Solutions to exercise 6.4.2 from strogatz nonlinear dynamics
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16. A certain certain electric electric motor receives receives 12 kW of electrical electrical power power and delivers delivers mechanical energy at the rate of 1000 kg m/s. What is its efficiency ?
Solution: ηm
power output 1000 kg m/s x 1 kJ/ 101.92 kg m = 0.8176 = = power input 12 kW
17. How many kg m of torque are given to a turbine shaft transmitting 30000 kW at 1800 rpm ? at 3600 rpm ? Solution:
P = 2πTN
@ N = 1800 rpm ;
@ N = 3600 rpm ;
30000 kJ/s x 101.92 kg m / kJ T = = 16221.072 kg m 2 π( 1800 rev/min x 1min/60 s) 30000(101.92) T = = 8110.5359 8110.5359 kg m 2π x 3600 x 1/60
18. What water horsepower horsepower (metric) is required to raise 1900 liters/min liters/min through a height of 38 m ? Solution: 38m
P = γ Q H = 9.807 kN/m3(1900 l/min x 1m3/1000 kg x 1min/60s) x
20. The overall efficiency of an electric system, system, coal file to lights, is 11 %. What fraction of a pound of coal containing 30250 J/g must be consumed to light a 100 W lamp 45 minutes ? Solution: Heat equivalent of one 100 W lamp lighted 45 min = 100 J/s(45min x 1 min/60s) = 270000 J Required heat input from fuel for 11 % efficiency = 270000 ÷ 0.11 = 2454545.5 J Fraction of 30250 J/g coal to supply 2454545.5 J = 2454545.5 J ÷ 30250 J/g = 81.1419 g ( 2.2 lb ÷ 1000 g ) = 0.1785 lb 21. A power dam creates a head of 25 m in a stream whose normal flow is 4200 cfs. Hydraulic turbines that can convert 92 % of the water power into shaft power are installed. They are connected to generators whose efficiency is 97.5 %. What is the kilowatt capacity of the plant ? Solution: Pg output of generator ηg = = Ps shaft power Ps = 0.92 water power = 0.92 γ Q H = 0.92(9.807 kN/m kN/m3 x 4200 ft3/s x 1 m3 / 35.31 ft3)(25 m) = 25552.082 kW Pg = 0.975 x 25552.082 = 24913.28 kW
= 11.8011 kW x 1 hp / 0.7355 kW = 16.045 hp
19. A power power plant serves serves a factory factory having having two 22 kW motors motors and ten 3.7 kW motors. motors. Assume Assume the efficienc efficiency y of motors 80 %, of transmi transmissi ssion on line 95 %, of generator 92 %. (A) What should the rated capacity of the generator be if it is assumed that all motors might be delivering their rated power simultaneously ? (b) what should the rated capacity of the engine be ? Solution: total input to motors (a) Transmission Transmission line efficiency efficiency , ηt = input to transmission ( or generator output) total motor output motor efficiency, ηm = total input to motors total motor output = 2 (22) + 10 (3.7) = 81 kW total input to motors = 81 kW ÷ 0.80 = 101.25 kW input to transmission lines = 101.25 ÷ 0.95 = 106.5789 kW (b) generator output generator efficiency, ηg = input to generator ( or engine output )
22. A farmer farmer has a small stream stream on his property property which which he thinks thinks might might supply supply enough power to light his buildings. The stream is gauged and found to be flowing 0.03 m 3/s. The available fall is 2.1 m. A small hydraulic turbine and generator will have a combined efficiency of 70 %. Taking the transmission transmission line efficiency as 80 %, How many 50 W lamps can be connected to the generator simultaneously ? Solution: let n = no. Of 50 W lamps that can be connected to the generator simultaneously power from transmission line n = power per lamp =
ηc ηt ( power available from the
stream
power / lamp 0.7(0.80)( 9.807 kN/m3 x 0.03 m3/s x 2.1 m ) = 50 W/Lamp x 1 kW/ 1000 W
1. If the mass of a typical neutron is 1.66 x 10-24 gram and the velocity is approximately 2200 mps. What is the kinetic energy of this neutron in electron ? 2. What power in kilowatt is represented by a flow of water from 5 cm diameter nozzle at a velocity of 135 mps ? 3. In a certain industrial process 85 kilogram of a liquid ( specific heat is 3.56 J/goC ) are required to be heated through 36oC in a 42 min period. If an immersed electric coil is used to accomplish this, how many watts will it draw while in operation, and what must be its working resistance in ohms ? 4. How much kinetic energy is released when a flywheel, as here described, is slowed from 250 rpm to 235 rpm ? mass of flywheel is 60 hyls; radius of gyration, 53 cm. 5. A locomotive is tested with a dynamometer car coupled between the tender and the train traveling over a level track at a speed of 130 kmph. The dynamometer registers 10615 kg. What horsepower is the locomotive then delivering ? on an extended run the dynamometer integrator gives 2945 kW hour and over the same stretch 6215 kg of 27920 J/g coal were consumed. Calculate the thermal efficiency of the locomotive. 6. Mechanical work in the form of a torque of 346 kg m is applied to an electric generator at a speed of 1800 rpm. The machine losses some of this energy in the form of a heat loss of 2640 kJ/min. The remainder of the energy is transformed to electrical form. Find the kilowatts of electrical power produced. 7. Waster drops perpendicularly 36.5 m in a certain waterfall. Assuming all available energy absorbed as heat after striking the lower pool> Find the rise in temperature of the water passing the pool. 8. If 2685200 kJ/hr are given to an engine which can convert 12 % of the heat into work, What is the power in kW developed by the engine ? 9. A plant containing one 600 kW and one 200 kW generating unit is being operated at rated capacity. The coal consumption at the same time is 860 kg/hr. Heating value of the coal is 22106 J/g. What fraction of heat in the coal is converted into useful energy ? 10. A truck being “freewheeling “ at 30 kmph is stopped in 90 m by the application of a uniform retarding force. Weight, 1800 kg. What is the magnitude of the force ?
