H Singh

R De

C Giordano

C A T Suazo

EVALUATING THE ROLE OF GEOMETRICAL FEATURES OF A TAYLOR VORTEX BIOREACTOR TO CULTURE ANIMAL CELLS BY USING COMPUTATIONAL FLUID DYNAMICS

RESUMO -Taylor bioreactor is fast becoming the next bioreactor to culture animal cells due to milder shear and homogenous flow structures through-out the bioreactor in comparison to the traditional stirred tank. However, there is not much information in the literature about how geometrical aspects of the Taylor bioreactor may affect the culture of animal cells which could result in poor efficiency. As a result, six different geometrical conditions of Taylor bioreactor are presented, related to off-bottom clearance and the external cylinders' bottom shape, after conducting a detailed grid and time-step independence study and validating with experimental results. Simulations were conducted at Reynolds number of 12077 and Taylor number of 5331 with a radius ratio of 0.84 and aspect ratio of 19.8. Through-out the bulk zone equivalent of 80 % of the gap-width, the mean velocity was found to be around 50 % of the maximum with similar gradients which is encouraging for cultivating of animal cells in Taylor bioreactor in comparison with spinner flask and stirred tank bioreactors where the gradients are much less uniform. Curved bottom surfaces of the external cylinder and higher off-bottom clearance areas adversely affect the flow structures creating dead-zone areas in the off-bottom clearance area.

INTRODUCTION

The interest in culturing animal cells has increased significantly in the last 20 years due to the need of large-scale production of monoclonal antibodies, hormones, vaccines, recombinant proteins and, more recently, the production of specialized cells for applications in tissue engineering and cell therapy [START_REF] Agarwal | Advent and maturation of Regenerative Medicine[END_REF][START_REF] Heathman | The translation of cell-based therapies: clinical landscape and manufacturing challenges[END_REF][START_REF] Mason | The global cell therapy industry continues to rise during the second and third quarters of 2012[END_REF] . Accordingly, a wide variety of cell lines have been cultivated in the reliable, well known and easy to scale-up stirred tank bioreactor with a design adapted from the fermenters used for microbial cultures (Stanbury et al., 1995).

In order to cultivate the animal cells in stirred tank bioreactor (Stanbury et al., 1995), some fine adjustments in the geometrical design of the fermenters were required to reduce the damage of cells due to shear stress, heat and contamination. In the design one of the most important but controversial item is the bottom geometry of the tank, which as per these authors should be hemispherical or concave to ensure better mixing at lower impeller rotational speed. On the contrary, Charles and Wilson (1994) stated that this requirement is not acceptable since there is no firm evidence to support this belief in addition to the fact that several stirred tank bioreactors with flat bottom have worked well in commercial applications and with the benefit of 50% lower cost of the tank. Reuveny (1990) and [START_REF] Smith | Large scale animal cell culture[END_REF] advocate the use of concave bottom, considering the usage of flat bottom to be inappropriate because of formation of stagnant regions in the bottom corners which can cause particle accumulation. In the case of animal cell culture, sedimentation problems can hinder the operation of the bioreactor during the cell adhesion process and during the actual cultivation by forming aggregates because of the large size of the particles (cells, cells clusters and microcarriers) (Ibrahim and Niewnow, 2004).

A study conducted by [START_REF] Chaudacek | Solids suspension behaviour in profiled bottom and flat bottom mixing tank[END_REF] showed that the accumulation of high density solids, such as silica, in flat bottom tanks can be eliminated by using axial impellers and implementing modifications in the geometry of the tank. [START_REF] Nienow | Suspension of microcarriers for cell culture with axial flow impellers[END_REF] applied this reasoning to low density solids, such as microcarriers, and validated the geometry modification principle using rounded corners in the bottom of the tank and a conical indentation in the region under the impeller. Although, stirred tank is the first bioreactor type for culturing animal cells from laboratory scale to industrial scale, it was not until the study of [START_REF] Nienow | Suspension of microcarriers for cell culture with axial flow impellers[END_REF] that there was a better definition of technical criteria demonstrating an advantage of modified bottom with respect to the flat bottom for bioprocesses.

As a result of such studies, in laboratory scale stirred tank bioreactors to culture animal cells, in general practice, the outer cylinder usually has a curved geometry on the bottom. The three well-known industrial suppliers of the stirred tank laboratory scale bioreactors, New Brunsswick, Infors HT and Applikon biotechnology, have curved (dished outwards) geometry of the outer cylinder. However, the demand of newer and more shear sensitive cell cultures require newer bioreactor types that generate milder hydrodynamic forces in comparison to the stirred tank [START_REF] Jain | Upstream process in antibody production: Evaluation of critical parameters[END_REF]. The Taylor bioreactor due to lower and uniform distribution of shear is fast becoming a strong candidate to replace the traditional stirred tank in the cultivation of animal cells by presenting a lower and more uniform shear (Curran and Black, 2004, 2005[START_REF] Gong | Age-related responses of suspension cultured taxus cuspidata to hydrodynamic shear stress[END_REF][START_REF] Haut | Hydrodynamics and mass transfer in a Couette Taylor bioreactor for the culture of animal cells[END_REF], O´Connor et al., 2002[START_REF] Santiago | Performance of a vortex flow bioreactor for cultivation of CHO-k1 cells on microcarriers[END_REF][START_REF] Tanzeglock | A Novel Lobed Taylor-Couette Bioreactor for the Cultivation of Shear Sensitive Cells and Tissues[END_REF][START_REF] German | Study of cell seeding on porous poly (d,l-lactic-co-glycolic acid) sponge and growth in a Couette Taylor bioreactor[END_REF].

However, in the case of Taylor bioreactor, there is empirical evidence of accumulation of particles at the bottom of the bioreactor which hinders the process of cell adhesion and proliferation. Considering the lack of information for bioreactors using the Taylor vortex principle to culture animal cells with different geometrical conditions, carrying out studies in this direction becomes extremely important and necessary since the geometry modifications can greatly influence the bioprocess efficiency.

Consequently, the shape of the outer cylinder and the height at which the inner cylinder is placed above the outer cylinder, or off-bottom clearance -obc, are considered as parameters which can lead to stagnant zones in a Taylor bioreactor. For example, if the inner cylinder is placed too high, there could be stagnant zones in the bottom of the tank because with only inner rotating cylinder tangential velocity has the biggest impact on the flow structures, thus implying that in the off-bottom clearance area there may not be sufficient amount of mixing due to weak axial flow.

However, as far as the authors are aware, the off-bottom clearance and shapes of the outer cylinders' base have not been investigated for Taylor bioreactor, but are of significant practical importance especially for culturing of anchorage dependent cells on microcarriers. Therefore, the main purpose of this article is to elucidate a favorable geometry for the particle suspension in a Taylor bioreactor at energy dissipation rates that are lower to those considered sub-lethal for the cells by providing an analysis of the flow structures and identifying the locations where the stagnation may occur using a computational fluid dynamics (CFD) validated model.

Best practice numerical methods are employed, and due attention is paid towards grid and time-step independence mesh. The shear-stress transport with curvature correction (SST-CC) turbulence model is employed in this study considering its more accurate prediction of the magnitude and location of the maxima of the turbulence kinetic energy and its dissipation in a stirred vessel [START_REF] Singh | An assessment of different turbulence models for predicting flow in a baffled tank stirred with a Rushton turbine[END_REF] in comparison to the k-ε, shear-stress transport (SST), Reynolds stress model (RSM) and scale adaptive simulation shear-stress transport (SAS-SST) turbulence models. The numerical model will be validated by comparison with the PIV & numerical study presented by [START_REF] Coufort | Analyse experimentale de la floculation en reacteur de Taylor-Couette: Influence de l'hydrodynamique sur les phenomenes d'agglomeration et de ruptura[END_REF] and [START_REF] Coufort | Flocculation related to local hydrodynamics in a taylorcouette reactor 10 and in a jar[END_REF] on the grounds of similar radius ratio of 0.87 and 0.84 (this study), consideration of only inner cylinder rotation and similar Reynolds number of 10995 and 12000 (this study).

