Optimization of post-column reactor radius in capillary high performance liquid chromatography Effect of chromatographic column diameter and particle diameter

Abstract

A post-column reactor consisting of a simple open tube (Capillary Taylor Reactor) affects the performance of a capillary LC in two ways: stealing pressure from the column and adding band spreading. The former is a problem for very small radius reactors, while the latter shows itself for large reactor diameters. We derived an equation that defines the observed number of theoretical plates (N_obs) taking into account the two effects stated above. Making some assumptions and asserting certain conditions led to a final equation with a limited number of variables, namely chromatographic column radius, reactor radius and chromatographic particle diameter. The assumptions and conditions are that the van Deemter equation applies, the mass transfer limitation is for intraparticle diffusion in spherical particles, the velocity is at the optimum, the analyte’s retention factor, k′, is zero, the post-column reactor is only long enough to allow complete mixing of reagents and analytes and the maximum operating pressure of the pumping system is used. Optimal ranges of the reactor radius (a_r) are obtained by comparing the number of observed theoretical plates (and theoretical plates per time) with and without a reactor. Results show that the acceptable reactor radii depend on column diameter, particle diameter, and maximum available pressure. Optimal ranges of a_r become narrower as column diameter increases, particle diameter decreases or the maximum pressure is decreased. When the available pressure is 4000 psi, a Capillary Taylor Reactor with 12 μm radius is suitable for all columns smaller than 150 μm (radius) packed with 2–5 μm particles. For 1 μm packing particles, only columns smaller than 42.5 μm (radius) can be used and the reactor radius needs to be 5 μm.

1. Introduction

It is well-known that post-column reactions can improve sensitivity and selectivity in liquid chromatographic analysis [1–3]. In just the past couple of years, papers using post-column reactions for a variety of bioanalytical applications have appeared, including amino acids and peptides [4,5], for fatty acid hydroperoxides in MEKC [6], for japonicus/polysaccharide [7], for glucosylated flavonoids [8], for acetyl-CoA esters [9], for phosphorylated peptides [10], for drugs (competitive binding) [11], for glucose and insulin [12], for bradykinin [13], for drugs [14], for lipoproteins [15], for tocopherols [16], for phosphorylated sugars [17], for peptides [3,18–21]. One should include the preparation of MALDI plates as a post-column reaction [22]. Recent applications in environmental chemistry using post-column reactors include glyphosate [23], iodate and bromate [24], thiourea [25], anions [26] and ion chromatography [27]. However, post-column reactors come with some disadvantages. The most obvious are the additional hardware and reagents required, and the band spreading associated with the additional mixing following the separation.

The band spreading issue, as it is treatable theoretically, has been addressed. Huber et al. [28] considered the problem of using rather slow chemical reactions following the separation. They concluded that a packed bed reactor was superior to an open tubular reactor. In their work, they assumed that mixing, as opposed to reaction, was instantaneous. On the other hand, Kucera and Umagat [29] considered the case of post-column reactors for microbore columns. With smaller peak volumes than in 4.6 mm I.D. (inside diameter) columns, microbore columns pose more of a challenge for post-column reactor design. These workers considered a fast reaction, o-phthalaldehyde with amino acids and primary aliphatic amines. They proposed a simple tubular reactor. Special attention was paid to creating a low-volume mixing ‘tee’. Peak variances depended linearly on the square of the reactor diameter, showing that the chromatography was compromised. In other words, they did not employ a reactor diameter small enough to make the additional band spreading from the reactor negligible. With 50 cm × 1 mm columns operating at 35 μL/min, and with an equivalent post-column reagent flow rate, they used post-column reactor tubing in the 180–250 μm I.D. range.

