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SONNTAG • BORGNAKKE • VAN WYLEN

FUNDAMENTALS

of

Thermodynamics

Sixth Edition

Sonntag, Borgnakke and van Wylen

CONTENT

SUBSECTION Correspondence table Concept problems Force displacement work Boundary work: simple one-step process Polytropic process Boundary work: multistep process Other types of work and general concepts Rates of work Heat transfer rates Review problems English unit concept problems English unit problems PROB NO. 1-19 20-30 31-46 47-58 59-70 71-81 82-94 95-105 106-116 117-122 123-143

Sonntag, Borgnakke and van Wylen

CHAPTER 4 6 ed. CORRESPONDANCE TABLE

The new problem set relative to the problems in the fifth edition. New 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 5th 1 2mod new New New 3 4 new New new New New 18 27 new new 5 new New 13 new new New New New 22 45 mod 8 12 14 New New New New 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 5th new 19 20 33 mod 37 36 15 30 6 New 32 7 9 34 10 New New 26 39 New 40 New New New New 58 59 60 61 New New New New New 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 5th new new new 43 new New new new New 47 HT 48 HT 49 HT 50 HT mod 51 HT mod 52 HT 53 HT 54 HT 55 HT 56 HT 57 HT 31 mod 11 16 17 23 21 mod 28 29 24 44 35

th

Sonntag, Borgnakke and van Wylen

The English unit problem set is New 117 118 119 120 121 122 123 124 125 5th new new new new new new new 68 64 New 126 127 128 129 130 131 132 133 134 5th New new 62 67 70 new 66 65 75 New 135 136 137 138 139 140 141 142 143 5th 69 73 72 76 63 new 77 78 79

The computer, design and open-ended problem set is: New 144 145 146 147 5th 80 81 82 83 New 148 149 150 151 5th 84 85 86 87 New 152 153 5th 88 89

Sonntag, Borgnakke and van Wylen

Concept-Study Guide Problems

4.1 The electric company charges the customers per kW-hour. What is that in SI units? Solution:

The unit kW-hour is a rate multiplied with time. For the standard SI units the rate of energy is in W and the time is in seconds. The integration in Eq.4.21 becomes

min s 1 kW- hour = 1000 W × 60 hour hour × 60 min = 3 600 000 Ws = 3 600 000 J = 3.6 MJ

Sonntag, Borgnakke and van Wylen

4.2 A car engine is rated at 160 hp. What is the power in SI units? Solution: The horsepower is an older unit for power usually used for car engines. The conversion to standard SI units is given in Table A.1 1 hp = 0.7355 kW = 735.5 W 1 hp = 0.7457 kW for the UK horsepower 160 hp = 160 × 745.7 W = 119 312 W = 119.3 kW

Sonntag, Borgnakke and van Wylen

4.3 A 1200 hp dragster engine has a drive shaft rotating at 2000 RPM. How much torque is on the shaft? Power is force times rate of displacement as in Eq.4.2 . . Power, rate of work W = FV=PV=Tω We need to convert the RPM to a value for angular velocity ω 2π 2π rad ω = RPM × 60 s = 2000 × 60 s = 209.44 s We need power in watts: 1 hp = 0.7355 kW = 735.5 W . 1200 hp × 735.5 W/hp T=W/ω= = 4214 Ws = 4214 Nm 209.44 rad/s

Sonntag, Borgnakke and van Wylen

4.4 A 1200 hp dragster engine drives the car with a speed of 100 km/h. How much force is between the tires and the road? Power is force times rate of displacement as in Eq.4.2 . . Power, rate of work W = FV=PV=Tω We need the velocity in m/s: V = 100 × 1000 / 3600 = 27.78 m/s We need power in watts: 1 hp = 0.7355 kW = 735.5 W . 1200 × 735.5 W Nm/s F =W/V= 27.78 m/s = 31 771 m/s = 31 771 N = 31.8 kN

4.5 Two hydraulic piston/cylinders are connected through a hydraulic line so they have roughly the same pressure. If they have diameters of D1 and D2 = 2D1 respectively, what can you say about the piston forces F1 and F2? For each cylinder we have the total force as: F1 = PAcyl 1 = P π D1/4 F2 = PAcyl 2 = P π D2/4 = P π 4 D1/4 = 4 F1 F1 The forces are the total force acting up due to the cylinder pressure. There must be other forces on each piston to have a force balance so the pistons do not move.

