Theoretical evaluation of the tritium extraction from liquid metal flows through a free surface and through a permeable membrane

Effective tritium extraction from PbLi flows is a requirement for the functioning of any PbLi based breeding blanket concept. For a continuous plant operation, the removal of the tritium dissolved in the PbLi has to be performed in line and sufficiently fast. Otherwise, tritium inventories in the liquid metal, start-up inventories and buffer inventories would be excessive from the safety point of view. Moreover, a slow response of the tritium extraction systems could also compromise the tritium self-sufficiency of the plant. A promising solution to this problem is to use highly permeable membranes in contact with the PbLi flow to promote the extraction via permeation. This technique is usually known as Permeation Against Vacuum (PAV). As an alternative, tritium could be extracted directly by permeation through a fluid free surface (FS) in contact with vacuum. In both configurations, the dynamics of tritium transport is ruled by a combination of convection, diffusion and surface recombination. In this paper, the tritium extraction processes in the FS and PAV configurations are studied in detail. For the first time, general analytical expressions for the extraction efficiency are derived for both techniques in a Cartesian geometry. These expressions are general in the sense that they do not impose any kind of assumption concerning the permeation regime of the membrane or the fluid boundary layer. The derived expressions have been used to analyze numerically the response of both configurations in a close loop system, such as the one of DEMO. The presented methodology allows comparing the FS and PAV configurations, assessing in which conditions one will be behave better than other.


Introduction
Liquid breeding blankets (BBs) are considered promising candidates for future fusion reactors. Among other advantages, metal flows are good tritium carriers that drag the tritium atoms generated inside them out of the reactor. The most developed concepts are based on eutectic PbLi, such as the Water Cooled Lithium Lead (WCLL) (e.g. [1]), the Dual Coolant Lithium Lead (DCLL)(e.g. [2]) or the Helium Cooled Lithum Lead (e.g. [3]). There are other advanced blanket concepts that are based on molten salts such as FLiBe or ClLiPb [4].
Any of these concepts requires a so-called tritium extraction and removal system (TERS) connected in line with the blanket in a closed loop configuration to effectively remove the tritium solved in the flow. Among the many different techniques identified in early reviews [5], the tritium permeation through solid membranes and the gas-liquid contactor (GLC) have attracted much of the attention. Indeed, the latter technology, which is used in many industrial applications, has been selected for the PbLi based European Test Blanket Module (TBM) within the International Thermonuclear Experimental Reactor (ITER) project [6]. GLC is being considered as a candidate for the TERS of the European Demonstration power plant (DEMO) as well. However, scaling this technique up to DEMO size is considered an important challenge [7]. For designing the DEMO TERS, the so called Permeation Against Vacuum (PAV) technique has been chosen as a reference for many years [8].
The PAV technique consist of using highly permeable solid membranes in contact with the liquid metal to promote the tritium transport from the liquid to a void cavity at the other side of the membrane. These systems, usually called permeators, have been experimentally tested in gas phase using Pd-Ag membranes [9]. However, the experimental validation of the technique using liquid metals is still pending. New PbLi facilities have been built for addressing this challenge in the near future [10][11][12].
From the theoretical perspective, the extraction efficiency of the PAV was firstly evaluated analytically by Humrickhouse and Merril within the framework of the US DCLL TBM [13]. The expressions derived in that work have been used for the conceptual design of the TERS for the EU-DCLL blanket for DEMO [14,15]. This first analytical model does not include any surface effects in the membrane-void interface, which are assumed to be much faster than diffusion through the membrane. This is usually referred to as diffusion-limited (DL) regime. This assumption is only satisfied for sufficiently high concentrations in the liquid metal which is not necessarily the case according with the results of system level models [16,17].
The so-called surface-limited (SL) approach has also been considered for designing PAV extractors [18]. This approach neglects the diffusion process in the mass transport, which is again not necessarily true in the real application. Indeed, both surface and diffusive equations should be taken into account together as they can be comparable. As a consequence, the PAV extraction efficiency will depend on the inlet concentration as stated by D'Auria et al [19]. This complicates the design of TERS whose response is now intrinsically coupled with the tritium dynamics in the blanket and other subsystem. In this paper, the extraction efficiency in a Cartesian geometry is solved analytically for the first time for any concentration scale at the TERS entrance.
Within the framework of the EUROfusion project, it has been recently suggested that removing the membrane could be beneficial for the extraction. Tritium could recombine fast enough in a free surface (FS) of liquid metal exposed directly to the vacuum. This approach is motivated by the relatively high recombination constant computed for the PbLi [20] and by recent measurements of the permeability of membrane candidates (niobium, vanadium and tantalum) which are some orders of magnitude below previous measurements [21,22].
Conceptually, the FS extractor is not different than the so called Vacuum Sieve Tray (VST) which has been proposed as an advanced alternative for PbLi extractors [23,24]. This technique is based on dividing the liquid metal into small falling droplets. This approach increases the permeation surface while decreases the diffusion length. The VST has shown a promising extraction efficiency however, scaling this configuration to the large PbLi flow rates needed even for low velocity blankets, such as the WCLL, is very challenging. Alternatively, the PbLi can circulate in horizontal channels with the top surface exposed to vacuum. Despite this avoids dividing the flow into many droplets, the difficulties associated with working with a free interface make scaling this technique up to DEMO size also difficult. The horizontal FS is one of the configuration studied in this paper.
Analytical expressions for the FS and PAV extractors are derived in sections 2-5. Section 6 shows the asymptotic regimes for the derived expressions (membrane SL and DL among others). It is shown that the membrane DL regime solution recovers the original expression of Humrickhouse and Merril [13]. Finally, in section 7 the analytic solutions are used to solve the problem in which the extractors are connected in line with an external tritium source following a close loop configuration. Numerical techniques are necessary for this final step. The conceptual response of PAV and FS is compared in relevant conditions for both high and low velocity blanket concepts.

