CSIRO PUBLISHING
www.publish.csiro.au/journals/mfr
Marine and Freshwater Research, 2003, 54, 271–285
Bubbler design for reservoir destratification
Goloka Behari SahooA,D and David LuketinaB,C
A
University of Hawaii, Department of Civil & Environmental Engineering, 2540 Dole Street, 383 Holmes Hall,
Honolulu, HI, 96822, USA.
B
Asian Institute of Technology, School of Civil Engineering, Water Engineering and Management Program,
PO Box 4, Klong Luang, Pathumthani, 12120, Thailand.
C
Water Resources Division, Ondeo Services, Technology and Research Center, Paris, 38 rue du President Wilson,
Le Pecq-sur-Seine, 78230, France.
D
Corresponding author. Email: goloka@hawaii.edu
Abstract. Two important bubbler-performance criteria, the mechanical efficiency, ηmech, and the destratification
time, Γ, were analysed as functions of two dimensionless parameters, G, the strength of stratification, and M, the
source strength. Equations to estimate the optimum airflow rate (via M) and corresponding ηmech and Γ for a known
linear stratification G in a reservoir were derived. Owing to difficulties in accurately determining the actual G, it
was demonstrated that it is appropriate practice to reduce the design G value by around 10%. It was shown that the
equivalent linear stratification method might lead to sub-optimal design for stratification profiles that deviate
substantially from a linear profile. Rather, a bubble-plume model should be applied. Finally, the effects of
incorporating changes in bubble radius in a bubble-plume model were examined. ηmech and Γ were found to be
relatively insensitive to bubble radius; however, the ideal bubble size for maintaining a suitable oxygen dissolution
efficiency is 1 mm.
MF02 45
GBu. bB.leS-aphluomaendbeDh.aLviuokertiand estraifcation
Extra keywords: gas transfer, mechanical efficiency, thermal stratification, water quality.
Introduction
Thermal stratification commonly occurs in deep lakes and
reservoirs. As a result, the denser bottom hypolimnetic water
gets isolated from the lighter surface epilimnetic water and
commonly undergoes substantial oxygen depletion. In turn,
the oxygen depletion leads to water quality problems. In such
cases, artificial mixing can be used to counter these effects.
Of the various artificial mixing methods used, air bubblers,
where compressed air is continuously injected through a
diffuser set at or near the bottom of the reservoir, are one of
the most common (McDougall 1978; Asaeda and Imberger
1993; Schladow 1993; McGinnis and Little 1998).
In a bubble plume, the rising air bubbles entrain the
ambient water and form an air–water mixture plume. As the
bubbles rise, there is a continuous increase of buoyancy
resulting from their volumetric expansion due to reduction of
hydrostatic pressure. However, as has been shown
experimentally by McDougall (1978) and Asaeda and
Imberger (1993) for a stratified ambient environment, the
© CSIRO 2003
entrained fluid bubble mixture eventually becomes heavy
compared with the surrounding ambient fluid as the plume
rises, resulting in the centreline velocity becoming zero. The
entrained fluid then forms a lateral outflow, which sinks to its
neutral-buoyancy level. The air bubbles, however, continue
to rise and form a new plume at the point of detrainment.
This entrainment and detrainment process, which is referred
to here as a cascade, may be repeated many times until the
bubbles reach the surface. In more technical terms, the
plume detrains all of the entrained water at the point where
the downward buoyancy dominates upward momentum flux.
This process of entraining and detraining water reduces the
stratification of the water column.
In the homogenous case, or the case of weak stratification
and high airflow rate, detrainment only takes place at the
surface. In the case of strong stratification and weak airflow
rate, more than one detrainment takes place.
McDougall (1978) examined bubble plume dynamics in
a stratified medium via a series of experiments and showed
10.1071/MF02045
1323-1650/03/030271
272
Marine and Freshwater Research
that mass, momentum and buoyancy conservation
equations could be applied to model this process. Based on
these equations, Schladow (1992) developed a bubble
plume model that took into account the fact that there is
potentially a terminal height to which any buoyant plume
will rise. By reinitializing the plume model at the height of
rise of each individual plume, he showed that it is possible
to explicitly represent the cascade and to analyse it in
detail. Consistent with previous experimental results
(McDougall 1978; and Asaeda and Imberger 1988),
Schladow (1992) demonstrated that more than one plume is
formed for a weak airflow rate and strong stratification.
Though Schladow’s (1992) model describes well the
hydrodynamic behaviour of plumes, the model does not
account for the oxygen and nitrogen transfer from the
gaseous air bubbles to the ambient water. Also, the model
does not include the change in slip velocity as the air
bubble size changes in response to the ambient pressure
decreasing as the bubbles rise.
Schladow (1993) developed design charts to optimize the
bubbler design parameters for linearly stratified reservoirs.
To use these charts for all non-linear types of stratification,
one must re-express the density stratification profiles in
terms of an equivalent linear stratification (ELS) where the
potential energies for both the actual stratification and ELS
are same. Schladow’s (1993) model, consistent with the
results of Asaeda and Imberger (1993), showed that the
maximum mechanical efficiency occurs when the airflow
rate is just sufficient to cause a single detrainment (outflow)
at the surface.
Wüest et al. (1992) demonstrated experimentally and
modelled, assuming top-hat profiles, a diffuser unit
consisting of 54 tubes. Of particular interest to Wüest et al.
(1992) was the gaseous mass transfer to the ambient water.
Though the model accounts for the change in slip velocity
with bubble size and gas transfer, it does not account for
more than one plume in cascade.
The present study uses a modified version of Schladow’s
(1992) model for three purposes.
䊉
䊉
䊉
To examine the sensitivity of the operating cost (i.e.
mechanical efficiency), destratification time and the
design airflow rate to uncertainty in the design input
parameters (mainly airflow rate and ambient
stratification).
To develop simple relationships that can be used to
apply
Schladow’s
(1993)
equivalent
linear
stratification method to determine the optimum airflow
rate. In doing so, we are mostly interested in the case
where the airflow rate may be adjusted as the reservoir
destratifies (i.e. some form of real time control). In
addition, we examine the range of validity for the
application of the ELS method.
Whether it is necessary for modelling purposes to
include changes in air bubble radius and slip velocity.
G. B. Sahoo and D. Luketina
Gas transfer to the ambient fluid is not discussed here
because it is addressed in a separate paper (G. B. Sahoo and
D. Luketina, unpublished data).
Bubble-plume model
A bubble plume model has been developed that combines the
features of the Schladow (1992) and Wüest et al. (1992)
models. This model is described in the remainder of this
section.
Features and assumptions
Previous investigators (McDougall 1978; Schladow 1992;
Wüest et al. 1992; Asaeda and Imberger 1993) have
presented the basic equations for bubble plume models based
on Gaussian or top-hat profiles for density and velocity
within the plume and discussed the values that should be
used for bubble plume parameters, such as the entrainment
coefficient α and the spreading ratio λ, which represents the
ratio of the effective buoyancy width to the effective
momentum width within the plume.
The features and principal assumptions made in
developing this model in a stratified environment are as
follows.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
The variation of density is only important in the
gravity (buoyancy) terms and not for mass fluxes
(i.e. the Boussinesq assumption as used in most
plume theories).
The density of gas is negligible compared with that
of water in the momentum equation.
Ambient currents are negligible compared with the
plume entrainment velocity.
The plume is fully turbulent.
The turbulent transport of momentum and scalar
quantities (heat, salinity, gases) are negligible
compared with advective transport.
The bubble source is assumed to release uniformly
sized bubbles at a constant rate.
Bubble coalescence and divisions are negligible. In
this model, the number of bubbles and the flux
number (number of bubbles per unit time rising
through a horizontal plane) are kept constant.
The bubbles are of uniform size at each height.
The properties of the initial plume water are those of
lake water at the depth of the diffuser.
Gas exchange between water and bubbles for gases
other than oxygen and nitrogen (e.g. argon, carbon
dioxide, methane and others) can be neglected.
However, such gases can readily be included in the
model.
We consider a continuous point source bubbler located at
a depth (h) below the surface of a density stratified water
column. Let Q(z) be the steady flow rate of gas at any height
(z) above the point source and Q0, the gas flow rate at
atmospheric pressure. By assuming isothermal expansion of
Bubble-plume behaviour and destratification
Marine and Freshwater Research
the gas, the volume flow rate of the gas can be expressed as
a function of height:
Q (z) =
Q0 p a
H
( − z ) ρr g
(
w ( r,z ) = w c ( z ) exp − r 2 b2
)
θ ′ ( r,z) = ( g ρr ) ρ0 ( z) - ρc ( z) exp  − r 2 ( λb)

