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Minimum entropy generation design method for the heat transfer process in the supercritical region
Energy Convers.Mgmt Vol. 32, No. 4, pp. 375-385, 1991
0196-8904/91 $3.00+ 0.00 Copyright © 1991 Pergamon Press pie
Printed in Great Britain. All rights reserved
MINIMUM ENTROPY GENERATION DESIGN METHOD FOR THE HEAT TRANSFER PROCESS IN THE SUPERCRITICAL REGION NILOFER
E~;RICAN
and S E Y H A N
UYGUR
Mechanical Engineering Department, Istanbul Technical University, Giimfi~suyu 80191, Istanbul, Turkey
(Received 3 March 1990; received for publication 12 December 1990) Abstract--In this study, a design method aiming for minimum entropy generation in a heat transfer process is derived and developed for a special condition: the heat transfer between the flows in the pipe that is in the supercritical region and the one across the pipe. Both of the flows have difficulties to be investigated because of their special characteristics. The method is applied to a fluidizcd bed boiler in a power cycle, and the effect of the method on the first and second law efficiencies of both the boiler and the cycle are examined. The operation is repeated for standard pipe geometries, and the results are also tabulated from the economic point of view. Entropy generation number
Exergy
Irreversibility
Second law
NOMENCLATURE AFB = Atmospheric fluidized bed boiler B = Exergy (kJ) CD = Drag coefficient Con = Condenser CT = Cooling tower Dia = Diameter E = Economizer FSH = Finishing superheater FWH = Feed water heater f = Friction factor G = Generator H = Enthalpy (k J) H ° = Ambient enthalpy (kJ) h = Specific enthalpy (kJ/kg) I = Irreversibility ID = Induced draft FD = Forced draft m = Material amount (tonnes) rh = Mass flow rate (kg/s) Ns -- Entropy production number Ns.Ap = Entropy production number resulting from pressure drops Ns.~r = Entropy generation number resulting from temperature differences Ns~ = Entropy generation number of first side Ns2 - Entropy generation number o f second side Ns.tota I ~ Total entropy generation number P -~ Pressure (kPa) PSHB - Presuperheater RH = Reheater
Q = Heat (k J) q' = Heat flux per length (kJ/m s) R ---Gas constant (kJ/kg K) S -~ Entropy (k J/K) S ' = Entropy per length (kJ/m K) T - Temperature (K) T® = Environmental temperature (K) U -- Internal energy (kJ) W ~ Work (kJ) Z = Compressibility factor
Greek letters A -- Difference operator E -- Exergy efficiency (second law effectiveness) 375
376
E~3RtCAN and UYGUR: SUPERCRITICAL REGION HEAT TRANSFER ~ = Exergy efficiencyof boiler E -~ Exergy efficiencyof cycle ~/1-- First law efficiencyof boiler r/2-- First law efficiencyof cycle 0 -- Portion of work spent because of irreversibilities /~ = Dynamic viscosity(kg/m s) v --Kinematic viscosity (m2/s) p -- Density (kg/m3) Superscripts
o = Ambient opt = Optimum Subscripts
act = Actual 0 = Environment Rev = Reversible
INTRODUCTION The law of conservation of energy mentions conservation of the quantity of the energy, n o t its quality. The quantity of energy contains the part of the energy which is not transformed and used during the conversion. This part is named as "loss" in the engineering field. By the second law point of view, it is the "unavailable part of energy". The available part of energy is never conserved, and it always decreases in every conversion. In other words, its quality decays. So, the objective in energy processes is to decrease this degeneration. The available part of energy is named "exergy", and the branch of thermodynamics that analyses the availability of the conversions is "exergy analysis". The various definitions of "exergy" can be summarized as the "potential of the system to give the maximum work during the process bringing it into an equilibrium state with its environment". A brief summary of the exergy analysis of various cycles and processes are widely present in the literature [1-5]. A new definition of efficiency results from the exergy analysis in the cycles (it is more realistic to name its effectiveness because it represents the effectiveness, in other words, the usefulness of the conversion, instead of its efficiency) defined as the availability of the energy gained from the cycle to the availability of the energy given to the cycle, while a new dimensionless number, the entropy generation number, has been obtained from the exergy analysis of the processes. This number is fundamental to the study because it has been derived and applied to the heat transfer process between the flows in the pipe in the supercritical region and the one across the pipes.
EXERGY AND E N T R O P Y G E N E R A T I O N N U M B E R The mathematical expression of exergy is given as follows [6] B --(U - U°) - T o ( S - S ° ) .
(1)
This definition can be modified for the cases of the system's expanding towards the environment and interacting with the environment by mass transfer as B = (U -
U °) - T o ( S -
S ° ) + P 0 ( V - V°)
(2)
and B = (H - n °) - To(S - S°).
It is proved that exergy is not with the entropy generation rate. minimize the entropy generation. in the literature [7-10]. For this number, is formed:
(3)
constant, and its decrease for every conversion is proportional So, to improve the availability of energy in a process means to Design methods based on this new approach are rarely present aim, a dimensionless number, called the entropy generation
actual entropy generation rate Ns = characteristic generation rate "
(4)
EGRICAN and UYGUR:
SUPERCRITICAL REGION HEAT TRANSFER
377
q'
r .
.
.
.
.
.
.
--
.
.
.
.
1T
t T 1
t
,
I ~
v
I
' ttttttttttttttttttttt
T2
o
Fig. 1. Flow inside pipe in supercriticalregion with prescribedheat flux through wall.
