Computational Modelling of the Hydride Generation Reaction in a Tubular Reactor and Atomization in a Quartz Cell Atomizer ()
1. Introduction
The hydride generation (HG) technique coupled with the spectrometric analysis is widely applied for detection and determination of the hydride forming elements (e.g. arsenic, antimony, bismuth, germanium, lead, selenium, tellurium, and tin) and has been applied recently for other group determination including transition and noble metals (e.g. platinum, cobalt, silver, copper, zinc, rhodium, palladium, osmium, chromium, gold, nickel, indium, thallium, and manganese) [1-3]. The working principle of the hydride generation technique relies on the reduction of the element from higher oxidation state to its lowest state (usually II or III), which appears as the volatile species (e.g. hydrides). Precisely speaking, HG technique converts aqueous species of some selected elements to volatile hydrides (As, Sb, Bi, Ge, Sn, Pb, Se, Te) or volatile species (Hg, Cd) or volatile, not yet identified, species (transition and noble metals). In some cases the reduction doesn’t take place; for example inorganic As (III) remains in the same oxidation state passing from H3AsO3 to AsH3. The generated hydrides are transferred with a carrier gas into an atomization cell, where the hydride molecules are dissociated into analyte atoms and atomized in order to be detected by spectrometric analysis. The mechanism of the hydride generation process has been studied thoroughly by different research groups [4-6] proposing different hypotheses and perspectives. The first hypothesis presumes the atomic hydrogen to be the active specie in the derivatization process. The atomic hydrogen which is also referred to as “the nascent hydrogen” is formed during the acidic hydrolysis of a hydroborate compound (e.g. NaBH4). The second hypothesis denies the need for intermediate reactive species and presumes the formation of the hydrides, in a pH range 4.7 to 12.7, occurs due to the action of borohydride compounds (BH4 or X-BH3 where X = Cl, Br, I) on the analyte. On the other hand, the results of another experimental study D’Ulivo et al. [7] showed that the molecular hydrogen and hydroboron species were formed during the hydrolysis of tetrahydroborate, and claimed that the hydrogen bonding to the boron species releases and recombines to form molecular hydrogen. Although the second hypothesis and the experimental evidence have presented solid arguments against the first one, the mechanism of the nascent hydrogen formation is still valid and accepted as a model by several researchers [8,9]. The most important link between the abovementioned hypotheses is: they agree that hydrogen gas is generated as one of the final products of the hydride generation reaction due to decomposition of hydroborate (THB). The decomposition of THB has been referred to be a second order reaction that might be last for few microseconds [1], whereas the generation and transfer of the hydrides from the liquid phase to the gaseous phase were considered to be a first order reaction [10]. On the other hand, the atomization process of the hydride has also been studied by many researchers [11-14] which raised several perspectives about the appropriate consistent mechanism. The general opinion supports the atomization by a thermal decomposition such as the case of electro-thermal atomization in a heated quartz tube. Nonetheless, this theory is found difficult to apply as a general case due to variations in the atomization temperature, which is strongly related to the equipment type. For instance, arsenic atomization is thought to occur at approximately 800˚C in a heated quartz tube, whereas arsenic is reported to atomize at 1700˚C - 1800˚C in a graphite tube furnace. In addition to the thermal decomposition theory, the atomization mechanism is attributed to the effect of free hydrogen radicals and the assistance of oxygen radicals in the atomizer. Generally speaking, the hydrogen radicals are found to be the most effective factor on the atomization process. Moreover, the oxygen radicals are also found to be an important factor, in which the hydride decomposition would occur in the presence of oxygen radicals alone; nevertheless, no atomization could be achieved with temperatures below 1700˚C [13]. Furthermore, the OH radicals which accompany the hydrogen radical formation are reported to play a powerful catalytic role on the hydride generation process. It is worth noting that the recombination of radicals is mentioned to be slower than the onward radical generation; therefore, the number of the hydrogen radicals is always expected to be above the equilibrium value. In practice, one of the most important parameters affecting the performance of such an analytical process is the position used to conduct the spectrometric data acquisition. Nonetheless, no clear guide is available which can show how to decide the optimum location for a fibre optics sensor along the atomization channel and whether an axial or radial data acquisition is optimal for such application. Alternatively, the theoretical investigations are the most practical way to give insight on the system performance even with some deviations in the results. In this sense, we have conducted theoretical investigations related to our work [15] to find the distribution of free arsenic atoms in a dielectric barrier discharge atomizer. For the DBD atomizer studied, the axial viewing at the end of the atomization channel was found the optimal. That result has been verified experimentally, where approximately 40% increase in the signal intensity was obtained upon applying the axial viewing.
