Laminar transport coefficients

The coefficients of transport properties considered here include the viscosity, diffusivity, and thermal conductivity of a gas. The transport coefficients vary with gas properties if the flow is laminar. When the flow is turbulent, the transport coefficients become strongly dependent on the turbulence structure. Here we only deal with the laminar transport coefficients the discussion of the turbulent transport coefficients is given in 5.2.4. 

In order for a model to be closured, the total number of independent equations has to match the total number of independent variables. For a single-phase flow, the typical independent equations include the continuity equation, momentum equation, energy equation, equation of state for compressible flow, equations for turbulence characteristics in turbulent flows, and relations for laminar transport coefficients (e.g., fJL = f(T)). The typical independent variables may include density, pressure, velocity, temperature, turbulence characteristics, and some laminar transport coefficients. Since the velocity of gas is a vector, the number of independent variables associated with the velocity depends on the number of components of the velocity in question. Similar consideration is also applied to the momentum equation, which is normally written in a vectorial form. 

Reactions carried in aqueous multiphase catalysis are accompanied by mass transport steps at the L/L- as well as at the G/L-interface followed by chemical reaction, presumably within the bulk of the catalyst phase. Therefore an evaluation of mass transport rates in relation to the reaction rate is an essential task in order to gain a realistic mathematic expression for the overall reaction rate. Since the volume hold-ups of the liquid phases are the same and water exhibits a higher surface tension, it is obvious that the organic and gas phases are dispersed in the aqueous phase. In terms of the film model there are laminar boundary layers on both sides of an interphase where transport of the substrates takes place due to concentration gradients by diffusion. The overall transport coefficient /cl can then be calculated based on the resistances on both sides of the interphase (Eq. 1) ... 

Usually, the values of the transport coefficients for a gas phase are extremely sensitive to pressure, and therefore predictive methods specific for high-pressure work are desired. On the other hand, the transport properties of liquids are relatively insensitive to pressure, and their change can safely be disregarded. The basic laws governing transport phenomena in laminar flow are Newton s law, Fourier s law, and Fick s law. Newton s law relates the shear stress in the y-direction with the velocity gradient at right angles to it, as follows ... 

The fluxes of mass, momentum, and energy of phase k transported in a laminar or turbulent multiphase flow can be expressed in terms of the local gradients and the transport coefficients. In a gas-solid multiphase flow, the transport coefficients of the gas phase may be reasonably represented by those in a single-phase flow although certain modifications... 

However, it should be noted that equality of shear stresses in lab and field will not produce exact equality of mass transport coefficients owing to different dependencies on flow velocity, as shown below (15). Clearly, the flow regimes must be the same in the laboratory and in the field (e.g., both laminar or both turbulent). 

The assumption of laminar flow can be used to substitute the gas transport coefficient by the ratio of the diffusion coefficient of the reaction species in the mixture and half of the mean... 

When the system under consideration is in laminar flow, it is often possible to give precise analytical expressions of the transport coefficients. In most other cases, including the important case of turbulent flow, the analytical approach generally fails and we must resort to semiempirical correlations, arrived at by the device known as dimensional analysis, which involves the use of dimensionless groups. 

To represent these facts in an organized fashion, we start our deliberations with a brief survey of the dimensionless groups pertinent to mass transfer operations. We next turn to transport coefficients that apply to systems in laminar flow and show how these coefficients are extracted from the solutions of the pertinent PDE models. This is followed by an analysis of systems in turbulent flow where the approach of dimensional analysis is used. We... 

Microscopic mechanics, statistical mechanics, kinetic theory laminar transport phenomena, phenomenological coefficients viscosity. 

Mesoscopic statistical theories of turbulence laminar and turbulent transport thermal conductivity, diffusivity effective transport coefficients... 

Data obtained for a smooth undisturbed flame by the spherical container method agree with the numerical simulation results of a spherical laminar flame propagation provided that an up-to-date detailed kinetic scheme with hydrogen oxidation and transport coefficients for describing multi-component diffusion of mass and heat are taken into account [27, 30, 43]. In some cases, when the agreement is not observed, it is necessary to analyze the arrangement of laminar flame velocity measurements. 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

One of the major interests of the HEX reactor is to offer a large ratio surface to reaction volume. Therefore, even if most of the time the laminar flow regime is not suitable to enhance transport phenomena with a moderate overall coefficient, the heat performances are expected to be high, since the compacity factor is always large. This fact is clearly exhibited in Table 12.4, where the results relative to the various HEX reactors studied in our laboratory have been plotted. 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

The RHSE has the same limitation as the rotating disk that it cannot be used to study very fast electrochemical reactions. Since the evaluation of kinetic data with a RHSE requires a potential sweep to gradually change the reaction rate from the state of charge-transfer control to the state of mass transport control, the reaction rate constant thus determined can never exceed the rate of mass transfer to the electrode surface. An upper limit can be estimated by using Eq. (44). If one uses a typical Schmidt number of Sc 1000, a diffusivity D 10 5 cm/s, a nominal hemisphere radius a 0.3 cm, and a practically achievable rotational speed of 10000 rpm (Re 104), the mass transfer coefficient in laminar flow may be estimated to be ... 

For small-scale, high-intensity turbulence, Damkohler reasoned that the transport properties of the flame are altered from laminar kinetic theory viscosity y0 to the turbulent exchange coefficient e so that... 

Chapter 3 Diffusion Coefficients. This chapter demonstrates how to estimate the diffusion coefficients of dilute chemical concentrations in water and air. The chapter is important any time that diffusion cannot be ignored in an application of chemical transport and fate. Some of these cases would be in laminar flows, in sediment, in groundwater transport, and close to an interface in turbulent flows. 

Tracer Determination of Longitudinal Dispersion Coefficient in Rivers. Tracers are generally used to determine longitudinal dispersion coefficient in rivers. Some distance is required, however, before the lateral turbulent diffusion is balanced by longitudinal convection, simitar to Taylor s (1953) analysis of dispersion in a laminar flow. This transport balancing distance, x is given by the equation... 

The gas film coefficient is dependent on turbulence in the boundary layer over the water body. Table 4.1 provides Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1,000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat, and mass transport is more precise for air than for water, and the techniques apphed to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are to the liquid side of the interface. 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

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