In this study, the process investigated is heat transfer, and the characteristic entropy generation rate is taken as the entropy generation rate that would occur if the heat transfer were a reversible one.
ENTROPY
GENERATION NUMBER FOR THE FLOW SUPERCRITICAL REGION
IN THE
The heat transfer phenomenon in the supercritical region is difficult to examine because most of the physical properties are dependent on temperature in this region. First, the actual entropy generation per unit length is derived from the first and second laws of thermodynamics. Consider a flow inside a pipe with the heat flux through its walls prescribed (Fig. 1). Equation (4) can be modified for this flow as follows: Sact , . Ns = qr~v
(5)
T+
Here, T~o is the temperature of the fluid flowing across the pipe. Using the first and second laws of thermodynamics, the actual entropy generation rate for this flow can be derived as: ,n /
dP'~
,/1
l'x
,~aet = ~-~ ~--~X) + q ~'~ -- -~--~) .
(6)
The T appearing in equation (6) is the instantaneous temperature of the fluid. Integrating both of the entropy generation rates (actual and characteristic) between the initial and the final temperatures of the fluid flowing in the pipe, the difficulty resulting from the dependence on temperature is defeated. The density appearing in the entropy generation equation, which strongly depends on temperature in the supercritical region, is written in terms of the compressibility factor, and this factor is accepted as constant at constant pressure for small temperature intervals. The validity of this assumption can be observed from the compressibility charts. The integrated actual entropy generation is put into dimensionless form by dividing with the characteristic one. fTt2Z2R2(TI + T 2 ) T ~ (#Re/2) 3 T~ T2 N~= + 2p2 Q r t- T2 - Ti ln ~Tz
(7)
Here Q = 2nrq'
(8)
2rn xr#"
(9)
and Re =
378
E(3RICAN and UYGUR:
SUPERCRITICAL REGION HEAT TRANSFER
The result is a dimensionless number, called the "entropy generation number integrated over temperature", containing two terms. One of these terms is a function of the friction factor, while the other is a function of the temperature difference: T
NS'Ae =
f r c 2 Z 2 R : ( T] + T2) To (#Re/2) 3 2P2Q r
= At/r,
(10a)
+ 1 - l
In
(10b)
(Air is defined as ir2 - ir] ). This result is well known from heat exchanger design, as the two main events that cause irreversibility, (a) pressure decrease in the pipe and (b) temperature difference between the finite areas. So, the connection between the second law and the actual event is obvious. The temperature difference appearing in the integrated entropy generation number is the temperature difference between the end states of the flow, but in fact, it represents the difference between the fluid temperature at an instant of time and the source temperature. When this difference decreases, the difference between the fluid and the source temperature increases along the flow, and the "losses" get bigger. By the operations described in detail in the study, the term of the integrated entropy generation number representing the temperature difference can be written in the form of the product of a logarithmic function with the inverse of the same argument. A short examination of this form shows that the entropy generation number increases with the decrease of the temperature difference between the end states, and this means that entropy generation increases with increasing temperature difference between the fluid and the source. Thus, it is proved that the method represents the actual event. The friction factor, on the other hand, is directly proportional with the other term of the N s number, proving a realistic approach of the second law. The combination of the two terms derived from the integrated entropy generation for the flow inside the pipe [equations (10a) and (10b)] is called "the entropy generation for the first side" which is: Nsl = Ns,T ap + Ns,ar. ENTROPY
GENERATION
NUMBER
FOR
THE
(11) FLOW
ACROSS
THE
PIPE
For the "second side", the entropy generation in a flow across the pipe is derived from a basic formulation (the integral formulation of the second law) without neglecting the work done by the system towards the viscous effects. Examining the flow separation event occurring during this type of flow (Fig. 2), this formulation is applied, and the actual entropy generation obtained is: q'
v2
1 4_CDRe # x,~o 2T~
S'aot = rckT 2 Nu
.,......-.--'4~ V 00
point Fig. 2. Separation of flow across cylinder.
(12)
EORICAN and UYGUR: SUPERCRITICALREGION HEAT TRANSFER
379
By defining
Q = 2nrq'
(13)
r Re = 2vx, ~ -
(14)
Nu = CRe"Pr ~/3
(15)
Y
and combining these equations with equation (5), the entropy generation number for the second side is obtained as QI~" CD 2v 3 ~ prtr l Ns=2"+lrt2kT~ C v" "Prl/3r "+1+" (16) x,~p Q' This number is also the combination of two terms: (a) Entropy production resulting from the temperature difference between the fluid and the outer side of the pipe (that is denoted by the average Nusselt number):
Q~" Ns2'ar = 2"+ lrt2kToo Cv"~,oopPrl/3r" + 1
(17)
and (b) Entropy production resulting from friction and pressure drag (denoted by the drag coefficient G,) Ns2,~p =
Co2v3x'~prcr2 Q
Ns2 = Ns2,A I, +
Ns2,A r .
(18) (19)
C O M B I N A T I O N OF E N T R O P Y G E N E R A T I O N N U M B E R S A N D O U T L I N E OF T H E D E S I G N M E T H O D
An algorithm is developed by taking into account two conditions, namely variable temperature difference and constant temperature difference. For both of the conditions, a total entropy generation number has been determined before beginning the design. This determination is to be done by means of the irreversibility concept:
1~To Ns = - - Ns, + Ns2.
Q/Too
(20)
Since the actual total entropy production equals the irreversibilitydivided by the ambient temperature, the entropy generation number, is written in terms of irreversibility.Multiplying both the denominator and the numerator of this modified number with the amount of work given to the process, a term which denotes the portion of the lost work resultingfrom the irreversibilities is obtained:
WToo
Ns=¢ - -
QTo
(21)
where I ~b = ~ .
