Mathilde Fouché

Cécile Robilliard

Stéphane Faure

Carlo Rizzo

Julien Mauchain

Marc Nardone

Rémy Battesti

Luc Martin

Anne-Marie Sautivet

Jean-Luc Paillard

A.-M Sautivet

J.-L Paillard

F Amiranoff

Search for photon oscillations into massive particles

Keywords: numbers: 12, 20, Fv, 14, 80, -j

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I. INTRODUCTION

Ever since the Standard Model was built, various theories have been proposed to go beyond it. Many of these involve, if not imply for the sake of consistency, some light, neutral, spinless particles very weakly coupled to standard model particles, hence difficult to detect.

One famous particle beyond the Standard Model is the axion. Proposed more than 30 years ago to solve the strong CP problem [1,2], this neutral, spinless, pseudoscalar particle has not been detected yet, in spite of constant experimental efforts [3,4,5,6]. Whereas the most sensitive experiments aim at detecting axions of solar or cosmic origin, laboratory experiments including the axion source do not depend on models of the incoming axion flux. Because the axion is not coupled to a single photon but to a two-photon vertex, axion-photon conversion requires an external electric or -preferentiallymagnetic field to provide for a virtual second photon [7].

At present, purely terrestrial experiments are built according to two main schemes. The first one, proposed in 1979 by Iacopini and Zavattini [8], aims at measuring the ellipticity induced on a linearly polarized laser beam by the presence of a transverse magnetic field, but is also sensitive to the ellipticity and, slightly modified, to the dichroism induced by the coupling of low mass, neutral, spinless bosons with laser beam photons and the mag- * Electronic address: mathilde.fouche@irsamc.ups-tlse.fr netic field [9]. The second popular experimental scheme, named "light shining through the wall" [10], consists of first converting incoming photons into axions in a transverse magnetic field, then blocking the remaining photonic beam with an opaque wall. Behind this wall with which the axions do not interact, a second magnetic field region allows the axions to convert back into photons with the same frequency as the incoming ones. Counting these regenerated photons, one can calculate the axionphoton coupling or put some limits on it. This set-up was first realized by the BFRT collaboration in 1993 [3].

Due to their impressive precision, optical experiments relying on couplings between photons and these hiddensector particles seem most promising. Thanks to such couplings, the initial photons oscillate into the massive particle to be detected. The strength of optical experiments lies in the huge accessible dynamical range: from more than 10 20 incoming photons, one can be sensitive to 1 regenerated photon! In fact, the "light shining through the wall" experiment also yields some valuable information on another hiddensector hypothetical particle [11]. After the observation of a deviation from blackbody curve in the cosmic background radiation [12], some theoretical works suggested photon oscillations into a low mass hidden sector particle as a possible explanation [13]. The supporting model for such a phenomenon is a modified version of electrodynamics proposed in 1982 [14], based on the existence of two U(1) gauge bosons. One of the two can be taken as the usual massless photon, while the second one corresponds to an additional massive particle usually called paraphoton. Both gauge bosons are coupled, giving rise to photon-paraphoton oscillations. Several years later, more precise observations did not confirm any anomaly in the cosmic background radiation spectrum [15] and the interest for paraphoton decreased, although its existence was not excluded. More recently, it was found out that similar additional U(1) gauges generally appear in string embeddings of the standard model [16], reviving the interest for experimental limits on the paraphoton parameters [17,18,19]. Some limits on the mass and the coupling constant of the paraphoton have already been obtained by a photoregeneration experiment [3]. Astrophysical limits on paraphoton parameters also exist. They have been derived from the agreement of the cosmic microwave background with the blackbody radiation [20], and more recently by the absence of distortions in the optical spectrum of distant Type Ia supernovae [21].

Our effort was motivated by the observation published by the PVLAS collaboration, and subsequently retracted [22], which they claimed could be explained by the existence of axions in the mass range 1-2 meV. We have therefore designed an apparatus optimised for that mass region to rapidly prove or disprove the interpretation in terms of axion-like particles of the PVLAS signal. Our preliminary results, excluding at a 3σ confidence level the existence of axions with parameters consistent with the PVLAS observation, have been published in November 2007 [23]. This paper is devoted to the final results of our "light shining through the wall" experiment, sensitive to axion-like particles and to paraphotons. We first derive the detection probabilities for both particles. We then detail our apparatus which strength lies in pulsed laser and magnetic field, thus reducing the demand on the detector noise. Finally, we present our latest experimental results and compare them with the limits obtained by other searches.

II. PHOTOREGENERATION PROBABILITY

A. Axion-Like Particle

The photon to axion-like particle conversion and reconversion transition probability (in natural units h = c = 1, with 1 T ≡ 195 eV 2 and 1 m ≡ 5 × 10 6 eV -1 ) after propagating over a distance z in the inhomogeneous magnetic field B writes [7,24]:

p a (z) = z 0 dz ′ ∆ M (z ′ ) × exp(i∆ a z ′ ) 2 , (1)
where

∆ M = B 2M and ∆ a = - m 2 a 2ω
, ω being the photon energy, m a the axion-like particle mass and M its inverse coupling constant with two photons. Note that this equation is valid for a light polarization parallel to the magnetic field since the axion has to be a pseudoscalar [1]. Finally, as we have two identical magnets, the photon regeneration probability due to axion-like particles is

P a = p 2 a (L) , (2)
with L the magnet length.

In order to have a number of regenerated photons as large as possible, the number of incident photons, the detection efficiency and the integral of the transverse magnetic field over the magnet length L have to be maximized. We define B 0 as the maximum field and L eq as the equivalent length of a magnet producing a uniform magnetic field B 0 such that

+L/2 -L/2 Bdz = B 0 L eq . (3)
On the other hand, p a (z) oscillates for too long magnets. Actually, for a homogeneous magnetic field B 0 , Eq. ( 1) gives:

p a = B 0 L 2M 2 sin 2 ( ∆osc 2 L) ( ∆osc 2 L) 2 , (4)
where ∆ osc = -∆ a . In our case, our search was focused on the 1 meV < m a < 2 meV, so that a length larger than 1 m would have been useless. Finally, very recently a detailed theoretical study of the photon to axion-like particle conversion probability pointed out that an enhancement of this probability is predicted at m a = ω [25]. In this particular condition, the probability of getting a photon after the wall is :

P a = 3β 4 16qm 4 a log 2qm 4 a β 4 , (5)
with β = B 0 /M and q = ∆/ω the quality factor of the laser source, ∆ being the laser bandwidth.

B. Paraphoton

In the modified version of electrodynamics developed in 1982 [14], the paraphoton weakly couples with the photon through kinetic mixing. Contrary to axion-like particles, photon-paraphoton oscillations are therefore possible without any external field and are independent on photon polarization.

Recently, the experimental signatures of paraphoton have been discussed in details in Ref. [19]. The conversion probability of a photon into a paraphoton of mass µ and vice-versa after a distance L is given by:

p γ = 4χ 2 sin 2 µ 2 L 4ω ( 6
)
where χ is the photon-paraphoton coupling constant, which arbitrary value is to be determined experimentally. This equation is valid for a relativistic paraphoton satisfying µ ≪ ω.

Comparing Eqs. ( 6) and ( 4), one notes that from a mathematical point of view the two are equivalent, µ corresponding to m a , and χ to B0ω Mm 2 a . This analogy originates from the fact that both formulas describe the same physical phenomenon, i.e. quantum oscillations of a two level system. Using this mathematical equivalence between paraphoton parameters and axion-like particle parameters, we were able to derive for the enhancement of the paraphoton conversion probability at µ = ω a formula equivalent to Eq. ( 5):

P γ = 3χ 4 16q log 2q χ 4 . (7)
In the case of a typical photoregeneration experiment, the incoming photons freely propagate for a distance L 1 and might oscillate into paraphotons before being stopped by a wall, after which the paraphotons propagate for a distance L 2 and have a chance to oscillate back into photons that are detected with efficiency η det . The photon regeneration probability due to paraphotons can therefore be written as:

P γ = p γ (L 1 )p γ (L 2 ) = 16χ 4 sin 2 µ 2 L 1 4ω sin 2 µ 2 L 2 4ω (8)
In our experiment, L 1 is the distance between the focusing lens at the entrance of the vacuum system, which focuses photons but not paraphotons, and the wall, which blocks photons only. Similarly, L 2 represents the distance separating the blind flange just before the regenerating magnet and the lens coupling the renegerated photons into the optical fibre (see Fig. 1). Note that Eq. ( 8) is a priori valid in the absence of magnetic field. If a magnetic field is applied, the formula remains valid provided that it can be considered as static during the experiment and its transverse spatial extent is larger than 1/µ [17], which is the case in our experiment for paraphoton masses larger than 2 × 10 -5 eV.

III. EXPERIMENTAL SETUP

As shown in Fig. 1, the experimental setup consists of two main parts separated by the wall. An intense laser beam travels through a first magnetic region (generation magnet) where photons might be converted into axionlike particles. The wall blocks every incident photon while axion-like particles would cross it without interacting and may be converted back into photons in a second magnetic region (regeneration magnet). The regenerated photons are finally detected by a single photon detector. The three key elements leading to a high detection rate are the laser, the generation and regeneration magnets placed on each side of the wall and the single photon detector. Each element is described in the following sections.

