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Energy & Fuels 2009, 23, 586–587
Communication Effects of Diffusion on Char-Desorption Profiles Wei-Yin Chen* and Shaolong Wan Department of Chemical Engineering, Anderson Hall, Post Office Box 1848, UniVersity of Mississippi, UniVersity, Mississippi 38677-9740 ReceiVed September 26, 2008. ReVised Manuscript ReceiVed December 5, 2008 Products from a laboratory reactor often travel a certain distance before an analytical instrument detects them. Questions therefore arise if diffusion during transport seriously distorted the signals attained by the analyzer. In this work, the dispersion of CO and CO2 from temperature-programmed desorption (TPD) of char before they reach the mass spectrometer (MS) is sequentially examined by a set of methods. The Taylor-Aris criterion assures that an axially dispersed, plug-flow model is sufficient to determine the dispersion effects. The effective axial dispersion coefficient reveals that dispersion is, however, not negligible and additional analysis is needed. Novel tracer experiments are performed to determine the residence time distribution (RTD) of the product, and the apparent kinetics recovered from mass spectrometer (MS)1-3 are compared to that with RTD correction. Finally, fast Fourier transform (FFT) is also conducted to reconstruct the actual product evolution. The analyses of Taylor,4,5 Aris,6 and Hunt7 concluded that a fully dispersed flow reduces to an axial-dispersed, plug-flow model if DM R 1 . ) Per uR 16L
(1)
where Per denotes the radial Peclet number, DM denotes the diffusivity, u denotes the mean fluid velocity, and L and R denote the length and radius of the tube. Under such conditions, the effective diffusivity in the reduced system, Ez, is the sum of molecular diffusivity and the apparent diffusivity contributed by the dispersion in the radial direction and the laminar flow, i.e., Ez ) DM +
R2u2 48DM
(2)
Substituting reactor and transfer line parameters1-3 into eq 1 suggested that the Taylor-Aris criterion is satisfied with 1/Per ) 0.128 and 0.503 at 100 and 1650 °C, respectively, and R/16L ) 1.172 × 10-3. The reciprocal Peclet numbers in the reactor section, Ez/uL, are 0.012 and 0.046 at 100 and 1650 °C, respectively. Compar* To whom correspondence should be addressed. Telephone: (662) 9155651. Fax: (662) 915-7023. E-mail:
[email protected]. (1) Chen, W. Y.; Wan, S.; Shi, S. Energy Fuels 2007, 21, 778–792. (2) Chen, W. Y.; Shi, G.; Wan, S. Energy Fuels 2008, 22, 3724–3735. (3) Wan, S.; Chen, W. Y.; Shi, S. Roles of mineral matter in the early stages of coal combustion. Energy Fuels, in press. (4) Taylor, G. Proc. R. Soc. A 1953, 219, 186–203. (5) Taylor, G. Proc. R. Soc. A 1954, 225, 473–477. (6) Aris, R. Proc. R. Soc. A 1956, 235, 67–77. (7) Hunt, B. Int. J. Heat Mass Transfer 1977, 20, 393–401.
Figure 1. Measured and regressed CO2 concentration because of the decomposition of CaCO3 at 1400 °C.
ing these values to the graphical representation of theoretical analysis8 suggested that dispersion is not completely negligible and additional analysis of the extent of its distortions are necessary on the measured rate constants, desorption, and oxidation, reported previously.1-3 These numbers are much smaller for the transfer line and capillary because of their relative long length and smaller diameter. Specific effects of dispersion are analyzed by two independent techniques. Both of them involve the use of RTD that characterizes the dispersion of species in the transfer line, including the reactor section after the gaseous product is formed. To obtain the response of a pulse production of a species, rapid decomposition of CaCO3 at high temperatures is performed at the flow conditions identical to our TPD experiments and CO2 is monitored by MS. A 4 mg mixture of 5.0 µM CaCO3 powder and Al2O3 particles of 120-250 µM at 1:49 weight ratio was injected into the preheated furnace at 900 and 1400 °C in two separate experiments. The heat-up (,1 s) and decomposition times (0.3 s at 900 °C) have been calculated to ensure that they are sufficiently short in comparison to the mean gas residence time in the reactor, 2.76 min. The CO2 profile monitored by MS is shown in Figure 1. This CO2 profile is adopted in recovering the characteristic parameter, Ez/uL, and the mean residence time of the system. For fluid in an open-open vessel with Ez/uL greater than 0.01, it is known that the solution of the axial dispersion model with a pulse tracer as its initial condition is8 (8) Levenspiel, O. Chemical Reaction Engineering, 3rd ed.; John Wiley and Sons, Inc.: New York, 1999; p 301.
