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NOTATION
NOTATION Note: Lower case symbols for properties represent specific quantities (i.e. per unit mass) Symbol
Meaning
Typical Units
A b, B B C Cp [CV]0 dh e, E Ea EUF f F g, G h, H h H i I /CR /a L m M
area steady flow availability Biot number capital cost specific heat capacity, at constant pressure calorific value at temperature To hydraulic diameter exergy work potential of heat transferred thermal exery energy utilisation factor fuel/air ratio; also friction factor fuel energy supplied Gibbs function enthalpy heat transfer coefficient plant utilisation interest or discount rate lost work due to irreversibility (total) lost work due to internal irreversibility lost work due to heat transfer to the atmosphere blade length mass fraction (e.g. of main steam flow) Mass flow; also fuel cost per annum; also molecular weight: also Mach number Ratio of air and gas specific heats, (Cpa)/(Cpg) non-dimensional compressor work non-dimensional turbine work non-dimensional net work non-dimensional heat supplied plant life annual operational maintenance costs pressure electricity cost per year heat supplied or rejected pressure ratio gas constant universal gas constant fuel costs per unit mass; also steam to air ratio entropy Stanton number time; also thermal barrier thickness temperature velocity specific work output, work output
me kJ/kg, kJ (-) £, $ kJ/kg K kJ/kg m kJ/kg, kJ kJ (-) (-);(-) kJ kJ/kg, kJ kJ/kg, kJ kW/m2K h/year (-) kJ kJ kJ m (-) kg/s; £, $ p.a.; (-); ( - )
n,n ~
NDCW NDTW NDNW NDHT N OM
p P q, Q r R /~ S s, S St t T V w, W
(--) (-) (-) (-) (-) years £, $ p.a. N/m 2 £, $ p.a. kJ/kg, kJ (-) kJ/kg K kJ/kmol/K £, $/kg, ( - ) kJ/kg K, kJ/K (-) s, m °C, K m/s kJ/kg, kJ (continued on next page)
xvii
xviii
Notation
(continued)
Symbol
Meaning
Typical Units
temperature difference ratios in heat transfer isentropic temperature ratio velocity ratio polytropic expansion index constants defined in text
(-),(-) (-) (-) (-)
3
proportions of capital cost -- r/c'rh0 = 1+ r/c (0 - 1); also capital cost factor
y
-- Cp/Cv
(-) (-) (-) (-) (-) (-)
W
+, W +
X
y Z
A,B,C,D,E,
various
~KK'
E
/J
~, so, o-, ~p T
4,
lOSS parameter heat exchanger effectiveness; also quantity defined in eqn. [4.24] cost of fuel per unit of energy efficiency - see note below ratio of maximum to minimum temperature area ratio in heat transfer; also CO2 performance parameter scaling factor on steam entropy, ratio of mass flows in combined cycle (lower to upper) non-dimensional heat supplied (Vs) or heat unused (vUN) parameters in cycle analysis density Tmin/Zmax; also corporate tax rate cooling air mass flow fraction _c_~. temperature function, J~ v , also turbine stage loading coefficient expansion index defined in text constant in expression for stagnation pressure loss
Subscripts
a, a ~, b, b ~, c, d, e, d, f, f'
states in steam cycle
a
air relating to heat rejection; artificial efficiency blade (temperature) boiler; relating to heat supply cooling air combustion (temperature) compressor (isentropic efficiency) Carnot cycle combustion chamber (efficiency or loss) combined plant (general) cogeneration plant control surface control volume debt dewpoint
A bl B
c cot C CAR CC CP CG CS CV d dp
£, S/kWh (-) (-) (-), kg/kWh
(-) (-) (-) kg/m 3
(-) (-) kJ/kg K, ( - )
(-) (-)
xix
Notation (continued)
Symbol
f g,G H HL HR JB i IJB k L LR max
min m NU O
O P P rit R REV S
S T U w x
X,Y 1, 1/,2,2/, 3, 3/ , 4, 4 / . . . . . . . 0
Superscripts CR
Q
(e.g 1VI,Q, W) / (e.g. x()
Meaning
Typical Units
demand maximum efficiency; also equity; also external electrical (unit price); also exit from turbine, and from first turbine stage fuel gas higher (upper, topping), relating to heat supply, work output between high and lower plants rejection from higher plant Joule-Brayton cycle inlet irreversible Joule-Brayton cycle product gas component; also year number (k= 1, 2 . . . . ) lower (bottoming), relating to heat supply, work output rejection from lower plant maximum minimum mixture non-useful (heat rejection) outlet overall (efficiency) polytropic (efficiency) product of combustion product of supplementary combustion rotor inlet temperature rational; also reactants reversible (process) steam; also state after isentropic compression or expansion; also surface area (As) state at entry to stack; also supplementary heating turbine (isentropic efficiency) useful (heat delivered) water; also maximum specific work cross-sectional flow area (Ax) states leaving heat exchanger; also states at entry and exit from component miscellaneous, referring to gas states conceptual environment (ambient state); also stagnation pressure
