Shad Valley MUN Mathematics of Iceberg Towing from

Shad Valley MUN Mathematics of Iceberg Towing from the Antarctica Dr. Leonard M. Lye, PEng, FCSCE, FEC Professor of Civil Engineering Program Director Shad MUN Shad Valley 1

Outline • Introduction • Basics of icebergs: some facts and figures • Iceberg towing problems: – – – – – What typical sizes are available? How far do we have to tow? Are ships available to do the towing? What velocity to tow to minimize losses? How long does it take? How much power is required? Size of steel cables required? How far will it drift after cables are disconnected? How long does it take for the drift to stop? • Other problems not answered • Conclusions Shad Valley 2

Introduction • Many places are desperate for fresh water – – – Deserts Population growth Industrial growth Famine areas Overuse and pollution • Iceberg water is cold, fresh, old, novelty – 12, 000 years old, fizzes as it melts – Temperature inside iceberg is – 15°C outside is about zero. • Ice caps holds most of the earth’s freshwater Shad Valley 3

So? • Why not capture some large icebergs from the Antarctica, tow them to say Long Beach California, let them melt, and then sell the nice fresh water to the thirsty people of Los Angeles, at a clear profit except for some expenses? • Pipe dream or feasible? ? Shad Valley 4

Harvesting icebergs on a small scale for iceberg vodka Shad Valley 5

World’s Freshwater Supply Shad Valley 6

Some water facts • Of the world’s water: – 96. 5% in ocean as salt water – 2. 53% is freshwater – Remainder are in the atmosphere • Of the freshwater resources: – 30. 1% is groundwater – 61. 7% in the Antarctica (90% of world’s ice) – 6. 68% in Greenland – Remainder in rivers, lakes, permafrost, Arctic Shad Valley 7

Antarctica and its major ice shelves Shad Valley 8

Some facts about Antarctica • 5 th largest continent (larger than Australia and Europe) • 14 million square kilometers in area • Only slightly less than 1. 5 times size of US • 13. 72 million square kilometers are icecovered (98% ice, 2% rock) • Coastline of 17, 968 km. • Coldest, windiest, driest, highest (on average), most inhabitable place on earth. Shad Valley 9

Basics of iceberg: some facts and figures Ice shelves begin to “calve”, large sections fracture, break off, and begin to drift in directions decided by water currents and winds. Shad Valley 10

North Atlantic’s iceberg alley Shad Valley 11

Types and sizes of icebergs Shad Valley 12

Shipping hazards Shad Valley 13

Some iceberg facts and figures • As many as 1, 600 icebergs drift past St. John’s each year • Drift from origin in Greenland to Newfoundland is about 1, 800 nautical miles (3, 333. 6 km) • Arctic icebergs have reached as far south as Bermuda (4000 km) and an Antarctica iceberg has reached as far north as Rio de Janeiro (5, 500 km) • Biggest iceberg ever recorded was 335 km long, 97 km wide (32, 500 sq. km) – larger than Belgium. Tallest recorded was 165 m high. Shad Valley 14

Iceberg Towing Problems • Availability? – Common Antarctica icebergs are tabular of about 1000 m long, 500 m wide, and a freeboard of 40 m. What is the mass and volume of this iceberg? • Is that large enough? – Water usage is about 0. 5 m 3/day/person. How long does it last for a city of 2 million? • How far do we have to tow? – To Long Beach, California along the great circle route is about 12, 340 km. Shad Valley 15

Routes of iceberg tows Shad Valley 16

More questions • Are ships available to do the tow? – Assume we can get two used powerful naval destroyers. L = 270 m, W = 33 m, Draft = 12 m, Weight = 50, 000 tons, top speed = 35 knots, horsepower = 200, 000. – Assume one iceberg is being melted while another iceberg being towed will arrive just in time • What is the optimal velocity of towing? – We want to minimize ice loss due to melting Shad Valley 17