Computational Model

Description of case set-up: The inner, r i , and outer, r o , cylinder radius of 54.5 and 65 mm, respectively, and a height, h, of 210 mm was used in this study. This leads to a gap width, b = r o -r i , radius ratio, η rr = r i /r o , and aspect ratio, Γ, of 10.6, 0.84 and 19.8, respectively. The angular velocity, ω, of 200 rpm or 20.94 rad/s was tested in this study with the Reynolds number, Re = r i ω(r o -r i )/ν, and Taylor number, Ta = r i 1/2 ω(r o -r i ) 3/2 /ν, of 12077 and 5331, respectively. Water at 20 ºC was used as the working fluid.

Three different bottom shapes of the outer cylinder were tested: flat, dished and dished inwards, as shown in Figure 1. The geometries flat bottom and dished bottom are commonly used shapes in laboratory scale bioreactors; while, dished inwards shape was tested based on the idea that with this configuration type accumulation of animal cells in the middle of the bottom of the bioreactor could be reduced. In terms of the off-bottom clearances, four different values were tested, 0, 5, 10 and 15 mm, with only the flat bottom geometry of the outer cylinder. Both the dished outwards and inwards geometries of the outer cylinder had an offbottom clearance of 15 mm. In total, six different geometries were tested: four for flatbottom at obc = 0, 5, 10 and 15 mm, and one each for dished and dished inwards at 15 mm obc.

The geometry for each of these six test cases was divided into multiple sections: three for flat bottom geometries and four for the other two cases. The idea behind this multiple geometry was to accommodate sliding-mesh technique and implement sweep mesh on the inner and outer surfaces. Because of the impossibility of implementing the sweep mesh in the complete off-bottom clearance area, another small section was created in the offbottom clearance area. Moreover, in the two geometries with curved surfaces on the bottom of the outer cylinder, another section was created covering the curved surfaces again in view of the implementation of the sweep mesh in the gap-width area. The top of the bioreactor is considered as symmetry because when working with animal cells the top is a free surface. Furthermore, due to the symmetrical flow conditions and to reduce the computational effort, only 25 % of the geometry was modelled using cyclic symmetry. two-equation turbulence models. This curvature correction clearly improved the predictions of the SST turbulence model in a tank stirred with Rushton turbine, as shown by [START_REF] Singh | An assessment of different turbulence models for predicting flow in a baffled tank stirred with a Rushton turbine[END_REF]. In addition, [START_REF] Poncet | Numerical modeling of fluid flow and heat transfer in a narrow Taylor-Couette poiseuille system[END_REF] stated that in the core region the turbulence is almost isotropic, thereby confirming the usage of two-equation turbulence model which assumes isotropic turbulence.

Although, these simulations were carried out with a geometrical mesh having a radial mesh value of 80 µm or in other words very close to the Kolmogorov scale of turbulence, the DNS or large eddy simulation (LES) models were not tested due to limited computational resources and time. A DNS or LES model would have required simulating a full-scale geometry instead of one-fourth geometry being used in this case, thereby increasing the computational effort by four times. In addition, it is known that DNS and LES models requires collection of largeamount of simulated time-steps to achieve statistical averaging of data [START_REF] Hartmann | Macroinstability uncovered in a Rushton turbine stirred tank by means of LES[END_REF]. Considering the number of geometries tested, it became impractical to use either DNS or LES because of this limitation of time and computational resources.

Computational mesh and time-step: In a Taylor bioreactor, rotating inner cylinder generates, mainly, tangential-velocity which is spread through the radial direction in the gapwidth. As a result the mesh was refined mainly in the radial direction and the final selected mesh had 96 nodes in the radial gap-width area, 432 nodes in the radial off-bottom clearance area and 2.4 million nodes in total. There were 210 and 102 nodes in the axial and tangential directions, respectively. The timestep of 1º rotation was selected after its comparison with 0.5º rotation showed little improvement in accuracy.

Computational aspects: The high resolution and second order backward Euler schemes were used to model the advection terms and transient terms, respectively, for all the equations The MRF technique was employed to study the transient nature of the flow field. The Convergence criterion of 10 -5 , which was achieved within a maximum of 5 iterations per time-step, for RMS scaled residuals was employed for the continuity, momentum and turbulence quantities. Double precision arithmetic was applied for all of the simulations. Once a pseudo steady state was achieved, the transient data was collected for only one more revolution since the acquired data would have been a repeat for each further revolution. It took around 7 revolutions to reach the pseudo steady state for each tested grid. All of these simulations were performed on a desktop with eight-core 3.2 GHz AMD Fx-8120 processor and 32 GB of memory RAM. The simulations were run using hpi based parallel processing with 4 cores.

Validation of the Computational Model

As mentioned earlier, the numerical results are validated by comparison with the PIV velocity results of the [START_REF] Coufort | Analyse experimentale de la floculation en reacteur de Taylor-Couette: Influence de l'hydrodynamique sur les phenomenes d'agglomeration et de ruptura[END_REF] on the grounds of similar radius ratio of 0.87 and 0.84 (this study) and the consideration of only inner cylinder rotation. It is important to note that the velocity flow profile data presented in their study was for 70 rpm and Re= 10995 in comparison to the 200 rpm and Re=12000 in this study in order to compare similar flow conditions.

Because of having similar geometrical and flow configurations, qualitative and quantitative comparisons should be considered valid. Figure 2 shows that both qualitatively and quantitatively the comparison of the tangential velocity flow structure is well depicted by the SST-CC turbulence model in this study. The SST-CC turbulence model not only predicts well the velocity flow structure and its magnitude in the boundary layer but also away from the walls in the centre of the gap-width. (2004).

Further validation is done by estimating torque numerically and empirically using the correlation of [START_REF] Wendt | Turbulente strmungen zwischen zwei rotierenden konaxialen zylindern [Turbulent currents between two rotating coaxial cylinders[END_REF] to compare the Power number obtained through these two methods. The Power number, N P = P b/(ρVω 3 r i 3 ), used by [START_REF] Coufort | Analyse experimentale de la floculation en reacteur de Taylor-Couette: Influence de l'hydrodynamique sur les phenomenes d'agglomeration et de ruptura[END_REF] and [START_REF] Douaire | A unique phenotypic modification of lactococcus lactis cultivated in a coquette bioreactor[END_REF] was employed for its caluculation, where P represents power, b represents gap-width, ρ represents density, V represents volume of the bioreactor, ω represents the rotational speed and r i represents inner cylinder radius. The Power number was found to be of value 1.4 × 10 -3 . For a Reynolds number of approximately 11000, similar to this study, [START_REF] Coufort | Analyse experimentale de la floculation en reacteur de Taylor-Couette: Influence de l'hydrodynamique sur les phenomenes d'agglomeration et de ruptura[END_REF] also observed a Power number of 1.4 × 10 -3 .

Results and Discussion

Outer cylinder´s bottom shape: Figure 3 presents vector profiles of the mean velocity for the different shapes of the outer cylinders' base, and Figure 4 shows the mean velocity profiles axially at R b = 0.009 and radially at 1mm below the inner cylinder, to demonstrate the impact of the variation in the shape of the bottom of the outer cylinder and off-bottom clearance area, obc. It can be seen that the dished and dished inward surfaces impact significantly the secondary vortex structure of the bioreactor towards the bottom. The Taylor-vortex pair which is nearer to the bottom of the bioreactor for Flat surfaces is pushed up for the dished surfaces. The location of first pair of Taylor-Vortex for these three different bottom surfaces is around 0.08, 0.15 and 0.23 Z L for flat, dished outward and dished inward surfaces types, respectively. For both dished surfaces before the formation of the Taylor vortex pair, a single vortex is formed and which is much longer for the dished inwards surface in comparison to the dished outwards, as can be seen in Figures 3a,3b. The axial profiles of velocity (Figure 4b) further confirms the formation of this single vortex and that the profiles for these three different shapes join up around Z L = 0.3 from bottom, in other words, the flow structure attains similar magnitude in the rest of the axial length. It can be seen that the dished surfaces also impacts significantly the magnitude of the velocity towards the bottom of the bioreactor. The velocity is below 20 % of the tip velocity for R b < 0.4 for dished surfaces in comparison to R b < 0.13 for the flat surface, as shown in the radial velocity flow profile located at 1 mm below the inner cylinder (Figure 4a). Furthermore, for dished surfaces the velocity magnitude remains below 30% in 70% of the area below the inner cylinder and never attains a velocity magnitude value above 40%. Thereby, indicating that the dead-zone, where mixing will not be efficient, is larger for dished surfaces, a pattern which can also be noticed in Figure 4.