Methodology
The problem to be analyzed is a pressure-driven fully developed horizontal flow of liquid metal with two configurations: the first considers the top surface of the flow in direct contact with vacuum (FS configuration) while the second considers a solid membrane in between the liquid and the vacuum (PAV configuration). Figure 1 depicts a sketch of the expected velocity and concentration profiles in the two configurations. The velocity profile differs in both cases because of the top boundary condition. The concentration profile can be divided into two different regions: the core and the boundary layer. The core concentration (c c ) is constant along the cross-section of the channel but in the boundary layer where the concentration drops from the core value to the interface value. The thickness and depth of the boundary layer depends on the liquid properties, the characteristics of the flow and the transport processes in the interface. In liquid metals, thin boundary layers are typically expected.
For simplicity, the problem is assumed to be 2 dimensional. This approximation is valid when the channel width is much larger than its height. This assumption is normally satisfied for the permeators that follow a Cartesian geometry (e.g. [14]) since this choice maximizes the permeation surface while minimize the diffusion length.
It can be argued, that the PAV configuration should include two membranes; one at the top and other at the bottom of the channel, not only one as depicted in figure 1. Despite this making perfect sense from a design perspective, the objective of the present analysis is to compare both configurations at the fundamental level. Therefore, it has been preferred to keep the same permeation surface in both configurations. In any case, it will be shown that including the second membrane in the final expression of the PAV extraction efficiency is straightforward.
In the steady state, the concentration field (c(x, z)) follows to the mass continuity equation: where u z (x) is the fully developed velocity field of the liquid metal and J k (x, z) is the tritium flux inside it. Arbitrarily, z is the coordinate along the flow direction and x the direction perpendicular to it. The tritium flux follows the first Fick's (J k = −D∂ k c), being D the tritium diffusivity inside the liquid. Solving this 2D equation analytically is normally complex. It can only be done for simple velocity profiles and boundary conditions (e.g. Couette flow and constant boundary flux [25]). Naturally, inside the solid domains (u = 0) the equation reduces to the second Fick's law. Inside an isothermal and rectangular membrane, such as the one considered for the PAV, the solution is simply a linear profile along the transversal coordinate x.
Instead of solving (1), using the same mass conservation arguments than Humrickhouse and Merrill [13], it is possible to obtain a simpler 1D axial transport equation. It is obtained by integrating the tritium flux through a rectangle of an infinitesimal thickness and whose side is the channel height (dashed rectangle in figure 1).
where U 0 is the average velocity in the channel (U 0 := h 0 u(x)dx/h), h is the height of the channel and J s (z) is the tritium flux normal to the surface J s (z) = J(x = h, z). To obtain (2), the advective tritium flux through a given cross-section is approximated using the core concentration: Equation (2) can be integrated along the total length (L) of the channel: Solving (3) provides an expression of the core concentration at the exit of the channel in terms of the inlet concentration and the parameters of the problem. Therefore, there is a three step methodology for each extractor configuration: first expressing the surface flux in terms of the core concentration, then integrate the inverse of the flux and finally solve the core concentration.
Once knowing the core concentration, it is immediate to obtain analytical expressions for the tritium extraction efficiency of the FS and PAV channel configurations. The extraction efficiency (η) is defined with the ratio between the tritium inlet concentration (c 0 ) and the tritium core concentration at the exit of the channel (c c (L)):

