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## P K NAG soltions

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**P K Nag Exercise problems - Solved**T hermodynamic s Contents Chapter-1: Introduction Chapter-2: Temperature Chapter-3: Work and Heat Transfer Chapter-4: First Law of Thermodynamics Chapter-5: First Law Applied to Flow Process Chapter-6: Second Law of Thermodynamics Chapter-7: Entropy Chapter-8: Availability & Irreversibility Chapter-9: Properties of Pure Substances Chapter-10: Properties of Gases and Gas Mixture Chapter-11: Thermodynamic Relations Chapter-12: Vapour Power Cycles Chapter-13: Gas Power Cycles Chapter-14: Refrigeration Cycles Solved by Er. S K Mondal IES Officer (Railway), GATE topper, NTPC ET-2003 batch, 12 years teaching experienced, Author of Hydro Power Familiarization (NTPC Ltd)**Benefits of solving Exercise ( unsolved) problems of P K**Nag • The best ways is to study thermodynamics is through problems, you must know how to apply theoretical concepts through problems and to do so you must solve these problems •It contains Expected Questions of IES, IAS, IFS and GATE examinations •It will enable the candidates to understand thermodynamics properly •It will clear all your doubts •There will be no fear of thermodynamics after solving these problems •Candidate will be in a comfortable position to appear for various competitive examinations •Thermodynamics- “the Backbone of Mechanical Engineering” therefore Mastering Thermodynamics is most important many of the subjects which come in later like Heat and Mass Transfer, Refrigeration and Air Conditioning, Internal Combustion Engine will require fundamental knowledge of Thermodynamics Every effort has been made to see that there are no errors (typographical or otherwise) in the material presented. However, it is still possible that there are a few errors (serious or otherwise). I would be thankful to the readers if they are brought to my attention at the following e-mail address: swapan_mondal_01@yahoo.co.in S K Mondal Page 2 of 265**Introduction**By: S K Mondal Chapter 1 1. Introduc tion S ome Important Notes ? ? ? Microscopic thermodynamics or statistical thermodynamics Macroscopic thermodynamics or classical thermodynamics A quasi-static process is also called a reversible process Intensive and Extensive Properties Intensive property: Whose value is independent of the size or extent i.e. mass of the system. e.g., pressure p and temperature T. Extensive property: Whose value depends on the size or extent i.e. mass of the system (upper case letters as the symbols). e.g., Volume, Mass (V, M). If mass is increased, the value of extensive property also increases. e.g., volume V, internal energy U, enthalpy H, entropy S, etc. Specific property: It is a special case of an intensive property. It is the value of an extensive property per unit mass of system. (Lower case letters as symbols) e.g: specific volume, density (v, ρ). Concept of Continuum The concept of continuum is a kind of idealization of the continuous description of matter where the properties of the matter are considered as continuous functions of space variables. Although any matter is composed of several molecules, the concept of continuum assumes a continuous distribution of mass within the matter or system with no empty space, instead of the actual conglomeration of separate molecules. Describing a fluid flow quantitatively makes it necessary to assume that flow variables (pressure, velocity etc.) and fluid properties vary continuously from one point to another. Mathematical descriptions of flow on this basis have proved to be reliable and treatment of fluid medium as a continuum has firmly become established. For example density at a point is normally defined as m ∀ ? ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ρ = lim ∀→ ? 0 Here ∀ If ∀ ? is very large ρ is affected by the in-homogeneities in the fluid medium. Considering another extreme if ∀ ? is very small, random movement of atoms (or molecules) would change their number at different times. In the continuum approximation point density is defined at the smallest magnitude of ∀ ? , before statistical fluctuations become significant. This is called continuum limit and is denoted by C ∀ ? . ? is the volume of the fluid element and m is the mass m ∀ ? ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ρ = lim ∀→ ∀ ? ? C Page 3 of 265**Introduction**By: S K Mondal One of the factors considered important in determining the validity of continuum model is molecular density. It is the distance between the molecules which is characterized by mean free path (λ). It is calculated by finding statistical average distance the molecules travel between two successive collisions. If the mean free path is very small as compared with some characteristic length in the flow domain (i.e., the molecular density is very high) then the gas can be treated as a continuous medium. If the mean free path is large in comparison to some characteristic length, the gas cannot be considered continuous and it should be analyzed by the molecular theory. A dimensionless parameter known as Knudsen number, Kn = λ / L, where λ is the mean free path and L is the characteristic length. It describes the degree of departure from continuum. Usually when Kn> 0.01, the concept of continuum does not hold good. In this, Kn is always less than 0.01 and it is usual to say that the fluid is a continuum. Other factor which checks the validity of continuum is the elapsed time between collisions. The time should be small enough so that the random statistical description of molecular activity holds good. In continuum approach, fluid properties such as density, viscosity, thermal conductivity, temperature, etc. can be expressed as continuous functions of space and time. Chapter 1 The Scale of Pressure Gauge Pressure Vacuum Pressure Absolute Pressure Local atmospheric Pressure Absolute Pressure Absolute Zero (complete vacuum) At sea-level, the international standard atmosphere has been chosen asPatm = 101.325 kN/m2 Page 4 of 265**Introduction**By: S K Mondal Some special units for Thermodynamics Chapter 1 kPa m /kg 3 Note: Physicists use below units Universal gas constant, Ru= 8.314 kJ/kmole− K R M u = R Characteristic gas constant, c =kJ/kmole-K kg/kmole For Air R =8.314 29 = 0.287 kJ/kg- K For water R =8.314 kJ/kmole-K kg/kmole 18 = 0.461 kJ/kg -K Units of heat and work is kJ Units of pressure is kPa 1 atm = 101.325 kPa 1 bar = 100 kPa 1 MPa =1000 kPa. Page 5 of 265**Introduction**By: S K Mondal Chapter 1 Questions with S olution P. K . Nag A pump discharges a liquid into a drum at the rate of 0.032 m3/s. The drum, 1.50 m in diameter and 4.20 m in length, can hold 3000 kg of the liquid. Find the density of the liquid and the mass flow rate of the liquid handled by the pump. Q1.1 (Ans. 12.934 kg/s) 2 π d 4 Solution: × Volume of drum = h 2 π×1 .50 4 3 ×4.2 = m 3 = 7.422 m 3000 kg 7.422 Vloume flow rate density mass Volume kg = = = density 404.203 3 3 m m = × mass flow rate kg = 0.032 404.203 × s kg = 12.9345 s Q1.2 The acceleration of gravity is given as a function of elevation above sea level by − 6 g = 980.6 – 3.086 × 10 H Where g is in cm/s2 and H is in cm. If an aeroplane weighs 90,000 N at sea level, what is the gravity force upon it at 10,000 m elevation? What is the percentage difference from the sea-level weight? g´ 980.6 3.086 10 10,000 100 = − × × × 2 2 977.514 9.77514 s s 90,000 W 90,000 N kgf 9.806 9178.054 kgf = ete W 9178.054 9.77514 N = × 89716.765 N = 90,000 89716.765 % less 100% 90,000 0.3147% less = Prove that the weight of a body at an elevation H above sea-level is given by mg d W g d H + ⎝ + ⎝ (Ans. 89,716.4 N, 0.315%) Solution: − 6 cm m = = = = sea − = × ( ) Q1.3 2 ⎛ ⎛ ⎜ ⎜ ⎞ ⎟ ⎠ ⎠ ⎞ ⎟ = = 2 0 Solution: Where d is the diameter of the earth. According to Newton’s law of gravity it we place a