Fundamentals of Modeling, Principles and MATLAB - Environmental Modeling - page 39

One has to extend the mass definition in situations, where the fluid does not fill
the entire volume. Then the total mass per space volume is expressed by yc with

porosity as additional factor. Porosity is dimensionless and measures the volu-

metric share of the fluid phase on the total volume. The physical dimension of the
product is thus further [M/L 3 ]. In aquifers groundwater usually fills approximately
25% of the volume; thus holds:

y

y¼

0.25.
The described concept can be extended to situations with several phases, where
each phase has its own

-value. In the unsaturated soil zone below the ground
surface, above the groundwater table there are three phases present: the soil as solid
phase, seepage water as liquid phase and soil air as gaseous phase.
The mass of a gas component is expressed in terms of partial pressure.
According to the ideal gas law the product of pressure and volume is a constant,
which changes only with temperature. Thus mass conservation can be formulated in
terms of pressures instead of volumes. According to Dalton's Law 2 the pressures of
a gas mixture have to be summed up to yield the total pressure.

y

2.3.2 Conservation of Momentum

The momentum of a fluid is expressed as the product v, where v denotes the

velocity. The physical unit of momentum is [M/(L 2 T)], where the letter 'T'
represents a time unit. As velocity is a vector, the momentum also is a vector,
with one vector component for each space dimension of the model.

r

2.3.3 Conservation of Energy

2 rv 2 with the physical unit [M/(LT 2 )].
If energy is measured in Joule, the given expression measures Joule per volume.
In problems, which include heat transfer, thermal energy is expressed in terms of
temperature T . The energy content per volume is given by the product

1

The kinetic energy of a fluid is expressed as

rCT , where
the new factor C is the specific heat capacity. The physical unit of C is [L 2 /(T 2 K)],
where 'K' represents the temperature measure unit, mostly Celsius or Kelvin. In
many tables values for the product rC can be found, which is addressed simply as

heat capacity. Heat capacity has the unit [M/(LT 2 K)]. If energy is measured in Joule
rC has the physical unit of J per volume and Kelvin, while C is measured in the
unit Joule per mass and Kelvin.

2 John Dalton (1766-1844), English chemist and physicist.