Mass transport through the boundary layer
In both extractor configurations, before being extracted tritium needs to go through the boundary layer. A widely used approximation is the linearization of the mass flux. This is used, for example, when imposing the so-called convective boundary conditions: where k t is the mass transfer coefficient and c s is the concentration at the surface (either free or in contact with a membrane). It is worth noting that both the core and surface concentrations depend on the longitudinal coordinate (z). The mass transfer coefficient depends, among other things, on the velocity profile. Therefore, this is expected to be different for both configurations. Since the velocity profile next to the membrane is zero (no slip condition), it is expected a higher convection transport (i.e. higher k t ) for the FS configuration than for the PAV.
It is common to express the influence of the mass transfer coefficient in terms of the Sherwood number which in these channels it is defined as: where D is the diffusivity constant of tritium in the liquid. The Sherwood number represent the ratio between convection transport and diffusion transport. Since diffusion is always present, Sh minimum value is unity. There are many empirical correlations available in literature which relates Sh with the Reynolds number (Re) and the Schmidt number (Sc). They usually adopt the following shape: where D h is the hydraulic diameter of the channel, ρ the liquid density and µ its dynamic viscosity. According with the experience of Komori et al [26], the mass transfer through the FS goes with Sc 1/2 . This contrast with the flows through horizontal plates, such as the PAV membrane, which goes with Sc 1/3 (e.g. [27]). The Re exponent depends on the flow regime. It is approximately 1/2 for laminar regimes and close to unity for turbulent regimes (e.g. 0.83 according to [28]).
It is worth noting that most of the empirical correlations were derived using other fluids rather than liquid metals. Theoretically, the differences between standard fluids and liquid metals are well represented by the value of the Sc number. However, dedicated experiments with liquid metals (specially PbLi) are definitely necessary to reduce the uncertainty introduced by the correlations.

FS configuration
In the FS configuration, the liquid metal is in direct contact with void (p T2 = 0). In this analysis, it is considered that there are no chemical species solved in the PbLi other than tritium. Therefore, in order to reach the vacuum, two atoms of tritium need to recombine and form a molecule of T 2 . In these conditions the flux of tritium atoms is proportional to the square of the concentration at the interface: where σk 2 is the recombination coefficient of the liquid. The factor 2 is present to account that for every molecule of T 2 that is recombined, two atoms of tritium leave the flow.
In the steady state, equations (5) and (8) has to be equal in every axial location. This leads to the following second order algebraic equation: whereĉ is the surface concentration normalized by the core concentration and W is the so called permeation number defined as: The permeation number represents the ratio between the time scale of diffusion transport over the time scale of the surface processes. Originally, the permeation number was introduced for gas-solid-gas systems and it is dependent on the gas pressure at one side of the solid [29]. In this case, despite defining W in terms of the core concentration, the conceptual meaning is exactly the same. Indeed, the well-known limiting cases of DL and SL regimes arises when W ≫ 1 and when W ≪ 1, respectively.
Equation (9b) reminds to the equation that appears when solving the problem of a solid membrane of thickness t in contact with a c 0 concentration in one side and vacuum conditions in the other side. The only difference is that k t plays the role of D/t.
The analytical solution of (9b) is the same solution than the one of the solid system [30] but with the mentioned substitution: We have called the dimensionless number W d as dynamic permeation number. It clearly represents the ratio between the time scale of the convection mass transport and the time scale of the surface processes. For a stagnant liquid in which convection is equal to diffusion (Sh = 1) the dynamic permeation number reduces to the regular permeation number in solids.
Likewise, it is possible to define two limits cases: the convection-limited (CL) regime in which convection is much slower than surface recombination (W d ≫ 1 and f(W d ) ∼ 1) and the dynamicsurface-limited (DSL) regime in which surface processes are much slower than convective transport through the boundary layers ( It is interesting noting that W d ⩽ W which means that the fluid motion will tend to move the regime towards DSL regime. Indeed, even in a stagnant condition in which W ≫ 1,  Once knowing the analytical expression of the tritium flux in terms of the core concentration, the next step is solving the integral (3). For this purpose, it is necessary to perform the following change of variables: This new variable allows writing the flux in a simpler way: With the previous equations, it is possible to solve (3): where ξ L and ξ 0 are ξ(c c = c(L)) and ξ(c c = c 0 ), respectively. Rearranging the variables: The dimensionless number τ fs was already identified by Humrickhouse and Merrill for the PAV configuration [13]. It can be called extractor number. It represents the ratio between the time scale of the liquid dynamics (L/u) and the time scale of the mass transport through the boundary layer (h/k t ). If τ fs is small, it means that the tritium is transported by the flow from the entrance to the exit of the channel in a time scale much smaller than the time needed to be extracted. Logically, the extractor will present poor efficiency in those conditions.
It is worth noting that the physical meaning of τ fs is not much different than the so called Stanton number (St). Indeed, both numbers are closely related: The factor D h /h appears because it has been chosen to use the hydraulic diameter for the definition of Re and the channel height for the definition of Sh.
It is possible to expresses ξ L in terms of only ξ 0 and τ fs by means of the so called Lambert ω function, in particular its principal branch ω 0 : The ω function can be seen as a generalization of the logarithm and it is defined as the inverse of the function f(ω) = ω exp ω. The principal branch (ω 0 ) is the only solution in the positive real domain. Since both τ fs and ξ 0 are strictly positive variables in the entire physical domain, the Lambert function is well defined.
Undoing the previous change of variables (using (12b)), the core concentration at the exit can be obtained: Introducing (18a) in the definition of the efficiency (4) and using (17), it is possible to obtain the following analytical solution for the extraction efficiency of the FS extractor: The extraction efficiency depends on only two dimensionless numbers: the extractor number τ fs and the dynamic permeation regime at the entrance of the extractor W d0 .   [13]. The asymptotic solutions for the FS are derived in section 6.
Analyzing the shape of the contours, it is possible to give an approximate design criterion for a Cartesian FS extractor. In order to have a minimum extraction efficiency of 20%, the extractor number τ fs needs to satisfy: This criterion is indicated with black dots in figure 3.