2 2 2

F = PAcyl = P π D2/4

F2 2 cb 1

Sonntag, Borgnakke and van Wylen

4.6 Normally pistons have a flat head, but in diesel engines pistons can have bowls in them and protruding ridges. Does this geometry influence the work term? The shape of the surface does not influence the displacement dV = An dx where An is the area projected to the plane normal to the direction of motion. An = Acyl = π D2/4 Work is dW = F dx = P dV = P An dx = P Acyl dx and thus unaffected by the surface shape. x Semi-spherical head is made to make room for larger valves. Ridge Bowl Piston

normal plane

Sonntag, Borgnakke and van Wylen

4.7 What is roughly the relative magnitude of the work in the process 1-2c versus the process 1-2a shown in figure 4.8? By visual inspection the area below the curve 1-2c is roughly 50% of the rectangular area below the curve 1-2a. To see this better draw a straight line from state 1 to point f on the axis. This curve has exactly 50% of the area below it. 4.8 A hydraulic cylinder of area 0.01 m2 must push a 1000 kg arm and shovel 0.5 m straight up. What pressure is needed and how much work is done? F = mg = 1000 kg × 9.81 m/s2 = 9810 N = PA P = F/A = 9810 N/ 0.01 m2 = 981 000 Pa = 981 kPa

W = ⌠F dx = F ∆x = 9810 N × 0.5 m = 4905 J ⌡

Sonntag, Borgnakke and van Wylen

4.9 A work of 2.5 kJ must be delivered on a rod from a pneumatic piston/cylinder where the air pressure is limited to 500 kPa. What diameter cylinder should I have to restrict the rod motion to maximum 0.5 m? π W = ⌠F dx = ⌠P dV = ⌠PA dx = PA ∆x = P 4 D2 ∆x ⌡ ⌡ ⌡ D= 4.10 Helium gas expands from 125 kPa, 350 K and 0.25 m3 to 100 kPa in a polytropic process with n = 1.667. Is the work positive, negative or zero? The boundary work is: P drops but does V go up or down? The process equation is: PVn = C P 1 2 W V W = ⌠P dV ⌡ 4W = πP∆x 4 × 2.5 kJ = 0.113 m π × 500 kPa × 0.5 m

so we can solve for P to show it in a P-V diagram P = CV-n as n = 1.667 the curve drops as V goes up we see V2 > V1 giving dV > 0 and the work is then positive.

Sonntag, Borgnakke and van Wylen

4.11 An ideal gas goes through an expansion process where the volume doubles. Which process will lead to the larger work output: an isothermal process or a polytropic process with n = 1.25? The process equation is: PVn = C The polytropic process with n = 1.25 drops the pressure faster than the isothermal process with n = 1 and the area below the curve is then smaller. P

1 n=1 2

W

V

4.12 Show how the polytropic exponent n can be evaluated if you know the end state properties, (P1, V1) and (P2, V2). Polytropic process: PVn = C Both states must be on the process line: P2V2 = C = P1V1 Take the ratio to get: P1 V2 P2 = V1 n n n n

and then take ln of the ratio P1 V2 V2 ln P = ln V = n ln V 1 1 2 now solve for the exponent n P1 V2 n = ln P ln V 2 1

/

Sonntag, Borgnakke and van Wylen

4.13 A drag force on an object moving through a medium (like a car through air or a submarine through water) is Fd = 0.225 A ρV2. Verify the unit becomes Newton. Solution: Fd = 0.225 A ρV2 Units = m2 × ( kg/m3 ) × ( m2/ s2 ) = kg m / s2 = N

4.14 A force of 1.2 kN moves a truck with 60 km/h up a hill. What is the power? Solution: W = F V = 1.2 kN × 60 (km/h)

3 3 × 60 × 10 Nm = 1.2 × 10

.

3600

s

= 20 000 W = 20 kW

Sonntag, Borgnakke and van Wylen

4.15 Electric power is volts times ampere (P = V i). When a car battery at 12 V is charged with 6 amp for 3 hours how much energy is delivered? Solution:

W = ⌠ W dt = W ∆t = V i ∆t ⌡ = 12 V × 6 Amp × 3 × 3600 s = 777 600 J = 777.6 kJ

.

.

