This document contains solutions to multiple physics problems involving heat transfer and temperature change.
Problem 46 calculates the time needed to raise the temperature of a large lake by 10°C when receiving a constant rate of incident solar energy.
Problem 51 finds the maximum mass of water that can be heated to a certain temperature by two heat sources operating at different power levels.
Problem 65 determines the equilibrium temperature reached when a hot iron object cools by transferring heat to cooler water until they equalize in temperature.
Problems 76 and 80 solve heat transfer differential equations to determine heat flow rates through cylindrical and conical objects with varying radii and temperatures across their surfaces.
This document contains solutions to multiple physics problems involving heat transfer and temperature change.
Problem 46 calculates the time needed to raise the temperature of a large lake by 10°C when receiving a constant rate of incident solar energy.
Problem 51 finds the maximum mass of water that can be heated to a certain temperature by two heat sources operating at different power levels.
Problem 65 determines the equilibrium temperature reached when a hot iron object cools by transferring heat to cooler water until they equalize in temperature.
Problems 76 and 80 solve heat transfer differential equations to determine heat flow rates through cylindrical and conical objects with varying radii and temperatures across their surfaces.
This document contains solutions to multiple physics problems involving heat transfer and temperature change.
Problem 46 calculates the time needed to raise the temperature of a large lake by 10°C when receiving a constant rate of incident solar energy.
Problem 51 finds the maximum mass of water that can be heated to a certain temperature by two heat sources operating at different power levels.
Problem 65 determines the equilibrium temperature reached when a hot iron object cools by transferring heat to cooler water until they equalize in temperature.
Problems 76 and 80 solve heat transfer differential equations to determine heat flow rates through cylindrical and conical objects with varying radii and temperatures across their surfaces.
This document contains solutions to multiple physics problems involving heat transfer and temperature change.
Problem 46 calculates the time needed to raise the temperature of a large lake by 10°C when receiving a constant rate of incident solar energy.
Problem 51 finds the maximum mass of water that can be heated to a certain temperature by two heat sources operating at different power levels.
Problem 65 determines the equilibrium temperature reached when a hot iron object cools by transferring heat to cooler water until they equalize in temperature.
Problems 76 and 80 solve heat transfer differential equations to determine heat flow rates through cylindrical and conical objects with varying radii and temperatures across their surfaces.
Incident energy should be same with the energy to increase the temperature of the lake. 200W/m2 (103 m)2 (time)
= (20 C 10 C) 4184J/kg K (103 m)2 10m 1g/cm3 106 cm3 /m3
(time) = 2.092 109 s 24213 days
51.
Mass of the water : m[kg]
4.184kJ/kgK m+1.1kJ/K 4.184kJ/kgKm 1600W (Tf Ti ) 675W (Tf Ti )
m 1.92 101 kg
65.
Equilibrium temperature : T [ C] Energy lost from the iron should be same with the energy to increase the temperature of the water. (550 C T ) 1.1kg 447J/kg K = (T 20 C) 15kg 4184J/kg K
T = 24.12 C
76.
Heat would flow from the side of the cylinder. Lets consider a hollow cylinder with length L, radius r, and infinitesimal thickness dr, as described in the figure. Then the area of the side is given by 2rL. H = kA dT dr = k(2rL) dT dr
Since r is the only variable in here; the others are constants, the differential equation can be easily solved by separation of variable. H 1r dr = 2kLdT R R2 R T2 R1 H 1r dr = 2kL T1 dT
H ln(R2 /R1 ) = 2kL(T1 T2 )
2kL(T1 T2 ) H = ln(R2 /R1 )
1 80.
Unlike the problem 76, in this problem, the heat flows through the bottom and top of the truncated cone. Imagine a disk with radius r(R1 < r < R2 ), and infinitesimal thickness dl. Then the heat flow is given as following.
H = k(r2 ) dT dl
However, there is a relation between r and l which can be obtained from the proportion relation. L l(r) = R2 R1 r
Then the heat flow can written as,
H = k(r2 ) R2 R L 1 dT dr
Similar to problem 76, this can be solved using separation of varialbes.