PAV configuration
In the case of the PAV configuration, the recombination of molecules takes place on the membrane interface exposed to the vacuum. The flux in this interface is given by (8) as well. However, in this configuration the recombination constant σk 2m is logically a membrane property. The steady state flux inside the membrane is described by the same solution than in the FS liquid (11a) but substituting the dynamic permeation number by the membrane permeation number (W m ). For consistency, this permeation number has to be defined using the membrane concentration at the interface PbLi/membrane (c i ): The sub-index m is used to name a property of the membrane. For example, D m is the membrane diffusivity and t m is the membrane thickness. The similarity between (21a) and (11a) is evident.
Assuming a continuity of the chemical potential, the concentration across the interface exhibits a discontinuity given by the tritium solubility at the liquid (K l ) and at the membrane (K m ): Equalizing the flow through the membrane (21a) with the flow through the boundary layer (5) and using (22), the following relation between the core concentration in the fluid and the membrane permeation number can be obtained: The variable ζ is called the membrane number and it is defined as: This dimensionless number represents the ratio between the diffusion transport through the membrane and through the liquid boundary layer. It was firstly identified in [13] when studying the PAV configuration in DL regime and cylindrical coordinates. Equation (24) is slightly different as it is adapted to the Cartesian coordinates.
Unfortunately, it is not trivial to inverse (23) and express c i as a function of c c . Nevertheless, this is not necessary to perform the integration of J −1 s (c c ). Indeed, it is enough to express the integral in terms of W m : where W m0 and W mL are the membrane permeation numbers at the entrance and exit of the channel, respectively. The derivative of (23) is written as follows: Using (26) on (25), equation (3) is written as follows: The obtained integral has three pieces. The first one is the same that has been already solved for the FS configuration. It is solved using the same change of variables: The other two pieces are immediate logarithmic integrals. Therefore: ) .
The left side of (28) is simply the product of the PAV extractor number τ pav and the membrane number ζ. Besides, taking into account (12c) it is possible to simplify (28): Like in the FS case, (29) can be solved by means of the Lambert ω 0 function: This expression could allow deriving a extraction efficiency based on the permeation number in the membrane. This is not considered optimal since the concentration scale at the entrance is given in terms of the concentration at solid side of the interface. However, it is much more convenient to give the extraction efficiency in terms of the concentration in the liquid core which should be the quantity measured and controlled in the real applications (either a reactor or an experimental facility).
For this reason, it has been considered convenient to introduce a new permeation number called mixed permeation num-berŴ which is defined using the properties of the membrane but with the concentration scale of the core. Moreover, the ratio between the solubilities has been introduced in the definition to keep the same concentration scale than W m : The variable ξ m can be expressed in terms ofŴ instead of W m using (12b), (12c) and (23). After solving the equation it can be proven that: The final step is undoing the change of variables by using (12a) and (12b) on (23): The extraction efficiency of the PAV configuration is then obtained using (30a) on (33a) for computing the core concentration at the exit of the channel: It can be checked that if a second identical membrane is added to the problem, the solution derived is maintained but with an extractor number twice as big as with one membrane (τ pav /2 → τ pav ). Figure 4 depicts the contours of (34a) as a function of τ pav and ζ for four different values of the entrance mixed permeation number (including the DL and SL regimes).
Analyzing figure 4 it can be observed that an extractor with a sufficiently permeable membrane give good extraction efficiencies at smaller values ofŴ 0 . Nevertheless, there is again a threshold for the extractor number τ below which the efficiency is very poor, no matter the value of ζ orŴ 0 .
The extraction efficiency of the PAV can therefore be expressed only in terms of three dimensionless variables: The PAV extractor number (τ pav ), the membrane number (ζ) and the mixed permeation numberŴ 0 at the entrance of the extractor. Nevertheless, from the mathematical point of view, the three can be grouped in only two independent variables which are: ζ ζ+1 τ pav and (1 + ζ)Ŵ 0 . Taking this last dependency into account it can be checked that the efficiency of the PAV and the FS configuration can be related as follows: Since η fs is a monotonic increasing function with respect both arguments, it is possible to ensure that η pav > η fs if: The opposite relation is also true. Unfortunately, in the real application it is expected that τ fs > ζ ζ+1 τ pav while W d0 < (1 + ζ)Ŵ 0 . Therefore, comparing both configurations in fusion conditions is not straightforward.
Nevertheless, using (35) it is trivial to derive a design criterion for a PAV extractor. Just by analogy with (20a) and (20b), it is possible to assert that for having a PAV extractor with an efficiency higher than 20% it is necessary to satisfy the following requirements:

