Kinetics of and Transport Phenomena in Gas–Solid Reactors

Santacesaria, Elio; Tesser, Riccardo

doi:10.1007/978-3-319-97439-2_6

Elio Santacesaria³ &
Riccardo Tesser⁴

1740 Accesses

Abstract

The fundamental laws of mass, heat, and momentum transport are briefly presented in this chapter and their relationships with chemical kinetics depicted. The approach presented starts with estimation of basic properties like viscosity, thermal conductivity, and mass diffusivity, all of which are of fundamental importance in describing transport phenomena. Transport phenomena originate gradients in temperature, pressure, and concentration that are the driving forces for transformations occurring in a system. Two scales of transformation can be considered, i.e., molecular and macroscopic. Molecular-transport phenomena are normally much slower than macroscopic ones and this can result in a limitation on chemical reaction rates. In this chapter, detailed examples of basic property calculations are reported as are examples of chemical gas–solid reactions limited by diffusional resistance (like ammonia oxidation). Finally, a great part of the chapter is dedicated to the evaluation of catalyst effectiveness factor. Matlab code associated with the examples in this chapter is available online.

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Authors and Affiliations

Eurochem Engineering s.r.l., Milan, Italy
Elio Santacesaria
Dipartimento di Scienze Chimiche, Complesso di Monte Sant’Angelo, University of Naples Federico II, Naples, Italy
Riccardo Tesser

Authors

Elio Santacesaria
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Riccardo Tesser
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Correspondence to Elio Santacesaria .

6.1 Electronic Supplementary Material

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOCX 45 kb)

Appendices

Appendix 1: Lennard–Jones Force Constants Calculated from Viscosity Data

Compound	$ \epsilon /k,\;^\circ {\text{K}} $	σ, A	Compound	$ \epsilon /k,\;^\circ {\text{K}} $	σ, A
Acetylene	185	4.221	Hydrogen	33.3	2.968
Air	97	3.617	Hydrogen chloride	360	3.305
Argon	124	3.418	Hydrogen iodide	324	4.123
Arsine	281	4.06	Iodine	550	4.982
Benzene	440	5.270	Krypton	190	3.61
Bromine	520	4.268	Methane	136.5	3.822
i-Butane	313	5.341	Methanol	507	3.585
n-Butane	410	4.997	Methylene chloride	406	4.759
Carbon dioxide	190	3.996	Methyl chloride	855	3.375
Carbon disulfide	488	4.438	Mercuric iodide	691	5.625
Carbon monoxide	110	3.590	Mercury	851	2.898
Carbon tetra-chloride	327	5.881	Neon	35.7	2.789
Carbonyl sulfide	335	4.13	Nitric oxide	119	3.470
Chlorine	357	4.115	Nitrogen	91.5	3.681
Chloroform	327	5.430	Nitrous oxide	220	3.879
Cyanogen	339	4.38	n-Nonane	240	8.448
Cyclohexane	324	6.093	n-Octane	320	7.451
Ethane	230	4.418	Oxygen	113	3.433
Ethanol	391	4.455	n-Pentane	345	5.769
Ethylene	205	4.232	Propane	254	5.061
Fluorine	112	3.653	Sulfur dioxide	252	4.290
Helium	10.22	2.576	Water	356	2.649
n-Heptane	282	8.88	Xenon	229	4.055
n-Hexane	413	5.909

See Satterfield and Sherwood (1963), Hirschfelder et al. (1949, 1954), Rowlinson and Townley (1953)

Appendix 2: Collision Integrals Ω_µ and Ω_D as a Function of T* = K_BT/ε for Apolar Molecules: Lennard–Jones Approach

As explained in the text, the following correlation was used for interpolating both of the collision integrals:

$$ \Omega _{i} = 10^{{\left( {ax^{6} + bx^{5} + cx^{4} + dx^{3} + ex^{2} + fx + g} \right)}} \quad {\text{where}}\;\;x = \log_{10} (T^{*} ) $$

The best-fitting coefficients for the polynomials related to Ω_D and Ω_μ are listed in the following table.

Integral	a	b	c	d	e	f	g
Ω_D	−0.0120	0.0877	−0.2146	0.1426	0.1948	−0.4848	0.1578
Ω_μ	−0.0165	0.1204	−0.3011	0.2360	0.1708	−0.4922	0.1997

Data to be interpolated are reported in the following table.