(2)
2

(3)
where ρc(z) is the centreline density of the plume, ρ0(z) is the
ambient density at height z, w(r,z) is the average vertical
velocity at height (z) and radial distance r from the plume
centreline, wc(z) is the vertical velocity at the centreline at
height (z) and b is the effective radius of the plume.
Conservation equations
Following McDougall (1978) and Schladow (1992), the
differential equation for continuity is:
(
)
d πb2 w
= 2παbw
dz
(πb2wρ0κT) can be reduced to the corresponding expression
for the temperature flux (πb2wT):
(
(4)
where the continuity equation has been derived using the
standard entrainment assumption ue = αw, where ue is the
entrainment velocity, w is the centreline velocity (the
subscripts ‘c’ being dropped), and α is the entrainment
coefficient (for a Gaussian profile). The usual value of the
entrainment coefficient for a buoyant plume is considered to
be 0.083 (List 1982; Milgram 1983; Poon 1985; Asaeda and
Imberger 1993; Schladow 1993). As explained by Morton
(1959) and Wüest et al. (1992), the entrainment coefficient
for a top-hat profile is 2 times the corresponding value
used for a Gaussian profile α. Furthermore, the plume width
of a top-hat profile is 2 times the plume width of a
Gaussian profile.
The heat flux equation is derived by assuming that the
coefficient of specific heat per unit water volume (ρ0κ) is
constant, and neglecting the heat content of the gas volume,
so that the conservation equation for the heat flux
)
d πb2 wT
= 2παbwTa
dz
(1)
where pa, the atmospheric pressure, is expressed in terms of
an equivalent water depth as ha = pa/ρrg, where ρr is the
equivalent reference density and g is gravitational
acceleration. Thus, the static pressure at the point source is H
= h + ha and the total pressure experienced by the gaseous
bubbles at any height (z) is H – z. At sea level ha ~
~ 10.2 m.
For modelling convenience, it is assumed that the
distributions of vertical velocity and density are Gaussian
shaped at all heights (McDougall 1978; Schladow1992).
This assumption of self-similarity has experimental support
in unstratified surroundings (Kobus 1968). Accordingly, the
equations for vertical velocity and density deficiency are:
273
(5)
where Ta is the ambient water temperature given by
measured temperature profiles and T is the plume water
temperature.
Following Schladow (1992), the momentum and
buoyancy conservation equations are:
d ( 12 πb2 w 2 )
= πλ 2b2θ′
dz
(6)

Q0Pa w
d  πλ 2b2 wθ′  = −πb2 wN 2 + d 
 (7)


2
dz  λ +1 
dz  ( H − z ) ρr ( w + u b ) 


Here, ub = ws(λ2+1), where ws is the slip velocity of air
bubbles relative to the liquid in the plume and the buoyancy
frequency (N) is defined as N = [–(g/ρr)dρ0/dz]1/2. The
bubble slip velocity ws is a function of air bubble radius and
can be calculated using the relationships developed by Wüest
et al. (1992) (see Table 1).
Poon (1985) conducted a series of experiments and
suggested that, with the possible exception of very large
bubble size, a constant value of λ = 0.3 is appropriate for the
spreading rate. McDougall (1978) and Schladow (1993)
reported similar assumptions. Wüest et al. (1992) used 0.8 as
the spreading rate for their top-hat-profile based model,
which corresponds to a spreading rate of 0.56 for a Gaussian
profile. A value of λ = 0.3 was used in this study and, in any
case, the results are not sensitive to the value of λ.
Neglecting the stratification close to the source,
Equation 7 can be simplified to the form (Schladow 1992):

θ′0 = 

( λ 2 +1)


w + u b b2 

Q0 p a
λ 2πρr ( H − z ) (
)
(8)
where, and θ´0 = gθ0, and θ0 is the density deficit between the
plume and the ambient water at the source.
Based on the assumption that the stratification can be
neglected close to the source, McDougall (1978) has
presented the power series solutions for the initial plume
Table 1. Parameter approximations for the bubble slip
velocity ws (m s–1) from Wüest et al. (1992) where R is the
bubble radius (m)
Approximations
Criteria
1.357
ws (R) = 4474(R)
ws (R) = 0.23
ws (R) = 4.202(R)0.547
R < 7.0 × 10–4
7.0 × 10–4 < R < 5.1 × 10–3
R > 5.1 × 10–3
274
Marine and Freshwater Research
G. B. Sahoo and D. Luketina
radius (b0) and the centreline velocity (w0) of a single bubble
plume:
1

 z 3


 MH 
b0 = 2αz 0.6 + 0.01719

−0.002527
G ≈ 0.776  − ∂ρ 
 ∂z 

2
3
 z


 MH 
1
3

 z 3


 MH 
 z 3


 MH 
+ z ( 0.4536 − 0.0105 M −1 ) +...
H

(10)
where M is a non-dimensional group that refers to the source
strength and is defined as (Schladow 1992):
)
(
4πα 2ρr H 2u 3b
V0 = πb20 w0, M0 = V0w0 and Φ0 = V0π0
(11)
The density (ρ) (kg m–3) of the reservoir water is
approximated as follows (UNESCO 1981):
ρ = 0.0001Τ3 –0.009095T2 + 0.0679T + 999.84
(12)
where T is the temperature in degrees centigrade.
Schladow (1992) defined a non-dimensional group, C,
which refers to stratification strength. However, the use of C
and M as the non-dimensional variables is somewhat
inconvenient because both contain the source strength
(airflow rate). For this reason, a new non-dimensional group
G, which is independent of airflow rate, is defined here and
we use G and M as the non-dimensional variables to describe
the plume behaviour:
 N 3H 3