(22)
Determining this portion before designing the system, the total entropy generation number is derived. Its separation for two sides is to be done according to the operating conditions. If the temperature difference is not constant, the entropy generation number [equation (10b)] resulting from the temperature difference can be minimized first. The result of the minimization is: ATOpt = 3 T,. (123)
380
E(~RICAN and UYGUR: SUPERCRITICALREGION HEAT TRANSFER Coal
Limestone
i
=-
i
.
~
=,
,,
C¢ .
I-
Fig. 3. Schematicdiagram of power cycle. Subtracting the minimized number from the total entropy generation number [equation (21)] and minimizing the entropy generation for the second side, the optimum radius and the outlet temperature can be determined. By the help of these two quantities, the pressure difference inside the pipe is obtained. If the outlet temperature is constant, then the minimization is only for the aim of finding the optimum radius and reaching the minimum entropy generation for the second side. Similar to the first condition, the entropy generation number for the first side is obtained, and this operation is followed by determination of the entropy generation number resulting from the pressure difference for the second side. Because the temperature difference is constant, the other term of Ns2 does not change.
APPLICATION
OF THE METHOD TO A FLUIDIZED O F A P O W E R CYCLE
BED
BOILER
The algorithm described above is applied to a power cycle in which the steam is generated by means of an atmospheric pressure fluidized bed boiler (Fig. 3). First, the effect of the boiler effectiveness (E~) on the overall cycle effectiveness (E2), where the exergy of the heat obtained in the boiler E~= the exergy of the heat of the coal + the auxiliary work and the net output work E2= the exergy of the heat of the coal + the auxiliary work has been examined, and it is shown that the latter one depends on the former one. The path of the flow in the boiler is explained in detail (Fig. 4). The heat transfer in this equipment is between a supercritical flow inside the pipe and the flow across the pipe. So, the definitions and the algorithm derived above can be applied to the cycle. The heat transfer in the boiler differs according to the section. The convection coefficient is higher than the one in the convection spaces above the bed because the heat transfer surfaces are extremely large inside the bed. The drag coefficient also changes. It depends on the radius of the particles used in the bed. To defeat all these difficulties, the heat transfer is examined separately, and a computer programme is prepared to find the entropy generation numbers and pressure differences for the determined irreversibilities.
CONCLUSIONS The result obtained is proof of the necessity of second law analysis. The pressure difference is directly proportional with the irreversibility and the entropy generation number for the first side (Figs 5-13 and Table 1). The exergy analysis of the boiler and the system shows that the exergy of the heat given by the combustion of the coal is able to be increased by modifying the design
E O R I C A N and UYGUR:
SUPERCRITICAL REGION HEAT T R A N S F E R
657 K 533 K
el -
I
381
_t -I
~ E1 _P§H_1B
_1 ~
II
) E2 i
F
SH_2B
.
I I
) E3 PSH3
_PSH_3_B _
tr=
_
--I
!
Fig. 4. Path of flow in atmospheric fluidized bed boiler of power cycle.
parameters. For the same irreversibility, the first law efficiency of both the boiler and the cycle change too much with entropy generation number for the flow inside the tube. The first law efficiency and the total heat transfer surface increase with the increase of the entropy generation number for the first side. Second law efficiency, on the other hand, depends mainly on the irreversibility. So, it changes due to the total entropy production number. Its change is not effected by the separation of the number. For this reason, it is possible to get an optimum solution in designing the pipes due to the minimum irreversibility (obtaining the high portion of the exergy saved) and maximum entropy generation number for the first side (obtaining the high first law efficiency). However, the pipe diameters must be standard for construction of the system. For this reason, the design of the system has been repeated for the standard average diameters of the ones between the minimum and maximum entropy generation numbers for constant irreversibility. It can be
382
E~RICAN and UYGUR: SUPERCRITICAL REGION HEAT TRANSFER
3000 -2500 --
/ e
Mln. dla. (max. N $I)
..I 2000 --
/
~" ~
• 15001000
. ~
St. mean dla.
~ m ./J/
_
/
,
/
+ / +
Max. dia.(min. Ns,)
5OO
0
0.2
0.4 Irreverslbillty
0.6
0.8
Fig. 5. Change of pressure drop with irreversibility in Economizer 1-2-3 (El, E2, E3).
1000 -800 --
t~
600
/ •
Min. dla. (max. N S1)
/ •
--
¢1.
•
400
/ .
--
200 --
a/~
a o ~ o ~
~ + o ~ t . _ . . _ . . _ +o . - /- - - - - - - ~ - 0
0.2
0.4
• St. mean dia. _+ Max. dia.j (rain. N el)
0.6
0.8
Irreverslbility Fig. 6. Change of pressure drop with irreversibility in Economizer 4-5-6 (E4, E5, E6).
350 -300 --
• MIn. die. (max, NSl)
250 -m
O..
• /
200 150
•
--
•
100 --
j e / m ~ : ~ . ; ~ e . / +
50 -0
St. mean die. Max. die. (mln. NsI)
0.2
0.4 I rreversibility
I
J
0.6
0.8
Fig. 7. Change of pressure drop with irreversibility in Economizer 7 (E7A).
E(3R|CAN and UYGUR:
SUPERCRITICAL REGION HEAT TRANSFER
3000 -2500 --
/ o
Min. dia. (max. NSl)
. , t
2000 D..
J
--
15oolOOO
/
.../ _
+
500 - -
o ~ m ~ m _=..._._.._+
,
0
0.2
0.4 Irreversiblllty
+
T
0.6
St. mean dla. Max. dla. (rain. NSt )
I
0.8
Fig. 8. Change of pressure drop with irreversibility in Economizer 7B (E7B).