man of m at an height of H then Page 6 of 265**Introduction**By: S K Mondal Chapter 1 m GMm d 2+ Force of attraction = … ( ) 2 H H (i) If we place it in a surface of earth then d GMm d2 = = Force of attraction mg ( ) o 2 GM d2 = or g ( ) o 2 1.3 GMm d 2 ( ) ∴ = Weight W ( ) ) ) 2 + H ( 2 d mg 2 o ( ) = from equation... i ( 2 d + H 2 2 d ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = mg Proved. o + d 2H Q1.4 The first artificial earth satellite is reported to have encircled the earth at a speed of 28,840 km/h and its maximum height above the earth’s surface was stated to be 916 km. Taking the mean diameter of the earth to be 12,680 km, and assuming the orbit to be circular, evaluate the value of the gravitational acceleration at this height. The mass of the satellite is reported to have been 86 kg at sea-level. Estimate the gravitational force acting on the satellite at the operational altitude. (Ans. 8.9 m/s2; 765 N) Their force of attraction = centrifugal force Solution: 2 mv = Centirfugal force r ⎛ ⎝ 2 × 28840 1000 60 60 × × + ⎞ ⎟ ⎠ ×⎜ 86 = N ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 3 12680 10 2 3 × 916 10 = 760.65 N (Weight) Q1.5 Convert the following readings of pressure to kPa, assuming that the barometer reads 760 mmHg: (a) 90 cmHg gauge (b) 40 cmHg vacuum (c) 1.2 m H2O gauge (d) 3.1 bar Solution: 760 mm Hg = 0.760 × 13600 × 9.81 Pa = 10139.16 Pa ? 101.4 kPa Page 7 of 265**Introduction**By: S K Mondal (a) (b) (c) (d) Q1.6 A 30 m high vertical column of a fluid of density 1878 kg/m3 exists in a place where g = 9.65 m/s2. What is the pressure at the base of the column. Chapter 1 90 cm Hg gauge = 0.90 × 13600 × 9.81 × 10-3 + 101.4 kPa = 221.4744 kPa 40 cm Hg vacuum = (76 – 40) cm (absolute) = 0.36 × 43.600 × 9.81 kPa = 48.03 kPa 1.2 m H2O gauge = 1.2 × 1000 × 9.81 × 10-3 + 101.4 kPa = 113.172 kPa 3.1 bar = 3.1 × 100 kPa = 310 kPa (Ans. 544 kPa) p = z ρg Solution: Q1.7 = 30 × 1878 × 9.65 Pa = 543.681 kPa Assume that the pressure p and the specific volume v of the atmosphere are related according to the equation abs and v is in m3/kg. The acceleration due to gravity is constant at 9.81 m/s2. What is the depth of atmosphere necessary to produce a pressure of l.0132 bar at the earth’s surface? Consider the atmosphere as a fluid column. 1.4 5 = = 2.3 10 × × , where p is in N/m2 pv (Ans. 64.8 km) Page 8 of 265**Introduction**By: S K Mondal Solution: Chapter 1 ρ = dp dh g Zero line H -h O 1 v = × × or dp dh g h gdh dp 2.3 10 × = or v p = h g ρ dh 1.4 3 = = pv 2300 p 1 n dh 2300 p 2300 p 1 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 1.4 = = = or v where n p + dp 1.4 h n g dh dp 2300 p ⎛ ⎝ ⎛ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ =⎜ or p = h g ρ n 2300 p =⎜ or g dh dp H ∫ 101320 ∫ n 2300 g 2300 g 1 dp p = or dh n 0 0 n ( ) )( ) ⎡ ⎣ ⎤ ⎦ 1 n − = − = = or h 101320 0 2420m 2.42 km ( − n Q1.8 The pressure of steam flowing in a pipeline is measured with a mercury manometer, shown in Figure. Some steam condenses into water. Estimate the steam pressure in kPa. Take the density of mercury as 13.6 10 kg/m × × kg/m3, the barometer reading as 76.1 cmHg, and g as 9.806 m/s2. 3 3 , density of water as 103 Solution: + ×ρ × = ×ρ × + p 0.50 g 0.03 g p o Hg H O 2 3 3 = = × × × + 0.5 13.6 10 × × × − 0.03 1000 9.806 Pa. × × or p 0.761 13.6 10 167.875 kPa 9.806 9.806 Q1.9 A vacuum gauge mounted on a condenser reads 0.66 mHg. What is the absolute pressure in the condenser in kPa when the atmospheric pressure is 101.3 kPa? (Ans. 13.3 kPa) Solution: Absolute = atm. – vacuum = 13.24 kPa − 3 3 = 101.3 – 0.66 × 13.6 × 10 × 9.81 × 10 kPa Page 9 of 265**Temperature**By: S K Mondal Chapter 2 2. Temperature S ome Important Notes Comparison of Temperature scale o o F 373K o 100C 30cm 80 212 Boiling Point Test Temperature K x C F Freezing Point o o 10cm o 273K 0 32F 0 C − − − ρ − − − C 0 F 32 32 − K 273 273 − 0 0 x 10 Relation: = = = = 30 10 − − 100 0 212 373 80 Questions with S olution P. K . Nag The limiting value of the ratio of the pressure of gas at the steam point and at the triple point of water when the gas is kept at constant volume is found to be 1.36605. What is the ideal gas temperature of the steam point? Q2.1 (Ans. 100°C) p p Solution: = 1.36605 t p p ∴ θ = × 273.16 ( ) v t = = × 273.16 1.36605 373.15 C ° Q2.2 In a constant volume gas thermometer the following pairs of pressure readings were taken at the boiling point of water and the boiling point of sulphur, respectively: Water b.p. 50.0 100 Sulphur b.p. 96.4 193 The numbers are the gas pressures, mm Hg, each pair being taken with the same amount of gas in the thermometer, but the successive pairs being taken with different amounts of gas in the thermometer. Plot the ratio of Sb.p.:H2Ob.p. against the reading at the water boiling point, and extrapolate the plot to zero pressure at the water boiling point. This Page 11 of 265 200 387 300 582**Temperature**By: S K Mondal gives the ratio of Sb.p. : H2Ob.p. On a gas thermometer operating at zero gas pressure, i.e., an ideal gas thermometer. What is the boiling point of sulphur on the gas scale, from your plot? Chapter 2 (Ans. 445°C) Solution : Water b.p. Sulphur b.p. 96.4 50.0 100 193 200 387 300 582 Extrapolating 1.926 S W T T p p T Ratio = 1.928 1.93 1.935 1.940 b.p b.p = = ∴ ° = 100 C ? 373K 1 0 50 100 200 300 2 = 1.926 1 2 ∴ = 373 1.926 × = = ° 718K 445 C 2 Q2.3 The resistance of a platinum wire is found to be 11,000 ohms at the ice point, 15.247 ohms at the steam point, and 28.887 ohms at the sulphur point. Find the constants A and B in the equation 2 = = + + + + R R 0(1 At Bt ) Solution: And plot R against t in the range 0 to 660°C. (3271, 1668628) R 36.595 11 y 0 660°C x t = Ω R 11.000 0 { } 2 = + A 100 × + B 100 × R R 1 100 0 4 = = + + 11 10 B × or or 3.861 10 15.247 × 11.000 1100A A 100B + ( ) − 3 ... i 2 = + 445 11A × + × 28.887 11.00 445 11B ( ) -3 3.6541×10 = A + 445B ... ii ( ) ( ) i gives. 6 10 3.921 10 + − equation ii − 7 = − = = × B A − 3 × { } − − 3 7 2 3.921 10 t × − 6 10 t × R 11 1 ( ) − − 3 7 2 = + 3.921 10 t × − 6 10 t × or Y 11 1 ( ) ( ) 2 − = − = × × − or t 3271 4 37922 Y 1668628 R 36.595 660 Page 12 of 265**Temperature**By: S K Mondal Q2.4 when the reference junction of a thermocouple is kept at the ice point and the test junction is at the Celsius temperature t, and e.m.f. e of the thermocouple is given by the equation at ε ε = = Where a = 0.20 mV/deg, and b = - 5.0 × 10-4 mV/deg2 (a) Compute the e.m.f. when t = - l00°C, 200°C, 400°C, and 500°C, and draw graph of ε ε against t in this range. (b) Suppose the e.m.f. ε ε is taken as a thermometric property and that a temperature scale t* is defined by the linear equation. t* = a'ε ε+ b' And that t* = 0 at the ice point and t* = 100 at the steam point. Find the numerical values of a' and b' and draw a graph of ε εagainst t*. (c) Find the values of t* when t = -100°C, 200°C, 400°C, and 500°C, and draw a graph of t* against t. (d) Compare the Celsius scale with the t* scale. Solution: Try please Q2.5 The temperature t on a thermometric scale is defined in terms of a property K by the relation t = a ln K + b Where a and b are constants. The values of K are found to be 1.83 and 6.78 at the ice point and the steam point, the temperatures of which are assigned the numbers 0 and 100 respectively. Determine the temperature corresponding to a reading of K equal to 2.42 on the thermometer. Chapter 2 2 + + bt (Ans. 21.346°C) t = a ln x + b 0 = a x ln 1.83 + b Solution: 100 = a x ln 6.78 + b … (i) … (ii) Equation {(ii) – (i)} gives 6.78 a ln 1.83 a b = − = − = = = ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⋅ ⋅ = 100 = or ∴ 76.35 a ln 1.83 46.143 76.35 lnk 76.35 21.33 C ° × ∴ ∴ − t 46.143 − * × t ln2.42 46.143 Q2.6 The resistance of the windings in a certain motor is found to be 80 ohms at room temperature (25°C). When operating at full load under steady state conditions, the motor