Application of inverse method to predict the breakthrough curve in fixed-bed adsorption

Figures & data

Figure 1. Schematic of the system under consideration.

Table 1. Base parameters of the solved examples (reference conditions: P = 82.425 kPa, T = 323.15 K).

	CO2	SO2	NO
b(Pa^–1)	12.22 × 10^−6	17.61 × 10^−6	48.52 × 10^−6
$q_m\left(\text{mole/kg}\right)$	2.9180	6.4730	0.1581
$k\left({\text{s}}^{-1}\right)$	0.0350	0.0690	0.0702

Figure 2. (a)–(c): The breakthrough curves for one adsorbing component on zeolite NaX.

Figure 3. (a)–(d): The breakthrough curves for the system with CO₂ and SO₂ as the adsorbing components on zeolite NaX.

Figure 4. (a)–(d): The breakthrough curves for the system with CO₂ and NO as the adsorbing components on zeolite NaX.

Figure 5. (a)–(d): The influence of adsorption parameters on breakthrough curve for the system with CO₂ as the adsorbing component on zeolite NaX.

Figure 6. Effect of temperature on the breakthrough curves of CO₂ adsorption on OXA-GAC (a) at a 50 mL min⁻¹ feed and (b) at a 100 mL min⁻¹ feed flow rate.

Figure 7. Comparison of the artificial experimental data and estimated values of the breakthrough curves of the system with CO₂ and SO₂ as the adsorbing components on zeolite NaX.

Table 2. The estimated overall mass transfer coefficients, percentage of errors, corresponding CPU times and number of iterations at different mole fractions for the system with CO₂ and SO₂ as the adsorbing components on zeolite NaX.

	$Y_{{\text{CO}}_2,0}$	YSO2,0	$k_{{\text{CO}}_2}^{\text{estimated}}$ (s^−1)	% Error	$k_{{\text{CO}}_2}^{\text{estimated}}$ (s^−1)	% Error	CPU time (s)*
σs = 0	0.06	0	0.0350	$3\times {10}^{-10}$	–	–	50 (9)**
	0.05	0.01	0.0350	$3\times {10}^{-10}$	0.0690	$1.6\times {10}^{-7}$	52 (9)
	0.04	0.02	0.0350	9 × 10^−10	0.0690	$1.6\times {10}^{-7}$	55 (9)
	0.03	0.03	0.0350	$2\times {10}^{-9}$	0.0690	$1.8\times {10}^{-7}$	55 (9)
	0.02	0.04	0.0350	$3\times {10}^{-9}$	0.0690	$2.1\times {10}^{-7}$	56 (9)
	0	0.06	–	–	0.0690	$1.5\times {10}^{-7}$	55 (9)
σs = 5 × 10^−4	0.06	0	0.0347	0.72	–	–	65 (7)
	0.05	0.01	0.0339	3.22	0.0675	2.18	135 (7)
	0.04	0.02	0.0353	0.98	0.0690	0.02	120 (7)
	0.03	0.03	0.0356	1.80	0.0674	2.27	134 (7)
	0.02	0.04	0.0334	2.86	0.0686	0.57	124 (7)
	0	0.06	–	–	0.0697	1.02	67 (7)
σs = 10^−3	0.06	0	0.0358	2.37	–	–	55 (6)
	0.05	0.01	0.0346	1.22	0.0641	7.06	113 (6)
	0.04	0.02	0.0347	0.94	0.0659	4.47	112 (6)
	0.03	0.03	0.0349	0.30	0.0705	2.23	121 (6)
	0.02	0.04	0.0369	5.44	0.0660	0.01	119 (6)
	0	0.06	–	–	0.0696	0.83	57 (6)

* Personal computers with Intel (R), Core(TM)2, Quad CPU Q8400 @ 2.66 GHz, RAM; 4.00 GB

** Number of iteration.

Table 3. The estimated overall mass transfer coefficients, percentage of errors, corresponding CPU times and number of iterations at different mole fractions for the system with CO₂ and NO as the adsorbing components on zeolite NaX.