The current work, organized in three sections, is dedicated to investigate the generation of the arsenic hydride in tubular reactor and the distribution of the free analyte in two designs of quartz cell atomizers; traditional flame in tube (FIT) atomizer and internally heated quartz tube atomizer (HQTA). In between these two stages, a simulation study is conducted to investigate the effect of adopting a helical tubular section for separating the generated hydrides and the side products from the liquid phase. The findings of the study should indicate the best reaction mechanism and the role of other parameters for the best compromise for design purposes.
2. Study of the Hydride Generation Process in a Tubular Reactor
2.1. Process Description
The hydride generation reaction is investigated in this study by assuming a sample of water contains 0.02 mg/l of arsenic, As (III), with the aim to be converted to arsenic hydride AsH3 upon being reacted with a reducing agent (NaBH4). The reactant streams are assumed to be injected into a tubular reaction section (2.4 mm diameter ) through a T-junction, provided that the reaction tube forms a configuration of helical coil. The inlet streams are assumed to have the following specification on the basis that the feed rate of the reagents (NaBH4 and HCl) to the HG generator is normally applied in a ratio 
as reported by Pohl et al. [16].
Stream 1: 2 ml/min H2O aqueous solution contains (0.2 % m/v) sodium tetrahydroborate (THB) stabilized by using (0.1% m/v) NaOH.
Stream 2: 2 ml/min H2O aqueous solution contains 0.02 mg/L As (III) and acidified by using 0.1 (M) HCl.
2.2. Chemical Reaction Mechanisms
2.2.1. Assuming the Evolution of Nascent Hydrogen from the Acidic Hydrolysis of THB
The mechanism of the hydride generation process was represented by the following reaction according to Dedina and Tsalev [1]:
(1)
where; A: the analyte, m+: the oxidation state of the analyte, n: the coordination number of the hydride,
: the nascent hydrogen.
The expected general form of the reaction which leads to the formation of the hydrides was described as follows:
(2)
The excess of un-reacted atomic hydrogen was mentioned to form the molecular hydrogen, which is one of the final products of the acidic hydrolysis of tetrahydroborate, as follows [7]:
(3)
(4)
Another general form was proposed as follows [13, 17]:
(5)
In this part of study, the reaction of arsenic hydride generation is represented by the following equations according to the nascent hydrogen hypothesis, (1st mechanism in the current study), assuming m = 3 and n = 3 and the hydrogen gas is generated directly:
(6)
(7)
(8)
where; K1: 2nd order decomposition rate constant = 1.22 ×108 L/mol/min = 2033.3 (m3/mol/sec) [1]; K2: the rate constant of the arsenic hydride formation. According to Van Wagenen et al. [10], the formation of arsenic hydride was found to be a first order reaction with a rate constant equal to 32 (1/sec); K3: 1st order rate constant of arsenic hydride release from the liquid phase, which was estimated equal to 3.5 (1/sec) by Van Wagenen et al. [10] in a three neck round bottom flask hydride generator.
An assumption made that the value and units of the rate constants mentioned above are applicable for the reactions described in the current case study, giving that no changes in the rate constants is expected to occur throughout the reaction series. Moreover, the concentration of (HCL) is considered to be equivalent to 10% of (NaBH4) concentration in the description of hydroborate decomposition rate (R1); this is to obtain the highest As signal to background ratio as demonstrated by Pohl et al. [16]. Thus; the reaction rates, for the current case, are described as follows:
The notations used to describe the species are shown as follows:
A = THB (i.e. NaBH4), B =
, C = As (III), D = H2, E = AsH3(aqueous), F = AsH3(gas).
The concentration gradients of the reactants and the products are described as shown in the following equations:
(9)
(10)
(11)
(12)
(13)
(14)
In order to find the optimal reactor length, further ordinary differential equations are added to the proposed system, which aims to simulate the velocity and the position required for the optimal conversions, as described by Zimmerman [18]:
(15)
(16)
where; u = velocity (m/sec), x = position (m), and 
= the initial concentration of the hydroborate.
The above system of eight ordinary differential equations has been analyzed using the ordinary differential equation solver (ode 23) in Matlab, which is principally based on the explicit Runge-Kutta method. The initial conditions were applied as follows: (CTHB = 1.347 mol/m3, CAs(III) = 2.66 × 10–4 mol/m3 , other species = 0 mol/m3 ).
The computations have produced the gradients of species concentration shown in Figure 1, which presents a comparison between the results obtained from the analytical solutions of both hypotheses.
