References and Bibliography

References

1. ES 526: 1961, Definitions of the calorific value of fuels, British Standards Institution, London

2. D. Fielding and J. E. C. Topps, Thermodynamic Datafor the Calculation of Gas Turbine Performance, R & M No. 3099, Ministry of Supply, H.M.S.O., London, 1959

3. Anon., Some Experiences in Combustion Scaling, AGARD Advanced Aero Engine Testing, AGARDOGRAPH 37, Pergamon Press, New York, 1959, pp. 177-204

4. E. M. Goodger, Hydrocarbon Fuels, Macmillan Press, London, 1975 5. G. Walker, Stirling-Cyc/e Machines, Clarendon Press, Oxford, 1973

Bibliography

G. F. C. Rogers and Y. R. Mayhew, Engineering Thermodynamics Work and Heat Transfer, Longman Group Ud, Harlow, 1980

A very comprehensive treatment of the subject, including heat transfer, with numerous worked examples and problems.

G. Boxer, Engineering Thermodynamics, Macmillan Press, London, 1976 G. Boxer, Applications of Engineering Thermodynamics, Macmillan Press, London, 1979

In each case, the subject is treated almost entire1y on worked examp1es that are representative of examination questions.

E. M. Goodger, Combustion Calculations, Macmillan Press, London, 1977 Provides a more detailed treatment of he at release and combustion efficiency, and develops a simplified method of determining combustion temperature.

209

Solutions to Test Questions

1. Non-chemieally reaeting fluid eomprising agas, vapour, liquid or any mixture of these.

2. (a) No heat transfer. (b) Neither heat nor work transfer. No transfer of matter in either ease.

3. (a) No transfer ofmatter with environment. (b) Fixed in shape, position and orientation relative to observer.

4. (a) Dependent on system mass. Direet1y additive. (b) Independent of system mass. (e) Expressed per unit mass. Indireet1y additive.

5. (a) Two. (b) Independent and at least one intensive.

6. Beeause the subjeet eoneerns systems in equilibrium.

7. By imagining the proeess taking plaee so slowly that the properties attain a eondition of quasi-statie equilibrium at eaeh instant.

8. Unresisted, irreversible, no work done.

9. Its summation around a eycle equals zero.

10. (a) Thermodynamic properties remain eonstant when system is isolated. (b) Temperature eommon.

11. Potential energy = mgz. Kinetie energy = -!mC2 •

12. (a) Measurement ofthermometrie volume ofliquid, requiring two referenee points for ealibration. (b) Measurement of pressure (or volume) ratio of a eonstant volume (or pressure ) of agas, extrapolated towards thermodynamie behaviour of ideal gas, and expressing temperature as an absolute value in terms of a ratio with one referenee point only, this being assigned an arbit-rary value in order to give numerieal equivalenee with the established thermometrie seale.

13. Two systems that are in thermal equilibrium with a third system are in thermal equilibrium with eaeh other.

210

14. (a) Property, exact differential, function of state. (b) N on-property, inexact differential, function of process path.

15. (a) 80th fonns of energy transfer, non-properties, and path functions. (b) Work - molecules moving in observable resultant direction, positive outflow. Heat - molecules moving completely randomly, positive inflow.

16. The work done in a reversible process.

17. Conduction, convection and radiation.

18. (a) For a system operating in a cyde, the net heat input equals the net work output. (b) qNET(+) = Wnet.

19. (a) For a process not comprising a complete cyde, and in the absence of effects of motion, gravity, etc., the difference between the energy input and output results in a change in the internal energy of the system. (b) nJ. (q-w) =~.

20. gzl +ic? +h 1 + lqZ =gzz +ic1 +h z + lWZ· 21. The sum of the internal energy and the flow work = (u + pv). Change in

enthalpy at constant pressure = heat transfer.

22. s.f. (q-w) = ßh.

23. Work transfer in a reversible flow process in the absence of effects of motion, gravity, etc.

24. The sum of enthalpy and kinetic energy, and thus the value of enthalpy when the system is brought to rest in the absence of both heat and work transfers.

25. A system operating in a cyde cannot convert all the heat supplied to it into work since some energy is always rejected as heat to a lower temperature sink.

26. (a) Sole effect a continuous output ofwork (violates first law). (b) Sole effects a continuous output of work and continuous input ofheat (violates second law).

27. Ratio of (desired heat flow/required work input). Signs of energy flow ignored.

28. Work ratio = (net positive work/gross positive work), and represents the extent of independence of perfonnance on component inefficiency.

29. Is a function only of the two limiting temperatures of operation, is the maximum possible between these two temperatures, and applies equally to all reversible cydes operating at these temperatures.

30. Fonner not dependent upon the properties of a thermometric material.

211

31. Efficieney of eonversion to work is higher.

32. Entropy s = qR/T. (a) Remains eonstant. (b) Rises.

33. T cis.

34. Reversible and adiabatie.

35. The entropy of a pure substanee in its most stable state approaehes zero as the temperature approaehes zero, that is, lim s = O.

T--+O

36. The portion of heat transfer unavailable for conversion to work transfer described as anergy

37. The energy available for work transfer less the expansion work absorbed in moving the environment.

38. The maximum useful work resulting when a system comes to equilibrium with its environment by means of work and heat transfer.

39. The maximum amount of work available from a system in thermal equilib-rium with its environment.

40. Relationship between the properties p, v, T and s for a pure substance in equilibrium.

41. Agas for which u and h are functions of T only, and which follows the law p V/T = constant. This constant is known as the universal gas constant when the volume of gas equals the molar volume V m at standard levels of p and T. The value ofthe universal gas constant is 8.3143 kJjkg K, and is common for all ideal gases.

42. (a) Each is a function of any two independent intensive properties. (b) Each is a function of temperature only.

43. (a) cp varies with temperature only. (b) cp eonstant.

44. The Mollier diagram is plotted as h-s. For an ideal gas h = F(T) and so the diagrams are similar, but this does not hold for real gases.

