of 20C at the surface of the sludge ( Ts ). This assump-
period. An equation for this configuration can be de-
veloped by conducting an energy balance across the
tion is conservative because the average temperature
of the effluent is 23C. The thawing period (Pth) is
pipe/sludge interface
2972 hours, which occurs during the months of Octo-
qk = e
(3)
ber, November, December, and January. A previous
study (Martel 1989) determined that the constants ks
and θ are 0.35 W/m C and 0.15, respectively. Substi-
where qk is the rate of heat transfer by conduction and
e is the rate of energy gain by the frozen sludge during
tuting these values into eq 8 results in a predicted thaw-
the phase change.
ing depth of 3.2 m.
Assuming that the pipe network will behave as a
Design depth
flat plate (i.e., the melting front will be uniform across
the surface of the frozen sludge), and the thermal con-
The depth of the freezing bed depends on the depth
ductivity of the pipe is infinite, we can express qk as
of sludge that can be frozen and thawed during an an-
(Kreith 1973)
nual freezethaw cycle. In this case, the energy bal-
ance equations predict that the depth of sludge that can
(
)
Kss A
qk =
Ts -Tf
(4)
be frozen will be significantly more than the depth of
∆
sludge that can be thawed. Therefore, the thawing depth
where A = surface area
will govern freezing bed design depth. Using solar
∆ = thickness of the settled solids
energy alone gives a design depth of 1.5 m. Using waste
T s = average temperature of the pipe surface.
heat from the effluent gives a design depth of 3.2 m. A
combination of both thawing methods would result in
The rate of energy transfer to the frozen sludge dur-
a design depth of 4.7 m.
ing the phase change can be calculated from (Kreith
From an economic point of view, it is tempting to
1973)
use a design depth of 4.7 m because of a resulting re-
duction in bed size. However, I am not aware of any
dy
e = ρt LA
(5)
freezing beds that have been built to that depth. The
dt
maximum design depth used in full scale beds that have
been built in Alaska is 2.0 m. Also, construction of a
where dy/dt is the rate change in the position of the
freezing bed with walls high enough to freeze 4.7 m of
freezethaw interface.
sludge may require special construction practices.
Substituting eq 4 and 5 into eq 3 results in the fol-
Therefore, I suggest that a freezing bed be built with a
lowing energy balance relationship
design depth not in excess of 2.0 m. For McMurdo, I
(T s - T f ) = ρf L dy .
K ss
recommend a design depth of 1.5 m with provision for
(6)
∆
dt
an additional 0.5 m provided by supplemental thaw-
If the solids are uniformly distributed in the frozen
ing with waste heat. This configuration will provide
sludge, then the thickness of the settled solids layer
the greatest flexibility of operation in that it allows
(∆) can be expressed as θy, where θ is the fraction of
more than one thawing method and provides additional
settled solids per unit depth of thawed sludge and y is
capacity.
the depth of thawed sludge. Making this substitution,
we see that eq 6 becomes
FREEZING BED SIZE
(T s - T f ) = ρf L dy .
K ss
(7)
θy
dt
The bed size can be calculated now that the design
Separating variables and integrating dt from 0 to
depth and annual sludge production rate are known. I
Pth (the thawing period) and dy from 0 to Y (the total
assumed a total bed depth of 2.0 m (1.5 m frozen sludge
depth of thawed sludge) gives the general equation for
plus 0.5 m additional capacity) and a bed width of 10
predicting the thawing depth in this case
m. Using an access ramp slope of 3:1, I calculated the
length of the bed to be 16 m (Fig. 4). Thus, the total
area occupied by the bed would be 160 m2. For rela-
(
)
12
2 K ss Pth Ts -Tf
(8)
tive scale, this bed size is approximately 30% larger
Y=
.
ρf Lθ
than the freezing bed at Fort McCoy. Of course, other
dimensional combinations are possible as long as the
design depth and annual sludge volume requirements
To calculate Y, I assumed an average temperature
are met.
5
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