(1+ λ2 )
pa 
4πα2gh a u3bρr



32
H5
Q0
(14)
Here, for a specific reservoir, the depth (h), and thus the total
head (H) is constant and the non-dimensional stratification G
depends only on the density gradient. The important aspect
of these two non-dimensional groups (source strength and
stratification strength) is that the plume variables, centreline
velocity, plume radius and the density difference are
identical for any two plumes in a linearly stratified ambient
fluid that have the same M and G values.
Differential Equations 4, 5, 6 and 7 were integrated on a
one-dimensional equi-spaced grid using a fourth order
Runge–Kutta scheme. The integrations were continued until
either the water surface was reached or the upward fluid flux
became zero. At this point, the water entrained by the plume
was detrained and the distance between the commencement
and detrainment point of the plume was considered as the
height of the rise of the plume. This water was then inserted
into the reservoir layer with the same density. If the
detrainment point was below the free surface, then a new
plume commences with the initial conditions being those at
the height of rise of plume. This process of plume formation
may be repeated many times until the surface is reached.
The plume density (ρn) at the height of rise of each plume
was found by integrating the entrained mass and volume:
hP
ρn, P =
h P-1
∫ 2πb(z)αw(z)ρ(z)dz
hP
h P-1
∫ 2πb(z)αw(z)dz
hP
=
∫ bwρ dz
h P-1
hP
h P-1
(15)
∫ bw dz
where hp is the height of rise of the Pth plume and h0 is the
diffuser depth.
Model results
(
Set-up and initial conditions
)
3 3
2
 3 4     N H 1+ λ pa   1 

C =  N H   H  = 
2
3
 
 gQ0   h a   4πα gh a u bρr   M 
or C = G
M

Numerical scheme
Q0pa λ 2 +1
Equations 8, 9 and 10 can be evaluated close to the source
to provide initial values b0 and w0. Thus the initial values of
volume (V0), momentum (M0) and buoyancy (Φ0) flux are:
G = 
H3 ,
1
1.609 − 0.3195

M=
32
C ≈ 9.71×10−6  − ∂ρ 
 ∂z 
Q
and M ≈ 79958.8 02
H
(9)
2
+0.06693


+ z ( −0.04609 + 0.000031 M −1 ) +...
H

w 0 = u b  MH 
 z 
Using the values λ = 0.3, α = 0.083 and the initial air bubble
radius (R) as 1 mm (this assumption is explained later), the
non-dimensional groups defined above can be simplified to:
(13)
For the numerical computations, a grid spacing of 5 cm was
used. For most scenarios, and unless specified otherwise, the
reservoir was considered to be 100 m deep giving 2000
equi-spaced grid points or layers. The bubbler was
positioned at the bottom of the reservoir and the
Bubble-plume behaviour and destratification
Marine and Freshwater Research
hydrodynamic behaviour of the plume was studied for
various flow rate and stratification conditions.
The temperature of the ambient water was set to vary
linearly from 10°C at the bottom to 25°C at the surface.
Based on the bubble-rise velocity experimental data of
Haberman and Morton (1954) and the mass-transfer
experimental data of Motarjemi and Jameson (1978), Wüest
et al. (1992) developed the relationships for the slip velocity
and the gas transfer coefficient for molecular oxygen and
nitrogen as a function of air-bubble size. They reported that
the gas-transfer coefficient for molecular oxygen and
nitrogen remains constant for bubble size greater than
0.67 mm and reduces steeply as bubble size reduces. It is
also clear from Table 1 that smaller bubble size produces
lower slip velocity and thus higher residence time within
water, which will result in higher gas-transfer rates. For these
reasons, and consistent with the recommendation of Wüest
et al. (1992) for efficient gas-transfer rate, the initial bubble
radius was set to 1 mm.
Mechanical efficiency and destratification time
The McDougall (1978) and Schladow (1992) studies
describe in detail the hydrodynamic behaviour of a bubble
plume for a linearly stratified case. This study examines
some of the practical aspects of optimum bubbler design for
different stratification scenarios. Of key importance are the
mechanical efficiency and destratification time, which
involve estimating the change in potential energy of water
column owing to bubbler operation. Assuming isothermal
compression, the instantaneous mechanical efficiency
(ηmech) can be estimated as:
=
( PEf
−PEi ) ∆PE ∆PE/∆t
=
=
Wiso
Wiso Wiso /∆t ∆t→0
∂PE ∂t
∂Wiso ∂t
∂PE = NP ∂PE P
∑ ∂t
∂t P=1
NP 
i=LP ρ − ρ
 i−1
i+1 
P=1  i=L
P−1 

