w a t e r r e s e a r c h 4 3 ( 2 0 0 9 ) 3 1 1 7 – 3 1 2 3

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Heat flux through a geothermally heated fluidized bed at thebottom of a lake

Xavier Sanchez*, Elena Roget, Jesus Planella

Department of Physics, University of Girona, Montilivi Campus, 17071 Girona, Spain

a r t i c l e i n f o

Article history:

Received 7 January 2009

Received in revised form

11 April 2009

Accepted 16 April 2009

Published online 3 May 2009

Keywords:

Convective plume

Fluidized bed

Microstructure measurements

Heat flux parameterization

Turbidity

Water quality

* Corresponding author.E-mail address: xavier.sanchez@udg.edu

0043-1354/$ – see front matter ª 2009 Elsevidoi:10.1016/j.watres.2009.04.027

a b s t r a c t

Heat fluxes and the underground inflow through a natural fluidized bed within the main

sub-basin of Lake Banyoles are studied and parameterized. In the upper part of this

fluidized bed, at a depth of about 30 m, the vertical gradients of particle concentration and

temperature are very sharply located within an interface a few centimeters thick. Within

this interface (lutocline), the depths where the temperature and the concentration gradi-

ents are maximum match exactly. On the other hand, the lutocline determines a flat,

horizontal surface dividing the water column into a hot, turbid medium at the bottom and

clear, colder, bulk water above. Through this interface the flow regime also varies from

being laminar just below it, to turbulent due to convective processes developing above it.

More precisely, in studied main sub-basin a buoyant plume develops above the lutocline,

as a result of the heat flux, and affects the lake’s water quality due to particles dragged

along by it. In this paper it is proposed to determine the temperature at the depth of

maximum gradient within the interface by means of measured temperature profiles, and

consider the stationary heat transport equation in the laminar region below it, in order to

obtain the water velocity and the heat flux. Heat flux parameterization is given based on

a large number of thermal high-resolution profiles, covering six campaigns in different

years and seasons. Furthermore, and in consideration of the fact that high-resolution

thermal profiles are not always available, some alternative parameterizations for the heat

flux are presented based only on the temperature of the fluidized bed and that of the lower

hypolimnion.

ª 2009 Elsevier Ltd. All rights reserved.

1. Introduction upward water inflow to the lake, small particles – in the order

Lake Banyoles, with a surface area of 1.2 km2, is the largest of

a series of five lakes (some intermittent) located in the same

hydrographic karstic basin in Catalonia (Spain), and the

second largest natural lake in Spain. About 95% of its total

water inflow (about 1 m3/s) comes through several warm

sources located in collapsed conic sub-basins formed on the

lake’s bottom. The main aquifer, a limestone layer under

gypsiferous rock, is corroded and, due to the underground

(X. Sanchez).er Ltd. All rights reserved

of micrometers – remain in suspension within a bottom layer

in the sub-basins. Inside these fluidized beds, and under

stationary conditions, the particle sedimentation velocity –

a decreasing function of the particle concentration – is coun-

terbalanced by the advective velocity of water inflow. So,

when the water inflow increases, the fluidized bed expands,

and its upper localized interface – the lutocline – rises (Casa-

mitjana and Roget, 1993). In the main basin, named BI, the

lutocline, under stationary conditions, is located at a depth of

.

w a t e r r e s e a r c h 4 3 ( 2 0 0 9 ) 3 1 1 7 – 3 1 2 33118

about 30 m and the inflow velocities are of the order of

magnitude of 10�6–10�5 m/s (Roget et al., 1994). In this basin,

the temperature of the underground inflow is constant all year

round at about 19 �C, and exceeds by up to 9 �C the bottom

temperature in winter, so the geothermal heat flow into the

lake is high (up to 24 MW in winter; Roget et al., 1993), and

greatly affects its heat budget (Casamitjana et al., 1988).

Moreover, as numerical simulations have shown (Casa-

mitjana et al., 1993), the warmer underground inflow greatly

modifies the thermal structure of the whole lake.

The dynamics above the lutocline are dominated by

a convective plume up to the depth where the plume reaches

its neutral buoyancy level (Colomer et al., 2001). Small parti-

cles (5–10 mm in diameter) are entrained from the lutocline

and dragged upwards, modifying the clarity of the water

column. Particle volume concentration, which ranges from 5

to 10 ml/l and depends on the convective strength, affects not

only the water quality but also the ecology of the lake, in terms

of fish distribution (Serra et al., 2002) and phytoplankton

photosynthetic activity.