1500 --
o/a/
1000 --
MIn.dla. (max.NSl)
o / St. meandla.
500 --
Max. dia. (rain. NSl) e
0
0.2
0.4 Irreversibillty
0.6
0.8
Fig. 9. Change of pressure drop with irreversibility in Presuperheater 1-2-3 (PSH 1A, PAH2-A, PSH3A).
4
3 --
1-
0
~ / o r e
e/.
. o~/ t 1 ~ / ~ / m . . ~/ I
Min'dla'(max'Nsl)
St. moandla. Max. dla. (min. NSl )
0.2
I
I
I
0.4
0.6
0.8
Irreverslbllity Fig. 10. Change of pressure drop with irreversibility in Presuperheater IB-2B-3B (PSHIB, PSH2B, PSH3B). ECM 32/4---F
383
384
EG1L[CAN and UYGUR: 2500
SUPERCRITICAL REGION HEAT TRANSFER
-
e~oj
2000 ~, 1500 -
MIn. dla. (max. NS1)
e J
500 -
Max. dla. (mln. N81) , 0.2
0
I 0.4
I 0.6
J 0.8
Irreverslbllity Fig. 11. Change of pressure drop with irreversibility in finishing superheater 4-5 (FSH4, FSH5).
200 --
150 -~" a. ~.
/ o / J ' j . .
100 --
0
MIn. dia. (max. NSl) St. mean dla.
~ . e
0.2
0.4
0.6
0.8
Irreverslbility Fig. 12. Change of pressure drop with irreversibility in Reheater 5-6 (RHS, RH6A).
3000 -2500 --
j e
MIn. dla. (max. NSl )
e
2000 --
e J
1500lOOO
500 --
__._..-.o St. mean dla.
I 0
0.2
0.4
0.6
0.8
Irreverslblllty Fig. 13. Change of pressure drop with irreversibility in Reheater 6B (RH6B).
E~IRtCAN and UYGUR:
SUPERCRITICAL REGION HEAT TRANSFER
385
Table 1. Effect of design parameters on first law efficiency, second law effectiveness and amount of pipe material I 0.2 0.2 0.2 0.5 0.5 0.5
Diameter (m)
Minimum (max. Standard mean Maximum (rain. Minimum (max. Standard mean Maximum (rain.
Nst) Ns~ ) Nsl) Nsl)
~1
172
El
E2
Ym (tonnes)
0.857 0.710 0.640 0.920 0.890 0.883
0.450 0.415 0.386 0.698 0.682 0.675
0.420 0.425 0.426 0.254 0.258 0.260
0.353 0.365 0.367 0.195 0.205 0.210
7.30 11.26 15.87 7.30 11.26 15.87
Table 2. Minimum, maximum and standard pipe diameters used for minimum entropy generation design
Section
Minimum diameter (ram)
Maximum diameter (ram)
Standard mean diameter (ram) (in.)
1=0,2 E123 EA56 E7A E7B FSH PSH123 PSHB RHA RHB
92 43 57 35 63 71 59 101 62
142 143 150 146 140 140 150 140 145
110 (4~) 90 (3~) 100 (4) 90 (3½) 100 (4) 100 (4) 100 (4) 120 (4½) 100 (4)
1=0.5 E123 E456 E7A E7B FSH PSH 123 PSHB RHA RHB
88 40 45 28 57 68 53 92 55
140 138 147 142 138 137 145 138 142
90 (3~) 80 (3) 90 (3½) 80 (3) 90 (3~) 90 (3~) 90 (3½) 110 (4) 90 (3½)
o b s e r v e d f r o m Figs 5-13 a n d T a b l e 1 t h a t the pressure d r o p , the first law efficiency, the second law effectiveness values a n d the a m o u n t o f the m a t e r i a l used for m i n i m i z e d e n t r o p y g e n e r a t i o n designs with the s t a n d a r d average d i a m e t e r s are between the intervals o f the limits. T h e values o f the s t a n d a r d average pipe diameters, t o g e t h e r with the m i n i m u m a n d m a x i m u m ones, are listed in T a b l e 2. A s a conclusion, the i m p o r t a n c e o f the second law has been p r o v e d in this study, a n d the o p i n i o n o f designing the t h e r m a l systems on the basis of, n o t only the first law, b u t also aiming at o p t i m i z a t i o n t h r o u g h the first law, the second law a n d the e c o n o m i c cost has been f o r m e d a n d developed.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
N. Lior and G. J. Rudy, Energy Convers. Mgmt 28, 327 (1988). S. U. Kumar and W. J. Minkowycz and K. S. Patel, Int. Commun, Heat Mass Transfer 16, 335 (1989). G. G. Rice, Stifling engine---availability criteria, lOth IECEC Conference, pp. 791-795 (1984). N. Eirican, Heat Recovery System and ClIP 8, 549 (1988). E. Michaelidcs, Energy Res. 8, 241 (1984). G. Lucca, On the Opportunityfrom the Didactic Point of View, To Derive Entropyfrom Exergy, Second Law Analysis of Thermal System. The American Society of Mechanical Engineers, pp. 183, 191 (1987). A. Bcjan, Int. J. Heat Mass Transfer 21, 655 (1978). A. Bejan, Trans. ASME 99, 374 (1977). A. Bejan, Trans. ASME I01, 718 (1979). A. Bejan, Adv. Heat Transfer 15, 1 (1982).
