The energy conservation principle can be formulated for a fluid element in the manner similar to the mass and momentum conservation as
Computational Fluid Dynamics
The transport is quantified by the vector field Ji (x, t ) of the flux of a species i , which denotes the direction and the rate of the mass flux of the species per unit area at the point x. In the same manner as in the derivation, we can find that the rate of change by diffusion of the mass content of species i in a fluid element of unit volume is ∇ · Ji .
The concentration of species can be expressed in terms of the mass fraction mi (x, t ), which is the ratio of the mass of species i to the total mass of the mixture in the same small volume. Another possibility is to use the concentration of species Ci = mi * ρ defined as the mass of species i per unit volume. The conservation law is
Let us consider the two-dimensional situation. The element has the sizes dx and L, volume δV = Ldx.
The velocity field is purely one-dimensional, but x-dependent with u = u(x).
During the time interval dt , the right-hand side boundary moves together with fluid molecules by the distance u(x + dx)dt.
The corresponding increase of volume is Ldtu(x + dx). At the same time, the volume decreases by Ldtu(x) due to the motion of the left-hand side boundary. The time rate of volume change per unit volume is
From the physical viewpoint, the equations describing fluid flows and heat and mass transfer are simply versions of the conservation laws of classical physics, namely:
• Conservation of chemical species (law of conservation of mass)
• Conservation of momentum (Newton’s second law of motion)
• Conservation of energy (first law of thermodynamics)
In some scenarios, additional equations are needed to account for other phenomena, such as, for example, entropy transport (the second law of thermodynamics) or electromagnetic fields.