is switched off and the resistance of the windings, immediately measured again, is found to be 93 ohms. The windings are made of copper whose resistance at temperature t°C is given by Page 13 of 265**Temperature**By: S K Mondal Chapter 2 R = R [1+ 0.00393 t] t 0 Where R0is the resistance at 0°C. Find the temperature attained by the coil during full load. (Ans. 70.41°C) R25 = R0 [1 + 0.00393 × 25] 80 R 1 0.00393 + { 93 72.84 1 0.00393 t t 70.425 C = ° Solution: ∴ = = Ω 72.84 [ ] 0 × 25 } ∴ = + × or A new scale N of temperature is divided in such a way that the freezing point of ice is 100°N and the boiling point is 400°N. What is the temperature reading on this new scale when the temperature is 150°C? At what temperature both the Celsius and the new temperature scale reading would be the same? Q2.7 (Ans. 550°N, – 50°C.) Solution: − − N − 150 100 or N = 550oN let N= C for C − 0 0 100 − = 400 100 o x N − 0 100 − then = − 100 0 400 100 100 300 x − x = or 100 x − 100 3 or x = or or or 3 x = x -100 2 x = -100 x = -50oC Q2.8 A platinum wire is used as a resistance thermometer. The wire resistance was found to be 10 ohm and 16 ohm at ice point and steam point respectively, and 30 ohm at sulphur boiling point of 444.6°C. Find the resistance of the wire at 500°C, if the resistance varies with temperature by the relation. 2 = = + + + + α α β β R R t t 0(1 ) (Ans. 31.3 ohm) Solution: 2 = = = + × + α × α × 444.6 β then α × β + 10 16 30 Solve R = (1 0 0 ) 100 ) β × β + × R R R 0 (1 100 + (1 + R , & α + 2 0 2 × α 444.6 ) 0 0 2 + × β 0(1 500 R 500 ) Page 14 of 265**Work and Heat Transfer**Chapter 3 By: S K Mondal 3. Work and Heat Transfer S ome Important Notes -ive W +ive W +ive Q -ive Q Our aim is to give heat to the system and gain work output from it. So heat input → +ive (positive) Work output → +ive (positive) vf f ∫ i ∫ vi −= i f = W pdV pdv d Q = du + dW ∫ i f f ∫ i = − + dW dQ u u f i vf ∫ vi − + pdV Q u u = − i f f i Questions with S olution P. K . Nag (a) A pump forces 1 m3/min of water horizontally from an open well to a closed tank where the pressure is 0.9 MPa. Compute the work the pump must do upon the water in an hour just to force the water into the tank against the pressure. Sketch the system upon which the work is done before and after the process. Q3.1 (Ans. 5400 kJ/h) (b)If the work done as above upon the water had been used solely to raise the same amount of water vertically against gravity without change of pressure, how many meters would the water have been elevated? (Ans. 91.74 m) (c)If the work done in (a) upon the water had been used solely to accelerate the water from zero velocity without change of pressure or elevation, what velocity would the water have reached? If the work had been used to accelerate the water from an initial velocity of 10 m/s, what would the final velocity have been? (Ans. 42.4 m/s; 43.6 m/s) Solution: (a) Flow rate 1m3/hr. Pressure of inlet water = 1 atm = 0.101325 MPa Pressure of outlet water = 0.9 MPa Page 15 of 265**Work and Heat Transfer**Chapter 3 Power pv ∴ = Δ ? By: S K Mondal 1 60 ( ) 3 m 3 = − × × 0.9 0.101325 10 kPa s kJ = 13.31 s (b) So that pressure will be 0.9 MPa ∴ ρ = × × 1000 9.81 ( 2 m V 2 1 or V 2 h g 0.9 MPa 0.9 10 6 = or h = m 91.743 m 1 ) 2 2 − = Δ = ρ (c) V pv where m v ? ? ? ? 1 ( ) 2 2 2 ρ − = Δ V p 1 Δ ρ p ρ 2 p 2 2 2 − = or V V 2 1 2Δ 2 2 2 = + or V V 1 ( ) 6 × − × 0.9 0.101325 1000 10 2 = + 10 = V 41.2 m/s. 2 Q3.2 The piston of an oil engine, of area 0.0045 m2, moves downwards 75 mm, drawing in 0.00028 m3 of fresh air from the atmosphere. The pressure in the cylinder is uniform during the process at 80 kPa, while the atmospheric pressure is 101.325 kPa, the difference being due to the flow resistance in the induction pipe and the inlet valve. Estimate the displacement work done by the air finally in the cylinder. (Ans. 27 J) Solution : = 80 kPa So work done = pΔV = 80 × 0.0003375 kJ = 0.027 kJ = 27 J Volume of piston stroke = 0.0045 × 0.075m3 = 0.0003375m3 ∴ΔV = 0.0003375 m3 as pressure is constant Final volume = 3.375×10 m -4 3 Initial volume = 0 Q3.3 An engine cylinder has a piston of area 0.12 m3 and contains gas at a pressure of 1.5 MPa. The gas expands according to a process which is represented by a straight line on a pressure-volume diagram. The final pressure is 0.15 MPa. Calculate the work done by the gas on the piston if the stroke is 0.30 m. (Ans. 29.7 kJ) Solution: Initial pressure ( Final volume (V1) = 0.12m2× 0.3m 1 p ) = 1.5 MPa Page 16 of 265**Work and Heat Transfer**Chapter 3 = 0.036 m3 p ) = 0.15 MPa By: S K Mondal Final pressure ( As initial pressure too high so the volume is neglected. Work done = Area of pV diagram ( ) 1 2 p p V 2 11.5 0.15 0.036 10 kJ 2 29.7 kJ = 2 1 = + × ( ) 3 = + × × p 1.5 M a P 0.15 MPa neg. V 0.36 m 3 Q3.4 A mass of 1.5 kg of air is compressed in a quasi-static process from 0.1 MPa to 0.7 MPa for which pv = constant. The initial density of air is 1.16 kg/m3. Find the work done by the piston to compress the air. (Ans. 251.62 kJ) Solution: For quasi-static process Work done = ∫ = pdV [ given pV C v ∫ dV V 2 = ∴ = = = p V p V pV p V C 1 1 1 1 2 2 v 1 ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ V V p V = ∴ = p V ln p 2 1 V 1 1 1 1 p p p p V V = ∴ = p V ln 1 1 2 1 1 2 2 1 0.1 0.7 = 0.1 1.2931 ln × × = MJ given p 0.1 MPa 1 m ρ = 1.5 1.16 3 = = = 251.63 kJ V m 1 1 1 0.7 MPa p 2 Q3.5 A mass of gas is compressed in a quasi-static process from 80 kPa, 0.1 m3 to 0.4 MPa, 0.03 m3. Assuming that the pressure and volume are related by pvn= constant, find the work done by the gas system. (Ans. –11.83 kJ) Given initial pressure( Initial volume( ) ) Solution: 1 p V = 0.1 m3 = 80kPa 1 Page 17 of 265**Work and Heat Transfer**Chapter 3 ) 2 p = 0.4 MPa = 400 kPa ) 2 V = 0.03 m3 n pV = C p V p V taking log both side lnp nlnV lnp nlnV or n lnV lnV lnp lnp − = − By: S K Mondal Final pressure ( Final volume( As p-V relation ∴ n n 2 = 1 1 2 e + = + 1 1 2 2 [ ] ⎛ ⎜ ⎝ 1 V V 2 2 1 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ p p = 1 2 or n ln ln 2 1 400 80 0.1 0.03 p V − − ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ p ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ ln ln 2 p 1.60944 1.20397 1 = = = ≈ ≈ or n 1.3367 1.34 V ln ln 1 V 2 p V ( ) ∴ = 1 1 n 1 80×0.1 2 2 Work done W − 400×0.03 1.34 1 − = = − 11.764 kJ Q3.6 A single-cylinder, double-acting, reciprocating water pump has an indicator diagram which is a rectangle 0.075 m long and 0.05 m high. The indicator spring constant is 147 MPa per m. The pump runs at 50 rpm. The pump cylinder diameter is 0.15 m and the piston stroke is 0.20 m. Find the rate in kW at which the piston does work on the water. (Ans. 43.3 kW) Area of indicated diagram ( Spring constant (k) = 147 MPa/m ) Solution: = 0.075 × 0.05 m2 = 3.75 × 10−m2 3 a d Page 18 of 265**Work and Heat Transfer**Chapter 3 By: S K Mondal Q3.7 A single-cylinder, single-acting, 4 stroke engine of 0.15 m bore develops an indicated power of 4 kW when running at 216 rpm. Calculate the area of the indicator diagram that would be obtained with an indicator having a spring constant of 25 × 106 N/m3. The length of the indicator diagram is 0.1 times the length of the stroke of the engine. (Ans. 505 mm2) Solution: Given Diameter of piston (D) = 0.15 m I.P = 4 kW = 4 × 1000 W Speed (N) = 216 rpm Spring constant (k) = 25 × 106 N/m Length of indicator diagram ( ) Let Area of indicator diagram = ( = 0.1 × Stoke (L) ) d a a k l dl p ) = ∴Mean effective pressure ( × d m d LAN 120 k l I.P l k L × p [ ] = m and I.P. as 4 stroke engine × a × × L A N = × ∴ d or I.P. 120 120 N × d × × = d or a d × A ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ 2 π D 4 = area A × × × I.P 0.1L k L × 120 4 D × = 2 ×π× N = and l 0.1L d × × × ×π× × × 4 0.1 120 4 1000 25 10 5.03 10 m 503mm 2 = m 6 2 × 0.15 216 − 4 2 = = × 2 Q3.8 A six-cylinder, 4-stroke gasoline engine is run at a speed of 2520 