Thierry Gousset

Jacob Lamblin

Sandra Valcares

Radioelectric fields from cosmic-ray air showers at large impact parameters

We discuss electric fields generated by cosmic ray air showers at large impact parameters b. An approximation relevant to this situation is given. The formulation makes explicit the relationship between the shower profile and the radio pulse shapes at large b, putting forward one important observational consequence, namely the decrease of the high-frequency cutoff ν c ∝ 1/b 2 when the impact parameter increases. The approximation is also used to give a detailed comparison between two emission models, the geosynchrotron model and the transverse current model.

Introduction

There is nowadays a renewed interest in the extensive air shower radio detection technique [1,2]. Results have been obtained which improved on both the experimental and theoretical exploratory studies that took place during the sixties. An illuminating review of this early phase is given in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF].

From the theoretical point of view, several comprehensive frameworks have been completed or initiated in the last five years [START_REF] Huege | [END_REF]5]. The most advanced approach (geosynchrotron approach) is by Huege and collaborators [START_REF] Huege | [END_REF]6,7]. It gives a wealth of details on the pattern of radio emission with different levels of sophistication for how the shower development is taken into account. More recently, based on the model first proposed by Kahn and Lerche [8], Scholten and Werner developed a different approach (transverse current approach) [5]. The present study can be seen as a follow-up of the latter work as it exploits a limiting behavior, the so-called point-like approximation, motivated in Ref. [5]. We give a somewhat different and complementary motivation for arriving at the same approximation. Our derivation follows a line of reasoning suggested in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. It makes it possible to extend the approximation to the cases of non vertical showers and also to get the corresponding expression for the electric field considered in Ref. [START_REF] Huege | [END_REF].

We believe that, however crude the approximation may be, it might be useful in the present stage of mastering the radio detection technique when progress is made using a delicate mix of physical prejudices and empirical facts. This is provided that the model captures some essential features of the phenomenon. For the approximation at hand a point that can be put forward is the direct relationship at large impact parameter between the time-development of an air shower and that of its radio component. This has two testable consequences. First, the decrease of the high-frequency cutoff of the electric field spectrum when the impact parameter rises: this has never been seen in experiment so far and is a crucial test for our understanding of radio emission. Next, the sensitivity of the radio component to the beginning of the shower development, making radio complementary to traditional techniques for that matter.

The derivation of the electric field expressions in the point-like approximation is given in Sect. 2 for both the transverse current and the geosynchroton approaches. This leads us to discuss the differences between the two approaches concerning on the one hand their phenomenologies and on the other hand their physical motivations. The section ends in a discussion of the domain of impact parameters and frequencies where the approximation makes sense. The relationship between the time pulse and shower evolution is investigated in Sect. 3, showing the importance of the Doppler effect. Sect. 4 is devoted to questions concerning a very large array, a natural topic for an application of large impact parameter calculations. Our main observations are collected in Sect 5.

The point-like model 2.1 Electric field expressions

A cosmic-ray air-shower is a charge and current system whose electromagnetic field carries genuine information on some important aspects of the shower. Because the shower front moves at about the speed of light, the electromagnetic signals emitted in the forward direction by the shower core at various stages of the shower history all travel with the particle front. Thus only at observation points located at sufficiently large impact parameters, i.e., away from the forward direction, the sequence of emissions results in a sequence of receptions, and the collected pulse is a reflection of the shower evolution. This is because the time scale at reception associated with the shower development becomes much larger than those coming from the shower extension. The hierarchy of time scales relevant at large impact parameters is thoroughly studied in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF].

This hierarchy suggests a model where all time scales but that due to shower evolution are set to zero. Setting time scales associated with the shower extension to zero corresponds to neglecting the shower extension altogether, hence taking the shower as a point-like system. Of course, this is only meant as a model for the densest part of the shower core where most of the ultra-relativistic charges stand. As discussed in Sect. 2.3, this model makes sense for impact parameters in the km range and at not too large frequencies.

The explicit expression for the electric field depends on both the origin of charge separation and the electric field emission mechanism. For electron-positron separation in the Earth magnetic field, two possibilities have been explored recently, considering that the electric field is generated either by charge acceleration (geosynchrotron emission) [START_REF] Huege | [END_REF] or by charge drift [5]. Both will be considered here in order to show the flexibility of our approach and to gauge the sensitivity of some results. For a given choice of charge separation and emission mechanism the expression of the electric field in the model is essentially an adaptation of the relevant textbook formula to the case where the number of elementary charges N ee evolves with time. Such a scheme was already followed in Ref. [9]. There it is also explained that at a given impact parameter b two effects are in competition for building the electric field. These are the inverse distance 1/R (notations in Fig. 1) between the charge and the observation point and the Doppler factor

∂t ∂t ′ -1 = 1 1 -nβ cos ϑ (t ′ is the emission time, t is the reception time, ϑ = ( v, -→ QA)
, n is the refraction index and β = v/c). The larger both factors are the larger is the field. For n very close to 1 (air) and v very close to c, the Doppler factor singles out very small angles (for n = 1, ϑ = O(1/γ)). The most favorable situation is therefore at small b. However, if we insist on having a somewhat larger b (so that the point-like model may apply) then the combination N ee /R will single out finite angles1 in contradistinction to the Doppler factor. The competition is illustrated in Fig. 2 as a function of Z (the relation between Z and ϑ is Z = b/ tan ϑ).

As a result of the above competition, the electric field picks up its largest magnitude in a region of small but not too-small angles where the following conditions are fulfilled:

• ϑ ≪ 1, the Doppler contraction plays a role, so that relativistic features of radiation are partly kept, (

) 1
This greatly simplifies the computation of the electric field since all source based time dependences can be expressed as function of t and b, i.e., observer based quantities. The transverse current field reads

E v (t, A) = (N ee (t ′ ) + t ′ Ṅee (t ′ )) e v T 4πε 0 c 1 (ct) 2 = e v T 4πε 0 c 4(ct ′ ) 2 (N ee (t ′ ) + t ′ Ṅee (t ′ )) b 4 . (2)
Eq. ( 2) follows from taking the derivative of the vector potential under the assumption that the average transverse velocity of all electrons and positrons in the shower, v T , is constant.

A complete derivation can be found in Ref. [5]. We give some details in the Appendix and also on how to arrive at the geosynchrotron field that reads

E a (t, A) = N ee (t ′ ) e a T ǫ 4πε 0 c 2 b 2 2 (ct) 3 = e a T ǫ 4πε 0 c 4(ct ′ ) 2 (-t ′ N ee (t ′ )) b 4 . ( 3
)
3: d dt ′ (t ′ N ee (t ′ ))/(t ′ N ee (t ′
)) in µs -1 for θ = 0 (right) and 45 • (left). Dashed curves show the respective behaviors of E a (t ′ ), rescaled so that their maxima = 0.1, to help focusing on the regions of interest.

a T is the transverse acceleration (transverse with respect to the shower axis) and ǫ is a unit vector showing the polarization which is perpendicular to the shower axis and such that (

B T , ǫ) = π 2 + 2 × ( B T , -→ BA). (4)
In Eqs. ( 2) and (3), the first expressions make it easy to recognize the one-particle field approximation in the relevant ϑ range, setting N ee (t ′ ) →const., whereas the second expressions are useful for discussing scaling relations with b, see Sect. 2.2.