The influence of suspended solids on water quality and

aquatic biota has recently been emphasized by Bilotta and

Brazier (2008). There are also various studies of lakes heated

by geothermal heat fluxes and how this affects water quality.

This is the case with Lake Vostok (Thoma et al., 2008), Lake

Baikal (Golubev, 2000) and Lake Rotomahana (Whiteford and

Graham, 1994). Other studies of geothermal heat flux in the

ocean are those carried out at the Cascadian Basin (Thomson

et al., 1995) and the Juan de Fuca ridge (Wheat et al., 2004).

How turbidity in the ocean is affected by submarine ground-

water discharges (jets) through permeable sediments has

been recently studied by Crusius et al. (2008).

The effects of heat and turbidity on water quality, and their

variability within the lake in the case of Lake Banyoles,

depend to a great extent on the magnitude of the water inflow,

which can be obtained for each sub-basin by analyzing water

samples taken just below the lutocline (Roget et al., 1994). In

Section 3.1 of this paper, we present a quick, alternative

method for determining inflow velocity at the different

underground springs and the associated heat flux. This

method is based on high-resolution temperature profiles,

which are easier to obtain than water samples. In Section 3.3

the phreatic heat flux (computed in Section 3.2) is parame-

terized using high-resolution temperature data. Section 3.4

presents an alternative heat flux parameterization based on

the temperature data available on regular temperature

profiles, which can be used when high-resolution thermal

profiles are not available, as is the case with historical data.

Knowledge of the mass inflow to the lake is not only rele-

vant for better understanding of the ecological dynamics of

the lake, it is also an indicator of the state of the aquifer

(Bonacci, 1993). Gathering new series of data with data from

old databases can also show long-term weather variations

and if such issues can be related to climate change.

2. Materials and methods

The results presented in this paper are based on data recorded

during six campaigns spanning 1999 and 2003, when 224

profiles of temperature, pressure and turbidity were recorded

above the main underground bottom sources of the lake,

namely basin BI, covering different seasons of the year. Data

were taken with an MSS profiler (Prandke and Stips, 1998) free

falling at 0.4 m/s and recording at 1024 Hz. Temperature was

measured with a microthermistor with a resolution of

0.002 �Cand a time constant of 7 ms, allowing a spatial reso-

lution of 3 mm. The particle concentration was obtained from

an optical backscatter turbidimeter incorporated into the

same profiler with a space resolution in the order of centi-

meters (Wolk et al., 2002). Because the optical backscatter

response of a turbidimeter depends on the size, shape and

different physical properties of the particle surface, we took

samples in situ and performed the calibration in the labora-

tory to obtain accurate particle concentration data within the

range 0–200 mg/l (Sanchez and Roget, 2007).

Fig. 1 shows an example of (a) the temperature and (b) the

particle concentration profiles above the underground spring

in basin BI. Different layers can be identified moving from top

to bottom in Fig. 1(a): the epilimnion (or mixed surface layer),

the seasonal thermocline, the hypolimnion, the lutocline and

the hot fluidized bed. In this example the fluidized bed is

approximately 1 �C warmer than the hypolimnion. In Fig. 1(b)

the particle concentration has been represented within a scale

up to 15 mg/l to have a better view of the particle distribution

in the hypolimnion. However, particle concentration within

the fluidized bed was of 130 g/l. From this figure it can be

observed how, due to the convective plume mentioned in

Section 1, small solid particles are dragged up to the seasonal

thermocline so that particle concentration in the hypolimnion

is higher than in the epilimnion, ranging from 3.5 to 4.5 mg/l.

A detailed view of the thermal and turbidity vertical

structure at the lutocline is presented in Fig. 2a. Specifically,

the temperature and the vertical temperature gradients are

presented in Fig. 2a and b, respectively, and the particle

concentration and the vertical particle concentration gradi-

ents in Fig. 2c and d. The temperatures highlighted in Fig. 2a

with the vertical dotted lines T0, T1 and T2 are, respectively, the

temperature of the homogeneous fluidized bed, the temper-

ature at the maximum thermal gradient depth within the

lutocline – a horizontal dashed line in the figure – and the

temperature in the lower hypolimnetic region. Note in Fig. 2b

that the temperature gradient varies from 20 �C/m at the

center of the lutocline, where temperature is T1, to 0 �C/m

immediately below the lutocline, where the temperature is T0.