0196-8904/91 $3.00+ 0.00 Copyright © 1991 Pergamon Press pie
Printed in Great Britain. All rights reserved
MINIMUM ENTROPY GENERATION DESIGN METHOD FOR THE HEAT TRANSFER PROCESS IN THE SUPERCRITICAL REGION NILOFER
E~;RICAN
and S E Y H A N
UYGUR
Mechanical Engineering Department, Istanbul Technical University, Giimfi~suyu 80191, Istanbul, Turkey
(Received 3 March 1990; received for publication 12 December 1990) Abstract--In this study, a design method aiming for minimum entropy generation in a heat transfer process is derived and developed for a special condition: the heat transfer between the flows in the pipe that is in the supercritical region and the one across the pipe. Both of the flows have difficulties to be investigated because of their special characteristics. The method is applied to a fluidizcd bed boiler in a power cycle, and the effect of the method on the first and second law efficiencies of both the boiler and the cycle are examined. The operation is repeated for standard pipe geometries, and the results are also tabulated from the economic point of view. Entropy generation number
Exergy
Irreversibility
Second law
NOMENCLATURE AFB = Atmospheric fluidized bed boiler B = Exergy (kJ) CD = Drag coefficient Con = Condenser CT = Cooling tower Dia = Diameter E = Economizer FSH = Finishing superheater FWH = Feed water heater f = Friction factor G = Generator H = Enthalpy (k J) H ° = Ambient enthalpy (kJ) h = Specific enthalpy (kJ/kg) I = Irreversibility ID = Induced draft FD = Forced draft m = Material amount (tonnes) rh = Mass flow rate (kg/s) Ns -- Entropy production number Ns.Ap = Entropy production number resulting from pressure drops Ns.~r = Entropy generation number resulting from temperature differences Ns~ = Entropy generation number of first side Ns2 - Entropy generation number o f second side Ns.tota I ~ Total entropy generation number P -~ Pressure (kPa) PSHB - Presuperheater RH = Reheater
Q = Heat (k J) q' = Heat flux per length (kJ/m s) R ---Gas constant (kJ/kg K) S -~ Entropy (k J/K) S ' = Entropy per length (kJ/m K) T - Temperature (K) T® = Environmental temperature (K) U -- Internal energy (kJ) W ~ Work (kJ) Z = Compressibility factor
Greek letters A -- Difference operator E -- Exergy efficiency (second law effectiveness) 375
376
E~3RtCAN and UYGUR: SUPERCRITICAL REGION HEAT TRANSFER ~ = Exergy efficiencyof boiler E -~ Exergy efficiencyof cycle ~/1-- First law efficiencyof boiler r/2-- First law efficiencyof cycle 0 -- Portion of work spent because of irreversibilities /~ = Dynamic viscosity(kg/m s) v --Kinematic viscosity (m2/s) p -- Density (kg/m3) Superscripts
o = Ambient opt = Optimum Subscripts
act = Actual 0 = Environment Rev = Reversible
INTRODUCTION The law of conservation of energy mentions conservation of the quantity of the energy, n o t its quality. The quantity of energy contains the part of the energy which is not transformed and used during the conversion. This part is named as "loss" in the engineering field. By the second law point of view, it is the "unavailable part of energy". The available part of energy is never conserved, and it always decreases in every conversion. In other words, its quality decays. So, the objective in energy processes is to decrease this degeneration. The available part of energy is named "exergy", and the branch of thermodynamics that analyses the availability of the conversions is "exergy analysis". The various definitions of "exergy" can be summarized as the "potential of the system to give the maximum work during the process bringing it into an equilibrium state with its environment". A brief summary of the exergy analysis of various cycles and processes are widely present in the literature [1-5]. A new definition of efficiency results from the exergy analysis in the cycles (it is more realistic to name its effectiveness because it represents the effectiveness, in other words, the usefulness of the conversion, instead of its efficiency) defined as the availability of the energy gained from the cycle to the availability of the energy given to the cycle, while a new dimensionless number, the entropy generation number, has been obtained from the exergy analysis of the processes. This number is fundamental to the study because it has been derived and applied to the heat transfer process between the flows in the pipe in the supercritical region and the one across the pipes.
EXERGY AND E N T R O P Y G E N E R A T I O N N U M B E R The mathematical expression of exergy is given as follows [6] B --(U - U°) - T o ( S - S ° ) .
(1)
This definition can be modified for the cases of the system's expanding towards the environment and interacting with the environment by mass transfer as B = (U -
U °) - T o ( S -
S ° ) + P 0 ( V - V°)
(2)
and B = (H - n °) - To(S - S°).
It is proved that exergy is not with the entropy generation rate. minimize the entropy generation. in the literature [7-10]. For this number, is formed:
(3)
constant, and its decrease for every conversion is proportional So, to improve the availability of energy in a process means to Design methods based on this new approach are rarely present aim, a dimensionless number, called the entropy generation
actual entropy generation rate Ns = characteristic generation rate "
(4)
EGRICAN and UYGUR:
SUPERCRITICAL REGION HEAT TRANSFER
377
q'
r .
.
.
.
.
.
.
--
.
.
.
.
1T
t T 1
t
,
I ~
v
I
' ttttttttttttttttttttt
T2
o
Fig. 1. Flow inside pipe in supercriticalregion with prescribedheat flux through wall.
In this study, the process investigated is heat transfer, and the characteristic entropy generation rate is taken as the entropy generation rate that would occur if the heat transfer were a reversible one.