Asymptotic solutions
The two solutions obtained from the extraction efficiency (19a) and (34a) can be further simplified when considering limit cases in which some of the dimensionless variables take extreme values. For doing so, it will be necessary to take into account the following properties of the Lambert ω 0 function: . (38d)

Low-efficiency (LE) regime
The first interesting regime is the case in which the advective transport through the channel is much faster than the extraction. This means τ fs ≪ 1 or equivalently τ pav ζ ζ+1 ≪ 1. It has been shown that in this regime the extractor will present poor efficiency, but this situation could easily happen for small length devices in laboratory scale. In the case of the PAV extractor, this regime can be also obtained for poorly permeable membranes.
Up to first order, the exponential function (19b) can be expressed as follows for low values of τ fs :  Figure 5 depicts a comparison between the general efficiency expression (19a) and the LE asymptotic regime (40a). It is clear that the validity of the low efficiency regime depends on the dynamic permeation regime.
Using relation (35), the LE regime efficiency for the PAV configuration can be directly obtained: It can be checked from the previous expressions that it is impossible to obtain decent extraction efficiencies for low τ values. Indeed, even in the most favorable scenario in the limit in which W d0 ≫ 1 orŴ 0 ≫ 1 the extraction efficiency goes linearly with the extractor number, which is low by assumption:

FS: convection-limited (CL) regime
This regime is applicable to the FS configuration when the convective transport is much faster that the surface recombination (W d0 ≫ 1). In these conditions: ξ 0 ∼ √ 8W d0 and the efficiency can be rewritten as follows: (43a) However, for sufficiently high values of W d0 it is possible to get to the point in which 8W d0 ≫ exp τ fs . Under this assumption Ω fs ≪ 1 and using (38a) the following asymptotic expression is obtained: η fs;CL (τ fs ) = 1 − exp(−τ fs ). (44)

FS: dynamic surface-limited (DSL) regime
This regime is applicable to the FS configuration when the convective transport is much slower that the surface recombination (W d0 ≪ 1). In these conditions: ξ 0 ∼ 4W d0 and Ω fs ≫ 1. Using (38b) and neglecting quadratic and logarithmic terms the following expression is obtained:

PAV: liquid-limited (LL) regime
This limit is obtained in the PAV configuration when the mass transport through the BL is much slower than the diffusion transport through the membrane (i.e. ζ ≫ 1). This regime is particularly interesting for the technological application since the membrane is selected by design to be as much permeable as possible.
The LL regime is characterized by: ζ ζ+1 τ pav ∼ τ pav and (1 + ζ)Ŵ 0 ∼ ζŴ 0 . There are not further simplifications that can be applied without making assumptions on the permeation regime inside the membrane.
However, if by assumption the mass transport through the BL is the absolute limiting process then, the mass transfer through the BL is much slower not only than the diffusion transport through the membrane but also much slower than the surface recombination rate in the vacuum interface. It can be checked that this means not only ζ ≫ 1 but also ζŴ 0 ≫ 1. Therefore, similarly than in the case of CL regime of the FS extractor, the extraction efficiency in LL regime can be written as follows: The previous expression presents the following asymptote when 8ζŴ 0 ≫ exp τ pav : (47) Figure 7 depicts the PAV extraction efficiency as a function of the membrane number together with the LL regime solution for some combinations of τ pav andŴ 0 .
It is worth noting that the opposite limit regime, sometimes called as membrane-limited or solid-limited regime (ζ ≪ 1) is a particular case of the LE regime introduced in section 6.1 when ζτ pav ≪ 1.

