Mass, Heat, and Momentum Transport Analogies

In Chapter 2, we used the control volume technique represented by equation (2.1) to transport mass into and out of our control volume. Inside of the control volume, there were source and sink rates that acted to increase or reduce the mass of the compound. Anything left after these flux and source/sink terms had to stay in the control volume, and was counted as accumulation of the compound. 

Flux rate - Flux rate + Source - Sink = Accumulation IN OUT rate rate 

Equation (2.1) will also apply to the transport of any fluid property through our control volume, such as heat and momentum. 

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Chapter 4 Mass, Heat, and Momentum Transport Analogies. The transport of mass, heat, and momentum is modeled with analogous transport equations, except for the source and sink terms. Another difference between these equations is the magnitude of the diffusive transport coefficients. The similarities and differences between the transport of mass, heat, and momentum and the solution of the transport equations will be investigated in this chapter. 

The mass transfer flux law is analogous to the laws for heat and momentum transport. The constitutive equation for Ja, the diffusional flux of A resulting from a concentration difference, is related to the concentration gradient by Pick s first law ... 

The vast majority of studies of turbulent transport have dealt widi momentum transport, so that an analogy between heat or mass transfer and momentum transport is highly desirable. Rather more stringent assumptions are required to demonstrate that such an anak is to be expected. Based on experimental observations such an analog appears to hold under high Reynolds number conditions, but it is not as successful as those relating heat transfer and mass transfer. 

The second milestone in chemical engineering came in 1960 with the publication of Transport Phenomena, by Bird et al. [2]. Their new approach emphasized the microscale processes and the analogy among mass, heat, and momentum transfer in different processes. 

The conventional parameterizations used describing molecular transport of mass, energy and momentum are the Fick s law (mass diffusion), Fourier s law (heat diffusion or conduction) and Newton s law (viscous stresses). The mass diffusivity, Dc, the kinematic viscosity, i/, and the thermal diffusivity, a, all have the same units (m /s). The way in which these three quantities are analogous can be seen from the following equations for the fluxes of mass, momentum, and energy in one-dimensional systems [13, 135] ... 

A transport of mass or diffusion of mass will take place in a fluid mixture of two or more species whenever there is a spatial gradient in the proportions of the mixture, that is, a concentration gradient. Mass diffusion is a consequence of molecular motion and is closely analogous to the transport of heat and momentum in a fluid. 

The radial dispersion coefficient for this case is, of course, the average eddy diffusivity as discussed in works on turbulence (H9). If the various analogies between momentum, heat, and mass transport are used. 

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. 

It is not difficult to observe that in all of these expressions we have a multiplication between the property gradient and a constant that characterizes the medium in which the transport occurs. As a consequence, with the introduction of a transformation coefficient we can simulate, for example, the momentum flow, the heat flow or species flow by measuring only the electric current flow. So, when we have the solution of one precise transport property, we can extend it to all the cases that present an analogous physical and mathematical description. Analogous computers [1.27] have been developed on this principle. The analogous computers, able to simulate mechanical, hydraulic and electric micro-laboratory plants, have been experimented with and used successfully to simulate heat [1.28] and mass [1.29] transport. 

The powerful analogy that exists among momentum, heat, and mass transport permits useful values of convective mass transfer coefficients to be calculated from known values of convective heat transfer coefficients. For a particular drying system with a specific geometry and flow characteristics, the following relationship is recommended. " ... 

The first model suggested for these dimensionless groups is named the Reynolds analogy. Reyuolds suggested that in fully developed turbulent flow heat, mass and momentum are transported as a result of the same eddy motion mechanisms, thus both the turbulent Prandtl and Schmidt numbers are assumed equal to unity ... 

An alternative interpretation may be ascribed to the Reynolds number, consistent with our earlier analogy of the similarity of momentum, heat, and mass transport. We may then interpret the dimensionless parameters appearing in the energy and diffusion equations in an analogous manner that is. 

It can be easily shown that for turbulent flows the heat-transfer coefficient and mass-transfer coefficient are related fairly simply to the friction factor /. This is so because the eddy which transports momentum (and thus increases the shear stress) also can transport heat and mass and thus increase the heat and mass transfer. This subject is discussed in heat-transfer and mass-transfer texts as the Reynolds analogy. 

Analogous to Newton s law of momentum transport and Fourier s law of heat transfer by conduction. Pick s first law for mass transfer by steady-state equimolar diffusion, is... 

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