A gas is contained in a closed rigid tank. An electric resistor in the tank tran

Question

A gas is contained in a closed rigid tank. An electric resistor in the tank transfers energy to the gas at a constant rate of 1000W. Heat transfer between the gas and the surroundings occurs at a rate of Q=-50t, where Q is in watts, and t is time, in min. a. Plot the time rate of change of energy of the gas for 0 to 20 min, in watts. b. Determine the net change in energy of the gas after 20 min, in kJ. c.If electricity is valued at $0.08 per kW*h, what is the cost of the electrical input to the resistor for 20 min of operation?

Explanation / Answer

Rate of change of the internal energy of the gas can be modeled as dU/dt = 1000*60-50t J/min or in watts as dU/dt = 1000-5/6*t J/s

a)

Plot the latter equation with dU/dt in W on the y-axis and time in seconds or minutes on the x-axis. It should be a straight line starting at (1000 W, 0 min) and ending at (0 W, 20 min) *Remember a minute is 60 seconds

b)

The net change in energy from 0 to 20 min will be the area under this curve.

U = (1/2)(1000 J/s)(20 min)(60 s/min) = 6*10^5 J = 600 kJ

c)

Cost = ($0.08)(s/kJ)(1/hr)(1 hr / 3600 s)(600 kJ) = ~$0.013

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