Remark: Volt times ampere is also watts, 1 W = 1 V × 1 Amp. 4.16 Torque and energy and work have the same units (N m). Explain the difference. Solution: Work = force × displacement, so units are N × m. Energy in transfer Energy is stored, could be from work input 1 J = 1 N m Torque = force × arm static, no displacement needed

Sonntag, Borgnakke and van Wylen

4.17 Find the rate of conduction heat transfer through a 1.5 cm thick hardwood board, k = 0.16 W/m K, with a temperature difference between the two sides of 20oC. One dimensional heat transfer by conduction, we do not know the area so we can find the flux (heat transfer per unit area W/m2). . . ∆T W 20 K q = Q/A = k = 0.16 m K × 0.015 m = 213 W/m2 ∆x

4.18 A 2 m2 window has a surface temperature of 15oC and the outside wind is blowing air at 2oC across it with a convection heat transfer coefficient of h = 125 W/m2K. What is the total heat transfer loss? Solution:

. Q = h A ∆T = 125 W/m2K × 2 m2 × (15 – 2) K = 3250 W as a rate of heat transfer out. 15 C o 2 C

o

Sonntag, Borgnakke and van Wylen

4.19 A radiant heating lamp has a surface temperature of 1000 K with ε = 0.8. How large a surface area is needed to provide 250 W of radiation heat transfer? Radiation heat transfer. We do not know the ambient so let us find the area for an emitted radiation of 250 W from the surface . Q = εσAT4 . 250 Q A= 4 = 0.8 × 5.67 × 10-8 × 10004 εσT = 0.0055 m2

Sonntag, Borgnakke and van Wylen

Force displacement work

4.20 A piston of mass 2 kg is lowered 0.5 m in the standard gravitational field. Find the required force and work involved in the process. Solution: F = ma = 2 kg × 9.80665 m/s2 = 19.61 N W= 4.21 An escalator raises a 100 kg bucket of sand 10 m in 1 minute. Determine the total amount of work done during the process. Solution: The work is a force with a displacement and force is constant: F = mg W=

∫ F dx

= F ∫ dx = F ∆x = 19.61 N × 0.5 m = 9.805 J

∫ F dx

= F ∫ dx = F ∆x = 100 kg × 9.80665 m/s2 × 10 m = 9807 J

Sonntag, Borgnakke and van Wylen

4.22 A bulldozer pushes 500 kg of dirt 100 m with a force of 1500 N. It then lifts the dirt 3 m up to put it in a dump truck. How much work did it do in each situation? Solution: W = ∫ F dx = F ∆x = 1500 N × 100 m = 150 000 J = 150 kJ W = ∫ F dz = ∫ mg dz = mg ∆Z = 500 kg × 9.807 m/s2 × 3 m = 14 710 J = 14.7 kJ

Sonntag, Borgnakke and van Wylen

4.23 A hydraulic cylinder has a piston of cross sectional area 25 cm2 and a fluid pressure of 2 MPa. If the piston is moved 0.25 m how much work is done? Solution: The work is a force with a displacement and force is constant: F = PA W= Units:

∫ F dx

=

∫ PA dx = PA ∆x

= 2000 kPa × 25 × 10-4 m2 × 0.25 m = 1.25 kJ kPa m2 m = kN m-2 m2 m = kN m = kJ

Sonntag, Borgnakke and van Wylen

4.24 Two hydraulic cylinders maintain a pressure of 1200 kPa. One has a cross sectional area of 0.01 m2 the other 0.03 m2. To deliver a work of 1 kJ to the piston how large a displacement (V) and piston motion H is needed for each cylinder? Neglect Patm. Solution: W = ∫ F dx = ∫ P dV = ∫ PA dx = PA* H = P∆V W 1 kJ ∆V = P = 1200 kPa = 0.000 833 m3 Both cases the height is H = ∆V/A 0.000833 H1 = 0.01 = 0.0833 m 0.000833 H2 = 0.03 = 0.0278 m F1

F2 2 cb 1

Sonntag, Borgnakke and van Wylen

4.25 A linear spring, F = ks(x − x0), with spring constant ks = 500 N/m, is stretched until it is 100 mm longer. Find the required force and work input. Solution: F = ks(x - x0) = 500 × 0.1 = 50 N W = ∫ F dx = ⌠ ks(x - x0)d(x - x0) = ks(x - x0)2/2 ⌡ N = 500 m × (0.12/2) m2 = 2.5 J

Sonntag, Borgnakke and van Wylen

4.26

n A nonlinear spring has the force versus displacement relation of F = kns(x − x0) . If the spring end is moved to x1 from the relaxed state, determine the formula for the required work. Solution: In this case we know F as a function of x and can integrate kns W = ⌠Fdx = ⌠ kns(x - xo)n d(x - xo) = n + 1 (x1 - xo)n+1 ⌡ ⌡

Sonntag, Borgnakke and van Wylen

4.27 The rolling resistance of a car depends on its weight as: F = 0.006 mg. How long will a car of 1400 kg drive for a work input of 25 kJ? Solution: Work is force times distance so assuming a constant force we get W = ⌠ F dx = F x = 0.006 mgx ⌡ Solve for x W x = 0.006 mg = 25 kJ = 303.5 m 0.006 × 1400 kg × 9.807 m/s2

