Introduction  (Page 4/5)

A differential element of volume in curvilinear coordinate system is related to differentials of the coordinates by the square root of the determinate of the metric tensor or the Jacobian, J .

Henceforth, Cartesian coordinates with subscript notation will be used.

Units

Dimensional quantities will be used in equations without explicit specification of units because it is understood that they will have the SI system of units. The SI units and mks units are similar with some exceptions as in electricity and magnetism. The following table lists the SI units of the quantities used in this course and the conversion factor needed to convert the quantity from some customary units to SI units. Multiply the quantity in customary units by the conversion factor to obtain the quantity in SI units. The following is taken from, The SI Metric System of Units and SPE METRIC STANDARD , Society of Petroleum Engineers.

Continuum approximation

The calculus of scalar, vector, and tensor fields require that these quantities be piecewise continuous down to infinitesimal dimensions. However, quantities such as density, pressure, and velocity become ambiguous or stochastic at the scale of molecular dimensions. Thus the fields discussed here are the average value of the quantity over a representative elementary volume, REV, of space that is large compared to molecular dimensions but small compared to the macroscopic variation of the quantities. The size of the REV depends on the scale that a problem is being investigated. For example, suppose one is investigating a fixed bed catalyst reactor. As a first order approximation for design purposes, the reactor may be modeled as a one-dimensional system with the cross-section of the reactor approximated as the REV. However, is one is investigating instabilities and channeling, the bed may be modeled in 2-D with the REV being a volume that is small compared to the macroscopic dimensions of the reactor but large compared to the size of the catalyst particles. If one is optimizing the transport-limited kinetics of the reactor, then the REV may be small compared to the size of the catalyst particle. If one is optimizing the balance between transport-limited and surface reaction rate limited kinetics, the REV may be small enough to describe the surface morphology of the catalyst particle. However, the molecular dynamics of the surface reaction is beyond the realm of transport phenomena.