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