Environmental Engineering Reference
In-Depth Information
2. Heterogeneity of MSW can be extreme, particularly
when there has not been prior sorting of the waste
materials.
3. The void ratios are commonly in the range of 2-4 for
MSW but are much lower for heap leach materials.
4. The specific gravity of MSW materials can vary
widely. The amount of organic matter can vary widely
from one site to another and the specific gravity of
organics can be quite low.
5. The natural water contents can be extremely high
for MSW. Special laboratory testing procedures are
required in order to obtain meaningful measures of
the water content in organic-rich MSW.
The mass of solids lost from the system is zero for most
conventional soil mechanics problems, and so Eq. 2.97
reduces to the form
M
t
M
t
M
w
+
M
a
=
(2.108)
M
t
A “volume-based” formulation is most commonly used in
soil mechanics rather than a “mass-based” formulation of the
derived partial differential equations that describe each pro-
cess. Consequently, it is possible to write the conservation
of mass in terms of a “volumetric requirement” that must
be satisfied. The volumetric requirement can be written as
These are a few of the similarities and differences asso-
ciated with engineering heap leach and MSW materials.
V
t
=
V
s
+
V
w
+
V
a
(2.109)
During a process, the sum of the volume changes asso-
ciated with any component of the system must equal the
change in the overall volume of the REV. Mathematically
this principle can be written as
2.6.3 Continuity When Mass Is Lost
The modeling of heap leach problems and municipal solid
waste problems is somewhat different from routine seepage
problem simulations in that the mass of solids can be removed
during the modeling period. From a continuum mechanics
standpoint, there are two primary issues to ensure: (i) that
the conservation of mass and energy is satisfied at all times
and (ii) that the physical model accounts for the removal of
solid mass from the system with time.
The conservation of mass is usually satisfied in soil mechan-
ics through use of a volumetric requirement referenced to a
referential elemental volume, REV. The REV is attached to a
fixed mass of soil solids. The mass of solids remains constant
during the processes under consideration.
Let us consider the case of an unsaturated material where
there can be a loss of solids from the system. The con-
servation of mass states that the sum of the components
constituting the REV must be equal to the total mass at all
times. The conservation of mass can be stated as follows:
V
t
V
t
V
s
V
t
V
w
V
t
V
a
V
t
=
+
+
(2.110)
Usually the change in volume of soil solids is zero for soil
mechanics problems, and as a result Eq. 2.110 reverts to the
following volumetric requirement that must be satisfied:
V
t
V
t
V
w
V
t
V
a
V
t
=
+
(2.111)
For heap leach problems and MSW problems, volume
(and mass) will be lost from the system with time. There-
fore, it is necessary to retain the change in volume of solids
term in the volumetric requirement. It may also be more
appropriate to directly satisfy the conservation of mass (i.e.,
Eq. 2.106).
M
t
=
M
s
+
M
w
+
M
a
(2.106)
2.6.4 Incorporation of Constitutive Relations into
Physical Processes
Constitutive relations, flow laws, compressibility laws, and
dissolution laws can now be used to determine how each of
the components in Eq. 2.106 can be represented.
where:
M
s
=
mass of the solids in the REV,
M
w
=
mass of water,
M
a
=
mass of air, and
2.6.4.1 Water Flow Processes
Changes in the amount of water in a soil (i.e., an REV) at
any time during a process can be determined by solving the
saturated-unsaturated seepage partial differential equation,
PDE. The seepage equation is solved by setting the time
derivative of the volumetric water content constitutive
equation equal to the spatial variation of water velocity.
The water-phase PDE is used to compute the water leaving
and entering the system. Solving the PDE provides informa-
tion on changes in hydraulic head, which can then be used
M
t
=
total mass of the overall REV.
When processes or changes occur with time, the sum of
the changes in any component of the system must also equal
the change in the overall mass of the REV. Mathematically,
this requirement can be stated as
M
t
M
t
M
s
M
t
M
w
M
t
M
a
M
t
=
+
+
(2.107)
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