Transport - Environmental Modeling - page 56

differential equation, with different meanings of the coefficients only. Thus we
refer to the temperature equation as transport equation as well.
The fundamental formulation of a conservation principle is expressed by the
general continuity equation. For mass the continuity equation was already derived
in Chap. 2. What follows next is a generalization of the mass conservation equation,
derived in Chap. 2. The conservation of variable A , which may represent mass,
momentum or energy, and which depends on time t and the three space directions x ,
y and z , is quantitatively expressed by the differential equation:

@t A ¼ @

@

@x j Ax þ @ @y j Ay þ @ @z j Az þ Q

(3.1)

where j Ax , j Ay and j Az represent fluxes in the three space directions. The three flux
terms are components of the flux vector j A corresponding to the three space
directions. Fluxes, as all other terms in the continuity equation, depend on the
independent variables x,y,z and t too. In the term Q all sources and sinks are
gathered. If Q ( x,y,z,t ) is positive, there is a source at time t at position r

¼ ( x,y,z );

if Q is negative, there is a sink.
The continuity equation states that the amount of change of variable A in time is
equal to the local flux budget. The continuity equation is derived from the budget of
a control volume, i.e. a volume of finite small extensions

D z (in 3D).
Figure 2.3 shows a control volume in 2D for a fluid filling the entire space, where
only two finite extensions

D x; D y and

D y are sufficient.
In the small but finite time interval

D x and

D t , the amount of A per volume unit changes from

A ( x, y, z, t )to A ( x, y, z, t +

D t ). The total amount of change in the control volume is thus
given by Aðx; y; z; t þ D tÞAðx; y; z; tð Þ D x D y D z . In Fig. 2.3 this corresponds to the
volume change D V . On the other hand, the total budget can be expressed by the fluxes,
the sources and the sinks. In each space dimension there are two surfaces, across which
mass, momentum or energy may enter or leave according to the corresponding flux
component. In x -direction the fluxes across the two faces are given by

D y D z D t ; the difference in flux terms of j Ax
from one side of the control volume to the opposite side has to be multiplied by the
face area of the control volume, which is here given by

j Ax ðx þ D 2 ; y; z; tÞj Ax ðx D 2

; y; z; tÞ

D y D z . In the notation of the
fluxes, visualized in Fig. 2.3, the A in the subscript is omitted and the + or sign denotes

the direction. Note the assumption that the time step t is small, so that the change of the

flux terms and also of the sinks and sources during that time can be neglected.
Both expressions of the change within the control volume with a time step have
to be equal, which is expressed in the detailed equation:

D

ð Aðx; y; z; t þ D tÞAðx; y; z; tÞ Þ D x D y D z

D x
2 ; y; z; tÞj Ax ðx

D x
2 ; y; z; tÞ

¼ j Ax ðx þ

D y D z D t

(3.2) D y

2

D y
2

þ j Ay ðx; y þ ; z; tÞj Ay ðx; y ; z; tÞ D x D z D t

D z
2 ; tÞj Az ðx; y; z

D z
2 ; tÞ

þ j Az ðx; y; z þ

D x D y D t þ Q D x D y D z D t