OPTIMIZATION OF ICEBERG TOWING VELOCITY FOR WATER SUPPLY

Simon Lefranvoisl, Philippe Doyon-Poulin2, Louis Gosselin], Marcel Lacroix2

IDepartement de genie mecanique, Universite Laval, Quebec City, Quebec, Canada, G lK 7P42Paculte de genie, Universite de Sherbrooke, Sherbrooke, Quebec, Canada, 11 K 7P4

Contact: Louis.Gosselin@gmc.ulaval.ca

Received July 2008, Accepted December 2008No. 08-CSME-27, E.LC. Accession 3065

ABSTRACT

A mathematical model for determining the optimum towing velocity of tabular icebergs ispresented. The optimization problem is formulated in terms of a benefit function that takes intoaccount the ice mass delivered and the total fuel consumption for the tow. Results indicate thatthe optimum towing velocity is mainly affected by the water-to-fuel cost ratio. It is shown thattowing icebergs from Antarctica to South Africa is a profitable way of supplying fresh-waterprovided that the towing velocity is optimized with the proposed method.

OPTIMISATION DE LA VITESSE DE REMORQUAGE D'UN ICEBERG POUR LAPRODUCTION D'EAU POTABLE

RESUME

Un modele mathematique pour determiner la vitesse de remorquage optimal d'un icebergtubulaire est presente. Le probleme d'optimisation est formule en termes d'une fonction beneficequi prend en consideration la masse de glace restant ala fin du parcours et la consommation decarburant requise pour Ie remorquage. Les resultats montrent que la vitesse optimale deremorquage est surtout influencee par Ie rapport du cout de l'eau sur celui du carburant. II estmontre que Ie remorquage de l'Antarctique vers Ie sud de l'Afrique peut etre profitable lorsquela vitesse de remorquage est optimisee par la methode proposee.

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1. INTRODUCTION

It is estimated that only 3% of the water on the planet is fresh-water [I]. About one-third of the fresh­water is in underground aquifers, lakes and rivers. The remainder of the fresh-water is locked up in icecaps and glaciers. Thus, it is not surprising that the idea of transporting large icebergs to arid regions toprovide a source of fresh-water is attractive. Several investigations have focused on problems associatedwith iceberg properties and natural dynamics, and their towing and melting in transit [2-5]. None of theseinvestigations has considered, however, the transport of icebergs as an optimisation problem. Indeed,while in transit, the iceberg suffers melting losses and the shorter the transit times, the larger the mass ofice delivered. Increasing the towing velocity augments, however, the drag and, consequently, the fuelconsumption. Therefore, the towing velocity should be carefully chosen so that the economic benefit ismaximized, that is a maximum amount of ice is delivered for a minimum amount of fuel consumed.Determining the optimum towing velocity of icebergs is the main objective of the present study.

In the next sections, a mathematical model for the towing of tabular icebergs is presented. Next, theoptimization problem is examined. A parametric study is then conducted and a case study for the towingof icebergs from Antarctica to South Africa [6] is presented.

2. PROBLEM STATEMENT

Icebergs of Antarctic origin are invariably tabular in shape, that is, they have flat horizontal topsurfaces. Therefore, for the sake of the present analysis, it is assumed that the iceberg is a largeparallelepiped. The mathematical model developed here could however be easily extended to othericeberg shapes. The iceberg is towed at an average velocity V over a distance d. Once it has reached itsfinal destination, the iceberg then serves as a source of fresh-water.

While in transit, the iceberg suffers melting losses. Ice melting is caused by two main actions. The firstcause of melting is absorbed solar radiation and convection heat transfer by the non wetted area of theiceberg, i.e. the top surface and the freeboard sides. The second cause is heat and mass transfer across thewetted area of the iceberg, i.e., the sides and bottom, due to the velocity of movement through the water.As a result, the final mass of the iceberg will be smaller than its initial mass. Now, the income associatedwith selling the fresh-water from the iceberg is proportional to its final mass, Mfina1 :

(1)

where XW is the price expected in dollars for I kg of ice. The fact that Cincome is proportional to the final icemass calls for a rapid tow of the iceberg in order to minimize the mass losses due to melting. On the otherhand, too fast a tow is costly since the fuel consumption increases with the towing velocity. The cost ofthe fuel Cfuel may be determined in the following manner:

(2)

where Xfuel is the price in dollars for I kg of fuel, SFC is the specific fuel consumption in kg/J, and Etrip isthe energy required to drive the iceberg over the entire tow trip. The total benefit of the iceberg tow isobtained by subtracting Eq. (1) from Eq. (2),

(3)

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C total = C income - C fuel

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One may therefore expect an optimum towing velocity that is a velocity which maximizes the totaleconomic benefit by providing an adequate trade-off between the amount of ice delivered and the cost ofthe fuel consumption. In the following sections, a mathematical model is presented for performing thesecalculations and optimizing the iceberg towing velocity.

