Catalysts thermal energy balance

The final scaled catalyst thermal energy balance is... 

THERMAL ENERGY BALANCE IN MULTICOMPONENT MIXTURES AND NONISOTHERMAL EFFECTIVENESS FACTORS VIA COUPLED HEAT AND MASS TRANSFER IN POROUS CATALYSTS... 

Answer Two. The thermal energy balance is not required when the enthalpy change for each chemical reaction is negligible, which causes the thermal energy generation parameters to tend toward zero. Hence, one calculates the molar density profile for reactant A within the catalyst via the mass transfer equation, which includes one-dimensional diffnsion and multiple chemical reactions. Stoichiometry is not required because the kinetic rate law for each reaction depends only on Ca. Since the microscopic mass balance is a second-order ordinary differential eqnation, it can be rewritten as two coupled first-order ODEs with split boundary conditions for Ca and its radial gradient. 

A to products by considering mass transfer across the external surface of the catalyst. In the presence of multiple chemical reactions, where each iRy depends only on Ca, stoichiometry is not required. Furthermore, the thermal energy balance is not required when = 0 for each chemical reaction. In the presence of multiple chemical reactions where thermal energy effects must be considered becanse each AH j is not insignificant, methodologies beyond those discussed in this chapter must be employed to generate temperature and molar density profiles within catalytic pellets (see Aris, 1975, Chap. 5). In the absence of any complications associated with 0, one manipulates the steady-state mass transfer equation for reactant A with pseudo-homogeneous one-dimensional diffusion and multiple chemical reactions under isothermal conditions (see equation 27-14) ... 

In conclusion, one solves two coupled first-order ODEs for the molar density profile of reactant A under isothermal conditions, without considering the thermal energy balance. Then, a volumetric average of the rate of conversion of reactant A to products due to multiple chemical reactions is obtained by focusing on the reactant concentration gradient at the external surface of the catalyst ... 

In addition to flow, thermal, and bed arrangements, an important design consideration is the amount of catalyst required (W), and its possible distribution over two or more stages. This is a measure of the size of the reactor. The depth (L) and diameter (D) of each stage must also be determined. In addition to the usual tools provided by kinetics, and material and energy balances, we must take into account matters peculiar to individual particles, collections of particles, and fluid-particle interactions, as well as any matters peculiar to the nature of the reaction, such as reversibility. Process design aspects of catalytic reactors are described by Lywood (1996). 

A complete description of the reactor bed involves the six differential equations that describe the catalyst, gas, and thermal well temperatures, CO and C02 concentrations, and gas velocity. These are the continuity equation, three energy balances, and two component mass balances. The following equations are written in dimensional quantities and are general for packed bed analyses. Systems without a thermal well can be treated simply by letting hts, hlg, and R0 equal zero and by eliminating the thermal well energy equation. Adiabatic conditions are simulated by setting hm and hvg equal to zero. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

Another potential model simplification involves assuming negligible energy accumulation in the gas phase as compared to that in the solid, which is equivalent to the earlier approximation [Eq. (66)] based on the relative magnitude of the energy accumulation in the gas and solid. For our system, the accumulation of energy in the solid is approximately 250 to 300 times that in the gas phase due to the relative thermal capacitance of the gas [Eq. (65)] and the similarity of the temporal behavior of the gas and catalyst temperatures (e.g., Fig. 19). Thus the accumulation term in the energy balance... 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

The thermal capacitance of the gas in the reactor is assumed negligible compared to that of the solid catalyst. Therefore a single dynamic energy balance is used for each lump, and the gas temperature is assumed to be equal to the catalyst temperature... 

Even when is low, the center and surface temperatures may differ appreciably, because catalyst pellets have low thermal conductivities (Sec. 11-5). The combined effect of mass and heat transfer on can still be represented by the general definition of the effectiveness factor, according to Eq. (11-41). Hence Eq. (11-42) may be used to find r, provided rj is the nonisothermal effectiveness factor. The nonisothermal 17 may be evaluated in the same way as the isothermal 77, except that an energy balance must be combined with the mass balance. 

Thermal effects constitute a significant portion of the study devoted to catalysis. This is true of electrochemical reactions as well. In general the reaction rate constants, diffusion coefficients, and conductivities all exhibit Arrhenius-type dependence on temperature, and as a rule of the thumb, for every 10°C rise in temperature, most reaction rates are doubled. Hence, temperature effects must be incorporated into the parameter values. Fourier s law governs the distribution of temperature. For the example with the cylindrical catalyst pellet described in the previous section, the equation corresponding to the energy balance can be written in the dimensionless form as follows ... 

