Generalized Balance Equations - thermofluids.net
Mass:
dm
dt
m

e
i
Rate of increase of mass
in an open system.
e
Mass flow rate
into the system.
where, m   AV 
 kg 
 s  ;
m

i
(1)
Mass flow rate
out of the system.
 kg 
 K 
AV
v
Energy:
dE
dt
m j


i i
i
Rate of increase
of stored energy
of the open system.
m
j
e e
e
Energy transported
by mass flow in.
Energy transported
by mass flow out.


Q
Rate of heat
transfer into
the system.
 kW 
Wext
(2)
Rate of external
work transfer
out of the system.
V2
gz  kJ 
where, j  h  ke  pe  h 

2000 1000  kg 
N
VI
and Wext  Wsh  Wel  WB ; Wsh  2 T ; Wel 
60
1000
W
WB   pdV  kJ  ; WB  lim B ;  kW 
t 0 t
 kW 
Entropy:
dS
dt
Rate of increase
of entropy for
an open system.

m s
i i
i
Entropy
transported by
mass flow in.

m s
e e
e
Entropy
transported by
mass flow out.

Q
TB
Entropy
transferred
by heat.
where, according to the second law, Sgen  0

Sgen
Rate of
generation of
entropy inside
the system
boundary.
 kW 
 K 
(3)
Customized Balance Equations - thermofluids.net
Closed Steady Systems (Wall, Light bulb, Laptop adapter, Gear box, closed cycles)
Mass Equation: m  constant
Energy Equation: 0  Q  Wext
Entropy Equation: 0 
(1)
kW
where, Wext  WB  Wsh  Wel
(2)
Q
 kW 
 Sgen 
; Second law asserts: Sgen  0
TB
 K 
(3)
Single-Flow Open-Steady Systems (pumps, turbines, nozzles, valves, pipes, etc.)
 kg 
 s 
Energy: mje  mji  Q  Wext
Mass: me  mi  m
(1)
kW 
V2
gz

 kJ/kg  , Wext  WB  Wsh  Wel
2000 1000
Q
 kW 
where, by second law, Sgen  0
Entropy: mse  msi   Sgen 
TB
 K 
where, j  h  ke  pe  h 
 kW 
(2)
(3)
Closed Processes (Heating water in a tank, piston-cylinder compression)
Mass: m  constant  kg
(1)
Energy: E  E f  Eb  Q  Wext or, me  Q  Wext
where, e  u  ke  pe  u 
Entropy:
V2
gz

2000 1000
 kJ/kg ;
 kJ 
Wext  WB  Wsh  Wel
S  S f  Sb  Q / TB  Sgen or, ms  Q / TB  Sgen
kJ/K
kJ 
where, Sgen  0
(2)
(3)
Open Processes (Filling an evacuated tank, filling a propane cylinder, discharge from a tank)
Mass: m  m f  mb  mi  me
kg
Energy: E  E f  Eb  mi ji  me je  Q  Wext
where, e  u  ke  pe  u 
and Wext  WB  Wsh  Wel
(1)
 kJ 
V2
gz
V2
gz

, j  h  ke  pe  h 

2000 1000
2000 1000
 kJ 
Entropy: S  S f  Sb  mi si  me se  Q / TB  Sgen where, Sgen  0
 kJ/kg 
(2)
(3)
Manual State Evaluation
thermofluids.net>Tables
General State Related Equations: ( applies to any substance)
V2
1
gz
; ke 
; pe 
; e  u  ke  pe ; j  h  ke  pe ; h  u  pv
v
1000
2000
E  me ;; S  ms ; KE  m  ke  ; PE  m  pe 
m  AV ; V  AV ; E  m e ; S  m s ;
m  V ;  
(1)
(2)
(3)
Tds  du  pdv  dh  vdp ; cv   u / T v ; c p   h / T  p
(4)
  constant cv =constant: see Tables>Table-A)
u  u2  u1  c(T2  T1 ) ;
cv  c p  c;
h  h2  h1  (u  pv)  u  (pv)  c(T2  T1 )  v(p2  p1 )
(5)
T
s  c p ln 2
T1
(7)
SL Model: (Assumptions:
PG Model: (Assumptions: p 
(6)
 RT ; cv =constant: see Tables>Table-C)
RT m
m R
m T
T
R
where R 
 RT 
T
R  nR ,
v
V
V M
M V
V
M
u  u2  u1  cv (T2  T1 ) , h  h2  h1  c p(T2  T1) , where
c p  (cv  R)
c
T
p
T
v
kR
R
s  c p ln 2  R ln 2 ; s  cv ln 2  R ln 2 ; also, k  p , c p 
; and cv 
T1
p1
T1
v1
k 1
k 1
cv
p   RT 
k
k
k
k
k 1
p  T  k 1     v   V  T  p  k  v 
s  con process: 2   2    2    1    1  ; 2   2    1 
p1  T1 
 1   v2   V2  T1  p1 
 v2 
For polytropic process replace k with n
IG Model: (Assumptions: p   RT ; cv is function of T : see Tables>Table-D)
k 1
(8)
(9)
(10)
k
p v 
; 2   1  ; (11)
p1  v2 
RT m
m R
m T
T
 RT 
T
R  nR
v
V
V M
M V
V
h  h T  , u  u T  s  s  p, T  (use ideal gas tables); c p  cv  R
p   RT 
(12)
(13)
The temperature dependent part of entropy is separated from the pressure dependent part:
T2
s   cp
T1
dT
p
p
0
 R ln 2  so (T2 )  so (T1 )  R ln 2 , where s (T ) is tabulated against T . (14)
T
p1
p1
PC Model: (see Tables>Table-B) Determine the phase, L, V or M, of the fluid. For vapor use superheated Table.
For mixture, use saturation table (if the quality is not known, your goal should be to evaluate the quality
first which is the key to finding all specific properties of a mixture). For liquid use the CL sub-model.
CL Sub-Model: v , u and s depend on T only. Therefore, use the temperature-sorted saturation table to
obtain v  v f @ T , u  u f @ T or s  s f @ T . To find h , use h  u  pv  u f @ T  pv f @ T .
RG Model: (see Tables>Table-E)
p  Z ( pr , Tr ) RT where Z, the compressibility factor, is obtained from a
chart.
pr and Tr are pressure and temperature normalized by the corresponding critical properties. Just
like entropy in the PG or IG model, h and u also have two parts, one temperature dependent and another
pressure dependent, in the RG model. The departure of these values from the corresponding IG values are
tabulated in the enthalpy and entropy departure charts as functions of pr and Tr . Therefore, the complete
state can be evaluated if
pr and Tr are given.