3. HEAT TRANSFER MODEL

The melting rate of the iceberg is determined by predicting the time evolution of the solid-liquidinterface. Due to the fact that the iceberg is tabular in shape, a shape amenable to towing, conduction heattransfer is assumed to be one-dimensional through the iceberg. The iceberg may then be considered as asemi-infinite body that is a body for which the core temperature is insensitive to the surface conditions.This assumption is supported by on-site observations that have revealed that the surface layers of icebergsexperience steep temperature gradients while the temperature of their core remains constant [3]. Themelting problem may then be split into two parts: First, the melting of the wetted area of the iceberg isanalysed and second the non wetted area is considered. It must be pointed out that in practice, waveerosion has shown to have an impact on iceberg deterioration [3, 7]. For the sake of simplicity, thisphenomenon is ignored here. As a result, the proposed mathematical model will tend to underestimate, tosome extent, the melting rate. In the future however, the model could be improved by taking into accountthe effect of wave erosion.

3.1. Wetted Area of the IcebergThe heat transfer phenomena that take place at the wetted area of the iceberg (the submerged portion)

are depicted in Fig. 1. Conduction heat transfer inside the iceberg is governed by the one-dimensionaldiffusion equation

(4)

The iceberg surface is subjected to convection with the surrounding water. As melting occurs, the meltmixes with the surrounding water and the iceberg interface moves upwards. The temperature of the solid­liquid interface (i.e., x = 0) is the melting temperature of the ice, i.e.,

(5) T(x = O,t) = Tm

According to the semi-infinite body assumption, the internal temperature far from x = 0 is assumedconstant and equal to the initial temperature ofthe body,

(6) T(x -) oo,t) = T(x,t = 0) = T", C

Performing an energy balance at the solid-liquid interface of the wetted area yields the followingdifferential equation for the time-varying position of the interface Su (the subscript u indicates theinterface for the wetted area),

(7)

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A dSu = h (T - T ) + k or IPs dt u "',W m 5 Ox X;O

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AirToo, a

Convection, he

DSolar Radiation, Psun

H

Conduction

Conduction

Too, c (core)

Fig. 1. Heat transfer mechanisms involved in the melting of icebergs.

Eq. (7) may be solved for Su once the temperature distribution within the iceberg has been determinedfrom Eq. (4). An analytical solution for Eq. (4) subjected to the boundary conditions (5) and (6) isavailable in reference [8]. Substituting this solution into Eq. (7) and performing the integration yields thefollowing dimensionless equation for the moving solid-liquid interface at the wetted area of the iceberg:

(8)

where

(9) Bi = huLu k

s

T -TR = OO,c m

II Too w - TmFa = ast

e

3.2. Non Wetted Area of the IcebergThe non wetted area of the iceberg (the emerged portion) gains heat by convection heat transfer from

the surrounding air and by the incoming solar radiation (Fig. 1). Performing an energy balance at thesolid-liquid interface for the non wetted area of the iceberg in a manner similar to the one exposed in theprevious section yields the following differential equation for the time-varying position of the interface Se(the subscript e indicates the interface for the non wetted area),

(10)

For simplicity, the absorbed solar heat flux is approximated with a sin2 -type equation,

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(11)

where the time period D is a full day.Once again, Eq. (10) may be solved for Se with the help of the temperature distribution T(x,t) reported

in Ref. [8]. The resulting dimensionless time-varying position of the solid-liquid interface for the nonwetted area is then

(12)

where

• [ sin (2PFo)] 2R Ste r;:;-Se(Fo)=SteeBieFo+y Fo 2P + j;, e",Fo

(13)

(14)

Bi = heLe k

s

cPoLy=

2APsUs

3.3. Heat Transfer CoefficientsThe convection heat transfer coefficients for the wetted and the non wetted areas of the moving

iceberg were determined using the following well known empirical correlations for forced convectionover flat plates [8],

(15)

with

(16)-h _ NUL,ekae-

L

The Prandlt number for air is Pra = 0.71 and, for water, Prw =9.4 .

