MAE118A: Introduction to Energy Systems
Midterm Exam Fall 2009
Kowsik Bodi
November 6, 2009
1.
The rate of consumption of useful work at
t
= 0
is given as
P
0
. This
work is provided by converting a primary energy source with an efficiency
η
0
.
The power demand,
P
, is growing at a fixed rate
r
per year and the conversion
efficiency is fixed. If at
t
= 0
there is a fixed quantity of energy,
Q
0
, available in
this primary energy resource which can be accessed, find how long the primary
energy resource will last.
25 pts.
Total energy reserve is at
Q
0
. Conversion efficiency of this resource for use
is
η
0
. So total available energy is
η
0
Q
0
.
Rate of consumption of energy is the Power demand. At
t
= 0, power demand
is
P
0
. If the fractional increase in power demand is
r
,
1
P
d
P
d
t
=
r
=
⇒
P
(
t
) =
P
0
e
rt
(1)
The total amount of consumed resources, at a given time
τ
, is the sum of
consumption from
t
= 0 till the year
t
=
τ
,
Q
(
τ
) =
t
=
τ
integraldisplay
t
=0
d
t
P
(
t
) =
t
=
τ
integraldisplay
t
=0
d
t
P
0
e
rt
=
P
0
r
[
e
rτ

1]
(2)
The resource is exhausted at a time
τ
=
T
if all available energy has been
consumed:
Q
(
T
) =
η
0
Q
0
,
η
0
Q
0
=
P
0
r
bracketleftbig
e
rT

1
bracketrightbig
T
=
1
r
bracketleftbigg
ln
parenleftbigg
rη
0
Q
0
P
0
+ 1
parenrightbiggbracketrightbigg
(3)
1
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2.
Consider a heat engine that functions by first compressing a gas from an
initial state 1 at pressure
p
1
to a state 2 with pressure
p
2
> p
1
via a reversible
adiabatic compression. This gas then has a quantity of heat,
q
h
, added to it via
an ideal combustion process. Let this state be denoted as state 3. The gas is
then goes through a reversible adiabatic expansion leaving it at state 4.
This
expansion leaves
p
4
=
p
1
. The gas at state 4 is then cooled down back to the
initial temperature
T
1
by rejecting a quantity of heat
q
l
to a cold reservoir. A
quantity of work,
w
c
, obtained from the expansion stage is used to operate the
compressor. The remainder of the work,
w
, is then available for useful purposes.
25 pts.
•
1
→
2: Adiabatic,
reversible
(isentropic) compression.
s
2
=
s
1
,
p
2
> p
1
,
c
p
r
(
T
2

T
1
) =
h
2

h
1
=
w
c
, where
c
p
r
is the specific heat at const.
pressure of the reactants, and
w
c
is the work done by the compressor.
•
2
→
3: Heat addition due to
ideal
(const. pressure) combustion.
p
3
=
p
2
,
c
p
pr
(
T
3

T
2
) =
h
3

h
2
=
q
h
, where
c
p
pr
is the specific heat at const.
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 Fall '09
 KowsikBodi
 Thermodynamics, Energy, QH QH QH

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