The Effects of Quartic Term Mathematical Model on The Concentration Profile of Fixed Bed Gas Adsorber

The need for a reliable mathematical model depicting the process inside a column adsorber has become a requisite in designing an effective gas adsorber. Even though this task can be done by using commercial software, it is still important to get an understanding of how the entire process happens. In this paper, we discuss a new way to approximate the concentration profile inside the porous solids. It is an extension of the work of Liaw et al., who adopted a parabolic (i.e., quadratic) profile, which is a function of pellet radius while retaining the spherical symmetry. We extend their work by adding the quartic term. The inclusion of this new term still preserves the form of linear driving force approximation with some correction to Glueckauf’s parameter (i.e., the effective diffusivity coefficient). The addition of the correction will affect the breakthrough curve so that it affects the saturation time. In the binary system of hydrogen/methane discussed in this study, we found that a negative correction to the diffusivity coefficient will make the saturation happen earlier compared to that of the parabolic case, and vice versa. This study may help us design an efficient gas purifier, in particular when we set out for the regeneration of the adsorbent.


A B S T R A C T
The need for a reliable mathematical model depicting the process inside a column adsorber has become a requisite in designing an effective gas adsorber. Even though this task can be done by using commercial software, it is still important to get an understanding of how the entire process happens. In this paper, we discuss a new way to approximate the concentration profile inside the porous solids. It is an extension of the work of Liaw et al., who adopted a parabolic (i.e., quadratic) profile, which is a function of pellet radius while retaining the spherical symmetry. We extend their work by adding the quartic term. The inclusion of this new term still preserves the form of linear driving force approximation with some correction to Glueckauf's parameter (i.e., the effective diffusivity coefficient). The addition of the correction will affect the breakthrough curve so that it affects the saturation time. In the binary system of hydrogen/methane discussed in this study, we found that a negative correction to the diffusivity coefficient will make the saturation happen earlier compared to that of the parabolic case, and vice versa. This study may help us design an efficient gas purifier, in particular when we set out for the regeneration of the adsorbent. Keywords: activated carbon; effective diffusivity coefficient; hydrogen; linear driving force; methane; parabolic profile A B S T R A K Kebutuhan model matematis yang dapat menggambarkan proses penyerapan dalam kolom adsorpsi telah menjadi kebutuhan yang tak terelakkan dewasa ini. Walaupun kini telah tersedia berbagai perangkat lunak komersial, namun tidak dapat dipungkiri bahwa memahami bagaimana proses tersebut terjadi tetap menjadi suatu hal yang berguna. Paper ini bertujuan untuk menampilkan cara baru dalam pemodelan konsentrasi adsorbat di dalam adsorben padat berpori. Kami memperluas metode yang dikembangkan Liaw et al. dengan menambahkan suku pangkat empat (kuartik). Penambahan ini akan mengoreksi koefisien difusivitas efektif dari persamaan linear driving force (LDF). Koreksi yang bernilai negatif, misalnya, akan mengurangi nilai koefisien difusivitas tersebut sehingga akan menghambat kemampuan adsorpsi. Hasil perhitungan kami pada sistem biner hidrogen/ metana menunjukkan bahwa suatu koreksi bernilai negatif dapat menyebabkan saturasi berlangsung lebih cepat dari kasus profil parabolik. Begitu pula sebaliknya, koreksi positif akan menambah daya adsorpsi sehingga saturasi dapat diperlambat. Studi ini kami harapkan dapat diterapkan untuk mendesain suatu kolom adsorpsi yang efisien, terutama dalam perencanaan proses regenerasi adsorben. Kata kunci: hidrogen; karbon aktif; koefisien difusivitas efektif; linear driving force; metana; profil parabolik

Introduction
The application of gas separation has been widely found in industry, especially in the refinery process. One method largely used is adsorption. In contrast to distillation, adsorption is more suitable for low-to medium-scale industries because it does not need large volatility, hence it uses relatively less energy (Ruthven, 1984). Considering its importance, it has become unavoidable that one needs a mathematical model to simulate the process. Such mathematical model is a set of differential equations describing the dynamics of physical variables, such as concentration and temperature of the system, as the functions of time and coordinate (i.e., adsorber length). This allows us to find a set of parameters that generates optimal results (e.g., product purity).
One of techniques found in literature is the approach of Liaw et al. (Liaw et al., 1979), in which the concentration profile inside the adsorbent was modeled by an isotropic with a parabolic (i.e., quadratic) function of radial coordinate. This comes from the fact that any well-behaved function can always be written in terms of power series of its variables, with higher-order terms being more and more negligible. With this assumption, the equation for gas inside the adsorbents can be eliminated, hence reducing the number of differential equations involved. The interesting thing with this approximation is that it could reproduce the Glueckauf's result of linear driving force (Glueckauf, 1955;Glueckauf and Coates, 1947), and more importantly, it could yield breakthrough curves in a good agreement with those derived by exact solution (Rosen, 1952), with considerably less computational time.
Therefore, it is not coincident that many authors have used this approach to a more complex system, see for example (Do and Rice, 1986;Patton et al., 2004;Tsai et al., 1983;Yang et al., 1997;Yang and Doong, 1985) and references therein.
In present work, we extend the work of This paper is organized as follows. In Sect.
2, we will explain our approach, especially the formalism of adding the quartic coupling into the concentration profile. The results will be presented and discussed in Sect. 3. The study is concluded in Sect. 4.