T*	Ω_D	Ω_μ	T*	Ω_D	Ω_μ	T*	Ω_D	Ω_μ
0.30	2.6620	2.7850	1.65	1.1530	1.2640	4.00	0.8836	0.9700
0.35	2.4760	2.6280	1.70	1.1400	1.2480	4.10	0.8788	0.9649
0.40	2.3180	2.4920	1.75	1.1280	1.2340	4.20	0.8740	0.9600
0.45	2.1840	2.3680	1.80	1.1160	1.2210	4.30	0.8694	0.9553
0.50	2.0660	2.2570	1.85	1.1050	1.2090	4.40	0.8652	0.9507
0.55	1.9660	2.1560	1.90	1.0940	1.1970	4.50	0.8610	0.9464
0.60	1.8770	2.0650	1.95	1.0840	1.1860	4.60	0.8568	0.9422
0.65	1.7980	1.9820	2.00	1.0750	1.1750	4.70	0.8530	0.9382
0.70	1.7290	1.9080	2.10	1.0570	1.1560	4.80	0.8492	0.9343
0.75	1.6670	1.8410	2.20	1.0410	1.1380	4.90	0.8456	0.9305
0.80	1.6120	1.7800	2.30	1.0260	1.1220	5.00	0.8422	0.9269
0.85	1.5620	1.7250	2.40	1.0120	1.1070	6.00	0.8124	0.8963
0.90	1.5170	1.6750	2.50	0.9996	1.0930	7.00	0.7896	0.8727
0.95	1.4760	1.6290	2.60	0.9878	1.0810	8.00	0.7712	0.8538
1.00	1.4390	1.5870	2.70	0.9770	1.0690	9.00	0.7556	0.8379
1.05	1.4060	1.5490	2.80	0.9672	1.0580	10.00	0.7424	0.8242
1.10	1.3750	1.5140	2.90	0.9576	1.0480	20.00	0.6640	0.7432
1.15	1.3460	1.4820	3.00	0.9490	1.0390	30.00	0.6232	0.7005
1.20	1.3200	1.4520	3.10	0.9406	1.0300	40.00	0.5960	0.6718
1.25	1.2960	1.4240	3.20	0.9328	1.0220	50.00	0.5756	0.6504
1.30	1.2730	1.3990	3.30	0.9256	1.0140	60.00	0.5596	0.6335
1.35	1.2530	1.3750	3.40	0.9186	1.0070	70.00	0.5464	0.6194
1.40	1.2330	1.3530	3.50	0.9120	0.9999	80.00	0.5352	0.6076
1.45	1.2150	1.3330	3.60	0.9058	0.9932	90.00	0.5256	0.5973
1.50	1.1980	1.3140	3.70	0.8998	0.9870	100.00	0.5130	0.5882
1.55	1.1820	1.2960	3.80	0.8942	0.9811	200.00	0.4644	0.5320
1.60	1.1670	1.2790	3.90	0.8888	0.9755	300.00	0.4360	0.5016
						400.00	0.4170	0.4811

Data from Hirschfelder et al. (1954)

The coefficients were obtained using a MATLAB program importing all data from an Excel file. By applying the correlations found to the collision integrals, average absolute percent errors of 0.1396 and 0.2343% are obtained for, respectively, Ω_D and Ω_μ. Plots of the fittings obtained with the same program are reported in the following figures.

Figures related to Appendix 2. Fittings obtained by mathematical regression analysis on the data of Ω_D and Ω_μ available in the literature and plots of the errors.

These results can be obtained using a MATLAB program available as Electronic Supplementary Material.