ENTROPY
GENERATION NUMBER FOR THE FLOW SUPERCRITICAL REGION
IN THE
The heat transfer phenomenon in the supercritical region is difficult to examine because most of the physical properties are dependent on temperature in this region. First, the actual entropy generation per unit length is derived from the first and second laws of thermodynamics. Consider a flow inside a pipe with the heat flux through its walls prescribed (Fig. 1). Equation (4) can be modified for this flow as follows: Sact , . Ns = qr~v
(5)
T+
Here, T~o is the temperature of the fluid flowing across the pipe. Using the first and second laws of thermodynamics, the actual entropy generation rate for this flow can be derived as: ,n /
dP'~
,/1
l'x
,~aet = ~-~ ~--~X) + q ~'~ -- -~--~) .
(6)
The T appearing in equation (6) is the instantaneous temperature of the fluid. Integrating both of the entropy generation rates (actual and characteristic) between the initial and the final temperatures of the fluid flowing in the pipe, the difficulty resulting from the dependence on temperature is defeated. The density appearing in the entropy generation equation, which strongly depends on temperature in the supercritical region, is written in terms of the compressibility factor, and this factor is accepted as constant at constant pressure for small temperature intervals. The validity of this assumption can be observed from the compressibility charts. The integrated actual entropy generation is put into dimensionless form by dividing with the characteristic one. fTt2Z2R2(TI + T 2 ) T ~ (#Re/2) 3 T~ T2 N~= + 2p2 Q r t- T2 - Ti ln ~Tz
(7)
Here Q = 2nrq'
(8)
2rn xr#"
(9)
and Re =
378
E(3RICAN and UYGUR:
SUPERCRITICAL REGION HEAT TRANSFER
The result is a dimensionless number, called the "entropy generation number integrated over temperature", containing two terms. One of these terms is a function of the friction factor, while the other is a function of the temperature difference: T
NS'Ae =
f r c 2 Z 2 R : ( T] + T2) To (#Re/2) 3 2P2Q r
= At/r,
(10a)
+ 1 - l
In
(10b)
(Air is defined as ir2 - ir] ). This result is well known from heat exchanger design, as the two main events that cause irreversibility, (a) pressure decrease in the pipe and (b) temperature difference between the finite areas. So, the connection between the second law and the actual event is obvious. The temperature difference appearing in the integrated entropy generation number is the temperature difference between the end states of the flow, but in fact, it represents the difference between the fluid temperature at an instant of time and the source temperature. When this difference decreases, the difference between the fluid and the source temperature increases along the flow, and the "losses" get bigger. By the operations described in detail in the study, the term of the integrated entropy generation number representing the temperature difference can be written in the form of the product of a logarithmic function with the inverse of the same argument. A short examination of this form shows that the entropy generation number increases with the decrease of the temperature difference between the end states, and this means that entropy generation increases with increasing temperature difference between the fluid and the source. Thus, it is proved that the method represents the actual event. The friction factor, on the other hand, is directly proportional with the other term of the N s number, proving a realistic approach of the second law. The combination of the two terms derived from the integrated entropy generation for the flow inside the pipe [equations (10a) and (10b)] is called "the entropy generation for the first side" which is: Nsl = Ns,T ap + Ns,ar. ENTROPY
GENERATION
NUMBER
FOR
THE
(11) FLOW
ACROSS
THE
PIPE
For the "second side", the entropy generation in a flow across the pipe is derived from a basic formulation (the integral formulation of the second law) without neglecting the work done by the system towards the viscous effects. Examining the flow separation event occurring during this type of flow (Fig. 2), this formulation is applied, and the actual entropy generation obtained is: q'
v2
1 4_CDRe # x,~o 2T~
S'aot = rckT 2 Nu
.,......-.--'4~ V 00
point Fig. 2. Separation of flow across cylinder.
(12)
EORICAN and UYGUR: SUPERCRITICALREGION HEAT TRANSFER
379
By defining
Q = 2nrq'
(13)
r Re = 2vx, ~ -
(14)
Nu = CRe"Pr ~/3
(15)
Y
and combining these equations with equation (5), the entropy generation number for the second side is obtained as QI~" CD 2v 3 ~ prtr l Ns=2"+lrt2kT~ C v" "Prl/3r "+1+" (16) x,~p Q' This number is also the combination of two terms: (a) Entropy production resulting from the temperature difference between the fluid and the outer side of the pipe (that is denoted by the average Nusselt number):
Q~" Ns2'ar = 2"+ lrt2kToo Cv"~,oopPrl/3r" + 1
(17)
and (b) Entropy production resulting from friction and pressure drag (denoted by the drag coefficient G,) Ns2,~p =
Co2v3x'~prcr2 Q
Ns2 = Ns2,A I, +
Ns2,A r .
(18) (19)
C O M B I N A T I O N OF E N T R O P Y G E N E R A T I O N N U M B E R S A N D O U T L I N E OF T H E D E S I G N M E T H O D
An algorithm is developed by taking into account two conditions, namely variable temperature difference and constant temperature difference. For both of the conditions, a total entropy generation number has been determined before beginning the design. This determination is to be done by means of the irreversibility concept:
1~To Ns = - - Ns, + Ns2.
Q/Too
(20)
Since the actual total entropy production equals the irreversibilitydivided by the ambient temperature, the entropy generation number, is written in terms of irreversibility.Multiplying both the denominator and the numerator of this modified number with the amount of work given to the process, a term which denotes the portion of the lost work resultingfrom the irreversibilities is obtained:
WToo
Ns=¢ - -
QTo
(21)
where I ~b = ~ .
