Suppose a vertical pipe is to be used as part of a system to manually cycle...

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.)

f.)Write an equation that expresses the heat current H of conduction through the ice as a function of the thickness h of the ice sheet (of area A) already formed. Assume the air temperature is a constant -10 ?C, and that the temperature of the bottom of the ice sheet is 0 ?C.

(g) (1 point) Since H = dQ dt , use your answer to (f) to express the amount of heat dQ conducted through the ice sheet in time dt.

(i) (2 points) 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).

(j) (2 points) 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.

(k) (2 points) 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

Expert Solution

samet mamat answered 2 years ago

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