45. (a) nJ. 1 W2 = cvCT1 - T2 )· (b) sJ. 1 W2 = cp(T1 - T2 ).

46. Gibbs-Dalton law. Behaves as an ideal gas.

47. Volume, pressure.

48. Liquefaetion by isothermal eompression.

49. T and p are no longer independent.

50. It is the only condition where a substance may exist in three phases in equilibrium.

51. By reducing the pressure, by flowing through a throttling orifice, without work or heat transfer, so that enthalpy is unchanged and the steam becomes

212

superheated. Values of enthalpy for saturated steam, water and superheated steam are then obtainable from the steam tables for the measured tempera-tures and pressures.

52. The saturation temperature corresponding to the partial pressure of the vapour in the mixture.

53. (a) Positive displacement compression and expansion. (b) Rotary duct com-pression and expansion.

54. Because the expression sJ. w = - f v dp holds only in the absence of the effects of motion.

55. (a) Reversible isothermal. (b) Isentropic.

56. (a) Polytropic. (b) Isentropic.

57. (a) Static pressure rises. (b) Total pressure constant, but falls in event of fluid friction.

58. In a compressor, the pressure level is moving away from that of the environ-ment, hence the flow tends to break away from the duct walls and reverse its direction.

59. (a) Light, small-scale, less expensive, robust. (b) Higher isentropic efficiency, smaller frontal area.

60. The overall work of compression is less, and of expansion more.

61. -&/ at constant volume. -flho at constant pressure. I:J{0 = AUo + R o T An.

Temperature rise of cold fluid 62. ------~--------------------

Maximum temperature difference available

63. Since the working fluid of open-circuit devices does not traverse a complete cycle, a closed-circuit reversible cycle, using air as the working fluid, and with corresponding heat transfer and pressure change processes, is devised for purposes of performance comparison and assessment.

64. It has the same thermal efficiency, but a higher work ratio.

65. (a) Joule; (b) Otto; (c) Otto; (d) Diesel.

66. Diesel efficiency is lower at the same compression ratio, but higher compres-sion ratios are used in practice, thus efficiencies are comparable with low-speed Diesel engines. High-speed Diesel engines are more efficient since Otto efficiency is appropriate.

67. (a) Increase compression ratio. (b) Increase mixture throughput by super-charging.

68. (a) Improves work output and work ratio. (b) Reduces thermal efficiency.

213

69. The Rankine cyde comprises reversible processes at constant pressure for heat supply and rejection, with isentropic expansion and isentropic pump-ing of liquid. Clockwise.

70. Elimination of a rotating precision component makes for mechanical simplicity, and the associated loss in refrigerating effect is not extensive.

71. By pumping a carrier fluid in which the low-pressure refrigerant vapour has previously been absorbed, and from which it is subsequently vaporised by heating.

214

Methods and Solutions to Problems

Chapter 3

3.1. No losses, therefore reversible, and Carnot efficieney applies.

QI = W/lICamot = 100J.

3.2. Minimum ratio ofheat supplied to heat absorbed = qI!q3 for Carnot

3.3. From eommon total-head enthalpy, h2 = 2.05 - 0.2 = 1.85 kJ/kg.

Chapter 4

4.1. h2 = 1.89 kJ/kg. C2 = (C2 )s y'0.8 = 17.89 m/s.

4.2. Q = fT dS (sinee reversible) = 2 (350) 0.2 = 140 kJ.

t:.U= MI - t:.(pv) = 96 kJ. Thus from first law, W = Q - t:.U = 44 kJ .

4.3. Maximum useful work = al - a2. Values of u from enthalpy expression, U = h - pv. Henee maximum useful work = 209.4 kJ.

4.4. (a) and (b) Let h = [(s, p). Express dh in two ways, and equate eoefficients. (e) h = v dp + T ds. Thus (dh)r and (oh/OP)T. Then use Maxwell. Similarly with (d), (e) and (f).

215

Chapter 5

5.1. (A) From equation of state

Vz = 0.375 m3 /kg and R = 0.25 kJ/kg K

Thus

Cp = 0.875 kJ/kg K

(Iqz)p = -87.5 kJ/kg (IWZ)p = -25 kJ/kg

(sz - SI)p = -0.252 kJ/kg K.

(B) From equation of state

pz = 150 kN/mz · Cv = 0.625 kJ/kg K

(lqz)v=-62.5kJ/kg and (IWZ)v=O

(sz - SI)v = -0.180 kJ/kg K.

(C) From adiabatic equation

pv'Y = constant

and from equation of state

Vz = 1.026 m3 /kg pz = 73 kN/m2

(I qz )ad = 0 by defmition (I Wz )ad = 62.5 kJ/kg

(sz - SI )ad = 0 since isentropic.

(D) Flom polytropic equation

pV1.2 = constant

and from equation of state

Vz = 2.106 m3 /kg pz = 36 kN/m2

Iq2 =62.5kJ/kg IWZ = 125kJ/kg (sz -st}=0.179kJ/kgK.

(E) From Boyle's law

pz = 47.5 kN/m2 (lqZ)T = (I WZ)T = 143.8 kJ/kg

(sz - sl)T = 0.359 kJ/kg K

(F) VZ. PZ and (sz - sd as in case (E)

Iqz = I Wz = O. 5.2. (A) Vgl = 0.1115 m3 /kg at 20 bar, 250°C

thus

Vz = 0.0836 m3 /kg

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At 20 bar

t s = 212.4°C vg = 0.0996 m3 jkg

thus

Xz = vz/vg = 0.839

(I wz)p = p(vz - v.) = -55.8 kJjkg

(Iqz)p=h z -h l =-409.3kJ/kg

(sz - SI)p = -0.834 kJjkg.

(B) vgz = vz/xz = 0.1329

hence

pz = 15 bar tz = 198.3°C (approximately)

(Iqz)v=uz -UI =-368.1kJjkg

(I wZ)v = 0 (sz - SI >v = -0.767 kJjkg K.