Such poor magnitude values of the velocity clearly confirm the empirical observations of accumulation of particles in the bottom of bioreactor for the dished outwards bottom structure. This aspect will demand the usage of higher rotational velocities to pick up particles settled at the bottom of bioreactor for these shapes. Usage of high rotational speeds could be detrimental to growth of animal cell cultures which are highly sensitive to the hydrodynamics of the fluid. The dished outwards shape which was found to be beneficial in the case of stirred tank bioreactor is clearly a disruptive factor in the case of Taylor bioreactor. Whereas, the idea that dished inward surface could have aided in reducing the dead zone are in the bottom of the bioreactor is proved wrong, as the velocity magnitude for this surface is smaller as compared to the dished outward. In contrast to the stirred tank, the flat bottom offers a significant advantage over the dished surfaces, especially at lower off-bottom clearance areas.

Off-bottom clearance: In contrast to the dished bottom surfaces, where the impact of the geometry is adversely effective up-to a certain height in the gap-width area, the offbottom clearance flow structures does not interfere in the flow structures of the gapwidth area. As soon as the off-bottom clearance height is finished, similar flow structures with similar magnitudes develop in the gap-width area instantly, as represented by the Figures 3(c), 3(d) and4(b). However, the off-bottom clearance area has a very big disadvantage of completely zero velocity magnitude values in the centre of the obc region for all geometry types presented here. Higher the off-bottom clearance height will be, larger the volume will be with such poor velocity magnitude values which lead towards the dead-zone area located in the middle of the off-bottom clearance area (Figures 3c 3d and4a).

The flow structure in the gap-width is generated due to the inner rotating cylinder, and is mainly tangential in nature. The axial flow is of secondary nature only, and is not strong enough to sustain flow in the offbottom clearance area. Unless there is a change in the geometry to help generate flow in the off-bottom clearance area, it will be important to keep it to a minimum height possible. Considering the high sensitivity of the animal cell lines and that the animal cell cultures are generally operated in batch operation mode, one possible change in the geometry could be the influx of axial flow through recirculation to generate some movement in the off-bottom clearance area. However, the idea of recirculation of the fluid requires significant amount of experimental and numerical study, and it is out of scope of this study.

Energy dissipation rate and Kolmogorov scale: In order to use the Taylor bioreactor, the most important constraint will be to keep the energy dissipation rate (EDR) within the sub-lethal responses of a cell-line in order to achieve the desired results. [START_REF] Godoy-Silva | Physiological Responses of CHO Cells to Repetitive Hydrodynamic Stress[END_REF] discovered that when cells are cultured in suspension around 60 W kg -1 = m 2 s -3 maximum EDR, the CHO cell line gave the sub lethal physiological responses that are critical to a bioprocess. CHO cell line is one of the most robust cell lines that exist in present time, but there are many cell lines, especially the ones requiring microcarriers, which are much more sensitive and their sub-lethal response should be found at a much lower value [START_REF] Mollet | Bioprocess equipment: Characterization of energy dissipation rate and its potential to damage cells[END_REF].

Ibrahim and Nienow (2004) suggested a volume average EDR value of 0.001 Wkg -1 to be considered as a base line for culturing cells on microcarriers in stirred vessels. However, [START_REF] Kaiser | Fluid Flow and Cell Proliferation of Mesenchymal Adipose-Derived Stem Cells in Small-Scale, Stirred, Single-Use Bioreactors[END_REF] and Hewitt et al., (2001) have successfully cultured mesenchymal stem cells at volume averaged EDR values of 0.0021 and 0.0034 W kg -1 in spinner flasks, a kind of stirred vessel. [START_REF] Santiago | Performance of a vortex flow bioreactor for cultivation of CHO-k1 cells on microcarriers[END_REF] have successfully cultured CHO-K1 cells on microcarriers in a Taylor bioreactor at 50 rpm for a Reynolds no 3660 and volume average EDR of 0.0037 Wkg -1 . [START_REF] Mollet | Bioprocess equipment: Characterization of energy dissipation rate and its potential to damage cells[END_REF] advocated the employment of local EDR instead of volume averaged EDR due to large difference between these two values in some vessels such as spinner flasks, where the difference between the maximum and volume average EDR is more than 100 times [START_REF] Hewitt | Expansion of human mesenchymal stem cells on microcarriers[END_REF][START_REF] Kaiser | Fluid Flow and Cell Proliferation of Mesenchymal Adipose-Derived Stem Cells in Small-Scale, Stirred, Single-Use Bioreactors[END_REF]. This could mean that the extremely high local values in the impeller regions of a stirred vessel should also be considered as an important factor as the cell damage could be due to these high local values in the impeller region instead of the small volume averaged EDR values. Moreover, when culturing cells on microcarriers the size of microcarriers also play a relevant role [START_REF] Hewitt | Expansion of human mesenchymal stem cells on microcarriers[END_REF][START_REF] Croughan | Hydrodynamic effects on animal cells grown in Microcarrier cultures[END_REF].

The analytical and numerical estimations of the Taylor bioreactor at 200 rpm are presented in Table 1 using the [START_REF] Wendt | Turbulente strmungen zwischen zwei rotierenden konaxialen zylindern [Turbulent currents between two rotating coaxial cylinders[END_REF] The numerical estimations of torque and its derivatives are in complete sync with the analytical estimations. As per global estimates of the EDR, the maximum value of 3.9 W kg -1 was obtained located at the inner wall for the rotational speed of 200 rpm. Based on the analytical estimations at 150, 100 and 50 rpm, the maximum and mean EDR value would be of 1.5 and 0.09, 0.45 and 0.033, and 0.06 and 0.006 W kg -1 , respectively as shown in Table 2.

In order to compare the data with a different geometry type Re/Re T ratio will be used as the criteria based on the study of [START_REF] Lathrop | Transition to shear driven turbulence in Couette-Taylor flow[END_REF]. Here Re T is the Reynolds number related to fully turbulent Reynolds number, and its value for stirred vessels is known to be 2×10 4 and in the case of Taylor reactor [START_REF] Lathrop | Transition to shear driven turbulence in Couette-Taylor flow[END_REF] have shown that Re T =1.3×10 4 . First of all, the impeller rotational speed is not used because it becomes irrelevant for different scales and geometry types. Reynolds number is relevant for the different scales of the same geometry, but when the geometry is changed the significance of Reynolds number also changes as Re T values are different for different geometry types: the Re T values for stirred vessels, Taylor-Vortex bioreactor, pipe flow and flat plate are 2×10 4 , 1.3×10 4 , 2.3×10 4 and 3.2×10 5 , respectively [START_REF] Lathrop | Transition to shear driven turbulence in Couette-Taylor flow[END_REF]. Therefore, for a better comparison between different geometry types, the Re/Re T ratio presents the option of comparing similar turbulence flow conditions.