Temperatures within the fluidized bed – which exceeds 20 m

(Casamitjana et al., 1996) – can however vary slightly in depth

but locally its gradient can be neglected as, within the preci-

sion of this study, it is zero. A comparison of Fig. 2a and

b reveals that at the depth where the maximum thermal

gradient occurs there is an inflexion point in the temperature

profile, which changes from concave to a convex behavior.

Hereafter, we will call this inflexion depth MG (Maximum

Gradient). These kinds of profiles, with an inflexion point, are

typical of cases where there is a discontinuity between two

different mediums at different temperatures (Incropera and

Dewitt, 1996). Accordingly, we can think of the lutocline as

a horizontal mathematical surface dividing the warm fluid-

ized bed below it from the cold, lowest hypolimnion above it.

Considering the lutocline as a warm surface at T1, we obtain

0 5 10 15Concentration (mg/l)

17 18 19 20 21 22 23 24

30

25

20

15

10

5

0

Temperature (ºC)

Dep

th (

m)

Seasonal thermocline

Epilimnion

Hypolimnion

Hot fluidized bed

Lutocline

a b

Fig. 1 – An example of temperature (a) and particle concentration (b) profiles.

w a t e r r e s e a r c h 4 3 ( 2 0 0 9 ) 3 1 1 7 – 3 1 2 3 3119

the convective fluxes from the lutocline to the hypolimnion at

T2 with the heat transfer coefficient approach used in engi-

neering sciences for convective heat fluxes above a hot plate

(Burmeister, 1993). The convective transfer parameter can be

thought of as the thermal resistance of a layer of fluid between

the heat transfer surface, at T1, and the hypolimnion layer, at

T2. This approach is discussed in Sections 3.3 and 3.4.

First, however, in Section 3.2 heat fluxes are obtained from

the temperature transport equation. Below the inflexion point

shown in Fig. 2a the flow regime is non-turbulent (Roget et al.,

1994) so that thermal transport is driven by the effects of

molecular diffusion plus a vertical advection of temperature

due to the water inflow. Note that the stability of the upper

interface of the fluidized bed (about 200 m in diameter) within

the entire spring basin, and the fact that the lutocline is very

localized in depth, support the idea of a laminar regime below

the lutocline. Above it, however, the existence of thermal

fluctuations – see the small scale variability in the thermal

gradient in Fig. 2b – shows the existence of a turbulent

convection regime (Lozovatsky et al., 2005). In this paper, we

will focus on the thermal profile in the laminar region in order

to determine the inflow velocity. Turbulent fluxes based on

microstructure data were obtained by Sanchez and Roget

(2007) in another sub-basin when a double diffusive interface

was present.

3. Results and discussion

3.1. Underground inflow velocity

Within the advective–diffusive region below the thermal

inflexion depth (MG), the laminar heat flux FQ is given by

(Kundu, 1990):

FQðzÞ ¼ �rcpkTdTðzÞ

dzþ rcpTðzÞU (1)

where z is the vertical coordinate pointing upwards, U is

the velocity of the inflow, r (103 kg/m3) is the density, cp

(4.2� 103 J/kg K) the heat capacity at constant pressure, kT

(1.4� 10�7 m2/s) is the thermal diffusivity and TðzÞ is the

temperature of the water column at depth z. The overbar

in T indicates that it refers to a mean profile, disregarding

the turbulence fluctuations we are measuring with the

microthermistor mounted in the profiler we used (see

Section 2).

If we evaluate Eq. (1) at the inflexion depth (MG), then (1)

can be written as:

FQðzMGÞ ¼ �rcpkTdTðzÞ

dz

����MG

þrcpT1U (2)

where according to Fig. 2a, TðzMGÞhT1 and U is the velocity at

the MG depth.