(C) Sz = Sfl + 0.839 SfgZ and SI = 6.547 kJjkg K

By trial and error

SI = Sz at pz = 0.5 bar tz = 81.3°C (approximately)

Thus

Vz = 0.839 vgz = 2.718 m3 jkg

(IWZ)ad =(UI -uz)=543kJjkg (lqZ)ad =0 (sz -s.)=O.

Note FrompVn = k, n = 1.155.

(D) From polytropic expression

Vz = 2.411 m3 jkg

thus steam is wet and Xz = 0.744

IWZ = PIVI -pzvz =512.3kJjkg n -1

Iqz = (uz - ud + I Wz = -234.4 kJ/kg

(sz - s.) = -0.619 kJ/kg K.

(E) At tz = 250°C pz = 40 bar vg = 0.04977 m3 /kg

Vz =xzvg = 0.0418 m3 /kg (sz - SdT = -1.004 kJ/kg K

(lqZ)T = T(sz - sd = -525 kJjkg

(I wZ)T = (lqZ)T - (uz - UI) = -201.3 kJ/kg.

217

5.3. Air 71, = 1.0 (T,,), = Tdp,,/pd'Y-l)!'Y = 298.6 K

From common total-head enthalpy

(C,,),= [C~ +2cp {Tl _(T,,),}],!2 =600.4mjs

71, = 0.9 T" = 316 K C" = (C,,), YO.9 = 569.6 m/s

(Lls)p = cp In (T" ),fT" = 0.0569 kJjkg K

Steam

71, = 1.0

steam superheated at 5 bar and 200°C, thus

SI = 7.060 kJjkgK= (s,,).I' and h l =2857kJjkg

At 1 bar

Sf = 1.303 kJjkg K and Sfg = 6.056 kJjkg K

Thus

x" = 0.95 and (h,,), = 2562 kJjkg

From common total-head enthalpy

(C,,), = 774.6 mjs

71, = 0.9 C" = (C,,), YO.9 = 734.9 mjs

From common total-head enthalpy

h" = 2592 kJjkg

Thus

X2 = 0.96 and (Lls)p = -0.06056 kJjkg K.

5.4. (hAI - h A ,,) = CpA (Tl - T,,) = - 25.125 kJjkg

P~l = 0.852 kPa, p~" = 6.638 kPa

Wl = 0.005 34, W2 = 0.04423

All values of hw and hv from Steam Tables. m A = 12909 kgjs Mass flow rate moist air = 12978 kg/s

Mass flow rate make-up water = m A (W2 - Wl) = 502 kg/s This is equivalent to 2.5 per cent of coolant flow rate.

Chapter6

6.1. Since design is optimal,p" = y(P1 P3)' (a) Total work donejunit massdelivered air = W = - 274.7 kJjkg.

218

(b) Heat transferred = - 82.4 kJ /kg delivered air. (c) Heat transferred = cp (TI - T2 ) = - 96.2 kJ/kg delivered air.

6.2. (a) S3 = Sgl ,hence X3 = 0.845. (b) VA and VI from given dimensions, hence m A and ml .

Vc = mc Vc = mA V3 = 0.04223.

From V2 = VI (~:-) ,and S2 = Sb h-s chart (or trial-and-error) gives P2 = 3.82 bar X2 = 0.9362, and U2 = 2401 kJ/kg.

Work per indicator diagram = AWI + I W2 - CW3 - AWC

=PA(VI - VA)+ml (UI -U2)--P3 (V3 - Vc) -mA(uC -uA)

= 41.6kJ.

(c) i.m.e.p. = W/(V3 - VA) = 6.93 bar. (d) Power = WN = 416 kW.

6 3 ( ) AT - 0.86 UCw - 0.86 UCa (t t) . . a ~ - - an (X2 - an (XI cp cp

= 0.86 UCa (tan ßI - tan ß2) = 19.6 K. cp

( ~T) 'Y/h-l) (b) rp = 1 + 11$ r; = 1.21. 6.4. (a) Ca = C2 sin (X2 = 42.26 m/s. U = 1.3 Ca = 54.94 m/s.

At 6 bar, vg = 0.3156 m3 /kg

Thus (radius)2 = m Vg = 0.3945 m , 'Ir (heigh t) Ca

N = ---.!!.- = 22.16 rev/s = 1330 rev/min. 2 'Irr

(b) w = U (2 C2 COS (X2 - U) = 6940 J/kg

Power = mw = 19085 W = 19.085 kW. (c) 11d = 2 U/C2 (2 COS (X2 - U/C2) = 0.8194.

1-(U/C2)2 +2(U/C2)coS(X2

( Vl - Vl) 2 2 (d) h l - h3 = 2 (h 2 - h3 ) = 2 2 = C2 - V2

= cl - (cl + U2 - 2 C2 U cos 25°) = 2 C2 U cos 25° - U 2 = 6940J/kg = 6.94kJ/kg.

219

6 5 E - (mA + mF) cpPr ItPr - (mA cpA ItA + mF S T F) = 0.992 . . nergy 71e -mFNSE

Temperature 71e = It,Pr - ItA = 0.994. (TtPr -ItA)

6.6. h2 = 1 (h le ) + Y (h lh ) hence l+y

Y = 0.2007 kg/s of bleed steam

= 1 [(h 2 - h le ) - To (S2 - Sie)] = 0.642. Y [(h 2 - h lh ) - To (S2 - Slh)]

Chapter 7

3600 7.1. (a) l1b = = 0.227. l1i = 0.284.

s.lc. (NSE)

(b) b.m.e.p. = -- ',val = 6.58 bar 3600 (n ) s.lc. vol air/kg fuel

i.m.e.p. = 8.23 bar. (c) 71atta = 0.541. (d) Relative efficiency = 0.525.