RPM. The area of the indicator card of one cylinder is 2.45 × 103 mm2 and its length is 58.5 mm. The spring constant is 20 × 106 N/m3. The bore of the cylinders is 140 mm and the piston stroke is 150 mm. Determine the indicated power, assuming that each cylinder contributes an equal power. (Ans. 243.57 kW) a l 2.45 10 58.5 837.607 kPa 0.150 m = × d p k Solution: m d 3 2 × × mm mm N m mm N m m × 1 ⎛ ⎝ ⎞ ⎟ ⎠ 3 2 = × 20 10 Pa × ∴ × ⇒ =⎜ N/m 3 2 1000 = = L Page 19 of 265**Work and Heat Transfer**Chapter 3 D 0.14 A 4 4 N 2520 = n 6 p LAN I.P. n as four stroke 120 0.14 2520 6 837.607 0.15 4 120 243.696 kW = A closed cylinder of 0.25 m diameter is fitted with a light frictionless piston. The piston is retained in position by a catch in the cylinder wall and the volume on one side of the piston contains air at a pressure of 750 kN/m2.The volume on the other side of the piston is evacuated. A helical spring is mounted coaxially with the cylinder in this evacuated space to give a force of 120 N on the piston in this position. The catch is released and the piston travels along the cylinder until it comes to rest after a stroke of 1.2 m. The piston is then held in its position of maximum travel by a ratchet mechanism. The spring force increases linearly with the piston displacement to a final value of 5 kN. Calculate the work done by the compressed air on the piston. By: S K Mondal 2 2 π π× = = = [ ] ∴ = × m 2 π× × = × × × kW Q3.9 (Ans. 3.07 kJ) Solution: Work done against spring is work done by the compressed gas φ 0.25m 1.2m 5000 2 + 120 = Mean force = 2560 N 1.2m 2560 1.2 N.m 3.072 kJ = = = Travel ∴ × Work Done By Integration At a travel (x) force (Fx) = 120 + kx At 1.2 m then 5000 = 120 + k × 1.2 ∴ Fx = 120 + 4067 x Page 20 of 265**Work and Heat Transfer**Chapter 3 By: S K Mondal 1.2 ∫ ∴ = W F dx x 0 1.2 ∫ [ ] = + 120 4067x dx 0 1.2 ⎤ ⎥ ⎦ ⎡ 2 x 2 = + × 120x ⎢ ⎣ 4067 0 2 1.2 = 120 1.2 × + × 4067 J 2 = = + 144 3072.24J 2928.24 J = 3.072 kJ Q 3.l0 A steam turbine drives a ship’s propeller through an 8: 1 reduction gear. The average resisting torque imposed by the water on the propeller is 750 × 103 mN and the shaft power delivered by the turbine to the reduction gear is 15 MW. The turbine speed is 1450 rpm. Determine (a) the torque developed by the turbine, (b) the power delivered to the propeller shaft, and (c) the net rate of working of the reduction gear. (Ans. (a) T = 98.84 km N, (b) 14.235 MW, (c) 0.765 MW) Power of the propeller = Power on turbine shaft Solution: Q 3.11 The net rate of working of the reduction gear = (15 – 14.235) MW = 0.7647 MW A fluid, contained in a horizontal cylinder fitted with a frictionless leak proof piston, is continuously agitated by means of a stirrer passing through the cylinder cover. The cylinder diameter is 0.40 m. During the stirring process lasting 10 minutes, the piston slowly moves out a distance of 0.485 m against the atmosphere. The net work done by the fluid during the process is 2 kJ. The speed of the electric motor driving the stirrer is 840 rpm. Determine the torque in the shaft and the power output of the motor. (Ans. 0.08 mN, 6.92 W) Page 21 of 265**Work and Heat Transfer**Chapter 3 By: S K Mondal Solution: Change of volume = A L 2 π d 4 = × L 2 π× 0.4 4 3 = × 0.485 m 3 = 0.061 m = Δ p V done As piston moves against constant atmospheric pressure then work φ = 0.4m M 0.485m = = × 101.325 0.061 kJ 6.1754 kJ Net work done by the fluid = 2 kJ ∴ Net work done by the Motor = 4.1754 kJ There for power of the motor 3 × 4.1754 10 10 60 × 6.96 W P W 6.96 60 2 840 π× = W = = Torque on the shaft × = = 0.0791mN Q3.12 At the beginning of the compression stroke of a two-cylinder internal combustion engine the air is at a pressure of 101.325 kPa. Compression reduces the volume to 1/5 of its original volume, and the law of compression is given by pv1.2 = constant. If the bore and stroke of each cylinder is 0.15 m and 0.25 m, respectively, determine the power absorbed in kW by compression strokes when the engine speed is such that each cylinder undergoes 500 compression strokes per minute. (Ans. 17.95 kW) Page 22 of 265**Work and Heat Transfer**Chapter 3 By: S K Mondal Solution: ( Initial volume V 2 π d 4 ) = × L 1 ( ) 2 π× 0.15 4 3 = × 0.25 m 3 = 0.00442 m Initial pressure p ( ) = V 5 p V 101.325 kPa. 1 ( ) 3 = = Final volume V 0.000884 m 1 2 1.2 1 1.2 2 = p V 1 2 1.2 1 1.2 2 p V = = ≈ Or p 699.41 700 kPa 1 V 2 ( ) − Work done/unit stroke unit cylinder W 1.2 p V 1.2 1 − ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ [ ] = × − p V 1 1 2 2 700 0.000884 1.2 1 − -ive work, as work done on the system W 500 2 1.2 Power 60 = 17.95 kW Q3.13 Determine the total work done by a gas system following an expansion process as shown in Figure. 101.325 0.00442 × − × ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = × 1.2 ( ) × × × = kW (Ans. 2.253 MJ) Solution: Area under AB = (0.4 – 0.2) × 50 × = 10 W 5 10 J = 6 1 MJ Page 23 of 265**Work and Heat Transfer**Chapter 3 By: S K Mondal A B 50 bar p pV = c 1.3 C 0.2 0.4 0.8 V m 1 3 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Area under BC p V − = 5 = = = = = 50 10 Pa × Here p V V p 0.4m 0.8m p V 50 bar B B p V 3 1 1 n 1 − × 2 2 B 3 C 5 5 × − × × 50 10 0.4 20.31 10 1.3 1 − 0.8W = 1.3 B 5 1.3 50 10 × × 0.4 = = B V p C 1.3 C 1.3 0.8 = 1.251MJ 5 = 20.31 10 Pa × Q3.14 Total work = 2.251MJ A system of volume V contains a mass m of gas at pressure p and temperature T. The macroscopic properties of the system obey the following relationship: a V ⎛ ⎜ ⎝ ⎝ ⎛ ⎜ ⎞ ⎟ ⎠ ⎠ ⎞ ⎟ − − p + (V b) = mRT 2 Where a, b, and R are constants. Obtain an expression for the displacement work done by the system during a constant-temperature expansion from volume V1 to volume V2. Calculate the work done by a system which contains 10 kg of this gas expanding from 1 m3 to 10 m3 at a temperature of 293 K. Use the values 4 2 3 15.7 10 Nm , 1.07 10 m a b = × = × = × = × , and R= 0.278 kJ/kg-K. − − (Ans. 1742 kJ) Solution: As it is constant temp-expansion then ( ) 2 V ⎝ ⎠ a ⎛ ⎜ ⎞ ⎟ ( ) ( ) + − = = p V b constant mRT k as T constant Page 24 of 265**Work and Heat Transfer**Chapter 3 By: S K Mondal ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ a a ( ) ( ) ( ) k ∴ + − = + − = p V b p V b 1 1 2 2 2 2 2 V V 1 ( ) constant k V 2 ∫ a V ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = ∴ + = W p dV p − b 1 2 ∫ k − a k − a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = − = − dV or p 2 2 V b V V b V 1 2 a V 1 V 1 V ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⎛ ⎜ ⎝ ( ) ∫ = − + − = + kln V b dv c 2 1 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ − − V V b b 1 V 1 V = + − k ln a 2 1 2 1 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − − V V b b a 1 V 1 V ( ) = + − + − p V b ln a 2 1 1 2 V 1 ⎞ ⎟ ⎠ 1 2 1 a ⎛ ⎜ ⎝ ( ) ( ) ( ) + − = = p V b constant mRT k as T constant 2 V Given m = 10 kg; T = 293 K; R = 0.278 kJ/kg. K ∴ Constant k = 10 × 293 × 0.278 kJ = 814.54 kJ a = 15.7 × 10 Nm4; b = 1.07 × 10-2m3 ⇒ V2 = 10m3, V1 = 1m3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − 2 − × 10 1.07 10 1 1.07 10 a 0.9 kJ × 1 1 1 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ∴ = + − W 814.54 ln a − 2 − × 10 ( ( 1742.14kJ ) = − 1883.44 ) = = 1883.44 157 0.9 kJ − × Q3.15 If a gas of volume 6000 cm3 and at pressure of 100 kPa is compressed quasistatically according to pV2= constant until the volume becomes 2000 cm3, determine the final pressure and the work transfer. (Ans. 900 kPa, – 1.2 kJ) 1v ) = 6000 cm3 = 0.006 m3 ) Initial volume ( Solution: Initial pressure ( 1p = 100 kPa 2 v ) = 2000 cm3 = 0.002 m3 If final pressure ( Final volume ( ) p ( 2 ) 2 × 100 0.006 2 p V ∴ = = = p 900 kPa 1 V 1 ( ) 2 2 2 2 0.002 Page 25 of 265**Work and Heat Transfer**Chapter 3 1 work done on the system p V p V n 1 1 900 0.002 100 0.006 kJ 2 1 1.2 kJ = By: S K Mondal = − ⎡ ⎣ ⎤ ⎦ 2 2 1 1 − = × − × ⎡ ⎣ ⎤ ⎦ − Q3.16 The flow energy of 