dome C.

Figure 1 .

 1 Figure 1. Polar firn structure as a function of depth and density. Scheme adapted from [5, 6].

Figure 2 .

 2 Figure 2. Linearization error for D = 0.05 D.

Figure 3 .

 3 Figure 3. Adjoint state with λ (z,t f ) = 1.

Theorem 3 . 2

 32 Consider the optimization problem (terminal cost, uncoupled states, input derivative coni (y i , y IC , u)dzdt

Figure 4 .

 4 Figure 4. Algorithm convergence: evolution of the gradient ∇ ∂ z D J .

Acknowledgements

The authors wish to thank Pr. Didier GEORGES for his fruitful remarks concerning the consideration of inequality constraints of the derivative type. This work was funded by CNRS program INSIS-PEPS-Automatique.


Florian Pasquet

Vincent Cottin

Nadia Sivova

Julie Scanff

Christiane Broussolle

MD Pascal Sève
email: pascal.seve@chu-lyon.fr.

Julie Le Scanff

Coexisting relapsing polychondritis and sarcoidosis: an unusual association

Keywords: Relapsing polychondritis, sarcoidosis, autoimmunity

Introduction

Relapsing polychondritis (RP) is an episodic and progressive inflammatory disease of the cartilaginous structures, including elastic cartilage of the ear and nose, hyaline cartilage of peripheral joints, fibrocartilage at axial sites, and cartilage of the tracheobronchial tree (1; 2). Inflammation of other proteoglycan-rich structures, such as the eyes, heart, blood vessels, and kidneys may also be present. Diagnostic criteria for RP are listed in Table 1 (1; 2). Although the etiology of RP is unclear, an autoimmune mechanism is considered to be most likely. RP is associated with autoimmune disorders in about one third of the patients, most frequently with connective tissue diseases and vasculitides (3; 4).

Sarcoidosis is a systemic granulomatous disease that most frequently involves the lungs, lymphs nodes, skin and eyes (5). An increase in polyclonal gamma-globulin serum level is frequent, with rarely autoantibodies We report here an unusual observation of coexisting RP and sarcoidosis. To our knowledge, this association in an immunocompetent patient has not been previously reported in the literature.

Case report

A 42-year-old woman was admitted to our hospital with complaints of swelling and pain of the left ear and arthralgia of the hands for 3 weeks. Her past medical history was significant for pulmonary sarcoidosis of 2 years. The diagnosis of sarcoidosis was based on the clinical picture of chronic cough, mediastinal lymphadenopathies, diffuse micronodular interstitial lung disease at chest imaging, and demonstration of typical granulomas at bronchial biopsy. Infectious causes were ruled out. As pulmonary function was normal and chest CT scan was stable, oral corticosteroids were not considered.

Upon examination the significant findings were: left pinna swelling, tenderness and warmth, sparing the non cartilaginous lobe (Fig. 1) and bilateral symmetrical polyarthritis involving wrists and metacarpophalangeal joints. The patient was apyretic. Heart and lung examinations were normal, and there was no evidence of peripheral lymphadenopathy, hepatosplenomegaly or eye involvement. The anamnesis revealed a first episode of chondritis of the right ear 3 years previously, and an episode of nasal chondritis 1 year before, both of which recovered spontaneously. Laboratory results revealed elevated serum level of C-reactive protein of 25 mg/L, and fibrinogen of 4.6g/L, with inflammatory profile of blood proteins electrophoresis. Blood count, liver enzymes, kidney function tests results and serum angiotensin-converting enzyme level were normal. Repeated blood cultures were sterile. Serologic investigations for HIV, hepatitis C and B viruses, Parvovirus B19, syphilis and Borrelia burgdorferi were negative. Rheumatoid factor, anti anticyclic citrullinated cyclic peptides, antinuclear, anti-double-stranded DNA and type-II anticollagen antibodies were not detected. Serum complement levels were normal. The HLA class II DR4 was not present. Bone radiographs of hands showed no erosions. Spirometry and chest CT results remained unchanged. Fiberoptic bronchoscopy showed absence of laryngeal or tracheal abnormality.