∂Wiso
p 
= 2.303pa QT log  h 
∂t
 pa 
2



i

zi  − ∑ q j  


 j=LP−1  
(17)
where zi is the height of the centre of the ith ambient layer
above the neutral-buoyancy layer of the Pth plume, ρi is the
density of the ith ambient layer, qj is the entrainment (m3 s–1)
from the jth layer of the reservoir, LP is the layer at the height
of rise of the Pth plume, L0 is the layer containing the diffuser
(18)
where pa and ph are the absolute pressure at the surface and
at the bubble-plume source (atmospheric pressure plus water
pressure), respectively, and QT is the total free airflow rate
required. If Ns is the number of buoyancy sources required,
then QT = NsQ0.
From Equations 16, 17 and 18, it is clear that the
instantaneous mechanical efficiency is independent of
bubbler-operation time and the surface area of the stratified
reservoir. However, it does depend on the airflow rate and
stratification strength.
The mechanical efficiency predicted by the model for M
values of 10, 1, 0.1 and 0.01 is plotted as a function of C
(Fig. 1) for comparison with Schladow’s (1992) result. As
the water depth and airflow rate were kept constant along
each of the M curves, the variation of C or G along any M
curve is purely a function of changing the density
stratification. All the curves of Fig. 1 are effectively the same
as Schladow’s (1992) result and, as explained by Schladow
(1992), are dominated by pronounced oscillatory behaviour,
where the mechanical efficiency ranges from 1% to 15%
with the highest M values producing the highest mechanical
efficiencies. The mechanical efficiency predicted by the
model for M values of 5, 1, 0.1 and 0.01 is plotted as a
function of G (Fig. 2), the new non-dimensional group used
(16)
where PEi is the initial potential energy of the water column,
PEf is the final potential energy of the water column after
bubbler operation for time (∆t), and ∂PE/∂t is the rate of
change of potential energy and can be estimated for a reservoir of any shape using (see Appendix 1 for the derivation):
≈ ∑ g ∑ 
and NP is the total number of plumes formed in cascade
within the water column of depth (h). The choice of the
neutral-buoyancy layer as the datum in Equation 17 is quite
important because it minimizes the numerical error in
determining ∂PE/∂t. This is explained further in Appendix 1.
The isothermal work of compression (Wiso) is defined
(Rogers and Mayhew 1974) as:
15
Mechanical efficiency η mech (%)
ηmech =
275
12
9
6
3
0
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
C
Fig. 1. Mechanical efficiency ηmech (%) as a function of C for M
equal to10 (thick black line), 1 (medium black line), 0.1 (thin black
line) and 0.01 (thick grey line) for a linearly stratified reservoir.
276
Marine and Freshwater Research
G. B. Sahoo and D. Luketina
15
2
12
4
3
7
6
5
5
1
20
4
3
9
Depth (m)
Mechanical efficiency η mech (%)
0
6
40
2
60
1
3
80
0
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04 1.E+05
1.E+06
G
Fig. 2. Mechanical efficiency ηmech (%) as a function of G for M
equal to 5 (thick black line), 1 (medium black line), 0.1 (thin black
line) and 0.01 (thick grey line) for a linearly stratified reservoir. G
values greater than 6300 (thick black dotted line) are not realizable in
practice and are only shown so that they can be compared with Fig. 1.
The mechanical efficiency peaks are numbered for the case M equal
to 1.
5
0
4
3
20
Depth (m)
2
40
1
60
80
100
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
G
Fig. 3. Maximum height of rise of plumes in a plume cascade in a
linearly stratified water column as a function of G for M equal to 1.
The number assigned to each line corresponds to the detrainment
points, with 1 being the detrainment point closest to the diffuser. G
values greater than 6300 (thick black dotted line) are not realizable in
practice.
for this study. Fig. 2 shows similar oscillatory behaviour, but
the curves are less scattered than in Fig. 1.
Asaeda and Imberger (1993) demonstrated via
experiments that bubbler operation would achieve maximum
efficiency when the supplied energy is just sufficient to lift
the heavy bottom water up to the water surface. The
oscillatory behaviour of all the curves in Figs 1 and 2 reflects
100
0.01
0.1
1
10
M
Fig. 4. Maximum height of rise of plumes in a linearly stratified
water column as a function of M for G equal to 3414. The number
assigned to each line corresponds to the detrainment points of that
numbered plume.
this important aspect. The heights of rise of plumes (i.e. the
detrainment points) are plotted as functions of G for M equal
to 1 in Fig. 3. The first mechanical efficiency peak (i.e. the
leftmost peak on any curve) of Fig. 2 corresponds to one
detrainment just forming at the surface, whereas the second
peak corresponds to one internal detrainment and a final
detrainment just forming at the surface (see Fig. 3).
Therefore, the peak number is linked to the number of
detrainment points. At other points the plume still has
positive momentum, which has not been fully utilized for
mixing. Fig. 4 is similar to Fig. 3 except that the detrainment
depths are plotted as a function of M for a fixed linear
stratification case of G equal to 3414 (this corresponds to a
bed and water-surface temperature of 10°C and 26.5°C,
respectively, in a 100-m deep reservoir). For sufficiently
large M values there is one detrainment – at the surface. As
the M value reduces, the number of detrainment points
increases. For the lowest M value examined here (0.01), six
internal detrainments and one final detrainment at the
surface are observed.
Before proceeding, it is instructive to determine the
practical range of G values that can occur in reservoirs. In
turn, these G values mostly depend on the thermal
stratification. Table 2 shows the most different surface and
bed temperatures recorded during a number of reservoir and
lake studies. These data show that the thermal stratification
does not tend to exceed 5°C as a top-to-bottom temperature
difference for any type of tropical reservoir or lake, and 15°C
and 25°C for any semi-tropical and temperate reservoirs (or
lakes) respectively. The associated G values were determined
by assuming that the temperature profile varied linearly from
Bubble-plume behaviour and destratification
Table 2.
Marine and Freshwater Research
277
Bed and surface temperatures recorded in a number of reservoirs and lakes and associated G values
Depth
(m)
Lake/Reservoir
Class
Valencia Lake, Venezuela
Upper Peirce Reservoir, Singapore
Babagaon Reservoir, Malaysia
Glennies Creek Reservoir, Australia
Teddington Storage, Australia
McCarrons Lake, USA
Calhoun Lake, USA
Martin Lake, Indiana, USA
DeGray Lake, Arkansas, USA
Tropical
Tropical
Tropical
Semi-tropical
Semi-tropical
Temperate
Temperate
Temperate
Temperate
35.0
22.0
57.0
55.2
08.6
14.0
24.0
16.5
48.0
the bed to the water surface and that R, α and λ are equal to
1 mm, 0.083 and 0.3 respectively. The maximum G value for
the data in Table 2 is around 3000. Table 3 shows the G
values associated with various temperature gradients. For a
deep reservoir (the deeper the reservoir, the larger the G
value), for example 100 m, having a bed-to-surface
temperature difference of 25°C gives an upper limit G value
of around 6300. G values of 1000 and 4100 are not likely to
be exceeded in tropical and semi-tropical reservoirs
respectively.
The first peak for M values of 10, 7.5, 5, 2.5 and 1 occurs
at G values of 18446, 13300, 8424, 3735, and 1257,
respectively, in a 100-m deep reservoir. Since the maximum
realizable G value is around 6300, the value of M in practice
is always less than 5 if the bubbler is to be operated at optimal
or near-optimal efficiency.
Of major importance when operating a bubbler is the time
T required to achieve a fully or near-fully mixed state. Rather
than using a time, Schladow (1992) used a mixing
effectiveness (Emix) expressed as the ratio of the change of
potential energy to the maximum change in potential energy
(i.e. that required to attain a fully mixed state) over a specific
time interval (set arbitrarily as one day). It is more useful to
express the destratification rate as a time per unit surface
area per buoyancy source (Γ) so that the results can be
Temperature (°C)
Bed
Surface
∆T
(°C)
G
27.0
27.2
24.0
11.1
17.0
00.0
07.0
03.6
06.0
01.2
03.8
05.0
14.9
12.0
22.0
19.0
22.4
24.0
0045
0173
0554
1950
0686
0519
1151
0996
3040
28.2
31.0
29.0
26.0
29.0
22.0
26.0
26.0
30.0
Source
Lewis (1983)
Yang et al. (1993)
Lamy-Contaret (2000)
Schladow (1993)
Burns (1998)
Ford and Stefan (1980)
Ford and Stefan (1980)
Wetzel (1982)
Martin and McCutcheon (1999)
readily applied to reservoirs with different surface areas (As)
– this is done as follows:
Γ=
T ∆PE mixed / As
=
As
∂PE/∂t
(19)
where the denominator is found using Equation 17 and the
numerator, the change in maximum potential energy per unit
surface area (∆PEmixed/As), is found using:
i=L
h
∆PEmixed As ≈ gh−ϕ ∑ ( ρm − ρi ) ziϕ+1∆z
i=L0
where γ and ϕ are constants for a reservoir with
cross-sectional area , A = γzϕ, Lh is the surface layer of the
reservoir, ∆z is the layer thickness and ρm is the homogenous
density of a fully mixed reservoir (see Appendix 1 for the
derivation). From Equation 19 and associated equations, it is
clear that mixing time per unit surface area of the water
column depends on the stratification profile, the airflow rate,
the reservoir shape (ϕ) and the number of bubblers per unit
surface area. Unless mentioned, the constant ϕ for the
reservoir shape is equal to 0 for the results that follow.
It is important to note that the mixing time per unit
surface area (Γ) estimated using Equation 19 is a lower
bound on the mixing time. This is because as mixing
Table 3. G values as a function of reservoir depth h and density gradient (∆ρ h–1) for R = 1 mm, α = 0.083 and
λ = 0.3, assuming a linear density gradient where ∆ρ is the density difference between the bottom and top layer;
corresponding temperature differences are also shown
∆T
(°C)
∆ρ
(kg m–3)
10
20
30
40
02.5
05.0
07.5
10.5
14.0
18.0
25.0
0.50
1.00
1.50
2.00
2.50
3.00
3.35
0071
0202
0371
0571
0797
1047
1235
0084
0239
0438
0674
0942
1237
1459
0108
0307
0563
0866
1209
1588
1873
0137
0388
0712
1095
1529
2008
2369
(20)
Reservoir depth h (m)
50
60
0169
0478
0878
1351
1887
2478
2923
0204
0577
1059
1630
2276
2990
3526
70
80
90
100
0242
0683
1254
1929
2693
3538
4172
0281
0795
1460
2246
3136
4119
4858
0323
0913
1677
2580
3603
4732
5581
0367
1038
1905
2930
4092
5375
6339
278
Marine and Freshwater Research
G. B. Sahoo and D. Luketina
Destratification time Γ (s m–2)
1.E+03
1.E+02
5
4
3
1.E+01
2
1
1.E+00
1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06
G
Fig. 5. Mixing time per unit surface area Γ (s m–2) as a function of
G for M equal to 5 (thick black line), 1 (medium black line), 0.1 (thin
black line) and 0.01 (thick grey line) for a linearly stratified reservoir.
G values greater than 6300 (thick black dotted line) are not realizable
in practice. The peaks are numbered for the case M equal to 1.
Destratification time Γ (s m–2)
1.E+03
1.E+02
of G. The important features of the staircase-like shape
depend on the number and location of the detrainment
points. Starting from the left side of a curve, the mixing time
per unit surface area remains nearly constant up to the first
peak. It then increases in a series of local peaks and troughs.
The trade-off between mechanical efficiency and mixing
time is demonstrated in Fig. 6. Since the mixing time per unit
surface area increases sharply as the peak number increases
without much change in the mechanical efficiency, it is
recommended that bubblers be designed to coincide with the
first peak (i.e. the medium grey line in Fig. 6). However,
lower airflow rates, and thus a smaller compressor, are
required to operate at the second peak compared with the
first peak. There are also other reasons why it may be
advantageous to design for the second peak. These reasons
are discussed in the next section.
Optimum operational conditions for linearly stratified cases
Relationships are presented in this section for determining
the mechanical efficiency ηmech, the airflow rate (via M) and
the partially normalized destratification time Γ associated
with optimal bubbler operation (i.e. the first peak). We also
consider the second peak because Schladow (1993) makes
use of this. Analysis of the first and second peak points
shows that the relationship between mechanical efficiency
and G for different reservoir depths (h) is well fitted
(correlation coefficient exceeding 0.99) by:
 4

