VARIOUS STEADY FLOW PROCESSES FOR AN IDEAL GAS
Assume that fluid a undergoes a steady flow process from inlet to exit, the energy equation for steady flow process on per unit mass basis is given as
q = (he - hi) + (Ce2/2 - Ci2/2) + (gZe - gZi )+ ws ;
If the kinetic energy and potential energy changes are negligible, the above equation reduces to
q = (he - hi) + ws ; ……………………(10.1)
Work done or shaft work, ws = -vdp - (Ce2/2 - Ci2/2) - (gZe - gZi )
If the kinetic energy and potential energy changes are negligible, then the above equation is reduced to
ws = -vdp ……………………(10.2)
10.1. Constant volume process (Isochoric process): vi = ve
v = constant = c
From eqn. (10.2), ws = - [v.(pe - pi)] = v.(pi - pe) kJ/kg ……….(10.3)
For an ideal gas (he - hi) = Cp (Te - Ti) ………...…………(10.4)
Substituting equation (10.3) and (10.4) in eqn. (10.1), we get
q = Cp (Te-Ti) + v (pi - pe)
10.2. Constant pressure process (Isobaric process): pi = pe
p = constant (c)
From eqn. (10.2), ws = 0 ……..………. (10.5)
For an ideal gas (he - hi) = Cp (Te-Ti) …..………… (10.6)
Put (10.5) and (10.6) in eqn. (10.1)
q = Cp (Te-Ti)
10.3. Constant temperature process (Isothermal process): Te = Ti
pv = constant (c)
and pv = RTi or v =
From eqn. (10.2), ws == = ……….(10.7)
For ideal gas, we have (he - hi) = Cp (Te -Ti)
Therefore, (he - hi) = Cp (Te -Ti) = 0 …………(10.8)
Using equations (10.7) and (10.8) in eqn. (10.1), we get
or
10.4. Adiabatic process
q = 0
or
From eqn. (10.2),
ws
= …….…. (10.9)
For ideal gas, (he hi) = Cp (Te-Ti) …………(10.10)
Using equations (10.9) and (10.10) in eqn. (10.1), we get
q = Cp (Te-Ti) + = 0
10.5. Polytropic process
pvn = c or
From eqn. (10.2),
ws
………....(10.11)
For ideal gas, (he - hi) = Cp (Te-Ti) …………(10.12)
put (10.11) and (10.12) in eqn. (10.1)
10.6. Throttling Process
It is a process in which fluid flows across some restriction, as shown in Fig. 10.1, in such a manner that there is always a drop of pressure without any change in kinetic energy, potential energy of the fluid.
During throttling process, there is
no heat transfer as the process is so fast that there is no time for the heat to transfer i.e. = 0
no work done i.e. = 0
Refer. Gas is made to pass through a restriction from high to low pressure side.
Fig. 10.1. The throttling process.
Energy Equation for steady flow process:
( hi + Ci2/2 + gZi) = ( he + Ce2/2 + gZe) + .............................……....(10.13)
By using throating conditions, the equation (10.13) is reduced to
h i = he
The enthalpy before and after throttling remain equal. Therefore, the throttling process is also called an equal-enthalpy process.
A coefficient called Joule-Thomson Coefficient which is a thermodynamic property is defined as
μJ =
As the value of change in pressure is always –ve because pe < pi ,
so the +ve value of Joule-Thomson Coefficient (μJ) means fall in temperature after throttling
and the –ve value of Joule-Thomson Coefficient (μJ) means rise of temperature after throttling.
10.6.1. Throttling Process for ideal gases:
dh = Cp dT
Therefore, (he - hi) = Cp (Te - Ti) ................................................…………(10.14)
As for throttling, he = hi
Therefore from eqn. (10.14), Ti = Te
From Joule-Thomson Coefficient definition, the μJ = zero.
Problem 10.1: 5 kg of air is compressed in a reversible steady flow polytropic process from 100 kpa and 40°C to 1000 kpa and during this process the law followed by the gas is pV1.25 = C. Determine the shaft work, heat transferred and the change in entropy CV = 0.717 kJ/kgK , R = 0.287 kJ/kgK.
Solution:
Given: m = 5 kg; Ti= 40°C = 40 + 273 = 313 K; Pi = 100 kPa = 1 x 105 N/m2 ;
Pe = 1000 kPa = 1 x 106 N/m2; n = 1.25
CP = R + CV or CP = 0.287 + 0.717 = 1.005 kJ/kgK
Determine shaft work, ws:
Formula: Shaft work ws =
=
Finding unknown, Te;
Equation of state
Therefore,
or Te = = 496 K
Answer: Shaft work, ws =
=
= 5 x 0.287 x (-183) = 262.6 kJ/kgK
Total shaft work, Ws = m ws = 5 x -262.6 = 1313 kJ
Determine heat transfer, iQe :
Formula: Apply eqn. of SSSF process
Hi + iQe = He + W
iQe = (He – Hi)+ W
= m CP(Te – Ti) + W
Answer: iQe = 5 x 1.005 x (496 – 313) – 1313 = 393.4 kJ
Problem 10.2: An axial flow compressor of a gas turbine plant receives air from atmosphere at a pressure 1 bar, temperature 300 K and velocity 60 m/s. At the discharge of compressor the pressure is 5 bar and the velocity is 100 m/s. The mass flow rate through the compressor is 20 kg/s. Assuming isentropic compressor, calculate the power required to drive the compressor. Also calculate the inlet and outlet pipe diameters.
Solution:
Given: pi = 1 bar; Ti = 300K; Ci = 60 m/s
Pe = 5 bar; Ce = 100 m/s
= 20 kg/s ; Isentropic compression
(a) Determine power required to drive compressor, ;
Formula: From first law of thermodynamics for steady flow process neglecting heat loss and change in P.E. and each term is expressed in kJ/kg, we have
or hi + Ci2/(2x1000) = he + Ce2/(2x1000) +
= - [(he - hi) + (Ce2- Ci2)/(2x1000)] = - [cp (Te - Ti) + (Ce2- Ci2)/(2x1000)]
Finding unknown, Te ;
For isentropic process, the final temperature is given by
= = 475.14 K
Answer: = - [cp (Te - Ti) + (Ce2- Ci2)/(2x1000)]
= - [1.005 (475.14 - 300) + (1002 - 602)/(2x1000)] = - 179.21 KJ/kg
(b) Determine the inlet and outlet pipe diameters:
Formula: Inlet diameter, Outlet diameter ,
Finding unknown Ai and Ae,
From continuity equation,
and
Finding unknown, ρi
or = 1.1614 kg/m3
Finding unknown, ρe
or = 3.67 kg/m3
Answer: = 0.604 m
= 0.263 m
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