Appendix 3: Parameters of the Stockmayer Equation for Some Polar Molecules

Substance	Dipol moment $ \mu_{\text{d}} ({\text{D}}) $	$ \sigma $ $ ({\AA}) $	$ \epsilon_{\text{o}} {/k}_{\text{B}} (k) $	$ \delta = \frac{{\mu_{\text{d}}^{2} }}{{2\epsilon_{\text{o}} \sigma^{3} }} $
H₂O	1.85	2.52	775	1.0
NH₃	1.47	3.15	358	0.7
HCl	1.08	3.36	328	0.34
HBr	0.80	3.41	417	0.14
HI	0.42	4.13	313	0.029
SO₂	1.63	4.04	347	0.42
H₂S	0.92	3.49	343	0.21
NOCl	1.83	3.53	690	0.4
CHCl₃	1.013	5.31	355	0.07
CH₂Cl₂	1.57	4.52	483	0.2
CH₃Cl	1.87	3.94	414	0.5
CH₃Br	1.80	4.25	382	0.4
C₂H₅Cl	2.03	4.45	423	0.4
CH₃OH	1.70	3.69	417	0.5
C₂H₅OH	1.69	4.31	431	0.3
n-C₃H₇OH	1.69	4.71	495	0.2
i-C₃H₇OH	1.69	4.64	518	0.2
(CH₃)₂O	1.30	4.21	432	0.19
(C₂H₅)₂O	1.15	5.49	362	0.08
(CH₃)₂CO	1.20	3.82	428	1.3
CH₃COOCH₃	1.72	5.04	418	0.2
CH₃COOC₂H₅	1.78	5.24	499	0.16
CH₃NO₂	2.15	4.16	290	2.3

1 D (Debye) = 1 × 10⁻¹⁸ statcoulomb × cm = 1 × 10⁻¹⁸ dine ^½ × cm²

Appendix 4: Collision Integral Ω_μ as a Function of T* = K_BT/ε and δ for Polar Molecules

The following correlations were used for determining both the collision integrals:

$$ \begin{aligned} f_{1} & = d_{1} + d_{2} \frac{\delta }{{T^{*} }} + d_{3} \frac{{\delta^{2} }}{{T^{{*^{2} }} }} + d_{4} \frac{{\delta^{3} }}{{T^{*} }} + d_{5} \delta^{4} \\ f_{2} & = \frac{{f_{1} \delta^{1.5} }}{{K^{{\log_{10} (T^{*} )}} }} \\ f_{3} & = a_{1} x^{6} + a_{2} x^{5} + a_{3} x^{4} + a_{4} x^{3} + a_{5} x^{2} + a_{6} x^{{}} + a_{7} \\\Omega _{i} & = 10^{{f_{2} + f_{3} }} \quad {\text{where}}\;\;x = \log_{10} (T^{*} )\;\;i = D{\text{ or }}\mu \\ \end{aligned} $$

The best-fitting coefficients for the functions related to Ω_D and Ω_μ are listed the following table.

	Collision integrals
Parameter	Ω_D	Ω_μ
d ₁	0.066225	0.067498
d ₂	−0.002888	−0.002375
d ₃	0.0000707	0.0000618
d ₄	−0.0000626	−0.0000865
d ₅	−0.0000785	−0.0001022
K	4.507	3.934
a ₁	0.010254	0.010948
a ₂	−0.033249	−0.039147
a ₃	−0.014026	−0.005066
a ₄	0.096320	0.105559
a ₅	0.068759	0.049660
a ₆	−0.434055	−0.425989
a ₇	0.163439	0.203885

These coefficients were determined by mathematical regression analysis made on the data reported by the literature and summarized in the following two tables.