0.124 m3/minof a fluid crossing a boundary to a system is 18 kW. Find the pressure at this point. (Ans. 8709 kPa) If pressure is p1 Area is A1 Velocity is V1 Volume flow rate (Q) = A1V1 ∴ Power = force × velocity = p1A1 × V1 = Solution: p1 1p× (Q) 0.124 60 V1 A1 ∴ = × 18 p 1 × 18 60 0.124 8.71 MPa = or p kPa 1 = Q3.17 A milk chilling unit can remove heat from the milk at the rate of 41.87 MJ/h.Heat leaks into the milk from the surroundings at an average rate of 4.187 MJ/h.Find the time required for cooling a batch of 500 kg of milk from 45°C to 5°C. Take the cpof milk to be 4.187 kJ/kg K. (Ans. 2h 13 min) Heat to be removed (H) = mst = 500 × 4.187 × (45-5) kJ = 83.740 MJ Net rate of heat removal H = ? Solution: − H rej leak 4.187 MJ / h − ( 37.683 MJ / h 83.740 37.683 2 hr.13 min. 20 sec. ) = = 41.87 ∴ = Time required hr = Q3.18 680 kg of fish at 5°C are to be frozen and stored at – 12°C. The specific heat of fish above freezing point is 3.182, and below freezing point is 1.717 kJ/kg K. The freezing point is – 2°C, and the latent heat of fusion is 234.5 kJ/kg. How much heat must be removed to cool the fish, and what per cent of this is latent heat? (Ans. 186.28 MJ, 85.6%) Solution: Heat to be removed above freezing point = 680 × 3.182 × {5 – (-2)} kJ = 15.146 MJ Page 26 of 265**Work and Heat Transfer**Chapter 3 By: S K Mondal Heat to be removed latent heat = 680 × 234.5 kJ = 159.460 MJ Heat to be removed below freezing point = 680 × 1.717 × {– 2 – (– 12)} kJ = 11.676 MJ ∴ Total Heat = 186.2816 MJ % of Latent heat = 159.460 = 85.6 % 186.2816× 100 Page 27 of 265**First Law of Thermodynamics**Chapter 4 By: S K Mondal 4. First Law of T hermodynamic s S ome Important Notes • dQis an inexact differential, and we write 2 ∫dQ Q = ≠ − Q or Q Q 1 2 − 1 2 2 1 1 • dWis an inexact differential, and we write 2 2 ∫ ∫ W −= = ≠ − W dW pdV W 1 2 2 1 1 1 ∫ ? ∫ ? δ = δ • Q W (ΣQ)cycle = (ΣW)cycle or • The summations being over the entire cycle. δQ – δW = dE •An isolated system which does not interact with the surroundings Q = 0 and W = 0. Therefore, E remains constant for such a system. •The Zeroth Law deals with thermal equilibrium and provides a means for measuring temperatures. •The First Law deals with the conservation of energy and introduces the concept of internal energy. •The Second Law of thermodynamics provides with the guidelines on the conversion heat energy of matter into work. It also introduces the concept of entropy. •The Third Law of thermodynamics defines the absolute zero of entropy. The entropy of a pure crystalline substance at absolute zero temperature is zero. Summation of 3 Laws •Firstly, there isn’t a meaningful temperature of the source from which we can get the full conversion of heat to work. Only at infinite temperature one can dream of getting the full 1 kW work output. •Secondly, more interestingly, there isn’t enough work available to produce 0K. In other words, 0 K is unattainable. This is precisely the Third law. Page 29 of 265**First Law of Thermodynamics**Chapter 4 By: S K Mondal •Because, we don’t know what 0 K looks like, we haven’t got a starting point for the temperature scale!! That is why all temperature scales are at best empirical. You can’t get something for nothing: To get work output you must give some thermal energy. You can’t get something for very little: To get some work output there is a minimum amount of thermal energy that needs to be given. You can’t get every thing: However much work you are willing to give 0 K can’t be reached. Violation of all 3 laws: Try to get everything for nothing. Page 30 of 265**First Law of Thermodynamics**Chapter 4 By: S K Mondal Questions with S olution P. K . Nag Q4.1 An engine is tested by means of a water brake at 1000 rpm. The measured torque of the engine is 10000 mN and the water consumption of the brake is 0.5 m3/s, its inlet temperature being 20°C. Calculate the water temperature at exit, assuming that the whole of the engine power is ultimately transformed into heat which is absorbed by the cooling water. (Ans. 20.5°C) = ω Power T. Solution: π× 2 1000 60 × ⎛ ⎝ ⎞ ⎟ ⎠ = ×⎜ 10000 6 = = 1.0472 10 W 1.0472MW = ° Let final temperature Heat absorb by cooling water / unit = m s t ∴ t C Δ ? ? ρ Δ = v s = 0.5 1000 4.2 × t ( ) × × − t 20 ( ) 6 ∴ ∴ ∴ × × × ≈ − = × 0.5 1000 4.2 t 20 t 20.5 C = t 0.5 20 1.0472 10 − = 0.4986 ° Q4.2 In a cyclic process, heat transfers are + 14.7 kJ, – 25.2 kJ, – 3.56 kJ and + 31.5 kJ. What is the net work for this cyclic process? (Ans. 17.34 kJ) -25.2kJ ∑ From first law of thermodynamics (for a cyclic process) Q W = ∑ ∑ ∑ ( ) Solution : = = + − − Q 14.7 17.44 kJ 31.5 25.2 3.56 kJ +14.7kJ -3.56kJ ∴ = W 17.44 kJ 31.5kJ Q4.3 A slow chemical reaction takes place in a fluid at the constant pressure of 0.1 MPa. The fluid is surrounded by a perfect heat insulator during the reaction which begins at state 1 and ends at state 2. The insulation is then removed and 105 kJ of heat flow to the surroundings as the fluid goes to state 3. The following data are observed for the fluid at states 1, 2 and 3. State v (m3) t (°C) 1 0.003 20 2 0.3 370 3 0.06 20 For the fluid system, calculate E2 and E3, if E1 = 0 (Ans. E2 = – 29.7 kJ, E3 = – 110.7 kJ) Page 31 of 265**First Law of Thermodynamics**Chapter 4 By: S K Mondal Solution: From first law of thermodynamics dQ = Δ = Δ + ∫ E pdV ∴ + Q E pdV 2 ∫ ( ) ∴ = − + Q E E pdV 1 2 − 2 1 1 0.1 10 (0.3 × [ ] ( ) 3 = − + − = or or E 29.7 kJ = − E 0.003) as insulated Q 0 2 3 − 2 1 E 2 3 ∫ ( ) = − + Q E E pdV 2 3 − 3 2 2 0.1 10 × ( E E ) ( ( ) ) 3 105 − = − + − or E E 0.06 0.3 3 2 3 − − = = = − + + + − × − or or or 105 105 E 29.7 29.7 − 0.1 10 24 + 24 0.06 0.3 3 3 105 110.7 kJ = − 29.7 3 Q4.4 During one cycle the working fluid in an engine engages in two work interactions: 15 kJ to the fluid and 44 kJ from the fluid, and three heat interactions, two of which are known: 75 kJ to the fluid and 40 kJ from the fluid. Evaluate the magnitude and direction of the third heat transfer. (Ans. – 6 kJ) From first law of thermodynamics dQ Q Q Q or 75 40 Q Q i.e. 6kJ from the system Solution: ∑ ∑ W = -15kJ 1 = dW W 15 6kJ ∴ + − + + = = − = − + W 44 + 1 2 3 1 2 Q = 75kJ 1 W = 44kJ 2 3 3 Q3 Q = -40kJ Q4.5 A domestic refrigerator is loaded with food and the door closed. During a certain period the machine consumes 1 kWh of energy and the internal energy of the system drops by 5000 kJ. Find the net heat transfer for the system. (Ans. – 8.6 MJ) = Δ + Q E W Solution: -W ( ) = − + Q E E W 2 1 − 2 1 − − 2 1 × 1000 3600 1000 = − + 5000kJ kJ = − 8.6MJ Page 32 of 265**First Law of Thermodynamics**Chapter 4 By: S K Mondal Q4.6 1.5 kg of liquid having a constant specific heat of 2.5 kJ/kg K is stirred in a well-insulated chamber causing the temperature to rise by 15°C. Find Δ Δ E and W for the process. (Ans. Δ E = 56.25 kJ, W = – 56.25 kJ) Solution: Q4.7 Heat added to the system = 1.5 × 2.5 × 15kJ = 56.25 kJ ∴ ΔE rise = 56.25kJ As it is insulated then dQ ∴ ΔQ= ΔE + W or 0 = 56.25 + W or W = – 56.25 kJ = 0 The same liquid as in Problem 4.6 is stirred in a conducting chamber. During the process 1.7 kJ of heat are transferred from the liquid to the surroundings, while the temperature of the liquid is rising to 15°C. Find Δ Δ E and W for the process. (Ans. Δ E = 54.55 kJ, W = 56.25 kJ) Solution: As temperature rise is same so internal energy is same ΔE = 56.25 kJ As heat is transferred form the system so we have to give more work = 1.7 kJ to the system So W = – 56.25 – 1.7 kJ = –57.95 kJ Q4.8 The properties of a certain fluid are related as follows: 196 u = = 0.287 ( pv = = + + 0.718 t + + t 273) Where u is the specific internal energy (kJ/kg), t is in °C, p is pressure (kN/m2), and v is specific volume (m3/kg). For this fluid, find cv and cp. (Ans. 0.718, 1.005 kJ/kg K) ∂ ∂ h T ( ⎛ ⎝ ⎞ ⎟ ⎠ =⎜ Solution: C p p + ∂ ) ⎡ ⎤ ⎥ ⎦ ∂ u pV T =⎢ ⎣ ⎡ p { } ( ) ⎤ ⎥ ⎥ ⎦ ∂ + + + 196 0.718t 0.287 t T ∂ 273 = ⎢ ⎢ ⎣ p + ∂ + ∂ + 0 0.718 t 0.287 t T ∂ 0 ⎡ ⎣ ⎤ ⎥ ⎦ =⎢ p = + =∂ T ∴∂ ⎣ t 273 t ⎡ ⎢ ⎤ ⎥ ⎦ ∂ ∂ t T ⎡ ⎤ ⎥ ⎦ = 1.005 ⎢ ⎣ T p = − 1.005 kJ / kg K Page 33 of 265**First Law of Thermodynamics**Chapter 4 u c T ∂ ⎝ ⎠ ⎡ ⎤ ∂ + =⎢ ⎥ ∂ ⎣ ⎦ ∂ ⎡ ⎤ = + ⎢ ⎥ ∂ ⎣ ⎦ = − A system composed of 2 kg of the above fluid expands in a frictionless piston and cylinder machine from an initial state of 1 MPa, 100°C to a final temperature of 30°C. If there is no heat transfer, find the net work for the process. By: S K Mondal ∂ ⎛ ⎞ ⎟ =⎜ v v ( ) 196 0.718t T v t T 0 0.718 v K 0.718 kJ / kg Q4.9 (Ans. 100.52 kJ) Solution: Heat transfer is not there so = Δ + = − Δ = − Δ Q E W E U W 2 ∫ = − C dT v 1 ( ( ) = − − 0.718 T T 2 1 30 ) = − = − − 0.718 100 50.26 kJ / kg ∴ Q 4.10 Total work (W) = 2 × (-50.26) = -100.52 kJ If all the work in the expansion of Problem 4.9 is done on the moving piston, show that the equation representing the path of the expansion in the pv-plane is given by pvl.4= constant. Solution: Let the process is Then pV = constant. n − p V p V [ ] = ∴ = Work done pV mRT 1 1 n 1 − 2 2 ( 1.005 0.287 kJ/kg ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = − R c c p v − − mRT mRT = = = 0.718 1 2 n 1 − − K mR n 1 − 2 0.287 × ( ) = − T T ( − 1 2 ) × − 100 30 = = or 100.52 n 1 n 1 − = = or or A stationary system consisting of 2 kg of the fluid of Problem 4.8 expands in an adiabatic process according to pvl.2= constant. The initial 0.39972 1.39972 ≈ n 1.4 Q4.11 Page 34 of 265**First Law of Thermodynamics**Chapter 4 By: S K Mondal conditions are 1 MPa and 200°C, and the final pressure is 0.1 MPa. Find W and Δ Δ E for the process. Why is the work transfer not equal to ∫ ∫ pdV ? ∫ (Ans. W= 217.35, Δ E = – 217.35 kJ, pdV = 434.4 kJ) − n 1 n − 1.2 1 1.2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ T T p p 0.1 1 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = = Solution: 2 2 1 1 0.2 1.2 ( ) ∴ = × T T 0.10 2 1 = = 322.251 49.25 C ° From first law of thermodynamics dQ ∴ = Δ = ∫ + E dW + 0 C dT dW ∫ v ∴ = − dW C dT v 2 ∫ ( ) = − × = − × − 0.718 dT 0.718 200 49.25 kJ/kg 1 = − × = − × = − = W 2 W 2 108.2356kJ 216.5kJ 216.5kJ p V p V n 1 − mRT n 1 mR T n 1 2 0.287 200 × = ∴ Δ E − ∫ = pdV 1 1 2 2 − − − mRT = 1 2 ( ) T 1 − 2 = ( ) − 49.25 ( ) − 1.2 1 = 432.65kJ ∫ As this is not quasi-static process so work is not pdV . Q4.12 A mixture of gases expands at constant pressure from 1 MPa, 0.03 m3 to 0.06 m3 with 84 kJ positive heat transfer. There is no work other than that done on a piston. Find DE for the gaseous mixture. (Ans. 54 kJ) The same mixture expands through the same state path while a stirring device does 21 kJ of work on the system. Find Δ E, W, and Q for the process. (Ans. 54 kJ, – 21 kJ, 33 kJ) Page 35 of 265**First Law of Thermodynamics**Chapter 4 ( ) pdV ( ) ( 1 10 0.06 30kJ = Heat added 89kJ Q E W or E Q W 89 30 54kJ Δ = − = − = By: S K Mondal Solution: Work done by the gas W =∫ p V = − V 2 1 ) 3 = × − 0.03 kJ = ∴ = Δ + Q4.13 A mass of 8 kg gas expands within a flexible container so that the p–v relationship is of the from pvl.2= constant. The initial pressure is 1000 kPa and the initial volume is 1 m3. The final pressure is 5 kPa. If specific internal energy of the gas decreases by 40 kJ/kg, find the heat transfer in magnitude and direction. (Ans. + 2615 kJ) − n 1 n n 1 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ n T T p p V V = = 2 2 1 Solution: 1 1 2 ⎛ ⎞ ⎟ ⎠ p p V V ∴ =⎜ 2 1 ⎝ 1 2 1 n ⎛ ⎞ ⎟ ⎠ V V p p =⎜ or 2 1 ⎝ 1 2 1 n ⎛ ⎞ ⎟ ⎠ p p = ×⎜ or V V 1 2 1 ⎝ 2 1 1000 5 p V − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 1.2 3 = × = 1 82.7m p V ∴ = W 1 1 n 1 − × − × 2 2 1000 1 5 82.7 − = − = − = = 2932.5kJ 1.2 1 Δ = − = Δ × E Q 8 40 E + 320 kJ 320 ∴ + = W 2932.5 2612.5kJ Q4.14 A gas of mass 1.5 kg undergoes a quasi-static expansion which follows a relationship p = a + bV, where a and b are constants. The initial and final pressures are 1000 kPa and 200 kPa respectively and the corresponding volumes are 0.20 m3 and 1.20 m3. The specific internal energy of the gas is given by the relation u = l.5 pv – 85 kJ/kg Where p is the kPa and v is in m3/kg. Calculate the net heat transfer and the maximum internal energy of the gas attained during expansion. (Ans. 660 kJ, 503.3 kJ) Page 36 of 265**First Law of Thermodynamics**Chapter 4 By: S K Mondal Solution: ( ) ( ) = = + + b 0.2 × × 1000 a .... i 200 a b 1.2 ... ii ( ) ii 800 − ∴ ∴ ( ) i gives b 1000 1160 = − = = + − × = a p 2 800 800V 1160 v ∫ 2 ∴ = W pdV v 1 1.2 ∫ ( ) = − 1160 800V dV 0.2 ⎡ ⎣ 1160 1.2 ⎤ ⎦ − 2 = − 1160V 400V 0.2 ( ) ( 560kJ ) 600kJ 2 2 = × − − 1.2 0.2 400 1.2 .2 kJ = − = 1160 0.2 1.5 1.2 1.5 215 60kJ 600 = 1.5 1000 × × − = u 85 215kJ/kg 1 = 1.5 200 × × − = u 85 155kJ/kg 2 ( × ) ∴ Δ ∴ Δ ∴ ⇒ = = = Δ = − = − = = u u m u U 1.5pv u = + 275 40kJ/kg 2 Δ 1 U Q u 40 1.5 W 85kJ/kg 1160 1.5 1160v − = + = 60 660kJ − − 800v ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = − 1.5 v 85kJ/kg 2 = − 800v 85kJ/kg ∂ ∂ u v = 1160 1600v − ∂ ∂ u v 1160 1600 ( 0.725 = ∴ = = for maximum u, 0 v 0.725 ) 2 ∴ = × − × − u 1160 0.725 335.5kJ/kg 1.5u 800 85kJ/kg max. = = = U 503.25kJ max max Q4.15 The heat capacity at constant pressure of a certain system is a function of temperature only and may be expressed as 41.87 2.093 100 t + + Where t is the temperature of the system in °C. The system is heated while it is maintained at a pressure of 1 atmosphere until its volume increases from 2000 cm3 to 2400 cm3 and its temperature increases from 0°C to 100°C. (a) Find the magnitude of the heat interaction. Page 37 of 265 = = + + C J/°C p**First Law of Thermodynamics**Chapter 4 By: S K Mondal (b) How much does the internal energy of the system increase? (Ans. (a) 238.32 J (b) 197.79 J) 373 ∫ = = − Q C dT t T 273 Solution: p 273 ∴ + = T 173 − t 100 373 ∫ 41.87 T 173 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = + 2.093 dT 273 2.093T ⎡ ⎣ 373 ⎤ ⎦ = + − 41.87ln T 173 273 ⎛ ⎜ ⎝ 200 100 ⎞ ⎟ ⎠ ( ) = − + 2.093 373 273 41.87ln = = + 209.3 238.32J = Δ 41.87ln2 +∫ −∫ Q E pdV E Δ = Q pdV ( ) = = = = − − Q p V V 2 1 ( ) 238.32 101.325 0.0024 − − − × × 0.0020 1000J ( 197.79J ) 238.32 40.53 J Q4.16 An imaginary engine receives heat and does work on a slowly moving piston at such rates that the cycle of operation of 1 kg of working fluid can be represented as a circle 10 cm in diameter on a p–v diagram on which 1 cm = 300 kPa and 1 cm = 0.1 m3/kg. (a) How much work is done by each kg of working fluid for each cycle of operation? (b) The thermal efficiency of an engine is defined as the ratio of work done and heat input in a cycle. If the heat rejected by the engine in a cycle is 1000 kJ per kg of working fluid, what would be its thermal efficiency? (Ans. (a) 2356.19 kJ/kg, (b) 0.702) Given Diameter = 10 cm Solution: 2 π× 10 4 Work 2 2 ∴ = = Area cm 78.54 cm p 2 3 ≡ × 300kPa 0.1m / kg 30kJ 78.54 30kJ / kg 2356.2kJ / kg Heat rejected 1000kJ = 1 cm = 30 cm dia ∴ = × Total work done = V 2356.2 η = × Therefore, 100% + 2356.2 1000 70.204% = Page 38 of 265**First Law of Thermodynamics**Chapter 4 By: S K Mondal Q4.17 A gas undergoes a thermodynamic cycle consisting of three processes beginning at an initial state where The processes are as follows: (i) Process 1–2: Compression with pV = constant to bar, U2= 690 kJ (ii) Process 2–3: W23= 0, Q23= –150 kJ, and (iii) Process 3–1: W31= +50 kJ. Neglecting KE and PE changes, determine the heat interactions Q12 and Q31. 