Other workers have optimized post-column reactors experimentally. The sheath-flow approach has proven suitable for CE of DNA [30]. Optimization of post-column conditions for chemiluminescence detection has been achieved [31]. Hollow fiber membrane systems have been optimized to minimize band spreading [32]. The virtue of inducing secondary flow for improving radial mixing has been shown by Selavka et al. [33]. Contributions to band spreading were evaluated in a post-column reactor for photochemical reactions [34] and for saccharides [35]. Much of this work is dated, and applies to columns with typical dimensions of 4.6 mm I.D., 10–20 cm long with 5 μm diameter particles.

Recently, our group has developed a mixer that is well-suited to mix reagents with effluent from a capillary column [1,36,37]. This mixer, in conjunction with a fused silica Capillary Taylor Reactor (a simple, open tube long enough to permit diffusional relaxation of radial concentration gradients [1,37]), creates a very simple post-column reaction device that does not increase band spreading under the conditions used [1]. In this system, diffusion-controlled reactions occurred in times on the order of 1 s, in physical volumes on the order of 10 s of nL [1,37]. Kinetically controlled reactions require longer reactors [37].

The magnitude of the inside diameter or radius of the post-column reactor is an important variable. It controls the reactor’s contribution to band spreading; smaller reactor radii are preferred from this perspective. However, as the HPLC column defines the flow rate, the pressure in the post-column reactor (of constant length) increases in inverse proportion to the fourth power of the radius. Thus, there is clearly an optimal value for reactor radius. This paper determines that optimum, and demonstrates its dependence on column diameter and particle diameter.

2. Assumptions and limitations

In chromatographic theory, there are many variables. If the number of variables can be reduced, the essence of the theory is often made clearer. However, there is also a sacrifice in the breadth of applicability of the results from putting limitations on the system. Thus, it is important to state and discuss the limiting nature of the assumptions.

We will set the value of three significant parameters by allowing them to depend on other freely chosen variables. One is the chromatographic velocity. We have set the velocity equal to the optimum velocity. Kennedy and Jorgenson [38] have shown that the optimum velocity in capillary columns is a function of the column diameter. In this paper, we have not taken that subtlety into account. Further, there are several formulations of H versus υ curves [39]. We have chosen to use the van Deemter equation (the expression is shown later), which works acceptably near the optimum velocity. Another parameter that is set is the post-column reactor length. Both theory [40] and experiment [1,37] show that if the reaction is fast enough, it will be completed in a length of cylindrical tubing that depends upon the reagents’ diffusion coefficients, the tubing radius and the velocity. Furthermore, the velocity in the reactor depends on the chromatographic velocity (which we fix) and geometrical parameters. Finally, we have taken the k′ (the retention factor) of a solute to be zero.

The limitations that accompany these decisions are as follows. Setting the velocity at the optimum means that the analysis will favor total number of theoretical plates over theoretical plates per unit time. We analyze both in this paper. Setting the length of the post-column reactor means that the detector will give a ‘full signal’ for fast enough reactions, but not for kinetically slow reactions. Finally, the restriction on k′ will make the reactor’s relative (to the column) contribution to band spreading larger than it would be for a retained solute. A retained solute has a k′-dependent variance (time units). By using k′ = 0 in the formulations, we ignore this. The result is that the chromatography seems more sensitive to the presence of extracolumn band spreading than it would be for k′ > 0.

Finally, we assume that the chromatographic system operates at the maximum pressure of the pumping system. The ultimate limitation to chromatographic throughput is pressure [41]. As the velocity is fixed at the optimum, we control the length of the chromatographic column to use the available pressure.

In this paper, we determine how the system dimensions influence chromatographic performance as judged by the number of theoretical plates, N, and the number of theoretical plates per time. The results will show that there is an optimum in reactor radius that is dependent on the particle diameter and the column radius. Very small reactor radii lead to excessive pressure in the reactor, diminishing chromatographic performance by forcing a shortening of the column (remember, total pressure is fixed). Very large reactor radii lead to excessive band spreading in the reactor.