Ω_D
T*	δ
T*	0.00	0.25	0.50	0.75	1.00	1.50	2.00	2.50
0.10	4.00790	4.00200	4.65500	5.52100	6.45400	8.21300	9.52400	11.31000
0.20	3.13000	3.16400	3.35500	3.72100	4.19800	5.23000	6.22500	7.16000
0.30	2.64940	2.65700	2.77000	3.00200	3.31900	4.05400	4.78500	5.48300
0.40	2.31440	2.32000	2.40200	2.57200	2.81200	3.38600	3.97200	4.53900
0.50	2.06610	2.07300	2.14000	2.27800	2.47200	2.94600	3.43700	3.91800
0.60	1.87670	1.88500	1.94400	2.06000	2.22500	2.62800	3.05400	3.47400
0.70	1.72930	1.73800	1.79100	1.89300	2.03600	2.38800	2.76300	3.13700
0.80	1.62200	1.62200	1.67000	1.76000	1.88600	2.19800	2.53500	2.87200
0.90	1.51750	1.52700	1.57200	1.65300	1.76500	2.04400	2.34900	2.65700
1.00	1.43980	1.45000	1.49000	1.56400	1.66500	1.91700	2.19600	2.47800
1.20	1.32040	1.33000	1.36400	1.42500	1.50900	1.72000	1.95600	2.19900
1.40	1.23360	1.24200	1.27200	1.32400	1.39400	1.57300	1.77700	1.99000
1.60	1.16790	1.17600	1.20200	1.24600	1.30600	1.46100	1.63900	1.82700
1.80	1.11660	1.12400	1.14600	1.18500	1.23700	1.37200	1.53000	1.69800
2.00	1.07530	1.08200	1.10200	1.13500	1.18100	1.30000	1.44100	1.59200
2.50	1.00060	1.00500	1.02000	1.04600	1.18000	1.17000	1.27800	1.39700
3.00	0.95003	0.95380	0.96560	0.98520	1.01200	1.08200	1.16300	1.26500
3.50	0.91311	0.91620	0.92560	0.94130	0.96260	1.01900	1.09000	1.17000
4.00	0.88453	0.88710	0.89480	0.90760	0.92520	0.97210	1.03100	1.09800
5.00	0.84277	0.84460	0.85010	0.85920	0.87160	0.90530	0.94830	0.99840
6.00	0.81827	0.81420	0.81830	0.82510	0.83440	0.85980	0.89270	0.93160
7.00	0.78976	0.79080	0.79400	0.79930	0.80660	0.82650	0.85260	0.88360
8.00	0.77111	0.77200	0.77450	0.77880	0.78460	0.80070	0.82190	0.84740
9.00	0.75553	0.75620	0.75840	0.76190	0.76670	0.78000	0.79760	0.81890
10.00	0.74220	0.74280	0.74460	0.74750	0.75150	0.76270	0.77760	0.79570
12.00	0.72022	0.72060	0.72200	0.72410	0.72710	0.73540	0.74640	0.76000
14.00	0.70254	0.70290	0.70390	0.70550	0.70780	0.71420	0.72280	0.73340
16.00	0.68776	0.68800	0.68880	0.69010	0.69190	0.69700	0.70400	0.71250
18.00	0.67510	0.67530	0.67600	0.67700	0.67850	0.68270	0.68840	0.69550
20.00	0.66405	0.66420	0.66480	0.66570	0.66690	0.67040	0.67520	0.68110
25.00	0.64136	0.64150	0.64180	0.64250	0.64330	0.64570	0.64900	0.65310
30.00	0.62350	0.62360	0.62390	0.62430	0.62490	0.62670	0.62910	0.63210
35.00	0.60882	0.60890	0.60910	0.60940	0.60990	0.61120	0.61310	0.61540
40.00	0.59640	0.59640	0.59660	0.59690	0.59720	0.59830	0.59980	0.60170
50.00	0.57626	0.57630	0.57640	0.57660	0.57680	0.57750	0.57850	0.57980
75.00	0.54146	0.54150	0.54160	0.54160	0.54180	0.54210	0.54240	0.54290
100.00	0.51803	0.51810	0.51820	0.51840	0.51840	0.51850	0.51860	0.51870