3.4. Time-Variation of the Iceberg MassThe dimensionless time-varying iceberg mass is determined with the following expression

(17)

where

(18)

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Fo3

.. ': :::::::::::::::~:::::::':::':{Stew=0.075

0.05

0.1

OL....J.---'-..J.l...--'--l....----'----'-----'----"----'o

0.8

0.6

0.4

.... :'.:'.:: {Re =7.5xI08....... ..... w

0.2 -..... - _.. _. ··········--··5xld

108 _..••.......••..••••••

Fig. 2. Time-varying iceberg mass.

S*(Fo) is calculated from the solid-liquid interfaces for the wetted and the non wetted areas of theiceberg, Su*(Fo) and Se*(Fo), provided by Eqs. (8) and (12) respectively.

As an example, Fig. 2 illustrates the time-varying iceberg mass calculated from Eq. (17) for differentflow and heat transfer conditions. Examination of this figure reveals that melting of the iceberg is stronglydependent on the conditions prevailing in the water, that is the towing velocity (the Reynolds number),and the underwater heat transfer conditions (the wetted Stefan number). On the other hand, the effect ofsolar radiation and of convection heat transfer with the surrounding air on the melting process isinsignificant. For instance, increasing the solar heat flux from 0 to 450 W/m2 augments the melting rateby a mere 0.1 %. This leads to the conclusion that convection heat transfer in the water is, by far, the mostinfluential phenomenon on the melting of the iceberg. Since the water temperature (the Stefan number forthe wetted area) is uncontrollable, the towing velocity (the Reynolds number) becomes the mainparameter on which the towing process can be optimized.

4. OPTIMIZATION OF THE TOWING VELOCITY

As mentioned in Section 2, the objective of the present study is to maximize the benefit of the icebergtow by determining the optimum towing velocity. On one hand, maximizing the iceberg mass at the finaldestination means reducing the heat transfer coefficients by diminishing the towing velocity. On the otherhand, diminishing the towing velocity means that the transit period is prolonged. Delivering the largestamount of ice is only half of the objective. The cost of the fuel consumption by the tow boat must also betaken into account. And this cost is, of course, a function of the towing velocity. The optimizationproblem may then be formulated in terms of a benefit function such as Eq. (3). The income, Cincome, isproportional to the mass of ice M* at time t = dIV, i.e., at Fo = aNd:

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(19) 2 *1M = L Hps M Fo;~Vd

The cost of fuel consumption, Cfuel, accounts for the total amount of energy consumed during the tow, thatis,

(20) EtJip = E ship + Eiceberg

An equation for the tow boat fuel consumption is reported in reference [6]:

(21)

For diesel powered tow boats, the fuel coefficient is Fe = 1.29x 106 m5/kgs2 and the ship displacement forsmall tow boats is estimated at W = 3800 m3 [9]. Thus, with a specific fuel consumption of SFC =

50x 10-6 kg/J [10], the energy required for the tow boat trip may be estimated. As for the energy spent fortowing the iceberg itself, it can be expressed as a function of the iceberg drag and the specific fuelconsumption of the tow boat,

(22)1 M* +1 2

Eiceberg ="2 PwalerLH 2 CoV d

The total drag force acting upon the iceberg may then be calculated with the drag coefficients proposed inreference [4] and the frontal area of the iceberg estimated from the height and the mean thickness of theiceberg in transit (M*+1)/2.

Substituting Eqs. (19)-(22) into Eq. (3), the dimensionless benefit function becomes:

(23) C* = C lotaltotal C

max

where C'total E (-00,1) and M* is given by Eq. (17). The dimensionless benefit function C*total is the ratio ofthe net benefit made from towing the iceberg (real economic benefit) to the maximum theoretical benefitthat is the benefit made ifno fuel were consumed. The towing strategy then is to adjust the velocity of thetow boat so as to maximize C*total'

The effect of the towing velocity on the benefit function estimated with Eq. (23) is illustrated in Fig. 3.Simulations were carried out for different Stefan numbers for the wetted area and for different water-to­fuel cost ratios. It is seen that for each case there is an optimum towing velocity (Reynolds number), i.e. avelocity that maximizes the benefit function. These optima are indicated on the figure. Moreover, theoptimum towing velocity is nearly independent of the Stefan number. The magnitude of the benefitfunction decreases, however, as the Stefan number increases (a higher Stefan number means a higherwater temperature). On the other hand, the water-to-fuel cost ratio affects the maximum benefit and theoptimum velocity. They both increase as this ratio augments.

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Ste =0.075 Xfu<JXw =0.2 .w 5 :

~,...--y------

C~0131

0.9

Ste =0.050···Jw

0.Q75-·····~

0.100·······

Fig. 3. Benefit function versus the Reynolds number.

