Reactor characteristic dimensionless number

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

A numerical study of the free-radical polymerization of styrene (Scheme 6.15) compared the behavior of an interdigital micromixer with a T-junction and a straight tube [37, 48], The diffusion coefficient of the reactive species was varied to simulate the viscosity increase during a polymerization. The performance of the polymerization turned out to be largely dependent on the radial Peclet number. This dimensionless number is defined as the ratio of the characteristic time of diffusion in the direction perpendicular to the main flow to the characteristic time of convection in the flow direction (i.e., the mean residence time) and, therefore, is directly proportional to the characteristic length of the reactor. 

The reactor length L is the characteristic dimension in the axial direction and the tube diameter IR is the characteristic radial dimension. Express your answers in terms of dimensionless numbers and parameters. 

Reynolds number for flow is a dimensionless number characterizing the turbulence in the reactor. When the Reynolds number is below 2500, the flow regime is described as laminar, and suitable scale-up approaches are used. The Reynolds nuiriber range, 2500 to 10,000, is considered by many as transitional, and engineers are very careful not to operate in that regime because of its imique characteristics. Reynolds numbers above 10,000 describe fully developed turbulence. 

The mass transfer in forced flow is trade-off between the attainable mass transfer time, t, and the energy input. The nonidentical geometries of the different devices complicate the direct comparison. Therefore, dimensionless numbers are used. The mass transfer performance is characterized by the ratio between the space time of the fluid and the characteristic mass transfer time. This ratio corresponds to the first Damkohler number for mass transfer. Whereas the porosity of a packed bed reactor is mostly in the order of 0.35 < < 0.45 depending on... 

In a catalytic membrane, the catalytic layer is usually well-defined and very thin (e.g., from 1 up to 30 pm) and its behaviour can be described in analogy with the catalytic slab reported in several chemical reaction engineering textbooks. Material balances on the thin catalytic layer of a membrane lead to the definition of a Thiele modulus (Cini et al, 1991a). Simple considerations on the Thiele modulus and the effectiveness factor in a catalytic membrane reactor have been given by Bottino et al. (2009) and Di Felice et al. (2010). The Thiele modulus, , is a dimensionless number composed of the square root of the characteristic reaction rate (e.g., for an n-order reaction), r ... 

Table 2.1 presents a non-exhaustive list of expressions of the characteristic times corresponding to the most commonly used phenomena involved in chemical reactors. The previous definitions unfortunately do not always enable one to build the expressions presented in Table 2.1. Various methods can be used such as a blind dimensional analysis, similar to the Buckingham method used for dimensionless numbers [7], which can be applied to the list of fundamental physical and chemical properties. Nevertheless, the most relevant method consists in extracting the expressions from a mass/heat/force balance.

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

As mentioned earlier, the characteristic behavior of a given reactor/operation is associated with a region in the multipara-metric space. This space has as many dimensions as the number of model parameters. For typical isothermal monolith reactor designs studied in Section 8.2, we found four relevant dimensionless numbers ... 

Number of pressure channels - the total number of pressure channels in the reactor. This characteristic is applicable to reactors where pressure channels are inside the reactor vessel to maintain the coolant under pressure and contain the nuclear fuel. Where applicable, a dimensionless number should be entered. 

The ratio of reaction and permeation rates is critical in designing an MR. Dimensionless numbers are important in parametric analysis of engineering problems. They allow comparison of two systems that are vastly different by combining the parameters of interest. Dimensionless numbers are used to simplify the meaning of the information in scaUng-up the reactor for real flow conditions and to determine the relative significance of the terms in the equations. The Damkohler number (Da) is the ratio of characteristic fluid motion or residence time to the reaction time, and the Peclet number (Pe) defines the ratio of transport rate by convection to diffusion or dispersion (Basile et al, 2008a Battersby et al., 2006 Moon and Park, 2000 Tosti et al., 2009). In the case of an MR, Da and Pe are defined in Equations [11.1] and [11.2]. 

Maintenance of proper temperature is a major aspect of reactor operation. The illustrations of several reactors in this chapter depict a number of provisions for heat transfer. The magnitude of required heat transfer is determined by heat and material balances as described in Section 17.3. The data needed are thermal conductivities and coefficients of heat transfer. Some of the factors influencing these quantities are associated in the usual groups for heat transfer namely, the Nusselt, Stanton, Prandtl, and Reynolds dimensionless groups. Other characteristics of particular kinds of reactors also are brought into correlations. A selection of practical results from the abundant literature will be assembled here. Some modes of heat transfer to stirred and fixed bed reactors are represented in Figures 17.33 and 17.18, and temperature profiles in... 

Effect of Compositional Nonuniformities on the Unifying Ability of Characteristic Time Ratios to Analyze the Dynamic State of Reactions Figure 11.10, plotting the dimensionless initial reactant concentration as a function of the Damkohler number, Da = ties/tr for both batch and continuous reactors. This analysis assumes a well-mixed reacting system, (a) What will the effects of poor mixing be and how will they influence this analysis (b) What is the maximum allowable striation thickness between the reacting species for the system to be considered well mixed ... 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

Now, all of the tools required to calculate the molar density of reactant A on the external surface of the catalyst are available to the reactor design engineer. It is important to realize that Ca, surface is the characteristic molar density, or normalization factor, for all molar densities within the catalyst. Hence, Ca, surface only appears in the expression for the intrapellet Damkohler number (i.e., excluding first-order kinetics) when isolated pellets are analyzed. Furthermore, intrapellet Damkohler numbers are chosen systematically to calculate effectiveness factors via numerical analysis of coupled sets of dimensionless differential equations. Needless to say, it was never necessary to obtain numerical values for Ca, sur ce in Part IV of this textbook. Under realistic conditions in a packed catalytic reactor, it is necessary to (1) predict Ca, surface and Tsurface, (2) calculate the intrapellet Damkohler number, (3) estimate the effectiveness factor via correlation, (4) predict the average rate of reactant consumption throughout the catalyst, and (5) solve coupled ODEs to predict changes in temperature and reactant molar density within the bulk gas phase. The complete methodology is as follows ... 

With the help of these characteristic numbers, mass and heat balances can now be derived in dimensionless form for the ideal reactors considered. 

To facilitate the discussion on the influence of the above-defined parameters (Equations 5.37-5.39) on the reactor behavior and the parametric sensitivity, Equations 5.34 and 5.36 are given in a dimensionless form. According to the studies of Barkelew [25] the mean residence time is referred to the characteristic reaction time and the temperature is given in the form of a relative temperature difference normalized with the Arrhenius number (Equation 5.41). 

Here, a, a and abuik are the atomic fractions of in the gas-phase oxygen, on oxide surface and in the oxide bulk, respectively /s4 is the fraction of molecule in the gas phase C02 is the gas-phase oxygen concentration (mol/mol) t is the residence time (s) b is the total number of surface sites (mol) per mole of gas molecules present in the catalyst section R, and RY are the rates of different types of exchange as calculated per active site of the surface (s" ) D is the diffusion coefficient of in the oxide bulk (m /s) h is the characteristic size of oxide particle (m) Ns and A buik are the quantities of oxygen atoms on the surface and in the oxide bulk, respectively is the dimensionless reactor length r is the dimensionless depth of the oxide layer. 

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