For similar Re/Re T ratio of ≈ 0.14 in comparison with the spinner vessel of [START_REF] Kaiser | Fluid Flow and Cell Proliferation of Mesenchymal Adipose-Derived Stem Cells in Small-Scale, Stirred, Single-Use Bioreactors[END_REF], the max and mean EDRs are around 50 and 2 times lower. In comparison with the spinner vessel study of [START_REF] Hewitt | Expansion of human mesenchymal stem cells on microcarriers[END_REF], at a higher Re/Re T value of 0.14 the mean EDR is practically half the value. In a Taylor bioreactor, it is important to note that the maximum value is only achieved in the viscous sub-layer very near to the wall where the cells may never reach due to the nature of the viscous layer of providing resistance to any kind of transfer. Away from the wall, the EDR decreases extremely rapidly and its magnitude is much smaller than the maximum value [START_REF] Coufort | Analyse experimentale de la floculation en reacteur de Taylor-Couette: Influence de l'hydrodynamique sur les phenomenes d'agglomeration et de ruptura[END_REF][START_REF] Tokgoz | Spatial resolution and dissipation rate estimation in Taylor-Couette flow for tomographic PIV[END_REF][START_REF] Van-Hout | Measurements of mean flow and turbulence characteristics in highreynolds number counter-rotating Taylor-Couette flow[END_REF]. It can be further confirmed by the fact that the average EDR, ‹ ε ›, was found to be 0.193 based on numerical simulations (Table 3). The animal cells are bound to spend the maximum amount of time within a Taylor-vortex in the bulk zone, between R b = 0.1 to 0.9, where the gradients are much smaller and have similar magnitude [START_REF] Gong | Age-related responses of suspension cultured taxus cuspidata to hydrodynamic shear stress[END_REF][START_REF] Hewitt | Expansion of human mesenchymal stem cells on microcarriers[END_REF][START_REF] Kaiser | Fluid Flow and Cell Proliferation of Mesenchymal Adipose-Derived Stem Cells in Small-Scale, Stirred, Single-Use Bioreactors[END_REF], i.e., similar hydrodynamic conditions in the major part of the reactor. This is a significant advantage for a Taylor bioreactor to be used for culturing animal cells in comparison to the spinner vessel where the probability of cells reaching the high gradient region near the impellers is high and the fluid flow conditions are less uniform with large difference between local and volume averages EDR values. These aspects clearly demonstrate the possibility of using the Taylor bioreactor not only for the suspended cell cultures but also in the case of the microcarrier systems.

Conclusions

The aim of this study was to understand the impact of the changes in the geometrical configuration, namely, off-bottom clearance area and different bottom shapes of the outer cylinder, of a Taylor bioreactor on its flow structure and its practicality as a bioreactor. The bulk zone comprising of the 80 % of the gap-width, where the cell cultures will spend most of the time, has a near constant velocity magnitude of around 50 % of the maximum and VEDR values which are around 10 times smaller in comparison to the boundary layer area, an aspect of great significance which demonstrates the possibility of culturing cells successfully.

The geometrical features of curved surface of outer bottom and off-bottom clearance area which are of practical importance in stirred vessels, impact adversely the flow structures in the Taylor bioreactor due to poor axial velocity component. In comparison with the spinner vessel for similar Re/Re T ratio, the maximum and mean EDRs were always found to be of lower magnitude values, and due to much less difference between the maximum and the mean values, the Taylor bioreactor presents more uniform structures in comparison to the spinner vessel. These attributes clearly confirm that the Taylor bioreactor is adequate for culturing of animal cells not only in suspension but also for the microcarrier culture systems. This study has provided the authors the guiding light in constructing such a bioreactor which is at present in testing phase with animal cells using the microcarriers.

Figure 1 :

 1 Figure 1: Three different shapes (Flat (a), Dished outwards (b) and Dished inwards (c) ) of the outer bottom cylinder at 15 mm offbottom clearance, respectively, for each geometry.

Figure 2 :

 2 Figure 2: Comparison of numerical tangential velocity results with the PIV results of Coufort (2004).

Figure 3 :

 3 Figure 3: Mean velocity vectors for different shapes and off-bottom clearances.

Figure 4 :

 4 Figure 4: Dimensionless mean velocity at 1mm below inner cylinder (a) and radial location of 0.1 mm away from the inner wall (b) for different geometry types.

Table 1 :

 1 Analytical and numerical estimations of the Taylor bioreactor at 200 rpm.

	´s

Table 2 :

 2 Summary of energy dissipation rate (EDR) for different geometries.

			Rotational	Reynolds No	EDR Max/Mean
	Group	Geometry	Speed
			Rpm	(-)	Re/Re T	W kg -1 , m 2 s -3
			200	12077 0.93	3.88/0.193
	Analytical	Taylor	150	9058	0.70	1.51/0.09
	Estimation	Bioreactor	100	6039	0.46	0.45/0.033
			50	3019	0.23	0.06/0.006
			30	1812	0.14	0.01/0.002
	Kaiser et		105	3014	0.15	0.58/0.004
	al., (2012)	Spinner	50	1435	0.07	0.062/0.0006
	Hewitt et al., (2011)	Flask	50	1690	0.08	(-)/0.0034

Acknowledgements


Dimitri Alexandre Rochman

Eric Bauge

Fission yields and cross sections: correlated or not?

Cross sections and fission yields can be correlated, depending on the selection of integral experimental data. To support this statement, this work presents the use of experimental isotopic compositions (both for actinides and fission products) from a sample irradiated in a reactor, to construct correlations between various cross sections and fission yields. This study is therefore complementing previous analysis demonstrating that different types of nuclear data can be correlated, based on experimental integral data.

Introduction

During the last decades, the importance of accurately estimating a degree of confidence in nuclear simulations has gathered an increasing amount of attention. Whereas measured values were historically often followed by uncertainties, calculated quantities were for too long provided alone. Nowadays, due to safety and economical pressures, increased computer power, and needs of informed decision making, it has become more common that uncertainties go along with best estimations. Nuclear data is no exception to this trend, and their recommended values (also called evaluations) now comes with a fair amount of covariance matrices in the latest national and international efforts [START_REF] Brown | ENDF/B-VIII.0: the 8th major release of the nuclear reaction data library with CIELO-project cross sections, new standards and thermal scattering data[END_REF][START_REF] Plompen | The joint evaluated fission and fusion nuclear data library, JEFF-3.3[END_REF][START_REF] Shibata | JENDL-4.0: a new library for nuclear science and engineering[END_REF][START_REF] Koning | TENDL: complete nuclear data library for innovative nuclear science and technology[END_REF]. Traditionally, such covariance matrices were first provided to the user community for neutron-induced reactions and their cross sections (called XS in the following), as being often the most sensitive quantities for many nuclear applications. At equal level of importance is the average number of neutrons emitted per fission (ν), which nowadays also comes with its own set of covariance matrices, and often the emitted spectra and angular distributions as well. Frequently, these matrices link different energy regions of standard deviations of the same quantity, and can also be provided for different reactions. But, they are restricted to the same quantities for a given isotope: for instance, for the energy dependence of the fission cross section of 235 U, or for the energy dependence between its capture and fission cross sections. Nevertheless, in modern evaluations, these correlations are not yet provided between uncertainties of different quantities, such as XS * e-mail: dimitri-alexandre.rochman@psi.ch and ν, or between isotopes. The reason mostly lies in the difficulty to construct such correlations based on selected differential measurements and reaction calculations. There is a noticeable exception, as the vast majority of measurements are performed relatively to a standard cross section (such as 197 Au(n,γ), or 235 U(n,f)); therefore almost all evaluated nuclear data must be correlated to some of these standards, and therefore also between themselves [START_REF] Carlson | Evaluation of the neutron data standards[END_REF]. But this is generally not yet considered in the evaluation efforts for the recommended uncertainties and correlations (from a practical aspect, the evaluation format was not defined to accommodate for such various correlations; an update of such format would therefore be necessary).

