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# Formula Sheet: Thermodynamics Mechanical Engineering Notes | EduRev

## Mechanical Engineering : Formula Sheet: Thermodynamics Mechanical Engineering Notes | EduRev

``` Page 1

Mechanical Engineering – GATE Exam

Thermodynamics

Symbol/Formula Parameter
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
);
Enthalpy per unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the
internal energy per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
Page 2

Mechanical Engineering – GATE Exam

Thermodynamics

Symbol/Formula Parameter
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
);
Enthalpy per unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the
internal energy per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
Mechanical Engineering – GATE Exam

2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume.(ml
2
t
-3
)
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
); we also have the enthalpy per
unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the internal energy
per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
Page 3

Mechanical Engineering – GATE Exam

Thermodynamics

Symbol/Formula Parameter
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
);
Enthalpy per unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the
internal energy per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
Mechanical Engineering – GATE Exam

2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume.(ml
2
t
-3
)
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
); we also have the enthalpy per
unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the internal energy
per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
Mechanical Engineering – GATE Exam

W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume.(ml
2
t
-3
)
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
); we also have the specific volume or volume per unit
mass, v (L
3
M
-1
) and the volume per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
); we also have the internal
energy per unit mass, u (L
2
T
-2
), and the internal energy per unit
mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
); we also have the enthalpy per
unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the internal energy
per unit mole h (ML
2
T
-2
?
-1
)
Page 4

Mechanical Engineering – GATE Exam

Thermodynamics

Symbol/Formula Parameter
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
);
Enthalpy per unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the
internal energy per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
Mechanical Engineering – GATE Exam

2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume.(ml
2
t
-3
)
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
); we also have the enthalpy per
unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the internal energy
per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
Mechanical Engineering – GATE Exam

W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume.(ml
2
t
-3
)
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
); we also have the specific volume or volume per unit
mass, v (L
3
M
-1
) and the volume per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
); we also have the internal
energy per unit mass, u (L
2
T
-2
), and the internal energy per unit
mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
); we also have the enthalpy per
unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the internal energy
per unit mole h (ML
2
T
-2
?
-1
)
Mechanical Engineering – GATE Exam

S Entropy (ML
2
T
-2
T
-1
); we also have the entropy per unit mass, s(L
2
T
-
2
T
-1
) and the internal energy per unit mole s (ML
2
T
-2
T
-1
?
-1
)
W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume. (ml
2
t
-3
)

Unit conversion factors
For metric units
? Basic:
o 1 N = 1 kg·m/s
2
;
o 1 J = 1 N·m;
o 1 W = 1 J/s;
o 1 Pa = 1 N/m
2
.
? Others:
o 1 kPa·m
3
= 1 kJ;
o T(K) = T(
o
C) + 273.15;
o 1 L (liter) = 0.001 m
3
;
o 1 m
2
/s
2
= 1 J/kg.
? Prefixes (and abbreviations):
o nano(n) – 10
-9
;
o micro( ?) – 10
-6
;
o milli(m) – 10
-3
;
o kilo(k) – 10
3
;
o mega(M) – 10
6
;
o giga(G) – 10
9
.
o A metric ton (European word: tonne) is 1000 kg.
For engineering units
Page 5

Mechanical Engineering – GATE Exam

Thermodynamics

Symbol/Formula Parameter
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
);
Enthalpy per unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the
internal energy per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
Mechanical Engineering – GATE Exam

2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume.(ml
2
t
-3
)
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
);
Specific volume or volume per unit mass, v (L
3
M
-1
) and the volume
per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
);
Internal energy per unit mass, u (L
2
T
-2
), and the internal energy per
unit mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
); we also have the enthalpy per
unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the internal energy
per unit mole h (ML
2
T
-2
?
-1
)
S Entropy (ML
2
T
-2
T
-1
);
Entropy per unit mass, s(L
2
T
-2
T
-1
) and the internal energy per unit
mole s (ML
2
T
-2
T
-1
?
-1
)
Mechanical Engineering – GATE Exam

W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume.(ml
2
t
-3
)
M Molar mass  (M/ ?)
m Mass (M)
M
m
n ?
Number of moles ( ?)
E Energy or general extensive property
m
E
e ?