A. Laser

In order to have the maximum number of incident photons at a wavelength that can be efficiently detected, the experiment has been set up at Laboratoire pour l'Utilisation des Lasers Intenses (LULI) in Palaiseau, on the Nano 2000 chain [26]. It can deliver more than 1.5 kJ over a few nanoseconds with ω = 1.17 eV. This corresponds to N i = 8 × 10 21 photons per pulse.

The nanosecond pulse is generated by a YLF seeded oscillator with a ∆ = 1.7 meV bandwidth. It delivers 4 mJ with a duration adjustable between 500 ps and 5 ns. Temporal shaping is realized with five Pockels cells. Then this pulse seeds single-pass Nd:Phosphate glass rods and disk amplifiers. During our 4 weeks of campaign, the total duration was decreased from 5 ns the first week to 4 ns and finally 3 ns while keeping the total energy constant. A typical time profile is shown in the inset of Fig. 6 with a full width at half maximum of 2.5 ns and a total duration of 4 ns.

The repetition rate of high energy pulses is imposed by the relaxation time of the thermal load in the amplifiers which implies wave-front distortions. Dynamic wave-front correction is applied by use of an adaptiveoptics system [27]. To this end a deformable mirror is included in the middle of the amplification chain. It corrects the spatial phase of the beam to obtain at focus a spot of about once or twice the diffraction limit, as shown in Fig. 2. This system allows to increase the repetition rate while maintaining good focusability although the amplifiers are not at thermal equilibrium. During data acquisition, the repetition rate has typically varied between 1 pulse per hour and 1 pulse every other hour.

At the end of the amplification chain, the vertically linearly polarized incident beam has a 186 mm diameter and is almost perfectly collimated. It is then focused using a lens which focal length is 20.4 m. The wall is placed at L 1 = 20.2 m from the lens in order to have the focusing point a few centimeters behind this wall. The beam is well apodized to prevent the incoming light from generating a disturbing plasma on the sides of the vacuum tubes.

Before the wall where the laser beam propagates, a vacuum better than 10 -3 mbar is necessary in order to avoid air ionization. Two turbo pumps along the vacuum line easily give 10 -3 mbar near the lens and better than 10 -4 mbar close to the wall. The wall is made of a 15 mm width aluminum plate to stop every incident photon. It is tilted by 45 • with respect to the laser beam axis in order to increase the area of the laser impact and to avoid retroreflected photons. In the second magnetic field region, a vacuum better than 10 -3 mbar is also maintained. Fig. 3 shows a histogram of laser energy per pulse for the 82 laser pulses performed during our campaign. The laser energy per pulse ranges from 700 J to 2.1 kJ, with a mean value of 1.3 kJ.

B. Magnetic field

Concerning the magnets, we use a pulsed technology. The pulsed magnetic field is produced by a transportable generator developed at LNCMP [28], which consists of a capacitor bank releasing its energy in the coils in a few milliseconds. Besides, a special coil geometry has been developed in order to reach the highest and longest transverse magnetic field. Coil properties are explained in Ref. [29]. Briefly, the basic idea is to get the wires generating the magnetic field as close as possible to the light path. As shown in Fig. 4, the coil consists of two interlaced race-track shaped windings that are tilted one with respect to the other. This makes room for the necessary optical access at both ends in order to let the laser in while providing a maximum B 0 L eq . Because of the particular arrangement of wires, these magnets are called Xcoils.

The coil frame is made of G10 which is a non conducting material commonly used in high stress and cryogenic temperature conditions. External reinforcements with the same material have been added after wiring to contain the magnetic pressure that can be as high as 500 MPa. A 12 mm diameter aperture has been dug into the magnets for the light path.

As for usual pulsed magnets, the coils are immersed in a liquid nitrogen cryostat to limit the consequences of heating. The whole cryostat is double-walled for a vacuum thermal insulation. This vacuum is in common with the vacuum line and is better than 10 -4 mbar. A delay between two pulses is necessary for the magnet to cool down to the equilibrium temperature which is monitored via the Xcoils' resistance. Therefore, the repetition rate is set to 5 pulses per hour. Furthermore the coils' resistance is precisely measured after each pulse and when equilibrium is reached, in order to check the Xcoils' non embrittlement. Indeed variations of the resistance provide a measurement of the accumulation of defects in the conductor material that occur as a consequence of plastic deformation. These defects lead to hardening and embrittlement of the conductor material, which ultimately leads to failure.

The magnetic field is measured by a calibrated pick-up coil. This yields the spatial profile shown in Fig. 5. The maximum field B 0 is obtained at the center of the magnet. Xcoils have provided B 0 ≥ 13.5 T over an equivalent length L eq = 365 mm. However, during the whole campaign a lower magnetic field of B 0 = 12 (0.3) T was used to increase the coils' lifetime.

A typical time dependence of the pulsed magnetic field at the center of the magnet is represented in Fig. 6. The total duration is a few milliseconds. The magnetic field reaches its maximum value within less than 2 ms and remains constant (±0.3%) during τ B = 150 µs, a very long time compared to the laser pulse.

C. Detector

The last key element is the detector that has to meet several criteria. In order to have a sensitivity as good as possible, the regenerated photon detection has to be at the single photon level. The integration time is limited by the longest duration of the laser pulse which is 5 ns. Since we expected about 100 laser pulses during our four week campaign, which corresponds to a total integration time of 500 ns, we required a detector with a dark count rate [START_REF]A dark count[END_REF] far lower than 1 over this integration time, so that any increment of the counting would be unambiguously associated to the detection of one regenerated photon.

Our detector is a commercially available single photon receiver from Princeton Lightwave which has a high detection efficiency at 1.05 µm. It integrates a 80 × 80 µm 2 InGaAs Avalanche Photodiode (APD) with all the necessary bias, control and counting electronics. Light is coupled to the photodiode through a FC/PC connector and a multimode fiber. When the detector is triggered, the APD bias voltage is raised above its reverse breakdown voltage V br to operate in "Geiger mode". A short time later -adjustable between 1 ns and 5 ns -the bias is reduced below V br to avoid false events. For our experiment, the bias pulse width is 5 ns to correspond with the longest laser pulse.

Typical output signals available on the detector are plotted in Fig. 7. Let's first consider Fig. 7a with no incident photon. The upper signal corresponds to the amplified APD output. The application of such a short pulse to a reverse-biased APD produces a capacitive transient. The first two transients temporally shifted by 5 ns correspond to the bias pulse. This signal enables to precisely determine the moment when detection starts. The last transients are due to an electronic reflection of the bias pulse.

When a photon is detected (Fig. 7b), the signal resulting from a photon-induced avalanche superimposes upon transients. The transient component may be much larger than the photon-induced component, making it difficult to discern. The detector uses a patented transient cancelation scheme to overcome this problem [30]. A replica of the unwanted transient is created and subtracted from the initial signal. The photon-induced signal will thus appear against a flat, low-noise background, as it is observed in Fig. 7b between the initial bias pulse and the reflected one. It can then be easily detected using a discriminator. To this end, this signal is sent to a fast comparator with adjustable threshold that serves as a discriminator and outputs a logic pulse, as shown by lower traces on Fig. 7.

To optimize the dark count rate and the detection efficiency η det , three different parameters can be adjusted: the APD temperature, the discriminator threshold V d set to reject electronic noise and the APD bias voltage V APD . The dark count rate is first optimized by choosing the lowest achievable temperature which is around 221 K. This rate is measured with no incident light, a trigger frequency of 5 kHz and an integration time of at least 1 s. Dark counts for a 5 ns detection gate as a function of V d is shown in Fig. 8a. It increases rapidly when V d is too low. On the other hand, η det remains constant for a large range of V d . We set V d to a value far from the region where dark count increases and where η det is still constant. This corresponds to less than 2.5 × 10 -2 dark count over 500 ns integration time.

The detection efficiency is precisely measured by illuminating the detector with a laser intensity lower than 0.1 photon per detection gate at 1.05 µm. The probability to have more than one photon per gate is thus negligible. Such a low intensity is obtained with the setup described in Fig. 9. A c.w. laser is transmitted through two supermirrors with a reflectivity higher than 99.98 % [45]. The angle of incidence is near normal in order to intercept the reflected beam and avoid spurious light without increasing transmission. This gives a measured transmission of 0.015 % for each mirror. Finally, to calculate the number of incident photons on the detector, we measure the laser power before the two supermirrors with a precise power sensor.

The detection efficiency as a function of the bias voltage is plotted in Fig. 8b. Our measurements show that η det slowly increases with V APD until a threshold where it increases dramatically for a value of V APD shortly below the dark count runaway value. The best compromise between detection efficiency and dark count rate is found at V APD = 78.4(0.05) V with η det = 0.48(0.025).

As said in the introduction, other similar experiments generally require long integration times which implies an experimental limitation due to the detection noise. Using pulsed laser, magnetic field and detection is an origi- nal and efficient way to overcome this problem. Photons are concentrated in very intense short laser pulses during which the detection background is negligible. This also means that if a photon is detected in our experiment in correlation with the magnetic field, it will be an unambiguous signature of axion generation inside our apparatus.

IV. EXPERIMENTAL PROTOCOL AND TESTS

A. Alignment

After the second magnet, the regenerated photons are injected into the detector through a coupling lens and a graded index multimode fiber with a 62.5 µm core diameter, a 0.27 numerical aperture and an attenuation lower than 1 dB/km. These parameters ensure that we can inject light into the fiber with a high coupling ratio, even when one takes into account the pulse by pulse instability of the propagation axis that can be up to 9 µrad.