Based on the clinical presentation, the diagnosis of RP was established and treatment with nonsteroid antiinflammatory agents and colchicine was initiated, with clinical improvement within one week. At 6 months follow-up, she was completely asymptomatic, and physical examination was normal. Fibrinogen and C-reactive protein levels were normal. The CT scan of the chest was unchanged.

Discussion:

According to Mc Adam et al. [START_REF] Mcadam | Relapsing polychondritis: prospective study of 23 patients and a review of the literature[END_REF], 25% to 35% of patients with RP have a concurrent disease (Table 2).

Coexisting RP and sarcoidosis is very unusual since it has been described only once. Zandman-Goddard et al. reported a case of coexisting RP and sarcoidosis in a patient with AIDS whose disease improved with antiretroviral drug therapy [START_REF] Zandman-Goddard | Combined autoimmune disease in a patient with AIDS[END_REF]. We report here the first observation in an immunocompetent patient. Sarcoidosis can lead to sclerouveitis, middle ear involvement, polyarthritis and saddle nose deformity that may mimic RP but does not cause chondritis [START_REF] Letko | Relapsing polychondritis: a clinical review[END_REF]. Moreover, hilar lymphadenopathies, interstitial lung disease, and tissue granulomas are very rare in RP. In our patient, the clinical findings described above were conclusive with RP.

Although the etiopathogenesis of RP remains unknown, there are many arguments to support the role of the immune system. Fisrt, as seen above, RP is frequently associated with autoimmune diseases, and responds to corticosteroids. Second, RP is linked to the HLA class II DR4 isotype, , which is frequently associated with other autoimmune diseases [START_REF] Lang | Susceptibility to relapsing polychondritis is associated with HLA-DR4[END_REF]. Third, autoantibodies to type II, IX and XI collagen and to matrilin-I, a cartilage protein, can be found in the sera of patients with RP (3; 9; 10). Moreover immunofluorescence studies of involved cartilages have shown deposits of Immunoglobulins G, M, A and C3 as well as the presence of CD4 + T lymphocytes and plasma cells [START_REF] Dolan | Relapsing polychondritis. Analytical literature review and studies on pathogenesis[END_REF]. Mice experimentally immunised with type 2 collagen or matrilin-1 develop auricular chondritis and symptoms resembling RP in humans [START_REF] Mccune | Type II collagen-induced auricular chondritis[END_REF].

Sarcoidosis is a systemic granulomatous disease that preferentially involves the lungs and lymph nodes. The etiopathogenesis also implicates the immune system. Sarcoidosis is the consequence of an exaggerated immune reaction mediated by CD4 + T lymphocytes in response to unknown antigens [START_REF] Iannuzzi | Sarcoidosis[END_REF]. T-cells present in alveoli and pulmonary interstitium secrete chemotactic factors to monocytes and macrophages, in particular monocyte chemotactic protein-1, which is also secreted by CD4 + T lymphocytes in patients with RP (5,[START_REF] Stabler | Serum cytokine profiles in relapsing polychondritis suggest monocyte/macrophage activation[END_REF].

Sarcoidosis and RP seem to share some etiopathogenic mechanisms: CD4 + T lymphocytes are implicated by an immune reaction through recruitement of the monocytes that contribute to inflammatory lesions. This could explain why these 2 diseases can coexist in the same patient.

In conclusion, sarcoidosis should be added to the list of diseases potentially coexisting with RP. Given the multitude of associated diseases, it has been proposed that RP be considered as a syndrome which can be primary or secondary. vertigo

Michet et al. criteria (one of two conditions recommended to be present)

1. Proven inflammation in two of three of the auricular, nasal or laryngotracheal cartilages 2. Proven inflammation in one of three of the auricular, nasal or laryngotracheal cartilages plus 2 other signs including ocular inflammation, vestibular dysfunction, seronegative arthritis and hearing loss

Figure 1 .

 1 Figure 1. Auricular chondritis of the patient

Table 2 Diseases associated with relapsing polychondritis

 2

	iseases

Acknowledgments: We thank Bonnie Possest for her administrative assistance