4
 i=0

i=0
ηmech =  ∑ a i h i  lnG + ∑ bi h i
1.E+01
1.E+00
1.E+00
1.E+01
1.E+02
(21)
where h is the depth of reservoir and the coefficients ai and
bi are presented in Table 4. Fig. 7 shows the fitted equations.
Similarly, M and the mixing time per unit surface area (Γ)
associated with the first peak for different reservoir depths
(h) are well fitted (correlation coefficients exceeding 0.99)
respectively by:
Mechanical efficiency η mech (%)
Fig. 6. Mixing time per unit surface area Γ (s m–2) as a function of
mechanical efficiency ηmech (%) for M equal to 5 (thick black line), 1
(medium black line), 0.1 (thin black line) and 0.01 (thick grey line) for
a linearly stratified reservoir. Only cases corresponding to G £ 6300
are shown. The medium grey line passes through the first mechanical
efficiency ηmech (%) peak for each M case.
proceeds and the stratification weakens, the rate of change of
potential energy ∂PRE/∂t reduces (similarly the mechanical
efficiency ηmech will reduce).
Fig. 5 shows the mixing time per unit surface area (s m–2)
as a function of G for M equal to 5, 1, 0.1 and 0.01 for a
linearly stratified reservoir. The curves take the form of
rising staircases, with the highest M values being associated
with the minimum destratification times for any given values


 B hi 
i 
M = ∑ ( Ai h i ) G 
4
(22)
i=0

( ϕ +1)  4 C hi G −0.3
Γ = 12  1 −
∑

Ns ϕ + 3 ( ϕ + 2 )2  i=0 ( i )

(23)