5. CASE STUDY

As mentioned in Section 1, towing tabular icebergs is of interest for providing a source of fresh-water toSouth Africa. Using the mathematical model presented above, the optimum towing velocity andmaximum benefit were determined for different towing scenarios. In accordance with the data reported inreference [3], the iceberg length and height were fixed at 500 m and 100 m respectively. The towingdistance was chosen at 5000 km which represents the approximate distance separating South Africa fromAntarctica. Sea water properties were taken from reference [11] and ice and air properties were retrievedfrom reference [8]. The iceberg core temperature was fixed to 261 K following the data reported inreference [12]. The air temperature was maintained at 278 K based on the South Indian Ocean islandstemperature data of reference [13]. A complete list of the parameters and data retained for the case studyis reported in Tables 1 and 2. The main results of the case study are summarized in Fig. 4. This figuredepicts the optimum benefit function and towing velocity in terms of the water-to-fuel cost ratio.Examination of this figure reveals that the maximum benefit decreases almost linearly with the watertemperature, that is for a fixed water-to-fuel cost ratio, the distance separating two consecutive C*-curvesfor different temperatures remains constant. Moreover, the optimum velocity increases with the watertemperature. As expected, both the maximum benefit and optimum velocity increase as the water-to-fuelcost ratio augments. This effect is however insignificant on the maximum benefit since the optimumvelocity, and thus, the fuel expense, remains small.

The price of tap-water is approximately 10-3 USD/kg. Considering the fact that bottled water is, ingeneral, much more expensive than tap water [13] and the fact that the cold of the ice is also ofcommercial value, a price tag of ~10-2 USD/kg was assumed for the fresh-water supplied by the iceberg.Also, as of March 2008, the cost of diesel fuel was ~0.73 USDIL (0.859 USD/kg) in South Africa [14].Using these values, the benefits reported in Fig. 4 were translated in terms of dollars with the followingrelation

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f bh . IT bl 1 Tha e ermopJ lysIca propertIes 0 su stances.Seawater Air Ice

density [kg/mil 1027.9 1.2781 920.9heat capacity [J/kgK] 3912.9 1006.5 1994.1

Thermal conductivity[W/mK] 0.5820 0.0243 2.3360dynamic viscosity [Ns/m2

] 0.0014 1.721x10-5 N/Alatent heat [J/kg] N/A N/A 333,316

melting temperature [K] N/A N/A 271absorptivity N/A N/A 0.95

temperature [K] variable 278 261

blf hT bl 2 M .a e am parameters 0 t e towmg pro emShip displacement [W] 3800 m3

Fuel coefficient [Fe] 1.29x 106 m5/kgs2

Specific fuel consumption [SFC] 50xl06 kg/J

Solar heat flux amplitude [Po] 58.887 W/mK

Iceberg length [L] 500m

Iceberg height [H] 100m

Iceberg Drag coefficient [CD] 0.90

T =273 K

278K

0.95

0.9

T =293 K ---------w

288 K-------

283 K -----:278 K----: i273 K-" : :

: : • I

0.5

V [mls]

Figure 4

xf""/xwOptimum towing velocity and maximum possible benefit.

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(24)

The results then show a maximum benefit of roughly 203x 106 USD for towing the tabular iceberg fromAntarctica to South Africa.

The calculations were refined by considering the round trip South Africa-Antarctica. In this case, thebenefit function is modified according to

d

(25)

where

(26)

C C with + Cwithout

t t With + twithout

twithout = Vwithout

In other words, the velocity of the tow boat when towing the iceberg and the velocity on the return trip toAntarctica may be different. Limiting the velocity on the return trip to a maximum of 15 knots (7.717m/s) (see Ref. [9]), the present study reveals a maximum benefit of l4lxl06 USD for a round trip thatlasts 479 days. One must bear in mind that these predictions result from a simple mathematical model foriceberg towing. Other elements, discussed in the next section, should be taken into consideration beforedrawing definitive conclusions on the profitability.

6. CONCLUSIONS

In the present context of global warming and of the ever increasing demand for fresh-water supplies inarid regions, iceberg towing appears to be an interesting alternative. A mathematical model fordetermining the optimum towing velocity of tabular icebergs was presented. The optimization problemwas formulated in terms of a benefit function that takes into account the ice mass delivered and the totalfuel consumption. Results have indicated that the optimum towing velocity is mainly affected by thewater-to-fuel cost ratio. It was shown, with a simple mathematical model, that towing icebergs fromAntarctica to South Africa is a profitable way of supplying fresh-water provided that the towing velocityis optimized. The authors recognize however that the model needs to be refined before definitiveconclusions on the profitability of iceberg towing can be reached.