Comparisons

Eq. (2) shows that the magnitude of the field writes as a function of t ′ divided by b 4 . As noticed in Ref. [5], because of Eq. ( 1) this means that the electric field pulse duration scales as b 2 and its peak value as 1/b 4 . Eq. (3) shows that the same properties hold for the geosynchrotron field.

The functional dependence on N ee is different for E v and E a . E a is proportional to t ′ N ee (t ′ ) whereas E v goes like the derivative with respect to t ′ of this quantity. As a consequence, it is easily checked that the time integral of E v (t, A) is zero, i.e., the E v pulse is strictly bipolar [5]. The E a pulse is predominantly monopolar. Sect. 3 is devoted to the discussion of the shower evolution dependence of the electric pulse.

As for comparison of magnitudes, the fields in the point-like approximation are in proportion of

v T × d dt ′ (t ′ N ee (t ′ )) vs a T × (t ′ N ee (t ′ )). ( 5
)
This amounts to comparing the ratio v T /a T to a shower development quantity. These are genuinely different quantities, which has a consequence on their orders of magnitude, but also on their parametric dependences. Fig. Consequently, the peak value of the E v field is one order of magnitude smaller than the E a field. The parametric dependence of this hierarchy can be discussed by looking at the involved scales. Physically, the shower development is best described in terms of slant depth, X, and a scale ∆X in this development, e.g., the width of N ee (X), can be transformed into a length scale by dividing ∆X by the air density at the point of interest, ∆X/ρ air . The ratio v T /a T ∼ τ is given by the lifetime of a particle (ultra-relativistic electron or positron) in the shower core [8], see also the Appendix B. Since c τ ∼ X 0 /ρ air the relative strength of E v and E a is given by the relative strength of X 0 and ∆X. The above discussed hierarchy can be seen as a consequence of ∆X ≫ X 0 for high-energy showers. However, no strong variation of this hierarchy was found when changing shower characteristics, such as inclination or energy.

3 displays d dt ′ (t ′ N ee (t ′ ))/(t ′ N ee (t ′ )) in µs -1 for B T B
Eq. (5) suggests that E v and E a are two components that derive from a single quantity. This fact may be connected to the occurrence of angular polarization in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF], which is the single quantity from which the total electric field derives in the latter formalism. In the present formulation, however, the relationship is less apparent when polarization is considered. Polarization of the velocity field is simply given by the direction of v T , i.e., perpendicular to the plan formed by the shower axis and the magnetic field. That of the geosynchrotron field, see Eq. ( 4) and Fig. 4, is more complex: it is still perpendicular to the shower axis, but not always in the direction of a T , hence giving another way to distinguish between E v and E a . The pattern shown in Fig. 4 is a purely relativistic effect. For a single charge this kind of polarization occurs at angle such that cos ϑ < nβ, i.e., precisely the regime where the point-like approximation is formulated (see Sect. 2.1). The polarization pattern obtained in the general geosynchrotron approach is even more complex. In the latter approach, at an observation point located at small b some particles are emitting at very small angles (cos ϑ > nβ) while others are emitting at larger angles resulting in a highly non trivial pattern. However, it is possible to understand qualitatively this pattern as a superposition of a small-angle component, opposite to the acceleration, and a large angle one with the pattern shown in Fig. 4: for antennas located at the north or the south of B (we refer to the cardinal points on the map with north to the top as usual) both fields add and the result is purely EW; for points east or west of B there is a partial cancellation but the superposition remains EW; for direction NE and the like both an EW and a NS components show up. (See Fig. 3 in Ref. [6].)

As a matter of fact E v and E a are genuinely different. The difference has its roots in the underlying physical picture for radio emission. For the E v field the picture is that of drifting charges much as in a conductor in a stationary state: charges are constantly accelerated by the external field but acceleration due to collisions is, on average, opposite to this external field acceleration, and coherence implies that any radiation is that of a system with no net acceleration. In the E a approach, on the contrary, collisions are considered to be inefficient to alter the radiation associated to the external field acceleration and are disregarded (in the conductor picture this would correspond to the transient regime). A dedicated study of shower evolution is necessary in order to see which picture is the right one (or which mix is effectively relevant). Such a study, e.g., based on shower simulations, goes beyond the scope of the present work. Naively, the scales involved have a tendency to promote the drifting regime since the timescale for collision is τ which is smaller than the timescale for evolution of the shower as a whole. In the following, when limiting a study to one field, the E v situation will be considered.

Validity

As was explained in Sect. 2.1, the point-like approximation requires that the shower evolution, as seen by an observer, has a time scale much larger than those due to the shower extension. The relevant time scales were studied in detail in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] and we will essentially reproduce the reasoning here, extending the consideration to somewhat larger b and also to inclined showers.

Allan makes his estimate for a vertical shower with energy around 10 17 eV. He first estimates a rise time for the pulse associated with the shower evolution (the so-called obliquity effect in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]) by computing the time elapsed between the reception of a signal emitted at a height 10 km (shower "start") and that at a height 5 km (shower maximum). At small b, the shower evolution alone, i.e., the function f 1 (t ′ ) = N ee (t ′ ), leads to the above choices for 'start' and 'max'. At larger b, Eqs. ( 2) and [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] of Z = -c t ′ at which f 1 , f 2 , and f 3 go through their first maxima are given in Table 1, for various inclinations and energies E p = 10 19 and 10 17 eV.

show that it is rather f 2 (t ′ ) = t ′2 |N ee + t ′ Ṅee | or f 3 (t ′ ) = |t ′ | 3 N ee (t ′ )
The rise time (multiplied by c) is, using Eq. ( 1),

∆ g = b 2 2Z max - b 2 2Z start ,
where Consideration of length scales is also important in the frequency domain. When ∆ g ≫ ∆ l and ∆ g ≫ ∆ r the loss of coherence is set by obliquity at a frequency ν c ∼ c/(2π∆ g ). Since ∆ g ∝ b 2 , an important signature of shower radio emission is provided by the observation of the scaling law ν c ∝ 1/b 2 at large b. There is also an upper bound for application of the point-like approximation in the frequency domain given by ν ′ c ∼ c/(2π∆ l ). Taking ∆ l = 3 -10 m gives ν ′ c = 5 -15 MHz. These figures are also put forward in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. It should be noted that the experimental situation is not clear concerning the loss of coherence at small b where the scales ∆ l and ∆ r are still relevant (∆ g → 0). It seems that a more or less flat spectrum has been observed around 50 MHz [10,11]. Such results call for confirmation but it may well be that the above figures for ν ′ c are underestimated making the range of applicability of the point-like model extending somewhat beyond 5 -15 MHz. In any case, observation of loss of coherence is one of the major goal to be achieved by the present radio detection experiments.

Z
3 Deciphering the shower profile

Pulse shapes and spectra

A selection of pulse shapes and spectra is shown in Figs. 5 and6, respectively for E v and E a , for showers of energy E p = 10 19 eV and X max = 770 g/cm 2 and an observer at b = 1 km.

