. 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.)
(a) What is the absolute pressure at the pump depth?
(b) What is the absolute pressure at the surface of the lake? (
c) What is the ratio of the flow speed at the pump to that at the nozzle?
(d) The design for the installation calls for propelling the water out of the nozzle 3 m into the air as a fountain. What gauge pressure in the pipe at the pump is required for this?
(e) When the air temperature is 0 ?C, the water at the surface freezes to form a sheet of ice. Explain why the entire volume of the lake doesn’t freeze.
(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) Since H = dQ dt , use your answer to (f) to express the amount of heat dQ conducted through the ice sheet in time dt.
(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.
(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).
(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.
(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.)
(l) Use your answer to (j) to calculate how long it will take to form an ice layer 20 cm thick.
(m) Use your answer to (j) to calculate how long it will take for the entire 59 m of the lake to freeze.
(n) Is it likely that the entire lake freezes? Explain.
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