7.2. (a) VI = 0.861 m3 /kg, P2 = 44.31 bar, T2 = 886 K

NSE 2q3 = = cp (T3 - T2 ), T3 = 2876 K, 0: = 3.25

1 + mA/mF

V4/V3 = 4.62, T4 = 1560 K, P4 = 5.20 bar. (b) 71Diesel = 0.548, i.m.e.p. = wnet/(VI - V2) = 13.64 bar.

7.3. T2 = T4 = 438.4 K, I W2 = 3W4 = 141.9 kJ/kg T7 = 738.6 K, 6 W7 = 283.8 kJ/kg, T9 = 884.7 K Ts = 773.1 K. Thus Wnet = 125.2 kJ/kg. Thermal efficiency = 0.215, and work ratio = 0.306.

7.4. (a) S5 at 6 bar and 250°C (superheated) = S6. Quality of steam at outlet from LP turbine = X6 = 0.959.

(b)x4 = 0.956. Rankine efficiency = 0.223. (c) s.s.c. = 5.76 kg/kW h.

7.5. h l = 1405.6 kJ/kg S2 = SI = S4 + cp In (T2 /T4 ), thus T2 = 311.7 K h2 = h4 + cp (T2 - T4 ) = 1564.3 kJ/kg h7 = h6 = hf at 34°C, thus X7 = 0.134 Mixing at entry to HP compressor gives h3 = 1548 kJ/kg

220

Thus T3 = 306.3 K and Ts = 399.8 K h s = 1749.2 kJ/kg, and h9 =hrs = 172 kJ/kg

1068.3 ß - =3.19

REF - 1403.7 - 1068.3

. RE Wnet = -- = 31.4 kW.

ßREF

221

Additional Problems with Solutions

1. A single-stage double-aeting reciproeating-type air eompressor operates at 300 rev/min with a free air delivery of 15 m3 Imin measured at inlet eonditions of 101.3 kPa and 288 K. The delivery pressure is 607.8 kPa, the clearanee ratio 0.05, and n = 1.3. Take R for air as 0.287 kJ/kg K. Determine: (a) the volumetrie efficieney, (b) the swept volume of the eylinder, (e) the mass of air delivered per eompression, (d) the delivery temperature, and (e) the indieated power input. (0.852; 0.0294 m3 ; 0.031 kg; 435.5 K; 56.2 kW)

2. Derive the equation for the stoiehiometrie eombustion of methanol, CH3 OH, with air, assuming no dissociation. Molar N2/02 ratio for air = 3.76. Caleulate the standard enthalpy of re action for liquid methanol burning stoiehiometri-cally to gaseous water and carbon dioxide, given

Ml~ • CH3 0H(Q) = - 238.6 kJ/mol

Ml~ • CO2 (g)

Ml~ • H2 0(Q)

Mlvap • H2 0

= - 393.8 kJ/mol

= - 285.7 kJ/mol

= 43.9 kJ/mol NOTE. Mlvap is negative when he at is released during eondensation. Convert the value of standard enthalpy of reaetion from kJ/mol to MJ/kg. Take molar masses in g/mol as C = 12, H = 1,0 = 16. (CH3 0H + 1.5 (02 + 3.76 N2 ) = CO2 + 2 H2 0 + 5.64 N2 ; - 638.8 kl/mol; - 19.96 MJ/kg)

3. (a) In an ideal Otto eycle, the initial pressure, temperature and volume are 1 bar, 300 K and 1 m3 respeetively. Sketch the p-V diagram showing the values of p and V at all four major points in the eycle, given the eompression ratio is 5.65, and the heat input 3390 kJ. Take "1 = 1.4,c" = I J/g K, and R = 0.287 J/g K. (b) If this diagram is now adapted to the Diesel eycle operating over the same range of pressure and with the same expansion eurve, insert any new values

222

in the p-V diagram, and ealeulate the new eompression ratio 'v, and the eut-off ratio, Q. (100, 1129,6623,586.6 kPa; 1,0.177,0.177, 1 m3 ; V2 = 0.05 m3 ; 20; 3.54)

4. (a) A elosed-cireuit gas-turbine plant operates between the temperatures of 1000 and 300 K, with apressure ratio of9. Determine the Joule eycle effieieney, given 'Y = 1.4. (b) The above plant is now redesigned to operate on an optimal two-stage basis over the same ranges of temperature and pressure. Sketch the T-s diagram, and determine both the speeifie work output and the new eycle efficieney, assuming isentropie eompressions and expansions, given cp = 1.005 kJ/kg K for both air and produets. What are the advantages of two-stage operation? (e) What would be the maximum eycle efficieney attainable if regeneration of unit temperature effeetiveness is employed? (0.466; 319.2 kJ/kg; 0.370; 0.589)

5. A simple throttled refrigerator operates with ammonia between evaporator and eondenser temperatures of -16 and 32° C respeetively, the refrigerant reaehing 50 K superheat during isentropie eompression, and being subeooled to 20°C be fore entering the throttle valve. (a) Sketch the flow diagram and a representative state diagram. (b) Determine the refrigerating effect and the coefficient of performance given the following data in units kJ, kg and K.

50 K superheat t hr hg Sr Sg

(oe) h

- 16 107.9 1425.3 0.440 5.563 1541.7 32 332.8 1469.9 1.235 4.962 1613.0

Take mean cp ofliquid ammonia between 32 and 20°C as 4.7 kJ/kg K. (1106.7 kJ/kg; 4.81)

223

S

5.978 5.397

Glossary

Adiabatic. No transfer of heat between system and environment. Air Standard Cycle. A theoretical closed·drcuit reversible cycle, using air as the working fluid, against which the performance of practical open-circuit plant can be compared. Anergy. The portion ofheat transfer not convertible to work transfer. A vailability. The maximum useful work obtained when a system comes to equilibrium with its environment. Coefficient of Performance. The ratio of desired heat flow to net work input in a heat pump or refrigerator. Compounding. Multiple staging of rotating-duct engines with fluid velocity either regained or redirected between stages. Control Region. Any defined region in space, contained within a control surface. Critical Point. A boundary state for a fluid above which no distinction is evident between the liquid and vapour phases, and below which liquefaction is possible by isotherm al compression. Dew Point. The temperature at which a constant pressure vapour commences to condense on cooling. Effectiveness. The gain of availability of one system expressed as a fraction of the loss of availability of another in thermal contact. Energy. The capacity of a system to change the state of its environment by interactions at the boundary. Enthalpy. A property of a fluid given by the sum of the internal energy and the flow work. Entropy. A property of a system equal in magnitude to the ratio of reversible heat transfer to absolute temperature, representing the degree of disorder in the system, and the unavailability of work transfer. Exergy. Alternative name for availability. Flow Work. Work required by a system to occupy a given region against a resistant pressure. Fluid. Agas, vapour, liquid or any mixture of these phases. Free Energy. The maximum work obtained when a system initially at environ-mental temperature comes to complete equilibrium with its environment.