The first boundary condition follows from symmetrical reasons. In practice, the effective heat conductivity of the catalyst, Tie, is often so high that the temperature gradients inside the particle are minor. On the other hand, there often emerges a temperature gradient in the fluid film around the catalyst particle, since the thermal conduction of the fluid is limited. The energy balance of the fluid film is reduced to... 

We now consider situations in which the catalyst particle is not isothermal. Given an exothermic reaction, for example, if the particle s thermal conductivity is not large compared to the rate of heat release due to chemical reaction, the temperature rises inside the particle. We wish to explore the effects of this temperature rise on the catalyst performance. We have already written the general mass and energy balances for the catalyst particle in Section 7.3. Consider the single-reaction case, in which we have Ra = and Equations 7.14 and 7.15 reduce to... 

The temperature of the catalyst is derived from various contributions of an energy balance at the interface. The conductive, convective, and diffusive energy transport from the gas phase adjacent to the surface as well as the chemical heat release at the surface, the thermal radiation and a possible external heating (here resistive heating) of the catalyst are included. This results in... 

In this system a reactant enters a chemical reactor at = 0 with a superficial velocity of v (cm/sec). The reactor is packed with catalyst particles. The heat of reaction is known as AH (cal/g mol). The reaction rate is zero-order in the reactant concentrtion and therefore is a constant down the reactor R (g mol/sec cm ). Because of the effect of the catalyst packing, both convective and dispersion thermal effects are present in this reactor. We want to compute the temperature profile down the reactor. We write a steady-state energy balance for the differential reactor volume, AAz, as... 

We now develop the solid phase catalyst bed thermal energy equation. Since the catalyst bed is stationary, the general energy balance becomes... 

In another reply Carassiti [14] argued that the catalytic connotation of photocatalysis is not consistent with Ostwald s idea that a catalyst changes the rate of a chemical reaction without any influence on the position of the equilibrium. If a photon is consumed in each cycle to generate an active species, light plays the role of a reactant. The catalyst may lower the activation barrier but the reaction is different from the thermal one in terms of a changed energy balance. In order to avoid unnecessary confusion Carassiti advised the application of the term photocatalysis only to such reactions which satisfy the conceptual requirements of catalytic processes. 

As it has been outlined in chapter 2 the widely accepted meaning of photocatalysis is that both light and a catalyst are necessary to bring about an appropriate reaction. The term catalyst is used somewhat different than in the classical thermal catalysis. First of all, here the term catalyst includes both catalytically active species able to effect repeatedly the conversion of a substrate S to a product P and nominal catalysts or initiators to be activated by light. Secondly, despite the classical meaning, a catalyst may well influence the free energy balance of the parent thermal reaction from S to P due to the participation of electronically excited species in photocatalytic reactions.The different features of photocatalysis when compared with thermal catalysis arise from the constituent "photo", indicating that a photon has to be absorbed by either the substrate or the catalyst prior to or within... 

Similar considerations apply to the thermal effects in catalyst particles. An energy balance in the spherical particle leads to a differential equation which is identical to Eq. (6.63), if the following relation holds ... 

Moving up into the reactor level, effects of convection, dispersion and generation are described in the conservation equations for mass and energy. The momentum balance describes the behavior of pressure. The interface between the reactor and the catalyst level is described by the external mass transfer conditions, most often represented in a Fickian format, i.e., a linear dependence of the rate of mass transfer on the concentration gradient. In cases where an explicit description of mixing and hydrodynamic patterns is required, the simultaneous integration of the Navier-Stokes equations is also conducted at this level. I f the reaction proceeds thermally, the conversion of mass and the temperature effect as a result of it are described here as well. 

The reformer takes an input flow rate of methane and computes the hydrogen output. The reformer module balances energy by combusting the reformate stream with air and exchanging the heat released to the catalyst reactor. Parameters on the reformer are the steam-to-carbon ratio and the outlet temperature of the exhaust products from the internal burner. The temperature at which the equilibrium reforming occurs depends on these parameters. Figure 1 shows the variation in thermal efficiency of the reformer with temperature and steam-to-carbon ratio. The minimum steam-to-carbon ratio is 2 however, reformers are often operated with excess steam to improve the efficiency and prevent coking problems. 

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