Injection is adjusted thanks to the fiber coupler, and by removing the wall and the blind flanges (see Fig. 1). As the high energy laser beam, the alignment beam comes from the pilot oscillator without chopping nor amplifying it. This procedure ensures that the pulsed kJ beam is perfectly superimposed to the alignment beam. During data acquisition, the mean coupling efficiency through the fibre was found to be η c = 0.85.

The alignment of the high energy beam is performed with a low energy 5 ns pulsed beam, allowing for a 10 Hz repetition rate. During alignment, several black crosses are distributed along the laser path to mark the optical axis. Mirrors mounted on stepper motors allow to align the beam very precisely on this axis. This procedure is carried out a few minutes before each high energy pulse.

The only remaining source of misalignment lies in thermal effects during the high energy pulse, which could slightly deviate the laser beam, hence generating supplementary losses in fibre coupling. This misalignment is mostly reproducible. This means that it can be corrected by a proper offset on the initial laser pointing. The far field of the high energy beam is imaged for each pulse at the output of the amplification chain (see Fig. 10). Since the focal length of the imaging system is similar to that of our focalisation lens, the position of the far field image on the alignment mark is a fair diagnosis of the alignment on the fiber coupler. The best offset was determined by trial and error method after a few high energy pulses.

B. Optical and electro-magnetic noise

In order to have the best sensitivity, a perfect optical shielding is necessary. As shown in Fig. 1, an aluminum blind flange closes the entrance to the regeneration magnet. A black soft PVC bellow placed between the exit of the magnet and the fibre coupler prevents stray light while mechanically decoupling the magnet which vibrates during its pulse and the fibre coupler which should stay perfectly still. Finally, another aluminum blind flange closes the exit of the generation area in order to stop any incident photon scattered inside the vacuum line.

A count on the single-photon receiver is most probably due to an incident photon on the photodiode but it may also originate from electro-magnetic noise during laser or magnetic pulses. To avoid such noise, the detector is placed in a Faraday shielding bay. In addition, a 30 m long fibre is used so that the detector can be placed far away from the magnets.

To test our protective device, laser and magnetic pulses were separately applied while triggering the detector. No fake signal was detected, validating the optical and electro-magnetic shielding.

C. Synchronization

Our experiment is based on pulsed elements which require a perfect synchronization : the laser pulse must cross the magnets when the magnetic field is maximum and fall on the photodiode during the detection gate.

The magnetic pulse is triggered with a TTL signal from the laser chain. The delay between this signal and the laser trigger is adjusted once and for all by monitoring on the same oscilloscope the magnetic field and the laser trigger. Then, the magnetic trigger has a jitter lower than 10 µs, ensuring that the laser pulse travels through the magnets within the 150 µs interval during which the magnetic field is constant and maximum. Synchronization of the laser pulse and the detector needs to be far more accurate since both have a 5 ns duration. The detector gate is triggered with the same fast signal as the laser, using delay lines. We have measured the coincidence rate between the arrival of photons on the detector and the opening of the 5 ns detector gate as a function of an adjustable delay. We have chosen our working point in order to maximize the coincidence rate (see Fig. 11). To perform such a measurement we used the laser pilot beam which was maximally attenuated by shutting off 4 Pockels cells along the amplification chain and chopped with a pulsed duration of 5 ns, which corresponds to the longest duration of the kJ beam. The best experimental limits are achieved when no fake signal is detected during the experiment. In this case, to estimate the corresponding upper conversion probability of regenerated photons, we have to calculate the upper number of photons that could have been missed by the detector for a given confidence level (CL).

The probability P n that n incident photons have been missed by the detector is P n = (1 -η det ) n when dark count is negligible. Therefore, the probability that n photons at most were missed by the detector writes

n k=0 P k ∞ k=0 P k = 1 -(1 -η det ) n+1
and has to be compared with the required confidence level CL. This yields the upper number of possibly missed photons n missed as the smallest integer n satisfying

1 -(1 -η det ) n+1 ≥ CL,
which writes

n missed = log(1 -CL) log(1 -η det ) -1. (9)
For example, with our value of η det , a confidence level of 99.7 % corresponds to less than 8 missed photons. The upper photon regeneration probability is then

P a or γ = n missed N eff , (10)
where N eff is the number of effective incident photons over the total number of laser shots, taking into account the losses described hereafter. Our experimental sensitivity limit for the coupling constant versus mass is finally calculated by numerically solving Eqs. ( 1) and (2) for axion-like particles, and Eqs. ( 6) and ( 8) for paraphotons.

B. Photon losses

The number of photons per laser pulse N i is measured at the end of the amplification chain with a calibrated calorimeter. Then the number of effective incident photons on the detector N eff should take into account every losses. The first source of losses is due to the coupling efficiency through the fibre. This is precisely calibrated once a day. Injection is checked before each pulse, just after the alignement of the high energy beam. The mean coupling efficiency is η c = 0.85.

As said before, the main source of misalignment lies in thermal effects during the high energy laser pulse, which mean value was corrected. Furthermore, using the c.w. alignment beam we calibrated the injection losses in the fibre as a function of the misalignment visible on the far field imaging. Thanks to this procedure, we were able to estimate the actual alignment losses for each pulse: they amounted to 30 % for a non-corrected pulse and varied between 0 and 10 % for corrected pulses, because of pulseto-pulse instabilities.

Possible jitter between the beginning of the detection and the arrival of the laser pulse on the detector is also taken into account. For each pulse, a single oscilloscope acquires the laser trigger, the detector trigger as well as the detection gate. Those curves allow to precisely calculate the moment t 0 when detection actually starts compared to the laser pulse arrival. Furthermore, the temporal profile of each laser pulse is also monitored. By integrating this signal from t 0 and during the 5 ns of detection, the fraction η f of photons inside the detection gate is calculated. This fraction has fluctuated between 0.6 and 1 at the beginning of our data acquisition with the 5 ns pulse, mainly due a 1 ns jitter that was then reduced to about 200 ps. Then, with the 4 ns and 3 ns laser pulses, jitter is less critical and η f = 1 is obtained almost all the time.

Finally, for axion-like particles the numerical solving of Eq.( 1) is performed with a fixed magnetic field B 0 . Variations of this magnetic field along data acquisition are taken into account by multiplying each number of incident photons by the factor (B 0,i /B 0 ) 4 , where B 0,i is the maximum field for the i th pulse.

Integration of every losses yields a total number of effective photons

N eff, a = i η c,i η p,i η f,i B 0,i B 0 4 N i , (11)
the sum being taken over the total number of laser and magnetic pulses. Concerning paraphotons, given that the magnetic fields has no effect on the oscillations, the formula writes FIG.

N eff, γ = i η c,i η p,i η f,i N i . (12)
12: 3σ limits for the axion-like particle -two photon inverse coupling constant M , as a function of the axion-like particle mass ma, obtained from our null result. The area below our curve is excluded.

VI. RESULTS

Data acquisition was spread over 4 different weeks. As shown in Fig. 3, 82 high energy pulses have reached the wall with a total energy of about 110 kJ. This corresponds to 5.9 × 10 23 photons. During the whole data acquisition, no signal has been detected.

A. Axion-Like Particles

The magnetic field was applied during 56 of those laser pulses, with a mean value of 12 T. The laser pulses without magnetic field aimed at testing for possible fake counts.

Our experimental sensitivity limits for axion-like particle at 99.7 % confidence level are plotted on Fig. 12. They correspond to a detection probability of regenerated photons P a = 3.3 × 10 -23 and give M > 9.1 × 10 5 GeV at low masses. The dark gray area below our curve is excluded. This improves the limits we have published in [23], which already excluded the PVLAS results [22].

We also compared our limits to other laboratory experiments in Fig. 13. They are comparable to other purely laboratory experiments [3,31,32], especially in the meV region of mass. On the other hand, they are still far from experiments which limits (stripes) approach models predictions [4,5,33,34].

Using Eq. ( 5), our experimental results correspond to M > 8 GeV at m a = 1.17 eV. Despite this enhancement, our limits are still very far from the inverse coupling constant of model predictions which is around 10 9 GeV for a 1 eV mass.

B. Paraphotons

In the case of paraphotons, we take into account the laser bandwidth ∆ by averaging P γ (ω) over ∆: FIG. 13: Limits on the axion-like particle -two photon inverse coupling constant M as a function of the axion-like particle mass ma obtained by experimental searches. Our exclusion region is first compared to other purely laboratory experiments such as the BFRT photon regeneration experiment [3], the GammeV experiment [31] and the PVLAS collaboration [32] with a 3σ confidence level. Those curves are finally compared to the 95 % confidence level exclusion region obtained on CAST [5] and the more than 90 % confidence level on microwave cavity experiments [4,33,34]. Model predictions are also shown as a dotted stripe between the predictions of the KSVZ model (lower line, E/N = 0) [35] and of the DFSZ model (upper line, E/N = 8/3) [36].