For E v the beginning of the electric field pulse is polarized along t ′ Ṅee (t ′ ) v T , i.e., opposite to v T ∝ v × B with v the unit vector along the shower axis and B the Earth magnetic field. For the latter we consider the situation in central Europe: the zenith angle is θ B = 153 • and the horizontal component goes from south to north. Consequently, the polarization is east-west for showers coming from north or south. Furthermore, the initial polarization is westward for a vertical shower while it is eastward for showers coming from south with θ ≥ 27 • . This explains the various pulse signs in Fig. 5. For other shower directions, the study of polarization can be made along the same lines. For E a , the electric field polarization depends on the observer location with respect to the shower (see Fig. 4). Here an observer standing east or west of the shower is considered. As in the E v case there is a flip of sign when crossing the magnetic lines at θ = 27 • .

The explicit relationships between pulse shape and time evolution of the shower charge number are given in Eqs. (2,[START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. In both cases, the Doppler-like effect, t ′ → t, distorts the time dependence. From Eq. ( 1):

∂t ∂t ′ = 1 2 b c t ′ 2 .
Early times (corresponding to t ′ negative and large) getting more contracted than later times. This explains the sharp rise, the slower decay, and in the E v case the even slower undershoot of the pulses shown in Figs. 5 and6.

The variation of rise time with θ was already discussed in Sect. 2.3. The probing of earlier shower stages when θ increases affects pulse shape and has the tendency to give larger fields. To get the complete change, this increase has to be combined with the overall amplitude modification coming from the variation of sin α(θ, ϕ), where α is the angle between the shower axis and the magnetic field. At fixed θ, the azimuth dependence

F (ν, b) = dt f (t ′ ) b 4 e 2iπνt = 1 b 2 dt ′ f (t ′ ) 2(ct ′ ) 2 e -iπν(b/c) 2 /t ′ which implies that F (ν, b) = b 0 b 2 F (νb 2 /b 2 0 , b 0 ).
As for the time pulse, it is thus possible to obtain a spectrum at another b from Figs. 5 or 6 taking into account that frequencies and spectrum magnitude are both ∝ 1/b 2 . A specific application of the rescaling law for frequencies is the scaling behavior of the high-frequency cutoff, ν c ∝ 1/b 2 , already discussed in Sect. 2.3.

For the E v field the spectrum is 0 at ν = 0 (bipolarity), goes through a maximum in the MHz range for b = 1 km and then falls off almost as a pure exponential (loss of coherence) at least at not-too-large θ. At θ = 60 • the loss of coherence starts beyond 10 MHz, i.e., a value comparable to the estimate given for ν ′ c , reminding us that the pointlike approximation is marginally applicable at b = 1 km for this large inclination (see Sect. 2.3). The loss of coherence for the E a spectrum follows the same trend. The main difference is at small frequencies, the E a spectrum being finite at ν = 0.

Sensitivity to shower rise

An interesting consequence of the Doppler distortion lies in the greater sensitivity of the radio signal beginning of the shower. illustration this feature is shown in Fig. 7. The pulse height difference is a signal of the difference between the two developments (here around 8 km). The region of greater sensitivity is thus different in radio detection and in fluorescence. Note that this property is generic in radio emission at large impact parameters [9]: in the present study it was observed for both the E v (Fig. 7) and the E a (not shown) emission models.

The effect is more visible in the frequency domain (right plot in Fig. 7). The shower growth being pushed at larger frequencies than the shower decay by the Doppler effect, the stiffening of the time pulse for the Gaisser-Hillas profile stands out in the spectrum at large frequencies. This may be important in practice, since large radio frequency interferences often render radio observation difficult at frequencies below 20 MHz. In such circumstances the greater sensitivity of radio detection to the first stage of shower evolution may be even more pronounced.

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Profile extraction

From Sect. 2.1, the electric field magnitude observed at the location of antenna i, A i , reads

E v (t, A i ) = K v (ct) 2 (N ee (t ′ ) + t ′ Ṅee (t ′ )), t ′ = -b 2 i 2c 2 t ,
for the transverse current component, and

E a (t, A i ) = K a b 2 i (ct) 3 N ee (t ′ ),
for the synchrotron one. K v and K a are both independent from the observation point of the event and are thus irrelevant. They can be set to 1 for the reasoning. Such simple expressions for electric fields make it possible to illustrate what knowledge can be gained from radio measurements. Imagine the full geometry of the shower is known, i.e., both the shower incidence and the impact point location, either by some joint particle detector array or by an analysis based on radio alone [START_REF] Gousset | contribution to the 30th International Cosmic Ray Conference[END_REF]. Then it is possible to obtain the full time evolution of N ee from the time pulse collected on antenna say i = 1 by looking at either

(ct) 2 × E v (t, A 1 ) = g 1 (-b 2 1 /(2ct)),
for the transverse current field, or

(ct) 3 × E a (t, A 1 ) = g 2 (-b 2 1 /(2ct)),
for the synchrotron field. In the latter case, the function g 2 (t ′ ) is directly proportional to N ee (t ′ ). In the former

g 1 (t ′ ) = N ee (t ′ ) + t ′ Ṅee (t ′ ).
g 1 (t ′ ) is the time derivative of t ′ N ee (t ′ ), so a time integration, followed by a division by t ′ , is necessary to obtain N ee (t ′ ). In any case, the unfolding of the shower profile is quite simple in the model, in spite of the presence of the Doppler distortion.

In principle, the signal collected on one single antenna is enough in order to make the aforementioned extraction, but several antennas may be used to enhance the signal over background ratio. In addition, the signal can only be extracted in practice in a limited frequency-range. For sure this is the more severe limitation to the suggested analysis. However, the shower evolution is not a complete unknown and various tools of signal analysis can be considered to obtain pieces of information on the N ee evolution. Having one such tool in mind, the point-like model expressions are sufficiently easy to manipulate in order to test the relevance of the foreseen method. It is not the purpose of the present study to elaborate on those methods which depend to a certain extent on the specifics of experiments and the noise situation at their locations.

Towards a large array

The radio detection technique is nowadays considered as a possible surrogate to more traditional techniques for future giant array, such as Auger north. One elementary detector per km2 is the typical scale for such an array, setting the scale of closest approach to 1 km. Since the phenomenology discussed above applies just to this regime, it may have consequences on the exploitation of a large array which we examine here.

Given a detection criterion it is straightforward to count the number of antennas that see a given cosmic ray shower event, once the distribution of antennas is fixed. The geometry considered is that of an hexagonal array on a horizontal ground located at sea level and the detection efficiency is studied as a function of the antenna spacing. Our detection criterion is that the electric field magnitude is above a threshold to be defined on at least three antennas belonging to the same elementary triangle. Following Ref. [START_REF] Falcke | [END_REF] the threshold is first fixed to the root mean square of the galactic noise below 100 MHz (see App. D): 2 E T = σ sky = 30 µV/m. This is a very crude model for the detection problem but it is sufficient to demonstrate some qualitative trends. The main drawback is the assumption that the full bandwidth, and especially the low frequency range 0 -20 MHz, might be used for radio detection at the location of the experiment. We come back to this question at the end of the section.

