The document discusses various thermodynamic processes including constant temperature, isothermal, isobaric, and adiabatic processes. It provides the equations of state and relationships between pressure, volume, temperature, internal energy, enthalpy, entropy, and work for both closed and open systems undergoing these processes. The summary focuses on defining the key thermodynamic processes and relating the relevant process variables using mathematical equations.

1. LECTURE UNIT 008
3. CONSTANT TEMPERATURE or ISOTHERMAL PROCESS (PV = c or T = c)
A. CONSIDERING A CLOSED OR NON-FLOW SYSTEM
ILLUSTRATION: (Piston-cylinder assembly perfectly cooled)
m
V
Q
P-v diagram T-s diagram
P T
1
1 2
2
v s
Equations:
1. PVT Relationships
a. General Gas Law Equation
(Equation of State)
PV = mRT
P1V1 P2V2
b. = = mR = constant
T1 T2
P1 V2
=
P2 V1
2. Work Non-Flow, WNF 2
General Formula: WNF = P dV
1
Where: PV = P1V1 = P2V2 = C
c
P=
2 V
WNF = c dV
V
1
= c (lnV2 - lnV1)
V2 P1
WNF = c ln = c ln
V1 P2
V2 P1
WNF = P1V1 ln = mRT1 ln
V1 P2
3. Heat Energy, Q
Using NFEE;
Q= U + WNF
Q = WNF
“The price of greatness is responsibility.”

2. 4. Change in Total Internal Energy, U
U = mCv(T2 - T1) = mCv( T)
U=0
5. Change in Total Enthalpy, H
H = mCp(T2 - T1) = mCp( T)
H=0
6. Change in Total Entropy, S
From; dQ = Tds
dQ
dS =
2
T
2
dS = 1 dQ = Q
T 1 T
1
V2
mRT1 ln
V1
S2 - S1 =
T
V2 P1
S = mR ln = mR ln
V1 P2
B. CONSIDERING AN OPEN OR STEADY-FLOW SYSTEM
ILLUSTRATION: WSF
PE1 PE2
KE1 KE2
TS
U1 U2
(PV = c)
Wf1 Wf2
Q
[Ein = Eout]
PE1 + KE1 + U1 + Wf1 + Q = PE2 + KE2 + U2 + Wf2 + WSF
Q = (PE1 - PE2) + (KE2 - KE1) + (U2 - U1) + (Wf2 - Wf1) + WSF
Q= PE + KE + U+ Wf + WSF
Recall: H= U+ Wf
Q= PE + KE + H + WSF
When: PE ˜ 0 (z1 ˜ z2)
˜ ˜
KE ˜ 0 (v1 ˜ v2)
˜
Q= H + WSF
But: H=0
Q = WSF
“There is nothing training cannot do. Nothing is above its reach.”

3. Gas Compressor
(Perfectly Cooled) P-v diagram
ma md
P
1 2
vc
TDC 3 2
v1
PV
vD
Wc
=
PV
c
Wc
=
c
BDC 1
EP 4
BP
Motor
IP
v
vc vD
v1
Theoretical Work or Power needed by the compressor
2
From:
WSF = - V dP
1
P2 P2
Wc = - P1V’1 ln = - maRaT1 ln , kW
P1 P1
4. ISENTROPIC PROCESS or REVERSIBLE ADIABATIC PROCESS (PVk = c or s = c), Q = 0
A. CONSIDERING A CLOSED OR NON-FLOW SYSTEM
ILLUSTRATION: (Piston-cylinder assembly perfectly insulated)
m
V
Q
P-v diagram T-s diagram
P T
1
1
PV
=
c
PV k
=
c
2 2
v s
Equations:
1. PVT Relationships
a. General Gas Law Equation
(Equation of State)
PV = mRT
k-1 k-1
b. T2 P2 k V1
= =
T1 P1 V2
“The achievements of an organization are the results of the combined efforts of
each individual.”