P γ = 1 ∆ ω-∆ 2 ω-∆ 2 P γ (ω)dω. (13)
The experimental sensitivity is then calculated by numerically solving

P γ = n missed N eff , (14)
where N eff is given by Eq. ( 12). In the regime of low mass µ ≪ ω/Lq, it is equivalent to P γ = P γ and the mixing parameter oscillates as a function of the paraphoton mass. For higher masses, oscillations are smoothed to a mean value. Note that the relevant mass ranges concerning axion-like particles are situated in the low mass regime, which explains why the averaging over the laser bandwidth was not useful. The deep gray area in Fig. 14 represents the parameters for paraphoton that our measurements exclude with a 95 % confidence level. It corresponds to a maximum FIG. 14: 95 % confidence level limits on photon-paraphoton mixing parameter as a function of the paraphoton mass obtained to our null result (deep gray area). Shaded regions are excluded. This is compared to excluded regions obtained on BFRT photon regeneration experiment [3] (light gray area), to searches for deviations of the Coulomb law [37] (points) and to comparisons of the Rydberg constant for different atomic transitions [38] (stripes).

photon regeneration probability P γ = 9.4 × 10 -24 . This sets a limit χ < 1.1 × 10 -6 for 1 meV< µ <10 meV (for higher masses, Eq.( 6) is not valid anymore). This improves by almost one order of magnitude the exclusion area obtained on BFRT photon regeneration experiment [3]. The enhanced probability at µ = ω given by Eq. ( 7) corresponds to χ < 1.9 × 10 -7 . For other ranges of mass, a more complicated calculation is required [25] which is beyond the scope of this article. Nevertheless, comparing to other laboratory experiments [37,38] (see [39] for review), we were able to constrain the paraphoton parameters in a region which had not been covered so far by purely terrestrial experiments.

VII. CONCLUSION AND OUTLOOKS

We have presented the final results of our photon regeneration experiment which exclude the PVLAS results.

Our null measurement leads to limits similar to other purely terrestrial axion searches, and improves the preceding limits by more than one order of magnitude concerning paraphotons [17].

As far as axion-like particles are concerned, improving the sensitivity of our apparatus in order to test the axion model predictions seems rather unrealistic, especially as the possible mass and two photon coupling constant ranges are still several orders of magnitude wide. In that respect, magnetic birefringence experiments like the one presently under development in Toulouse [40] seem more promising: aimed at measuring for the first time the long predicted QED magnetic birefringence of vacuum [41], it will improve by one to two orders of magnitude the precision of purely terrestrial axion searches.

More generally, let us argue that such precision optical experiments may prove useful for experimentally testing the numerous theories beyond standard model in the low energy window, a range in which the large particle accelerators are totally helpless. For example, our apparatus can be modified to become sensitive to chameleon fields [42].

Finally, very intense laser beams such as those planned at ELI [43] will become available in the forthcoming years. Such facilities should open new exciting opportunities for our field.

FIG. 1 :

 1 FIG.1: Sketch of the apparatus. The wall and the blind flanges are removable for fibre alignment.

FIG. 2 :

 2 FIG. 2: Focal spot without correction(a) and with wave-front correction (b). This correction allows to maintain a spot of one or two diffraction limits despite the amplifiers' not being in thermal equilibrium.

FIG. 3 :

 3 FIG. 3: Number of high energy pulses versus laser energy during the four weeks of data acquisition.

FIG. 4 :

 4 FIG.4: Scheme of XCoil. Magnetic fields B1 and B2 are created by each of the race-track shaped windings. This yields a high transverse magnetic field B while allowing the necessary optical access for the laser photons γ.

FIG. 5 :

 5 FIG. 5: Transverse magnetic field inside the magnet along the laser direction. At the center of the magnet we have a mean maximum magnetic field B0 = 12 T. Integrating B along the optical path yields 4.38 T.m.

FIG. 6 :

 6 FIG.6: Magnetic field B0 at the center of the magnet as a function of time. The maximum is reached within 1.75 ms and can be considered as constant (±0.3%) during τB = 150 µs. The 3 to 5 ns laser pulse is applied during this interval. Inset: temporal profile of a 4 ns laser pulse.

FIG. 7 :

 7 FIG. 7: Amplified APD output (upper curve) and logic signal (lower curve) of the detector as a function of time. The capacitive transients on the APD output signals are due to the gated polarisation of the photodiode in Geiger mode. (a) Signals with no incident photon. (b) Signals when a photon is detected.

FIG. 8 :FIG. 9 :

 89 FIG. 8: Detection efficiency (•) and dark count per 5 ns bias pulse (△) as a function of the discriminator threshold (a) (VAPD fixed to 78.4 V) or APD bias voltage (b) (V d fixed to 0.760 V). The APD temperature is fixed to the lowest achievable value 221.5 K. Dashed lines indicate the chosen working point.

FIG. 10 :

 10 FIG.10: Monitoring of the optical path followed by the high energy beam. Losses due to misalignment are estimated by comparing the centre of the beam to the centre of the black cross. The upper image corresponding to an uncorrected laser beam pointing exhibits 30% injection losses, while the lower one is perfectly corrected.

FIG. 11 :

 11 FIG. 11: Coincidence rate between the arrival of photons on the detector and its 5 ns detection gate as a function of an arbitrary delay time. The dashed line indicates our working point, chosen in order to maximize the coincidence rate.

ACKNOWLEDGMENTS

We thank the technical staff from LCAR, LNCMP and LULI, especially S. Batut, E. Baynard, J.-M. Boudenne, J.-L. Bruneau, D. Castex, J.-F. Devaud, P. Frings, M. Gianesin, P. Guéhennec, B. Hirardin, J.-P. Laurent, L. Polizzi, W. Volondat, and A. Zitouni. We also thank B. Girard and G. Rikken for strongly supporting this project. This work has been possible thanks to the ANR-Programme non thématique (Contract ANR -BLAN06-3-139634).


Alessandro Di Pretoro

Ludovic Montastruc

Flavio Manenti

Xavier Joulia

Exploiting Residue Curve Maps to Assess Thermodynamic Feasibility Boundaries under Uncertain Operating Conditions

The very first step of almost any separation process design procedure is the thermodynamic feasibility analysis. In the case of distillation, residue curve maps (RCMs) represent an essential tool to assess whether the separation is feasible or not. However, the analysis is generally carried out by referring to nominal operating conditions and product purities as specification. This means that, when process parameters are likely to undergo fluctuations, the prediction of the system response is not that obvious. An ABE/W (acetone-butanol-ethanol/water) mixture was then selected as a case study since it allows us to discuss several non-ideal thermodynamic behaviors and because of the renewed interest in biorefinery and sustainable processes during recent years. Residue curve mapping was then exploited to determine the thermodynamic feasibility range for multicomponent distillation processes as well as for distillation trains and process-intensified solutions taking into account both product purity and product recovery specifications. The final product of this study is a thorough procedure to determine the flexibility boundaries of feed and product compositions as well as an immediate and intuitive graphical representation from a binary standard distillation column to a complex multicomponent dividing wall column application.

■ HIGHLIGHTS

(1) An a priori thermodynamic flexibility assessment could avoid a backward investigation on the system unfeasibility root causes when perturbations occur. (2) Biofuel purification is a challenging operation due to the high non-ideality of water-alcohols interactions and the seasonality of the feedstock. (3) Residue curve mapping has proved to be an effective methodology worth exploited even when uncertainty should be taken into account.

INTRODUCTION

Feasibility analysis is a key point and the very first step of any process design procedure independent of the process nature. It is usually carried out according to given operating conditions in order to proceed to the following design phases. After the solution to the initial problem has been preliminarily designed, it is usually tested by varying design variables or operating parameters. 1 However, whether the process results to be unfeasible, it could be very difficult to understand, which is the constraining aspect that prevents the operation to achieve the desired specifications. The inability of a system to withstand operating conditions different from the ones according to which it has been designed indeed can be either due to tight constraints, such as physical laws, or loose constraints, such as available materials or design choices. 2 In the latter case, those limitations can be usually overcome by means of higher investments or at the cost of a lower profitability. On the contrary, in the former cases, it does not exist in any way to bypass the system failure, and a different design solution should be employed. Moreover, in all those cases, an a priori thermodynamic flexibility assessment would have avoided the backward investigation on the unfeasibility root cause.

These considerations are obviously dependent on the uncertain variable deviation range as well as the deviation likelihood. 1 The petrochemical industry for instance is based on widely studied and well-established processes, and crude oil supplies are ruled by multiyear contracts involving oil blends whose properties are known and are almost constant in time. However, the growing interest toward sustainable feedstocks during recent years implies considerable modifications in the process design approach. The biomass nature indeed is ruled by the cycle of seasons and cannot ensure constant chemicophysical properties during the year; 3,4 this causes the definition of "nominal operating conditions" to be of little significance and an a priori design under uncertainty to be extremely useful.

A relevant part of biobased processes indeed involves the use of a fermenter upstream the product separation section; the obtained fermentation broth is rich in water and alcohols in prevalent quantities that were not that present in petroleum feedstocks and considerably complicate the thermodynamic behavior due to the high non-ideality of their interactions. It would not be then surprising if, during an operation, an external operating condition perturbation or malfunctioning of an upstream unit led the separation section conditions outside the feasibility boundaries with relevant implications on the delicate profitability of a fermentation process.

Among the most common operations in chemical plants, distillation is by far the most used and the one processing huge feedstock capacities. It involves relatively high energy costs and its separation effectiveness is of critical importance for the cost-effectiveness of the entire process.

For all these reasons, this study deals with the thermodynamic flexibility assessment of different typologies of distillation processes involving compounds coming from an upstream microbial fermentation process as thoroughly described in the following chapters.