Sonntag, Borgnakke and van Wylen

4.28 A car drives for half an hour at constant speed and uses 30 MJ over a distance of 40 km. What was the traction force to the road and its speed? Solution: We need to relate the work to the force and distance W = ⌡ dx = F x ⌠F W 30 000 000 J F = x = 40 000 m = 750 N L 40 km km 1000 m V = t = 0.5 h = 80 h = 80 3600 s = 22.2 ms−1

Sonntag, Borgnakke and van Wylen

4.29 The air drag force on a car is 0.225 A ρV2. Assume air at 290 K, 100 kPa and a car frontal area of 4 m2 driving at 90 km/h. How much energy is used to overcome the air drag driving for 30 minutes? 1 P 100 kg ρ = v = RT = = 1.2015 3 0.287 ×290 m km 1000 m V = 90 h = 90 × 3600 s = 25 m/s ∆x = V ∆t = 25 × 30 × 60 = 45 000 m F = 0.225 A ρV2 = 0.225 ×4 ×1.2015 ×252

2 2 kg × m = 676 N = 675.8 m 3 m s2

W = F ∆x = 676 N × 45 000 m = 30 420 000 J = 30.42 MJ

Sonntag, Borgnakke and van Wylen

4.30 Two hydraulic piston/cylinders are connected with a line. The master cylinder has an area of 5 cm2 creating a pressure of 1000 kPa. The slave cylinder has an area of 3 cm2. If 25 J is the work input to the master cylinder what is the force and displacement of each piston and the work out put of the slave cylinder piston? Solution: W = ∫ Fx dx = ∫ P dv = ∫ P A dx = P A ∆x W 25 ∆xmaster = PA = = 0.05 m 1000×5×10-4 A∆x = ∆V = 5 ×10-4× 0.05 = 2.5 ×10-5 m = ∆Vslave = A ∆x ∆xslave = ∆V/A = 2.5 × 10-5 / 3 ×10-4 = 0.0083 33 m Fmaster = P A = 1000× 5 ×10-4 ×103 = 500 N Fslave = P A = 1000 ×103× 3 ×10-4 = 300 N Wslave = F ∆x = 300 × 0.08333 = 25 J

Master

Slave

Sonntag, Borgnakke and van Wylen

Boundary work simple 1 step process

4.31 A constant pressure piston cylinder contains 0.2 kg water as saturated vapor at 400 kPa. It is now cooled so the water occupies half the original volume. Find the work in the process. Solution: Table B.1.2 v1= 0.4625 m3/kg V1 = mv1 = 0.0925 m3 V2 = V1 / 2 = 0.04625 m3 v2 = v1/ 2 = 0.23125 m3/kg

Process: P = C so the work term integral is W = ∫ PdV = P(V2-V1) = 400 kPa × (0.04625 – 0.0925) m3 = -18.5 kJ

P C.P. 2 1 T cb T

C.P. P = 400 kPa

400

144 2 v

1 v

Sonntag, Borgnakke and van Wylen

4.32 A steam radiator in a room at 25°C has saturated water vapor at 110 kPa flowing through it, when the inlet and exit valves are closed. What is the pressure and the quality of the water, when it has cooled to 25oC? How much work is done? Solution: Control volume radiator. After the valve is closed no more flow, constant volume and mass. 1: x1 = 1, P1 = 110 kPa ⇒ v1 = vg = 1.566 m3/kg from Table B.1.2 2: T2 = 25oC, ? Process: v2 = v1 = 1.566 m3/kg = [0.001003 + x2 × 43.359] m3/kg x2 = State 2 : T2 , x2 1.566 – 0.001003 = 0.0361 43.359 P2 = Psat = 3.169 kPa

From Table B.1.1

1W2 = ⌠PdV = 0 ⌡

Sonntag, Borgnakke and van Wylen

4.33 A 400-L tank A, see figure P4.33, contains argon gas at 250 kPa, 30oC. Cylinder B, having a frictionless piston of such mass that a pressure of 150 kPa will float it, is initially empty. The valve is opened and argon flows into B and eventually reaches a uniform state of 150 kPa, 30oC throughout. What is the work done by the argon? Solution: Take C.V. as all the argon in both A and B. Boundary movement work done in cylinder B against constant external pressure of 150 kPa. Argon is an ideal gas, so write out that the mass and temperature at state 1 and 2 are the same PA1VA = mARTA1 = mART2 = P2( VA + VB2) => VB2 =

