Question

The flat iceberg drifts over the ocean, as it is driven by the wind that blow over the top. The iceberg may be modeled as a block of frozen fresh water at 0 oC. The temperature of surrounding sea water is 10 oC, and the relative velocity between it and iceberg is 10 cm/s. The length of iceberg in the direction of drift is L=100 m. The relative motion between the sea water and the flat bottom of the iceberg produce a boundary layer of length L. The 10 oC temperature difference across this boundary layer drives a certain heat flux into the bottom surface of the iceberg. This heating effect causes the steady erosion (thinning) of the flat piece of ice. If H(t) is the instantaneous high of the ice slab, calculate the ice melting rate dH/dt average over the swept length of the iceberg.

Answer 1

This problem can be thought of as convective heat transfer transfer from an isothermal plate. The iceberg can be regarded as and isothermal flat plate.

Temperature of iceberg = 0 oC

Temperature of surrounding water = 10 oC

Relative velocity = 10 cm/s = 0.1 m/s

Length of iceberg = 100 m

For this case; the length averaged Nusselt Number (Nu) is:

Nu = 0.664 ReL0.5 Pr0.33

where

ReL = (density X velocity X length) / viscosity

For water:

Density = 1000 kg / m3

Velocity = 0.1 m/s

Viscosity = 0.001 Pa.s

which gives:

ReL = (1000 X 0.1 X 100) / 0.001 = 10000000

Pr = 7.56 (water)

Substituting these values:

Nu = 0.664 X (10000000)1/2 X 7.560.33 = 4093

we know:

Nu = h L / k

k = Thermal conductivity of water = 0.58 W / m oC

Substituting this in the above expression:

4093 = h X 100 / 0.58

h = 23.74 W / m2K

From Newton's Law of cooling heat transfet rate is given by:

Q = h A (T w- Ti) = h X L X W X (T w- Ti)

Q = 23.74 X 100 X W X (10 - 0) = 23740W J /s

where W is the width of iceberg.

The heat trannsfer rate can also be expressed as

Q = latent heat X mass transfer rate

23740W = latent heat X mass transfer rate

Latent heat of ice = 334000 J / kg

Therefore;

23740W = 334000 X mass transfer rate

Mass transfer rate = 0.071W kg/s

From mass balance we get:

V = LWH ; which gives

Assuming L, W and ρ is constant;

which on simplification gives:

Substituting value of ρ = 1000 kg/m3 and L = 100 m :

we get

which is the required melting rate.