METHODOLOGY

Distillation feasibility is a widely spread and deeply studied research domain since the very beginning of the 20th century. Several methodologies exist in order to assess whether a given mixture split is possible or not on the basis of the components' volatility. Some shortcut methods have been outlined as well for the preliminary design phase accounting for a constant relative volatility between the species and the heavy-light key components' assumption. However, when thermodynamics gets complicated, equilibrium diagrams are the reference to be employed. A useful and well-established tool to assess the thermodynamic feasibility of a multicomponent mixture distillation process based on equilibrium diagrams is residue curve mapping. Despite the robust theory behind them and their effective graphical representation allowing for an immediate understanding of the equilibrium phenomena, residue curve maps (RCMs) are seldom used as the main tool to show the feasibility study results by process engineers.

Those curves were defined first at the beginning of the 20th century by Ostwald (1900) 5 and Schreinemakers (1901) 6 in Germany to describe ternary mixtures with the presence of azeotropic species. However, the country where they have seen the highest development after approximately 50 years is Russia; Gurikov (1958) 7 indeed classified all the possible ternary diagrams and Zharov and Serafimov 8-12 completed their work for mixtures with a higher number of components. All those studies remained untranslated from Russian for a long time.

The main research work concerning residue curves during recent years was performed by Petlyuk, 13,14 and it will be the mainly referenced author in this chapter. The residue curve represents the evolution of the mixture composition during the open evaporation process as shown in Figure 1.

From a numerical point of view, the mass balance on the ith component can be reconducted, in correspondence to each infinitesimal amount of the vaporized mixture dL → 0, to the differential equation:

ξ = - x x y d d i i i (1)
where dξ represents the infinitesimal dimensionless time span, i.e., the evaporation extent. A residue curve is then given by the locus of points in the composition domain satisfying the ordinary differential equation corresponding to a certain moment of time and to a portion of evaporated liquid.

Another physical interpretation that could result to be more appealing from a practical point of view concerns distillation columns. In particular, at total reflux, composition profiles of packed columns are described by residue curves, whereas distillation lines coincide with the composition profiles of staged columns. [START_REF] Laroche | Homogeneous azeotropic distillation: separability and flowsheet synthesis[END_REF] A close correspondence between RCMs and distillation trajectories at infinite reflux was also proved. Distillation trajectories represent the composition profile along the column equilibrium stages; at infinite reflux, as first discussed by Thormann, [START_REF] Thormann | Destillieren und Rektifizieren[END_REF] they are given by

= = ̅ ̅ • + x y K T P x y x ( , , , ) i k i k i k k k k i k ( 1) ( ) ( ) ( ) ( ) ( ) ( ) (2)
If we interpolate those broken lines with a continuous one, the so-called distillation c-lines can be obtained. Distillation lines and (equilibrium) tie-line curves are a unique function of the vapor-liquid equilibrium, independent of their analogies with the column composition profile. Thus, distillation lines are equivalent to residue curves as characteristics of the VLE in a thermodynamic sense although they do not coincide as discussed by Kiva et al. (2003). [START_REF] Kiva | Azeotropic phase equilibrium diagrams: a survey[END_REF] Residue curves indeed are oriented curves; they move from the unstable node that is the light component or azeotrope (the star in Figure 2a,b) to the stable node that is the heavy component or azeotrope (the square in Figure 2a,b) passing Whenever two or more distillation bundles are present, except for binary distillation, one or more particular residue curves called "separatrices of saddle stationary points" delimiting them exist as well. Different from the other RCs, the separatrices begin or come to an end not in the node points but in the saddle points (cf. Figure 2b). They separate two distillation regions, i.e., concentration bundles where all RCs have the same stable and unstable nodes in common. The main outcome related to the analysis of distillation regions is that separatrices cannot be crossed, which means that we cannot move between two points belonging to different distillation regions during a single distillation process.

In the light of the above, the important conclusion that can be drawn is that, in order to be part of the same distillation column, the characteristic points representing lateral feeds and product withdrawals should therefore have the same stable and unstable nodes, i.e., the RCs passing through them should belong to the same distillation region. This is a rule of general validity with the exception of a few particular cases when the VLE equilibrium behavior exhibits a nonideality resulting in inflections of the total reflux trajectories. Those cases are not specifically treated in this study, but a more accurate discussion can be found in Wahnschafft et al. (1992) [START_REF] Wahnschafft | The product composition regions of single-feed azeotropic distillation columns[END_REF] and Petlyuk (2004). 14 Thus, the first goal of this study is to take advantage of the RCM theory robustness in order to discuss distillation of complex multicomponent mixtures under uncertain operating conditions, i.e., the flexibility range of the operation. Flexibility will be evaluated according to the resilience index introduced by Saboo et al. (1985) [START_REF] Saboo | Design of resilient processing plantsVIII. A resilience index for heat exchanger networks[END_REF] defined as the largest total disturbance load independent of the direction of the disturbance, a system able to withstand without becoming unfeasible.

An additional purpose of this study is then to provide graphic tools to ease the understanding of the multicomponent mixtures' behavior both for a single distillation column and for more complex distillation configurations using the product purity and its recovery ratio as specifications.

Calculations have been performed by coupling Matlab for the numerical and graphical part with Simulis Thermody-namics software providing the component databank as well as the thermodynamic models and parameters.

CASE STUDY

It is widely agreed that sustainability can be identified as the worldwide central issue of the last (and the next) several years. According to the IEA Bioenergy Annual Report, 20 biofuels will represent around 30% of energy consumption in transport by 2060. Their role is particularly important in sectors that are difficult to decarbonize, such as aviation, shipping, and other long-haul transport. That is why several bioprocesses have seen renewed interest in recent years both from a research and an industrial point of view.

For this reason, the ABE/W (acetone-butanol-ethanol/ water) mixture was selected as the case study analyzed in this paper. It comes from an upstream microbial fermentation process characterized by product inhibition [START_REF] García | Challenges in biobutanol production: How to improve the efficiency?[END_REF] and the recovery of at least the most valuable products, n-butanol and acetone, is necessary for the profitability of the process. After a preliminary dewatering operation, usually liquid-liquid extraction or pervaporation, [START_REF] Ezeji | Bioproduction of butanol from biomass: from genes to bioreactors[END_REF][START_REF] Xue | Integrated butanol recovery for an advanced biofuel: current state and prospects[END_REF][START_REF] Errico | Multiobjective Optimal Acetone-Butanol-Ethanol Separation Systems Using Liquid-Liquid Extraction-Assisted Divided Wall Columns[END_REF] the remaining products can be separated by means of different distillation configurations. The fluctuation related to the feedstock nature requires a dedicated design procedure taking into account the inlet composition uncertainty and aiming to a flexible design choice.

An additional reason why this multicomponent mixture is particularly suitable for this flexibility assessment is that the complex and highly non-ideal thermodynamics allows us to analyze interesting singular points such as homogeneous and heterogeneous azeotropes as well as to include liquid-liquid equilibrium in the distillation feasibility assessment and optimization discussion thanks to the aqueous-organic phase demixing.

From the most significant binary case studies to more complex multicomponent mixtures, both simple distillation and energy integrated configurations will be discussed by referring to these four compounds.

The process specifications accounted for in the case of a single distillation column are (1) n-Butanol recovery ratio in the bottom: 0.9604.

(2) Bottom n-butanol mass fraction: 0.99.

Industrial & Engineering Chemistry Research

In the case of the water-ethanol mixture, the recovery specifications will be applied to water since, in order to obtain high-purity ethanol with a single distillation column, the feed composition should be at least 95.63% w/w pure (azeotropic composition) with a consequent poor flexibility interest.

More realistic specifications for the ethanol-water separation will be considered as listed here below:

(1) Water recovery ratio in the bottom: 0.95.

(2) Bottom water mass fraction: 0.95. Furthermore, in the case of two distillation columns or a dividing wall column, the following constraints should be included in addition to the n-butanol ones:

• Acetone recovery ratio in the distillate: 0.985.

• Distillate acetone mass fraction: 0.995. The single distillation column case studies will be then discussed first, and the distillation trains as well as the DWC case studies will be presented at the end of the following chapters. All the distillation columns presented in this paper will refer to an atmospheric operating pressure.

The most suitable thermodynamic model to describe the ABE/W mixture equilibrium behavior is the non-random two liquids (NRTL). [START_REF] Renon | Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures[END_REF] It results indeed to be the most accurate model able to describe water-alcohols as well as biofuel equilibria in general. [START_REF] Errico | Multiobjective Optimal Acetone-Butanol-Ethanol Separation Systems Using Liquid-Liquid Extraction-Assisted Divided Wall Columns[END_REF][START_REF] Kiss | Novel applications of dividing-wall column technology to biofuel production processes[END_REF][START_REF] Okoli | II Design of dividing wall columns for butanol recovery in a thermochemical biomass to butanol process[END_REF][28] 5. THE BINARY SYSTEMS Among the six possible binary systems of an ABE/W mixture, two in particular have been selected to conduct the flexibility limit analysis, namely, the water-ethanol and water-butanol mixtures. The first reason of this choice was that it would be obviously meaningless to analyze mixtures with no saddles, e.g., acetone-ethanol, since it is always possible to perform a sharp separation by distillation with an infinite number of stages (that is the characteristic condition of RCMs). The second reason why it is worth discussing these two mixtures is that they allow us to show analogies and differences between homogeneous and heterogeneous azeotropes, i.e., whether the liquid phase can undergo demixing or not.