224

Gas. A fluid at a temperature above its criticallevel. It cannot be liquefied by pressure alone. Beat. The thermal form of energy transfer across the boundary of a system. Beat Engine. A device that converts heat transfer into work transfer. Beat Pump. A device that converts work and heat transfer into high-temperature heat transfer. Ideal Gas. Agas that follows theequation of state. Impulse. Equal relative velocity , pressure and enthalpy across rotor, giving re action of zero. Internal Energy. The intrinsic energy of translation, rotation, vibration and bonding of the particles comprising the internal structure of the system. Irreversibility. The loss of work transfer when a process is carried out irrever-sibly instead of reversibly. Isobaric. Constant pressure. Isochoric. Constant volume. Isothermal. Constant temperature. Perfect Gas. An ideal gas with constant specific heats. Permanent Gas. The original term for agas whose critical temperature is well below ambient and which therefore could not be liquefied by compression at the lowest temperatures obtainable at the time. Phase. A homogeneous physical state of matter. Polytropic. Pressure and volume related at all times by the expression pvn = constant. Pure Substance. A system whose chemical composition is both homogeneous and constant. Quality. The mass fraction of vapour present in a vapour-liquid mixture. Reaction. Proportion of overall enthalpy change occurring in rotor ducts. Refrigerator. A device that converts work transfer into low-temperature he at transfer. Regeneration. Utilisation of some of exhaust heat to reduce input heat require-ment. Reversible. The characteristic of a process that proceeds by infinitesimally small steps and can therefore operate in either direction. Superheat. Condition of a vapour at a temperature above its boiling level for the given pressure. Surge. Unstable flow in rotary-duct compressor as a result of mismatch of angles of flow with blades. System. Any fIXed quantity of matter contained within a boundary. Temperature. The capacity of a system to transfer energy in the form of heat to any other system in thermal contact. Thermal Efficiency. The ratio of net work output to gross he at input. Thermal Equilibrium. The characteristic common to a number of systems when their temperatures are equal such that no energy transfers between them when placed in thermal contact. Thermodynamic Equilibrium. The condition of a system when its thermo-

225

dynamic properties do not change spontaneously following isolation from the environment. Thermodynamic Process. The path taken by the properties when a system changes from one equilibrium state to another. Thermodynamic Property. Any characteristic of a system that depends only on the thermodynamic state and not on the process by which the state was reached. Thermodynamic State. The condition of a system at any instant, as determined by its thermodynamic properties. Thermodynamic Temperature. The level on a theoretical scale of temperature devised from the concept of reversible engines operating in a cycle. Thermodynamies. An experimentally based science dealing with the properties of matter, and the transfer and conversion of energy . Throttling. A process of reducing the pressure of a flowing fluid without overall change in enthalpy. Tripie Point. The state where a material can exist in all three phases at the same time. Vapour. A fluid at a temperature below its criticallevel. It can be liquefied by pressure alone. Whirl. Component of absolute velocity of fluid tangential to rotor. Work. The mechanical form of energy transfer across the boundary of a system.

226

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ener

al)

(s c

onst

.)

(T c

onst

.)

(p c

onst

.)

(v c

onst

.)

n.f.

,q

, ,W

, +

(U,

-u

,)

0 , W

, +

(U

, -

u,)

(h

,-h

,)

(u,

-u

,)

s.r.

1 q

2·

, W,

+ (

h,

-h

, )

0 , w

, + (

h 1

-h

,)

(h

,-h

,)

(U,

-u

,)

n.t.

1 W

l ,q

,-(U

,-U

,)

(U,

-U

,)

,q,

-(u

, -

u,)

P

(V,

-v,)

0

S.r.

1 W

2·

,q,

-(h

, -

h,)

(h

,-h

,)

,q,

-(h

, -

h,)

0

v(p

, -P

l)

(b)

Pe

r!e

ct G

ases

Law

pv

" co

nst.

pv

'Y c

onst

. pv

con

st.

v/T

con

st.

p/T

con

st.

,q,

c. (

1 -

n)

(T,

-T

,) 0

n

-I

RT

In (

p,/

p,)

cp

(T,

-T

,)

c.(

T,

-T

,)

"l

(c.

-~)

In (

P,/

P')

0

R I

n (p

,/p

,)

c p I

n (T

,/T

,)

c. In

(T

,/T

,)

n.e. .

Wl ~ (

T,

-T

,)

n -

I c.(

T,

-T

,)

RT

in (

P'/

Pl)

R

(T,

-T

,)

0

S.r

. 1

Wl*

~ (

T,

-T

,)

n -

I cp

(T,

-T

,)

RT

ln (

p,/

p,)

0

R(T

, -

T,)

·Cha

nges

in E

.p a

nd E

K n

eglig

ible

.

The

der

ivat

ion.

of

all

thes

e ex

pres

sion

s ar

e gi

ven

in s

ectio

ns 3

.3,

3.4,

5.5

, 5.

6 an

d 5.