Outside the major evaluated libraries, it was recently shown that it is possible to correlate different quantities, for instance XS and ν, and even different isotopes (e.g. 235 U and 238 U) using so-called integral experimental data. Such experiments, based on various isotopes and complex geometries, or providing measured quantities sensitive to many reactions, allow to construct a vast amount of new correlations when using Bayesian methods such as the so-called BMC (Bayesian Monte Carlo), BFMC (Backward-Forward Monte Carlo), or MOCABA [START_REF] Bauge | Evaluation of the covariance matrix of 239 Pu neutronic cross sections in the continuum using the backward-forward Monte-Carlo method[END_REF][START_REF] Koning | Bayesian Monte Carlo method for nuclear data evaluation[END_REF][START_REF] Hoefer | MOCABA: a general Monte Carlo-Bayes procedure for improved predictions of integral functions of nuclear data[END_REF][START_REF] Bauge | Full model nuclear data and covariance evaluation process using TALYS, Total Monte Carlo and backward-forward Monte Carlo[END_REF] (other methods as presented in Refs. [START_REF] Pelloni | Resonance parameter adjustment in the resolved region based upon an Asymptotic Generalized Linear Least-Squares methodology in conjunction with the Monte Carlo method[END_REF][START_REF] De Saint Jean | On the use of Bayesian Monte-Carlo in evaluation of nuclear data[END_REF][START_REF] Pelloni | Comparison of progressive incremental adjustment sequences for cross-section and variance/covariance data adjustment by analyzing fast-spectrum systems[END_REF][START_REF] Pelloni | Cross-section adjustment in the fast energy range on the basis of an Asymptotic Progressing nuclear data Incremental Adjustment (APIA) methodology[END_REF][START_REF] Salvatores | Methods and issues for the combined use of integral experiments and covariance data: results of a NEA International Collaborative Study[END_REF][START_REF] De Saint Jean | Evaluation of cross section uncertainties using physical constraints: focus on integral experiments[END_REF][START_REF] Capote | A new formulation of the Unified Monte Carlo Approach (UMC-B) and cross-section evaluation for the dosimetry reaction 55 Mn(n,γ) 56 Mn[END_REF][START_REF] Smith | A unified Monte Carlo approach to fast neutron cross section data evaluation[END_REF][START_REF] Helgesson | Combining total Monte Carlo and Unified Monte Carlo: Bayesian nuclear data uncertainty quantification from auto-generated experimental covariances[END_REF][START_REF] Watanabe | Cross section adjustment method based on random sampling technique[END_REF][START_REF] Kawano | Evaluation and propagation of the 239Pu fission cross-section uncertainties using a Monte Carlo technique[END_REF] could also be used, even if developed towards different goals). These methods allow to take into account integral data during an evaluation process, based on an adjustment of specific quantities. The experimental integral data used for these studies were mostly criticality benchmarks, with the provided k eff or reaction rates, but also reactor data such as boron concentrations [START_REF] De Saint Jean | Evaluation of neutron-induced cross sections and their related covariances with physical constraints[END_REF][START_REF] Siefman | Data assimilation of post-irradiation examination data for fission yields from GEF[END_REF][START_REF] Alhassan | Bayesian updating for data adjustments and multi-level uncertainty propagation within Total Monte Carlo[END_REF][START_REF] Rochman | Correlation nu-sigma for U-Pu in the thermal and resonance neutron range via integral information[END_REF][25][START_REF] Rochman | Monte Carlo nuclear data adjustment via integral information[END_REF][START_REF] Bauge | Cross-observables and crossisotopes correlations in nuclear data from integral constraints[END_REF][START_REF] Rochman | Nuclear data correlation between different isotopes via integral information[END_REF][START_REF] Rochman | Correlation nu-sigma-chi in the fast neutron range via integral information[END_REF]. It was also shown that still using differential data with oxide targets, it is possible to provide correlations between oxygen, hydrogen and the other compound isotopes [START_REF] Ch | Neutron-induced damage simulations: beyond defect production cross-section, displacement per atom and iron-based metrics[END_REF].

These various new results indicate that it is possible to build covariance matrices for cross sections between isotopes, or between ν, XS and fission spectra for a specific isotopes, or finally between XS of different isotopes. A larger number of reactions, quantities and isotopes could therefore be considered as correlated. The question of recommending such correlation matrices in evaluations is still debated among specialists.

Following this line of research, some quantities have still been left aside from this global correlation effort, namely fission yields. In the following, the term "fission yield" will designate without distinction independent or cumulative fission yields (called FY). It is a known deficiency that current evaluated libraries do not provide correlation matrices for fission yields, neither for a specific fissioning isotope (such as all fission yields from 235 U(n,f)), or between fissioning isotopes. This represents a practical problem for nuclear data users, as uncertainties on quantities dependent of FY (e.g. long-lived fission products from spent fuel) are believed to be too large, due to the lack of FY correlations [START_REF] Leray | Uncertainty propagation of fission product yields to nuclide composition and decay heat for a PWR UO2 fuel assembly[END_REF][START_REF] Rochman | Nuclear data uncertainties for typical LWR fuel assemblies and a simple reactor core[END_REF][START_REF] Mills | Uncertainty propagation of fission product yield data in spent fuel inventory calculations[END_REF][START_REF] Frosio | Fission yields and cross section uncertainty propagation in Boltzmann/Bateman coupled problems: Global and local parameters analysis with a focus on MTR[END_REF]. As mentioned before, the current evaluation format is not defined to propose such correlations for FY; there are nevertheless efforts to update such format.

To help resolving this problem of missing correlations, the present work demonstrates that specific integral data can be used to construct correlations between fission yields, but also between XS and FY. Such correlations were not yet put into lights and this paper presents an example of XS-FY correlations using measured and simulated isotopic concentrations from a specific sample irradiated in various reactor cycles. In the following, we first describe the considered experimental data and simulation tools (sample, irradiation, measurements), the method used to build such correlations (BFMC) and finally the results. It will hopefully be clear that fission yields and cross sections can be correlated, thus extending again the dimension of correlated nuclear data.

On the use of a PIE sample

As presented in the introduction, many types of nuclear data correlations can be constructed using integral data as constraints. Most common integral data used during the evaluation process are the criticality state of a system (or criticality benchmark) [START_REF] Briggs | Integral benchmark data for nuclear data testing through the ICSBEP & IRPhEP[END_REF], called k eff , sometimes accompanied by spectral indexes (ratios of capture over fission reaction rates) or reaction rates for capture, (n,2n), (n,p) or (n,α) reactions in more complex systems [START_REF] Briggs | An overview of the international reactor physics experiment evaluation project[END_REF][START_REF] Kodeli | SINBAD shielding benchmark experiments status and planned activities[END_REF].

In the present work, we are proposing to take advantage of the so-called Post Irradiation Examination samples (or PIE samples), which consist in the destructive analysis of part of an irradiated fuel assembly. Such analysis provides measurements for the isotopic concentrations of specific isotopes (in mg/gU, for actinides and fission products) which live long enough to be measured a few years after the last assembly irradiation. Typically, such PIE data are used in the validation of transport and depletion codes (as presented for instance in Refs. [START_REF] Tardy | First burnup credit application including actinides and fission products for transport and storage cask by using French experiments[END_REF][START_REF] Rochman | Improvement of PIE analysis with a full core simulation: the U1 case[END_REF][START_REF] Rochman | Consistent criticality and radiation studies of Swiss spent nuclear fuel: the CS2M approach[END_REF]), but can also be used in nuclear data-related activities. Such samples are made of UO 2 or MOX fuel, are irradiated in conventional water reactors for various cycles, and can reach burnup values of 50 to 70 MWd/kgU (and sometimes more). Therefore, a number of minor actinides can be measured (up to 245 Cm) with a number of fission products, from the most produced during fission (e.g. 99 Tc) to the less produced (e.g. Gd). Additionally, the neutron energy of relevance is mostly in the thermal region.

The irradiation and cooling of such samples can be simulated with a variety of codes, and it was demonstrated that nuclear data, both cross sections and fission yields, can greatly influence isotopic concentrations [START_REF] Rochman | Improvement of PIE analysis with a full core simulation: the U1 case[END_REF]: if the concentration of actinides mainly depends on capture and fission cross sections, fission yields can be the most influential parameters for specific fission products (such as Cs isotopes). PIE samples then become very suited integral experiments to correlate different cross sections themselves, as well as cross sections and fission yields.

