ERGY8070. Week 3: Practice Problems

Week 3 ERGY8070 Energy Storage Practice Problems 1. Consider the pumped hydro system below: a. What is the maximum amount of energy stored in MWh in the scenario below? Note that the density of water is 1000 kg/m3 and acceleration due to gravity is 9.81 m/s2. b. What is the energy stored in MWh if the system is 75% efficient? c. Is this system open or closed loop? 1,100,000 m3 92m 800,000 m3 2. A pumped storage system has a 50,000 m3 upper reservoir that is located 40m above a lower reservoir. Assuming no losses: a. What is its energy density in MWh/m3? b. What is it its power capacity (in MW) if water can flow through the turbine at a rate of 500 m3/h? 3. What are some of the energy losses present in a flywheel energy storage system and how are they mitigated? 4. True or False: mechanical energy storage can be both stored potential energy and stored kinetic energy? 5. A thermal energy storage system heats a substance until it changes from solid to liquid to store heat in. Heat is then released when the material solidifies. Is this unit using latent heat, sensible heat or both? 6. True or False: in an asymmetrical supercapacitor, both electrodes are porous carbon? 7. What is the energy stored in a 300 kg rotating flywheel (in J) with a 4m radius using solid uniform discs. The flywheel is rotating at 5,000 RPM. Week 3 ERGY8070 8. An isothermal compressed air energy storage system has a volume of 450,000 m3 and a compressed air pressure range from 1 to 75 bars. Assuming 30% efficiency: a. What is the amount of stored energy (in MJ)? b. What is the average power output over a 2 hour period? 9. A pumped storage facility has a height of 450 m and an efficiency of 93%. The density of water is 1000 kg/m3 and acceleration due to gravity is 9.81 m/s2. a. What is the flowrate needed to generate 100 MW for 4.5 hours per day? b. What is the required working volume per day? Week 3 ERGY8070 ANSWERS 1. Pumped hydro example: a. What is the maximum amount of energy stored: 𝐸𝑝 𝑖𝑛 𝑀𝑊ℎ = 𝜌𝑉𝑔𝐻 1000 𝑘𝑔/𝑚" ∗ 1,100,000 𝑚" ∗ 9.81 𝑚/𝑠 # ∗ 92 𝑚 = = 275.8 𝑀𝑊ℎ 3.6 ∙ 10! 3.6 ∗ 10! Note: 3.6 * 109 = 3,600,000,000 b. What is the energy stored in MWh if the system is 75% efficient? 𝐸𝑝 𝑖𝑛 𝑀𝑊ℎ = 𝜂𝜌𝑉𝑔𝐻 0.75 ∗ 1000 𝑘𝑔/𝑚" ∗ 1,100,000 𝑚" ∗ 9.81 𝑚/𝑠 # ∗ 92 𝑚 = = 206.8 𝑀𝑊ℎ 3.6 ∙ 10! 3.6 ∗ 10! Note: 3.6 * 109 = 3,600,000,000 c. open or closed loop? – this is a closed loop system, as there is no naturally flowing water source 2. A pumped storage system has a 50,000 m3 upper reservoir that is located 40m above a lower reservoir. Assuming no losses: a. What is its energy density in MWh/m3? 𝑘𝑔 9.81𝑚 1000 ∗ ∗ 40𝑚 𝑀𝑊ℎ 𝜌𝑔𝐻 𝑚3 𝑠2 𝑒$ 𝑖𝑛 = = = 0.000109 𝑀𝑊ℎ/𝑚3 ! ! 𝑚3 3.6 ∙ 10 3.6 ∙ 10 b. What is it its power capacity (in MW) if water can flow through the turbine at a rate of 500 m3/h? 𝑘𝑔 9.81𝑚 1000 𝑚3 ∗ 𝑠2 ∗ 40𝑚 ∗ 500𝑚3/ℎ 𝜌𝑔𝐻𝑄 𝑃𝑐 𝑖𝑛 𝑀𝑊 = = = 0.05 𝑀𝑊 3.6 ∙ 10! 3.6 ∙ 10! 3. Flywheels have air friction and mechanical friction, as per Week 3 slide 33 4. True 5. Both. Since it involves a temperature change it has sensible heat and since it involves a phase change, it has latent heat. Week 3 ERGY8070 6. False 7. First calculate rotational velocity in rad/s: 𝝎 = 5,000 RPM / 9.55 = 523.6 rads/s Next, calculate energy stored (note that k = 0.5 for solid uniform disks): 𝟏 𝑬𝒓 = 𝟐 𝒌 𝒎 𝒓𝟐 𝝎𝟐 = 0.5 x 0.5 x 300 kg x 42 x (523.6)2 =328,988,352 J 8. a. ) * *-. 𝐸 = 𝜂𝑃( 𝑉( ln L) ! M=0.3 ∗ 7,500,000 𝑃𝑎 ∗ 450,000𝑚3 ∗ ln O+,P ∗ *,000,000.= -4,371,600 MJ " The negative symbol indicates that the work (energy) is being done to the gas by the surroundings. b. Recall that a W is a J per second and a MW is a MJ per second. Therefore: P = 4,371,600 MJ / (2 hours * 60min/hr * 60sec/min) = 607.2 MW 9. a. 𝑃1 = 𝜌𝑔𝐻𝑄𝜂 Rearrange for Q: 𝑄 = )1 2345 = #,%%%,%%% ' # (' )* + 0.!"∗*000 ∗!.:* ∗;,0< +, -. *00 -7∗ = 24.383 m3/s b. The volumetric flowrate we calculated is a rate of volume per time. Because we know the flow rate and that generation is needed for 4.5 hours/day, we can calculate working volume as: V = Qt = 24.383 m3/s * 4.5 hours/day * 60min/hour * 60min/s = 395,000 m3/day