(22)
Determining this portion before designing the system, the total entropy generation number is derived. Its separation for two sides is to be done according to the operating conditions. If the temperature difference is not constant, the entropy generation number [equation (10b)] resulting from the temperature difference can be minimized first. The result of the minimization is: ATOpt = 3 T,. (123)
380
E(~RICAN and UYGUR: SUPERCRITICALREGION HEAT TRANSFER Coal
Limestone
i
=-
i
.
~
=,
,,
C¢ .
I-
Fig. 3. Schematicdiagram of power cycle. Subtracting the minimized number from the total entropy generation number [equation (21)] and minimizing the entropy generation for the second side, the optimum radius and the outlet temperature can be determined. By the help of these two quantities, the pressure difference inside the pipe is obtained. If the outlet temperature is constant, then the minimization is only for the aim of finding the optimum radius and reaching the minimum entropy generation for the second side. Similar to the first condition, the entropy generation number for the first side is obtained, and this operation is followed by determination of the entropy generation number resulting from the pressure difference for the second side. Because the temperature difference is constant, the other term of Ns2 does not change.
APPLICATION
OF THE METHOD TO A FLUIDIZED O F A P O W E R CYCLE
BED
BOILER
The algorithm described above is applied to a power cycle in which the steam is generated by means of an atmospheric pressure fluidized bed boiler (Fig. 3). First, the effect of the boiler effectiveness (E~) on the overall cycle effectiveness (E2), where the exergy of the heat obtained in the boiler E~= the exergy of the heat of the coal + the auxiliary work and the net output work E2= the exergy of the heat of the coal + the auxiliary work has been examined, and it is shown that the latter one depends on the former one. The path of the flow in the boiler is explained in detail (Fig. 4). The heat transfer in this equipment is between a supercritical flow inside the pipe and the flow across the pipe. So, the definitions and the algorithm derived above can be applied to the cycle. The heat transfer in the boiler differs according to the section. The convection coefficient is higher than the one in the convection spaces above the bed because the heat transfer surfaces are extremely large inside the bed. The drag coefficient also changes. It depends on the radius of the particles used in the bed. To defeat all these difficulties, the heat transfer is examined separately, and a computer programme is prepared to find the entropy generation numbers and pressure differences for the determined irreversibilities.
CONCLUSIONS The result obtained is proof of the necessity of second law analysis. The pressure difference is directly proportional with the irreversibility and the entropy generation number for the first side (Figs 5-13 and Table 1). The exergy analysis of the boiler and the system shows that the exergy of the heat given by the combustion of the coal is able to be increased by modifying the design
E O R I C A N and UYGUR:
SUPERCRITICAL REGION HEAT T R A N S F E R
657 K 533 K
el -
I
381
_t -I
~ E1 _P§H_1B
_1 ~
II
) E2 i
F
SH_2B
.
I I
) E3 PSH3
_PSH_3_B _
tr=
_
--I
!
Fig. 4. Path of flow in atmospheric fluidized bed boiler of power cycle.
parameters. For the same irreversibility, the first law efficiency of both the boiler and the cycle change too much with entropy generation number for the flow inside the tube. The first law efficiency and the total heat transfer surface increase with the increase of the entropy generation number for the first side. Second law efficiency, on the other hand, depends mainly on the irreversibility. So, it changes due to the total entropy production number. Its change is not effected by the separation of the number. For this reason, it is possible to get an optimum solution in designing the pipes due to the minimum irreversibility (obtaining the high portion of the exergy saved) and maximum entropy generation number for the first side (obtaining the high first law efficiency). However, the pipe diameters must be standard for construction of the system. For this reason, the design of the system has been repeated for the standard average diameters of the ones between the minimum and maximum entropy generation numbers for constant irreversibility. It can be
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E~RICAN and UYGUR: SUPERCRITICAL REGION HEAT TRANSFER
3000 -2500 --
/ e
Mln. dla. (max. N $I)
..I 2000 --
/
~" ~
• 15001000
. ~
St. mean dla.
~ m ./J/
_
/
,
/
+ / +
Max. dia.(min. Ns,)
5OO
0
0.2
0.4 Irreverslbillty
0.6
0.8
Fig. 5. Change of pressure drop with irreversibility in Economizer 1-2-3 (El, E2, E3).
1000 -800 --
t~
600
/ •
Min. dla. (max. N S1)
/ •
--
¢1.
•
400
/ .
--
200 --
a/~
a o ~ o ~
~ + o ~ t . _ . . _ . . _ +o . - /- - - - - - - ~ - 0
0.2
0.4
• St. mean dia. _+ Max. dia.j (rain. N el)
0.6
0.8
Irreverslbility Fig. 6. Change of pressure drop with irreversibility in Economizer 4-5-6 (E4, E5, E6).
350 -300 --
• MIn. die. (max, NSl)
250 -m
O..
• /
200 150
•
--
•
100 --
j e / m ~ : ~ . ; ~ e . / +
50 -0
St. mean die. Max. die. (mln. NsI)
0.2
0.4 I rreversibility
I
J
0.6
0.8
Fig. 7. Change of pressure drop with irreversibility in Economizer 7 (E7A).
E(3R|CAN and UYGUR:
SUPERCRITICAL REGION HEAT TRANSFER
3000 -2500 --
/ o
Min. dia. (max. NSl)
. , t
2000 D..
J
--
15oolOOO
/
.../ _
+
500 - -
o ~ m ~ m _=..._._.._+
,
0
0.2
0.4 Irreversiblllty
+
T
0.6
St. mean dla. Max. dla. (rain. NSt )
I
0.8
Fig. 8. Change of pressure drop with irreversibility in Economizer 7B (E7B).