Among available PIE samples, the ARIANE GU1 sample is selected in this work. It is part of the open-literature database SFCOMPO [START_REF]ARIANE International Programme Final Report[END_REF][START_REF] Michel-Sendis | SFCOMPO-2.0: an OECD NEA database of spent nuclear fuel isotopic assays, reactor design specifications, and operating data[END_REF] and consists of a UO 2 sample, enriched at 3.5% in 235 U, and was irradiated during 4 consecutive cycles in the Gösgen power plant (approximately 3.5 years of irradiation, followed by 6 years cooling time). The sample burnup is estimated to be close to 58.6 MWd/kgU, and a number of isotopic concentrations are reported (details can be found in the ARIANE report). The consideration of a unique experiment is certainly not satisfactory from a comprehensive perspective. The correlations extracted in the following are therefore solely presented for illustration; in a more realistic evaluation work, a larger number of experiments needs to be selected, to be representative enough of the current experimental knowledge. For readers interested in details and agreements between calculated and measured isotopic contents for the GU1 sample, we refer to the study presented in reference [START_REF] Rochman | Analysis for the ARIANE GU1 sample: isotopic compositions and decay heat[END_REF] 3 Method to build correlations Before defining the method applied to construct correlations, the definition of the correlation itself in the context of this paper can be useful. Mathematical definitions of the correlation coefficient can be found in references [START_REF] Bauge | Cross-observables and crossisotopes correlations in nuclear data from integral constraints[END_REF][START_REF] Rochman | Analysis for the ARIANE GU1 sample: isotopic compositions and decay heat[END_REF], and for the present work, its interpretation will be the following. A correlation coefficient ρ between two quantities A and B indicates that if A is changing by 1×σ A (σ A being the standard deviation of A), then B will be changing by ρ × σ B . ρ can be positive or negative, and varies between -1 and 1. In the following, if one cross section A (at a given energy) is positively correlated with another one B, it means that if A increases, B also increases. In the case of negative correlation, A and B vary in opposite directions. This definition is useful to understand the following representative case: if a measured quantity E is created by the A process and disappears because of the B process, keeping E constant implies that A and B vary in same direction; A and B therefore become positively correlated.

If E is created by both A and B processes, then keeping E constant implies that A and B become negatively correlated. As presented in the following, posterior correlations (created by taking into account a specific measured E value) can be understood with these simple steps in mind, as Bayesian methods such as BFMC tend to produce posterior calculated quantities closer to experimental data.

The simulation of such sample is presented in reference [START_REF] Rochman | Analysis for the ARIANE GU1 sample: isotopic compositions and decay heat[END_REF] and the calculated results (nominal and perturbed) of this reference are used in the present work.

In short, a model with the deterministic code CASMO5 is used for simulating the irradiation and cooling times [START_REF] Rhodes | CASMO-5 development and applications[END_REF], and the isotopic concentrations of a number of isotopes are calculated. It allows to calculate simple C/E ratios for each isotope (Calculation over Experimental values). Additionally, a number of nuclear data can be varied, and for each perturbed set of cross sections and fission yields, new C values are obtained, leading to calculated uncertainties ∆C, due to nuclear data.

In these calculations, the nominal values are based on the ENDF/B-VII.1 library; the cross section covariance matrices are also obtained from the ENDF/B-VIII.0 library, whereas in the case of fission yields, an in-house tool is used for prior correlations between some yields (all uncertainties also come from ENDF/B-VIII.0) [START_REF] Leray | Nuclear data uncertainty propagation on spent fuel nuclide compositions[END_REF], mainly because of the lack of correlation in all evaluated libraries. The covariance information from the ENDF/B-VIII.0 library are therefore associated with the best-estimate XS from the ENDF/B-VII.1 library. While this is theoretically not a recommended practice, this approach accommodates for pragmatic approach due to limited accessibility to CASMO5 libraries at the time of this study. The varied nuclear data (all perturbed together) are the following:

-capture cross sections: 103 235 U, 239,241 Pu, -fission yields for fast neutron-induced fission: 238 U.

All cross section covariance matrices are provided in 19 energy groups, from 0 to 20 MeV, as indicated in reference [START_REF] Rochman | Improvement of PIE analysis with a full core simulation: the U1 case[END_REF]. In total, more than 4000 CASMO5 simulations are performed with the above perturbed nuclear data. In reference [START_REF] Rochman | Analysis for the ARIANE GU1 sample: isotopic compositions and decay heat[END_REF], it was demonstrated that other nuclear data do no significantly affect the isotopic concentrations of interest.

The method used to calculate correlations between different nuclear data is the BFMC method. It was presented in reference [START_REF] Bauge | Evaluation of the covariance matrix of 239 Pu neutronic cross sections in the continuum using the backward-forward Monte-Carlo method[END_REF] and applied in some of the previous citations. A short summary is provided here. For each of the random calculations i based on perturbed nuclear data, calculated isotopic concentrations are obtained. For a specific isotope (for instance 99 Tc), the calculated values C i (i varying from 1 to 4000) can be compared with the measured value E. The comparison can be used to calculate a weight w i , representing the degree of agreement between both quantities:

w i = exp - C i /E (C j /E) min 2 .
(1)

In this equation, the ratio with the "min" label corresponds to the case j leading to the best agreement between the calculated and experimental value (ideally, C j and E are equal). The use of this normalization factor is the main difference between the BMC and BFMC method: it helps to compensate for the so-called "model defects" that the CASMO5 simulations contain. The use of weights w i favors the samples which agree with the experiment with the effect that the posterior distribution is in better agreement with E, compared to the prior value. They are then used to calculate posterior distributions, for instance using weighted averages, weighted standard deviations, or weighted correlations. Equations can for instance be found in reference [START_REF] Rochman | Monte Carlo nuclear data adjustment via integral information[END_REF]. In the following, only the prior and posterior correlation values will be presented, but such Bayesian approach also provides posterior central values and variances.

Equation ( 1) could be generalized to the use of many experimental data at once in the calculation of weights (ideally coming both integral and differential experiments), but in the following, experimental data will only be considered one by one to understand the origin of the constructed correlations.

As mentioned, the irradiation of the GU1 sample is simulated and the reported experimental values E from the ARIANE report are used [START_REF]ARIANE International Programme Final Report[END_REF]. 50 isotopes were measured (18 actinides, 32 fission products), but only a limited number of them present an acceptable C/E ratio: only 21 C/E cases among 50 are within a 1σ, with σ being (∆E) 2 + (∆C) 2 . This is not uncommon in PIE validation work, for instance due to approximations in the simulations, (very) small experimental uncertainties, or partial isotopic dissolution. Therefore, only a limited number of isotopic calculations will be used in the following for equation [START_REF] Brown | ENDF/B-VIII.0: the 8th major release of the nuclear reaction data library with CIELO-project cross sections, new standards and thermal scattering data[END_REF].

In the next section, specific representative results will be presented and detailed, and to simplify the interpretation of the correlations, only single isotopic composition will be analyzed at once.

Results

Different cases are presented in the following. First, only the constructed correlations between cross sections (XS) are analyzed, followed by the correlations between fission yields (FY) only. Finally the mixed cases of correlations between FY and XS are discussed.

All presented figures of correlations between nuclear data follow the same structure. They are made of a number of square blocks, each block representing a specific nucleus and reaction (for instance 235 U(n,f)). They are separated by solid black lines. In the case of cross sections, such blocks are divided in ten zones (possibly filled with blue or red squares), each one representing one neutron incident energy group, in total ranging from 0.01 eV to 6 eV. In the case of fission yields, each of the ten zones represents one specific fission yield. Both x and y axes are similar.

Example 1: Correlations among cross sections, for the same reaction channel

This example presents the case of the correlations between the capture cross sections of two different isotopes: 242 Pu and 244 Cm. Both of these isotopes were not present in the sample at the beginning of the irradiation, and are built-up by neutron capture on 239 Pu (for 242 Pu) and on 242 Pu (for 244 Cm). Both of these actinides have a small fission cross section in the thermal range, so that their disappearance is mainly due to their capture cross sections. Examples of different correlations, obtained by taking into account different isotopic compositions E in the BFMC weighting process are presented in Figure 1. The left matrix represents the prior correlation matrix, as obtained from the ENDF/B-VIII.0 covariance matrices. As observed, energy-energy correlations within the same reaction exist, but not between isotopes. This is a common feature of all evaluated libraries, where cross-isotope correlations are rarely provided.

The middle matrix presents the posterior correlations when using the experimental isotopic composition of 244 Cm as the measured value E. As mentioned, 244 Cm is mainly produced from neutron capture on 242 Pu, and disappears due its own radiative capture. Therefore the positive posterior correlation observed between both isotopes in this matrix can typically be explained by the example of reactions A and B from Section 3: keeping the 244 Cm constant implies that both capture cross sections are positively correlated, as observed.