3. Equation derivation

3.1. Reactor requires pressure

The total pressure in the system is given by Eq. (1).

$$P_{\text{m}}=\frac{\phi \eta {\upsilon }_{\text{c}}{L}_{\text{c}}}{d_{\text{p}}^2}+\frac{8\eta {\upsilon }_{\text{r}}{L}_{\text{r}}}{a_{\text{r}}^2}$$ (1)

Here, P_m is the maximum pressure of the HPLC pump, η is the viscosity of the mobile phase, L_c and L_r represent the length of the column and the reactor, respectively, υ_c and υ_r are the average linear velocities in the column and the reactor, respectively, d_p is the diameter of the packing particles, a_r is the radius of the capillary used to make the reactor and φ is the column’s permeability. The left-hand term on the right side is the pressure drop in the column [41] while the right-hand term is the pressure drop in the reactor. We will now use the assumptions described above with the van Deemter equation to remove some of the variables.

3.2. Average velocity is the optimum velocity

$$H_{\text{c}}=\frac{2D}{\upsilon }+\lambda d_{\text{p}}+\frac{d_{\text{p}}^2}{gD}\upsilon$$ (2)

In the van Deemter equation, Eq. (2), H_c is the height equivalent to a theoretical plate for the chromatographic column, D is the solute’s diffusion coefficient in the mobile phase, υ is the average mobile phase velocity, d_p is the packing particle diameter, and λ and g are numerical, geometry-dependent constants (values of such parameters are discussed below). We choose the optimal velocity as the column velocity.

$${\upsilon }_{\text{c}}={\upsilon }_{\text{opt}}=\sqrt{2g}\frac{D}{d_{\text{p}}}$$ (3)

υ_r is a function of υ_c and the ratio of a_c, the radius of the column and a_r. The factor b in Eq. (4) below is equal to ɛ_e + (1 − ɛ_e)ɛ_i, where ɛ_e is extraparticle porosity and ɛ_i is the intraparticle porosity.

$${\upsilon }_{\text{r}}={\upsilon }_{c}b{\left(\frac{a_{\text{c}}}{a_{\text{r}}}\right)}^2$$ (4)

3.3. Reactor length is the length required for significant diffusional mixing

L_r, the length of the reactor, is given by Eq. (5) [1,37,40]. We fix L_r based on capillary Taylor theory [1,37]. The numerical factor represents the degree of mixing by diffusion. A larger number means more complete mixing. We choose 2.5 here, which means the length of the reactor is just enough for complete mixing. Kinetic factors are not considered here, thus extent of mixing is equivalent to the extent of reaction.

$$L_r=\frac{2.5{\upsilon }_{\text{r}}{a}_{\text{r}}^2}{D}$$ (5)

3.4. Revised pressure expression

After substituting Eqs. (3)–(5) into Eq. (1), we obtain Eq. (6):

$$P_{\text{m}}=\frac{\sqrt{2g}\phi D\eta L_{\text{c}}}{d_{\text{p}}^3}+\frac{40gD\eta b^2{(a_{\text{c}}/a_{\text{r}})}^4}{d_{\text{p}}^2}$$ (6)

The first term on the right-hand side of Eq. (6) is still the pressure drop across the column (P_c) while the second term is that across the post column reactor (P_r). This equation can be recast into a dimensionless ratio of column pressure to available pressure to give Eq. (7). In Eq. (7), the dimensionless second term of the right-hand side is the fraction of the available pressure needed by the reactor.

$$\frac{P_{\text{c}}}{P_{\text{m}}}=\frac{P_{\text{m}}-P_{\text{r}}}{P_{\text{m}}}=1-\frac{40gD\eta b^2{(a_{\text{c}}/a_{\text{r}})}^4}{d_{\text{p}}^2{P}_{\text{m}}}$$ (7)

3.5. Reactor adds band broadening

Band spreading occurs in the reactor, but it may not be significant if the chromatographic peaks are broad. Thus, we must compare the extent of broadening from both sources. We have already shown experimentally that the mixer portion of the mixer/reactor adds no measurable broadening [1]. The variance (in units of time squared) for the column (at the optimum velocity) is given by Eq. (8).

