ME 330 PS8 Solutions Fall 2010

ME 330 PS8 Solutions Fall 2010 - Problem Set 7 Problem#1...

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Problem Set 7, Problem #1 Given: Pumped water energy storage system using a reservoir at height z and a piping system with major and minor losses totaling D fL and . A dual purpose pump/turbine with head curve shown below is used for charging and discharging the energy. L K 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 h p,T /h m Q p,T /Q m Find: a) Derive the equations for pump head ( h p ) and turbine head ( h T ) produced versus flow rate Q in terms of the maximum head ( h m ) and maximum flow rate ( Q m ) b) Derive and expression for the flow rate Q p delivered by the pump in terms of h m , Q m , z , f , L , D , and K L c) Derive and expression for the flow rate Q T used by the turbine in terms of h m , Q m , z , f , L , D , and K L and compare this result to Q p (are they the same or different) d) What is efficiency η , defined as the power extracted by the turbine to the power added by the pump, of this energy storage system? Express your answer in terms of h m , Q m , z , f , L , D , and K L . Assumptions: Steady state, incompressible, neglect efficiency of pump and turbine, large reservoirs such that the surface velocity and pressure are zero and atmospheric Analysis: This is a mechanical energy balance problem Pump/turbine head curve: Any parabolic curve can be represented by C Bx Ax y + + = 2
0 0 ) ( ) 0 ( 0 , , , , , 2 , = = = = = + + = = Q T h m T P T P m T P T P T P dQ dh Q Q h h Q h C BQ AQ h = 2 , , 1 m T P m T P Q Q h h Flow rate Q p : =

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ME 330 PS8 Solutions Fall 2010 - Problem Set 7 Problem#1...

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