PAV: diffusion-limited (DL) regime
This regime is obtained in the PAV configuration when the diffusion processes in the membrane are much slower than the surface recombination in the vacuum side (Ŵ 0 ≫ 1). The DL regime is probably the one that has received more attention in previous works (e.g. [13,14]). For high values ofŴ 0 it is always true that (1 + ζ)Ŵ 0 ≫ 1. Therefore, relation (35) can be directly applied to (44) to obtained the following asymptotic expression: Equation (48) is the original equation firstly deduced by Humrickhouse and Merrill [13].

PAV: surface-limited (SL) regime
This regime is obtained in the PAV configuration when the diffusion processes in the membrane are much faster than the surface recombination in the vacuum side (Ŵ 0 ≪ 1). Under these conditions little simplification can be made on (34a) other than changing a bit (34c): It is necessary to assume that surface processes are also much slower than the mass transfer through the boundary (i.e. ζŴ 0 ≪ 1) to obtain a simplified equation equivalent to (44):

Closed loop configuration
It has been observed that the extraction efficiency of both FS and PAV configurations depends strongly on the concentration scale at the entrance of the channel. Therefore, from a mathematical perspective, the problem is not closed only with the design specifications of the extractor channel.
In the real application, the TERS will be placed in a closed loop configuration in which the blanket will act as an external source of tritium. The blanket tritium source is pulsed in a tokamak but for the sake of simplicity, in this work it is assumed a continuous time-averaged generation rate (G (mol s −1 )).
In the steady state, the sources and sinks of the whole system must balance. Therefore, the generation rate in the blanket must be equal to the extraction rate the TERS (E (mol s −1 )) plus the tritium loss rate(P (mol s −1 )). In this work, this last term is neglected since loss rate are needed to be much smaller than the generation rate in a working fusion plant.
Under this assumption and using the mass balance equation (2), it is obtained the extra equation needed to close the system: where N is the total number of extraction channels and Q is the total volumetric rate of PbLi: Q = N · U 0 A (A is the channel cross section). The concentration scale c * = G/Q is interpreted as the minimum allowable concentration at the entrance of the extractor (or the exit of the blanket). The extra equations of the FS and PAV configurations can be written as follows: Or in dimensionless form: where W * d andŴ * are the dynamic permeation number (11c) and the mixed permeation number (33b), respectively, defined using the c * concentration scale.
Unfortunately, equations (53a) and (53b) are not analytically solvable except in some of the simple asymptotic cases described in section 6. For this reason, they have been solved numerically. For this matter it is convenient to use relation (35) which allows the solving of only one system of equations since: The numerical solution obtained for equations (53a) and (53b) is depicted in figure 8. It has been chosen to express the solution in terms of c 0 /c * . This is motivated by the fact that the technological challenge of the TERS is minimizing c 0 and not only maximizing the extraction flux. Indeed, in the steady state, every set of parameters of both extractor configurations would provide very similar extraction fluxes (equal when neglecting the system loss rate). The difference would be the concentration scale (and therefore the tritium inventory) needed to achieve such fluxes. If this scale is too high the permeation losses might even be comparable to the generation rates, which is logically unacceptable for the design perspective.
The numerical results have been fitted to the following power function in the range of W d * ⩽ 1 following the same methodology applied in [31]: For the PAV configuration, in the range of (1 + ζ)Ŵ * ⩽ 1 the fitted solution is: For values of W d * > 1 (or (1 + ζ)Ŵ * > 1) the ratio c 0 /c * shows a very weak dependence on W d * (or (1 + ζ)Ŵ * ) and the data fits quite well to the following simpler function of τ : The extraction efficiency can be recovered by simply making c * /c 0 . For most design parameters (e.g the membrane properties or the channel geometry), minimizing c 0 is equivalent to maximizing the extraction efficiency. However, the total PbLi flow rate is an interesting exception. Increasing the total flow rate would decrease the efficiency by decreasing the extractor number τ . However, increasing Q also decreases the minimum allowable concentration c * which decreases also c 0 . Equations (55a) and (56a) shows that the latter effect is stronger than the former. Therefore, increasing the total flow rate is always beneficial for having smaller concentration in PbLi.
This effect explains the observed result in which high velocity blanket concepts such as the DCLL does not need a high extraction efficiency in order to keep the tritium inventory low [16]. In comparison, low velocity blanket concepts are expected to require of higher efficiencies for having the same inventories.
The generation rate follows the opposite behavior. Increasing the generation in the blanket would mean high extraction efficiency but at the cost of having a high concentration scale in PbLi.