On the other hand, Eq. (1) at the zone located immediately

below the lutocline and within the fluidized bed, reduces to:

FQ ¼ rcpT0U (3)

where the flow velocity, U, can be considered to be equal to

that in the core of the lutocline (MG depth) because of its small

extension – for the six campaigns analyzed the thinness of the

lutocline ranged from 10 to 30 cm. That is, the U in Eqs. (2) and

(3) are the same. The temperature at Eq. (3), T0, is the

temperature of the fluidized bed immediately below the

lutocline where, as discussed in Section 2, the thermal

gradient can be neglected; that is why the diffusive term in the

general Eq. (1) was not considered in Eq. (3).

Finally, assuming stationary conditions, the heat flux

immediately below and within (at MG depth) the lutocline

18 19−29.75

−29.7

−29.65

−29.6

−29.55

Dep

th (

m)

−40 −20 0 −2 −1 0 0 100 200

a b c d

MG

T0T1T2

ºC ºC/m mg/l ×104 mg/l/m

C. gradientConcentrationT. gradientTemperature

Fig. 2 – Magnification of the lutocline depth of Fig. 1, with the temperature (a), gradient (b), particle concentration

(c) and vertical gradient (d) profiles. The MG depth and the T0, T1 and T2 temperatures are shown.

w a t e r r e s e a r c h 4 3 ( 2 0 0 9 ) 3 1 1 7 – 3 1 2 33120

must be the same so that, equating Eqs. (2) and (3), we obtain

the inflow velocity at the lutocline level:

U ¼ �kTdT=dzjMG

ðT0 � T1Þ(4)

For the six campaigns considered in this study, we have calcu-

lated T0, T1, T2, DT ¼ ðT0�T2Þ, DT0 ¼ ðT1 � T2Þ and dT=dzjMG as

mean values of all the profiles recorded for each campaign.

These results are summarized in Table 1 (where the first column

contains the campaign dates). In addition, in column 8 the

values of U obtained from Eq. (4) for each campaign are pre-

sented. As observed, the values of U range from 4.0� 10�6 to

6.4� 10�6 m/s, with a mean and standard deviation of

(5� 1)� 10�6 m/s. These values are in agreement with those

obtained by Roget et al. (1993, 1994) using a different approach.

Moreover, the goodfit ofour data to the standard Newton law for

the heat flux, which will be presented in Section 3.3, also indi-

rectly corroborates the computed inflow velocities based on Eq.

(4). In Table 1 there are three more columns containing the

Table 1 – The columns show the campaign dates, the temperatat the MG depth, and the calculated values, U, FQ and Pe.

T0 [�C] T1 [�C] T2 [�C] DT [�C] DT0 [

July 13, 1999 19.27 16.70 15.44 3.83 1.2

March 3, 2000 19.18 12.65 10.25 8.93 2.4

May 25, 2000 19.27 15.76 14.07 5.20 1.6

June 21, 2002 19.25 16.65 15.82 3.43 0.8

September 16, 2003 19.02 18.17 17.94 1.08 0.2

October 21, 2003 19.04 18.37 18.19 0.85 0.1

computednet heat fluxes and the corresponding Peclet number.

These magnitudes will be commented on in Section 3.2.

3.2. Heat flux

Following Eq. (3) and considering the bulk temperature of the

lake T2, the net heat inflow (Roget et al., 1994) through the

main underground source (BI) in Lake Banyoles during

the campaigns studied ranges between 14 and 170 W/m2

depending on the season and the year. This can be observed

from Table 1, where FQ¼ rCrU(T0� T2) for individual

campaigns are presented in columns 9. These values are in the

range of those obtained by Casamitjana et al. (1988) and Roget

et al. (1993) for the same lake and sub-basin, using another

methodology. They are also of the same order as those of

the geothermal heat fluxes in Lake Rotomahana, where

they range from 10 to 48 W/m2 (Whiteford and Graham, 1994).

