Unsteady-State Mass and Energy Balances

Inventory Rate Equation

Chemical species, mass, momentum, and energy are concepts that form the basis of science and engineering and are all conserved quantities.

For any conserved quantity, φ, an inventory rate equation can be written to describe its transformation. Inventory of the conserved quantity is based on a specified unit of time, which is reflected as a rate and it takes the form:

The area in the last two equations above is perpendicular to the direction of flux. The generation rate per unit volume and depletion rate can be expressed as follows:

The rate of accumulation of φ is the time rate of change of total φ contained within the volume of the system. Let ρ be the mass density. Thus,

Unsteady State Mass Balance

For a single component system, if ρ is the mass density and v is the velocity vector, then the mass flux (mass per unit time per unit area) is given by:

Unsteady State Energy Balance

The conservation statement for total energy under unsteady-state conditions is given by:

[Rate of energy in] - [Rate of energy out] = [Rate of energy accumulation]

The rate of energy entering or leaving the system by a flowing stream is given by:

Flow Work

To proceed one step further, it is necessary to introduce the concept of flow work. This is a type of work in a flow system.

Consider a fluid of mass (dm) flowing into an open system through a circular duct of cross-sectional area A. As this differential mass crosses the boundary of the system, the resisting pressure P may be assumed to remain constant since the differential volume of the mass (dm) is very small compared to the volume of the system.

The work done on the system to push the fluid of mass (dm) against a constant force F = P A is given by:

dW = P A dx = P dV

This is the flow work. Note that the flow work is positive when fluid enters the system, and negative when fluid leaves the system.

Development of the General Energy Equation

The rate of work done on the system by the surroundings and it is composed of the three terms:

Where:

A = Work associated with the expansion or compression of the system boundaries

B = Shaft Work

C = Flow Work

Substituting the above equation into the equation for rate of energy entering or leaving the system by a flowing stream and using the definition of enthalpy results in the general energy equation which is expressed below:

In most practical applications, changes in kinetic and potential energies are negligible thus the above equation can simplify as:

The term dW in above equations includes the work associated with the displacement of system boundaries and the shaft work.