In a two-component system, the residue curve moves from the unstable to the stable node along a 1-D trajectory, i.e., the liquid phase composition curve. Since distillation regions for N component mixtures are N -1 dimensional spaces, in a binary system, the distillation region is part of the x 1 domain.

The following chapter will then concern the well-known water-ethanol equilibrium under a flexibility perspective; after that, a chapter related to the water-butanol binary system will be presented to show how introducing an additional operation different from distillation will affect the residue curve shape and the flexibility of the separation.

5.1. Homogeneous Azeotrope. Among all binary mixtures showing a feasibility limit, water-ethanol is by far the most popular due to its several uses and its ancient history. At atmospheric pressure, this mixture presents a minimum boiling point azeotrope at 78.2 °C and an ethanol composition of 95.63% w/w corresponding to approximately 89.5% mol/mol. 29 The y versus x and T versus xy equilibrium diagrams are reported in Figure 3a,b respectively. As it can be noticed, the residue curves, i.e., the composition of the residual liquid phase, in a binary plot are just represented by the diagonal of the y versus x diagram (since they correspond to the column profile at infinite reflux) or by the bubble curve in T versus xy.

The azeotrope divides then the 1-D space into two distillation regions, x eth ε[0,0.895] and x eth ε[0.895,1]. In the first distillation region, the sharp split that can be obtained is pure water as a bottom product and at the azeotropic composition as a distillate product; in terms of residue curve vocabulary, it can be stated that pure water is the stable node and the azeotrope is the unstable node in the first distillation region, while, in the second distillation region, the azeotrope is still the unstable node but pure ethanol becomes the stable one.

Therefore, since 0.95 w/w water purity is desired, the first trivial conclusion can be immediately drawn: the feed ethanol molar fraction should be lower than the azeotropic one, i.e., the operating line will lie on the left side of the equilibrium diagram. The operating point corresponding to the bottom product is fixed according to the given specification x eth w = 0.05 (x eth mol = 0.02). The other constraint related to the recovery ratio affects the feed split into the distillate and bottom streams; this coincides, from a geometrical point of view, with the application of the so-called lever rule: Therefore, for any given feed composition, the bottom point location is fixed by means of the bottom composition specification, and the distillate point can be located by taking into account the mass balance.

̅ -̅ ̅ -̅ = z x x z D B B D (3)
With the exception of particular cases, the necessary and sufficient conditions to state that a separation by distillation is possible is that the feed, distillate, and bottom composition points lie in the same distillation region, i.e., they all should have the same stable and unstable nodes (for further details about the theoretical background, please refer to Petlyuk (2004) 14 ). In the case of binary distillation, then, they all should lie on the same side of the azeotrope.

By applying these rules to the ethanol-water case study under analysis, it can be deduced that the maximum ethanol content in the feed stream is the one for which the distillate characteristic point lies on the distillation region boundary (x eth D = x eth azeo = 0.895). Thus, to satisfy the process specification (the water recovery ratio in the bottom equal to 0.95), the maximum ethanol molar fraction in the feed is z eth = 0.308. It means that, if the process operating conditions correspond to a lower value, there is a flexibility range z eth ε[0,0.308] within which the feed ethanol content can fluctuate without compromising the desired separation feasibility with a single distillation column. On the contrary, if the feed composition perturbations above the limit value can be attained, the thermodynamic feasibility boundary is breached and there is no way to achieve the process specifications no matter the investment that we would be willing to afford; in those cases, the only solution is to design a different separation process.

In conclusion, it is worth remarking that, although this result is strictly related to the process specifications, the same methodology can be nevertheless applied to any distillation case study.

5.2. Heterogeneous Azeotrope. Though, at first glance, the water-butanol mixture could look similar to the waterethanol one, it needs a considerably different procedure. Indeed, the main difference can be immediately noticed by means of the equilibrium diagrams in Figure 4a,b. If making the assumption of simple distillation (no decanter), the operating points are represented by the solid markers (inverted triangle for the bottom, diamond for the feed, and triangle for the distillate). Even for this case study, the heteroazeotrope x wat azeo = 0.763 28 divides the 1-D space into two distillation regions, x wat ε[0,0.763] and x wat ε[0.763,1], and it is the unstable node for the region on the left-hand side (pure nbutanol, heteroazeotrope) as well as for the region characterized by the heteroazeotrope-pure water split. Then, in order to obtain pure butanol with the desired recovery ratio, the feed stream should be located on the left of the heteroazeotrope.

The thermodynamic flexibility problem can then be solved analogously to the water-ethanol mixture (Figure 5 left); the bottom stream molar fraction is fixed according to the purity specification, the lever rule can be applied according to the recovery ratio, and, with the maximum water content in the distillate being x wat D = x wat azeo = 0.763, the maximum water content in the feed flow rate results to be z wat = 0.1434.

However, in the case of the heterogeneous azeotrope, there is a well-established column configuration [START_REF] Luyben | Control of the Heterogeneous Azeotropic n-Butanol/Water Distillation System[END_REF] aimed at increasing the operation effectiveness based on the use of a decanter instead of a conventional reflux drum (cf. Figure 5 right). For the analysis we are trying to carry out, the addition of a decanter implies considerable modifications worth discussing before providing the corresponding results.

First of all, the distillation column with a decanter as a reflux drum (as shown in Figure 5 right) has one degree of freedom less with respect to the standard configuration since the reflux flow rate is a direct consequence of the equilibrium demixing and cannot be managed by a dedicated control system to achieve a desired specification. Thus, the separation problem should be differently posed: for a given feed flow rate, indeed, either the butanol purity or the butanol recovery ratio can be imposed. This means that one process specification must be loosened and transformed from equality into inequality constraint. It can then be stated that a 99% w/ w purity is required and an n-butanol recovery ratio must be equal to at least 0.9604.

The second main difference between the two configurations concerns the overcooling degree in the condenser. In the standard distillation column, the reflux overcooling is unprofitable because it involves higher-duty requirements both in the condenser and reboiler since a liquid phase further from the equilibrium conditions is recycled without affecting the distillate (and then the reflux) composition. On the contrary, when demixing occurs, the aqueous and organic phase compositions depend on temperature. The water content of the distillate stream can be then modified according to the overcooling degree enhancing the separation with a relatively poor cost increase since the condenser temperature allows the use of ambient water as a cooling duty.

However, as evidenced by well-established experimental data, [START_REF] Stockhardt | Vapor-Liquid Equilibria and Boiling-Point Composition Relations for Systems n-Butanol-Water and Isobutanol-Water 1[END_REF][START_REF] Orr | Versatile Vapor-Liquid Equilibrium Still[END_REF] the aqueous phase composition in the temperature range [0 °C, 91 °C] can only vary from 0.9727 to 0.9838 with an x wat aq = 0.98 value at the boiling point without considerably affecting the thermodynamics of the system. On the contrary, the water content in the organic reflux can significantly change; even though this parameter could improve the separation efficiency from an energetical point of view due to the fact that a lower amount of water circulates through the column, it does not modify the separation thermodynamic feasibility since the reflux stream does not take part in the overall mass balance. For all these reasons, the thermodynamic feasibility analysis has been carried out without condenser overcooling.

The residue curve analysis results can be noticed in Figure 4a,b by referring to the solid markers (diamond for the feed and triangles for the bottom and distillate). The distillate water molar fraction shifted from x wat D = x wat azeo = 0.763 to x wat D = x wat aq = 0.98 with a relevant increase of the maximum water content allowed in the feed stream to z wat max = 0.6647. Note that, if the feed composition is included in the range [x wat org ,0.6647], it is worth feeding directly in the condenser in order to take advantage of the feed demixing and reduce the amount of water introduced in the column as discussed in the work of Luyben (2008). [START_REF] Luyben | Control of the Heterogeneous Azeotropic n-Butanol/Water Distillation System[END_REF] The trend of the butanol recovery ratio in the bottom stream is represented in Figure 6 ]. It can be noticed that the butanol recovery specification is satisfied for any water content in the feed lower than the maximum value.

MULTICOMPONENT SYSTEMS

Process feedstocks are usually composed of several different components even though sometimes only a few of them are worth recovering. Multicomponent mixtures equilibria can be visualized by means of a multiphase diagram. The diagrams' dimensionality is related to the number of compounds, namely, N component mixtures require an (N -1) D diagram. For this reason, the thermodynamic flexibility assessment procedure will be graphically discussed here below up to four components, i.e., the limit of a 3D representation. These diagrams allow nevertheless the ABE/W mixture behavior analysis. The procedure explained in the following paragraphs is nonetheless of general validity independent of the number of mixture components. The analytical procedure in the case of an N higher than 4 is discussed in a dedicated chapter at the end of this section.

The mixture water content will still be used as an uncertain variable since it has been revealed to be the most critical parameter for the separation.

6.1. Ternary System. Since butanol-water and ethanolwater binary mixtures have been analyzed in the previous chapter, the ternary mixture butanol-ethanol-water obtained combining them will be discussed first while acetone will be added later as the fourth compound.