A look at Fig. 5 indicates that an antenna located at 1 km receives an electric field magnitude above this threshold for a vertical shower. For such vertical events and an antenna spacing d = 1.5 km, Fig. 8 depicts the area spanned by impact parameters of E p = 10 19 eV events seen by 3 antennas. The proportion of detected events is simply given by the ratio of the area of one filled region to that of a basis triangle. For 10 19 eV vertical showers and d = 1.5 km this is about 35%. The behavior of this ratio as a function of antenna spacing is given in Fig. 9, left panel, for 10 19 eV vertical showers. The efficiency τ is a function of the impact parameter at which the threshold is reached, b T , and the antenna spacing, d, only through the ratio of these quantities:

τ ≡ τ (d/b T ).
b T appears as a natural unit for d and is in addition the value of d at which τ departs from 1.

From this it is easy to see the effect of a change in the threshold level: for E T = 30 µV/m, b T = 1.15 km; doubling E T leads to (using E max ∝ b -4 ) b ′ T = 1.15/2 1/4 = 0.97 km. It is also easy to obtain the efficiency if geosynchrotron emission is assumed. Using the figures given in Sect. 2.2 we get b T = 2.2 km for E T = 30 µV/m. This is a large increase for the detection capability which is of course a direct consequence of the hierarchy between E v and E a discussed in Sect. 2.2.

The right panel in Fig. 9 displays efficiency integrated over arrival directions from vertical to 60 • -inclined showers with a weight sin θ cos θ, in order to incorporate changes both in solid angle and in the effective area of the antenna array. As already explained in Sect 3 (see also [9]) inclined showers are in general more effective at large impact parameters, making them easier to detect than vertical ones: for d = 1.5 km, the efficiency is now 90 %. This is illustrated in Fig. 10 (left panel) where the detection efficiency as a function of zenith angle in the north direction for an antenna spacing of 2 km and an energy of 10 19 eV is presented. The effect of the zenith angle is strong. Vertical showers are never detected whereas all showers with inclination bigger than 40 degrees are detected. Of course, in practice, this would be moderated for antennas with maximum directivity pointing towards the vertical direction.

In Fig. 10, the efficiency versus azimuth for a zenith angle of 30 degrees (same antenna

Outlook

In the present study we showed how extensive air shower radio electric fields can be evaluated at large impact parameter with an analytical formulation. Expressions were derived for two geomagnetic effects currently under investigations in the literature, thus helping pointing out aspects for which both approaches are at variance. The relationship between radio pulse shape and shower profile, and some considerations on the sizing of a giant array could be easily discussed.

The specific figures are certainly no more than illustration, not only because the pointlike approximation is too crude, but also because of their sensitivities on some shower and emission mechanism parameters. However, the approximation may be sufficient to describe tendencies and we think to make a point on some issues in the field.

As it has been understood so far in every approach to radio emission, the radio component of an air shower is driven by its time-development at large impact parameter. This is a simple consequence of propagation laws once it is assumed that the radio component is generated by the bulk of relativistic electrons and positrons in the shower. Let us collect two essential predictions that result from this fact.

The high-frequency cut in the spectrum of the shower radio-component decreases when the impact parameter increases. This prediction is not in any sense new as it was already at the heart of the discussion in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. Recent studies [START_REF] Huege | [END_REF]5] show specific realizations of this universal phenomenon. Only in the point-like model developed here, it is easy to isolate the origin of the phenomenon and because of its universal nature to emphasize the need to look for it. As a byproduct, we stressed that triggering at high frequency has the consequence of biasing radio detection towards small impact parameter and thus making the corresponding device blind to the phenomenon. We also emphasized that the interrelation between impact parameters and accessible frequency range is one of the main topic to be addressed when designing a large radio detection array.

The radio signal is sensitive to the beginning of the shower development, much more so than the fluorescence signal is whose sensitivity is directly proportional to N ee . Thus radio is complementary to the fluorescence technique for the purpose of measuring the shower development. Such a sensitivity may help to discriminate between various high-energy hadronic scenarios, hence providing a rich potential for the radio technique.

A Derivation of electric fields in the point-like approximation A.1 Point-Like expressions

For a point-like charge, ρ(t, x) = qδ (3) ( x -vt),  = ρ v a direct application of Maxwell equations in a medium of refractive index n leads for the scalar and vector potentials in Lorentz gauge to

V (t, x) = 1 4πε 0 n 2 q R|1 -n v • n/c| , A(t, x) = 1 4πε 0 c 2 q v R|1 -n v • n/c| ,
with n = R/R. For an accelerating charge, the derivation of the acceleration field is in textbooks. A positron and an electron at the same location and with the same speed have opposite accelerations and therefore produce the same electric field. Then, in the point-like approximation, N ee positrons and electrons generate 3 , where a is the positron acceleration.

E a = N ee e 4πε 0 c 2 n ∧ ( n -n v/c) ∧ a R|1 -n v • n/c|
For E v , the derivation from the potentials with a time varying charge q → N ee (t ′ ) × e leads to (see also [5])

E v = N ee e v T 4πε 0 nc n 2 v 2 /c 2 -v • n/c R 2 |1 -n v • n/c| 3 -Ṅee e v T 4πε 0 c 2 1 -n v • n/c R|1 -n v • n/c| 3 ,
where v T is the positron drift velocity.

A.2 Small angle approximation

A further simplification is motivated in Sect. 2.1, relevant in the region of small angles such that |1 -n v/c| ≪ ϑ ≪ 1.

Applying this simplification to the above electric field expressions makes use of the following set of rules

R → b/ϑ, 1 -n v • n/c → ϑ 2 /2, ϑ → 2 ct/b.
The numerator of E a is given by

n ∧ ( n -nR v/c) ∧ ˙ v ≈ ϑ 2 2 2( b • a T ) b -a T ≡ ϑ 2 2 a T ǫ,
where a T is the acceleration transverse to v, or equivalently to n since this is the same at small ϑ. These replacements (together with n v/c → 1) lead to Eqs. ( 2) and (3).

B Acceleration and drift velocity

In order to get quantitative values for the fields we need estimates for a T and v T which we collect from various sources in the literature. For a shower whose axis makes an angle α with the geomagnetic field B, the acceleration of an electron or a positron in the shower core is ecB sin α/(γm e ). Averaging this over γ (1/γ = 1/γ ) leads to an estimate for a T in Eq. ( 3) a T = ecB sin α γm e .

Taking γ = 60, which corresponds to the median value [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] and to a mono energetic approximation in Ref. [START_REF] Huege | [END_REF], thus enabling us to make contact with computation there, together with B sin α = 20 µT we get a T /c = 0.06 µs -1 .

After time t the electron or positron has a transverse velocity (1/γm e )ecB sin α t. Kahn and Lerche [8] use this with replacements t → τ = 1 µs (τ is the lifetime of the particle in the shower) and γ → γ. With γ = 60, this gives v T /c = 0.06. 3 A detailed approach to more properly estimate the drift velocity taking into account both energy loss and energy dependence of the lifetime τ is given in Ref. [5]. It leads to v T /c = 0.04, and this is the value which is used in the present paper for vertical showers.

Figure 1 :

 1 Figure 1: Geometry and notations: R = -→ QA and ϑ = ( v, -→ QA).

Figure 2 :

 2 Figure 2: Shape of N ee /R (θ = 45 • ) [thick line] and that of 1/(1 -nβ cos ϑ) (b = 1.5 km, n = 1, β = 1 [solid], n = 1, γ = 60 [long dashes], n = 1.0003, β = 1 [short dashes]).

