Dimensionless numbers, physical interpretation

Dimensionless numbers are very simple to calculate, although proper interpretation is dependent on the choice of variables such as which length to use for the characteristic length. The magnitudes of the dimensionless numbers are predictors of physical behavior. In addition, the functional relationships written in terms of the appropriate dimensionless numbers are extremely powerful because they will be valid for that geometry and boundary conditions for a wide range of variable combinations. In chemoreception there are six dimensionless numbers that are particularly important (Re, Pe, Sh, Fo, Sc, and Wo). 

Thus 1 is the total heat energy required to bring a solid from an initial temperature Tso to Tm and to melt it at that temperature. Sundstrom and Young (33) solved this set of equations numerically after converting the partial differential equations into ordinary differential equations by similarity techniques. Pearson (35) used the same technique to obtain a number of useful solutions to simplified cases. He also used dimensionless variables, which aid in the physical interpretation of the results, as shown below ... 

This method of compiling a complete set of dimensionless numbers makes it clear that the numbers formed in this way cannot contain numerical values or any other constant. These appear in dimensionless groups only when they are established and interpreted as ratios on the basis of known physical interrelations. Examples ... 

This dimensionless group, introduced for conciseness in rate correlations, has no simple physical interpretation. It is the product of several ratios A/kT represents the ratio of the characteristic London interaction energy to the thermal energy of the particle, R is the aspect ratio, while the Peclet number may be considered as the ratio of a characteristic energy for drag losses to the thermal energy possessed by the particle. This interpretation for the Peclet number becomes evident by using the relation D = tnkT to write... 

The case of the Reynolds number discussed above sho vs that the physical interpretation of one dimensionless group is not unique. Generally, the interpretation of dimensionless groups used in the flo v area in terms of different energies involved in the process, can be obtained starting vith the Bernoulli flovr equation. The relationship existing between the terms of this equation introduces one dimensionless group. 

This section will present one of the possible physical interpretations of these important dimensionless numbers. First, to show the meaning of Nusselt number, we consider the heat transfer flux in the x direction in the case of a pure molecular mechanism compared with the heat transfer characterizing the process when convection is important. The corresponding fluxes are then written as ... 

The dimensionless source term essentially represents the ratio of generation to convection. For various generation terms, several additional dimensionless numbers may be defined. For example, if the generation of momentum due to gravitational forces is considered, a dimensionless number, called as the Froude number (Fr), is defined as the ratio of convection to gravitational factors. The dimensionless numbers discussed here along with other dimensionless numbers are listed in Table 2.1 together with their physical interpretation. 

The Grashof number can be interpreted physically as a dimensionless number that represents the ratio of the buoyancy forces to the viscous forces in free convection and plays a role similar to that of the Reynolds number in forced convection. 

To reduce the number of ODE even further, a physically motivated transformation is applied. The idea is that for a given composition of the feed gas (CEU/I-hO-mix-ture, characterized by the S/C ratio), the composition of the gas at any point in the anode channel or in any of the reforming units is described by only two states the extent of the reforming reaction, assigned re/, and the extent of the oxidation reaction, assigned fo,x. These variables are made dimensionless and normalized to unity, so they can be interpreted as follows ... 

The Grashof number may be interpreted physically as a dimensionless group representing the ratio of the buoyancy forces to the viscous forces in the free-convection flow system. It has a role similar to that played by the Reynolds number in forced-convection systems and is the primary variable used as a criterion for transition from laminar to turbulent boundary-layer flow. For air in free convection on a vertical flat plate, the critical Grashof number has been observed by Eckert and Soehngen [1] to be approximately 4 x 10". Values ranging between 10" and 109 may be observed for different fluids and environment turbulence levels. ... 

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