2

250 × 0.4 - 0.4 = 0.2667 m3 150

3 ⌠ 1W2 = ⌡ PextdV = Pext(VB2 - VB1) = 150 kPa (0.2667 - 0) m = 40 kJ 1

Sonntag, Borgnakke and van Wylen

4.34 A piston cylinder contains air at 600 kPa, 290 K and a volume of 0.01 m3. A constant pressure process gives 54 kJ of work out. Find the final volume and temperature of the air. Solution: W = ∫ P dV = P∆V 54 ∆V = W/P = 600 = 0.09 m3 V2 = V1 + ∆V = 0.01 + 0.09 = 0.1 m3 Assuming ideal gas, PV = mRT, then we have P2 V2 P2 V2 V2 0.1 T2 = mR = P V T1= V T1 = 0.01 290 = 2900 K 1 1 1

Sonntag, Borgnakke and van Wylen

4.35 Saturated water vapor at 200 kPa is in a constant pressure piston cylinder. At this state the piston is 0.1 m from the cylinder bottom and cylinder area is 0.25 m2. The temperature is then changed to 200oC. Find the work in the process. Solution: State 1 from B.1.2 (P, x): State 2 from B.1.3 (P, T): v1 = vg = 0.8857 m3/kg v2 = 1.0803 m3/kg (also in B.1.3)

Since the mass and the cross sectional area is the same we get v2 1.0803 h2 = v × h1 = 0.8857 × 0.1 = 0.122 m 1 Process: P = C so the work integral is W = ∫ PdV = P(V2 - V1) = PA (h2 - h1) W = 200 kPa × 0.25 m2 × (0.122 − 0.1) m = 1.1 kJ

P C.P. 1 2 T cb T

C.P. P = 200 kPa

200

200 120 1 v

2

v

Sonntag, Borgnakke and van Wylen

4.36 A cylinder fitted with a frictionless piston contains 5 kg of superheated refrigerant R-134a vapor at 1000 kPa, 140°C. The setup is cooled at constant pressure until the R-134a reaches a quality of 25%. Calculate the work done in the process. Solution: Constant pressure process boundary work. State properties from Table B.5.2 State 1: v = 0.03150 m3/kg , State 2: v = 0.000871 + 0.25 × 0.01956 = 0.00576 m3/kg Interpolated to be at 1000 kPa, numbers at 1017 kPa could have been used in which case:

1W2 =

v = 0.00566 m3/kg

∫ P dV = P (V2-V1) = mP (v2-v1)

T C.P. P = 1000 kPa

= 5 × 1000 (0.00576 - 0.03150) = -128.7 kJ

P C.P. 2 1 T cb 1000

140 39 2 v

1

v

Sonntag, Borgnakke and van Wylen

4.37 Find the specific work in Problem 3.54 for the case the volume is reduced. Saturated vapor R-134a at 50oC changes volume at constant temperature. Find the new pressure, and quality if saturated, if the volume doubles. Repeat the question for the case the volume is reduced to half the original volume. Solution: R-134a Table B.4.1:

1W2

50oC v1 = vg = 0.01512 m3/kg, v2 = v1 / 2 = 0.00756 m3/kg

= ∫ PdV = 1318.1 kPa (0.00756 – 0.01512) m3/kg = -9.96 kJ/kg

P C.P. 2 1 T cb T

C.P. P = 1318 kPa

1318

50 v

2

1 v

Sonntag, Borgnakke and van Wylen

4.38 A piston/cylinder has 5 m of liquid 20oC water on top of the piston (m = 0) with cross-sectional area of 0.1 m2, see Fig. P2.57. Air is let in under the piston that rises and pushes the water out over the top edge. Find the necessary work to push all the water out and plot the process in a P-V diagram. Solution: P1 = Po + ρgH = 101.32 + 997 × 9.807 × 5 / 1000 = 150.2 kPa ∆V = H × A = 5 × 0.1 = 0.5 m3 1W2 = AREA = ∫ P dV = ½ (P1 + Po )(Vmax -V1) = ½ (150.2 + 101.32) kPa × 0.5 m3 = 62.88 kJ Po H2O P1 P0 Air

V1 Vmax

P 1 2 V

cb

Sonntag, Borgnakke and van Wylen

4.39 Air in a spring loaded piston/cylinder has a pressure that is linear with volume, P = A + BV. With an initial state of P = 150 kPa, V = 1 L and a final state of 800 kPa and volume 1.5 L it is similar to the setup in Problem 3.113. Find the work done by the air. Solution: Knowing the process equation: P = A + BV giving a linear variation of pressure versus volume the straight line in the P-V diagram is fixed by the two points as state 1 and state 2. The work as the integral of PdV equals the area under the process curve in the P-V diagram. State 1: P1 = 150 kPa Process: P = A + BV ⇒