$${\sigma }_{\text{c}}^2=\frac{H_{\text{c}}{L}_{\text{c}}}{{\upsilon }_{\text{c}}^{\text{2}}}$$ (8)

To get an expression for the variance, we will use the column length defined by pressure conservation (Eq. (9), from Eq. (6)),

$$L_{\text{c}}=\frac{P_{\text{m}}{d}_{\text{p}}^3}{\sqrt{2g}\phi D\eta }-\frac{20\sqrt{2g}{b}^2{(a_{\text{c}}/a_{\text{r}})}^4{d}_{\text{p}}}{\phi }$$ (9)

the optimal velocity (Eq. (3)) and the minimum value of H_c (H_min, Eq. (10)) derived from the optimum velocity and the van Deemter equation (Eq. (2)).

$$H_{\text{c}}=H_{min}=\left(2\sqrt{\frac{2}{g}}+\lambda \right)d_{\text{p}}$$ (10)

After substituting Eqs. (3), (9) and (10) into Eq. (8), we get the final description for the column variance.

$${\sigma }_{\text{c}}^2=\frac{2\sqrt{2/g}+\lambda }{{\text{(2}g\text{)}}^{\text{3}/\text{2}}{D}^3\phi \eta }{d}_{\text{p}}^6{P}_{\text{m}}-\frac{20(2\sqrt{2/g}+\lambda )b^2}{\sqrt{2g}{D}^2\phi }{\left(\frac{a_{\text{c}}}{a_{\text{r}}}\right)}^4{d}_{\text{p}}^4$$ (11)

3.6. Variance in the reactor

The variance in the same units for the reactor is given by Eq. (12) [1].

$${\sigma }_{\text{r}}^2=\frac{H_{\text{r}}{L}_{\text{r}}}{{\upsilon }_{\text{r}}^{\text{2}}}=\frac{{\upsilon }_{\text{r}}{a}_{\text{r}}^2}{\text{24}D}\frac{2.5{\upsilon }_{\text{r}}{a}_{\text{r}}^2}{D}\frac{1}{{\upsilon }_{\text{r}}^2}=\frac{2.5a_{\text{r}}^4}{24D^2}$$ (12)

L_r is defined above, and H_r is the height equivalent to a theoretical plate in a reactor [1],

$$H_{\text{r}}=\frac{{\upsilon }_{\text{r}}{a}_{\text{r}}^2}{24D}$$ (13)

3.7. Total peak variance

The total peak variance is the sum of the column and reactor variances.

$$\begin{array}{l}{\sigma }^2={\sigma }_{\text{c}}^2+{\sigma }_{\text{r}}^2=\frac{2\sqrt{2/g}+\lambda }{{(2g)}^{3/2}{D}^3\phi \eta }{d}_{\text{p}}^6{P}_{\text{c}}+\frac{2.5}{24D^2}{a}_{\text{r}}^4\\ =\frac{M}{D^3\eta }{d}_{\text{p}}^6{P}_{\text{m}}-\frac{40gM{b}^2}{D^2}{(\frac{a_{\text{c}}}{a_{\text{r}}})}^4{d}_{\text{p}}^4+\frac{2.5}{24D^2}{a}_{\text{r}}^4\end{array}$$ (14)

Here, we define M as $\frac{2\sqrt{2/g}+\lambda }{{(2g)}^{3/2}\phi }.$

The band broadening arising from the reactor can be considered negligible when it contributes no more than 10% extra to the peak width in the absence of the reactor. This is the case if σ_r²/σ_c² ≤ 0.21.