It is worth remarking that we are still dealing with a single distillation column whose specifications are related to the n- Nominal operating conditions should be defined even though they have little importance with respect to the feasibility boundary composition. In order to do that, the limit composition of the water-butanol mixture without a decanter can be used and ethanol in the same amount as water can be added. Finally, by renormalizing, the composition z ̅ = [z but , z wat , z eth ] = [0.75,0.125,0.125] is obtained. It is worth remarking that these values have been selected both to compare the water/butanol ratio that can be achieved in the presence of a further component and to avoid the new component addition drastically displacing the operating point with respect to the distillation boundary in the composition space.

In ternary diagrams, residue curves can move from the unstable to the stable node in a bidimensional space. The mass balances are still represented by the lever rule.

RCMs related to nominal operating conditions are plotted in Figure 7a. The color notation is green for the feed, red for the distillate, and blue for the bottom. The dark green slice shows the region where the liquid phase demixing occurs. In order to account for the eventual presence of two liquid phases, the option including liquid split calculation in Simulis Thermodynamics has been ticked.

As it can be noticed, under nominal operating conditions, all the distillation column streams belong to the same distillation bundle where n-butanol is the stable node while the water-ethanol azeotrope is the unstable one. In order to be conservative, the temperature used to outline the immiscibility region is the boiling temperature of the unstable node since it corresponds to the lowest value achieved along the residue curve.

Starting from this operating point, water can be added as usual in order to assess its maximum allowed molar fraction. The addition of water in a ternary diagram, keeping unchanged the quantity of the other compounds, consists of the operating point displacement toward the water vertex as shown by the black arrow in Figure 7a. The distillate and the bottom product compositions can be then calculated based on the specifications and the equilibrium. Since the bottom product composition is fixed by the required butanol purity, by shifting the feed composition, the distillate characteristic point location changes accordingly in order to satisfy the lever rule. The limit feasibility conditions are shown by the dotted lines in Figure 7a and correspond to the feed composition z ̅ = [0.709,0.173,0.118]. For a further water content increase, indeed, the distillate stream characteristic point crosses the distillation boundary and falls into the distillation region where the stable node is represented by water instead of nbutanol as shown in Figure 7b.

Beside the effective graphical visualization, there are a few remarks worth doing from a quantitative point of view. The first observation refers to the water-butanol ratio at the feasibility limit; in the case of a binary mixture without decantation, this ratio is approximately 1:6.14, while with the addition of ethanol, the separation feasibility is enhanced until a 1:4.1 ratio. This means that, for this case study, the maximum allowed amount of water is higher if a certain third compound is present.

The second remark states that the increase in this ratio quantitatively depends on the amount of the third species. In particular, from a graphical point of view, a higher ethanol addition reflects in a rotation toward the ethanol vertex [0, 0] of the operating line.

Moreover, as it can be noticed from the plot, the residue curve related to the distillate stream enters the immiscibility region. In this case, its calculation is based on the VLLE and accounts for the liquid phase demixing. The dotted line represents the boundary compositions in case the decanter is exploited to enhance the separation as previously discussed for the binary case study. The same remarks concerning the distillate overcooling keeps being valid for the ternary mixture as well due to the correlation between the immiscibility region and operating temperature.

However, for this case study, the characteristic points are always far from this region; thus, the separation is not concerned by heterogeneous phases in the column.

It is finally worth highlighting that all those results are strictly related to the system thermodynamics and do not take into account operating costs and controllability of the process that could affect the design choice.

6.2. Quaternary System. According to the procedure followed in the ternary case study, the previous boundary composition z ̅ = [0.709,0.173,0.118] can be taken as an initial value. Then, an amount of acetone equal to the water molar The so-obtained quaternary mixture can then be represented in a corresponding quaternary diagram. This diagram consists of a 3D pyramid whose vertices correspond to each pure component; the distillation region in a quaternary diagram is topologically defined by a 3D space and its boundary is a surface, while the mass balance still lies on a line connecting the inlet/outlet characteristic points. Finally, since both the BWE and BWA mixtures show liquid- phase demixing, the immiscibility region is a 3D space included between those two surfaces.

Residue curve mapping can be then performed under nominal operating conditions as shown in Figure 8a where pure ethanol corresponds to the origin of the axes [0,0,0]. All the three curves lie in the same distillation region and an additional water perturbation is possible with respect to the ternary case study. In a quaternary diagram, the addition of a single species keeping unchanged the proportion between the other ones is carried out by shifting the operating point toward the corresponding vertex as usual. Thus, a water increase in the distillation column feed is graphically represented by the black arrow in Figure 8a.

The extreme condition is obtained for a feed composition equal to z ̅ = [0.594,0.162,0.099,0.145] as shown in Figure 8b. For a further water content increase in the feed, the residue curve passing through the distillate characteristic point (the red one in the figure) overcomes the distillation boundary in analogy with the ternary case study, and its stable nodes become pure water instead of pure butanol. Figure 8c shows the unfeasible conditions and the separatrix, which, in this case is a surface, is represented by dashed contour lines.

In analogy with the previous case, it can be concluded that the addition of acetone improved the separation by increasing the maximum allowed water/butanol ratio in the feed from 1:4.1 to 1:3.67.

Even in this case, the result quantitatively depends on the amount of the other species; the maximum allowed water content could indeed increase further for different values of acetone and ethanol content due to the convexity of the separatrix surface.

If the operating points lied in the pure water distillation region, indeed, the addition of acetone and/or ethanol would have reduced the operation flexibility due to the inverse separatrix shape experienced from that side.

The same remarks about convexity apply to the ternary case as well as to mixtures with a higher number of components even though they cannot be graphically represented in a physical space.

Even though calculations have been performed for a given value, the separatrix shape depends on thermodynamics only, and the detailed study of the process feasibility boundaries allows us to assess which maximum allowed perturbation could withstand the distillation system, i.e. its thermodynamic flexibility.

Moreover, it is worth remarking that this analysis refers to one purity constraint and one recovery constraint on the same product (i.e., the separation yield); for different specifications, a dedicated study should be performed.

Finally, it should be highlighted that this study does not account for economic or controllability aspects but it just establishes the physical distillation constraints that cannot be violated independently on the technological solution.

6.3. Systems with More than Four Components. All the examples presented so far are related to mixtures whose equilibrium diagrams can be visualized in a three-dimensional (or lower) space.

When more than four components are present, the composition domain is a N -1 D space and the graphical approach cannot be exploited for a more immediate physical understanding. However, from a computational point of view, the way residue curves can be outlined does not change. They are still one-dimensional curves in the composition hyperspace whose coordinates can be obtained by the integration of eq 1.

In order to have a practical example of the analytical approach for the feasibility boundaries' estimation, let us consider the ABE/W mixture case study if some methanol is present as well in the fermentation broth. According to the usual procedure, given the boundary composition z ̅ = [0.594,0.162,0.099,0.145], methanol can be added in the same amount as water and the final composition after n o r m a l i z a t i o n r e s u l t s t o b e z ̅ = [0.511,0.139,0.086,0.125,0.139].

Given the separation specifications and the mass balances, bottom and distillate streams compositions are then estimated, the residue curve passing through each one of the three characteristic points can be calculated as usual, and stable and unstable nodes can be easily determined.

Under nominal operating conditions, the residue curves corresponding to the feed and the products have the same stationary points; thus, the separation still results to be feasible even after the addition of methanol. The feed water content can be then perturbed in order to assess the new boundary conditions.

The flexibility assessment results corresponding to the unfeasible condition are presented in Table 1. Although the presence of methanol implies one additional singular point in the composition space, i.e., the acetone-methanol azeotrope, we are able to further enhance the separation. As it can be noticed, the boundary composition for a water perturbation is z ̅ = [0.498,0.162,0.083,0.121,0.136], corresponding to a maximum molar water to butanol ratio in the feed of about 1:3. 25.

In general, in the feasible region, the unstable node shifts from "acetone" to the "acetone-methanol azeotrope", while the stable one is still pure butanol. On the contrary, for higher water perturbations, the residue curve passing through the distillate characteristic point ends at the "pure water" node as already experienced in the previous chapters. In any case, the lack of a geometrical representation makes the understanding of the newly generated distillation bundles substantially more difficult.

The additional computational effort due to the presence of a fifth compound is comparable with that related to the previous scale-up.

In conclusion, it is worth remarking that the procedure introduced here above may lead to a feasible domain underestimation when the VLE equilibrium behavior exhibits a nonideality resulting in inflections of the total reflux trajectories as anticipated in the Introduction. Those cases, which are not discussed in this research study, require an

DISTILLATION TRAINS AND DIVIDING WALL COLUMN

When more than two pure components should be recovered from multicomponent mixtures or if more than two specifications should be satisfied in order to have a profitable separation process, distillation columns can be combined in distillation trains. There exist several distillation train configurations named after the sequencing of the product purification. The most popular are the direct, indirect, and midsplit configurations. The first two consist of several columns in series, and the separation starts from the most volatile compound to the heaviest for the direct and the opposite for the indirect. The midsplit configuration is a parallel configuration where pure compounds are obtained downstream after previous partial splits.