Figure

  Figure 3: d dt ′ (t ′ N ee (t ′ ))/(t ′ N ee (t ′ )) in µs -1 for θ = 0 (right) and 45 • (left). Dashed curves show the respective behaviors of E a (t ′ ), rescaled so that their maxima = 0.1, to help focusing on the regions of interest.

Figure 4 :

 4 Figure 4: E a polarization map for a shower going through the map at point B. Acceleration of positrons is to the right but the polarization is not always opposite to it. (Polarization of E v would be to the left everywhere.)

  start gives an estimate for the signal start. In Allan's estimate, Z start = 2Z max and ∆ g ≈ b 2 /(4 Z max ). This gives the main scaling behavior of ∆ g . More precisely, defining a starting point as the instant at which the relevant function (f 2 or f 3 ) reaches 10% of its maximal value, ∆ g is 90% of b 2 /(4 Z max ) for vertical showers and 2/3 of b 2 /(4 Z max ) for 60 • -inclined showers.Following estimates of Ref.[START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] the timescale criterion for the point-like approximation reads∆ g ≫ ∆ r = b r Z , ∆ g ≫ ∆ l = l,where r and l are characteristic lateral and longitudinal displacements in the shower. For electrons and positrons of critical energy[START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] The criterion ∆ g ≫ ∆ r gives b ≫ 4 r. For a 10 19 eV shower, the maximum of f 2 or f 3 occurs around a 10 km height where p = 0.3 atm, leading to 4 r ≈ 500 m. The second criterion, ∆ g ≫ ∆ l , is equivalent to b ≫ √ 4 Z max l. Plugging f 2 maxima for the 10 19 eV case gives b ≫ 2 Z max l = 0.6, 0.7, 0.8, 1 km, respectively, for inclinations 0, 30, 45 et 60 • . The figures show that it is the second criterion that is the most relevant.

Figure 5 :Figure 6 :

 56 Figure 5: Transverse current electric fields at b = 1 km (component towards east) for showers coming from south at various zenith angles: θ = 0 • , 30 • , 45 • , 60 • .

Figure 7 :

 7 Figure 7: Sensitivity to shower rise: left plot shows two 10 19 eV vertical-shower profiles (Gaisser-Hillas and Greisen) as a function of altitude; the corresponding E v field pulses and spectra (at b = 1 km) are shown respectively on the right and bottom plots. The effect is similar for E a (not shown).

Figure 8 :

 8 Figure 8: Hexagonal array and area of detection.

Figure 9 :

 9 Figure 9: Detection efficiency vs antenna spacing for 10 19 eV showers assuming transverse current emission: vertical configuration (left) and integrated over inclination (right).

Figure 11 :

 11 Figure 11: Detection efficiency assuming transverse current emission as a function of energy for antenna spacing d = 2 km.

Table 1 :

 1 Z max in km: E p = 10 19 eV (left) and E p = 10 17 eV (right).

				3	θ max f 1 max f 2 max f 3
	0	2.5	7.5	6	0	4	9	8
	30	4.5	9.5	8	30	6.5	12	10
	45	8	13.5	11.5	45	10.5	16.5	14
	60	17	24.5	21	60	20.5	28.5	24.5

that drive the pulse behaviors. Values ACCEPTED MANUSCRIPT θ max f 1 max f 2 max f

This is not true for very inclined showers if b is kept in the km range. For this reason inclinations were limited to θ ≤ 60 • in the present study.

The assumed bandwidth is fixed by the radio experiment, i.e., not necessarily restricted to observations at large impact parameters for which the effective bandwidth is much more limited (see Sect.

3.1).

In Ref.[8], γ = 200 is quoted, which would lead to v T /c = 0.02.

Acknowledgments

This work was supported in part by grant ANR-NT05-2 42808 (CODALEMA) from the French Agence Nationale de la Recherche.

spacing and energy) is also shown. Fixing θ singles out the geomagnetic field effect, otherwise entangled with the zenith angle effect discussed above, since the change of efficiency in this case can only be attributed to the variation of the field strength with α(θ, ϕ). The efficiency falls down when approaching the geomagnetic field direction towards the south. Here the effect is large because the choice θ = 30 • is close to the magnetic lines at 27 • .

Obviously, all these results depend on shower energy. Fig. 11 shows the detection efficiency as a function of energy for an antenna spacing of 2 km, integrated over all arrival directions as in Fig. 9. The variation is not very strong as it takes three energy decades to reach 100 % efficiency.

The above detection condition requires very favorable conditions which may be approached only at exceptionally quiet location. Seemingly the site of the Pierre Auger Observatory in Malargue is of this kind, as the study of Ref. [14] shows. Even there, however, the atmospheric noise is likely to take over the galactic noise at low frequency [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. This is not included in the computation of σ sky . A dedicated study of the noise at the experimental site and as a function of time (day-night variation) would be mandatory to obtain a more realistic estimate of the background. More often, radio frequency interferences show strong variations with frequency. A detection criterion is thus best formulated in the frequency domain. In a rather quiet radio environment, a detection threshold of 1 µV/m/MHz above 20 MHz in a 1 µs time window can be achieved [10]. Such a detection condition leads to similar results with typically a factor 0.5 on the antenna spacing. The antenna spacing for a large array should be around 1 km rather than 2 km.

C Shower characteristics

For shower evolution we primarily used the Greisen parametrization [START_REF] Stanev | High energy cosmic rays[END_REF] N(X) = 0.31

with E C = 89 MeV and X 0 = 36.7 g/cm 2 . Throughout, X max was chosen as 770 g/cm 2 (respectively 700, 630) for E p = 10 19 eV (respectively 10 18 , 10 17 ) [5].

For comparison, the Gaisser-Hillas parametrization [START_REF] Gaisser | Cosmic rays and particle physics[END_REF] was also considered

In addition, the observation level was always chosen as that of sea-level and the atmosphere thickness as a function of altitude, for a shower coming from the direction of zenith angle θ, was taken as X(z, θ) = 1020 g/cm 2 cos θ exp(-z/(8.8 km)).

D Detection threshold

To be visible a shower radio signal should be above the contribution from the galactic noise, whose power spectral density (for one polarization) is given by [START_REF] Kraus | Antennas[END_REF] P ν = k B T sky (ν).

This can be related to the electric field ǫ ν to which an antenna with gain G (effective area A e = Gλ 2 /(4π), λ = c/ν) is exposed for a time ∆t

where Z 0 = 1/(ε 0 c) is the intrinsic impedance of free space. The variance of an electric field can be computed from its spectrum by

where E ν is the Fourier transform of E(t) restricted to the time window [t 0 , t 0 + ∆t]. E ν is defined for positive and negative frequencies, while ǫ ν is a positive-frequency quantity such that ǫ 2 ν = E 2 ν + E 2 -ν . For the sky noise, this gives


Gianfranco Angelino
email: gianfranco.angelino@polmi.it

Costante M Invernizzi
email: costante.invernizzi@unibs.it

Carbon dioxide power cycles using liquid natural gas as heat sink

Keywords: Carbon Dioxide, Liquefied Natural Gas, Power Cycles, Supercritical Thermodynamic Cycles

Liquefied natural gas (LNG) is recognized as a source of usable cryogenic exergy for power cycles. The performance of conventional cycles are calculated. A binary steam-Organic Rankine Cycle (ORC) at 550 • C has an efficiency of about 52 %, somewhat higher than that of a nitrogen Brayton cycle (50.7 % at 700 • C). Carbon diox-