Specific molar energy (energy per unit mass) or general extensive
property per unit mass
eM
n
E
e ? ?
Specific energy (energy per unit mole) or general extensive
property per unit mole
P Pressure (ML
-1
T
-2
)
V Volume (L
3
); we also have the specific volume or volume per unit
mass, v (L
3
M
-1
) and the volume per unit mole v (L
3
?
-1
)
T Temperature ( T)
? ? Density (ML
-3
); ? = 1/v.
x Quality
U Thermodynamic internal energy (ML
2
T
-2
); we also have the internal
energy per unit mass, u (L
2
T
-2
), and the internal energy per unit
mole, u (ML
2
T
-2
?
-1
)
H = U + PV Thermodynamic enthalpy (ML
2
T
-2
); we also have the enthalpy per
unit mass, h = u + Pv (dimensions: L
2
T
-2
) and the internal energy
per unit mole h (ML
2
T
-2
?
-1
)
Mechanical Engineering – GATE Exam

S Entropy (ML
2
T
-2
T
-1
); we also have the entropy per unit mass, s(L
2
T
-
2
T
-1
) and the internal energy per unit mole s (ML
2
T
-2
T
-1
?
-1
)
W Work (ML
2
T
-2
)
Q Heat transfer (ML
2
T
-2
)
u
W
?
:
The useful work rate or mechanical power (ML
2
T
-3
)
m ? : The mass flow rate (MT
-1
)
2
2
V
?
:
The kinetic energy per unit mass (L
2
T
-2
)
gz: The potential energy per unit mass (L
2
T
-2
)
E
tot
:
The total energy = m(u +
2
2
V
?
+ gz)  (ML
2
T
-2
)
Q
?
:
The heat transfer rate (ML
2
T
-3
)
dE
cv
dt
:
The rate of change of energy for the control volume. (ml
2
t
-3
)

Unit conversion factors
For metric units
? Basic:
o 1 N = 1 kg·m/s
2
;
o 1 J = 1 N·m;
o 1 W = 1 J/s;
o 1 Pa = 1 N/m
2
.
? Others:
o 1 kPa·m
3
= 1 kJ;
o T(K) = T(
o
C) + 273.15;
o 1 L (liter) = 0.001 m
3
;
o 1 m
2
/s
2
= 1 J/kg.
? Prefixes (and abbreviations):
o nano(n) – 10
-9
;
o micro( ?) – 10
-6
;
o milli(m) – 10
-3
;
o kilo(k) – 10
3
;
o mega(M) – 10
6
;
o giga(G) – 10
9
.
o A metric ton (European word: tonne) is 1000 kg.
For engineering units
Mechanical Engineering – GATE Exam

? Energy:
o 1 Btu = 5.40395 psia·ft
3
= 778.169 ft·lb
f
= (1 kWh)/3412.14 = (1 hp·h )/2544.5  =
25,037 lb
m
·ft
2
/s
2
.
? Pressure:
o 1 psia = 1 lb
f
/in
2
= 144 psfa = 144 lb
f
/ft
2
.
? Others:
o T(R) = T(
o
F) + 459.67;
o 1 lb
f
= 32.174 lb
m
·ft/s
2
;
o 1 ton of refrigeration = 200 Btu/min.
Concepts & Definitions

Formula Units
Pressure F
P
A
?
Pa
? Units
2
1 1 / Pa N m ?
5
1 10 0.1 bar Pa Mpa ??

1 101325 atm Pa ?

Specific Volume V
v
m
?
3
/ m kg
Density m
V
? ?    ?
1
v
? ?
3
/ kg m
Static Pressure Variation
P gh ? ??

, ?? ? ? ? ?
Pa
Absolute Temperature ( ) ( ) 273.15 T K T C ? ? ?
Properties of a Pure Substance

Formula Units
Quality
vapor
tot
m
x
m
? (vapour mass fraction)
1
liquid
tot
m
x
m
?? (Liquid mass fraction)

Specific Volume
f fg
v v xv ??

3
/ m kg

Average Specific Volume
(1 )
fg
v x v xv ? ? ? (only two phase mixture)
3
/ m kg
Ideal –gas law
c
PP ??

c
TT ??

1 Z ?

? Equations
Pv RT ?

PV mRT nRT ??

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