Thermodynamics speed of a weight bar through ice

[Liquidxlax]

Homework Statement

A steel bar of rectangular cross-section, with height a and width b (into the paper), is placed on a block of ice with its ends extending a little beyond the ice, (ice has a length c). A weight of mass m is hung from each end of the bar. The entire system is at 0 degrees celcius.

As a result of the pressure exerted by the bar, the ice melts beneath the bar and refreezes above the bar. Heat is therefore liberated above the bar, conducted through the metal, and then absorbed by the ice beneath the bar. Given the latent heat of fusion per kilogram of ice (L), the density of ice ρ_i, the density of water ρ_w, the thermal conductivity of steel (Kappa) which relates the heat q crossing a unit area per unit time to the temperature gradient (dT/dz) in a direction perpendicular to the plane through the relationship q=-(kappa)(dT/dz), the temp T(=0 degrees celcius) of the ice, the accelertation due to gravity g, the mass m, and the dimensions a, b, and c, where c is the length of the block of ice, show that the speed v with which the bar sinks through the ice is


v= [(2mg(kappa)T)/(abcL²ρ_i)]*[(1/ρ_i)-(1/ρ_w)]

Homework Equations

A=bc

dP/dT = L/TΔV ~ ΔP/ΔT

dq = -(kappa)(ΔT/Δz) ( my prof wrote it as q with a dot over top so i assume it's some sort of dq)

(dQ_fusion/dt) = (dM/dt)(L/A)

The Attempt at a Solution

Everytime i look at it, i don't see a constant velocity since the pressure should be increasing therefore the temperature below the bar would be increasing. anyways

I thought to try and relate the equations with ΔT to get rid of the ΔT

so,

dq= -(kappa)(ΔT/Δz) = -(kappa)([ΔPTΔV]/LΔz)

then for the dQ_fusion i thought to multiply through by dt and equate the above equation dq to dQ_fusion

so

dz= -(kappa)*([ΔP*T*ΔV*A]/[dM*L²)


then i thought that ΔP has to be 2(dM)g/A or 2mg/A (since it's in the top of the answer)

ΔV = Adz


that's all i have and i don't think it's right.

any help would be appreciated

 

[Liquidxlax]

guess i could answer my own question, for anyone who has a similar one.

first of all Δv is the molar density

and dq is supossed to be (dq/dt) which is equal to the dQ_fusion/dt

so

dq/dt=dQ_fusion/dt

=> -(kappa)(ΔT/Δz) = (dM/dt)(L/A)

and (ΔT/Δz)=(ΔT/a) since Δz=a-0

move L/A over

(dM/dt)= -(kappa)(ΔTA/aL)

next you want to solve for ΔT from the dP/dT

ΔT = (ΔPTΔv)/L

where ΔP = -2mg/bc

so then we have

dM/dt = [2mg(kappa)TΔvA]/[L²abc]

now you need to related the change in mass with respect to time with the change in position of the bar with respect to time

dM/dt = ρ_ibc(dy/dt)

therefore

dy/dt = (dM/dt)(1/(ρ_ibc))

then

dy/dt = [2mgT(kappa)Δv]/[L²abcρ_i]

the disappears because A=bc

molar volume is equal to 1/ρ

which gives the final answer of


v= [(2mg(kappa)T)/(abcL2ρi)]*[(1/ρi)-(1/ρw)] = dy/dt