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ing also a gas compressor, exhibit an improved thermodynamics by reducing the temperature difference within the regenerator, with the result of increasing the overall efficiency at values better than those of both binary and Brayton cycles. At 600 • C top temperature, for example, a compound cycle at 100 bar maximum pressure has an efficiency of 55.3 % (52.3 % for a binary steam-ORC cycle at 550 • C, 150 bar steam parameters; 46.5 % for the nitrogen cycle at 600 • C top temperature).

ide is recognized as an almost ideal medium for implementing single fluid condensation cycles. Its proven practical use both at low temperature (by the refrigeration industry) and at high temperature (by the nuclear reactor industry) makes it suitable for direct utilization without any extended preliminary research. A carbon dioxide cycle in its basic configuration featuring a pump, a regenerator, a heater and a condenser is much simpler than the binary steam-ORC cycle but has a lower efficiency (around 47 %). All condensing cycles (ORC, CO 2 ...) exhibit a limited capability of exploiting the whole cryogenic exergy of LNG in that they cannot heat the natural gas at temperatures above the condensation temperature. This drawback is fully overcome in nitrogen Brayton cycles which can heat LNG up to ambient temperature. Slightly modifying the basic CO 2 cycle so that it can partially use free thermal energy from sea water increases efficiency to 51 %. Multiple condensation cycles allow a better overall performance at the cost of a more complex layout. Compound CO 2 cycles, featur-

Introduction

All over the world natural gas transportation is increasingly relying on liquefaction and conveyance by specialized tankers. For structural reasons LNG is shipped basically at atmospheric pressure at a temperature of about -160

• C. At the receiving terminal a large amount of cryogenic cold is available which, in general, is lost since LNG is directly vaporized using sea water as heat source.

Sometimes, mainly in Japan, LNG physical exergy is used for technological (air separation, CO 2 liquefaction, ...) or for power uses. In the latter case a propane or a refrigerant Rankine cycle using sea water as heat source and LNG as heat sink generates electricity without any fuel consumption.

All existing power plants of this kind (around fifteen) employ a small fraction of the overall LNG cooling capability available at the site (net outputs limited to a maximum of 9400 kW). Often the net power obtained from the thermodynamic cycle is supplemented by an additional contribution due to the direct expansion of methane which is heated at a pressure greater than that requested by the final user and eventually expanded [START_REF] Otsuka | Evolution of an LNG Terminal: Senboku Terminal of Osaka Gas[END_REF][START_REF]Anononymus Possible use of the cold energy of LNG at the Taranto regasification terminal[END_REF].

Envisaging the use of a high temperature conversion cycle (for example a closed nitrogen Brayton cycle, ref. [START_REF] Krey | Utilization of the cold by LNG vaporization with closed-cycle gas turbine[END_REF]) the amount of power which can be generated increases greatly. As a first approximation a typical LNG import terminal of 8 Gm 3 /year gaseous methane capacity (which could feed at least 5000 MWe combined cycle plant) has the potential of generating about 150

MWe in a high efficiency cycle using the LNG as heat sink.

It is not surprising that most existing energy recovery plants employ an organic fluid Rankine cycle since, besides direct expansion, this is a practical, straightforward way of exploiting the LNG thermal exergy.

Owing to the cryogenic temperatures envisaged the direct use of steam is banned. With reference to condensing cryogenic cycles both hydrocarbons and refrigerants exhibit a limited thermal stability besides some technical and environmental drawback. Carbon dioxide, on the contrary, while lending itself to the implementation of a condensing cycles, is stable up to very high temperatures, is non flammable, non toxic, non corrosive, inexpensive and was used extensively in the past, for decades, by the power industry in the Advanced Gas Cooled Reactors (AGR) at a top temperature of 650 • C. In principle it could represent an ideal option for low temperature uses as water it is at ambient temperatures. A limit of CO 2 as a working fluid is its triple point at -57 • C which, in general, prevents its use at even lower temperatures.

Carbon dioxide power cycles were extensively investigated in the past for applications where the final cooling of the working medium in a condenser takes place just below the critical temperature of the substance (31 • C, ref. [START_REF] Angelino | Perspectives for the liquid phase compression gas turbine[END_REF][START_REF] Feher | The supercritical thermodynamic power cycle[END_REF][START_REF] Angelino | Method for obtaining mechanical energy from a thermal gas cycle with liquid phase compression[END_REF][START_REF] Angelino | Carbon dioxide condensation cycles for power production[END_REF]). In these conditions waste heat rejection takes place near the top of the saturation dome and the power cycle extends almost completely in the gas region. When LNG is used as a cooling medium condensation can be performed at a much lower temperature and pressure thus enhancing the condensing nature of the power cycle and dramatically reducing optimum operating pressures.

In recent years there was a revival in CO 2 cycles studies mainly connected with their perpective use in medium temperature advanced nuclear reactors ([8, 9, 10]). A detailed analysis shows that supercritical CO 2 cycles could be more efficient than steam or closed nitrogen cycles at temperatures above 600 • C and that turbomachinery design exhibits definitive advantages over both steam and nitrogen. As a consequence of the rising interest in CO 2 as a working medium some hardware development was performed relating to compressors, heat transfer equipment and plant layouts ( [START_REF] Wang | Compressor design for the supercritical CO 2 Brayton cycle[END_REF][START_REF] Nikitin | Printed circuit heat exchanger thermalhydraulic performance in supercritical CO 2 experimental loop[END_REF]).

Having made clear that carbon dioxide features a number of qualities to be proposed as working fluid at cryogenic temperatures the aim of this study is the evaluation in detail of its merits in the exploitation of the LNG cooling capability.

With reference to typical capacity of LNG import terminals a power plant output in the range 50-80 MW was considered in order to estimate the turbomachinery and other components performance both in conventional and in CO 2 conversion systems.

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For the LNG to be heated two situations of practical interest are considered: vaporization at 70 bar i.e. at the supercritical pressure needed by long distance conveyance pipelines and vaporization at 30 bar as requested by power station supply lines.

Conventional power plants

In order to obtain a consistent picture of the merits of conventional and of new solutions a steam/ORC and a closed nitrogen gas cycle were firstly examined. Both these systems can be considered as "conventional" (even if none is in operation in connection with an LNG power plant), in that the related hardware was widely implemented in a number of power stations built in the past.