3.8. Column efficiency

N is defined here as the number of theoretical plates from the column when using an ideal reactor that requires pressure but adds no band spreading. The column variance, Eq. (11), is used to generate Eq. (15). M is defined above.

$$N=\frac{t_{0\text{c}}^2}{{\sigma }_{\text{c}}^2}=\frac{L_{\text{c}}}{H_{\text{c}}}=\frac{P_{\text{m}}{d}_{\text{p}}^2}{4g^2{\phi }^2\eta DM}-\frac{10b^2{(a_{\text{c}}/a_{\text{r}})}^4}{{\phi }^{2}Mg}$$ (15)

However, with a post-column reactor, the retention time and the peak variance depend on more than just the column. The observed number of plates, N_obs, includes all effects of the reactor.

$$\begin{array}{l}{N}_{\text{obs}}=\frac{t_0^2}{{\sigma }^2}=\frac{{(t_{0\text{c}}+t_{0\text{r}})}^2}{{\sigma }_{\text{c}}^{\text{2}}+{\sigma }_{\text{r}}^2}\\ =\frac{{[J{d}_{\text{p}}^2-40b^2{(a_{\text{c}}/a_{\text{r}})}^4+5\phi {(a_{\text{r}}/d_{\text{p}})}^2]}^2}{\text{4}gMJ{\phi }^2{d}_{\text{p}}^2-160gM{\phi }^2{b}^2{(a_{\text{c}}/a_{\text{r}})}^4+0.4{\phi }^2{(a_{\text{r}}/d_{\text{p}})}^4}\end{array}$$ (16)

where t_0c and t_0r are the dead time in the column and the reactor, respectively; M has the same definition as in Eq. (14); J represents the ratio P_m/gDη.

Eq. (16) looks unwieldy. However, N_obs is only dependent on P_m and three size-related variables, d_p, a_c and a_r. The figures based on these equations are much clearer. First, we will show the effect of column radius on the reactor choice. Second, we will find the relationship between N_obs and the variables d_p and a_r to get the optimal reactor radius for P_m equal to 4000 psi and a_c equal to 50 μm. Finally, the effects of P_m on the reactor choice will be shown.

To display the effect of the reactor, we also calculate the number of theoretical plates of the column under the same conditions but without the reactor (N₀) as below. Here, the pressure available for the column is the maximum offered by the pump.

$$N_0=\frac{P_{\text{m}}{d}_{\text{p}}^2}{\phi \eta D(4+\lambda \sqrt{2g})}$$ (17)

3.9. Description of some empirical parameters

There are many parameters used in the calculations. Here, we list the parameters and discuss their origin and value.

φ, the column’s permeability depends upon the method of packing and the particles for a packed column. For various columns, it ranges from 500 to 1000 [41]. We routinely obtain a value of 750 for 2.6–5 μm particle packings in 100 μm diameter columns. The effect of changes in φ on the results is negligible. λ and g are numerical, geometry-dependent constants. Typically, λ is 2 and g is 30 [39]. The value 2 comes from experiment and the simple random walk approach to eddy dispersion. The number 30 comes from assuming that the C-term is governed by diffusional relaxation within the spherical particle. The value for λ will never be far from 2. The value for g would give a C-term in the Knox equation of 0.033, which is close to the observed value from many systems of C ~ 0.05. D is the diffusion coefficient of the solute in the mobile phase. η is the viscosity of the mobile phase. In our final expressions of N_obs, N and N₀, D and η always appear as the product Dη for a given solute and at constant temperature, this product, sometimes called the Walden product (Dη = RT/6πa where R is the gas contstant, T is the temperature and a is the radius of the diffusing molecule), will remain fairly constant as the solvent changes. We used 5.0 × 10⁻¹⁰ m²/s as D and 0.001 N s/m² as η. These values are typical for solutes of a few hundred Daltons molar mass and aqueous eluents. As the Walden product is in the final expressions, the product of these values is suitable for other solvents (but not dramatically larger or smaller solutes). b is equal to ɛ_e + (1 − ɛ_e)ɛ_i, where ɛ_e is extraparticle porosity and ɛ_i is the intraparticle porosity. Usually, both porosities are near 0.4 [39], so b is 0.64. This term is appropriate if the mean, not the extraparticle, velocity is used in the van Deemter equation. Of all of the parameters, the value of g is the most dependent on the particular system (particle shape, intra- versus interparticle mass transport limitation, diffusional versus kinetic stationary phase mass transfer limitation).

