Diffusivity Dimensionless numbers

The dimensionless numbers in tlris equation are the Reynolds, Schmidt and the Sherwood number, A/ sh. which is defined by this equation. Dy/g is the diffusion coefficient of the metal-transporting vapour species in the flowing gas. The Reynolds and Schmidt numbers are defined by tire equations... 

The dimensionless number Le is called the Lewis number (m Russian literature it is called the Luikov number). The Lewis number incorporates the specific heat capacity of humid air pCp (J/m C), the diffusion factor of water vapor in... 

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

At a close level of scrutiny, real systems behave differently than predicted by the axial dispersion model but the model is useful for many purposes. Values for Pe can be determined experimentally using transient experiments with nonreac-tive tracers. See Chapter 15. A correlation for D that combines experimental and theoretical results is shown in Figure 9.6. The dimensionless number, udt/D, depends on the Reynolds number and on molecular diffusivity as measured by the Schmidt number, Sc = but the dependence on Sc is weak for... 

From an analysis of the electrochemical mass-transfer process in well-supported solutions (N8a), it becomes evident that the use of the molecular diffusivity, for example, of CuS04, is not appropriate in investigations of mass transfer by the limiting-current method if use is made of the copper deposition reaction in acidified solution. To correlate the results in terms of the dimensionless numbers, Sc, Gr, and Sh, the diffusivity of the reacting ion must be used. 

In the discussion above, we have considered only the velocity field in a turbulent flow. What about the length and time scales for turbulent mixing of a scalar field The general answer to this question is discussed in detail in Fox (2003). Here, we will only consider the simplest case where the scalar field 4> is inert and initially nonpremixed with a scalar integral length scale that is approximately equal to Lu. If we denote the molecular diffusivity of the scalar by T, we can use the kinematic viscosity to define a dimensionless number in the following way ... 

L.Dq/u.Rq dimensionless number Mark-Houwink-Sakurada constant Axial diffusion coefficient... 

Mark-Houwink-Sakurada constant Mass transfer coefficient around gel Fractional reduction in diffusivity within gel pores resulting from frictional effects Solute distribution coefficient Solvent viscosity nth central moment Peak skewness nth leading moment Viscosity average molecular weight Number of theoretical plates Dimensionless number... 

It is seen that we are comparing kinematic viscosity, thermal diffusivity, and diffu-sivity of the medium for both air and water. In air, these numbers are all of the same order of magnitude, meaning that air provides a similar resistance to the transport of momentum, heat, and mass. In fact, there are two dimensionless numbers that will tell us these ratios the Prandtl number (Pr = pCpv/kj = v/a) and the Schmidt number (Sc = v/D). The Prandtl number for air at 20°C is 0.7. The Schmidt number for air is between 0.2 and 2 for helium and hexane, respectively. The magnitude of both of these numbers are on the order of 1, meaning that whether it is momentum transport, heat transport, or mass transport that we are concerned with, the results will be on the same order once the boundary conditions have been made dimensionless. 

PRANDTL NUMBER. A dimensionless number equal to the ratio of llie kinematic viscosity to the tlienuoiiielric conductivity (or thermal diffusivity), For gases, it is rather under one and is nearly independent of pressure and temperature, but for liquids the variation is rapid, Its significance is as a measure of the relative rates of diffusion of momentum and heat m a flow and it is important m the study of compressible flow and heat convection. See also Heat Transfer. 

In Equation (4.9) the balance between convective transport and diffusive transport in the membrane boundary layer is characterized by the term JvS/Di. This dimensionless number represents the ratio of the convective transport Jv and diffusive transport Dj/8 and is commonly called the Peclet number. When the Peclet number is large (./ 5>> D,/S), the convective flux through the membrane cannot easily be balanced by diffusion in the boundary layer, and the concentration polarization modulus is large. When the Peclet number is small (Jv <5C D,/8), convection is easily balanced by diffusion in the boundary layer, and the concentration polarization modulus is close to unity. 