The left correlation matrix exhibits the opposite behavior, simply by considering the 245 Cm as the measured value E instead of 244 Cm. In this case, 245 Cm is solely produced by the 244 Cm(n,γ) reaction, and indirectly also by the 242 Pu(n,γ) reaction. Therefore both reactions contribute to the build-up of 245 Cm: to keep 245 Cm constant, if one increases, the other one needs to decrease. They are therefore negatively correlated. This example illustrates that different types of correlations can be built, depending on the choice of the considered experimental value E. It clearly indicates that a correlation factor is only meaningful within a given context which depends on the choice of the experimental value E used to calculate the weighted posterior distribution. As detailed in the discussion section, these correlations depend on the characteristics of the irradiation of the sample, and other correlations can be found for other irradiation conditions.

Example 2: Correlations among cross sections, for different channels

As previously indicated, correlations between specific cross sections are usually obtained and included in evaluated nuclear data libraries, based on theoretical reaction calculations and differential data. But similarly to the previous example, a number of actinide cross sections can be correlated due to specific PIE measurements and Figure 2 illustrates another of such cases. The left matrix is the prior nuclear data correlation matrix, as obtained from sampling specific reactions, based on the covariance matrices from the ENDF/B-VIII.0 library. One can observe that cross sections for four reactions are presented, namely 235 U(n,f), 238 U(n,γ), 239 Pu(n,f) and 239 Pu(n,γ). Different isotopes are not correlated, but strong energyenergy correlations can appear for a specific isotope, and also between its reactions (note the positive correlations between the 239 Pu capture and fission cross sections). The structure of these correlations, which is not the subject of this paper, comes mainly from the physics of resonances in this low-energy range.

The right matrix represents the posterior correlation matrix obtained with the BFMC method, using one single PIE measurement as the E value, namely the isotopic concentration of 239 Pu. 239 Pu was not present in the fuel at the beginning of the sample irradiation, but it is builtup from 238 U during the irradiation cycles. The longer the irradiation, the more 239 Pu is produced by neutron capture on 238 U, but it also disappears because of its own fission and capture cross sections. Therefore, the remaining concentration of 239 Pu after irradiation (and cooling time) is increased by the 238 U(n,γ) reactions, and These simple explanations allow to understand the correlations appearing between 238 U and 239 Pu in the case of the right matrix of Figure 2. As in the case of Cm and Pu capture cross sections, it is important to keep in mind the choice of the experimental value E, used to calculate the weighted posterior correlations.

Example 3: Correlations among fission yields

The next example concerns the correlations between fission yields of different actinides, and does not involve cross sections. During the irradiation of the GU1 sample, 235 U is being burned mainly due to its fission cross section, and at the same time, the plutonium isotopes are being built-up. Two of them are also contributing to the fission rates: 239 Pu and 241 Pu, as their thermal fission cross sections are relatively large (at the end of irradiation, the integral fission rates are 44%, 8%, 38% and 10% for 235 U, 238 U, 239 Pu and 241 Pu, respectively). Therefore, the three isotopes with an important thermal fission cross section, as well as their fission yields contribute to the creation of fission products. One can expect that by considering the measured concentration E from a given fission product, correlations between the fission yields from these three isotopes should be observed. This is the case presented in Figure 3 where E is the measured concentration for 133 Cs.

Again, the left matrix is the prior correlation matrix. Only a limited number of fission yields is presented here, 10 for each actinide. As observed, the prior matrix contains a very limited number of correlations, being a common feature of all evaluated nuclear data libraries. The prior correlations are due to the in-house tool, as explained in the introduction. The right matrix presents the posterior correlations obtained by using the measured isotopic composition of 133 Cs as E to calculate the weighted posterior distribution. This fission product is produced by the three mentioned actinides, with different contributions. It is nevertheless produced indirectly, as for mass 133, the main isotope directly produced by fission is 133 I (with the highest fission yield). This iodine isotope, having a short half-life of about 8 days, decays to 133 Cs during the irradiation and cooling times. Therefore three fission reactions lead to the creation of this isotope, via 133 I. In order to keep it constant, the 133 I fission yields for the three main fissioning actinides are negatively correlated, as observed in Figure 3. It is interesting to note that a very strong negative correlation appears between the 133 I fission yields of 235 U and 239 Pu, since they are the two main contributors to mass 133 by fission.

Example 4: Correlations between fission yields and cross sections

The example presented in this section is representative of a number of cases. Similar correlations between fission yields and capture cross sections were also observed considering the isotopic compositions of 145 Nd, 154 Sm, or 134 Cs. As mentioned, the production of a measured fission product can depend on many fission yields. These fission products can either be directly the isotope of interest, or they can be isotopes subsequently decaying to the observed isotopes. Moreover, neutron capture cross sections can also contribute to the decrease of the target nucleus population (as well as to the increase of the daughter nucleus population). In this case, positive correlations are expected between specific fission yields and the capture cross section(s) which depopulate them, as explained in the example of Section 3. The example presented in Figure 4 illustrates such effects. It represents the case of correlations between specific fission yields and a capture cross section. Again, the left figure is the prior correlation matrix, between the 147 Pm(n,γ) cross section and a selection of fission yields of 235 U and 239 Pu. As observed before, very limited correlations among fission yields exist, except for the Nd isotopes, presenting a negative correlation between 147 Nd and other Nd isotopes.

The right correlation matrix is obtained from the posterior distribution computed using the measured isotopic concentration of the 147 Sm fission product as the experimental constraint E. As in the previous example, it is not directly produced by fission. The short-lived 147 Nd (half-life of 11 days) is first produced, decaying to 147 Pm (half-life of 2.6 years) by beta emission, which finally produces 147 Sm also by beta decay. Because the halflife of 147 Nd is short compared to the irradiation time, its capture rate is small compared to the one of 147 Pm. Additionally, 147 Sm does not disappear much because of its own capture cross section, as it is only slowly produced by decay over the irradiation and cooling periods (each being 3.5 and 3 years). In the present case, the 147 Pm(n,γ) cross section and the 147 Nd fission yields are removing and creating 147 Sm, respectively. This explains the observed positive correlations between the 147 Pm(n,γ) cross section and 147 Nd fission yields for 235 U and 239 Pu on Figure 4. Additionally, because 147 Nd is negatively correlated with the other Nd isotopes in the prior matrix, variations of 147 Nd are also in opposite directions with the other Nd isotopes in the posterior matrix, although the negative correlation is weaker than in the prior case.

This example illustrates that correlations between XS and FY can be built, depending on the specific analyzed case. As mentioned, this example is not unique, and other ones were found. It is nevertheless the first time that these correlations are highlighted and even quantified. From the point of view of the evaluation of nuclear data, it is an additional evidence that it is worthwhile to consider correlations between different types of quantities across various isotopes: cross sections, ν (number of emitted neutron per fission), χ (energy spectra of these neutrons), fission yields, but also single and double differential spectra. For the time being, only correlations from theoretical reaction calculations and from the analysis of differential data are explicitly included in general purpose nuclear data libraries.

Discussion

The above examples illustrate that it is possible to generate correlations between virtually all the nuclear data that contribute to a given measured experimental value E. Criticality benchmarks, which have traditionally been used to select candidates for inclusion in evaluated nuclear data libraries, have been shown to induce correlations between cross sections, fission neutrons (both nubar and prompt fission neutron spectra), across energies and isotopes [START_REF] Rochman | Monte Carlo nuclear data adjustment via integral information[END_REF]. Those correlations are already present in the central values of evaluated nuclear data libraries since the "right" combination of cross sections, fission neutron observables were carefully selected (within the evaluated uncertainties) to optimize agreement with a large array of criticality benchmarks. The ENDF/B-VIII.0 library [START_REF] Brown | ENDF/B-VIII.0: the 8th major release of the nuclear reaction data library with CIELO-project cross sections, new standards and thermal scattering data[END_REF] acknowledges the calibration of the nubar observable for major actinides in order to "optimize k eff criticality" and so does the JEFF3.3 library [START_REF] Plompen | The joint evaluated fission and fusion nuclear data library, JEFF-3.3[END_REF]. So far the correlations stemming from use of integral constraints are only reflected in the central values, and not in the covariance matrices, leading to the apparent paradox of those libraries allowing good accurate (within 1 or 2σ of the experiment) calculation of experimental values, while producing propagated uncertainties that are significantly larger. That issue was discussed at length in reference [START_REF] Bauge | Cross-observables and crossisotopes correlations in nuclear data from integral constraints[END_REF] but, at the time, it was not settled.