In other lakes, however, geothermal fluxes are much lower –

0.07 W/m2 in the case of Lake Baikal (Golubev, 2000).

ures T0, T1, T2, DT, DT0, the modulus of the thermal gradient

�C] jdT=dzjMG [�C/m] Uð�10�6Þ [m/s] FQ ½W=m2� Pe

6 117 6.4 102 0.49

0 213 4.6 170 0.37

9 157 6.3 136 0.48

3 81 4.4 63 0.32

3 25 4.1 19 0.27

8 19 4.0 14 0.27

w a t e r r e s e a r c h 4 3 ( 2 0 0 9 ) 3 1 1 7 – 3 1 2 3 3121

The Peclet number at the MG depth was computed for all

the campaigns and their values are shown in the last column

of Table 1. The Peclet number was defined as the ratio

between the net advective and the diffusive components of

the heat flux within the lutocline at the level of the maximum

temperature gradient (MG depth):

Pe ¼ UDT0

�kTdT=dzjMG

(5)

As can be observed in Table 1, Peclet values range between 0.3

and 0.5, indicating that at the MG depth the diffusive contri-

bution is at least twice the advective one.

3.3. Heat flux parameterization based on high-resolution temperature data profiles

Classically, convective transfer has been parameterized by

power laws such as

FQ ¼ CDT0p (6)

where C is a constant and p is the scaling exponent (Turner,

1973). Following this approach, we have parameterized the

heat flux as a function of DT0 ¼ T1 � T2, which is the temper-

ature difference between the MG depth within the lutocline –

considered a warm surface at T1 (see Section 2) – and the

hypolimnion. Note that T1, and therefore DT0, depend on the

forcing conditions, i.e. the velocity U, and the temperature of

the underground inflow T0.

In order to verify the potential law (6), in Fig. 3 we have

represented, in logarithmic scales, the heat flux FQ versus DT0

for the six campaigns studied. As observed, the correlation

between both variables follows a linear trend very closely

(r2¼ 0.998), thus confirming the potential law model. The

lineal fit (in logarithmic representation) within a 95% confi-

dence interval, is:

log10ðFQÞ ¼ ð0:98� 0:06Þlog10ðDT0Þ þ ð1:875� 0:029Þ (7)

and the heat flux therefore follows a power law, as established

in Eq. (6):

FQ ¼ ð75� 5ÞDT0ð0:98� 0:06Þ (8)

10−1 100

101

102

ΔT’ (ºC)

FQ

(W

/m2 )

Experimental data

Linear fit: r2=0.998

October 21, 2003

September 16, 2003

June 21, 2002

July 13, 1999

May 25, 2000March 3, 2000

p=0.98±0.06

Fig. 3 – Heat flux FQ as a function of the MG relative

temperature DT0, with logarithmic scales. Also shown is the

best linear fit to obtain a potential law, with exponent p.

where the numeric factors in (7) and (8) are in standard SI

units.

It is interesting to see that the obtained exponent

p¼ 0.98� 0.06 is close to 1, which is a value widely used in

engineering for convective heat transfer from a hot plate to its

surroundings. This approach is also known as Newton’s Law

(Incropera and Dewitt, 1996).

If we assume that our data follows Newton’s Law, that is if

we consider p¼ 1 to be a good approach, and force a linear

trend ( p¼ 1), we obtain:

FQ ¼ ð73� 6ÞDT0 (9)

with a good correlation coefficient r2¼ 0.990. The multiplica-

tive term in Eq. (9) is generally called the convective heat transfer

coefficient. This coefficient, which in general depends on fluid

properties (the Prandtl number) and the Reynolds number for

forced convection is equal to (73� 6) W/m2 K in our study.

Relatively low values of the coefficient – in the order of under

100 – are usually found in natural convection (Incropera and

Dewitt, 1996).

That small variation of the convective heat transfer coef-

ficient (6/73) indicates that the background dynamic processes

above the lutocline in basin BI are similar throughout the year.

(The campaigns included in this study cover all four seasons

of the year.) Generalization of this law to other sources

cannot, however, be taken for granted as they are located at

different sub-basins within the lake where the background

dynamics can be different (Casamitjana and Roget, 1990).

3.4. Heat flux parameterization based on standardtemperature data profiles

In Eqs. (8) and (9) the heat flux has been parameterized as

a function of the temperature jump between the hypolimnion

and the temperature at the depth of the maximum thermal

gradient within the lutocline, which is DT0. However, high

spatial resolution temperature data allowing the DT0 to be

known are often not available. This being the case, it would be

interesting to be able to obtain the heat flux as a function of

DT ¼ T0 � T2, i.e. the net thermal jump between the fluidized

bed and the hypolimnion.

