Modulus Hatta

Mu stands for the Hatta modulus, in recognition of the scientist who first dealt with this problem, Hatta (1932). 

Here we only treat the reaction A g- l) + bB l) products (/). We assume that the rate is fast enough so that no unreacted A enters the main body of the liquid. This assumes that the Hatta modulus is not very much smaller than unity. 

These parameters, such as the coefficient of diffusion, D, mass-transfer coefficient in the gas and liquid phase or film, kg and k], Thiele modulus, Hatta number, Ha, and enhancement factor, E, are all dependent on the pressure. 

The mass-transfer coefficient with a reactive solvent can be represented by multiplying the purely physical mass-transfer coefficient by an enhancement factor E that depends on a parameter called the Hatta number (analogous to the Thiele modulus in porous catalyst particles). 

Biofilters are chemically enhanced absorbers, and therefore mass transfer limited (see Absorption with Chemical Reaction in Sec. 14). The magnitudes for the Hatta [= Damkohler II = (Thiele modulus)2] numbers are quite low, perhaps below 5. Nevertheless, for design simplicity, mass-transfer limitation is generally assumed to be in the liquid phase (the biofilm). For a single-component biofilter, the simplified biofilter model and design equation is... 

Solving the diffusion-reaction equation in the liquid, the enhancement factor can be related to the Hatta number Ha, which is similar to the Thiele modulus defined for heterogeneous gas-solid catalysts. Thus, the Hatta number and its relation to the controlling regime are... 

Fluid-fluid reactions are reactions that occur between two reactants where each of them is in a different phase. The two phases can be either gas and liquid or two immiscible liquids. In either case, one reactant is transferred to the interface between the phases and absorbed in the other phase, where the chemical reaction takes place. The reaction and the transport of the reactant are usually described by the two-film model, shown schematically in Figure 1.6. Consider reactant A is in phase I, reactant B is in phase II, and the reaction occurs in phase II. The overall rate of the reaction depends on the following factors (i) the rate at which reactant A is transferred to the interface, (ii) the solubihty of reactant A in phase II, (iii) the diffusion rate of the reactant A in phase II, (iv) the reaction rate, and (v) the diffusion rate of reactant B in phase II. Different situations may develop, depending on the relative magnitude of these factors, and on the form of the rate expression of the chemical reaction. To discern the effect of reactant transport and the reaction rate, a reaction modulus is usually used. Commonly, the transport flux of reactant A in phase II is described in two ways (i) by a diffusion equation (Pick s law) and/or (ii) a mass-transfer coefficient (transport through a film resistance) [7,9]. The dimensionless modulus is called the Hatta number (sometimes it is also referred to as the Damkohler number), and it is defined by... 

The Flatta number is the ratio of the reaction in the liquid surface/mass transfer into the bulk phase or a modified Thiele modulus for GL systems to correct the mass transfer for chemical reaction. Because the ratio involves the reaction rate, the actual form of the Hatta number depends on the reaction kinetics. For first order reactions. 

Mass transfer with simultaneous chemical reaction is characterized using the reaction-diffusion modulus known as the Hatta number, defined as the ratio of the reaction in the film to the mass transfer rate through it. Thus, for a first-order reaction the Hatta number is given by... 

The connection between chemical reaction engineering and transport phenomena also stems from multiphase reactions. For solid catalyzed gas or liquid reactions, mass transfer in the bulk or on the surface may become a problem. For gas-liquid reactions, the transport of the species to the reaction zone has to be considered. Similar problems arise for liquid-liquid reactions. Thns, we intend to give a brief introduction to these problems and, in the process, introdnce dimensionless qnantities such as the Thiele modulus, Damkohler number, Hatta modulus, effectiveness factor, and enhancement factor, and nse them in designing reactors. 

Define, evaluate, and use the concepts of enhancement factor and Hatta modulus. 

We will use these characteristic times to define some dimensionless numbers, such as Damkohler number, Thiele modulus, and Hatta modulus. Here, we will give the broad definitions, and the true meanings will be clear as we move along the chapter ... 

Hatta modulus represents the ratio of the kinetic rate in the absence of transport effects to maximum diffusional rate of species A into a liquid. 

If the rate is slow, Hatta modulus is small. If the rate is fast, the Hatta modulus is large. For very large Hatta modulus, the enhancement factor is equal to Hatta modulus, since tanh oo = i. Finally, for instantaneous reactions, the enhancement factor has a limiting value given by Equation 6.80. A qualitative plot of enhancement factor versus Hatta modulus is given in Figure 6.12. The merits of the figure and how it can be used for equipment selection will be discussed in Chapter 11. 

Figure 6.12 Enhancement factor as a function of the Hatta modulus.

Another condition for regime 2 is that the amount of A that reacts in the film before reaching the bulk be negligible. For the very slow and slow reactions, the kinetic term resides along with the differential equation describing the transport of the species in the film, such that the definition of Hatta modulus is necessary as was derived in Chapter 6 for a pseudo-first-order reaction. In Chapter 6, we defined the Hatta modulus as... 

Arrhenius frequency factor, same units as the rate constant Percolation rate constant, appropriate units (usually 1/s) Reactor length, m Mass order of reaction Hatta modulus... 

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