Solving Thermodynamics Problems – Part II (Entropy)
This is a follow on to the previous problem solving notes posted previously. These
notes are focused on the additional analysis tools afforded by the 2nd law. Keywords that
will imply a 2nd law solution (although it does not guarantee a 2nd law analysis is needed)
are:
“minimum”, “maximum”, “ideal”, “reversible”, component “’s”, cycle
efficiencies, cycle COP’s, “isentropic”, etc. In general, the 2nd law will be involved with
problems that have a defined path, but one of the end states is not readily known and not
all the heat or work interactions are known.
The solution process should follow the same steps from the previous notes, except
you now have the entropy balance that can be applied along with the mass balance and
the energy balance. The entropy balance is:

Q
j
ds
  m
 m
  isi   m ese  s gen
dt
Tj
for an open system, and:
s  
Qj
m s
gen
Tj
for a closed system.
In addition to the entropy balance expressions above, 2nd law analysis brings with it
many new definitions/principles and substance models for finding entropy. See the
catalogs in the appendix for ways to find entropy, definitions of cycle metrics, definitions
of component metrics, and definitions of some cycles.
2nd Law Principles
Clausius Statement: Heat can only flow from hot to cold.
Kelvin-Planck Statement: A heat engine must exchange heat with at least 2 thermal
reservoirs of different temperatures.
Clausius Inequality
Q j 
 Tj   0
 cycle
The limiting case of 0 is for a reversible process.
Definition of Entropy:
Q 
dS 

T  rev
Gary L. Solbrekken
ME3324, Fall 2002
10/01/02
For a Real Process with Irreversibilities:
Q
dS 
 Sgen
T
where:
Sgen > 0  possible, irreversible process
Sgen = 0  limiting, reversible process
Sgen < 0  impossible process
Note: Sgen is ALWAYS your metric to determine if a process is possible or not!
Air Standard Assumptions:
1) working fluid is air that can be treated as an ideal gas
2) all processes are internally reversible
3) combustion processes are modeled as heat addition processes
4) exhaust processes are modeled as heat rejection processes
Other Useful Definitions:
Pressure ratio:
Compression ratio:
Gary L. Solbrekken
rp = p2/p1 (usually used for open system processes)
r = Vmax/Vmin (usually used for closed system processes)
ME3324, Fall 2002
10/01/02
Appendix
This appendix contains a series of catalogs for common parameters that are needed in
solving thermodynamics problems. The lists are not intended to be exhaustive, nor is the
information contained in the catalogs complete. For a complete discussion of a particular
entry, a reference from the Cengel and Turner (C&T) book is included with the chapter
and section identified. Note that the symbols used take on the context of the problem.
Again, the user should consult C&T for details.
Catalog of Cycle Metrics - Real
Cycle Type
Heat Engine (C & T ch 6.4)
Metric

W
W
Q
net
 net  1  out

Qin
Qin
Qin

Q
Q
Qcold
COPR  cold  cold 

W
W
Q Q
th 
Refrigerator (C & T ch 6.5)
in
Heat Pump (C & T ch 6.5)
COPHP 
in
hot
cold

Q hot Q
Q hot
 hot 

Win W
Q hot  Qcold
in
Catalog of Cycle Metrics – Ideal/Carnot
Cycle Type
Heat Engine (C & T ch 6.11)
Metric
th,Carnot  1
Tcold
Thot
Refrigerator (C & T ch 6.12)
COPR ,Carnot 
Tcold
Thot  Tcold
Heat Pump (C & T ch 6.12)
COPHP,Carnot 
Thot
Thot  Tcold
Note: All temperatures must be absolute
Catalog of Component Metrics
Device
Nozzle (C&T ch 7.12)
Turbine (C&T ch 7.12)
Compressor
Pump (C&T ch 7.12)
Regenerator (C&T ch 8.8)
Gary L. Solbrekken
Device Metric
nozzle 
2
actual
V
2
adiabatic, rev
V
turb,ad ,rev 

Wactual
W
actual


Wadiabatic,rev W
adiabatic, rev
comp / pump ,ad ,rev 
 regen 
Wadiabatic,rev
Wactual


