University of Plymouth
School of Engineering
Key Formulae: BEng 2 Thermodynamics (THER205)
GAS TURBINES
Compressor Isentropic Efficiency:  C 
Turbine Isentropic Efficiency:
T 
Tisen
Isentropic Power Input

Actual Power Input
Tactual
T
Actual Power Output
 actual
Isentropic Power Output
Tisen
For isentropic compression/expansion:
T final
Tinitial
 p final 

 
 pinitial 
 1

For a simple single shaft Gas Turbine with constant cp and γ throughout:
1
 1
T
w
1
 3 T (1  rp  )  (rp   1)
c pT1 T1
C
Specific work:

maximum specific work occurs when:
rp  (CT
T3 2( 1)
)
T1
1
T3
{1  T (1  rp  )}  1
T
 th  1  1
 1
T3
1
 {1 
(rp   1)}
T1
C
Thermal efficiency:
Regenerator thermal ratio =
Tactual
Tmax imum
REFRIGERATION AND HEAT PUMPS
  hevaporator
Refrigeration Effect = m
COPchart 
h evaporator
h compressor
(refrigeration)
Compressor Isentropic efficiency =
  hcondenser
Heating Effect = m
COPchart 
h condenser
(heat pump)
h compressor
hisen
Isentropic Power Input

Actual Power Input
hactual
where m is refrigerant mass flow rate
RECIPROCATING COMPRESSORS
Vclearance  1
 r p n  1

Vswept 
n 1
n

p inletVinduced (r p n  1)
n 1
Volumetric efficiency:
 vol  1 
Net Indicated Work per cycle:
Wnet
Stage pressure ratio for minimum work:
1
rp  R p N
where Rp= overall pressure ratio, and N is the number of stages
ENERGY EFFICIENCY
Energy Conversion Efficiency =
Net (useful ) EnergyOutput
GrossEnergyInput
Annual heating cost for a heating plant:
24Q np
where
16.5 heat
Q  design heat load (kW )
n  number of degree days
p  fuel cost(£/kWh )
 heat  average heating efficiency
STEADY STATE CONDUCTION
T
plane
Q   
x
2T
Q   
r 
ln  o 
 ri 
2-D steady: at an internal node: T0=(T1+T2+T3+T4)/4
1-D steady:
on boundaries: isothermal
insulated (or symmetry)
cylindrical
Tw = const
Tw-1 = Tw+1
convective (plane surface)
T0 = (T1/2+T2+T3/2+BTf)/(2+B)
convective (external corner)
T0 = (T1/2+ T3/2+BTf)/(1+B)
convective (internal corner)
T0 = (T1/2+T2/2+T3+T4+BTf)/(3+B)
General conduction equation:
T
Q 
 2T 
t
c p
B  Grid Biot Number 
  thermal diffusivit y 
ha


c p
CONVECTION
Q  hA(Tw  T f )
Basic equation:
Stanton Number: St 
h
Vc p
Reynolds Analogy: St 
f
2
Grashoff Number: Gr 
Prandtl-Taylor modification: St 
g 2l 3 T
2

f
1


2  1  rv (Pr  1) 
where: rv = velocity ratio sub-layer:free stream = 1.99 Re -0.125 for smooth tubes
Nux  0.664 Re x 2 Pr
1
On a flat plate:
1
4
For fully developed turbulent flow in tubes: Nu  0.023 Re 0.8 Pr 0.4
HEAT EXCHANGERS
Basic design equations:
Q  UATlog
1
  thermal resistance s hot  to  cold
UA
Effectiveness of a counter flow heat-exchanger:
E
1  e  NTU (1C )
1  Ce  NTU (1C )
Effectiveness of a parallel flow heat-exchanger:
E
1  e  NTU (1C )
1 C
FINS
For long ‘thin’ fins:
 cosh m( L  x)

0
cosh mL
For short ‘fat’ fins:
 h 
cosh m( L  x)  
 sinh m( L  x)

m 


0
 h 
cosh mL  
 sinh mL
 m 
 fin 
and
 h 
tanh( mL)  

m 

 fin 
  h 
 h 

 tanh( mL)
  mL
1  
  m 
 m 

Area weighted fin efficiency:    1   (1   fin )
On the finned side:
Q   hA 0
where
tanh( mL)
mL
and
where m 

A fin
A
hp
Ax
IC ENGINES
Air Standard thermal efficiencies:
For Dual Combustion Cycle :

1 
   1
th  1   1 

rc    1     1
For Otto Cycle :
 th  1 
For Diesel Cycle : th
1
rc 1
1    1 
 1   1 

rc     1
Volume at BDC : VBDC 
rc
Vs
rc  1
Mean effective pressure (MEP) =
(where Vs is the stroke volume)
Wnet
Vs
Brake or Indicated Power = PLAN
P = BMEP or IMEP
L = stroke Length
A = piston face Area
N = No. of power strokes per sec.
BSFC =
Fuel Consumptio n
Brake Power
Volumetric efficiency (for 4-stroke naturally aspirated engines)
v 
vol. of free air trapped per cycle
trapped mass

swept vol. of cylinder
theoretica l mass
i.e.
v 
1
2
a
m
 aVs N