Comparison between FS and PAV configurations
The previous fitted expressions can be used to compare the response of the FS and PAV configurations. It is important mentioning that the comparison between both configurations is made assuming the same geometrical and operational conditions for both kinds of channels. That means that this analysis does not value the technological limitations associated to any of the technologies. For example, it is likely to be more difficult to spread the flow across several channels in the case of the FS configuration than in the PAV configuration. However, this kind of design dependent aspects are not considered in this work which is focused on comparing both configurations at a fundamental level.
As explained in section 3, for the same flow rate and the same geometry, the mass transfer coefficient of the FS configuration is expected to be higher than the one in the PAV configuration: τ fs > τ pav > ζ 1+ζ τ pav . Therefore, taking into account (54), a first criterion can be obtained by comparing the other arguments of the efficiency: W d * and (1 + ζ)Ŵ * : where the sub-index l denotes the properties of the liquid and the sub-index m denotes the properties of the membrane. Using the definition of the membrane number (24) and the relation between the solubility and the surface constants: K = √ σk1 σk2 : It is possible to assert that the FS configuration would always response better (i.e. with lower c 0 ) than the PAV configuration if W d * > (1 + ζ)Ŵ * which can be written as: Condition (60) is not necessarily satisfied for highly permeable membranes (e.g. vanadium or niobium) which have high tritium solubilities and likely high dissociation coefficient (σk 1 ). Besides, the high dispersion found in the literature concerning the tritium solubility in PbLi [32] also makes it difficult to evaluate if condition (60) is satisfied or not.
It is worth noting that if the PAV configuration is considered to have two membranes per channel then it is not guaranteed that τ fs > ζ 1+ζ τ pav . Therefore, relation (60) is not enough to guarantee a better performance of the FS over the PAV configuration.
If (60) is not satisfied then the situation is more complex to analyze. The threshold between both configurations is obtained equalizing equations (55a) and (56a) which allows to write the following expression for the critical value of τ pav : The previous equation is valid for (1 + ζ)Ŵ * ⩽ 1 and W d * ⩽ 1. However, for a sufficiently permeable membrane, it should be possible to achieve the regime in which (1 + ζ)Ŵ * > 1. In this case and assuming that still W d * ⩽ 1, the critical value of τ is simpler: If τ pav is above this critical value τ cr , then the PAV configuration would present a better response (lower c 0 ) than the FS configuration. Figure 9 depicts a comparison between the c 0 /c * of a PAV in LL regime (ζ ≫ 1) and the FS for different values of W d * . A black arrow illustrates the difference between τ fs and τ cr . This difference (∆τ ) is plotted in figure 10 as a function of τ fs for different values of W d * . Figure 9 shows that the LL PAV and FS responses would be very similar in case of sufficiently high τ or sufficiently high W d * . However, in the regime in which W d * ⩽ 1, the membrane number seems to be an effective way of reducing the concentration scale in the system.
The difference between τ fs and τ cr showed in figure 10 is a measure of the difference in the design requisites between the FS and LL PAV configurations. Indeed, if the PAV were able to provide the same response (in terms of c 0 ) than the FS with lower τ , the constrains on the extractor design would be lighter (e.g. less extractor channels or shorter channels). Since  ∆τ grows fast with τ fs the requirements of the LL PAV configuration are comparatively reduced for effective extractors. Nevertheless, this situation is only true if the FS is working in the regime in which W d * < 1 which is expected to be true in most blankets designs as shown in the following subsection.
Naturally, the critical value of τ pav for other membrane parameters can be derived using a similar approach but in this case the τ cr would be dependent of ζ.
A scenario with a poorly permeable membrane (ζ = 1) is depicted in figure 11. Even if assuming (ζ + 1)Ŵ 0 > 1, the FS configuration behaves better (except for very low values of W d * ).