1

V1 = 1 L = 0.001 m3

P 2

State 2: P2 = 800 kPa V2 = 1.5 L = 0.0015 m3 linear in V 2 P1 + P2 W2 = ⌠ PdV = (V2 - V1) 1 ⌡ 2

(

)

1

W

V

1 = 2 (150 + 800) kPa (1.5 - 1) × 0.001 m3 = 0.2375 kJ

Sonntag, Borgnakke and van Wylen

4.40 Find the specific work in Problem 3.43. Saturated water vapor at 200 kPa is in a constant pressure piston cylinder. At this state the piston is 0.1 m from the cylinder bottom. How much is this distance if the temperature is changed to a) 200 oC and b) 100 oC. Solution: Process: P=C ⇒ w = ∫ Pdv = P1(v – v1) v1 = vg (200 kPa) = 0.8857 m3/kg va = 1.083 m3/kg

State 1: (200 kPa, x = 1) in B.1.2: CASE a) State a: (200 kPa, 200oC) B.1.3:

1wa

= ∫ Pdv = 200(1.0803 – 0.8857) = 38.92 kJ/kg vb ≈ vf = 0.001044 m3/kg

CASE b) State b: (200 kPa, 100oC) B.1.1:

1Wb

= ∫ PdV = 200(0.001044 – 0.8857) = -176.9 kJ/kg

T C.P. P = 200 kPa

P C.P. b bW1 cb

200

1 a T a 1W

200 120 100 b v 1

a

v

Sonntag, Borgnakke and van Wylen

4.41 A piston/cylinder contains 1 kg water at 20oC with volume 0.1 m3. By mistake someone locks the piston preventing it from moving while we heat the water to saturated vapor. Find the final temperature, volume and the process work. Solution 1: v1 = V/m = 0.1 m3/1 kg = 0.1 m3/kg 2: Constant volume: v2 = vg = v1 V2 = V1 = 0.1 m3

1W2 = ∫ P dV = 0

0.1 - 0.10324 T2 = Tsat = 210 + 5 0.09361 - 0.10324 = 211.7°C

Sonntag, Borgnakke and van Wylen

4.42 A piston cylinder contains 1 kg of liquid water at 20oC and 300 kPa. There is a linear spring mounted on the piston such that when the water is heated the pressure reaches 3 MPa with a volume of 0.1 m3. a) Find the final temperature b) Plot the process in a P-v diagram. c) Find the work in the process. Solution: Take CV as the water. This is a constant mass: m2 = m1 = m ; State 1: Compressed liquid, take saturated liquid at same temperature. B.1.1: v1 = vf(20) = 0.001002 m3/kg, State 2: v2 = V2/m = 0.1/1 = 0.1 m3/kg and P = 3000 kPa => Superheated vapor close to T = 400oC Interpolate: T2 = 404oC Work is done while piston moves at linearly varying pressure, so we get:

1 1W2 = ∫ P dV = area = Pavg (V2 − V1) = 2 (P1 + P2)(V2 - V1)

from B.1.3

= 0.5 (300 + 3000)(0.1 − 0.001) = 163.35 kJ

P C.P. T 2 C.P. 2 300 kPa T 20 v 1 v

300

1

Sonntag, Borgnakke and van Wylen

4.43 A piston cylinder contains 3 kg of air at 20oC and 300 kPa. It is now heated up in a constant pressure process to 600 K. a) Find the final volume b) Plot the process path in a P-v diagram c) Find the work in the process. Solution: Ideal gas PV = mRT State 1: T1, P1 ideal gas so P1V1 = mRT1 V1 = mR T1 / P1 = 3 × 0.287 × 293.15/300 = 0.8413 m3 State 2: T2, P2 = P1 W2 and ideal gas so P2V2 = mRT2 V2 = mR T2 / P2 = 3 × 0.287 × 600/300 = 1.722 m3

1

= ⌠ PdV = P (V2 - V1) = 300 (1.722 – 0.8413) = 264.2 kJ ⌡

P 2 T1 T2 293 v 600 1 v T 2 300 kPa

300

1

Sonntag, Borgnakke and van Wylen

4.44 A piston cylinder contains 0.5 kg air at 500 kPa, 500 K. The air expands in a process so P is linearly decreasing with volume to a final state of 100 kPa, 300 K. Find the work in the process. Solution: Process: P = A + BV From the process:

1

(linear in V, decreasing means B is negative)

1

W2 = ⌠ PdV = AREA = 2 (P1 + P2)(V2 - V1) ⌡

V1 = mR T1/ P1 = 0.5 × 0.287 × (500/500) = 0.1435 m3 V2 = mR T2/ P2 = 0.5 × 0.287 × (300/100) = 0.4305 m3

1

W2 = 2 × (500 + 100) kPa × (0.4305 - 0.1435) m3 = 86.1 kJ

P 500 T2 100 cb 1

T 1 T1 2 v 300 2 v 500 1

Sonntag, Borgnakke and van Wylen

4.45 Consider the nonequilibrium process described in Problem 3.109. Determine the work done by the carbon dioxide in the cylinder during the process. A cylinder has a thick piston initially held by a pin as shown in Fig. P3.109. The cylinder contains carbon dioxide at 200 kPa and ambient temperature of 290 K. The metal piston has a density of 8000 kg/m3 and the atmospheric pressure is 101 kPa. The pin is now removed, allowing the piston to move and after a while the gas returns to ambient temperature. Is the piston against the stops? Solution: Knowing the process (P vs. V) and the states 1 and 2 we can find W. If piston floats or moves: P = Plift = Po + ρHg = 101.3 + 8000 × 0.1 × 9.807 / 1000 = 108.8 kPa Assume the piston is at the stops (since P1 > Plift piston would move) V2 = V1 × 150 / 100 = (π/4) 0.12 × 0.1× 1.5 = 0.000785× 1.5 = 0.001 1775 m3 For max volume we must have P > Plift so check using ideal gas and constant T process: P2 = P1 V1/ V2 = 200/1.5 = 133 kPa > Plift and piston is at stops.

1W2 =

∫ Plift dV = Plift (V2 -V1) = 108.8 (0.0011775 - 0.000785)

= 0.0427 kJ

Remark: The work is determined by the equilibrium pressure, Plift, and not the instantaneous pressure that will accelerate the piston (give it kinetic energy). We need to consider the quasi-equilibrium process to get W.

Sonntag, Borgnakke and van Wylen

4.46 Consider the problem of inflating the helium balloon, as described in problem 3.79. For a control volume that consists of the helium inside the balloon determine the work done during the filling process when the diameter changes from 1 m to 4 m. Solution : Inflation at constant P = P0 = 100 kPa to D1 = 1 m, then P = P0 + C ( D* -1 - D* -2 ), D* = D / D1,

to D2 = 4 m, P2 = 400 kPa, from which we find the constant C as: 400 = 100 + C[ (1/4) - (1/4)2 ] => C = 1600 kPa π The volumes are: V = 6 D3 => V1 = 0.5236 m3; V2 = 33.51 m3

2

WCV = ⌠ PdV ⌡

1

= P0(V2 - V1) + ⌠ C(D* -1 - D* -2)dV ⌡

1

2

π V = 6 D3,

π π dV = 2 D2 dD = 2 D13 D* 2 dD* D2*=4 ⌠ ⌡ (D*-1)dD*

⇒ WCV = P0(V2 - V1) + 3CV1

D1*=1

4 D2* 2 - D1* 2 * * = P0(V2 - V1) + 3CV1[ - (D2 - D1 )] 2 1 16-1 = 100 × (33.51 – 0.5236) + 3 × 1600 × 0.5236 [ 2 – (4–1)] = 14 608 kJ

Sonntag, Borgnakke and van Wylen

Polytropic process

4.47 Consider a mass going through a polytropic process where pressure is directly proportional to volume (n = − 1). The process start with P = 0, V = 0 and ends with P = 600 kPa, V = 0.01 m3. The physical setup could be as in Problem 2.22. Find the boundary work done by the mass. Solution: The setup has a pressure that varies linear with volume going through the initial and the final state points. The work is the area below the process curve. P 600 W = ⌠ PdV = AREA ⌡ = 2 (P1 + P2)(V2 - V1) W 0 0 0.01 V = 2 (P2 + 0)( V2 - 0) = 2 P2 V2 = 2 × 600 × 0.01 = 3 kJ

1 1 1 1

Sonntag, Borgnakke and van Wylen

4.48 The piston/cylinder shown in Fig. P4.48 contains carbon dioxide at 300 kPa, 100°C with a volume of 0.2 m3. Mass is added at such a rate that the gas compresses according to the relation PV1.2 = constant to a final temperature of 200°C. Determine the work done during the process. Solution: From Eq. 4.4 for the polytopic process PVn = const ( n = 1 ) / 2 P2V2 - P1V1 W2 = ⌠ PdV = 1 ⌡ 1-n 1 Assuming ideal gas, PV = mRT mR(T2 - T1) , 1W2 = 1-n P1V1 300 × 0.2 kPa m3 But mR = T = 373.15 K = 0.1608 kJ/K 1