7. T

he r

eade

r is

re

com

men

ded

to c

heck

the

val

idit

y o

f ea

ch e

xpre

ssio

n, a

nd t

hen

to u

se t

his

tabl

e as

a c

onve

nien

t re

fere

nce

for

foll

owin

g th

e de

velo

pmen

t o

f ex

pres

sion

s fo

r pr

oces

s an

d ey

c1e

perf

orm

ance

in

ehap

ters

6 a

nd 7

.

N

N

00

Ap

pen

dix

B:

The

rmod

ynam

ic V

alue

s fo

r R

epre

sent

ativ

e Su

bsta

nces

The

uni

ts i

n th

e ta

ble

are:

s a

nd C

in k

J/m

ol K

, han

d g

in k

J/km

ol,

R a

nd c

in

kJ/k

g K

, M i

n g/

mol

, T

in K

, an

d p

in M

Pa.

CO

z(g)

C

O(g

) H

zO(g

) H

zO(l

) C

(gr)

H

z(g)

O

z(g)

N

z(g)

A

ir(g

)

sO

213.

68

197.

54

188.

72

70.0

1 5.

69

130.

58

205.

03

191.

50

194.

34

~so

2.96

89

.33

-44

.38

-1

63

.09

0

0 0

0 0

f.

~hf.

-3

93

52

0

-11

0 5

30

-24

1 8

30

-28

58

20

0

0 0

0 0

~gf

-39

44

02

-1

37

160

-2

28

59

0

-23

7 1

90

0 0

0 0

0 M

4

4.0

10

28

.010

18

.015

12

.011

2.

016

31.9

99

28.0

13

28.9

6 R

0.

189

0.29

7 0.

461

4.12

4 0.

260

0.29

7 0.

287

Cv

0.63

8 0.

739

1.52

9 10

.06

0.65

1 0.

739

0.71

8

~ 0.

832

1.03

8 2.

024

14.1

9 0.

912

1.03

7 1.

005

v 28

.09

20.7

1 27

.55

20.2

8 20

.82

20.7

0 20

.75

Cp

36

.64

29

.07

36.4

9 28

.60

29.1

7 29

.06

29.1

0 'Y

1.30

4 1.

404

1.32

4 1.

410

1.40

1 1.

404

1.40

3 T C

R

30

4.2

13

3.0

647.

3 33

.0

154.

8 12

6.2

PC

R

7.38

3.

50

22.1

2 1.

29

5.08

3.

39

Val

ues

of

Cv

to 'Y

are

at

stan

dard

con

diti

ons

of

1 at

mos

pher

e an

d 15

°C, e

xcep

t fo

r H

z O

(g)

at 1

00°C

. T

he a

bove

val

ues

are

the

mos

t re

pres

enta

tive

of

thos

e fo

und

from

a n

umbe

r o

f so

uree

s. F

or

a m

ore

accu

rate

wor

k, r

efer

ence

to

the

mos

t re

cent

and

con

sist

ent

set

of

data

is r

ecom

men

ded.

Appendix C: Functions of Ratio 'r'

Certain functions of pressure ratio (r p) and of volume ratio (r v) are seen to occur frequently in thermodynamic cycles appropriate to the following applications

Reciprocating expanders and compressors

In rp , r/ 1n and rp (n-l)/n where n ranges from 1.2 to 1.3

Reciprocating-piston engines

r v"'(-l where 'Y = 1.4 Gas turbine engines

r p (",(-1)/-r where 'Y = 1.4 for air undergoing compression and

'Y = 1.32 for products undergoing expansion

Table C.1 summarises these ratios.

229

Tabl

e C

.1

Fun

ctio

ns o

f ,at

io ,

n =

1.2

n

= 1

.3

1 =

1.3

2 1

=1

.4

1 -=

0.8

33

n -1

=0

.16

7

1 -=

0.7

69

n -

1 =

0.2

31

1 -

1 =

0.2

42

1-1

=0

.4

1 -

1 =

0.2

86

n n

n n

1 1

Rat

io

N ~

, In

, ,l

ln

,(n

-l)/

n

,l/n

,(

n-l

)/n

,(

'r-l

)/"1

,"

1-1

,(

'r-l

)/"1

C

1.5

0.40

55

1.40

2 1.

070

1.36

6 1.

098

1.10

3 1.

176

1.12

3 2.

0 0.

6931

1.

782

1.12

2 1.

704

1.17

3 1.

183

1.32

0 1.

219

2.5

0.91

63

2.14

6 1.

165

2.02

4 1.

235

1.24

9 1.

443

1.29

9 3.

0 1.

0986

2.

498

1.20

1 2.

328

1.28

9 1.

305

1.55

2 1.

369

3.5

1.25

28

2.84

0 1.

232

2.62

1 1.

335

1.35

5 1.

651

1.43

0 4

.0

1.38

63

3.17

5 1.

260

2.90

5 1.

377

1.39

9 1.

741

1.48

6 4.

5 1.

5041

3.

502

1.28

5 3.

180

1.41

5 1.

440

1.82

5 1.

537

5.0

1.60

94

3.82

4 1.

308

3.44

9 1.

450

1.47

7 1.

904

1.58

4 5.

5 1.

7047

4.

140

1.32

9 3.

711

1.48

2 1.

512

1.97

8 1.

628

6.0

1.79

18

4.45

1 1.

348

3.96

8 1.

512

1.54

4 2.

048

1.66

9 6.

5 1.

8718

4.

758

1.36

6 4.

220

1.54

0 1.

574

2.11

4 1.

707

7.0

1.94

59

5.06

1 1.

383

4.46

8 1.

567

1.60

3 2.

178

1.74

4 7.

5 2.

0149

5.

361

1.39

9 4.

711

1.59

2 1.

630

2.23

9 1.

778

8.0

2.07

94

5.65

7 1.

414

4.95

1 1.

616

1.65

6 2.

297

1.81

1

8.5

2.14

01

5.95

0 1.

429

5.18

7 1.

639

1.68

0 2.

354

1.84

3 9.

0 2.

1972

6.

240

1.44

2 5.

420

1.66

0 1.

703

2.40

8 1.