In future studies, the model should be improved by relaxing some of the underlying assumptions. Forinstance, wave erosion of the iceberg should be accounted for [3, 7]. The effect of ocean currents shouldalso be considered in the optimization method since the predicted optimum velocities reported here are ofthe same order of magnitude as that ofthese currents. Other effects such as the operation and maintenancecosts of the equipments and salaries of the crew should also be included in the economic study. All of theabove will undoubtedly affect the optimization study and ultimately diminish the profitability of icebergtowing.

ACKNOWLEDGEMENTSThe authors are grateful to the National Science and Engineering Research Council of Canada for theirfinancial support.

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REFERENCES

[1] S.A Kalogirou, Seawater desalination using renewable energy sources, Progress in Energy andCombustion Science 31-3 (2005) 242-281.

[2] W.F. Weeks and W.J. Campbell, Icebergs as a fresh-water source: an appraisal, Journal ofGlaciology, vol. 12, No. 65, pp. 207-233, 1973.

[3] G.R. Bigg, M.R. Wadley, D.P. Stevens, J.A Johnson, Modelling the dynamics andthennodynamics of icebergs, Cold Regions Science and Technology, vol. 26, pp. 113-135, 1997.

[4] J.G. Job, Numerical modelling of iceberg towing for water supplies- a case study, Journal ofGlaciology, vol. 20, No. 84, pp. 533-542, 1978.

[5] S. Loset, Numerical modelling of the temperature distribution in tabular icebergs, Cold RegionsScience and Technology, vol. 21, pp. 103-115, 1993.

[6] V. Smakhtin, P. Ashton, A Batchelor, R. Meyer, E. Murray, B. Barta, N. Bauer, D. Naidoo, J.Olivier and D. Terblanche, Unconventional water supply options in South Africa - A review ofpossible solutions, Water International 26-3 (2001) 314-334.

[7] T.AM. Silva, G.R. Bigg, K.W. Nicholls, Contribution of giant icebergs to the Southern Oceanfreshwater flux, Journal of Geophysical Research-Oceans 111 (2006) C03004.

[8] F.P. Incropera, D.P. Dewitt, T.L. Bergman, AS. Lavine, Fundamentals of Heat and MassTransfer Sixth Edition, New-York: Wiley (2001)

[9] Canadian Coast Guard - Newfoundland and Labrador Region, Fleet: CCGS Ann Harvey,Fisheries and Ocean$ Canada, http://www.nf1.dfo-mpo.gc.ca/ccg/fleet.htm.

[10] C.B. Barras, Ship Design and Perfonnance for Masters and Mates, Oxford: Butterworth­Heinemann (2004) 132-135.

[11] Chemical Hazards Response Infonnation System (CHRIS), Selected Properties of Fresh-water,Sea Water, Ice and Air, United States Coast Guard, http://www.chrismanual.com.

[12] Climat of the Crozet Islands, Meteo France,http://www.meteofrance.comIFRiclimat/clim_crozet.jsp.

[13] Drinking Water Quality in South Africa - A Consumer's Guide, Department of Water Affairs andForestry Republic of South Africa,http://www.dwaf.gov.zaiDocuments/Other/DWQM/DWQConsumerPamphletJul05.pdf.

[14] Basic Fuel Price 2008, Department of Minerals and Energy Republic of South Africa,http://www.dme.gov.za/energy/ib1c08.stm.

NOMENCLATUREBic

CoCruel

Cincome

Ctotal

DdEFeFoHhk

Biot numberspecific heat capacity, J/kgKdrag coefficientFuel cost, USDincome from water, USDtotal benefit, USDperiod, sdistance, mfuel expenses, kgfuel coefficient, mS/kgs2

Fourier numberheight, mheat transfer coefficient, W/m2Kthennal conductivity, J/mK

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L length, mM mass, kgNu Nusselt numberP heat flux, W/m2

Psun solar heat flux, W/m2

Pr Prandlt numberR temperature difference ratioRe Reynolds numberS phase-change interface position, mSFC specific fuel consumption, kg/JSte Stefan numberT temperature, KUSD US$t time, sV velocity, m/sW ship displacement, m3

x position, m

Greek Symbolsa thermal diffusivity, m2/s~ area and period to thermal diffusivity ratio£ absorptivityy solar heat absorbed to ablation speed ratioA latent heat of fusion, J/kgIJ. dynamic viscosity, Ns/m2

p density, kg/m3

Xfuel price of fuel, USD/kgXw price of water, USD/kg

Subscriptsa aIrc coree emerged (non wetted area)f final

initialm meltings solidu underwater (wetted area)w water

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