where Ns is the number of buoyancy sources and the values
of the coefficients Ai, Bi and Ci are presented in Table 4.
When determining Equation 23, the effect of ϕ on the mixing
time per unit surface area (i.e. the term in the square
brackets) was derived theoretically (see Appendix 1).
Knowledge of the reservoir stratification via G and the
reservoir depth (h) allow the optimum mechanical efficiency
to be computed using Equation 21, whereas the source
Bubble-plume behaviour and destratification
Table 4.
M (Equation 22)
Ai
Bi
–1.170
0.149
–3.539 × 10–3
3.624 × 10–5
–1.347 × 10–7
–6.517 × 10–4
2.178 × 10–4
–4.381 × 10–6
4.048 × 10–8
–1.401 × 10–10
0.819
–1.414 × 10–3
2.663 × 10–5
–2.360 × 10–7
0.768 × 10–9
451.924
–27.335
0.667
–7.060 × 10–3
2.690 × 10–5
1.063
0.750 × 10–2
–1.272 × 10–4
1.549 × 10–6
–7.421 × 10–9
–3.124
0.145
–3.364 × 10–3
3.285 × 10–5
–1.160 × 10–7
–0.813 × 10–4
0.221 × 10–4
–0.412 × 10–6
0.350 × 10–8
–0.110 × 10–10
0.840
–1.156 × 10–3
2.514 × 10–5
–2.575 × 10–7
0.970 × 10–9
5774.925
–357.388
8.751
–92.517 × 10–3
35.207 × 10–5
1.E+02
12
10
8
1.E+01
6
4
Destratification time Γ (s m–2)
Mechanical efficiency η mech (%)
14
2
0
1.E+00
1.E+01
Γ (Equation 23)
Ci
1.091
1.294 × 10–2
–2.391 × 10–4
2.097 × 10–6
–7.232 × 10–9
16
1.E+02
G
1.E+03
279
Values of coefficients in Equations 21, 22 and 23
ηmech (Equation 21)
ai
bi
i
First Peak
0
1
2
3
4
Second Peak
0
1
2
3
4
Marine and Freshwater Research
1.E+00
1.E+04
Fig. 7. First peak mechanical efficiency ηmech (%) (lines rising from
left to right) and corresponding destratification time per unit surface
area Γ (s m–2) (lines falling from left to right) as a function of G (thick
black line fitted to solids squares), G – 10%G (medium black line
fitted to asterisks) and G + 10%G (thin black line fitted to solid
circles) for a reservoir of 50 m depth (h).
strength (i.e. the airflow rate) and the mixing time Γ can be
estimated using Equations 22 and 23 respectively.
The preceding design equations rely on knowledge of G –
however, a suitable design value of G can be difficult to
determine owing to natural variability. Schladow (1993)
stated that ‘the two dimensionless parameters that describe
bubble plumes, M and C, contain three variables – water
depth, airflow rate and the strength of the stratification. For
a given water depth and initial strength of the stratification,
the selection of a suitable airflow rate (Q0) that will yield a
desirable position on the M–C curve is difficult. Based on
many computer simulations; the most efficient design is that
one that has its initial value of C between the first and second
peaks of the M–C curve, but very close to the second peak.
As destratification proceeds, the efficiency approaches
towards the first peak. Equations 21–23 can be used to
conveniently obtain the second peak values for ηmech, M and
Γ. Here, however, we are mostly concerned with the case
where the airflow rate may be adjusted as the reservoir
destratifies (i.e. some form of real-time control). In this case,
the first peak is the best choice and Equations 21–23 with
first peak coefficients should be used. However, the issues of
measurement accuracy (of the stratification), evolution of
the stratification prior to the airflow rate being updated and
the accuracy of the model prediction need to be considered.
This can be examined by looking at Fig. 7, which shows the
trend lines of peak mechanical efficiency ηmech (%) and
destratification time per unit surface area Γ (s m–2) as
function of G for a 50-m deep reservoir. Also shown are the
mechanical efficiency and destratification time that would
result from the design G being 10% less than actual (G –
0.1G in Fig. 7) and vice-versa. For the case of the design G
being 10% less than the actual G, the mechanical efficiencies
do not vary much from the peak mechanical efficiencies.
However when the design G is 10% greater than the actual G,
the mechanical efficiency at which the bubbler operates will
decrease substantially, particularly at high G values. This is
because the mechanical efficiency drops steeply after the
peaks, whereas values before the peak rise at a lesser rate.
The trend lines of destratification time per unit surface area
also show a similar sensitivity. This infers that the designer
should be careful about overestimating the design G. A
conservative practice would be to reduce the design value of
G by 10% if attempting to operate at the first peak.
All of the bubble plume operational design criteria
discussed so far are intended for the linear stratification case.
However, the thermal stratification in a reservoir does not
often follow a linear pattern. Schladow (1993) reported that
non-linear step stratifications could be treated as linear
stratifications by finding the linear stratification that has the
same potential energy relative to the base of the water body.
The following section examines in more detail the use of
equivalent linear-stratification algorithms for various step
stratifications.
Marine and Freshwater Research
G. B. Sahoo and D. Luketina
0
(a)
Depth (m)
20
0
a2, b2, c2
d3, e3, f3
20 (b) e2
b1
40
a1
60
c1
Depth (m)
280
40
d1
f1
e1
80
(b)
100
0.01
0.1
1
10
M
0
(c)
100
0.01
0
g3
20
Depth (m)
Depth (m)
(d)
40
g2
60
80
100
0.01
0.1
g1
(d)
40
1
M
10
10
1
10
h3
h2
60
80
0.1
1
M
20
(c)
f2
60
80
(a)
d2
100
0.01
h1
0.1
M
Fig. 9. Height of rise of individual plumes in a plume cascade as
function of M for (a) single-step stratification cases a–c, (b)
double-step stratification cases d–f, (c) double-step stratification case
g and (d) double-step stratification case (h). Subscripts 1, 2 and 3
refer to the first, second and third detrainment points respectively.
(e)
(g)
(f)
(h)
Fig. 8. Eight step-stratification cases (a–h) and equivalent linear
stratifications.
Optimum operational conditions for step-stratification cases
Eight different step stratifications of equal potential energy
and identical bottom density, along with their equivalent
linear stratification (the equivalent linear stratification is the
same for all cases), were examined (see Fig. 8). The height
of rise of plumes (i.e. detrainment locations) for each step
case and the equivalent linear stratification case are plotted
in Figs 9 and 3 respectively. At sufficiently large M values
there is one detrainment, which occurs at the surface, for all
cases. For the lowest M value (0.01) examined here for the
step-stratification cases, the number of detrainment points
found in a step-stratified profile corresponds to the number
of steps of the stratified profile and a final detrainment at the
surface. It is concluded from this behaviour, as expected, that
the step stratification approaches the characteristics of the
linear case as the number of steps increases. The mechanical
efficiency for both cases were examined for different source
strength M = 10 to 0.01 as shown in Fig. 10. For the cases of
single-step stratification (Figs 8a–c), the potential energy
change owing to mixing can only occur at the step, whereas
the second plume, which detrains at the surface, does not
contribute to mixing. This results in very low mechanical
efficiency as shown in Fig. 10a. The double-step
stratification cases have considerably higher efficiencies. In
particular, cases e and h have efficiencies close to that for the
linear stratification (Fig. 10b).
Zic et al. (1992) conducted a series of experiments similar
to the typical lake thermal stratification and reported that the
energy efficiencies for typical lake destratification set-ups
are much lower (<3%) in temperature (weakly) stratified
water, whereas the energy efficiencies are higher by up to
12% for the case of linearly stratified saline (strong)
stratification set-ups of Asaeda and Imberger (1988). In this
study too, mechanical efficiency efficiencies are found to be
very low (<1%) in the case of typical single-step
stratification, wheres it is higher for the linear stratification
case by up to 13%.
The first peak mechanical efficiency ηmech (rightmost in
Fig. 10a) corresponds to the case of a plume in a linear
stratification having a single detrainment or outflow. The
subsequent peaks (i.e. 2 to 5) correspond to plumes having
progressively more outflows as M is reduced. Quite clearly,
the highest mechanical efficiency for step cases corresponds
Marine and Freshwater Research
14
1
(a)
12
2
10
3
4
8
5
6
4
Step cases a to c
2
0
0.01
0.1
1
10
Mechanical efficiency η mech (%)
M
14
12
1
(b)
2
10
8
3
5
h
e
4
6
g
4
d
2
0
0.01
0.1
1
f
10
M
Fig. 10. Mechanical efficiency ηmech (%) as function of M for (a)
single-step stratification cases of a (thick black line), b (medium
black line), c (thin black line) and equivalent linear stratification
(thick grey line) and (b) double-step stratification cases of d (thin
grey line), e (medium grey line), f (thin black line), g (medium black
line), h (thick black line) and equivalent linear stratification case
(thick grey line). The mechanical efficiency peaks of the linear