4. 2. Work Non-Flow, WNF 2
General Formula: WNF = P dV
1
Where: PVk = P1V1k = P2V2k = C
P = c V-k
2
WNF = cV-k dV
1
V2
V-k+1
=c
-k +1
V1
k.
P2V2 V2-k+1 - P1V1k . V1-k+1
=c
-k + 1
P2V2 - P1V1 = mR (T2 - T1)
WNF =
1-k 1-k
3. Heat Energy, Q
2
Q = Tds
1
Q=0
Hence, if we using NFEE, WNF can also be solved by:
Q= U + WNF
WNF = - U
4. Change in Total Internal Energy, U
U = mCv(T2 - T1) = mCv( T)
5. Change in Total Enthalpy, H
H = mCp(T2 - T1) = mCp( T)
6. Change in Total Entropy, S
2
From; Q = Tds
1
2
dQ
S=
1
T
S=0
B. CONSIDERING AN OPEN OR STEADY-FLOW SYSTEM
ILLUSTRATION: WSF
PE1 PE2
KE1 KE2
TS
U1 U2
(PVk = c)
Wf1 Wf2
Q=0
[Ein = Eout]
PE1 + KE1 + U1 + Wf1 + Q = PE2 + KE2 + U2 + Wf2 + WSF
“Leaders always find a way to make things happen.”

5. PE1 + KE1 + U1 + Wf1 + Q = PE2 + KE2 + U2 + Wf2 + WSF
Q = (PE1 - PE2) + (KE2 - KE1) + (U2 - U1) + (Wf2 - Wf1) + WSF
Q= PE + KE + U+ Wf + WSF
When: PE ˜ 0 (z1 ˜ z2)
KE ˜ 0 (v1 ˜ v2)
˜ ˜
Q= H + WSF
But: Q = 0
WSF = - H
From P-v diagram (Area under P-v Curve)
From the General Formula
2
WSF = - V dP Where: PVk = c
1 c
Vk =
P
1/k
V = c 1/k
P
2 V = c P-1/k
WSF = - cP-1/k dP
1
P2
P(-1/k + 1)
= -c
(-1/k + 1)
P1
(-1/k + 1)
P2 - P1(-1/k + 1)
= -c
(-1/k + 1)
But: C = P1/k V = Pa1/kV1 = P21/kV2
Substituting and solving simultaneously, we have:
P2V2 - P1V1
= -k
k-1
P2V2 - P1V1 mR (T2 - T1)
=k =k
1-k 1-k
Or;
WSF = k (WNF)
Gas Compressor
(Perfectly Insulated)
ma md
P-v diagram
2
P
1
vc
TDC 3 2
v1 W
PV =
PV =
vD
PV
k
c
PV =
=
c
k
c
c
Wc
c
1
BDC 4
EP
BP
Motor
v
IP
vc vD
Theoretical Work or Power needed by the compressor v1
2
From: WSF = - V dP
1
k P V’
WSF = P2V’1 - P1V’1 * 1 1
1-k P1V’1
k-1 k-1
k k
kP1V’1 P2 kmaRaT1 P2
WC = -1 = -1
1-k P1 1-k P1
“What happens to a man is less significant than what happens within him.”

6. 5. SPECIAL POLYTROPIC PROCESS (PVn = C)
A. CONSIDERING A CLOSED OR NON-FLOW SYSTEM
ILLUSTRATION: (Piston-cylinder assembly air-cooled)
m
V
Q
P-v diagram T-s diagram
P T
1
1
PV
=
PV c
n
=
c
PV k
=
c
2 2
v s
Equations:
1. PVT Relationships
a. General Gas Law Equation
(Equation of State)
PV = mRT
n-1 n-1
n
T2 P2 V1
b. = =
T1 P1 V2
Derivation:
P1V1n = P2V2n
mRT1 mRT2
(V1n) = (V2n)
V1 V2
T1(V1n-1) = T2(V2n-1)
n-1
T2 V1
=
T1 V2
And;
P1V1n = P2V2n
n n
mRT1 = P2 mRT2
P1
P1 P2
(mR)nT1n (mR)nT2n
P1 = P2
P1n P2n
T1n T2n
n-1 =
P1 P2n-1
n-1
n
T2 P2
=
T1 P1
“Unless we think of others and do something for them, we miss one of the
greatest sources of happiness.”