2.2.2. Assuming the Stepwise Decomposition of Hydroborate and Formation of Hydroboron Intermediate Species
Other analytical evidences [8,9,19] supported the hypothesis of forming the hydroboron species with life time longer than the life time of the hydroborate (
). The borane complex [BH3Y]n , where Y represents neutral or anionic ligand, is hydrolysed in the aqueous phase and catalyzed by acid and eventually decomposes to give molecular hydrogen according to the following reaction path [9]:
(17)
(18)
The overall hydrolysis rate constant KHyd of THB (second-order reaction) is estimated equal to 1.6 × 106 (L/mol/sec) at 25˚C; where H+ applied in the range (0.2 M - 10 M) [8]. The following reactions are adopted in the current case study to investigate the arsenic hydride generation according to the hydroboron intermediates theory, considering the formation of arsenic hydride occurs due to the reaction of the analyte with the hydrogen atoms that released from the intermediate species, 
.
(19)
(20)
(21)
where; KHyd = 1.6 × 106 (L/mol/sec) =1600 (m3/mol/sec) and K2 = 3.5 (1/sec). As mentioned previously in Section (2.2.1), the value of (K1 = 32 sec–1) has been utilized to describe the reaction rate by assuming the value and units of K1 are applicable for the current case; moreover, [H+] is assumed equal to 10% of NaBH4 concentration; this is in order to envisage the case where a stable acidity undertaken throughout the reaction series. In light of the hydroboron intermediates hypothesis (2nd mechanism in the current study), the reaction rates contribute in the arsenic hydride production can be approximated according to the following equations:
The notations used for the species are shown as follows:
B =
, C = 
, D = As(III), E = AsH3(aqueous)F = AsH3 (gas), G = H2, J = 
.
The concentration gradients of the reactants and products are described as follows, where the last two equations are added to estimate the optimal reactor length:
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
The above system of nine ordinary differential equations was analyzed by using the ordinary differential equation solver (ode 23) in Matlab. The initial conditions were considered to be (CTHB = 1.347 mol/m3, CAs(III) = 2.66 × 10–4 mol/m3 , other species = 0 mol/m3 ). The results obtained from the analytical solution of the two hypotheses are shown in Figure 1, in which the schemes to the left represent the gradients according to the nascent hydrogen hypothesis whereas the schemes to the right refer to the gradients according to the hydroboron intermediates hypothesis. It can be observed in the schemes shown in Figure 1 that a time slot (0 to 2 seconds) has been selected for the undertaken analysis including all activities. A general overview on the analytic-cal solution shows that the second mechanism reveals that a faster decomposition process occur in comparison with the first mechanism by approximately two orders of magnitude, as presented in Figure 1(a). Since the hydroborate decomposition in both reaction mechanisms was considered to be a second order reaction; therefore the results of both mechanisms have shown a decomposition rate conducted within 0.1 and 0.005 seconds for the first and second mechanism respectively. The results of the second mechanism were found to be in agreement with the data shown in the literature (e.g. 14 × 10–6 second for 0.2 M [H+]) [8].
The results also show that the intermediate species, represented by the atomic hydrogen in the first mechanism and the intermediate complex 
in the second mechanism, saturates to a maximum and become stable within 1.0 second in the first mechanism, whereas it reached the stable concentration value within less than 0.01 second in the second mechanism, as shown in Figure 1(b). Other differences between the examined mechanisms were found in the consumption of arsenic concentration, the generated hydrogen, the formation of the intermediate arsenic hydride (in the aqueous phase), and their release to the gas phase, as presented in Figures 1(c)-(e). In this regard, the arsenic concentration is shown to be totally depleted from the reaction bulk within 0.03 second in the first mechanism whereas it takes approximately 0.1 second according to the second mechanism. In contrast, the peak value of the generated hydrogen was reached after 0.03 second in the first mechanism compared with 0.003 second in the second mechanism. Moreover, the second mechanism shows that a higher amount of hydrogen is generated due to the hydroborate decomposition by approximately 4 orders of magnitude and this is in agreement with the finding of Pohl et al. [16], who reported a hydrogen gas value of 0.6 ml/min is generated from only 0.1 m/v NaBH4 decomposition. Furthermore, both mechanisms show that aqueous arsenic hydride would be generated after 0.001 second of the reaction start up and totally depleted within approximately 1 second. However, the first mechanism showed that 0.03 second is required to reach the peak AsH3(aqueous) concentration, whereas the second mechanism disclosed that more time (about 0.1 second) is required. On the other hand, both mechanisms exhibit arsenic hydride in the gaseous phase generated after 0.01 second of the reaction start up, increasing relatively with time, and reaching its maximum value after approximately 1.8 seconds, as presented in Figures 1(f)-(g). This result indicates that the first part of the reaction tube, which is approximately 5 cm length, is required to achieve the full conversion of arsenic to arsenic hydride. The result might also denote the next portion of the reaction tube would be devoted to conduct the separation of the produced gaseous from the liquid phase. In conclusion, the results of the second mechanism have shown greater agreement with the observations reported in the literature; thus, the results from the second mechanism are adopted to study the separation of the gaseous hydrides in the helical tubular section.
