Question

Suppose a vertical pipe is to be used as part of a system to manually cycle nutrients upward from the floor of a lake. (Many lakes do this naturally, some do not. Green Lake, near Syracuse, NY, is one such lake.) A pump is to be installed on the lake floor at the base of the pipe. The base of the pipe will have a diameter of 9 cm. The nozzle of the pipe at the top will have a diameter of 4 cm. The lake is 59 m deep at the installation point. (Ignore any viscosity.)

Part G

Since H = dQ dt , use your answer to (f) to express the amount of heat dQ conducted through the ice sheet in time dt.

Answer: dQ = 10(KA/n)dt

Part H

Consider a short time interval dt, and let an additional thickness dh be formed in that time. Express the mass dm that freezes during this time in terms of dh, the area A, and the density of water.

Answer: dm = rhoAdh

Part I

Express the amount of heat dQ that must be removed from the water at the bottom of the ice sheet to freeze the mass dm you found in (g).

Part J

Based on (g) and (i), set the expressions for dQ equal to each other to obtain a differential equation relating the heat that must be removed to freeze a new layer to the heat conducted through the ice sheet.

Part K

Separate variables, and integrate to find the thickness h of the ice sheet as a function of time t. (Note that h = 0 when t = 0.)

Need help with Part I through K

Answer 1

PART I

Let the latent heat of fusion for water at 0 ^oC be L J/kg.

Thus to freeze mass dm we need to remove dQ amount of heat given by:

.......EQ1

We found out that;

putting value of dm in EQ1, we get

........EQ2

PART J

now we have already found out that the heat conduction is given by:

.........EQ3

equating EQ3 and EQ2

....EQ4

PART K

We can integrate the EQ 4 to het the raltion between h and t. we are given that at t=0, h=0 and let at time t the instantaneous thickness of ice sheet be h.