1500 --
o/a/
1000 --
MIn.dla. (max.NSl)
o / St. meandla.
500 --
Max. dia. (rain. NSl) e
0
0.2
0.4 Irreversibillty
0.6
0.8
Fig. 9. Change of pressure drop with irreversibility in Presuperheater 1-2-3 (PSH 1A, PAH2-A, PSH3A).
4
3 --
1-
0
~ / o r e
e/.
. o~/ t 1 ~ / ~ / m . . ~/ I
Min'dla'(max'Nsl)
St. moandla. Max. dla. (min. NSl )
0.2
I
I
I
0.4
0.6
0.8
Irreverslbllity Fig. 10. Change of pressure drop with irreversibility in Presuperheater IB-2B-3B (PSHIB, PSH2B, PSH3B). ECM 32/4---F
383
384
EG1L[CAN and UYGUR: 2500
SUPERCRITICAL REGION HEAT TRANSFER
-
e~oj
2000 ~, 1500 -
MIn. dla. (max. NS1)
e J
500 -
Max. dla. (mln. N81) , 0.2
0
I 0.4
I 0.6
J 0.8
Irreverslbllity Fig. 11. Change of pressure drop with irreversibility in finishing superheater 4-5 (FSH4, FSH5).
200 --
150 -~" a. ~.
/ o / J ' j . .
100 --
0
MIn. dia. (max. NSl) St. mean dla.
~ . e
0.2
0.4
0.6
0.8
Irreverslbility Fig. 12. Change of pressure drop with irreversibility in Reheater 5-6 (RHS, RH6A).
3000 -2500 --
j e
MIn. dla. (max. NSl )
e
2000 --
e J
1500lOOO
500 --
__._..-.o St. mean dla.
I 0
0.2
0.4
0.6
0.8
Irreverslblllty Fig. 13. Change of pressure drop with irreversibility in Reheater 6B (RH6B).
E~IRtCAN and UYGUR:
SUPERCRITICAL REGION HEAT TRANSFER
385
Table 1. Effect of design parameters on first law efficiency, second law effectiveness and amount of pipe material I 0.2 0.2 0.2 0.5 0.5 0.5
Diameter (m)
Minimum (max. Standard mean Maximum (rain. Minimum (max. Standard mean Maximum (rain.
Nst) Ns~ ) Nsl) Nsl)
~1
172
El
E2
Ym (tonnes)
0.857 0.710 0.640 0.920 0.890 0.883
0.450 0.415 0.386 0.698 0.682 0.675
0.420 0.425 0.426 0.254 0.258 0.260
0.353 0.365 0.367 0.195 0.205 0.210
7.30 11.26 15.87 7.30 11.26 15.87
Table 2. Minimum, maximum and standard pipe diameters used for minimum entropy generation design
Section
Minimum diameter (ram)
Maximum diameter (ram)
Standard mean diameter (ram) (in.)
1=0,2 E123 EA56 E7A E7B FSH PSH123 PSHB RHA RHB
92 43 57 35 63 71 59 101 62
142 143 150 146 140 140 150 140 145
110 (4~) 90 (3~) 100 (4) 90 (3½) 100 (4) 100 (4) 100 (4) 120 (4½) 100 (4)
1=0.5 E123 E456 E7A E7B FSH PSH 123 PSHB RHA RHB
88 40 45 28 57 68 53 92 55
140 138 147 142 138 137 145 138 142
90 (3~) 80 (3) 90 (3½) 80 (3) 90 (3~) 90 (3~) 90 (3½) 110 (4) 90 (3½)
o b s e r v e d f r o m Figs 5-13 a n d T a b l e 1 t h a t the pressure d r o p , the first law efficiency, the second law effectiveness values a n d the a m o u n t o f the m a t e r i a l used for m i n i m i z e d e n t r o p y g e n e r a t i o n designs with the s t a n d a r d average d i a m e t e r s are between the intervals o f the limits. T h e values o f the s t a n d a r d average pipe diameters, t o g e t h e r with the m i n i m u m a n d m a x i m u m ones, are listed in T a b l e 2. A s a conclusion, the i m p o r t a n c e o f the second law has been p r o v e d in this study, a n d the o p i n i o n o f designing the t h e r m a l systems on the basis of, n o t only the first law, b u t also aiming at o p t i m i z a t i o n t h r o u g h the first law, the second law a n d the e c o n o m i c cost has been f o r m e d a n d developed.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
N. Lior and G. J. Rudy, Energy Convers. Mgmt 28, 327 (1988). S. U. Kumar and W. J. Minkowycz and K. S. Patel, Int. Commun, Heat Mass Transfer 16, 335 (1989). G. G. Rice, Stifling engine---availability criteria, lOth IECEC Conference, pp. 791-795 (1984). N. Eirican, Heat Recovery System and ClIP 8, 549 (1988). E. Michaelidcs, Energy Res. 8, 241 (1984). G. Lucca, On the Opportunityfrom the Didactic Point of View, To Derive Entropyfrom Exergy, Second Law Analysis of Thermal System. The American Society of Mechanical Engineers, pp. 183, 191 (1987). A. Bcjan, Int. J. Heat Mass Transfer 21, 655 (1978). A. Bejan, Trans. ASME 99, 374 (1977). A. Bejan, Trans. ASME I01, 718 (1979). A. Bejan, Adv. Heat Transfer 15, 1 (1982).