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Questions and solutions. 1-Brayton cycle. 2-combustion 3_Rocket Thrust
Typology: Quizzes
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Deadline: 16.12.
Consider air standard ideal Brayton cycle. The cycle parameters are given as θ=2 and α=6.
Compare the increase in the net work output and in the thermal efficiency when we double θ as
Compare net work output and thermal efficiency
Assumptions
Figure 1: Ideal Brayton Cycle
1 - 2 Isentropic compression (in a compressor)
2 - 3 Constant-pressure heat addition
3 - 4 Isentropic expansion (in a turbine)
4 - 1 Constant-pressure heat rejection
The T-s and P-v diagrams of an ideal Brayton cycle are shown in Fig. 1. Notice that all four
processes of the Brayton cycle are executed in steady-flow devices; thus, they should be
analyzed as steady-flow processes. When the changes in kinetic and potential energies are
neglected, the energy balance for a steady-flow process can be expressed, on a unit-mass basis,
as
q in^ ^ qout^ ^ win^ ^ wout^ ^ hexit^ hinlet Eq.1.
Therefore, heat transfers to and from the working fluid are
qin h 3 h 2 c (^) P T 3 T 2 Eq.1.
And
qout h 4 h 1 c (^) P T 4 T 1 Eq.1.
First case
wnet 1.005 (kJ / kg K. ) 298( K )(6 2)(1 0.5)
wnet 598.98( kJ / kg)
Second case
wnet 1.005 (kJ / kg K. ) 298( K)(6 4)(1 0.25)
wnet 449.235( kJ / kg)
General case
Discussion
will be value of for which has maximum value. For becomes wnetmaximum. For
compressor is doing more work. So there is an optimum level for wnet. This is the case where
Ethane (C 2 H 6 ) at 25°C is burned in a steady-flow combustion chamber at a rate of 5 kg/h with
the stoichiometric amount of air, which is preheated to 500 K before entering the combustion
chamber. An analysis of the combustion gases reveals that all the hydrogen in the fuel burns to
H 2 O but only 95 percent of the carbon burns to CO 2 , the remaining 5 percent forming CO. If
the products leave the combustion chamber at 800 K, determine the rate of heat transfer from
the combustion chamber.
Ethane gas is burned with stoichiometric amount of air during a steady-flow combustion
process. The rate of heat transfer is to be determined.
Assumptions
Steady operating conditions, Ideal gases, Kinetic and potential energies are negligible,
Combustion is complete
Solving the problem
2 6 (25 )
800 500 1
out
C H C Combustion products Air chamber K K atm
Q
The molar mass of C H 2 6 is 30 kg /kmoland theoretical combustion equation of C H 2 6 is
C H 2 6 k O( 2 3.76 N 2 ) 2 CO 2 3 H O 2 3.76k N 2
Where k is the stoichiometric coefficient and is determined from the O 2 balance,
k 2 1.5 3.
Then equation becomes,
The heat transfer for this process is determined from the energy balance
out P (^ f f )^ R (^ f f) P R
Q (^) N h h h (^) N h h h Eq.2.
Assuming the air and the combustion products to be ideal gases. From the above table
we can determine the enthalpies of formation of gases at different temperatures.
A rocket motor burns propellant at a rate of 50 kg/s. The exhaust speed is 3500 m/s and the
nozzle is perfectly expanded. Calculate
(a) the rocket thrust in kN
(b) the rocket motor specific impulse Isp (s).
a) Rocket Thrust
The rocket produces thrust, T, by expelling propellant mass from a thrust chamber with a
nozzle. The propellant (fuel + oxidizer) mass flow rate is mand the ambient pressure of the
surrounding air is P 0.
Figure 4: Rocket thrust schematic.
Ae = nozzle exit area
Pe = area averaged exit gas pressure
e = area averaged exit gas density
Ve = area averaged x − component of velocity at the nozzle exit
Force balance of the rocket is;
2
Then our rocket thrust formula is;
2
The propellant mass flow is;
e e e
and the rocket thrust formula is often written
T mVe ( Pe P 0 )Ae
If the nozzle exit pressure Pe< P 0 the nozzle is over expanded.
Pe =P 0 Perfectly Expanded
Pe >P 0 Under Expanded
In this question the nozzle perfectly expanded so the rocket thrust equation becomes;
T mV
T 50 kg / s (3500 m / s)
T 175, 000 N 175 kN
b) Specific Impulse
The specific impulse defined as the thrust per unit weight flow of propellant and so the
gravitational acceleration at the surface of the Earth is always inserted. The specific impulse is
defined as
(Rocket Thrust Eq.) T mVe ( Pe P 0 )Ae
Equivalent Velocity
e e e
m m
In this question C Ve because the nozzle is perfectly expanded
I Total ımpulse F t F dt mC dt mC
Total Impulse Specific Impulse: Weight
0 0
sp
mg g
In this question, the rocket is assumed at sea level. Therefore, the gravitational acceleration is
taken as 9.81.
2
sp
m s I s m s