1p= 1 bar, V1= 1.5 m3 and U1= 512 kJ. 2 p = 2 (Ans. 74 kJ, 22 kJ) +∫ − = Δ Solution: Q E pdV 1 2 v ∫ dV V 2 ( ) −= − + Q u u p V 1 2 2 1 1 1 v 1 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ p p ( ) = − + 100 1.5 ln × × 690 512 1 2 = = − 178 103.972 74.03kJ As W2-3 is ZERO so it is constant volume process. As W31 is +ive (positive) so expansion is done. u u 150 540kJ Q u u W E W 540 512 50 28 50 22kJ = − + = ∴ ∴ = = = Δ − − + = 3 2 + 31 1 3 ( ) = − − + Q4.18 A gas undergoes a thermodynamic cycle consisting of the following processes: (i) Process 1–2: Constant pressure p = 1.4 bar, V1 = 0.028 m3, W12 = 10.5 kJ (ii) Process 2–3: Compression with pV = constant, U3 = U2 (iii) Process 3–1: Constant volume, U1 – U3 = – 26.4 kJ. There are no significant changes in KE and PE. (a) Sketch the cycle on a p–V diagram (b) Calculate the net work for the cycle in kJ (c) Calculate the heat transfer for process 1–2 (d) Show that cycle cycle ∑ ∑ ∑ ∑ = = Q W. (Ans. (b) – 8.28 kJ, (c) 36.9 kJ) ( ) b W = 10.5 kJ Solution: 12 Page 39 of 265**First Law of Thermodynamics**Chapter 4 (a) By: S K Mondal 3 ∫ = W pdV 3 23 u 3 2 3 ∫ dV V ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ pV = C = p V p 2 2 2 1.4 bar u 2 2 ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ V V V V 1 W = 10.5kJ 12 u1 = 3 p V ln 2 2 3 0.028m 2 V = 1 p V ln 2 2 2 0.028 0.103 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = 1.4 100 0.103 ln × × × ( × ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = − as W p V V 12 2 1 ( ) = − = − 18.783kJ 10.5 1.4 100 V 0.028 2 3 ∴ = V 0.103 m 2 = ( ) ∴ c Q W Net work output U 26.4 10.5kJ = 0 as constantvolume 31 = − 8.283 kJ ans.(b) = − + U + W 12 2 1 12 = 36.9kJ = = = − + (d) Q U 0 18.783kJ U U − Q = ∑ U W 23 3 − 2 23 = − = − Q 18.783kJ 26.4kJ Q + + Q 0 + 31 ∴ 2 Q 3 = − − 36.9kJ 18.783 8.283kJ = − 26.4 12 23 31 ∑ ∑ ∴ = W Q Proved. Page 40 of 265**First Law Applied to Flow Process**By: S K Mondal Chapter 5 5.First Law Applied to Flow Proc ess S ome Important Notes •S.F.E.E. per unit mass basis V h 2 [h, W, Q should be in J/kg and C in m/s and g in m/s2] 2 1 1 1 h 2000 1000 [h, W, Q should be in kJ/kg and C in m/s and g in m/s2] •S.F.E.E. per unit time basis V h g + + ⎜ ⎟ ⎝ ⎠ ⎛ = + + ⎜ ⎝ Where, w = mass flow rate (kg/s) 2 2 V dQ dm dW dm + + + = + + + gZ h gZ 1 2 1 1 2 2 2 2 V d V d gZ Q gZ 1000 W + + + = + + + h 2 2 2 dm 2000 dm ⎛ ⎞ 2 dQ d + w Z 1 1 1 1 τ 2 ⎞ ⎟ ⎠ 2 dW d V + w Z h g x 2 2 2 2 τ 2 •Steady Flow Process Involving Two Fluid Streams at the Inlet and Exit of the Control Volume Mass balance w w A V A V A V v v + = + w w A 1 2 3 4 V + = + 3 3 1 1 2 2 4 4 v v 1 2 3 4 Where, v = specific volume (m3/kg) Page 41 of 265**First Law Applied to Flow Process**By: S K Mondal Energy balance V w h Z g + + ⎜ ⎟ ⎝ ⎠ ⎛ = + + ⎜ ⎝ Questions with S olution P. K . Nag Q5.1 A blower handles 1 kg/s of air at 20°C and consumes a power of 15 kW. The inlet and outlet velocities of air are 100 m/sand 150 m/srespectively. Find the exit air temperature, assuming adiabatic conditions. Take cpof air is 1.005 kJ/kg-K. Chapter 5 ⎛ ⎞ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 2 V dQ d + + + + w h Z g 1 2 1 1 1 2 2 2 τ 2 2 ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 2 V dW d V + + + + w h Z g w h Z g x 3 4 3 3 3 4 4 4 τ 2 2 (Ans. 28.38°C) Solution: 2 1 t = 20°C V = 100 m/s 1 V = 150 m/s t = ? 2 1 2 1 2 dW dt = – 15 kN From S.F.E.E. ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 2 2 V gZ 1000 V gZ 1000 dQ dt dW dt + + + = + + + w h w h 1 1 2 1 1 2 2 2000 2000 dQ dt = = = = Here w w 1 kg / s ; Z Z ; 0. 1 2 1 2 2 2 100 2000 150 2000 100 2000 ∴ + + = + − h 0 h 15 1 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 2 150 2000 ∴ − = + − h h 15 2 1 ( ) − = or C t t 8.75 p 2 1 8.75 1.005 = + = .7 C ° or t 20 28 2 Q5.2 A turbine operates under steady flow conditions, receiving steam at the following state: Pressure 1.2 MPa, temperature 188°C, enthalpy 2785 kJ/kg, velocity 33.3 m/sand elevation 3 m. The steam leaves the turbine at the following state: Pressure 20 kPa, enthalpy 2512 kJ/kg, velocity 100 m/s, and elevation 0 m. Heat is lost to the surroundings at the rate of 0.29 kJ/s. If the rate of steam flow through the turbine is 0.42 kg/s, what is the power output of the turbine in kW? (Ans. 112.51 kW) Page 42 of 265**First Law Applied to Flow Process**By: S K Mondal Solution: 1 2 w w 0.42kg/s = = Chapter 5 dQ dt 1 p = 1.2 MPa t = 188°C h = 2785 kJ/kg V = 33.3 m/s = 3 m Z1 = – 0.29 kJ/s 1 1 1 1 dW dt 1 = ? 3 m 2 2 p = 20 kPa h = 2512 kJ/kg V = 100 m/s = 0 Z2 2 2 2 By S.F.E.E. ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 2 2 V gZ 1000 V gZ 1000 dQ dt dW dt + + + = + + + w h w h 1 1 2 1 1 2 2 2000 2000 ⎧ ⎫ ⎬ ⎭ ⎧ ⎫ ⎬ ⎭ 2 2 × 33.3 2000 9.81 3 1000 dW dt 100 2000 dW dt + + − = + + + or 0.42 2785 ⎨ ⎩ 0.29 0.42 2512 ⎨ ⎩ 0 = + or 1169.655 1057.14 dW dt = or 112.515 kW Q5.3 A nozzle is a device for increasing the velocity of a steadily flowing stream. At the inlet to a certain nozzle, the enthalpy of the fluid passing is 3000 kJ/kg and the velocity is 60 m/s. At the discharge end, the enthalpy is 2762 kJ/kg. The nozzle is horizontal and there is negligible heat loss from it. (a) Find the velocity at exists from the nozzle. (b) If the inlet area is 0.1 m2 and the specific volume at inlet is 0.187 m3/kg, find the mass flow rate. (c) If the specific volume at the nozzle exit is 0.498 m3/kg, find the exit area of the nozzle. (Ans. (a) 692.5 m/s, (b) 32.08 kg/s (c) 0.023 m2) ( ) 2 a Find V i.e. Velocityatexit from S.F.E.E. V gZ V gZ dQ h h 2000 1000 dm 2000 1000 1 Data for a h = 3000 kJ/kg 1 V = 60 m/s 1 Solution: 2 2 2 dW dm + + + = + + + 1 1 2 1 2 2 h = 2762 kJ/kg 2 For data for c v = 0.498 m /kg 2 Data for b A = 0.1 m 1 v = 0.187 m /kg 1 3 1 2 2 3 Page 43 of 265**First Law Applied to Flow Process**By: S K Mondal dQ Here Z Z and dm V h 2000 V V or 2000 or V V = Chapter 5 dW 0 and 0 dm V h 2000 = = = 1 2 2 2 2 ∴ + = + 1 1 2 2 2 2 − ( ) = − h h 1 1 2 ( ) 2 2 2 + − 2000 h h 1 1 2 ( ( ) 2 = + − or V V 2000 h h 2 1 1 2 ) 2 = + − 60 2000 3000 2762 m / s = 692.532 m / s A V ( ) b ( ) = Mass flow rate w 1 v 1 1 × 0.1 60kg / s 0.187 me so A 692.532 0.498 = = 32.1kg / s ( ) c Mass flow rate is sa × = 32.0855613 2 2 = or A 8.023073 m 2 Q5.4 In oil cooler, oil flows steadily through a bundle of metal tubes submerged in a steady stream of cooling water. Under steady flow conditions, the oil enters at 90°C and leaves at 30°C, while the water enters at 25°C and leaves at 70°C. The enthalpy of oil at t°C is given by h = 1.68 t + 10.5 × 10-4 t2 kJ/kg What is the cooling water flow required for cooling 2.78 kg/s of oil? (Ans. 1.47 kg/s) + + 2 H o O + + + + + + w (h 0 0) w (h 0 0) 0 + + + + Solution: w (h 0 0) w (h 0 0) 0 o oi H O H O o o,o H O 2 2 i 2 1 2 90°C 30°C Oil Water 25°C 70°C 1 2 ∴ − − w (h h ) = w (h h ) o o,o hoi = 1.68 × 90 + 10.5 × 10–4 × 902 kJ/kg = 159.705 kJ/kg ho,o = 1.68 × 30 + 10.5 × 10–4 × 362 kJ/kg = 51.395 kJ/kg oi H 0 H O H O 2 2 o 2 i × H o = 2.78 108.36 4.187 (70 = 1.598815 kg/s ? 1.6 kg/s W ∴ kg/s − 25) 2 Q5.5 A thermoelectric generator consists of a series of semiconductor elements (Figure) heated on one side and cooled on the other. Electric current flow is produced as a result of energy transfer as heat. In a Page 44 of 265**First Law Applied to Flow Process**By: S K Mondal particular experiment the current was measured to be 0.5 amp and the electrostatic potential at (1) Was 0.8 volt above that at (2) Energy transfer as heat to the hot side of the generator was taking place at a rate of 5.5 watts. Determine the rate of energy transfer as heat from the cold side and the energy conversion efficiency. Chapter 5 (Ans. Q2 = 5.1 watts, η = 0.073) • • • + Solution: Q = E Q 1 2 • × + or 5.5 = 0.5 0.8 Q 2 • Q η = 5.5 or = 5.1 watt 2 − 5.1 × 100% = 7.273% 5.5 Q5.6 A turbo compressor delivers 2.33 m3/s at 0.276 MPa, 43°C which is heated at this pressure to 430°C and finally expanded in a turbine which delivers 1860 kW. During the expansion, there is a heat transfer of 0.09 MJ/s to the surroundings. Calculate the turbine exhaust temperature if changes in kinetic and potential energy are negligible. (Ans. 157°C) Solution: 3 V = 2.33 m /s ; p = 0.276 1 Pa ; t = 930°C M t = 93°C 1 C.C. 1 1 dW dt = 1860 kW 2 2 dQ dt = – 0.09 × 1000 kJ/s = – 90 kW dQ dt dW dt dQ dt + + w h = w h 1 1 2 2 dW dt w (h1 – h2) = ∴ − 1 or = 1860 – (–90) = 1950 kW Page 45 of 265**First Law