General microscopic balance

From Chemepedia

The general microscopic balance gives point-to-point variations of a specific (intensive) property in a defined space. Properties of interest include momentum (velocity), internal energy (temperature), and mass or mole fraction of a species in a multicomponent mixture.

Introduction

The microscopic balance is formulated as a differential equation on a "per volume" basis of a transport quantity . If we specify as the value of the transport quantity per unit volume, and the convective flux of as , then the differential equation describing how varies in a continuum has the general form

            (1)

where

  • is the Del or nabla operator
  • is the time variation of the transport quantity per unit volume
  • is the net convective flow per unit volume of the transport quantity due to both advective (bulk flow) and molecular (diffusive) mechanisms.
  • is the generation/consumption of the transport quantity per unit volume.

Convective flux and flow rate

Convective flux is the movement of a property due to both advective and molecular mechanisms. It has dimensions of "property per unit time per unit area" and has the general form


            (2)

where

  • is the advective flux of due to a velocity field
  • is the molecular flux of due to the gradient in .

The flow rate of the transport property, or , is related to the convective flux by

            (3)

where

  • is the unit normal that defines

Molecular flux

The different types of molecular flux are

Type of transfer Notes
Mass Fick's law of diffusion
Momentum Newton's law of viscosity
Energy Fourier's law of conduction

Types of microscopic balances

The most common types of microscopic balances include total mass, component mass in a mixture, momentum, and internal energy. The various parts of Eq. 1 are provided in the table below

Property Notes
Total mass 0 Also known as the continuity equation.
Mass (of component in a mixture) Also known as the convection–diffusion equation. The generation/consumption term is the rate of chemical reaction.
Momentum Also known as the equation of motion. The generation/consumption term is the conversion of hydrostatic pressure to momentum.
Internal energy Also known as the heat equation. The generation/consumption term is reversible generation/consumption of internal energy due to compression/expansion plus the irreversible generation of internal energy due to viscous dissipation.

It is to be noted that the continuity equation and the convection diffusion-equation can also be expressed on a mole basis. Here, Eq. (1) becomes

            (4)
Property Notes
Total moles The generation/consumption term is non-zero as the number of moles in not necessarily conserved in a reaction, i.e. if
Moles (of component in a mixture)

See Also