fundamentals of engineering supplied-reference handbook - Ventech!

Special Cases of Closed Systems
Constant Pressure (Charles' Law): wb = P∆v
(ideal gas) T/v = constant
Constant Volume: wb = 0
(ideal gas) T/P = constant
Isentropic (ideal gas), Pv k = constant:
Constant Temperature (Boyle's Law):
w = (P2v2 – P1v1)/(1 – k)
= R (T2 – T1)/(1 – k)
(ideal gas) Pv = constant
wb = RTln (v2 / v1) = RTln (P1 /P2)
Polytropic (ideal gas), Pv n = constant:
w = (P2v2 – P1v1)/(1 – n)
Open Thermodynamic System
Mass to cross the system boundary
There is flow work (PV) done by mass entering the system.
The reversible flow work is given by:
wrev = – ∫ v dP + ∆KE + ∆PE
First Law applies whether or not processes are reversible.
FIRST LAW (energy balance)
2
i
2
e
Σm � [ h + V 2 + gZ ] − Σm�
[ h + V 2 + gZ
i
Qin net s s
i
+ � −W�
= d ( m u ) dt , where
Wnet � = rate of net or shaft work transfer,
ms = mass of fluid within the system,
us = specific internal energy of system, and
Q� = rate of heat transfer (neglecting kinetic and
potential energy).
Special Cases of Open Systems
Constant Volume: wrev = – v (P2 – P1)
Constant Pressure: wrev = 0
Constant Temperature: (ideal gas) Pv = constant:
wrev = RTln (v2 /v1) = RTln (P1 /P2)
Isentropic (ideal gas): Pv k = constant:
wrev = k (P2v2 – P1v1)/(1 – k)
= kR (T2 – T1)/(1 – k)
w
rev
e
⎡
( k −1) k
k ⎛ P ⎞ ⎤
⎢
⎥
⎢
⎜ 2
= RT1
−
⎟
k −
⎥
⎣ ⎝ P1
⎠ ⎦
1
1
Polytropic: Pv n = constant
wrev = n (P2v2 – P1v1)/(1 – n)
e
]
57
THERMODYNAMICS (continued)
Steady-State Systems
The system does not change state with time. This
assumption is valid for steady operation of turbines, pumps,
compressors, throttling valves, nozzles, and heat
exchangers, including boilers and condensers.
∑ m�
i
2
2
( h + V 2 + gZ ) − ∑ m�
( h + V 2 + gZ )
i
∑ m�
= ∑ m�
i
i
e
i
e
+ Q�
in
e
−W�
e
out
= 0
e
and
m� =
where
mass flow rate (subscripts i and e refer to inlet and
exit states of system),
g = acceleration of gravity,
Z = elevation,
V = velocity, and
W� = rate of work.
Special Cases of Steady-Flow Energy Equation
Nozzles, Diffusers: Velocity terms are significant. No
elevation change, no heat transfer, and no work. Single mass
stream.
hi + Vi 2 /2 = he + Ve 2 /2
Efficiency (nozzle) =
V
2
2
e
( h − h )
i
−V
2
i
es
, where
hes = enthalpy at isentropic exit state.
Turbines, Pumps, Compressors: Often considered adiabatic
(no heat transfer). Velocity terms usually can be ignored.
There are significant work terms and a single mass stream.
hi = he + w
hi
− he
Efficiency (turbine) =
h − h
Efficiency (compressor, pump) =
i
e
es
hes
− hi
h − h
Throttling Valves and Throttling Processes: No work, no
heat transfer, and single-mass stream. Velocity terms are
often insignificant.
hi = he
Boilers, Condensers, Evaporators, One Side in a Heat
Exchanger: Heat transfer terms are significant. For a singlemass
stream, the following applies:
hi + q = he
Heat Exchangers: No heat or work. Two separate flow
m� :
rates 1 m� and 2
�
( h − h ) = m�
( h − h )
m1 1i
1e
2 2e
2i
i