referring to internal irreversibility referring to thermal exergy (associated with heat transfer); also to lost work due to external irreversibility associated with heat transfer rate of (mass flow, heat supply, work output, etc) new or changed value (e.g. of efficiency) (continued on next page)
xx
Notation
(continued)
Symbol
Meaning
/ (e.g. a~, b~, 1~, 2', 3I, 4/)
states in feed heating train, in reheating or intercooling
- (e.g. 'i')
mean or averaged (e.g. temperature)
Typical Units
Note on efficiencies
ri is used for thermal efficiency of a closed cycle, but sometimes with a subscript (e.g. riH for thermal efficiency of a higher cycle); rio is used for (arbitrary) overall efficiency of a plant. A list of efficiencies is given below. Plant Thermal Efficiencies 77
rir~ riL riCP ric~ riCAR
higher cycle lower cycle combined cycle cogeneration plant Carnot cycle
Plant (Arbitrary) Overall Efficiencies rio
(rio)H
higher plant
(riO)CP combined plant (riO)L lower plant Rational Efficiencies riR Component Efficiencies riB boiler
ric rtr rip
Cycle
compressor, isentropic turbine, isentropic polytropic
Descriptions
The nomenclature originally introduced by H a w t h o r n e and Davis is followed, in which compressor, heater, turbine and heat exchanger are denoted by C, H, T and X respectively and subscripts R and I indicate reversible and irreversible. For the open cycle the heater is replaced by a burner, B. In addition subscripts U and C refer to uncooled and cooled turbines in a cycle and subscripts 1, 2 . . . . indicate the n u m b e r of cooling steps. Thus, for e x a m p l e [CBTX]Ic2 indicates an open irreversible regenerative cycle with two steps of turbine cooling.
Meaning
Typical Units
A b, B B C Cp [CV]0 dh e, E Ea EUF f F g, G h, H h H i I /CR /a L m M
area steady flow availability Biot number capital cost specific heat capacity, at constant pressure calorific value at temperature To hydraulic diameter exergy work potential of heat transferred thermal exery energy utilisation factor fuel/air ratio; also friction factor fuel energy supplied Gibbs function enthalpy heat transfer coefficient plant utilisation interest or discount rate lost work due to irreversibility (total) lost work due to internal irreversibility lost work due to heat transfer to the atmosphere blade length mass fraction (e.g. of main steam flow) Mass flow; also fuel cost per annum; also molecular weight: also Mach number Ratio of air and gas specific heats, (Cpa)/(Cpg) non-dimensional compressor work non-dimensional turbine work non-dimensional net work non-dimensional heat supplied plant life annual operational maintenance costs pressure electricity cost per year heat supplied or rejected pressure ratio gas constant universal gas constant fuel costs per unit mass; also steam to air ratio entropy Stanton number time; also thermal barrier thickness temperature velocity specific work output, work output
me kJ/kg, kJ (-) £, $ kJ/kg K kJ/kg m kJ/kg, kJ kJ (-) (-);(-) kJ kJ/kg, kJ kJ/kg, kJ kW/m2K h/year (-) kJ kJ kJ m (-) kg/s; £, $ p.a.; (-); ( - )
n,n ~
NDCW NDTW NDNW NDHT N OM
p P q, Q r R /~ S s, S St t T V w, W
(--) (-) (-) (-) (-) years £, $ p.a. N/m 2 £, $ p.a. kJ/kg, kJ (-) kJ/kg K kJ/kmol/K £, $/kg, ( - ) kJ/kg K, kJ/K (-) s, m °C, K m/s kJ/kg, kJ (continued on next page)
xvii
xviii
Notation
(continued)
Symbol
Meaning
Typical Units
temperature difference ratios in heat transfer isentropic temperature ratio velocity ratio polytropic expansion index constants defined in text
(-),(-) (-) (-) (-)
3
proportions of capital cost -- r/c'rh0 = 1+ r/c (0 - 1); also capital cost factor
y
-- Cp/Cv
(-) (-) (-) (-) (-) (-)
W
+, W +
X
y Z
A,B,C,D,E,
various
~KK'
E
/J
~, so, o-, ~p T