Steam-ORC system

With reference to the steam cycle in principle it would have been possible to assume a standard vacuum condensing cycle supplemented by a bottoming ORC unit. However since the low pressure turbine is the most expensive plant component it seemed more reasonable to "cut" the steam cycle at a pressure slightly above one atmosphere and to commit to an ORC bottoming cycle the task of exploiting the full thermal potential from about 100

• C to the cryogenic LNG temperatures. In this way reheating of partially expanded steam becomes unnecessary since the steam quality at the end of the expansion is sufficiently high (above 0.92 at 150 bar and 550

Brayton cycle system

The successfull more than decennial experience of closed cycle gas turbines ( [START_REF]Various Authors Closed cycle gas turbines for all fuels: coal[END_REF][START_REF] Frutschi | Closed-Cycle gas turbines. Operating experience and future potential[END_REF]) represents a sound basis for the extension of this concept to cryogenic temperatures. A detailed solution for one such plant serving an LNG terminal is reported in [START_REF] Krey | Utilization of the cold by LNG vaporization with closed-cycle gas turbine[END_REF]. Nitrogen with a very small oxygen addition is believed to be the best working fluid. Top temperatures are limited to 700-800

• C owing to material problems in the primary heater. The cooling characteristics of LNG which must be heated from -160 • C to ambient temperature suggest the adoption of a simple, non-intercooled, regenerative cycle. The overall maximum to minimum temperature ratio (τ = T max /T min ) which was around 3.4 in historical plants [START_REF]Various Authors Closed cycle gas turbines for all fuels: coal[END_REF] in the peculiar LNG application rises to All the assumptions made to evaluate the performance of closed gas (nitrogen) cycles are given in Table 1.

System efficiency and parameter SP P for super-and sub-critical LNG are given in Figure 2a and 2b respectively at various compressor pressure ratio and turbine inlet temperatures. For supercritical LNG at 700 of the supercritical CO 2 (average heat capacity of 2.64 kJ/kg K) and of the low pressure gas after the turbine expansion (average heat capacity of 0.96 kJ/kg K). This substantially irreversible heat transfer represents the main efficiency limit of the simple CO 2 conversion cycle.

Assuming the components characteristics reported in Table 2 (which are consistent with those stipulated for the steam and for the Brayton cycles), In the case of CO 2 , however all the compression is performed by a pump and the total power involved in both compression and expansion is only slightly larger than the plant net output (1.12 power ratio against 2.22 for Brayton cycle).

As it happens in all condensing cycles a compromize must be found between cycle efficiency (which request the minimum achievable condensation temperature) and the LNG exergy utilization which requires a high LNG final temperature which can only be obtained with a high condensation temperature and a reduced cycle efficiency.

Owing to these conflicting requirements the specific power output SP P

ACCEPTED MANUSCRIPT

reaches a maximum at a definite condensation temperature while cycle efficiency increase regularly at decreasing t cond . Such a behaviour is illustrated in Figure 5. With reference to a LNG mass rate of 1 k/s at 150 bar top pressure, for example, an optimum SP P value of 0.43 MWe is achieved at

t cond = -10 • C
with a cycle efficiency of 38 % (bold lines).

Two condensation levels

The situation outlined above can be improved with a cycle featuring two condensation levels (see Figure 6) in which the LNG heating is performed in two steps with a reduction of the average temperature difference between condensing CO 2 and LNG (Figures efficiency reaches 43 %. Obviously multiple condensation cycles could also be considered with a further benefit in system performance.

One condensation level with a partial input of free energy extracted from sea water

As it is clear from Figure 3 the pre-heating of CO 2 after condensation starts at a very low temperature, for example around -50 • C. Such preheating could be performed partially using sea water heat at about ambient temperature. Obviously in this case the regenerative pre-heating is shifted toward higher temperatures as shown in Figure 7. As a consequence the whole thermodynamic cycle is modified with a substantial gain in efficiency † while parameter SP P turns out to be practically constant (Figure 5, light lines).For example at t cond = -50 • C, p max = 150 bar t max = 600 • C cycle efficiency increases from 46.3 % to 51.0 %. Owing to the simplicity of the technical intervention involved the partial use of the free heat from sea water seems, in general, advisable. † Free energy transferred from sea water is not considered as part of the cycle heat consumption.

High pressure compound cycles

With reference to An advantage of the high pressure compound cycle with respect to the nitrogen cycle is that the power input for the mechanical compressor is comparatively small since it handles only about half of the total mass rate. At a top temperature of 700 • C a remarkable efficiency of 57.2 % is achieved (50.7 % for the nitrogen cycle).

Low pressure compound cycles

In the high pressure compound cycle both pump and compressor pressurize the working fluid to the same pressure which, aiming to a good overall efficiency, must be comparatively large (for example 150 bar). Since compressor losses are more severe than pump losses such a high pressure generates within the compressor a larger than desiderable fluid-dynamic penalty which could be reduced if compressor discharge pressure is substantially lowered. For the pump, on the contrary, discharge pressure can be kept high. In other words the gas cycle and the condensation cycle can be partially de-coupled generating a new cycle configuration, see ref. [START_REF] Angelino | The Use of Liquid Natural Gas as Heat Sink for Power Cycles[END_REF], which is illustrated in the T-S plane in

Conclusions

The thermodynamic analysis reported in the previous sections, with reference to Figure 12 too, suggests the following conclusions:

1. conventional power cycles, both Rankine and Brayton, can be adapted to the exploitation of the LNG thermal exergy with conversion efficiencies in the range 50 to 54 %. Rankine cycles must have a binary (steam-organic fluid) configuration. Nitrogen Brayton cycle, although less efficient than Rankine cycle at a fixed top temperature, has the

ACCEPTED MANUSCRIPT

capability of heating the LNG up to ambient temperature.

2. Carbon dioxide exhibits several merits to be considered as a working fluid at cryogenic temperatures: it is inert, stable, non toxic, inexpensive, readily available and was extensively used in the past both at low temperatures (by the refrigeration industry) and at high temperatures (mainly in the AGR nuclear reactors). Furthermore carbon dioxide cycles in the condensing mode, have the potential of covering, with a single fluid, the whole available temperature range, for example from -50 to 600 • C. The recent revival of the studies related to fast nuclear reactors using CO 2 as working fluid for the power cycle produced a wide literature covering thermodynamics, material science and harware development.

3. Simple CO 2 condensing cycles, featuring a gas turbine, a regenerator, a condenser and a pump, offer a moderately good efficiency (for example, 46.3 % at 600 • C, 150 bar) combined with a very simple layout. The simple cycle can be adapted to partially use free heat extracted from sea water with a significant efficiency gain (from 46.3 % to 51.0 %).

More complex compound cycles, which require also a compressor, allow much higher efficiencies at moderate top pressures. The low pressure compound cycle for example at a maximum pressure of 100 bar exhibit

For the bottoming cycle

  n-butane (n -C 4 H 10 ) was selected as the working fluid since its critical temperature of 152.0 • C approximately optimizes the cycle performance. A one stage extraction was assumed for the regeneration process. As it is typical when dealing with organics at the end of the expansion the vapour is slightly superheated which favours a good turbine efficiency. In principle a hydrocarbon with a more complex molecule such as n-pentane (n -C 5 H 12 ) could be used. In this case however, at a condensation temperature of -50 • C vapour pressure at the turbine exhaust would be impractically low (0.013 bar against 0.096 bar for n-butane). For a steam/n-butane binary plant configuration (see Figure 1.a) the computed overall efficiency, obtained by the use of the AspenPlus program, is 52.3 %. A more straightforward solution consisting of a standard, re-heat steam cycle condensing at 30 • C followed by a n-butane cycle vaporizing at 20 • C and condensing at -50 • C according to the same computing program would exhibit a slightly better overall efficiency (53.7 %) at the cost of a complex vacuum section and of a second steam superheating (42.3 % steam, 19.7 % n-butane cycle efficiency).

about 7 *

 7 with a strong reduction in compressor work and a substantial gain in efficiency (from historical values of about 30 % to well above 50 %). The ability of closed gas cycle to burn various coal grades and other low quality fuels represents a distinctive feature in comparison with open cycle gas turbines. With reference to a supercritical LNG vaporization the heat rejection mode of a closed gas cycle matches almost perfectly with the LNG heat absorption law. This character is almost unique and makes the gas cycle a privileged conversion system.As it is clear from Figure1.a and 1.b, in contrast with the ORC cycle behaviour, the closed gas cycle is capable of heating the LNG up to the ambient temperature thus exploiting the entire cooling capability of the liquefied natural gas. Condensing cycles can reach a similar target only with a complex configuration featuring a plurality of condensation levels. As a consequence a typical Specific Power Performance Parameter (SP P ) defined as the power output to the LNG mass rate ratio, (MW/(kg/s)) reaches for the closed gas cycles unrivalled values.