stratification case are numbered.
to the first peak. Applying the design method of Schladow
(1993) to the second peak (i.e. two detrainment points)
would result in very low efficiencies. Rather, it is
recommended to use a simple bubbler model to find the
optimum bubbler operational conditions for step
stratification. Using the design charts for the equivalent
linear method as proposed by Schladow (1993) for all
step-stratification cases may lead to sub-optimal design.
Effect of bubble size on bubbler design
The final objective to be examined here is the necessity of
inclusion of variable air-bubble radius and corresponding
slip velocity for modelling purposes.
In practice, the diameter of air bubbles have been reported
in the range 1.5–3.7 mm (Goossens 1979), 2–5 mm (Zic et
al. 1992) and about 2 mm and 4 mm for ceramic and nozzle
bubblers respectively (Asaeda and Imberger 1993). Zic
(1990) reported that as the mean observed diameter of the
bubbles in an air-bubble plume is about 2 mm, a commonly
chosen value for slip velocity is 0.3 m s–1 (also see Kobus
1968; McDougall 1978; Goossens 1979; Milgram 1983).
However, Wüest et al. (1992) reported – ‘since the gas
exchange rate (fractional volume changes per unit time)
281
depends on the surface-to-volume ratio of the bubbles, the
oxygen dissolves faster in the lake if the bubbles are small.
In addition, smaller bubbles have lower slip velocity and thus
longer contact time with the water before reaching the depth
of maximum plume rise.’ Based on this, Wüest et al. (1992)
recommended 0.8 mm ≤ R ≤ 1 mm as the best choice for the
case of efficient oxygenation and artificial mixing of the
Baldeggersee Lake during summer using oxygen as the
source. The same logic applies if air is used as the source
instead of oxygen (i.e. a bubble size between 0.8 and 1 mm
is appropriate).
The mechanical efficiency ηmech (%) and destratification
time per unit area Γ (s m–2) for M equal to 1 and R values
equal to 1 mm, 1.5 mm, 3 mm and 6 mm are presented in
Figs 11 and 12 as a function of G.
The peak mechanical efficiency ηmech is found to be
relatively insensitive to the changes in bubble radii
considered here as the destratification time per unit area Γ
(s m–2). Demonstrating that changes in the air-bubble radius
and slip velocity can be neglected when modelling ηmech
and Γ.
However, the oxygen-dissolution efficiency is somewhat
more sensitive and reduces as the bubble radius R increases
(Sahoo and Luketina 2003). Since the gas-transfer
coefficient for oxygen and nitrogen is independent of bubble
size greater than 0.67 mm (Wüest et al. 1992), and it is
difficult in practice to maintain exact initial bubble size
through out the operation, an average bubble radius of
around 1 mm should be considered for design purposes.
12
10
Mechanical efficiency ηmech (%)
Mechanical efficiency η mech (%)
Bubble-plume behaviour and destratification
8
6
4
2
0
1.E+00
1.E+01
1.E+02
G
1.E+03
1.E+04
Fig. 11. Mechanical efficiency ηmech (%) as a function of G for
bubbler radii R equal to 1 mm, 1.5 mm, 3 mm (thick black line since
these cases have identical results) and 6 mm (medium grey line) for a
linearly stratified reservoir for M equal to 1. G values greater than
6300 (thick black dotted line) are not realizable in practice.
282
Marine and Freshwater Research
G. B. Sahoo and D. Luketina
Destratification time Γ (s m–2)
1.E+02
1.E+01
1.E+00
1.E+00
1.E+01
1.E+02
G
1.E+03
1.E+04
Fig. 12. Mixing time per unit surface area Γ (s m–2) as a function of
G for bubbler radii R equal to 1 mm, 1.5 mm, 3 mm (thick black line
since these cases have identical results) and 6 mm (medium grey line)
for a linearly stratified reservoir for M equal to 1. G values greater than
6300 (thick black dotted line) are not realizable in practice.
Conclusions
The two dimensionless parameters, M, the source strength,
and C, the stratification strength, are known to govern
bubble-plume behaviour. However, the use of C and M as the
non-dimensional variables is somewhat inconvenient as both
contain the airflow rate. Here, a new dimensionless group for
the stratification strength known as G is defined. G is
independent of the airflow rate and this means that the range
of practicable G values in lakes and reservoirs can be
determined. For example, G values of 1000 and 4100 are not
likely to be exceeded in tropical and semi-tropical reservoirs,
respectively, and any thermally stratified reservoir or lake
less than 100-m deep is not likely to have a G value in excess
of 6300.
The primary objective of bubbler design is to find the
optimum design conditions. Two indicators of bubbler
performance, the mechanical efficiency ηmech (%) and the
mixing time per unit area Γ (s m–2) were analysed as
functions of M and G by using a one-dimensional
bubble-plume model. The numerical error in the model when
estimating the rate of change of potential energy ∂PE/∂T is
considerably reduced if the neutral-buoyancy layer is
selected as the datum.
General equations (Equations 21 and 23) to estimate
optimum bubbler design conditions i.e. airflow rate Qo (via
M) and corresponding mechanical efficiency ηmech and
destratification time per unit area Γ (s m–2) for a known
linear or an equivalent linear stratification G in a reservoir of
any shape have been derived. In cases where the airflow rate
may be adjusted (i.e. some form of real time control) as the
reservoir destratifies, it is recommended that the bubbler be
designed to operate at the first peak mechanical efficiency
ηmech (%).
While the use of G simplifies the design process, a
suitable design value of G can be difficult to determine
owing to natural variability. Examining the sensitivity of
ηmech and Γ (s m–2) to changes in G, it is found that the
mechanical efficiencies are lower and destratification times
are longer if the design G is higher than the actual G. Since
it can be difficult to accurately determine the actual G, a
conservative practice would be to reduce the design values of
G by nearly 10% if attempting to operate at the first (or
second) peak.
The mechanical efficiency is found to be substantially
less for a single-step stratification case than its equivalent
linear stratification. This is because the potential energy
change due to mixing can occur at the step (i.e. the interface
between the upper and lower layers). A double-step
stratification will have higher mechanical efficiencies that
can, in some cases, approach the efficiency of its equivalent
linear stratification. Therefore, it is inferred that use of the
equivalent linear method with design charts as proposed by
Schladow (1993) is valid only for linear or approximately
linear stratification cases and may lead to sub-optimal design
if applied to step-stratification cases. In a step-stratification
case, a simple bubble-plume model is more useful in finding
the optimum design criteria.
The mechanical efficiency ηmech (%) and destratification
time per unit surface area Γ (s m–2) are analysed as a function
G for different bubble radii and it is found that ηmech (%) and
Γ (s m–2) are relatively insensitive to the bubble radii.
However, in order to maintain a suitable oxygen dissolution
efficiency, a bubble size of 1 mm is recommended.
Acknowledgments
This work was carried out under the Asian Institute of
Technology–ONDEO Services collaborative agreement. A
PhD scholarship awarded to the first author by the French
Ministry of Foreign Affairs is gratefully acknowledged.
References
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Manuscript received 9 April 2002; revised and accepted 10 April 2003.
284
Marine and Freshwater Research
G. B. Sahoo and D. Luketina
Appendix 1
The rate of change of potential energy (∂PE/∂t)
During instantaneous bubbler operation, each plume detrains at the top of the plume and entrains water below this. The detrained fluid enters the
ambient layer that has the same density, referred to as the neutral-buoyancy layer. Due to this entrainment and detrainment process, the ambient
neutral-buoyancy layer becomes thicker, ambient layers below the neutral-buoyancy layer shift downward and all ambient layers above the
neutral-buoyancy layer shift upwards, causing an increase in the potential energy of the water column. There is no change in potential energy of the
neutral-buoyancy layer itself. The rate of change of the potential energy of the water column is formulated in the remainder of this sub-section.
The potential energy of a water column of depth h consisting of stratified water layers is:
h
PE = ∫ ρAgξdz
(A -1)
0
where ρ is the density of the fluid in the layer of thickness dz at height ξ above the reservoir bed and z is positive upwards. For an Eulerian reference
frame, ξ is invariant with time and the rate of change of potential energy of a water column of depth h having a cross-sectional area A = γζϕ with
height ξ where γ and ϕ are constants, is then:
h ∂ρ
h ∂ρ