873

9.5

2.25

13

6.52

8 1.

455

5.65

1 1.

681

1.72

6 2.

461

1.90

3 10

.0

2.30

26

6.81

3 1.

468

5.87

8 1.

701

1.74

8 2.

512

1.93

1 10

.5

2.35

14

7.09

6 1.

480

6.10

3 1.

721

1.76

8 2.

561

1.95

8

11.0

2.

3979

7.

376

1.49

1 6.

325

1.73

9 1.

788

2.60

9 1.

984

11.5

2.

4423

7.

654

1.50

2 6.

545

1.75

7 1.

808

2.65

6 2.

009

12.0

2.

4849

7.

931

1.51

3 6.

763

1.77

4 1.

827

2.70

2 2.

034

12.5

2.

5257

8.

205

1.52

3 6.

979

1.79

1 1.

845

2.74

6 2.

058

13.0

2.

5649

8.

478

1.53

3 7.

192

1.80

7 1.

862

2.79

0 2.

081

13.5

2.

6027

8.

749

1.54

3 7.

404

1.82

3 1.

879

2.83

2 2.

104

14.0

2.

6391

9.

018

1.55

2 7.

614

1.83

9 1.

896

2.87

4 2.

126

14.5

2.

6741

9.

286

1.56

2 7.

823

1.85

4 1.

912

2.91

4 2.

147

15.0

2.

7081

9.

552

1.57

0 8.

029

1.86

8 1.

928

2.95

4 2.

168

N

15.5

2.

7408

9.

816

1.57

9 8.

235

1.88

2 1.

943

2.99

3 2.

188

~ -

16.0

2.

7726

10

.079

1.

587

8.43

8 1.

896

1.95

8 3.

031

2.20

8 16

.5

2.80

34

10.3

41

1.59

6 8.

640

1.91

0 1.

973

3.06

9 2.

228

17.0

2.

8332

10

.602

1.

604

8.84

1 1.

923

1.98

7 3.

106

2.24

7 17

.5

2.86

22

10.8

61

1.61

1 9.

040

1.93

6 2.

001

3.14

2 2.

265

18.0

2.

8904

11

.119

1.

619

9.23

8 1.

948

2.01

5 3.

178

2.28

4

18.5

2.

9178

11

.376

1.

626

9.43

5 1.

961

2.02

9 3.

213

2.30

2 19

.0

2.94

44

11.6

31

1.63

4 9.

631

1.97

3 2.

042

3.24

7 2.

319

19.5

2.

9704

11

.886

1.

641

9.82

5 1.

985

2.05

5 3.

281

2.33

7 20

.0

2.99

57

12.1

39

1.64

8 10

.018

1.

996

2.06

7 3.

314

2.35

4

Index

absolute humidity 102 absolute zero of temperature 12,42,43,52 adiabatic process 28,32,39,47,85, 120 adiabatic saturator 103 adiabatic system 3 air preheater system 186 alr standard cycle 153 Amagat-Leduc law 91 anergy 57 availability 53,55,63,145,179 available work 53 Avogadro's law 74 axial-flow machine 126,132

Bernoulli equation 27, 143 binary cycle 186 blade speed ratio 133 blow down 117 Boltzmann's constant xix, 51 boundary 2,13,144 Boyle's law 72 Brayton cycle 167

calorificvalue 141 Carnot cycle 39,47,54,151,155,169,176 Celsius scale 12, 43 characteristic equation 73 Charles' law 72 Clausius inequality 46 clearance ratio 112,117 closed circuit 153 closed system 3 coeffident of performance 36, 37, 190 combined cycles 187 combustion effidency 141 compounding 133 compressibility 74, 120 compression ratio 157, 160 compressor

fixed duct 129 positive displacement 115 redprocating 11 0 rotary duct 122,126,174

condensation 11 7, 177, 187 conduction 19 control body 2 control region 3, 25, 11 0 control surface 3, 123 convection 20 cooling tower 104 corollaries of thermodynamic laws 27,33 critical point 94, 187 critical pressure ratio 122 cut-off 117, 157 cyclic process 8,23,34,37,39, 151

Dalton's law 91 Darrieu 's function 57 dead sta te 56 degree of reaction 128, 135 degree of superheat 96 device 3,35 dew point 100 diagram effidency 132,133,136 Diesel cycle 153,154,155,159 diffuser 120, 143 dryness fraction of steam 96 dual cycle 158 ducted flow 120, 122

economiser system 186 effectiveness 60,145,146,173,180 effidency

burner 37 Carnot 40,43,44,46,54,57,152,155,

172,184 combustion 141 diagram 132,133, 136 Diesel 157, 160 exergetic 58,59 isentropic 51,119,121,123,129,173 isothermal 111 Joule 168 mechanical 111,163,175 Otto 157, 160 overall 37

232

overall isotherm al 111 Rankinc 178 relative 36 thermal 34,35,37,40,42,46,60,145,

151,162,171,172,175,179,184 volumetrie 113, 163

efficiency ratio 36 energy 10,11,14,25 energy conversion 23,35 energy density 141 energy equation

non-flow 25,26,27,28,31,84 steady-flow 27,31,77,87,154,190

energy transfer 13 enthalpy 25,53,63,77,82,89

total head 33,89,120 enthalpy asf(D only 77 enthalpy of formation 139 enthalpy of reaction 140 entropy 46,49,52,54,60,63,90 entropy of formation 52 environment 2, 13, 49 equation of state 12,72,83 equilibrium

thermal 13,61 thermodynamie 3,52

Ericsson cycle 153,155,159,172 exact differentialS, 14,76,78 exergy 53,55,57,60 expansion work 56 extensive property 4,10,47 ex ternal energy 10