Definition sketch for iceberg towing Assume we have captured an iceberg of the following dimensions: L = 1, 200 m, B = 400 m, ρi = 900 kg/m 3, ρw = 1030 kg/m 3, h 1 = 25 m, H = ? Shad Valley 18

Minimizing melting • Melting is caused by two main factors: – Absorbed solar radiation by the non-wetted area of the iceberg – Heat and mass transfer across wetted area of the iceberg due to movement through the water • Melting due to solar radiation is proportional to time of exposure, t* • Melting due to heat and mass transfer is proportional to square of velocity i. e. U 2. Shad Valley 19

Model • Assuming air and water temperature do not change, t* = D/U where D = distance towed • Equation to describe relative loss of ice mass (k 1 and k 2 are constants): What U will minimize DM/Mo? Shad Valley 20

Plot of Equation of Melting Shad Valley 21

Solve by calculus • Answer seems to be around 1 m/s. • We can also obtain the answer by differentiating LHS w. r. t U and equating it to zero and solve for optimum U. With D =12. 34 x 106 m, k 1 = 5. 57 x 10 -9 s-1 k 2 = 0. 0314 s 2 m-2 Uopt = 1. 03 m/s (2 knots) DM/Mo = 10% Shad Valley 22

Time taken • t* = 12, 340, 000/1. 03 = 11. 98 x 106 s which is about 139 days or roughly 5 months. • Twice a year delivery per ship would be 4 icebergs per year. • After each delivery, ship has a few weeks to rest up and for repairs. • At 30 knots top speed, ship can get down to the Antarctica in less than 10 days. Shad Valley 23

How much power is required to tow the iceberg? • We need to provide enough power to tow the iceberg at 1. 03 m/s. • That is, we need enough power (P) to provide thrust T (a force) so that iceberg will move. • What are the forces involved? What forces are resisting movement? • We need Newton’s second law of motion here. Shad Valley 24

Good old Newton With CF=0. 01 and CD=1. 20, and L = 1, 200 m, B = 400 m, U = 1. 03 m/s, h 2 = 175 m, rw = 1, 030 kg/m 3, 47. 9 x 106, T = F = 50. 8 x 106 N Shad Valley 25

Not done yet! • • • Power = work done/time, or Force x distance/s Effective power P = T U (k. W or k. J/s) P is proportional to U 3 (2 U requires 8 P) Power required = 52, 300 k. W Shaft power = 52, 300/efficiency of propellers If efficiency is 50%, shaft power required would be 105, 000 k. W or 140, 000 hp. • Ship can provide 200, 000 hp, hence we are okay. Shad Valley 26

What size of cables do we need? • Towing force T = 50. 8 x 106 N • Assuming steel cables can withstand a tensile stress s of 2. 41 x 108 N/m 2, cross-sectional area required = T/s = 0. 21 m 2 • That is, about 0. 5 m in diameter! • What length of cable is required? • What is the total weight of the cables? Density of steel is about 8, 000 kg/m 3 • Can the ship handle the additional weight? • How are we going to put it around the iceberg? Shad Valley 27

Assuming all goes well … • After 139 days of towing, the iceberg and ship approaches Long Beach. Cable is disconnected, ship goes to the shipyard. • Massive iceberg without the cable thrust force will continue to drift towards the harbour. • What distance will it travel before it comes to a stop, and how long will that take? Shad Valley 28

Good old Newton again Assuming a 10% shrinkage in C. Shad Valley 29

Time to stop and then what? • If we solve for the time taken for the iceberg to stop, t = 8. 9 hours. • What would have happened if we did not disconnect in time? • How are we going to melt the iceberg, collect the melt water, and distribute the water? • Any ideas? Shad Valley 30

Conclusions • While most of the world’s fresh water is locked up in ice, the great idea of transporting icebergs to places that need the water maybe just an idea. • There are many staggering engineering problems as can be seen from our back of the envelope calculations. • In addition, there are political, environmental, and economic problems with such a venture. Who owns the icebergs? Who has jurisdiction? Who is going to pay for it? Shad Valley 31

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