Mass balance

From Chemepedia

A mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique.

Variables used on this page

Definition Variable Additional Information
Flow velocity Velocity is a vector. Average velocity in a single component system often reported as a scalar and is . Often the brackets are not used for average velocity. In multi-component systems, both a mass average velocity vector and a molar average velocity vector can be defined .


Mass flow rate or more generally
Component density The mass of species dissolved per volume of solution or mixture
Density For a multicomponent mixture,
Component molar concentration The moles of a solute dissolved per volume of solution or mixture
Molar concentration For a multicomponent mixture,
Diffusive flux The mass flux of species due to diffusion. denotes a molar diffusion rate.
Reaction rate The reaction rate of in terms of mass per unit volume. denotes a molar reaction rate per unit volume, and denotes a reaction rate at a surface.
Unit normal The unit normal is orthogonal (or normal, or perpendicular) to a differential surface of area dA.

Overall mass balance

The mass that enters a system must, by conservation of mass, either leave the system or accumulate within the system. In other words,

or

            (1)

In the case of control volume with well-defined inlets and outlets, the mass balance can be written in a simpler form

            (2)

Relationship between mass flow rate and average flow velocity

If the cross-sectional area at point is , the average velocity can be related to the mass flow rate via


            (3)

Overall steady-state mass balance with one entrance and one exit

For a mass balance without only one entrance and one exit at steady-state we can express the mass balance as


            (4)


Component mass balance for mixtures

A mixture is a material made up of two or more different substances in which the individual identities are retained. The mixture may be two or more different gasses, a gas dissolved in a liquid, two or more liquids, or solids dissolved in a liquid. The total mass of the system (even in the presence of a reaction) is conserved according to the above equations, i.e. the sum of all the chemical species of the system. However, if we consider each species or each component of the mixture individually, a mass balance on each component must be amended to allow for the generation or depletion (consumption) of each chemical species due to a reaction as well as transfer of the species across a control surface due to diffusion.

For instance, a balance on species A of a multi-component mixture would have the following form:

            (5)

Note that (3) can also be expressed in terms of the molar concentration of A by dividing each term by the molar mass of species A, note also, when unit normal is in the same direction of the diffusion flux, this value is positive and when unit normal is in the opposite direction of the flux, this value is negative, if the unit normal and flux are perpendicular to one another, this value is zero,


            (6)