Fahrenheit scale 43 first law 23,60,179 flow work 17, 25, 56 fluid 1 formation reaction 64,139 free energy 60,61,63 free expansion 8,75,86,91 free gas delivery 112 free-piston engine 164 friction 8,50,86,98,121,123,162 fundamental equation 67

gas constant 74, 228 gas dynamics 120,175 gas tables 80,82 gas thermometer 12,42 gas turbine engine 173,187 Gibbs free energy function 62,64, 65 Gibbs-Dalton law 5, 92 gravitational acceleration xvii, xix, 10

he at capacity 78 he at engine 34,35,37,39,58 heat exchange 37,144,173 heatpump 35,37,39,41,190 heat release 138

heat transfer 2,5, 11, 14, 19,34,47 heat transfer coefficient 19,147 Helmholtz free energy function 62,65 h-s diagram 82, 99 humidity 100 hygrometry 100

ice point xix, 11,43 ideal cycies 151 ideal gas 12,72,74,75,78,81,92,94 ideal gas law 73 impulse 132, 135 independent property 5,77 index of process 83 indicated mean effective pressure 114,119,

161 indicator diagram 110,112,119,161 inexact differential 14, 19 intensive property 4, 77 intercooling 114, 129 internal constraint 3 internal energy 5,10,53,75,82 internal energy as f(D only 77 internal kinetic energy 10 internal potential energy 10 internally reversible process 8 interstage cooling 114, 129 irreversibility 8, 16,46,50,60, 180 irreversible process 7,50,98 isentropic process 47,49,54,80,84,88,

89,119,123,171 isobaric process 29, 32, 85, 88 isochorie process 29,32,86,89 isolated system 3, 11, 49 isotherm al process 29,32,39,47,81,85,

88,94

Joule cycle 153, 167 Joule experiment 75 Joule-Thomson experiment 77,98,194

Keenan function 57 Kelvin xv,43 kinetic energy xviii, 5, 8,10,32,89, 120,

136

latent heat of vaporisation 52,63,94,96 load ratio 157

mass continuity 25, 31 maximum useful work 55,63 maximum work 55,62,65 Maxwell's relationships 69,76 mean effective pressure 114,119,161,163 mixtures of ideal gases 91 mixtures of liquid and vapour 94 molar mass xix, 74, 140 molar volume xix, 74 Mollier diagram 82

233

momentum equation 143 multi-stage process 114,129,136,172,191

non-flow availability 56 non-flow energy equation 25,26,27,28,

31,84 non-flow work 18 nozzle 121, 132

open circuit 153, 160 open system 3 Otto cycle 153,154, 155, 159

paddle work 8, 17, 86 partial pressu re 91, 92, 1 00 partial volume 91 path function 14,19 path of process 7, 16 perfect gas 80,123,145 permanent gas 12 perpetual motion 27, 35 p-h diagram 191 phase 1,5,94 pinch point 186 plant 3 point function 5,14 polytropic process 83,84,87,111,119,179 porous plug 8,77,98 positive displacement 109, 158, 165 potential energy xviii, 5, 10 pressure ratio 111, 125, 129, 131, 168 primary property 6 process 9,37 property 9,69 psychrometric chart 105 psychrometry 100, 105 pure substance 5,7,52,75,78,94,97 p-v diagrams 16,83, 118 p-v-T diagrams 95

quality of steam 6,96, 179 quasi-sta tic process 7

radial-flow machine 126,131,132 radiation 13, 20 Rankine cycle 153, 176 Rankine scale 43 reaction, degree of 128, 135 reaction, enthalpy of 140 reciprocating devices 110,117 reduced pressure 80 reduced steady-flow energy equation 27, 33 reduced volume 80 refrigerating effect 190 refrigerator 35,37,39,190 regenerative cycle 171,174,184 reheating 172, 175, 179 relative efficiency 36, 163 relative humidity 102,103

relative vapour pressure 102 reservoir 20,33,41,42 resisted process 7 reversed Carnot cycle 190 reversed Joule cycle 194 reversible process 7, 8, 109, 151 rotary devices 115, 123, 131

saturation 94, 160 secondlaw 8,33,34,41,43,60,179 secondary heat transfer 186 sink 20,35,39,43 source 20,35,39 specific energy 37,141, 162 specific fuel consumption 163 specific heat capacity 78,81,120,173,228 specific humidity 102 specific property 4,47 specific steam consumption 119, 178 specific volume 6 stagnation enthalpy 33 standard conditions 52 state 10,65 state diagram 112 state function 4 state principle 6 static pressure 120 stationary system 3 steady-flow availability 56 steady-flow energy equation 27, 31, 77,

87,154,190 steady-flow work 18 steam point 11,12,43 steam tables 96 Stefan-Boltzmann law 20 Stirling cycle 153,154, 158, 166 stirring work 17 subcooled liquid 94 subcooling 191 supercharging 163 superheated vapour 94, 100 superheating 178, 191 surge 130 surroundings 3 system 2,3,25

temperature empirical 11 thermodynamic 12,42

temperature equivalent of velo city 89 thermal conductivity 4, 19 thermal efficiency 34,35,37,40,42,46,60,

145,151,162,171,172,175,179,184 thermal equilibrium 3,52 thermodynamic gradients 66 thermodynamic potentials 65,69 thermodynamic processes 6, 109 thermodynamic property 3,4,5,9 thermodynamic relationships 66

234

thermodynamie state 3,65 thermodynamie temperature 12,42 thermodynamies 1 thermometrie material 11,42 third law 52 throttling 8,32,98, 191, 194 total pressure 120 total-head enthalpy 33,89, 120 tripie point xix, 43, 95 T-s diagram 48,82,99, 101, 152, 173 turbine 123, 174, 179 turbulenee 8,86, 122, 123 two-property rule 5,75,78

universal gas eonstant xix, 74 unresisted proeess 8 useful work 54

van der Waals' equation 74 velocity pressure 90

Wankel engine 165 wet-and-dry bulb thermometer 103 whirl 125 work done faetor 129 work funetion 65 work ratio 37,38,40,151,169,171

175, 178 ' work transfer 2,5,13,14,15,35

expansion 56 flow 17,25,56 paddle 8, 17, 86

zeroth law 13

235