An alternative solution to multicomponent-mixture separation is a dividing wall column, i.e., a single distillation column whose central section is divided by one or more walls and with one or more side withdrawals. It represents the arrangement in a single distillation column shell of the Petlyuk column configuration. [START_REF] Petlyuk | Thermodynamically Optimal Method for Separating Multicomponent Mixtures[END_REF] This configuration is equivalent to the distillation train even though only two external duties are present. The reason why it has been studied in depth is that it is the only large-scale process intensification case where both CAPEX and OPEX as well as the required installation space can be drastically reduced. [START_REF] Glinos | Optimality Regions for Complex Column Alternatives in Distillation Systems[END_REF][START_REF] Schultz | Reduce costs with dividing-wall columns[END_REF] However, from a thermodynamic point of view, distillation trains and the DWC can be considered equivalent configurations. The Petlyuk arrangement of the DWC, i.e., pre-fractionator and column, and the equivalent indirect configuration are shown in Figure 9 for an infinite number of equilibrium stages. The indirect configuration has been selected as a case study since Di Pretoro et al. [START_REF] Di Pretoro | Flexibility Assessment of a Distillation Train: Nominal vs Perturbated Conditions Optimal Design[END_REF][START_REF] Di Pretoro | Flexibility assessment of a biorefinery distillation train: Optimal[END_REF] already proved it to be particularly convenient for n-butanol recovery in ABE separation. However, in this paper, the thermodynamic assessment has been extended including both the columns of the distillation train configuration. Furthermore, the following procedure can be scaled up to any number of columns, i.e., any number of walls and side streams in a DWC.

The residue curve interpretation does not considerably change. It still represents the column concentration profile for an infinite number of trays at an infinite reflux ratio. The main consequences of this theoretical consideration is that the distillation feasibility condition still applies: in order to belong to the same distillation column profile, the residue curves passing through the characteristic point of every side stream should belong to the same distillation region, i.e., it should have the same stable and unstable nodes. Thus, the main change with respect to the single column case study is that all the streams entering or exiting the distillation train (or the DWC) should be included in the analysis.

As already anticipated in Section 4, the additional specifications to be included will refer to the acetone purity and recovery ratio; this means that the side stream will contain the remaining components, i.e., mainly water and ethanol. While the specification related to a product purity directly provides its coordinate in the diagram, the recovery ratio constraint requires a reinterpretation.

By performing the overall mass balance on the entire system, the following equation can be easily obtained:

= + + F D S B (4)
that is:

= + + D F S F B F 1 (5)
On the other hand the partial mass balance states as:

• ̅ = • ̅ + • ̅ + • ̅ F z D x S x B x D S B (6)
dividing everything by F the equation of a plane is achieved: Industrial & Engineering Chemistry Research

̅ = • ̅ + • ̅ + • ̅ z D F x S F x B F x D S B (7)
This means that, for given product compositions (x ̅ D and x ̅ B ) and recovery ratios (directly related to D F and B F ), the side stream molar fractions and flow rate can be assessed and the corresponding characteristic point lies on the same plane as the other three.

In conclusion, the mass balance in a quaternary diagram is translated into a coplanarity condition and the recovery ratios establish the relative position of the side stream characteristic point with respect to the distillate and the bottom ones.

T h e r e f o r e , a g i v e n i n l e t c o n c e n t r a t i o n z ̅ = [0.660,0.120,0.100,0.120] similar to the ones in the previous chapter is assumed. Residue curves under nominal operating conditions are calculated as usual for the four streams and plotted in Figure 10a. The orange curve refers to the side stream, while the color notation for the other curves is unchanged. Water content, which is the most critical parameter, can be then perturbed as usual until feasibility limits are achieved for z ̅ = [0.641,0.146,0.097,0.116] as plotted in Figure 10b. For a further increase in the water molar fraction, the residue curve related to the side stream crosses indeed the distillation boundary and does not attain the pure butanol vertex but the pure water one instead.

Even though the case study is different from the one in the previous chapter since a second compound, i.e., acetone, is recovered as well, the feasibility boundary values can be commented in analogy to the single distillation column.

First of all, it is worth noticing that the separatrix surface is the same as the single column case study since it is a function of the system thermodynamics, i.e., of the components only.

While, for the single column, the distillate stream is the one crossing the distillation boundary, in the case of indirect distillation train and the DWC, the second column bottom stream and the side stream are those who cause the process unfeasibility.

For this case study, the limit water to butanol ratio is 1:4.4 resulting in a substantial decrease with respect to the previous cases where acetone was not purified. Due to the acetone recovery, a consistent amount of ethanol is indeed withdrawn in the distillate stream. This causes the side withdrawal to be richer in water both because acetone is not present anymore and because it carries with it part of the ethanol causing a faster shifting of the side stream operating point toward the pure water vertex.

As anticipated, this procedure has no upper limits concerning the number of species in the feed and the number of column/walls, but at least three components are required in order to have a useful employment of distillation trains or DWC.

CONCLUSIONS

Residue curve mapping has been deeply explored in this analysis. Three main outcomes can be identified from this study and are worth commenting on.

The first one is the definition of a thorough procedure to assess thermodynamic flexibility of distillation processes when the product recovery ratio is used as a separation specification. This parameter indeed can be directly correlated to the separation productivity that is the main income item in process design. Depending on the number of components, the recovery ratio fixes the relative position of the distillation column streams over the N -2 dimensional space constrained by the mass balances.

The second useful tool described in this paper is the thermodynamic feasibility assessment and definition of flexibility boundaries in the case of distillation trains and integrated distillation systems such as the DWC as well as its graphical representation. Since, at infinite reflux and number of equilibrium stages they are equivalent configurations, there is no difference concerning the thermodynamic feasibility limits. However, when dealing with costs and control, they are likely to show considerably different behaviors.

Finally, according to the shape of the N-dimensional separatrix boundary, RCM graphical representations allow us to predict the effect of the presence of a further component in the feed stream from a flexibility point of view. The convexity/concavity of the separatrix would result respectively in a positive/negative impact on the separation effectiveness under uncertainty.

In conclusion, this study provides a useful graphical and numerical tool to the decision maker in order to evaluate the potentialities and the limitations of the most common distillation processes applied to a given multicomponent mixture. Even though the presented results are case-specific, the proposed procedure has a general validity and can be performed with any mixture and any of the most-used flexibility indexes.

Residue curve mapping has proved indeed to be an effective methodology worth exploiting even when uncertainty should be taken into account that is particularly the case with biobased processes.

Figure 1 .

 1 Figure 1. Open evaporation process.

Figure 2 .

 2 Figure 2. (a, b) RCM examples (solid square: stable, solid circle: saddle, and solid star: unstable).

Figure 3 .

 3 Figure 3. (a, b) Ethanol-water equilibrium diagrams: solid inverted triangle -bottom, solid diamond -feed, and solid triangle -distillate.

Figure 4 .

 4 Figure 4. (a, b) n-Butanol-water equilibrium diagrams: solid inverted triangle -bottom, solid diamond -feed, and solid triangle -distillate.

  in the range z wat ε[x wat B , z wat max

Figure 5 .

 5 Figure 5. Standard distillation column vs distillation column with a decanter.

Figure 6 .

 6 Figure 6. Butanol recovery ratio in the bottom stream.

Figure 7 .

 7 Figure 7. (a, b) Distillation RCM on ternary diagrams.

Figure 8 .

 8 Figure 8. (a-c) Distillation RCM on quaternary diagrams in a lateral (left) and top (right) view.

Figure 9 .

 9 Figure 9. Infinitely staged indirect distillation train (left) and DWC (right).

Table 1 .

 1 Analytical Results for Five-Component RCMs

	stream	z But	z Wat	z Eth	z Ace	z Met	stable node	unstable node
	feed	0.498	0.162	0.083	0.121	0.136	n-butanol	acetone-MeOH
	bottom	0.972	0.020	0.008	0	0	n-butanol	acetone-MeOH
	distillate	0.039	0.299	0.156	0.239	0.267	water	acetone-MeOH

accurate separatrix hypersurface estimation in order to check if it can be crossed or not in columns operating at total reflux.

  Laboratoire de Geńie Chimique, Universitéde Toulouse, CNRS/INP/UPS, Toulouse 31100, France; Email: ludovic.montastruc@ensiacet.fr Authors Alessandro Di Pretoro -Laboratoire de Geńie Chimique, Universitéde Toulouse, CNRS/INP/UPS, Toulouse 31100, France; Dipartimento di Chimica, Materiali e Ingegneria Chimica ≪Giulio Natta≫, Politecnico di Milano, Milano 20133, Italy; orcid.org/0000-0001-7394-5396 Flavio Manenti -Dipartimento di Chimica, Materiali e Ingegneria Chimica ≪Giulio Natta≫, Politecnico di Milano, Milano 20133, Italy Xavier Joulia -Laboratoire de Geńie Chimique, Universitéde Toulouse, CNRS/INP/UPS, Toulouse 31100, France Complete contact information is available at: https://pubs.acs.org/10.1021/acs.iecr.0c02383 Notes The authors declare no competing financial interest.

	Industrial & Engineering Chemistry Research
	■ AUTHOR INFORMATION Corresponding Author
	Ludovic Montastruc -■ ABBREVIATIONS ABE/W acetone-butanol-ethanol/water
	B	bottom flow rate (mol/s)
	n C ij CAPEX capital expenses ($) excess Gibbs energy coefficients (J/(mol•K n ))
	D	distillate flow rate (mol/s)
	DWC dividing wall column
	F	feed flow rate (mol/s)
	g i,j	NRTL excess Gibbs energy (J/mol)

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https://dx.doi.org/10.1021/acs.iecr.0c02383 Ind. Eng. Chem. Res. 2020, 59, 16004-16016


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