4,
lOSS parameter heat exchanger effectiveness; also quantity defined in eqn. [4.24] cost of fuel per unit of energy efficiency - see note below ratio of maximum to minimum temperature area ratio in heat transfer; also CO2 performance parameter scaling factor on steam entropy, ratio of mass flows in combined cycle (lower to upper) non-dimensional heat supplied (Vs) or heat unused (vUN) parameters in cycle analysis density Tmin/Zmax; also corporate tax rate cooling air mass flow fraction _c_~. temperature function, J~ v , also turbine stage loading coefficient expansion index defined in text constant in expression for stagnation pressure loss
Subscripts
a, a ~, b, b ~, c, d, e, d, f, f'
states in steam cycle
a
air relating to heat rejection; artificial efficiency blade (temperature) boiler; relating to heat supply cooling air combustion (temperature) compressor (isentropic efficiency) Carnot cycle combustion chamber (efficiency or loss) combined plant (general) cogeneration plant control surface control volume debt dewpoint
A bl B
c cot C CAR CC CP CG CS CV d dp
£, S/kWh (-) (-) (-), kg/kWh
(-) (-) (-) kg/m 3
(-) (-) kJ/kg K, ( - )
(-) (-)
xix
Notation (continued)
Symbol
f g,G H HL HR JB i IJB k L LR max
min m NU O
O P P rit R REV S
S T U w x
X,Y 1, 1/,2,2/, 3, 3/ , 4, 4 / . . . . . . . 0
Superscripts CR
Q
(e.g 1VI,Q, W) / (e.g. x()
Meaning
Typical Units
demand maximum efficiency; also equity; also external electrical (unit price); also exit from turbine, and from first turbine stage fuel gas higher (upper, topping), relating to heat supply, work output between high and lower plants rejection from higher plant Joule-Brayton cycle inlet irreversible Joule-Brayton cycle product gas component; also year number (k= 1, 2 . . . . ) lower (bottoming), relating to heat supply, work output rejection from lower plant maximum minimum mixture non-useful (heat rejection) outlet overall (efficiency) polytropic (efficiency) product of combustion product of supplementary combustion rotor inlet temperature rational; also reactants reversible (process) steam; also state after isentropic compression or expansion; also surface area (As) state at entry to stack; also supplementary heating turbine (isentropic efficiency) useful (heat delivered) water; also maximum specific work cross-sectional flow area (Ax) states leaving heat exchanger; also states at entry and exit from component miscellaneous, referring to gas states conceptual environment (ambient state); also stagnation pressure
referring to internal irreversibility referring to thermal exergy (associated with heat transfer); also to lost work due to external irreversibility associated with heat transfer rate of (mass flow, heat supply, work output, etc) new or changed value (e.g. of efficiency) (continued on next page)
xx
Notation
(continued)
Symbol
Meaning
/ (e.g. a~, b~, 1~, 2', 3I, 4/)
states in feed heating train, in reheating or intercooling
- (e.g. 'i')
mean or averaged (e.g. temperature)
Typical Units
Note on efficiencies
ri is used for thermal efficiency of a closed cycle, but sometimes with a subscript (e.g. riH for thermal efficiency of a higher cycle); rio is used for (arbitrary) overall efficiency of a plant. A list of efficiencies is given below. Plant Thermal Efficiencies 77
rir~ riL riCP ric~ riCAR
higher cycle lower cycle combined cycle cogeneration plant Carnot cycle
Plant (Arbitrary) Overall Efficiencies rio
(rio)H
higher plant
(riO)CP combined plant (riO)L lower plant Rational Efficiencies riR Component Efficiencies riB boiler
ric rtr rip
Cycle
compressor, isentropic turbine, isentropic polytropic
Descriptions
The nomenclature originally introduced by H a w t h o r n e and Davis is followed, in which compressor, heater, turbine and heat exchanger are denoted by C, H, T and X respectively and subscripts R and I indicate reversible and irreversible. For the open cycle the heater is replaced by a burner, B. In addition subscripts U and C refer to uncooled and cooled turbines in a cycle and subscripts 1, 2 . . . . indicate the n u m b e r of cooling steps. Thus, for e x a m p l e [CBTX]Ic2 indicates an open irreversible regenerative cycle with two steps of turbine cooling.