7. 2. Work Non-Flow, WNF
P2V2 - P1V1 = mR (T2 - T1)
WNF =
1-n 1-n
3. Heat Energy, Q
Using NFEE;
Q= U + WNF
Q = mCv T + mR T
1-n
(1 - n)mCv T + mR T
=
1-n
= mCv T - nmCv T + mR T
1-n
mCp T - nmCv T
=
1-n
Cp
Where: k=
Cv
Cp = k Cv
= m Cv k - n T
1-n
Q = mCn T
Cn = Cv k - n
1-n
4. Change in Total Internal Energy, U
U = mCv(T2 - T1) = mCv( T)
5. Change in Total Enthalpy, H
H = mCp(T2 - T1) = mCp( T)
6. Change in Total Entropy, S
From; dQ = Tds
2 2
dQ
dS =
1 1
T
s2
mCn dT
S =
s1 T 2
dQ
(S2 - S1) = mCn
1
T
T2
S = mCn ln T = mCn (lnT2 - lnT1)
T1
n-1 n-1
n
T2 P2 V1
S = mCn ln = mCn ln = mCn ln
T1 P1 V2
B. CONSIDERING AN OPEN OR STEADY-FLOW SYSTEM
WSF
ILLUSTRATION:
PE1
KE1 PE2
U1 TS KE2
Wf1 (PVn = c) U2
Wf2
Q
“Champions don’t become champions in the field-they are merely recognized there.”

8. [Ein = Eout]
PE1 + KE1 + U1 + Wf1 + Q = PE2 + KE2 + U2 + Wf2 + WSF
Q = (PE1 - PE2) + (KE2 - KE1) + (U2 - U1) + (Wf2 - Wf1) + WSF
Q= PE + KE + U+ Wf + WSF
When: PE ˜ 0 (z1 ˜ z2)
KE ˜ 0 (v1 ˜ v2)
˜ ˜
Q= H + WSF
WSF = Q - H
From P-v diagram (Area under P-v Curve)
From the General Formula 2
WSF = - V dP
1
P2V2 - P1V1 mR (T2 - T1)
=n =n
1-n 1-n
Or;
WSF = n (WNF)
Gas Compressor
ma md
(Air-Cooled) P-v diagram
1 2
P
vc
TDC
3 2
v1
vD
PV =
n
PV =
PV
Wc Wc
PV
k
c
=c
PV =
=
c
k
PV
c
n
BDC
EP
=
c
c
1
BP 4
Motor
IP
v
vc vD
Theoretical Work or Power needed by the compressor
v1
2
From: WSF = - V dP
1
k P1V’1
WSF = P2V’1 - P1V’1 *
1-n P1V’1
n-1 n-1
nP1V’1 n n
P2 nmaRaT1 P2
WC = -1 = -1
1-n P1 1-n P1
Output 1. Motor Efficiency
Efficiency = BP 100%
Input mo=n EP *
2. Mechanical Efficiency
BP
M=n EP *
100%
3. Compression Efficiency
Wc
C=n IP *
100%
Wc = depends on the process involved.
WcT=c = Isothermal process
Wcs=c = Isentropic process
4. Overall Efficiency WcPV n =c = Polytropic process
n = Wc * 100%
o
BP
n = n n * 100%
o C M
“Whoever gives heed to instruction prospers, and blessed is he who trusts in the Lord.”
Proverbs 16:20