Mass macro-model

This involves knowledge of chemistry, by the factors distinguishing the micro-kinetics of chemical reactions and macro-kinetics used to describe the physical transport phenomena. The complexity of the chemical system and insufficient knowledge of the details requires that reactions are lumped, and kinetics expressed with the aid of empirical rate constants. Physical effects in chemical reactors are difficult to eliminate from the chemical rate processes. Non-uniformities in the velocity, and temperature profiles, with interphase, intraparticle heat, and mass transfer tend to distort the kinetic data. These make the analyses and scale-up of a reactor more difficult. Reaction rate data obtained from laboratory studies without a proper account of the physical effects can produce erroneous rate expressions. Here, chemical reactor flow models using matliematical expressions show how physical... 

Steam-liquid flow. Two-phase flow maps and heat transfer prediction methods which exist for vaporization in macro-channels and are inapplicable in micro-channels. Due to the predominance of surface tension over the gravity forces, the orientation of micro-channel has a negligible influence on the flow pattern. The models of convection boiling should correlate the frequencies, length and velocities of the bubbles and the coalescence processes, which control the flow pattern transitions, with the heat flux and the mass flux. The vapor bubble size distribution must be taken into account. 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

Here follows a section outlining the heat and mass transport phenomena of the thermochemical conversion on both the micro- and the macro-scale of the fuel bed. Knowledge about the heat and mass transport phenomena on micro-scale is very important to be able to understand and model, for example, the mass flow of conversion gas. 

The long-term goal in the science of thermochemical conversion of a solid fuel is to develop comprehensive computer codes, herein referred to as a bed model or CFSD (computational fluid-solid dynamics). Firstly, this CFSD code must be able to simulate basic conversion concepts, with respect to the mode, movement, composition and configuration of the fuel bed. The conversion concept has a great effect on the behaviour of the thermochemical conversion process variables, such as the molecular composition and mass flow of conversion gas. Secondly, the bed model must also consider the fuel-bed structure on both micro- and macro-scale. This classification refers to three structures, namely interstitial gas phase, intraparticle gas phase, and intraparticle solid phase. Commonly, a packed bed is referred to as a two-phase system. 

Mathematical modeling is the science or art of transforming any macro-scale or microscale problem to mathematical equations. Mathematical modeling of chemical and biological systems and processes is based on chemistry, biochemistry, microbiology, mass diffusion, heat transfer, chemical, biochemical and biomedical catalytic or biocatalytic reactions, as well as noncatalytic reactions, material and energy balances, etc. 

S //Asa mediator between CFD calculations and macro-scale process simulations, the reactor geometry is represented by a relatively small number of cells which are assumed to be ideally mixed. The basic equations for mass, impulse and energy balance are calculated for these cells. Mass transport between the cells is considered in a network-of-cells model by coupling equations which account for convection and dispersion. The software is capable of optimizing a process in iterative simulation cycles in a short time on a standard PC, but it also requires experimentally-based data to calibrate the software modules to a specific micro reactor. 

The pore connectivity r of two types of silica (highly porous beads, monolithic silicas) was calculated according to the pore network model proposed by Meyers and Liapis. Nt was proportional to the particle porosity in the case of highly porous beads. The differences in the pore connectivity for both types of silica were reflected in the mass transfer kinetics in liquid phase separation processes by measuring the theoretical plate height-linear velocity dependencies. In a future study, monolithic silicas possessing different macro- as well as mesopores will be investigated and compared with the presented results. 

The internal mass transfer is modeled with Fick s diffusion inside the (macro) pores (Eq. 6.82) as well as surface or micropore diffusion in the solid phase (Eq. 6.83). Note that Eqs. 6.82 and 6.83 no longer include the number of particles (Eq. 6.20) and therefore represent the balance in one particle. 

However, a solution in the Laplace domain has been derived by Kucera [30] and Kubin [31]. The solution cannot be transformed back into the time domain, but from that solution, these authors have derived the expressions for the first five statistical moments (see Section 6.4.1). For a linear isotherm, this model has been studied extensively in the literature. The solution of an extension of this model, using a macro-micropore diffusion model with external film mass transfer resistance, has also been discussed [32]. All these studies use the Laplace domain solution and moment analysis. 

Even if the influence of macro-kinetic factors (heat- and mass-transfer) is taken into account, the models of this type are usually called micro-kinetic . This is due to the principal importance of the set of elementary reactions. In all cases these models are based on a guess (more or less experimentally and/or theoretically grounded) about the factors—chemical and physical—important for the behavior of the systems, as well as about the factors which can be ignored for certain purposes. 

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