10.1021/ef8008212 CCC: $40.75 2009 American Chemical Society Published on Web 01/05/2009
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Energy & Fuels, Vol. 23, 2009 587
Figure 2. TPD profiles with and without the influences of dispersion in the down stream.
1
(
(1 - θ)2 4θ(Ez/uL)
)
(3)
Figure 3. Experimentally observed TPD profile and TPD profile restored by RTD and FFT.
where θ ) t/τ, t, θ, and Eθ,00 denote the time, the dimensionless time, and the RTD of a pulse tracer in an open-open vessel, respectively. The experimental data were regressed against eq 3 in Mathcad, and τ and Ez/uL were found to be 2.76 min and 0.02, respectively; they are very close to the calculated values based on the characteristics of our flow system discussed earlier. The predicted RTD based on the recovered parameters is shown in Figure 3 for comparison. The RTD is used to examine the TPD profile by two different methods. In the first method, the TPD profiles are examined on the basis of a set of representative rate constants for desorption of surface oxides with and without the influences of dispersion. For the system without the influence of dispersion, the rate of production of a desorption product under a constant heating rate can be simulated on the basis of the following equation:9
in Mathcad where the y-axis c represents dV/dt. The diffusion shifts the TPD profile slightly to the right. However, regression based on the predicted TPD profile involving dispersion gives the activation energy, E ) 51 kcal mol-1, a 2% departure from the assumed value of 50 kcal mol-1. In the second approach, the “true” TPD profile was obtained by taking numerical FFT inversion of the ratio of the FT of the apparent TPD profile recorded by MS y(s) to the FT of the weight function that characterizes the dispersion w(s) Mathematically, it can be expressed as x(s) ) y(s)/w(s), where x(s) denotes the FT of the restored TPD without the interference of dispersion. These computations are conducted by programs in Matlab format; Matlab provides built-in functions for both FFT and its inverse transform, IFFT. Figure 3 illustrates the comparison of the experimentally observed TPD profile of the bituminous coal char oxidized by 2% O2 at 1000 °C and the TPD restored by the FFT procedure discussed above. The response of CaCO3 decomposition at 1400 °C presented in Figure 1 was adopted as the weight function w(t). Computational work is also conducted on the basis of the responses of CaCO3 decomposition at 900 °C; both simulations involve only a small shift of 2.5 min, which is essentially the mean residence time of CO in the transfer line after their desorption from the char. This analysis suggests that the dispersion does not severely distort the measured desorption and oxidation rate constants reported previously.2,3
Eθ,00 )
√4π(Ez/uL)
exp -
[
( )]
K0R 2 dV K0V0 E E ) exp T exp (4) dT m RT mE RT where V, T, K0, V0, m, E, and R denote the moles of desorption product, temperature, frequency factor, overall yield of desorbed product, heating rate, activation energy, and ideal gas constant, respectively. Under the influence of dispersion, the TPD curve altered by diffusion can be obtained by taking the convolution integral of eqs 3 and 4,8 i.e., dV ) dtdiffusion
mE ∫ dV(t′) dT t
0
θ,00(t - t′)dt′
(5)
Figure 2 shows the results of these two simulations generated (9) Juntgen, H.; van Heek, K. Fuel 1968, 47, 103–117.
Acknowledgment. The authors acknowledge the financial support of the National Science Foundation under Grant CTS-0122504. EF8008212
