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Mean temperature of heat reception asapplied to the Rankine cycle

P. E. Liley

3608 Mulberry Drive, Lafayette, IN. 47905-3937, USA

E-mail: eandpliley@insightbb.com

Abstract An alternative, possibly simpler, thermodynamic analysis of Rankine cycles is proposed.

Keywords Rankine cycle; vapor cycles; irreversibility; availability; thermodynamic efficiency

Nomenclature

f saturated liquid

h specific enthalpy, kJ/kg

i specific irreversibility, kJ/kg

P pressure, bar

Q heat transfer, kJ/kg

s specific entropy, kJ/kg.K

t Celsius temperature, C

T absolute temperature, K

h mean temperature of heat reception, Kv specific volume, m3/kg

W specific work, kJ/kg

x quality of liquidvapor mixture

Subscripts

c condition at condenser

C condition at dead state surrounding

h condition at boiler inletH condition at source

o dead state reference condition

h thermodynamic efficiency

Introduction

This paper considers the application of the concept of the mean temperature of heat

reception, i.e., the effective temperature of energy input in a cycle, as contrasted

with the varying temperatures which actually occur. It will be presumed that thereader will have a knowledge of the basic Rankine cycle and of superheating and

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Mean temperature of heat reception 109

Analysis

Fig. 1 illustrates a Rankine cycle in which the fluid is isentropically compressed

from point 1 to point 2, heated to condition 3 and isentropically expanded from point

3 to point 4, finally being condensed from point 4 back to the original condition,

point 1. The heat added in the cycle is 23 Tds, which is the area under the Ts

diagram from point 2 to point 3. In the present analysis this is equated to the area

of the rectangle of height h and width Ds, where h is defined as the mean tem-perature of heat input. The use of this concept has, to the best of the authors knowl-

edge, been considered by only two authors. Haywood [1] gives a definition of hsimilar to that above but then remarks that it would not be sensible to use it in numer-

ical computation. Surprisingly, in another publication [2] he uses it in two calcula-

tions. Cole [3] gives a descriptive account of the concept without any numericalcalculations.

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Fig. 1 Temperatureentropy diagram for Rankine cycles.

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110 P. E. Liley

vf(P2 - P1) = 4.01kJ/kg so that h2 = h1 + Wp = 125.3kJ/kg.1 The turbine work is WT

= h4 - h3 = 973.4kJ/kg, so that the net work output per cycle is 969.4kJ/kg. Theenergy input in the boiler is Qh = h3 - h2 = 2675.3kJ/kg and the heat removed in thecondenser is Qc = h1 - h4 = 1705.9kJ/kg for a net heat transfer of 969.4kJ/kg intothe system. The thermodynamic efficiency of the cycle is h = Wnet/Qh = 0.3624. Allthe above is the standard cycle analysis.

Alternatively, as the heat supplied in the cycle is Qh = Tds, if a meantemperature of heat reception, h, is assumed, then Qh = hDs = h(s3 - s2). Usings3 = 6.0689kJ/kg.K and s2 = s1 = 0.4220kJ/kg.K with Qh = h3 - h2 = 2675.3kJ/kgenables h to be calculated as 473.76K. Since the cycle is now a rectangle on

the Ts diagram, the cycle efficiency, using an absolute temperature of 302.12K for

Tc, is 1 - Tc/Th = 0.3623. This result agrees with that obtained above. Calculationsmade in this manner only require a knowledge of the cycle parameters at points 2

and 3.

Table 1 lists values of thermodynamic properties for points 19 of Fig. 1 and of

values derived therefrom. Points 14 refer to the cycle analyzed above. The cycle

1256 is for the steam to be superheated to 40 bar, 500C, and cycle 125789 is

for the steam to be superheated and then reheated to 500C. These results may be

compared to those which will be obtained by the standard analysis and excellent

agreement is obtained.

As a further check on the validity of using the mean temperature of heat addition,

similar calculations were made for boiler pressures of 20, 60, 80 and 100 bar,

and all the results are summarized in Table 2. Except possibly for the cases

of superheat with and without reheat at 100 bar, the agreement seems to be

satisfactory.

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1 After this paper was prepared recalculation using the IAPWS formulation 1995 [4] steam tables

TABLE 1 Thermodynamic properties and results for Rankine cycles

Point 1 2 3 4 5 6 7 8 9

Cycle all all 1 1 2 2 3 3 3

t(C) 28.97 28.97 250.39 28.97 500.00 28.97 123.71 500.00 28.97T(K) 302.12 302.12 523.54 302.12 773.15 302.12 396.86 773.15 302.12

P (bar) 0.04 40 40 0.04 40 0.04 2.24 2.24 0.04

h (kJ/kg) 121.28 125.29 2800.6 1827.2 3445.4 2135.9 2711.5 3486.9 2 551.1

s (kJ/kg.K) 0.4220 0.4220 6.0689 6.0689 7.0902 7.0902 7.0902 8.4643 8.4643

x 0.0 1.0 0.7014 0.8283 1.0 0.9990

Cycle 1234 1256 125789

h (K) 473.76 497.90 509.24

h 0.3623 0.3932 0.4067

Cycle identification: 1 = simple Rankine cycle, points 14; 2 = Rankine cycle with superheat, points 1, 2, 5, 6;3 = Rankine cycle with superheat and reheat, points 1, 2, 79.