	$Y_{{\text{CO}}_2,0}$	$Y_{\text{NO,0}}$	$k_{{\text{CO}}_2\left(s^{-1}\right)}^{\text{Estimated}}$	% Error	$k_{\text{NO}\left(s^{-1}\right)}^{\text{Estimated}}$	% Error	CPU time s*
σs = 0	0.05	0.01	0.0350	$1.4\times {10}^{-6}$	0.0702	$4.5\times {10}^{-5}$	46 (8)**
	0.04	0.02	0.0350	$1.3\times {10}^{-6}$	0.0702	$4\times {10}^{-5}$	45 (8)
	0.03	0.03	0.0350	$1.4\times {10}^{-6}$	0.0702	$3.8\times {10}^{-5}$	45 (8)
	0.02	0.04	0.0350	$1.5\times {10}^{-6}$	0.0702	$3.9\times {10}^{-5}$	46 (8)
	0	0.06	–	–	0.0702	$3.2\times {10}^{-5}$	44 (8)
σs = 5 × 10^−4	0.05	0.01	0.0346	1.05	0.0699	0.39	170 (8)
	0.04	0.02	0.0345	1.31	0.0697	0.74	177 (8)
	0.03	0.03	0.0351	0.38	0.0697	0.68	161 (7)
	0.02	0.04	0.0356	1.69	0.0700	0.17	155 (7)
	0	0.06	–	–	0.0696	0.67	69 (7)
σs = 10^−3	0.05	0.01	0.0356	1.88	0.0667	4.93	150 (7)
	0.04	0.02	0.0357	2.19	0.0659	6.11	147 (7)
	0.03	0.03	0.0331	5.54	0.0710	1.18	141 (7)
	0.02	0.04	0.0339	2.76	0.0721	3.17	151 (7)
	0	0.06	–	–	0.0693	1.32	55 (7)

* Personal computers with Intel (R), Core(TM)2, Quad CPU Q8400 @ 2.66 GHz, RAM; 4.00 GB

** Number of iteration.

Table

bi	affinity constant of i compound
C	concentration in the gas phase
${\overline{C}}_i$	concentration of i compound in the gas phase
Dz,i	axial dispersion coefficient of i compound
ki	overall mass transfer coefficient of i compound
Lz	length of the adsorption column
Mci	dimensionless parameter, (Equation (15c))
P	pressure of adsorption column
${\overline{P}}_i$	partial pressure of i compound
qi	dimensionless concentration of i compound in the adsorbed phase
qm,i	maximum concentration of i compound in the adsorbed phase
${\overline{q}}_i$	concentration of i compound in the adsorbed phase
${\overline{q}}_i^{\ast }$	equilibrium concentration of i compound in the adsorbed phase(Equation (4(4) $${\overline{q}}_i^{\ast }=q_{m,i}\frac{b_i{\overline{P}}_i}{1+{\sum }_{j=1}^2{b}_j{\overline{P}}_j}$$(4) ))
ri	radius of the adsorption column
V	dimensionless superficial gas velocity
$\overline{V}$	superficial gas velocity
t	dimensionless time
tf	final time of adsorption
$\overline{t}$	time
T	temperature of column bed
Yi	mole fraction of i compound in the gas phase
Yi,0	inlet mole fraction of i compound in the gas phase
Yi,out	estimated outlet mole fraction of i compound
$Y_i^{exp}$	measured outlet mole fraction of i compound
z	dimensionless coordinate variable
$\overline{z}$	coordinate variable
Greek symbols
ɛb	bed void fraction
ɛt	overall void fraction
$\widehat{\mathit{\epsilon }}$	a prescribed upper limit of the optimized functional J
${\mathit{\lambda }}_{q_i}$	Lagrange multiplier of concentration of i compound in the adsorbed phase
${\mathit{\lambda }}_V$	Lagrange multiplier of velocity
${\mathit{\lambda }}_{Y_i}$	Lagrange multiplier of mole fraction of i compound in the gas phase
ρb	adsorbent apparent density
σs	standard deviation
${\mathit{\omega }}_{Y_i}$	the random error of the field variable Yi