Heat transfer and its counterpart diffusion mass transfer are in principle not correlated with a scale or a dimension. On a molecular level, long-range dimensional effects are not effective and will not affect the molecular carriers of heat. One could say that physical processes are dimensionless. This is essentially the background of the so-called Buckingham theorem, also known as the n-theorem. This theorem states that a product of dimensionless numbers can be used to describe a process. The dimensionless numbers can be derived from the dimensional numbers which describe the process (for example, viscosity, density, diameter, rotational speed). The amount of dimensionless numbers is equal to the number of dimensional numbers minus their basic dimensions (mass, length, time and temperature). This procedure is the background for the development of Nusselt correlations in heat transfer problems. It is important to note that in fluid dynamics especially laminar flow and turbulent flow cannot be described by the same set of dimensionless correlations because in laminar flow the density can be neglected whereas in turbulent flow the viscosity has a minor influence [144], This is the most severe problem for the scale-up of laminar micro results to turbulent macro results. 

Schmidt number a dimensionless number, characteristic of each gas, which varies strongly with temperature and weakly with salinity, and is used to account for viscosity effects on the diffusion of gases. 

Often, the same expressions from this model are applied to the reductions of pellets, in which cases such structural factors as particle size distribution, porosity and pore shape, its size distribution, etc. should really affect the whole kinetics. Thus, the application of this model to such systems has been criticised as an oversimplification and a more realistic model has been proposed [6—12,136] in which the structure of pellets is explicitly considered to consist of pores and grains and the boundary is admitted to be diffusive due to some partly reduced grains, as shown in Fig. 4. Inevitably, the mathematics becomes very complicated and the matching with experimental results is not straightforward. To cope with this difficulty, Sohn and Szekely [11] employed dimensional analysis and introduced a dimensionless number, a, given by... 

The question concerning the correlation between the roasting time and the meat size can now be easily answered, even without explicitly knowing the function / To reach the same temperature distribution T/T0 or (T0 -T)/T0 in differently sized bodies, the dimensionless number Fo = aQ/A must display the same (= idem, identical) numerical value. Because thermal diffusivity a remains unaltered in the meat of same species (o = idem), this leads to... 

Let us now consider a catalytic packed bed reactor , i.e. a tubular reactor filled with a grained catalyst through which the gas mixture flows. With the particle diameter of the catalyst, dp, an additional dimensionless number dp/d is added to the pi-space the Reynolds number is now expediently formed with dp. The reaction rate is related to the unit of the bulk volume and characterized by an effective reaction rate constant ko,eff = k . The thermal conductivity (k) also has to be valid for the gas/bulk solids system and diffusion can be considered as being negligible (Sc is irrelevant). The complete pi-space is therefore ... 

In the general case of a solute B in a plastic matrix P the parameter AP is a function of temperature and produces a more or less significant deviation from the activation energy, EA - 86.923 kJ mol-1 in reference Eq. (6-20). Consequently we can write Ap = Ap -Xp/T, with the athermal, dimensionless number AP and the parameter Xp with the dimension of a temperature, respectively. Both values, AP and Xp can be obtained from two diffusion measurements at different temperatures, using a reference solute B in matrix P (see Chapter 15). 

It follows from Eqn. 7.95 that the concentration profile depends on a single dimensionless number which is equal to the ratio of the time scale for internal diffusion ... 

For a membrane thickness of Ax, dimensionless number A/Ax is closely related to the Thiele modulus used for the characterization of heterogeneous reaction columns. This dimensionless quantity is also related to the relaxation time of chemical reaction rr and the average relaxation time of diffusion processes rd as follows ... 

Thiele was one of the first to use the concept of an effectiveness factor [5]. To calculate this factor he introduced a dimensionless number, nowadays called the Thiele modulus. Thiele defined the effectiveness factor as the amount of a certain component that is actually converted, divided by the amount that could have been converted if retardation by diffusion had not occured. The dimensionless modulus he used was defined as... 

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