In Fig. 4, like in Fig. 3, we have represented the heat flux FQ

versus DT for the six campaigns on a logarithmic scale. As can

be observed and as was expected, the DT parameterization

has a worse correlation (r2¼ 0.973) than the DT0 parameteri-

zation (r2¼ 0.998; Fig. 3). It is interesting, however, that even in

this case, the correlation is still good. We can, thus, obtain an

alternative parameterization to estimate the heat flux based

on measurements that are not high-resolution. Specifically,

exact lineal regression between the logarithms of FQ and DT,

within the 95% confidence level, is:

log10ðFQÞ ¼ ð1:1� 0:2Þlog10ðDTÞ þ ð1:23� 0:23Þ (10)

and so

FQ ¼ ð17� 9ÞDTð1:1�0:2Þ (11)

where the multiplying prefactor in (10) and (11) are in standard

SI units.

100 101

101

102

ΔT (ºC)

FQ

(W

/m2 )

Experimental data

Linear fit: r2=0.973

October 21, 2003

September 16, 2003

June 21, 2002

July 13, 1999

May 25, 2000March 3, 2000

p=1.1±0.2

Fig. 4 – As in Fig. 3, but with the relative temperature of the

homogeneous part of the fluidized bed DT.

w a t e r r e s e a r c h 4 3 ( 2 0 0 9 ) 3 1 1 7 – 3 1 2 33122

If we force a perfect linear trend between the heat flux FQ

and the total thermal increment DT (that is, we assume an

exponent p¼ 1), we still find a fairly good dependence

although with a lower correlation coefficient (r2¼ 0.920).

Therefore, in this case, the Newtonian law will be

FQ ¼ ð20� 11ÞDT (12)

where the convective heat transfer coefficient in Eq. (12) is equal to

(20� 11) W/m2 K.

In fact, comparison of Eq. (12) with FQ¼ rCvUDT corroborates

that, within an error of 55% for the heat flux, we could consider

a constant velocity inflow for basin BI of U¼ 5� 10�6 m/s. This is

in accordance with the data used for the parameterizations we

present here, from which the mean values and standard devi-

ation of U and DT are, respectively (5� 1)� 10�6 m/s and

(4� 3) �C. Note that the relative variability of U with respect to

the mean is only 20% and that of DT is 75%.

In fact is has been widely documented that inflow velocity

at source BI remains fairly constant. See Soler et al. (2007) for

a description of isolated non-stationary events.

4. Conclusions

Detailed analysis of high-resolution vertical thermal profiles at

the depth of the upper interface of a fluidized bed – the lutocline

– has allowed us to obtain the velocity of the water inflow from

U ¼ �kTdT=dzjMG=ðT0 � T1Þ, where kT is the thermal diffusivity,

dT=dzjMG is the thermal vertical gradient within the lutocline at

the depth where it is maximum (MG), and T0 and T1 are the

temperatures inside the fluidized bed and in the MG depth.

This method allows the obtention of the inflow of the

underground sources in Lake Banyoles which keeps a fluid-

ized bed at the bottom of the sub-basins where they are

located. Because recording temperature profiles is easier than

taking the water samples that are needed with the method

used up to now to determine the velocity inflow, results pre-

sented here will facilitate the control of the system and of the

state of the aquifer. This result is completely general and it

can be applied at any of the 11 sources of Lake Banyoles or any

fluidized bed under stationary conditions.

Net phreatic heat fluxes at basin BI have been parameter-

ized as FQ ¼ ð75� 5ÞðT0 � T1Þð0:98�0:06Þ which, within the vari-

ability of our results, can be considered to have a power

coefficient equal to one. This approach corresponds to New-

ton’s law of convection, which is largely used in engineering.

In our case the convective heat transfer coefficient was found

to be within the characteristic range of the natural convective

processes. Although similar approaches can be used for the

other underground sources in Lake Banyoles, the different

dynamics of the sub-basins of the lake might affect the heat

transfer coefficient.

All in all, the possibility of exactly determining the

underground inflow based on the formula for the inflow

velocity which has been presented in this paper greatly facil-

itates the fitting and validation of the numerical models

implemented for Lake Banyoles (see the example given in

Section 1) and will help improve forecasts of particular

features in karstic lakes.

Acknowledgements

This work has been partly supported by the FIS2008-03608

project of the Spanish Science and Innovation Ministry.

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