3. Release of the Generated Arsenic Hydride
The second reaction mechanism presumes the arsenic hydride and other side products (H2, and the intermediate borane complex) would be produced at the end of the reaction series. As mentioned earlier, the second part of the reaction tube, which begins after 5 cm, is for separation. Due to a system of two phases, a helical tubular section has been studied to perform the separation process. This configuration has been adopted to utilize the beneficial effect of the secondary flow to enhance the releasing efficiency. The secondary motion (flow perpendicular to the main flow direction) is generated in the curved section of the tube due to the centripetal action which forces the liquid phase towards the tube wall. This phenomenon occurs due to a greater density and inertia of the liquid phase compared to the gaseous phase. As a result, the attraction forces between the two phases would be reduced and finally lead to the gaseous species separation. The effect of using two turns of a helical coil was studied by assuming the following conditions, taking into consideration the main products from the hydride reaction (AsH3 and H2) and neglecting other side products to simplify the computations:
Ø 2 ml/min water stream contains 2.66 × 10–4 (mol/m3) AsH3 and 5.38 (mol/m3) H2 is introduced to the helical coil. The indicated values refer to the species concentration after passing the first part of the reaction tube, Figures 1(d) and (g).
Ø Two turns of a helical coil (2.4 mm ID) with a top and bottom turn radius of 1 cm, is applied. The total height of the spiral is 2 cm and the total length of the assembled tubular section is 12.56 cm.
The coil configuration is shown in Figure 2.
A geometry of three dimensions represents the helical tube was built in Autocad software and exported to Comsol Multiphysics 3.5 a software to conduct the computations. Two models were used in Comsol to investigate the separation process inside the helical section. The first model is the laminar flow model (incompressible NavierStokes) which was coupled with a mass transfer model (convection and diffusion). The fluid flow is described by incompressible Navier-Stokes equations, in which a laminar flow regime and a constant fluid density were assumed to describe the system in a steady state condition according to the following equations [20]:
(31)
(32)
Here, η denotes the dynamic viscosity (Ns/m2), u the velocity vector (m/s), ρ the density of the fluid (kg/m3), p the pressure (Pa), and F is a body force term (N/m3).
The boundary conditions were taken as follows: at the inlet of the tubular section, the velocity vector is normal to the boundary, i.e. u∙n = u0; whereas the pressure at the outlet boundary is set (p = p0 = 0). Moreover, the flow is considered running down along the bottom half of the tubular section with a specific velocity, u, whereas the top half of the tubular section is considered as a no-slip
Figure 2. The geometry of the helical tubular section.
boundary (u = 0).
The mass transfer process in the studied helical section is modeled as a convection-diffusion equation:
(33)
where Ci and Di represents the concentration (mol/m3) of species (i) and the diffusion coefficient (m2/s) respecttively, whereas Ri denotes the reaction term (mol/m3/sec). At the inlet section of the tubular helical section, the boundary condition is assumed (Ci = Ci0) which equals the initial concentration. The outlet boundary conditions presume that no mass flux occurs due to diffusion, hence dominated by convection, therefore the total flux is described by:
(34)
Moreover, zero mass transfer is imposed at the interior surfaces due to the assumption of impermeable boundaries. The computations were conducted numerically by applying the finite element method treated by (Lagrange – P2P1) elements for pressure stability. The mesh was refined manually by changing the element size, which consequently result in increasing the accuracy. The accuracy of the solution was inferred by estimating a specific parameter (concentration of AsH3) at a specific point along the helical tube. A numerical error estimation study was conducted to infer the appropriate grid size for higher solution accuracy. Eventually, a very fine mesh of 84,194 elements and 826,978 degrees of freedom were applied in the computations, which produced the lowest relative error. The computation results of the finest grid size are adopted for discussion.
Figure 3 illustrates the simulation results represented by the velocity field distribution and the change of species concentration along (z) coordinate. The results show that a gradual increase in the velocity field along (z) coordinate occurs, which could be attributed to the effect of the secondary motion on the momentum transfer. Moreover, concentration depletion occurs for all species through the helical section which indicates higher probability for species transfer from the liquid phase to the gaseous phase as a result of high diffusion rate. The results also show that both AsH3 and H2 are completely depleted from the liquid phase within the tested length of the helical coil. However, the hydrogen gas is shown to be depleted faster than the arsenic hydride, and this is clearly attributed to a higher diffusion coefficient of hydrogen (1.32 × 10–7 cm2/sec) compared with the arsenic hydride diffusion coefficient (3.45 × 10–10 cm2/sec) [21, 22]. The diffusion coefficients also result in big differences between the estimated Peclet numbers of the examined species at a specific point along the tubular section (1.5 × 108 for AsH3 and 4 × 105 for H2 estimated at x = 1.5675, y = 0.745, and z = 0.0115), where Peclet number is defined as the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient.