MECHANICAL ENGINEERING

THERMODYNAMICS
 Ideal Gas Equation
PV  nRT Values for Universal Gas
constant
P  MW 
 ft  lb f
RT R  8.314
J
 1545
mol  K lbmol  R
V RT
v   0.08206
Li  atm
 10.73
ft 3  psia
m P(MW ) mol  K lbmol  R
cal BTU
 1.98  1.98
mol  K lbmol  R
 Universal Gas Constants

Value of Universal Gas Constant:

R = 8.314 J/mol-K = 1545 ft-lbf/lbmol-R

3
= 0.08206 Li-atm/mol-K = 10.73 ft -psi/lbmol-R
= 1.986 cal/mol-K = 1.986 BTU/lbmol-R

For steam:
R = 0.462 kJ/kg-K = 85.8 ft-lbf/lb-R

For air:
R = 0.286 kJ/kg-K = 53.3 ft-lbf/lb-R
 PROPERTIES OF WATER AND AIR

Values of Specific Heats:

For water: C p (liquid)  4.186 kJ/kg-K = 1 BTU/lb-R

C p (solid)  2.1 kJ/kg-K = 0.5 BTU/lb-R
C p (vapour )  1.9 kJ/kg-K = 0.45 BTU/lb-R
For air: C p (air )  1 kJ/kg-K = 0.24 BTU/lb-R

Values of Latent Heat for Water:

at 0C, h fusion  80 cal/g = 33.5 kJ/kg

at 100C, hvaporization  540 cal/g = 2255 kJ/kg
 FIRST LAW OF THERMODYNAMICS

Work: W  V2 PdV
 V1

Heat: constant P: Q  nC p T2  T1 

constant V: Q  nCv (T2  T1 )

First Law of Thermodynamics U  Q  W

NON-FLOW PROCESSES
 NON-FLOW PROCESSES
Process W Q ΔU
Isobaric Pext V2  V1  Q  U  W nCv T2  T1 

Isothermal V
nR ln V2  nR ln
1
P2
P1
Q W 0

Isochoric 0 U  Q nCv T2  T1 

nR T2  T1  0
Adiabatic W U  W
1
 Polytropic and Adiabatic Processes

For polytropic processes

k 1 1k k
T2  P2  k T2  V2  P2  V2 
     
T1  P1  T1  V1  P1  V1 
For adiabatic processes, k = γ
Cp
Mayer relation:  C p  Cv  R
Cv
Values of Cv, Cp and Υ
 Pure Substances
Properties of Pure Substances

Quality of steam: mg
x
mg  ml

h  xhg  (1  x)hl s  xsg  (1  x)sl

u  xug  (1  x)ul v  xvg  (1  x)vl
 STEM TABLE SHORTHANDS

Steam Table Short-cuts: hf  4.186(T  273) kJ/kg

hg  1.636T  2064.2 kJ/kg all T in Kelvins,K

hfg  3207  2.55T kJ/kg

To convert to BTU/lb multiply by 0.43

h in BTU/lb = 0.43  (h in kJ/kg)
 FLOW PROCESSES

Continuity Equation (Mass Conservation): m  vA

where 𝐴𝑐 = cross-sectional area of pipe.

Engineering Bernoulli Equation:

v22  v12 g  z2  z1  P2  P1
    F  Q  Ws
2gc gc 

v22  v12 g  z2  z1 
  H2  H1  Q  Ws
2gc gc
Note: the kinetic and potential energy terms must be divided by g c
in all cases. That is gc = 1 for mks(SI) and gc = 32.174 for
fps(English)
Important Conversions!
2
m
1 2  1 kgJ
s
ft lb f
1 hp  746 W 1 hp  550 s
1 BTU  778 ft  lb f
 COMPRESSIBLE FLOW
Isothermal Flow: Adiabatic Flow:

v max   RT v max   RT

T0  k  1 2
 1 Ma
T 2

k /( k 1)
P0   k  1 2
 1  Ma 
P  2 
 ENTROPY AND SECOND LAW
Q
ENTROPY, S: S 
T

 T2   V2 
Entropy of Ideal Gas: S  nCV ln    nR ln  
 T1   V1 
 T2   P2 
S  nCP ln    nR ln  
 T1   P1 
T 
Entropy of Solids/Liquids: S  mC p ln  2 
 T1 
 CARNOT ENGINE
Ideal Carnot Engine:
QH  W  QC

W QH  QC TH  TC
  
QH QH TH
S  0
For non-ideal engines:
Availability(Exergy)  Wmax  QH  TH S
actual Wactual
Second Law Efficiency  II  
Carnot QH  TH S
 CARNOT REFRIGERATOR AND HEAT PUMP

Ideal Carnot Refrigerator & Heat Pump: Q  W  Q

H C

QC QC TC
Refrig: COPR   
W QH  QC TH  TC
QH QH TH
Heat Pump: COPhp   
W QH  QC TH  TC
 PROBLEM 1
Convert 750°R to K.

A. 390.33 K
B. 395.33 K
C. 410.33 K
D. 416.33 K
 PROBLEM 2
Determine the density of air at 760 mmHg
absolute and 22°C?

A. 1.014 kg/m3 B. 1.316 kg/m3

C. 1.197 kg/m3 D. 1.266 kg/m3
 PROBLEM 3
Determine the quality of steam in a vessel
containing 2 kg of saturated vapor and 8 kg of
saturated liquid.

A. 100% B. 20% C. 80% D. 60%

 PROBLEM 4
What is the resulting pressure when one
kilogram of air at 104 kPa and 98°C is heated at
constant volume to 450°C?
A. 202.67 kPa B. 194.67 kPa
C. 186.53 kPa D. 198.65 kPa
 PROBLEM 5
Find the enthalpy of water at 212°F and 14.7 psi
if the dryness factor is 30%. Use the
approximate enthalpy formula of liquid.

A. 461 Btu/lb B. 471 Btu/lb

C. 481 Btu/lb D. 491 Btu/lb
 PROBLEM 6
How much heat is required in order to vaporize
25 lbs of water from 104°F to steam at 257°F
with quality of 70%?

A. 14,500 BTU C. 23,400 BTU

B. 20,300 BTU D. 32, 500 BTU
 PROBLEM 7
120 kg of water at 30°C in a boiler is provided
with 272 MJ of heat. The final temperature is
140°C. Determine the quality of steam
produced:
A. 60% B. 80% C.90% D. 45%
 PROBLEM 8
Consider 1 kg of air at 32°C that expanded by a
reversible polytropic process with n = 1.25 until the
pressure is halved. Determine the heat transfer.
Specific heat at constant volume for air is 0.1786
kJ/kg-K.

A. 17.02 kJ heat rejected

B. 17.02 kJ heat added
C. 7.05 kJ heat rejected
D. 7.05 kJ heat added
 PROBLEM 9
Two kilogram of air in a rigid tank changes its
temperature from 32°C to 150°C. Find the work
done during the process.

A. 236 B. 170 C. 195 D. 0

 PROBLEM 10
A perfect gas has a value of R = 58.8 ft-lb/lb-°R
and k = 1.26. If 20 Btu are added to 10 lbs of this
gas at constant volume when initial temperature
is 90°F, find the final temperature.

A. 97°F B. 104°F C. 154°F D. 185°F

 PROBLEM 11
A tank contains 90 ft3 of air at a pressure of 350
psig; if the air is cooled until its pressure and
temperature decreases at 200 psig and 70°F
respectively, what is the decrease in internal
energy?
A. 6232.09 Btu B. -5552 Btu
C. 5552 Btu D. -6232.09 Btu
 PROBLEM 12
A mixture of 0.4 lbm of helium and 0.2 lbm of
oxygen is compressed polytropically from 14.7
psia and 60oF t0 60 psia according to n = 1.4.
Determine the polytropic work.
A. 139 Btu B. 239 Btu C. 339 Btu D. 539
Btu
 PROBLEM 13
Steam at 2 MPa and 250°C in a rigid cylinder is cooled
until the quality is 30%. Find the heat rejected from the
cylinder.

@ 2 Mpa and 250°C: υ = 0.11144 m3/kg u = 2679.6 kJ/kg

@ 2 Mpa, (saturated): υf = 0.0011767 m3/kg υg =
0.09963 m3/kg, uf = 906.44kJ/kg, ufg = 1693.8 kJ/kg

A. -423.23 kJ/kg B. -926.26 kJ/kg

C. -1265.02 kJ/kg D. 1082.34 kJ/kg
 PROBLEM 14
At 1.3 MPa, a mixture of steam and water has an
entropy of 3 kJ/kg-K, Find the enthalpy of the mixture.
@ 1.3MPa: sf = 2.2515 kJ/kg-K sg = 6.4952 kJ/kg-K
hf = 814.93 kJ/kg hfg = 1972.7 kJ/kg pl

A. 1627.71 kJ/kg B. 1533.33 kJ/kg

C. 1234.45 kJ/kg D. 1162.40 kJ/kg
 PROBLEM 15
A mixture with 70% quality at 500 kPa is heated isothermally
until its pressure is 300 kPa. Find the heat added during the
process.
@ 600 kPa: sf = 1.8607 kJ/kg-K sfg = 4.9606 kJ/kg-K
@ 300 kPa and 151.86°C: s = 7.0888 kJ/kg-K

A. 745.92 kJ/kg B. 535.16 kJ/kg pl

B. 982.44 kJ/kg D. 765.34 kJ/kg
 PROBLEM 16
A tank contains exactly one kilogram of water
consisting of liquid and vapor in equilibrium at 1 MPa. If
the liquid contains one-third and the remaining is vapor
of the volume of the tank, what is the enthalpy of the
contents of the tank?
@ 1MPa: υf = 0.0011273 m3/kg υfg = 0.19444 m3/kg
hf = 762.81 kJ/kg hfg = 2015.3 kJ/kg
Pl
A. 644.40 kJ/kg B. 774.40 kJ/kg pl
C. 785.92 kJ/kg D. 435.29 kJ/kg
 PROBLEM 17
Fifty kilograms of cooling water per second
enter the condenser at 25°C and leaves at 50°C.
Find the heat carried away by water.

A. 1234.45 kW B. 5233.75 kW
C. 2340.53 kW D. 3140.25 kW
 PROBLEM 18
The gain of entropy during isothermal nonflow
process of 5 lb of air at 60°F is 0.462 Btu/°R. Find
the V1/V2.

A. 3.85 B. 0.259 C. 1.0 D. 0.296

 PROBLEM 19
Air enters a diffuser with a velocity of 200 m/s.
Determine the velocity of sound if air
temperature is 30°C.

A. 349 m/s B. 359 m/s

C. 369 m/s D. 379 m/s
 PROBLEM 20
Steam enters a turbine stage with an enthalpy of
3700 kJ/kg and a velocity of 80 m/s and leaves
with an enthalpy of 2864 kJ/kg with a velocity of
12.8 m/s. If the rate of a steam flow through the
turbine is 0.44 kg/s, what is the work done in
kW?
A. 365 kW B. 365.64 kW
C. 366.0 kW D. 366.50 kW
 PROBLEM 21
A turbine receives 150 lbm/s of air at 63 psia
and 2450°R and expands it polytropically to 14.7
psia. The exponent n is equal to 1.45 for the
process. Determine the power.

A. 23,856 BTU/s C. 41,700 HP

B. 52,343.16 kW D. 53,343.16 ft-lb/s
 PROBLEM 22
A nozzle receives 0.5 kg/s of air at a pressure of
2700 kPa and a velocity of 30 m/s and with an
enthalpy of 923 kJ/kg, and the air leaves at a
pressure of 700 kPa and with an enthalpy of 660
kJ/kg. Determine the exit velocity from the
nozzle.
A. 923 m/s B. 726 m/s
C. 700 m/s D. 660 m/s
 PROBLEM 23
Air flows through a nozzle at a speed of 350 m/s.
Find the stagnation temperature if the entrance
temperature is 200°C.

A. 241.25°C B. 251.25°C
C. 261.25°C D. 271.25°C
 PROBLEM 24
Air flows through a nozzle with temperature of
entrance of 420 K stagnation temperature of
468 K. Find the Mach number.

A. 0.744 B. 0.755 C. 0.764 D. 0.774

 PROBLEM 25
A Carnot cycle has a sink temperature of 100°F
and a cycle efficiency of 70%. Find the
temperature of the heat source.

A. 1306.70°F B. 1406.70°F
C. 1506.70°F D. 1606.70°F
 PROBLEM 26
A Carnot cycle has a maximum temperature of
550°F and minimum temperature of 100°F. If the
heat added is 4200 Btu/min, find the
horsepower output of the engine.

A. 34.53 B. 40.56 C. 44.13 D. 65.40

 PROBLEM 27
Calculate the approximate enthalpy of water at
90°C.
A. 366.83 kJ/kg B. 376.83 kJ/kg pl
C. 386.83 kJ/kg D. 396.83 kJ/kg
 PROBLEM 28
Determine the degrees of superheat of steam at
101.325 kPa and 170°C.

A. 50°C B. 70°C C. 60°C D. 80°C

 PROBLEM 29
What is the mass of acetylene gas, V = 0.94 cu.
ft., R = 59.35 ft-lb/lb-°R, T = 90°F, P = 200 psia?

A. 0.816 lb B. 0.841 lb
C. 0.829 lb D. 0.852 lb
 PROBLEM 30
A reverse Carnot cycle requires 3 Hp and
extracts energy from a lake to heat a house. If
the house is kept at 70°F and requires 2000 Btu
per minute, what is the temperature of the lake?

A. 35.29°F B. 36.29°F
C. 39.29°F D. 40.29°F
 PROBLEM 31
A heat engine receives heat from a source at
1200 K at a rate of 500 kJ/s and rejects the
wasted heat to a sink at 300 K. If the power
output of the engine is 200 kW, the second law
efficiency of this heat engine is?

A. 35% B. 40% C. 53% D. 75%

 PROBLEM 32
Two kilogram of gas is confined in a 1 m3 tank at
200 kPa and 88°C. What type of gas is in the
tank?
A. Helium B. Ethane
C. Methane D. Ethene
 PROBLEM 33
Find the heat needed to raise the temperature
of water from 30°C to 100°C with 60% quality.
Consider an atmospheric pressure of 101.325
kPa.
A. 293.09 kJ/kg B. 1,772.90 kJ/kg
C. 1,547.90 kJ/kg D. 1,647.29 kJ/kg
 PROBLEM 34
Ammonia weighing 22 kg is confirmed inside a
cylinder equipped with a piston has an initial
pressure of 413 kPa at 38°C. If 3200 kJ of heat is
added to the ammonia until its final pressure
and temperature is 413 kPa and 100°C,
respectively. What is the amount of work done
by the fluid in kJ?
A. 667 B. 304 C. 420 D. 502
 PROBLEM 35
Work done by a substance in reversible non flow
manner in accordance with V = 100/P ft3, where
P is in psia. Evaluate the work done on or by the
substance as the pressure increases from 10 psia
to 100 psia.
A. 33,157.22 ft-lb C. 43,157.22 ft-lb
B. -33,157.22 ft-lb D. -43,157.22 ft-lb
 PROBLEM 36
Calculate the change in enthalpy as 1 kg of
nitrogen is heated from 1000 K to 1500 K,
assuming the nitrogen is an ideal gas at a
constant pressure. The temperature dependent
specific heat of nitrogen is Cp = 39.06 – 512.79
T1.5 + 1072.7 T2 – 820.4T3 where Cp is in kJ/kg-
mol, T is in K.
A. 600 kJ C. 800 kJ
B. 697.27 kJ D. 897.27 kJ
 PROBLEM 37
Wet saturated steam at 16 bar (hf = 859 kJ/kg ,
hgf = 1935 kJ/kg) reducing valve and is throttled
to a pressure of 8 bar (hr = 721 kJ/kg, hfg = 2048
kJ/kg). Find the dryness fraction of the reduces
pressure steam.
A. 0.8833 C. 0.9933
B. 0.7733 D. 0.6633
 PROBLEM 38
During a reversible process, there are abstracted
317 kJ/s from 1.134 kg/s of a certain gas while
the temperature remains constant at 26.7°C. For
this gas cp = 2.232 and cv = 1.713 kJ/kg-K. The
initial pressure is 586 kPa. Determine the final
volume flow rate.
A. 0.301 m3/s B. 0.03 m3/s
C. 0.5 m3/s D. 0.05 m3/s
 PROBLEM 39
A reciprocating compressor handles 1,400 cfm
of air measured at intake where P1 = 18 psia and
T1 = 90°F. The discharge pressure is 92 psia.
Calculate the work if the process of the
compression is isothermal.
A. -180.5 hp B. -179.5 hp
C. -227.6 hp D. -228.6 hp
 PROBLEM 40
Steam is expanded through a nozzle and the
enthalpy drop per kg of steam from the initial
pressure to the final pressure of 60 kJ. Neglecting
friction, find the velocity of discharge and the exit
area of the nozzle to pass 0.20 kg/s if the specific
volume of the steam at exit is 1.5 m3/kg.
A. 346.4 m/s, 879 mm2 C. 765.6 m/s, 467 mm2
B. 356.7 m/s, 278 mm2 D. 346.4 m/s, 866 mm2
 PROBLEM 41
A liquid with a specific gravity of 1.26 is being
pumped in a pipeline from A to B. At A, the pipe
diameter is 60 cm and the pressure is 300
kN/m2. At B, the pipe diameter is 30 cm and the
pressure is 330 kN/m2. Point B is 1.0 m lower
than A. Find the flow rate if the pump puts 16
kW into the flow. Neglect head loss.
A. 4.2 m3/s B. 0.42 m3/s
C. 2.4 m3/s D. 0.24 m3/s
 PROBLEM 42
1 ton of steam at 0.2 Mpa and 300degC is compressed in
a cylinder isobarically in order to condense some 30% of
the vapour to liquid water. Compute the heat removed
from the system:
At 0.2 Mpa, 300degC, u = 2808.6 kJ/kg-K, v = 1.3162
m3/kg
For sat’d vapour at 0.2 Mpa, uf = 504.5, ug = 2529.5, vf =
0.001061, vg = 0.8857

A. -2.333 MJ C. -0.5064 MJ
B. -1.025 MJ D. -0.1024 MJ
 PROBLEM 43
Helium gas inside a tank at 120F is leaking
through a nozzle to a 77F atmosphere. Compute
the Mach number of flow:
A. 0.56, subsonic
B. 0.49, subsonic
C. 1.33, supersonic
D. 1.67, supersonic
 PROBLEM 44
What is the stagnation pressure of the helium
gas inside the tank in Prob no. 43?

A. 19.2 psia
B. 11.6 psia
C. 21.8 psia
D. 33.7 psia
 PROBLEM 45
Air is compressed in a cylinder such that the volume
changes from 100 to 10 in3. The initial pressure is
50 psia and the temperature is held constant at 100
F. Calculate the work.

A. 1.67 Btu
B. 3.78 Btu
C. 1.23 Btu
D. 1.99 Btu
 PROBLEM 46
Determine the amount of heat necessary to
heat 0.2 lb/s water at 83F to steam at 212F with
quality of 67%?

A. 914,600 btu/hr
B. 255,800 btu/hr
C. 816,400 btu/hr
D. 180,333 btu/hr