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Mean temperature of heat reception 111

Availability (exergy) and irreversibility analysis

With the usual assumption of negligible kinetic or potential energy changes, the flow

or stream availability may be defined as y = (h - ho) - To(s - so), where the sub-script o refers to the surrounding or dead state. If a heat transfer occurs with a

system at temperature T, the availability is y = q(1 - To/T). For further details thesummary by Black and Hartley [6] seems to be particularly cogent.

For a simple Rankine cycle operating between 0.04 and 40 bar, cycle 1234, with

heat flowing from a heat reservoir at 750C and the sink or dead space temperature

being 20C, Dyboil = (h3 - h2) - To(s3 - s2) = 1019.9kJ/kg, where To = 293.15K. Dycond= (h4 - h1) - To(s4 - s1) = 50.5kJ/kg. Since s3 - s2 = s4 - s1, Dyboil - Dycond = (h3 - h2) -(h4 - h1) = Qh - Qc. This is also (h3 - h4) - (h2 - h1) = WT - Wp. So that 1019.9 - 50.5= 969.4kJ/kg, which checks. Since the pump and the turbine are considered to bereversible, the total irreversibility in the cycle is icyc = iboil + icond = To(s3 - s1 - Qh/TH) -To(s1 - s3 - Qc/TC) = To(Qc/TC - Qh/TH) = 939.37kJ/kg, where TH = 1023.15K and TC

= 293.15K. The available part of the heat from the high-temperature reservoir isQh(1 - To/TH) = 1908.8kJ/kg. Adding the net work output per cycle, 969.4kJ/kg, to

TABLE 2 Rankine cycle data for cycles operated between condenser pressure of 0.04 bar

and stated boiler pressures

Boiler pressure (bar)

Item Case 20 40 60 80 100

qh (kJ/kg) 1a 2675.4 2675.3 2656.6 2628.5 2593.2

Wnet (kJ/kg) 1a -887.6 -969.4 -1005.0 -1020.9 -1024.6h 1a 0.3318 0.3623 0.3783 0.3884 0.3951

(K) 1b 452.11 473.76 485.96 493.98 500.67

h 1b 0.3318 0.3624 0.3783 0.3884 0.3951

qh (kJ/kg) 2a 3344.4 3320.1 3295.0 3269.2 3242.7

Wnet (kJ/kg) 2a -1226.6 -1305.5 -1349.8 -1365.2 -1377.1h 2a 0.3668 0.3939 0.4078 0.4176 0.4278

(K) 2b 477.10 497.90 510.18 518.73 525.13h 2b 0.3668 0.3932 0.4078 0.4176 0.4247

qh (kJ/kg) 3a 4167.2 4095.5 4040.2 3990.0 3945.1

Wnet (kJ/kg) 3a -1576.1 -1665.7 -1732.7 -1714.8 -1732.7h 3a 0.3782 0.4067 0.4220 0.4298 0.4392

(K) 3b 489.40 509.24 521.48 529.82 536.33

h 3b 0.3827 0.4067 0.4206 0.4298 0.4367

P7 (bar) 3 0.04 2.235 4.192 6.671 9.682

t7 (C) 3 113.39 123.71 145.36 168.05 178.51

Case identification: 1 = simple Rankine cycle, 2 = Rankine cycle with superheat, 3 = Rankine cycle withsuperheat and reheat. a = actual cycle parameters from usual calculation, b = Mean temperature of heatreception parameters.

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Using the mean temperature of heat input concept, iboil = To(1 - h/TH)Ds =293.15(1 - [473.76/1023.15])5.6469 = 888.88kJ/kg and icond = To(Tc/TC - 1)Ds =50.65kJ/kg for a total irreversibility of 939.57kJ/kg. An additional check is provided

by the relation icyc = Qh (hCarnot - hactual) = 2675.3(1 - 0.2865 - 0.3624) = 939.25kJ/kg, again in agreement. To summarize the proposed method, determine the initialenthalpy, h1, and entropy, s1, values, also the enthalpy, h2, after isentropic compres-

sion, the maximum enthalpy value, hm, and the maximum entropy, sm, reached in the

cycle. Then the heat added in the cycle is Qh = hm - h2, the mean temperature ofheat reception is h = (hm - h2)/(sm - s1), the thermodynamic efficiency is 1 - Tc/Th,where Tc is the condenser temperature = T1, the net work per cycle is ( h - T1)(sm- s1) or hQh. The irreversibility in the boiler is To(1 - h/TH)(sm - s1) where To andTH are the temperatures of the heat sink and the heat source respectively. The irre-

versibility of the condenser is To(Tc/TC - 1)(sm - s1). The simplicity of the proposed

method does not hold where reheating occurs, as one has to determine interstagepressures, enthalpies and entropies, but the method can be used as a check.

References

[1] R. W. Haywood,Equilibrium Thermodynamics for Engineers and Scientists (Wiley, New York, 1980),

section 14.1.7, pp. 205206. (This book has been reprinted by Krieger Publishing, Malabar, FL.)

[2] R. W. Haywood,Analysis of Engineering Cycles. Worked Problems (Pergamon Press, Oxford, 1991),

see problem 7.6.

[3] G. H. A. Cole, Thermal Power Cycles (Arnold, London, 1991), section 11.6, pp. 149150.

[4] A. H. Harvey, Thermodynamic Properties of Water(NISTIR 5078, Boulder, CO, 1998).[5] L. Haar, J. S. Gallagher, et al.,NBS/NRC Steam Tables (Hemisphere, Washington, DC, 1984).

[6] W. Z. Black and J. G. Hartley, Thermodynamics, 3rd edn (Harper Collins, New York, 1996), see table

6.1, p. 324, and relevant text.

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