WCLL and DCLL conditions
It is of interest to give some numbers under conditions relevant for the WCLL (low velocity blanket) and the DCLL (high velocity blanket). The same DEMO relevant generation rate of 10 −3 mol s −1 (around 300 g d −1 ) is assumed but there are differences between the operational conditions of the WCLL (ṁ ∼ 10 3 kg s −1 and T ∼ 600 K) the DCLL (ṁ ∼ 2.5 × 10 4 kg s −1 and T ∼ 823 K). For making the comparison, the PbLi properties recommended in the review made Martelli et al [33] have been used together with tritium diffusivity of Reiter [34] and the recombination coefficient calculated by Pisarev et al [20].
A sufficiently permeable membrane (ζ = 10 and ζŴ * > 1) is considered in case of the PAV configuration. It should be noticed that this condition might be too restrictive for available materials. With the high hydrogen diffusivity of vanadium and niobium from some references this should be achievable [35]. However, both the dispersion in the PbLi solubility and the membrane properties [21] can jeopardize this assumption, specially at low temperatures (WCLL). Moreover, very recent measures of the dissociation constant in vanadium [36] has shown moderate values.
It is also necessary to assume a certain geometry for the channel extractors. The same design proposed in [14] has been used (channels of 80 × 5 mm 2 and 5 m length). A total of 500 extractor channels have been selected as a tentatively amount. It should be noted that this extractor design is not optimized for these conditions and it should be seen as a framework for comparison purposes.
The final piece needed is the correlation for the Sherwood number (7a). For the FS configuration a correlation based on the experiments of Komori has been used [37]. In the case of the PAV configuration the correlation of Berger and Hau [28] has been used. Both correlations assume turbulent regime.
The results of the comparison are depicted in table 1. The first set of rows are the input parameters used for the comparison. The second and third set of rows are the results of the FS extractor system obtained when using the correlation (55a) and when solving numerically (53a), respectively. There is a deviation of around 10% between them.
The fourth set of rows depicts the critical value of τ pav necessary to design a LL PAV-based extractor system equivalent to the described FS-based one. N cr is the number of extractor channels necessary to obtain the same concentration scale than 500 FS extractor channels.
It is clear that the PAV configuration with a sufficiently permeable membrane (ζ = 10 and ζŴ * > 1) have much lighter design requisites than the FS configuration in both WCLL and DCLL conditions which in this example is translated into less channels. This important difference is caused by the value of W * d which is below 1.
The results of this comparison show also the already mentioned behavior in which a high velocity blanket requires a lower efficiency to work with lower tritium concentrations. The obtained concentration scales are in the order of magnitude of the ones computed by system level models [16,17].

Conclusions
This paper presents a comprehensive theoretical study of the tritium extraction process in rectangular liquid metal channels. Two configurations have been considered which involve different physical interactions: the permeation through an FS and through a permeable membrane (PAV).
For the first time, general, independent of the permeation regime, analytical expressions of the extraction efficiencies at the steady state have been derived for both configurations. Despite the high amount of design parameters and material properties involved in the system, it has been proven that the solution can be expressed in terms of few dimensionless numbers. The solutions found follow asymptotically the known permeation-dependent solutions (e.g. diffusion-limited solution).
The analytical solutions have been used for evaluating the suitability of both configurations for the TERS system of the EU-DEMO. It has been necessary to use numerical techniques to solve the final algebraic equation in which the extractor channels are connected in line with a tritium generation source (the BB) following a close loop configuration.
The capability of both configurations to minimize the tritium concentration in PbLi have been compared. The comparison has been performed at a fundamental level, assuming the same channel geometry and operational conditions. This means that possible technological and design constrains of any of the two configurations have not been taken into account.
It has been found that the FS configuration has the potential to response better than any PAV design. However, the requisites on the PbLi properties and blanket operational conditions to do so are quite strict for fusion applications (W d * > (1 + ζ)Ŵ * ). In fact, with typical operational values of the WCLL and DCLL blankets and the available PbLi properties, the necessary conditions are not satisfied. Nevertheless, the uncertainty on the PbLi properties could change this fact.
Likewise, it has been found that a PAV configuration performing a sufficiently permeable membrane would require one order of magnitude less extractor channels (less permeation surface) than the equivalent FS configuration for both WCLL and DCLL conditions. Nonetheless, this assertion is based on the fact that good permeable membranes are available. Conditions for the optimal membrane properties have been deduced in this work (ζ > 10 and ζŴ * > 1). It is also uncertain if the typical material candidates such as vanadium or niobium would fulfill them, specially at low temperatures (WCLL conditions).
For high PbLi temperatures (DCLL conditions), the requisites of the membrane properties are lighter and more likely acceptable. However, for WCLL conditions the FS configuration might be more suitable, depending mainly on the transport properties of PbLi and membrane at the operational temperature (∼600 K). Complementary, it has been proven that the efficiency requisites on the TERS are reduced when increasing the total PbLi flow rate. Therefore, a high velocity blanket would exhibit lower tritium inventories even with smaller extraction efficiencies.
Finally, it should be mentioned that the whole analysis is based on the steady state behavior of the system. Transient effects have not been analyzed. In equivalent conditions, it is expected that FS configuration has a shorter transient period than the PAV configuration. This difference can be small depending on the diffusivity of tritium in the PAV membrane.