W
adiabatic, rev

W
actual
Q actual
Q max
ME3324, Fall 2002
10/01/02
Catalog of Substance Models
Substance Model
Application Domain
Characteristics
Property Tables – solid, Whenever
experimental Real data, so this is ideal as
liquid, vapor (C&T ch 7.3 data is available for long as the experimental
and ch 7.7)
substance of interest
conditions used to make
table are broadly applicable
to the application. Can also
use Tds eqns.
Tds = du + pdv
Tds = dh + vdp
Incompressible – liquid Most liquids and processes Properties are approximated
(C&T ch 7.8)
where volume expansion is by the saturated liquid
not of interest (an example properties at the system
of a mis-application would temperature. Specific heats
be a natural convection are temperature dependent
process)
only.
s(T,p)  sf(T)
s = ∫C(T)dT/T  Cavln(T2/T1)
Incompressible
(C&T ch 7.8)
–
solid Most solids and processes
where volume expansion is
not of interest (an example
of a mis-application would
be
a
deformation/bar
expansion process)
Ideal Gas (C & T ch 7.9)
Special case vapor where PR
(P/Pcrit) is nearly 0.
Specific
heats
are
temperature
dependent
only.
s = ∫C(T)dT/T  Cavln(T2/T1)
Allows use of ideal gas
equation of state, PV =
mRT.
When combined
with Tds eqns and fact that
specific heats are a function
of temperature only, the
following 2 forms arise:
s = ∫Cv(T)dT/T + Rln(v2/v1)
Cv,avln(T2/T1) + Rln(v2/v1)
s = ∫Cp(T)dT/T - Rln(p2/p1)
Cp,avln(T2/T1) - R ln(p2/p1)
Ideal Gas – special case of Special cases of entropy
isentropic process (C & T change when the device of
ch 7.9)
interest executes a process
that is adiabatic and
reversible in nature.
Gary L. Solbrekken
ME3324, Fall 2002
The following arise when
s from above is set to 0
and manipulated.
Variable specific heats:
(p2/p1)|s=Pr2/Pr1=f(T2)/f(T1)
(v2/v1)|s=vr2/vr1=f(T2)/f(T1)
Constant specific heats:
(T2/T1)|s = (v1/v2)k-1
(T2/T1)|s = (p1/p2)(k-1)/k
(p2/p1)|s = (v1/v2)k
10/01/02
Catalog of Cycles
Otto Cycle (C&T ch 8.5):
- uses air standard assumption
- air in piston/cylinder assembly is modeled as a closed system
Process Step
1–2
2–3
3–4
4–1
Process description
Adiabatic, reversible, compression
Constant volume heat addition with piston at TDC
Adiabatic, reversible, expansion
Constant volume heat rejection with piston at BDC
Brayton Cycle (C&T ch 8.7 – 8.8)
- uses air standard assumption
- air through engine is modeled as an open system
- can be modified with component ’s and regeneration
Process Step
1–2
2–3
3–4
4–1
Process description
Adiabatic, reversible, compression (compressor)
Constant pressure heat addition (combustor)
Adiabatic, reversible, expansion (turbine)
Constant pressure heat rejection (exhaust/heat exchanger)
Rankine Cycle (C&T ch 8.10 – 8.13)
- uses water as working fluid
- water through engine is modeled as an open system
- can be modified with component ’s and reheat
Process Step
1–2
2–3
3–4
4–1
Process description
Adiabatic, reversible, compression (pump, state 1 is sat liq)
Constant pressure heat addition (boiler)
Adiabatic, reversible, expansion (turbine)
Constant pressure heat rejection (condenser)
Vapor-Compression Refrigeration Cycle (C&T ch 8.16 – 8.17
- uses refrigerant as working fluid
- refrigerant through engine is modeled as an open system
- can be modified with component ’s
Process Step
1–2
2–3
3–4
4–1
Gary L. Solbrekken
Process description
Adiabatic, reversible, compression (compressor, state 1 is sat vapor)
Constant pressure heat rejection (condenser)
Constant enthalpy expansion (expansion valve, state 3 is sat liq)
Constant pressure heat absorption (evaporator)
ME3324, Fall 2002
10/01/02