The examples shown in the present paper help clarifying this debate. In our example 1, correlations between the 242 Pu(n,γ) and 244 Cm(n,γ) cross-sections, are of opposite directions depending on which measured isotope E ( 244 Cm or 245 Cm) was used for the calculation the weighted posterior. It is obvious that those correlations are not universal enough to be included in a general purpose evaluated nuclear data library, and belong to adjusted application libraries. Similarly, example 3, which exhibits strong negative correlations between the 133 I fission yields of three actinides when the measured 133 Cs production is used to calculate the posterior, is only relevant if 133 Cs is measured. Therefore such correlations do not belong to a general purpose evaluated nuclear data library. Example 4 shows that the use of integral experimental constraints in the calculation of a posterior distribution ( 147 Sm in this case) can even generate correlations between capture cross-sections and fission yields. Such correlations also do not belong to a general purpose evaluated nuclear data library.

On the other hand, those correlations are essential in order to account for the measured quantities with the precision needed by a number of applications. They can help in reducing calculated uncertainties on integral quantities, without necessarily changing the uncertainties on the relevant differential quantities. One of the underlying assumptions is that all parameters of the integral experiments are enough under control for not affecting the posterior quantities. Consequently, such correlations need to find their place in adjusted application libraries. Conversely, for the same reason, general purpose libraries should not be expected to be fully usable in applications without some adjustment (and inclusion of the needed correlations).

The path forward from there lies in the careful organization of the interface between the general purpose and the adjusted applications libraries. Future general purpose evaluated nuclear data libraries should make sure that they rely on as little integral information as possible during their evaluation (e.g. by including FY correlations based on differential measurements), and explicitly quantify the effect of this integral information in the associated covariance files. Additionally, automatic mechanisms (algorithms and codes systems) for nuclear data adjustment should be associated to evaluated nuclear data libraries, so that adjusted libraries can be easily generated and in a reproducible way. The BFMC, BMC, MOCABA methods [START_REF] Bauge | Evaluation of the covariance matrix of 239 Pu neutronic cross sections in the continuum using the backward-forward Monte-Carlo method[END_REF][START_REF] Koning | Bayesian Monte Carlo method for nuclear data evaluation[END_REF][START_REF] Hoefer | MOCABA: a general Monte Carlo-Bayes procedure for improved predictions of integral functions of nuclear data[END_REF][START_REF] Bauge | Full model nuclear data and covariance evaluation process using TALYS, Total Monte Carlo and backward-forward Monte Carlo[END_REF] as well as the other methods presented in references [START_REF] Pelloni | Resonance parameter adjustment in the resolved region based upon an Asymptotic Generalized Linear Least-Squares methodology in conjunction with the Monte Carlo method[END_REF][START_REF] De Saint Jean | On the use of Bayesian Monte-Carlo in evaluation of nuclear data[END_REF][START_REF] Pelloni | Comparison of progressive incremental adjustment sequences for cross-section and variance/covariance data adjustment by analyzing fast-spectrum systems[END_REF][START_REF] Pelloni | Cross-section adjustment in the fast energy range on the basis of an Asymptotic Progressing nuclear data Incremental Adjustment (APIA) methodology[END_REF][START_REF] Salvatores | Methods and issues for the combined use of integral experiments and covariance data: results of a NEA International Collaborative Study[END_REF][START_REF] De Saint Jean | Evaluation of cross section uncertainties using physical constraints: focus on integral experiments[END_REF][START_REF] Capote | A new formulation of the Unified Monte Carlo Approach (UMC-B) and cross-section evaluation for the dosimetry reaction 55 Mn(n,γ) 56 Mn[END_REF][START_REF] Smith | A unified Monte Carlo approach to fast neutron cross section data evaluation[END_REF][START_REF] Helgesson | Combining total Monte Carlo and Unified Monte Carlo: Bayesian nuclear data uncertainty quantification from auto-generated experimental covariances[END_REF][START_REF] Watanabe | Cross section adjustment method based on random sampling technique[END_REF][START_REF] Kawano | Evaluation and propagation of the 239Pu fission cross-section uncertainties using a Monte Carlo technique[END_REF] all qualify for such a goal to some extent (a similar new paradigm is suggested by [START_REF] Herman | New paradigm for nuclear data evaluation[END_REF]). If automated adjustment of libraries becomes easily achievable and a quantification of the "distance" between two nuclear data files (such as the overlap between the multivariate Gaussian distributions representing the libraries and their associated covariance matrices) is implemented, then a possible quality criterion for general purpose nuclear data libraries could be that the distance between the general purpose library and the multiple derived adjusted applications libraries is minimum.

Conclusion

This paper highlights cases of correlations between cross sections and fission yields due to selected integral measurements. Specifically, the use of the measured isotopic compositions from an irradiated sample with the BFMC method leads to nuclear data correlations not previously observed. It is an additional example that various types of nuclear data can be correlated together by the use of integral experimental constraints. If taken into account in specific evaluated libraries, such correlations can improve calculations of integral quantities and answer requests from a number of users, wishing for smaller calculated uncertainties. Because such correlations are constructed with specific measurements, and are case dependent, it is advocated that such correlations (and adjusted nuclear data) find their place in dedicated adjusted libraries. This possibility can improve the user's satisfaction, but also emphasizes the fact that current nuclear data evaluations do not lead to a unique set of cross sections, nubar or fission yields. These sets should be validated on a case by case basis, being very careful to clearly define their domain of validation and exercising extreme caution when crossing the boundaries of these domains. In the end, the quantitative agreement between the experimental values and the simulated ones, on both the central values and uncertainties constitutes a strong validation of the nuclear data used in simulations. For the uncertainties, such comparisons constitute the only nonintrusive1 way to validate an existing evaluated covariance file [START_REF] Bauge | Coherent investigation of nuclear data at CEA DAM: Theoretical models, experiments and evaluated data[END_REF].

Fig. 1 .

 1 Fig. 1. Case of correlations between cross sections for 242 Pu(n,γ) and 244 Cm(n,γ). Left: prior correlation matrix without PIE data; middle: posterior correlation matrix built using the PIE measurement from 244 Cm; right: posterior correlation matrix built using the PIE measurement from 245 Cm.

Fig. 2 .

 2 Fig. 2. Case of correlations between various actinide cross sections. Left: prior correlation matrix without PIE data; right: posterior correlation matrix using the PIE measurement from 239 Pu.

Fig. 3 .

 3 Fig. 3. Case of correlations between fission yields from 235 U, 239 Pu and 241 Pu. Left: prior correlation matrix without PIE data; right: posterior correlation matrix using the PIE measurement from 133 Cs.

Fig. 4 .

 4 Fig. 4. Case of correlations between 147 Pm(n,γ) and fission yields from 235 U and 239 Pu. Left: prior correlation matrix without PIE data; right: posterior correlation matrix using the PIE measurement from 147 Sm.

  Rh, 109 Ag, 133 Cs, 143,145 Nd, 147 Pm, 149,151,152 Sm, 155 Eu, 235,238 U, 237 Np, 239-242 Pu, 241 Am, 242-244 Cm, -fission cross sections: 235,238 U, 237 Np, 239-242 Pu, 241 Am, 242-244 Cm, -fission yields for thermal neutron-induced fission:

In this context, nonintrusive means that only the evaluated covariance data file is being examined for validation, without relying on knowledge of the methods and data used to generate it.

The work of D. Rochman was partly supported by swissnuclear, the association of the Swiss nuclear power station operators, with the COLOSS project. It was also partly funded by the European Union's Horizon 2020 Research and Innovation Programme under grant agreement No 847593.

Author contribution statement

Both authors defined the necessary steps to make correlations visible between fission yields and cross sections. D. Rochman performed the various calculations. E. Bauge and D. Rochman performed the analysis of the results. The paper was written by both co-authors.