4. Results and discussion

4.1. Effect of column diameter: is a Capillary Taylor Reactor useful for large columns?

The introduction discussed cases in which the simple Capillary Taylor Reactor added measurable band spreading with 4.6 and 1 mm I.D. columns. Table 1 provides a view of this problem. Table 1 shows the effect of a Capillary Taylor Reactor on the maximum number of observed theoretical plates, Max N_obs, as a function of d_p and a_c (Eq. (16)). Values for N₀, the number of theoretical plates generated by the capillary column without the reactor (Eq. (17)), are also given. The ‘a_r range’ is the range of reactor radii at which N_obs is at least 90% of N₀. The reactor can have a significant deleterious effect on the column efficiency. For a column having a radius of 150 μm, the Max N_obs/N₀ ratio increases from 19.5 to 99.9% as the diameter of packing particles increases from 1 to 5 μm. The influence of the reactor is small for the 2–5 μm particles but significant for 1 μm particles. The larger the column diameter is, the more effect the reactor has on the column efficiency. The Max N_obs/N₀ ratio decreases as column radius increases. The a_r range in which N_obs/N₀ is larger than 90% becomes narrow or even disappears as column radius increases. Therefore, a small column is preferred when used with a Capillary Taylor Reactor especially when the column is packed with small particles. Specifically, the effect of the reactor is negligible when column radius is smaller than 150 μm for particle diameters greater than or equal to 2 μm.

Table 1.

Effect of the column radius on N_obs/N₀

dp (μm)		1	2	3	4	5
N0		3770	15100	34000	60400	94300
	Max Nobs	735	13600	33400	60100	94200
ac = 150 μm	ar range (μm)	–	12	9–25	8–38	7–52
	Max Nobs/N0	19.5%	90.1%	98.2%	99.5%	99.9%
	Max Nobs	94	3600	25700	55600	91400
ac = 500μm	ar range (μm)	–	–	–	26–35	22–52
	Max Nobs/N0)	2.5%	23.8%	75.6%	92.1%	96.9%
	Max Nobs	68	535	10000	41600	82000
ac = 1000μm	ar range (μm)	–	–	–	–	–
	Max Nobs/N0	1.8%	3.5%	29.5%	68.9%	86.6%

P_m = 4000 psi; D = 5.0 × 10⁻¹⁰ m²/s; η = 0.001 N s/m²; b = 0.64; φ = 750; g = 30; λ = 2.

4.2. Optimization of the reactor radius for capillary columns with various particle sizes

Fig. 1 gives the relationship between the number of theoretical plates (N or N_obs) and the radius of the reactor a_r for a 100 μm I.D. column (a_c = 50 μm). N is the number of theoretical plates observed with a hypothetical reactor that requires pressure but causes no extracolumn band spreading. N_obs is defined by Eq. (16). N and N_obs depend similarly on a_r at small values of a_r. N and N_obs sharply increase and then reach a plateau or maximum as a_r increases. Larger particles exhibit a significant plateau, while the smallest particle size, 1 μm, shows a maximum. As a_r increases beyond the maximum, N does not change at all while N_obs gradually decreases. The shape of Fig. 1 can be understood qualitatively by considering systems for large and small values of a_r.

Fig. 1.

Plot of N (dashed line) and N_obs (solid line) against the radius of reactor. Each curve corresponds to a different particle diameter. The highest curve is for d_p = 5 μm, and the lowest is for d_p = 1 μm. a_c = 50 μm; P_m = 4000 psi; D = 5.0 × 10⁻¹⁰ m²/s; η = 0.001 N s/m²; b = 0.64; φ = 750; g = 30, λ = 2.