Applied to Flow Process**By: S K Mondal 1p 1 1 1 p RT 0.287 kJ/ kg Or h1 – h2 = 275 ∴ Cp (t1 – t2) = 275 t1 – t2 = 275 1.005 273.60 ∴ t2 = 430 – 273.60 = 156.36º C Q5.7 A reciprocating air compressor takes in 2 m3/min at 0.11 MPa, 20°C which it delivers at 1.5 MPa, 111°C to an aftercooler where the air is cooled at constant pressure to 25°C. The power absorbed by the compressor is 4.15 kW. Determine the heat transfer in (a) The compressor (b) The cooler State your assumptions. Chapter 5 V = m R T V 3 276 kPa ×2.33 m / s • ∴ m = = = 7.091 kg/s 1 1 1 × 316K 1 or (Ans. – 0.17 kJ/s, – 3.76 kJ/s) Solution: dQ dt dW dt ∴ + + + + (a) w (h 0 0) = w h 1 1 1 2 dQ dt 0.0436 (111.555 – 20.1) – 4.15 = ⎛ ⎞ ⎟ ⎠ ∴ ⎜ ⎝ dQ dt = –0.1622 kW i.e. 1622 kW loss by compressor dW dt = – 4.15 kW 1 2 3 • 3 V = 2 m /min p = 0.11 t = 20°C 1 1 MPa 1 Cooles t = 111°C p = 1.5 Pa 2 2 M 1 2 3 n n − − Compressor work = (p V - p V ) = (mRT mRT ) 2 1 2 2 1 1 n -1 n 1 = 1.4 0.4× × − 0.0436 0.287(111 20)kW = 3.9854 kW ∴dQ = 3.9854 – 4.15 = –0.165 kW dt dQ dt (b) For cooler Page 46 of 265**First Law Applied to Flow Process**By: S K Mondal Chapter 5 • − = = 0.0436 × 1.005 × (111 – 25) kJ/s = 3.768348 kW to surroundings m c (t t ) P 2 1 Q5.8 In water cooling tower air enters at a height of 1 m above the ground level and leaves at a height of 7 m. The inlet and outlet velocities are 20 m/s and 30 m/s respectively. Water enters at a height of 8 m and leaves at a height of 0.8 m. The velocity of water at entry and exit are 3 m/s and 1 m/srespectively. Water temperatures are 80°C and 50°C at the entry and exit respectively. Air temperatures are 30°Cand 70°Cat the entry and exit respectively. The cooling tower is well insulated and a fan of 2.25 kW drives the air through the cooler. Find the amount of air per second required for 1 kg/s of water flow. The values of cpof air and water are 1.005 and 4.187 kJ/kg K respectively. (Ans. 3.16 kg/s) a 1 Z Solution: Let air required is w kg/s ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ a2 1 a 1 w 1 w 1 Z V 2000 ⎛ ⎜ ⎝ w = w (say) and dQ g 1000 V 2000 V 2000 ⎛ ⎜ ⎝ g 1000 V 2000 dQ dt Z a 1 a 1 w 1 w 1 ∴ + + + + + + w w h h ⎞ ⎟ ⎠ 2 ⎞ ⎟ ⎠ a2 2 a 2 w 2 w 2 Z g 1000 g 1000 dW dt a 2 a 2 w 2 w 2 + + + + + + w w = h h a 1 a 2 w 1 w 2 ∴ w = w = w = 1 kg/s = 0 dt a w w V = 30 m/s t = 70°C 2 V = 3 m/s, t 1 = 80°C 2 1 a 7 m 8 m a w w V = 20 m/s t = 30°C 1 = 1.005 kJ/kg – K V = 1 m/s, t 2 = 1 kg/s = 4.187 kJ/kg – K = 50°C 1 2 a w w w w = 1 m 0.8 m 1 2 a pc w pc dW dt = – 2.25 kW ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ 2 2 a a 2 − V V g a 1 a 2 a 1 a 2 ∴ − + + − Z Z (h h ) ( ) 1 2000 1000 V + ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ 2 2 w 2 w 1 − V g dW dt w 2 w 1 w 1 w 2 − + − + (h h ) ( Z Z ) = 2000 2 30 1000 ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ 2 − 20 9.81 1000 × − + + − Or w 1.005 (30 70) (1 7) 2000 Page 47 of 265**First Law Applied to Flow Process**By: S K Mondal Chapter 5 1 3 9.81 80) (0.8 8) 2000 1000 2 2 − − + + × − − = 4.187 (50 2.25 or – w × 40.509 = –127.9346 w = 127.9346 ∴ = 3.1582 kg/s ≈ 3.16 kg/s 40.509 Air at 101.325 kPa, 20°Cis taken into a gas turbine power plant at a velocity of 140 m/sthrough an opening of 0.15 m2 cross-sectional area. The air is compressed heated, expanded through a turbine, and exhausted at 0.18 MPa, 150°C through an opening of 0.10 m2 cross- sectional area. The power output is 375 kW. Calculate the net amount of heat added to the air in kJ/kg. Assume that air obeys the law pv = 0.287 (t + 273) Where p is the pressure in kPa, v is the specific volume in m3/kg, and t is the temperature in °C. Take cp = 1.005 kJ/kg K. Q5.9 (Ans. 150.23 kJ/kg) • Volume flow rate at inlet = V1A1 m3/s = 21 m3/s Solution: (V) 1 • p V × = 101.325 0.287 RT w p 21 1 w ) = Inlet mass flow rate ( = 25.304 kg/s 1 1 × R T 293 1 • Volume flow rate at outlet = (V ) = 2 2 2 2 × × = 25.304 0.287 180 423 = 17 m3/s dW dt = 375 kW 1 CC 2 2 M p = 101.325 kPa t = 20°C V = 140 m/s A = 0.15 m 1 p = 0.18 Pa = 180 kPa t = 150°C A = 0.1 m V = 171 m/s 2 1 1 1 2 2 2 1 2 2 • V A = 17 Velocity at outlet = 0.1 = 170.66 m/s 2 2 ∴ Using S.F.E.E. 2 1 1 h 2000 1 w = ∴ dQ dt ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 2 V V dQ dt w dW dt + + + + + + w 0 = w h 0 1 2 2 2000 = w = 25.304 kg/s 2 ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ 2 2 2000 V 2000 2 − V V dW dt − + + = w (h h ) 1 2 1 w⎧ ⎫ ⎬ ⎭ 2 2 2 − V dW dt − + + = C (t p ⎨ ⎩ t ) 1 2 1 Page 48 of 265**First Law Applied to Flow Process**By: S K Mondal Chapter 5 ⎫ − − + + ⎬ ⎭ = 3802.76 kW ⎧ 2 2 171 140 = 25.304 1.005(150 ⎨ ⎩ 20) 375 kW 2000 dQ d dQ d 3802.76 25.304 tw kJ m= = = 150.28 kg Q5.10 A gas flows steadily through a rotary compressor. The gas enters the compressor at a temperature of 16°C, a pressure of 100 kPa, and an enthalpy of 391.2 kJ/kg.The gas leaves the compressor at a temperature of 245°C, a pressure of 0.6 MPa, and an enthalpy of 534.5 kJ/kg. There is no heat transfer to or from the gas as it flows through the compressor. (a) Evaluate the external work done per unit mass of gas assuming the gas velocities at entry and exit to be negligible. (b) Evaluate the external work done per unit mass of gas when the gas velocity at entry is 80 m/s and that at exit is 160 m/s. (Ans. 143.3 kJ/kg,152.9 kJ/kg) 2 2 2 V g 1000 Z V g 1000 Z dQ dm dW dm + + + + + + (a) h h = Solution: 1 1 2 1 2 2000 2000 = Z Z so For V1 and V2 negligible and dW dm = h1 – h2 = (391.2 – 5345) kJ/kg = –143.3 kJ/kg i.e. work have to give 1 2 1 2 t = 16°C p = 100 kPa h = 391.2 kJ/kg 1 t = 245°C p = 0.6 mPa = 600 kPa h = 534.5 kJ/kg 2 1 1 2 2 1 2 dQ dt ∴ = 0 (b) V1 = 80 m/s; V2 = 160 m/s dW dm = 2 2 2 − V V − + So (h h ) 1 2000 2 1 2 2 − 80 160 = –143.3 + kJ/kg = (–143.3 – 9.6) kJ/kg 2000 = –152.9 kJ/kg i.e. work have to give The steam supply to an engine comprises two streams which mix before entering the engine. One stream is supplied at the rate of 0.01 kg/s with an enthalpy of 2952 kJ/kgand a velocity of 20 m/s. The other stream is supplied at the rate of 0.1 kg/s with an enthalpy of 2569 kJ/kg and a velocity of 120 m/s. At the exit from the engine the fluid leaves as two Page 49 of 265 Q5.11**First Law Applied to Flow Process**By: S K Mondal streams, one of water at the rate of 0.001 kg/s with an enthalpy of 420 kJ/kg and the other of steam; the fluid velocities at the exit are negligible. The engine develops a shaft power of 25 kW. The heat transfer is negligible. Evaluate the enthalpy of the second exit stream. Chapter 5 (Ans. 2402 kJ/kg) dQ dt ∴ = 0 Solution: By mass balance dW dt = 25 kW w h V = 0.01 kg/s = 2952 kJ/kg = 20 m/s 11 11 11 h w V = ? = ? = 0 22 22 22 = 0.1 kg/s = 2569 kJ/kg = 120 m/s 12 w h V w h V = 0.001 kg/s = 420 kJ/kg = 0 21 12 12 21 21 W11 + W12 = W21 + W22 ∴ W22 = 0.01 + 0.1 – 0.001 kg/s = 0.109 kg/s 2 11 11 11 W h 2000 ⎝ ⎠ ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 V V dQ dt dW dt 2 ⎞ ⎟ ⎠ ∴ + + + + W h 12 12 12 2000 × + = W (h ) + W h 21 21 22 22 ⎛ ⎞ ⎟ ⎠ ⎛ 2 20 2000 = 0.001 × 420 + 0.109 × h22 + 25 29.522 + 257.62 = 0.42 + 0.109 × h22 + 25 286.722 = 0.109 × h22 + 25 h22 = 2401.2 kJ/kg 120 2000 ∴ + + + + 0.01 2952 ⎜ ⎝ 0.1 2569 ⎜ ⎝ 0 Q5.12 or or or The stream of air and gasoline vapour, in the ratio of 14: 1 by mass, enters a gasoline engine at a temperature of 30°Cand leaves as combustion products at a temperature of 790°C.The engine has a specific fuel consumption of 0.3 kg/kWh. The net heat transfer rate from the fuel-air stream to the jacket cooling water and to the surroundings is 35 kW. The shaft power delivered by the engine is 26 kW. Compute the increase in the specific enthalpy of the fuel air stream, assuming the changes in kinetic energy and in elevation to be negligible. (Ans. – 1877 kJ/kg mixture) In 1 hr. this m/c will produce 26 kWh for that we need fuel = 0.3 × 26 = 7.8 kg fuel/hr. ∴ Mass flow rate of fuel vapor and air mixture Solution: Page 50 of 265