Formalism
The adsorber is a fixed bed column with a binary mixture of hydrogen and methane, while the adsorbent is activated carbon. The sketch of the column is depicted in Fig. 1.
Here denotes the gas concentration side the bulk/column, is the flow velocity (measured at the inlet and assumed to be constant along the column), ‾ is the mean gas concentration within the porous solids, and ≡ /(1 − ) with being the void parameter of the column.
Throughout this article, we will consider several assumptions regarding the column.
First, it is a plug-flow system, and there is no significant axial diffusion flow. This explains the absence of the second derivative term in with and denoting the diffusion coefficient and the adsorbate radial coordinate, respectively. These two equations are related to each other through Eq. (3).
where , as the symbol suggests, is the mole fraction at the surface of the adsorbent with radius , is the mass transfer coefficient, and is the adsorption equilibrium constant.
One can see that in Eq. (1), (or ∂ / ∂ ) is expressed in average form, which is defined as: It is then straightforward to find ∂ ‾/ ∂ . That is, where we have used Eq. (2). Equation (4) suggests that should we know the dependence on radial coordinate, ∂ ‾/ ∂ can be determined. This is the essence of Liaw et al. (Liaw et al., 1979).
We assume that be expressed as a power series expansion (Eq. (6)).
with +1 ≪ . Thanks to the finiteness of , there is no term with negative power of , and therefore such series can be truncated at some power of . We further assume that is spherically symmetric function, meaning that all odd powers vanish or 2 +1 = 0. In the present case we will keep up to ( 4 ) term, so takes the form of: Now by using Eq. (7), we find (∂ / ∂ ) = = 2 2 (1 + 2 4 2 / 2 ). Similar to (Liaw et al., 1979), it would be more useful if one can state this derivative in terms of physical quantities and ‾, namely: q R = a 0 + a 2 2 + 4 4 (8) Taking into consideration that ( 4 / 2 ) 2 is much less than 1, so it can be treated a perturbation parameter, we then obtain Eq.
(10).  Next, we would like to see how enters the differential equation and subsequently affects the breakthrough curves. Apart from , Eq.
(12) pretty much looks like the one discussed in (Liaw et al., 1979). So we follow their procedure to eliminate and to simplify Eq.
(1). We start by transforming variables and .
By taking derivative with respect to on both sides followed by using Eqs. (3) and (1), we can express ‾ in terms of and as Eq. (17).
Now, we eliminate by taking the derivative with respect to followed by using Eq. (15). We finally get Eq. (18). (1 − )) (19) The boundary conditions are given by Eq. (20) = 0 for ≤ 0 and ∀ = 0 for > 0 and = 0, where 0 is the initial gas concentration fed at the inlet.
What we want from solving the aforementioned differential equation is to observe how the adsorption for each molecule occurs. In particular, we would like to see when the breakpoint (i.e., saturation) starts to occur. By knowing it, we can avoid the buildup of the unwanted gas inside the column, so we can start the regeneration process of the adsorbent. For our purpose, the breakpoint is defined to occur when / 0 = 1%.

Model parameters
Having elaborated the required formalism, we now address the parameters used in this article. We first discuss the effective diffusivity coefficient given in Eq.
where is the molecular diffusivity, is the Knudsen diffusivity, and is the tortuosity, i.e., integral, respectively. The procedure of how to calculate these two parameters is outlined in literature, see for example Bird et al. (Bird et al., 2006). As for the Knudsen diffusivity, it is given by Eq (23).
= 9.7 × 10 −3 √ , with being the pore radius, which is much smaller than the adsorbent radius .
The values of all parameters used throughout this article are given in Tables 1 and 2.   (Yang and Doong, 1985) Quantity Value Adsorbent radius 0.028 cm Pore radius 1.6 nm Inter-particle void fraction ( ) 0. and following the procedure of Yang and Doong (Yang and Doong, 1985), we obtain the effective diffusivity coefficients of hydrogen and methane to be 4.1 × 10 −4 cm 2 /s and 1.9 × 10 −4 cm 2 /s, respectively.
Despite having relatively larger , it is not always true to say that the activated carbon will adsorb hydrogen better than it does on methane. It turns out that the equilibrium constant (see Eq. (18)) proves to play the upper hand. From the Langmuir relation, the equilibrium constant of hydrogen is found to be 2 ≃ 0.00012, which is almost two orders of magnitude smaller than 4 ≃ 0.02. Such equilibrium constants resemble the lowpressure regime of Park et al. (Park et al., 1998). Higher indicates that the corresponding molecule is more likely to occupy the sorption sites, and vice versa.
Considering these values, it is expected that hydrogen will get saturated earlier than methane, so this column can be considered as a hydrogen purifier.
Our conjecture above is supported from This can be understood by recalling that the negative will lower the effect of (see Eq. However, it should be noted that the value of is limited by perturbativity; that is, it cannot be much grater or even very close to one. The value of | | = 0.6 in our example is barely considered perturbative, and thus it is presented here only for the sake of illustration. In reality, it seems more plausible to have much smaller | | whose precise value can only be determined through data fitting.

Conclusions
We have extended the work of (Liaw et al., 1979