There is a minimum value of a_r that can be derived from Eq. (7). This equation shows that all the pressure available is used by the reactor when both sides of the equation equal zero. As a_r is decreased towards this limit, (40gDηb²/(d_p² P_m))^1/4 a_c, more of the available pressure is used to push fluid through the reactor. To compensate, the column’s length approaches zero, so N_obs approaches zero as well. Thus, the effect of the reactor on the column efficiency as a_r/a_c approaches (40gDηb²/(d_p² P_m))^1/4 mostly comes from the pressure it steals. The pressure available to the column is increased with a larger diameter reactor, according to Eq. (7). For a large enough value of a_r, it will approach P_m. However, when a_r becomes large, the variance from dispersion within the reactor becomes significant and cannot be neglected compared to the variance caused by the column. N_obs gradually decreases as a_r increases as shown on the right-hand side of Fig. 1 because of the effect of the reactor variance (Eq. (12)).

Obviously, what we are most interested in is the middle part of Fig. 1 where the effect from the reactor is minimal and N_obs is the largest. Columns with larger packing particles have a larger column variance, so there is a wider region of conditions that allow the band broadening from the reactor to be negligible compared to that from the column. Therefore, columns with larger packing particles have a wider plateau.

Table 2 lists the range of reactor radii that give a value of N_obs that is larger than 90% of N₀, which we used to define the acceptable region. The parameters ‘Max N_obs’, ‘a_r range’ and N₀ have the same definitions as in Table 1. t₀ is the retention time of an unretained solute. In addition, to determine whether it is possible to use a post-column reactor without significant sacrifice, Table 2 also compares Max N_obs with N₀. Values of Max N_obs for particle sizes from 2 to 5 μm are all larger than 90% of N₀. This is not the case with a 1 μm column, where Max N_obs is only 86.2% of N₀. This means that there is no reactor size that will yield an observed number of theoretical plates greater than or equal to 90% of N₀. The variance from a 1 μm column is quite small and the variance from the reactor is never negligible.

Table 2.

Range of useful reactor radii with maximum N_obs as a criterion

dp (μm)	Max Nobs	ar range (μm)	N0	Max Nobs/N0 (%)	t0 (s)
1	3250	–	3770	86.2	2.4
2	15000	4–14	15100	99.3	39
3	34000	3–25	34000	100	199
4	60400	3–38	60400	100	628
5	94400	3–53	94400	100	1533

a_c = 50 μm; other conditions are the same as in Table 1.

Large values of N are often accompanied by long experimental time which is also shown in Table 2. To get a different perspective, we also looked at N_obs per unit time (N_obs/t₀) (Fig. 2). Fig. 2 displays how N_obs/t₀ changes with a_r. From Fig. 2, with a proper size reactor, the N_obs/t₀ of a column with 1 μm particles can be more than four times larger than that of a 2 μm column, and over 20 times greater than a 5 μm column. Therefore, smaller particles are better in faster separations, even with a post-column reactor.

Fig. 2.

N_obs/t₀ (s⁻¹) as a function of a_r. Conditions are the same as in Fig. 1.

Table 3 is analogous to Table 2, but for the number of observed plates per time. It displays the range of a_r that will yield N_obs/t₀ that is at least 90% of N₀/t₀. Note that the range is only slightly different from that for N_obs. Similar to Table 2, columns with large particles have a wider plateau for N_obs/t₀.

Table 3.

Range of useful reactor radii with maximum N_obs/t₀ as a criterion

dp (μm)	Max Nobs/t0 (s−1)	N0/t0 (s−1)	ar range (μm)
1	1530	1540	3–5
2	390	390	2–13
3	170	170	2–24
4	100	100	2–37
5	60	60	2–51

Conditions are the same as listed in Table 2.