7 .

 7 Making reference to the actual CO 2 physical properties the simplest configuration for a power cycle extended to cryogenic temperatures turns out to be the one shown in Figure 3. The working fluid is condensed at -50 • C from point 5 to point 1, pressurized by pump to point 2, regeneratively pre-heated to point 6, super-heated to point 3, expanded in a turbine to point 4 and regeneratively pre-cooled to point 5 in a counterflow heat exchanger. With reference to Figure 3.c even if the minimum temperature difference within the regenerator is small (for example 10 • C) the computed hot-end temperature difference is rather large (212 • C) owing the strong difference in the heat capacity of the high and of the low pressure fluid i.e.

Figure 4

 4 Figure 4 reports the overall cycle efficiency as a function of turbine inlet

6 .b and 6 .

 66 c). The definite gain in SP P parameter is clear from Figure 5, dotted lines. For a fixed lower condensation temperature of -50 • C at an upper condensation temperature of -10 • C, at 150 bar, SP P rises to 0.53 MWe/(kg/s) (from 0.43) while cycle

Figure 3 . 5 %

 35 c, as previously observed, the large temperature difference in the simple cycle regeneration process (t 4t 6 = 212 • C in the diagram reported) is responsible of a severe entropy production and efficiency loss. To get around this drawback in the regeneration process heat capacities of the cold and of the hot medium can be balanced reducing the mass rate of the condensed gas by compressing, in a mechanical device (without condensation), a fraction of the pre-cooled gas. Figure 8 reports the new cycle configuration and component layout as originally suggested in [7]. As shown in Figure 8.b heat transfer in the cold regenerator becomes much less irreversible (average temperature difference 17 • C) with an important gain both in efficiency and in the SP P parameter. As shown in Figure 9 at a top temperature of 600 • C, for example, maximum efficiency is 53.3 % (47.for the simple cycle, 46.5 % for the nitrogen Brayton cycle) and maximum SP P is 0.69 MWe/(kg/s) (0.50 for the simple cycle, 0.72 for the nitrogen Brayton cycle). It must be observed, however, that for CO 2 in opposition to what happens in the nitrogen cycle, top efficiency and top SP P are reached at different condensation temperatures (-50 • C for the best efficiency, 0 • C for the best SP P ).

Figure 10 .

 10 As in the cycle of Figure 8 in the low temperature regenerator the low pressure CO 2 of both the gas and the condensation cycle pre-heat jointly the condensed liquid from point 1 to point 15, while cooled from point 20 to point 23. Two independent high temperature regenerators preheat the high pressure gas from point 15 to point 16 in the condensation cycle and from point 2 to point 4 in the gas cycle. As a consequence of this de-coupling optimum top pressures fall to about 100 bar for the condensation cycle and to about 40 bar for the gas cycle which implies obvious structural benefits and a simplified design for the compressor and the turbine of the gas cycle. Figure 11 reports cycle efficiency as a function of compressor discharge pressure. At 600 • C top temperature optimum efficiency of the low pressure compound cycle is 55.3 % at 100 bar pump, 40 bar compressor discharge pressure which compares with 53.3 % of the high pressure compound cycle. At 700 • C top temperature efficiency reaches a remarkable 59.0 %.

Figure 1

 1 Figure 1 Binary steam-butane cycle (Figure 1.a) and nitrogen closed gas

Figure 2

 2 Figure 2 Efficiency and specific power output of Brayton nitrogen cycles

Figure 3

 3 Figure 3 Carbon dioxide simple cycle at -50 • C condensation temperature.

Figure 4

 4 Figure 4 Carbon dioxide cycle efficiency at 600 • C (Figure 4.a) and 700 • C

Figure 5

 5 Figure 5 Carbon dioxide simple cycle efficiency and specific power output

Figure 6

 6 Figure 6 Temperature-entropy diagram and heat rejection mode of a two

Figure 7

 7 Figure 7 Carbon dioxide simple cycle with partial use of sea water heat:

Figure 8

 8 Figure 8 High pressure compound cycle with improved low temperature

Figure 9

 9 Figure 9 Efficiency and specific power output for high pressure compound

Figure 10

 10 Figure 10 Low pressure compound cycle: T-S diagram and component lay-

Figure 11

 11 Figure 11Comparison of cycle efficiency of low-and high-pressure com-

Figure 12

 12 Figure 12Cycle efficiency for the various options considered as a function

Figure 1

 1 Figure 1.a

  Figure 3

Table 1 :

 1 Characteristics of steam, n-butane and nitrogen cycles.

	a) steam cycle
	type of cycle	superheated with four regenerative extractions
	turbine inlet temperature	550 • C
	turbine inlet pressure	150 bar
	condensation temperature	111.4 • C
	condensation pressure	1.5 bar
	turbine adiabatic efficiency	0.85
	pump hydraulic efficiency	0.70
	mechanical efficiency	0.985
	alternator efficiency	0.98
	overall fractional pressure drop	20 %
	computed cycle efficiency	0.317
	b) n-butane bottoming cycle
	type of cycle	saturated with one regenerative extraction
	turbine inlet temperature	103 • C
	turbine inlet pressure	16 bar
	condensation temperature	-50 • C
	condensation pressure	0.096 bar
	turbine adiabatic efficiency	0.90
	pump hydraulic efficiency	0.70
	mechanical efficiency	0.985
	alternator efficiency	0.98
	overall fractional pressure drop	20 %
	computed cycle efficiency	0.301
	c) binary cycle
	overall efficiency	0.523
	specific power output, SP P	0.44 MW/(kg/s) of LNG
	d) nitrogen closed gas cycle
	turbine inlet temperature	600 to 800 • C
	compression inlet temperature	-120 • C
	turbine adiabatic efficiency	0.90
	compressor adiabatic efficiency	0.85
	regenerator effectiveness	0.90
	mechanical efficiency	0.985
	alternator efficiency	0.98
	overall fractional pressure drop	13 %

Table 2 :

 2 Assumed component efficiencies for CO 2 cycles.

	turbine inlet temperature

* In the case of an open combined cycle with an LNG heat sink τ exceeds 11.0.

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an efficiency of 55.3 % at 600 • C and of 59.0 % at 700 • C 4. With respect to a binary steam-ORC solution, CO 2 cycles are characterized by a much simpler layout, higher efficiencies and smaller and cheaper turbomachinery. With respect to a nitrogen Brayton cycle at a given top temperature, CO 2 cycles exhibit a better fuel economy but with the drawback of not being able to heat the LNG up to ambient temperature.