Calculation of External Temperature Differences

When a fixed-bed reactor operates at steady state, an amount of energy equal to the heat released by reaction on the catalyst pellet must be transferred to the bulk fluid. In Sec. 10-2 this requirement was used to relate the concentration and temperature differences between pellet and fluid. Here we want to develop a method for predicting the magnitude of the temperature 

The heat transferred to the bulk fluid, Q, can be simply expressed in terms of 7] — Tj, as 

Equating Q and Qr -slI steady state determines 7 — 7], in terms of AH, the rate parameters A and E for the.reaction, h, and the unknown surface concentration Q. The requirement that Qr = Q introduces interesting questions about stable operating conditions. The problem is very similar to the stability situation in stirred-tank reactors, discussed in Sec. 5-4. We consider this problem and the evaluation of E — E, for two cases negligible and finite external-diffusion resistance. 

Negligible Diffusion Resistance With this restraint C in Eq. (10-24) becomes Q . Then equating Qr and 0 establishes E — Tj, in terms of the heat of reaction — AH, A, E, and h. The resultant equation cannot be solved analytically for 7 — 7),. However, the intersection of curves for Q and Qr plotted against 7 — will give the solution. According to Eq. (10-24), Qr will be an exponential curve in 6, as shown in Fig. 10-4 for an exothermic reaction. Equation (10-25) is linear in 6. It could intersect Qr as indicated by curves Qi or Q2, or it could be below Qr, as shown by curve 23, depending on the magnitude of h and the location of Qr.. 

The curves in Fig. 10-4 were originally proposed by Frank-Kamenet-ski and are useful for describing regions where multiple values of 7 are 

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Thus the mass-transfer rate between particle and fluid for a fluidized bed is approximately two orders of magnitude greater than for a fixed bed. With this large transport rate, it is evident that Q — Q, calculated by the methods described in Sec. 10-3, will be negligible. A similar result applies for external temperature differences. 

Calculations of the temperature fields for the target reactor power of 500 kW of thermal energy were performed to scope the differences between cooling at the reactor core boimdary and cooling at the external boundary of the reflector. These scoping calculations were performed for the preliminary core size of 100-cm-diam by 100-cm-long cylinder with a 15-cm-thick reflector on all surfaces. Two computational approaches were used to model the reactor—a simplified infinite-cylinder 1-D model and a detailed 3-D finite element. 

A typical CC-measurement procedure (for a pH-sensitive LAPS structure) is depicted in Fig. 6.3. From the raw data material of the CC-mode measurement of all measurement spots under the pH-sensitive layer, a calibration plot can be derived. For example, for the above example, an average pH sensitivity of 54.2 mV/pH with a standard deviation of 0.5 mV/pH between the different measurement spots can be calculated. This initial calibration measurement allows furthermore the determination of different measurement parameters, e.g., the hysterisis, overall drift, stability, selectivity and the influence of external disturbances such as light and temperature. These parameters are important to evaluate the performance of the complete LAPS-based measurement system. 

The interpretation of measured data for Z(oi) is carried out by their comparison with predictions of a theoretical model based either on the (analytical or numerical) integration of coupled charge-transport equations in bulk phases, relations for the interfacial charging and the charge transfer across interfaces, balance equations, etc. Another way of interpretation is to use an -> equivalent circuit, whose choice is mostly heuristic. Then, its parameters are determined from the best fitting of theoretically calculated impedance plots to experimental ones and the results of this analysis are accepted if the deviation is sufficiently small. This analysis is performed for each set of impedance data, Z(co), measured for different values of external parameters of the system bias potentials, bulk concentrations, temperature... The equivalent circuit is considered as appropriate for this system if the parameters of the elements of the circuit show the expected dependencies on the external parameters. 

The measurements of external and internal specific surface area have already been discussed in Chapter 1, Section 1.1.3. The principles and the isotherm equation of the BET method to measure external specific surface area, including macro- and mesopores, have been presented in Chapter 1, Section 1.3.4.1.5. The external specific surface area is usually determined by nitrogen gas adsorption at the temperature of liquid nitrogen. Both static (one-point) and dynamic (five-point) methods are applied. The calculations are made by Equation 1.75 (Chapter 1), using one or five different pressure values. The external specific surface area is calculated from the maximum number of surface sites, that is, monolayer and the cross-sectional area of nitrogen molecules. 

The complex electric permittivity, k = k + k , where k = C/C o is the real, and k = tan(8) / K is the complex part of the permittivity, was measured in the frequency interval 300 Hz - 1 MHz at different temperatures by a Solartron 1200 inq>edance gain analyser, using a parallel plate capacitor made of stainless steel. From the capacitance, C, and the tangent loss, tan(6), the values of k and k were calculated [2]. The temperature was controlled within O.IK using a platinum resistor Pt(lOO) as a sensor and a K30 Modinegen external cryostat coupled with a N-180 ultra-cryostat. 

Example 10-1 Experimental, global rates are given in Table 10-2 for two levels of conversion of SOj to SO3. Evaluate the concentration difference for SO2 between bulk gas and pellet surface and comment on the significance of external diffusion. Neglect possible temperature differences. The reactor consists of a fixed bed of x -in. cylindrical pellets through which the gases passed at a superficial mass velocity of 147 lb/(hr)(ft ) and at a pressure of 790 mm Hg. The temperature of the catalyst pellets was 480°C, and the bulk mixture contained 6.42 mole % SOj and 93.58 mole % air. To simplify the calculations compute physical properties on the basis of the reaction mixture being air. The external area of the catalyst pellets is 5.12 ft /lb material. The platinum covers only the external surface and a very small section of the pores of the alumina carrier, so that internal diffusion need not be considered. 

The reactor consisted of a fixed bed of x --in. cylindrical pellets. The pressure was 790 mm Hg. The external area of catalyst particles was 5.12 ft /lb, and the platinum did not penetrate into the interior of the alumina particles. Calculate the partial-pressure difference between the bulk-gas phase and the surface of the catalyst for SOj at each mass velocity. What conclusions may be stated with regard to the importance of external diffusion Neglect temperature differences. 

For use in reactor design the global rate should be calculable at all locations in the reactor. We suppose that the intrinsic rate equation is available. The problem is to evaluate the global rate corresponding to possible bulk concentrations Q, bulk temperatures 7, and flow conditions. If external and internal temperature differences can be neglected, the problem is straightforward and is essentially the reverse of the stepwise solution outlined in Sec. 12-1. The double-trial procedure is not necessary, because /(C) is known. The effective diffusivity of the catalyst pellet is required. The equations we need are Eq. (10-1) for external diffusion,... 

The calculation of the relative characteristic peak areas on the chromatograms of the volatile pyrolysis products, using an external standard irrespective of the pyrolysis procedure, permits one to take into account the sensitivity of the detector, with easy computation of the ratio between the peak areas of the component of interest and the standard which, under normal conditions (sample size, carrier gas flow-rate, pyrolysis temperatures, etc.) are proportional to the absolute amounts of the pyrolysis products. This method of calculation is essentially a modification of the absolute calibration method in gas chromatography, which had never been used before in Py—GC.To facilitate comparison of the results obtained at different times or on different instruments, the results of individual measurements should preferably be presented in terms of specific yields (or relative characteristic peak areas), i.e., the yield of the volatile pyrolysis products must be calculated per 1 mg (or g or ng) of the pyrolysed sample with respect to 1 mg (or g or Mg) of the external standard. Such a calculation makes sense in the range of sample sizes which affect only insignificantly the specific yield of light pyrolysis products. 

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