4.3. The special case of 1 μm particles: is a smaller column better when using a reactor?

As mentioned before, the reactor always affects the observed chromatographic efficiency when d_p is 1 μm (as shown in Fig. 1, there is no plateau there at d_p = 1 μm and Table 2, N_obs/N₀ always smaller than 90%). This is because the column variance from a 1 μm column is extremely small so that the variance from the reactor cannot be neglected. The only way to minimize the variance from the reactor is to use small radius reactors, which will be accompanied by a sacrifice in available column pressure. However, Eq. (7) tells us that the pressure drop for the column will not change if a_c/a_r and d_p are constant. Therefore, for a given particle diameter, if the column diameter is scaled down as the reactor diameter is scaled down, the fraction of the maximum pressure available for the column should remain constant. That means when the column diameter is reduced, the reactor diameter should be reduced proportionally. Therefore, using a smaller column diameter might be helpful.

Fig. 3 shows curves of N_obs for 1 μm particle packing plotted against reactor radius with column radius as a parameter. In Fig. 3, the curves of N_obs versus reactor radius increase sharply, reach a maximum and then gradually decrease at larger reactor radii. The maximum value of N_obs increases more than two times and the optimum value of a_r becomes smaller as the column diameter changes from 100 to 25 μm. When the column radius is equal to or less than 42.5 μm, the Max N_obs/N₀ ratio starts to approach 90%, which means the reactor’s effect is negligible at the optimal radius of the reactor, which is 5 μm.

Fig. 3.

Plot of N_obs against the radius of the post-column reactor. d_p = 1 μm; a_c is altered from 25 to100 μm. The solid line represents N_obs while the dashed line represents N₀. Other conditions are the same as in Fig. 1.

4.4. Maximum pressure (P_m) and optimal reactor radius

Results shown in Table 4 indicate the effect of P_m. Max N_obs, the maximum value of N_obs under optimal conditions, increases almost proportionally as P_m increases. The ‘a_r range’ increases slightly as P_m increases. However, a_r, the reactor radius at which the maximal N_obs is achieved, is independent of P_m.

Table 4.

Effect of the P_m on the acceptable range of reactor radii when a_c = 50 μm

dp (μm)		1	2	3	4	5
	Max Nobs	1400	7500	17000	30200	47200
Pm = 2000 psi	ar (μm)	5	7	10	12	14, 15
	ar range (μm)	–	5–12	4–22	3–32	3–45
	Max Nobs	3250	15000	34000	60400	94400
Pm = 4000 psi	ar (μm)	5	7	10	12	14, 15
	ar range (μm)	–	4–14	3–25	3–38	3–53
	Max Nobs	7000	30100	67900	121000	189000
Pm = 8000 psi	ar (μm)	5	7	10	12	14, 15
	ar range (μm)	5	3–16	3–29	3–44	2–61

Other conditions are the same as in Table 1.

4.5. Conclusion

In this paper, the influence of the diameter of a Capillary Taylor Reactor on the performance of capillary liquid chromatography was studied. The reactor steals pressure from the column which in turn decreases the numbers of theoretical plates when small reactor diameters are used. At the same time, it adds some band broadening. The choice of the reactor radius actually depends on several factors such as packing particle size, maximal available pressure and column diameter. For packing particles larger than or equal to 2 μm, the Capillary Taylor Reactor is suitable for columns with radii not larger than 150 μm (for “suitable”, we defined as if N_obs/N₀ > 90%). A reactor with about a 12 μm radius (i.e., the commercially available 25 μm I.D. capillary) works well under such conditions. For 1 μm particles, however, the requirements are more stringent. Only columns with smaller than 42.5 μm radii can be used and the optimal reactor radius is 5 μm, which is much smaller compared to those coupled with larger particle packing columns. The results above are obtained under a maximal pressure of 4000 psi. The limitations in the column diameter become less restricted when a higher maximum pressure is available.