General macroscopic balance

From Chemepedia

Macroscopic balances are necessary to account for the transfer of mass, energy, and momentum into and out of a control volume. A control volume is a fixed volume in space. The surface enclosing the control volume is referred to as the control surface. A macroscopic balance applied states that the accumulation (or time derivative) of the property must equal the influx and outflux of the property across the control surface plus generation (and/or) consumption:

The influx and outflux across the control surface can be due to both bulk fluid flow (defined by the velocity field and due to non-fluid flow mechanisms, such as diffusion or conduction (defined by the diffusive flux field ).

Macroscopic balances can be applied to both conserved properties and non-conserved properties. The difference between the two types of balances is that with a conserved property, all terms in the macroscopic balance can be determined without any knowledge of processes inside the control volume. With a non-conserved property, details inside of the control volume must be known to determine one or more terms of the macroscopic balance. These are so-called reaction terms that must be added to the balance to account for generation and consumption of the property due to reasons other than influx and outflux. These types of balances include mechanical energy, internal energy, and a component mass or mole balance on a multicomponent mixture or solution.

Variables used on this page

Definition Variable Additional Information
Transport quantity Represents a transport quantity of interest, such as mass or energy. is the amount of per unit volume, is the amount of per unit mass, and is the amount of per unit mole. is the flow rate of , or per unit time.


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 .


Molecular flux A vector that describes the transport of in response to a gradient. The total amount of transported into and out of a control volume due to diffusion is . is positive if is being transported into the control volume. There are two types of diffusion relationships: Fourier's law of conduction and Fick's law of diffusion.
Density Mass per unit volume
Volumetric flow rate


Mass flow rate or more generally


Unit normal The unit normal is orthogonal (or normal, or perpendicular) to a differential surface of area dA.

Control volume integrals

The properties within a control volume tend to vary with position. If we specify as a quantity on a per mass basis, the total amount of the property within the control volume is determined by an integral of the product over the volume:

            (1)

The time derivative of the total amount of the property is then

            (2)

For instance, if we consider mass as the conserved quantity, , and and

General Macroscopic Balance

The time derivative (or the rate change) of a transport quantity in a control volume is

            (3)

where

  • is the influx/outflux of per unit area due to bulk fluid flow
  • is the influx/outflux of per unit area due to non-bulk flow mechanisms such as diffusion or conduction
  • is the generation/consumption of per unit volume, such as by a homogenous reaction
  • is the generation/consumption of per unit area at the system boundary, such as by a heterogenous reaction
Figure 1. The unit normal is perpendicular to a surface

While the influx/outflux terms look complicated, they are written in terms of a unit normal in order to generalize the application of the macroscopic balance to any type of control volume. We consider a differential surface area dA on the control surface and denote its unit normal by . If we consider just the bulk-flow term, the bulk flow rate of property through dA is since the dot product gives the normal component of the velocity.

Figure 2. A velocity field on a surface
  1. If the velocity vector is tangent to the unit normal, the associated flux is zero because the fluid just flows parallel to the control surface and neither in or out.
  2. If the unit normal is in the same direction as the velocity vector, , and this denotes outflow.
  3. If the unit normal is in the opposite direction of the velocity vector, , and this denotes inflow.

In most situations, fluid crosses the control surface at a finite number of well-defined inlets and outlets (see Figure 3). In such situations, it is convenient to cut the control surface directly across each inlet and outlet and replace the surface integral with terms based on the average velocity (which is strictly only true as long as the density and the conserved property does not vary across the inlet or outlet):

Figure 3. A control volume with two well-defined inlets and one outlet

            (4)

where the summations are over the inflow and outflow streams. The average velocity is defined as

            (5)

So, for instance, if the control volume has two well-defined inlets and one exit:

            (6)

Using mass as an example, and applying it to the control volume in Figure 3, Equation (3) becomes

            (7)

Alternatively, the above equation can also be written in terms of volumetric flow rates, :

            (8)

or mass flow rates, :

            (9)

See also