∂PE ∂  h
∂PE
∂PE
=  ∫ ρAgξdz  ⇒
= g ∫ γξϕ+1dz ⇒
= g∫
uξdz
∂t
∂t  0
∂
t
∂
t
∂
t
0
0 ∂z

(A - 2)
where u, the velocity at which fluid in an ambient layer at height ξ falls is given by:
u = ∂z ∂t z=ξ =
1 ξ
∫ ( −q m )dz
γξϕ 0
(A - 3)
where qm is the entrainment flow rate per unit height (m2 s–1).
Equation A-2 shows that the instantaneous rate of potential energy is independent of the shape of the reservoir.
Equations A-2 and A-3 are invariant to the choice of datum, however, there is less numerical error in evaluating Equation A-2, if the datum is
selected to coincide with the neutral-buoyancy layer since u changes rapidly in the vicinity of the neutral layer. It can be shown that Equation A-2
can be applied separately to each plume in a cascade and that the rate of change of potential energy due to each plume can be summed to find the
rate of change of potential energy of the water column. Further, the neutral-buoyancy layer of each plume is used as the datum for that plume.
Equation A-2 can be written in discrete terms for the Pth plume extending from layer LP-1 to the layer LP at the height of rise of the plume as:
i=LP  ρ − ρ  

i
∂PE P
≈ g ∑  i−1 i+1  ξi  − ∑ q j  

2
∂t
i=LP−1 
  j=LP−1  

(A - 4)
where ξi is the mid-depth height of the ith ambient layer above the neutral-buoyancy layer, ρi is the density of the ith ambient layer, qj is the entrainment
(m3 s–1) from the jth layer of the reservoir, and L0 is the layer containing the diffuser. In order to satisfy continuity, the entrainment qj, is determined
from:
q = q
plume, i
 j

q j = q j = q plume, i − q plume, L
P

q j = 0

∀ i ≠ Ln , LP
∀ i = Ln
(A - 5)
∀ i = LP
where Ln, is the neutral-buoyancy layer, qplume,i and qplume,LP represent the plume entrainment at the ith layer and detrainment at the top of the Pth
plume from layer LP respectively.
It can be shown that it is valid to apply Equation A-4 separately to each plume in a cascade and the rate of change of potential energy due to each
plume can be summed to find the rate of change of potential energy of the whole water column so that:
N  i=L
NP ∂PE

 
i
ρ − ρi+1 
∂PE
P ≈ ∑P g ∑P  i−1
= ∑

 ξi  − ∑ q j  



2
∂t
P=1  i=L
P=1 ∂t

 j=LP−1  
P−1 

(A - 6)
where NP is the total number of plumes formed in the plume cascade within the water column h.
The maximum change in potential energy (∆PEmix)
Considering the reservoir bed as the datum, the initial and fully mixed state potential energy per unit surface area of any layer can be calculated.
Therefore, the maximum change in potential energy (∆PEmix) per unit surface area (As = γhϕ) over the whole water column h is:
i=L
h
∆PE mixed
h
= gh −ϕ ∫ ( ρm − ρ) ξϕ+1 dz ≈ gh −ϕ ∑ ρm − ρi
As
i=L0
0
(
)
ξiϕ+1∆z
(A - 7)
Bubble-plume behaviour and destratification
Marine and Freshwater Research
285
where Lh is the surface layer of the reservoir, ∆z is the layer thickness and ρm, the homogenous density of a fully mixed reservoir (i.e. after sufficient
bubbler operation), is given by:
ρm =
i=L
h ϕ
1h
1
∑ γξ ρ ∆z
∫ ρAdz ≈
V0
V i=L0 i i
(A - 8)
where V is the volume of the reservoir.
The maximum change in potential energy (∆PEmix) for linear stratification
For a linear stratification of buoyancy frequency, N2 = –(g/ρr)∂ρ/∂z, the density ρ (kg m-3) can be approximated by, ρ = ρb(1–N2ξ/g), (where ρr has
been approximated by the density at the bed ρb. The volume of a reservoir having a cross-sectional area A = γξϕ is:
h
h
0
0
V = ∫ Adz = ∫ γξϕ dz =
γ
h ϕ+1
ϕ +1
(A - 9)
Substituting ρ = ρb(1–N2ξ/g) and Equation A-9 into Equation A-8 yields:
 ( ϕ +1) N 2 h 
ρm = ρb 1 −

 ( ϕ + 2) g 
(A -10)
Substituting ρ = ρb(1–N2ξ/g) and Equation A-10 into Equation A-7 gives the maximum change in potential energy (∆PEmix) per unit surface area
(As = γhϕ) over the whole water column h as:
 1
∆PE mixed
( ϕ +1) 
= h 3ρb N 2 
−
As
ϕ
+
3

( ϕ + 2) 2 

(A -11)

Combining Equation 14, the definition of the buoyancy frequency and Equation A–11 gives:
∆PE mixed
G2 3  1
( ϕ +1) 
= 1.185 h 3g 2 
−
As
 ϕ + 3 ( ϕ + 2) 2 
H


(A -12)
For a specific linear stratification of a reservoir of depth h, H is constant and thus 1.185h3gG2/3/H2 is constant so that ∆PEmixed/ As is a function of
ϕ only.
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