Ω_μ
T*	δ
T*	0.00	0.25	0.50	0.75	1.00	1.50	2.00
0.10	4.10050	4.26600	4.83300	5.74200	6.62900	8.62400	10.34000
0.20	3.26260	3.30500	3.51600	3.91400	4.43300	5.57000	6.63700
0.30	2.83990	2.83600	2.93600	3.16800	3.51100	4.32900	5.12600
0.40	2.53100	2.52200	2.58600	2.74900	3.00400	3.64000	4.28200
0.50	2.28370	2.27700	2.32900	2.46000	2.66500	3.18700	3.72700
0.60	2.08380	2.08100	2.13000	2.24300	2.41700	2.86200	3.32000
0.70	1.92200	1.92400	1.97000	2.07200	2.22500	2.61400	3.02800
0.80	1.79020	1.79500	1.84000	1.93400	2.07000	2.41700	2.78800
0.90	1.68230	1.68900	1.73300	1.82000	1.94400	2.25800	2.59600
1.00	1.59290	1.60100	1.64400	1.72500	1.83800	2.12400	2.43500
1.20	1.45510	1.46500	1.50400	1.57400	1.67000	1.91300	2.18100
1.40	1.35510	1.36500	1.40000	1.46100	1.54400	1.75400	1.98900
1.60	1.28000	1.28900	1.32100	1.37400	1.44700	1.63000	1.83800
1.80	1.22190	1.23100	1.25900	1.30600	1.37000	1.53200	1.71800
2.00	1.17570	1.18400	1.20900	1.25100	1.30700	1.45100	1.61800
2.50	1.09330	1.10000	1.11900	1.15000	1.19300	1.30400	1.43500
3.00	1.03880	1.04400	1.05900	1.08300	1.11700	1.20400	1.31000
3.50	0.99630	1.00400	1.01600	1.03500	1.06200	1.13300	1.22000
4.00	0.96988	0.97320	0.98300	0.99910	1.02100	1.07900	1.15300
5.00	0.92676	0.92910	0.93600	0.94730	0.96280	1.00500	1.05800
6.00	0.89616	0.89790	0.90300	0.91140	0.92300	0.95450	0.99550
7.00	0.87272	0.87410	0.87800	0.88450	0.89350	0.91810	0.95050
8.00	0.85379	0.85490	0.85800	0.86320	0.87030	0.89010	0.91640
9.00	0.83795	0.83880	0.84140	0.84560	0.85150	0.86780	0.88950
10.00	0.82435	0.82510	0.82730	0.83080	0.83560	0.84930	0.86760
12.00	0.80184	0.80240	0.80390	0.80650	0.81010	0.82010	0.83370
14.00	0.78363	0.78400	0.78520	0.78720	0.78990	0.79760	0.80810
16.00	0.76834	0.76870	0.76960	0.77120	0.77330	0.77940	0.78780
18.00	0.75518	0.75540	0.75620	0.75750	0.75920	0.76420	0.77110
20.00	0.74364	0.74380	0.74450	0.74550	0.74700	0.75120	0.75690
25.00	0.71982	0.72000	0.72040	0.72110	0.72210	0.72500	0.72890
30.00	0.70097	0.70110	0.70140	0.70190	0.70260	0.70470	0.70760
35.00	0.68545	0.68550	0.68580	0.68610	0.68670	0.68830	0.69050
40.00	0.67232	0.67240	0.67260	0.67280	0.67330	0.67450	0.67620
50.00	0.65099	0.65100	0.65120	0.65130	0.65160	0.65240	0.65340
75.00	0.61397	0.61410	0.61430	0.61450	0.61470	0.61480	0.61480
100.00	0.58870	0.58890	0.58940	0.59000	0.59030	0.59010	0.58950

A mathematical regression analysis was performed using a MATLAB program available as Electronic Supplementary Material.

By applying the correlations found in the calculation of the collision integrals, average absolute percent errors of 1.62 and 1.85% are obtained for, respectively, Ω_D and Ω_μ. The obtained fittings can be appreciated in the plots reported in the following figures.

Figures 1 related to Appendix 4. Fittings obtained by mathematical regression analysis on the data of Ω_D available in the literature and parity plot .

Figures 2 related to Appendix 4. Fittings obtained by mathematical regression analysis on the data of Ω_μ available in the literature and parity plot .

Appendix 5: Additive Volume Increments for the Estimation of the Molar Volume V_b at Normal Boiling Point

Substance	V_b increment, cm³/g mol
Air	29.9
Ammonia	25
Bromine	27
Carbon	14.8
Chlorine, terminal, as R–CI	21.6
Medial, as R–CHC1–R	24.6
Fluorine	8.7
Helium	1.0
Hydrogen (in compound)	3.7
Hydrogen (molecular)	14.3
Mercury	15.7
Nitrogen	31.2
In primary amines	10.5
In secondary amines	12.0
Oxygen, molecular	14.8
Doubly bound	7.4
Methyl esters and ethers	9.1
Ethyl esters and ethers	9.9
Higher esters and ethers	11.0
Acids	12.0
Joined to S, P, or N	8.3
Phosphorus	27
Sulfur	25.6
Rings: 3-membered	−6
4-membered	−8.5
5-membered	−11.5
6-membered	−15
Naphthalene	−30
Anthracene	−47.5

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Santacesaria, E., Tesser, R. (2018). Kinetics of and Transport Phenomena in Gas–Solid Reactors. In: The Chemical Reactor from Laboratory to Industrial Plant. Springer, Cham. https://doi.org/10.1007/978-3-319-97439-2_6

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DOI: https://doi.org/10.1007/978-3-319-97439-2_6
Published: 05 November 2018
Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-97439-2
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