1W2

=

0.1608(473.2 - 373.2) kJ K 1 - 1.2 K = -80.4 kJ

Sonntag, Borgnakke and van Wylen

4.49 A gas initially at 1 MPa, 500°C is contained in a piston and cylinder arrangement with an initial volume of 0.1 m3. The gas is then slowly expanded according to the relation PV = constant until a final pressure of 100 kPa is reached. Determine the work for this process. Solution: By knowing the process and the states 1 and 2 we can find the relation between the pressure and the volume so the work integral can be performed. Process: PV = C ⇒ V2 = P1V1/P2 = 1000 × 0.1/100 = 1 m3 For this process work is integrated to Eq.4.5 ⌠ = ∫ P dV = ⌡ CV-1dV = C ln(V2/V1) P 1 2 W V

1W2

V2 1W2 = P1V1 ln V = 1000 × 0.1 ln (1/0.1) 1 = 230.3 kJ

Sonntag, Borgnakke and van Wylen

4.50 Helium gas expands from 125 kPa, 350 K and 0.25 m3 to 100 kPa in a polytropic process with n = 1.667. How much work does it give out? Solution: Process equation: PV = constant = P1V1 = P2V2

1/n n n n

Solve for the volume at state 2 V2 = V1 (P1/P2) 1250.6 = 0.25 × 100 = 0.2852 m3

Work from Eq.4.4 P2V2- P1 V1 100× 0.2852 - 125× 0.25 W2 = = kPa m3 = 4.09 kJ 1 1 - 1.667 1-n

Sonntag, Borgnakke and van Wylen

4.51 Air goes through a polytropic process from 125 kPa, 325 K to 300 kPa and 500 K. Find the polytropic exponent n and the specific work in the process. Solution: Process: Ideal gas Pvn = Const = P1v1 = P2 v2 Pv = RT so RT 0.287 × 325 = 0.7462 m3/kg v1 = P = 125 n n

RT 0.287 × 500 v2 = P = = 0.47833 m3/kg 300 From the process equation => ln(P2/ P1) = n ln(v1/ v2) ln 2.4 n = ln(P2/ P1) / ln(v1/ v2) = ln 1.56 = 1.969 The work is now from Eq.4.4 per unit mass P2v2-P1v1 R(T2 - T1) 0.287(500 - 325) = = = -51.8 kJ/kg 1w2 = 1-n 1-n 1-1.969 (P2/ P1) = (v1/ v2) n Sonntag, Borgnakke and van Wylen

4.52 A piston cylinder contains 0.1 kg air at 100 kPa, 400 K which goes through a polytropic compression process with n = 1.3 to a pressure of 300 kPa. How much work has the air done in the process? Solution: Process: Pvn = Const. T2 = T1 ( P2 V2 / P1V1) = T1 ( P2 / P1)(P1 / P2 )

(1 - 1/1.3) 1/n

= 400 × (300/100) = 515.4 K Work term is already integrated giving Eq.4.4 1 mR = (P V - P1V1) = (T -T ) 1W2 1−n 2 2 1−n 2 1 = 0.2 × 0.287 × (515.4-400) = -477 kJ 1 − 1.3 P 2 1 W n=1 V

Since Ideal gas,

Sonntag, Borgnakke and van Wylen

4.53 A balloon behaves so the pressure is P = C2 V1/3, C2 = 100 kPa/m. The balloon is blown up with air from a starting volume of 1 m3 to a volume of 3 m3. Find the final mass of air assuming it is at 25oC and the work done by the air. Solution: The process is polytropic with exponent n = -1/3. P1 = C2 V1/3 = 100 × 11/3 = 100 kPa P2 = C2 V1/3 = 100 × 31/3 = 144.22 kPa P 1 W V

2

1W2 = ∫ P dV =

P2V2 - P1V1 1-n

(Equation 4.4)

144.22 × 3 - 100 × 1 = 249.5 kJ 1 - (-1/3) P2V2 144.22 × 3 = 5.056 kg m2 = RT = 0.287 × 298 2 =

Sonntag, Borgnakke and van Wylen

4.54 A balloon behaves such that the pressure inside is proportional to the diameter squared. It contains 2 kg of ammonia at 0°C, 60% quality. The balloon and ammonia are now heated so that a final pressure of 600 kPa is reached. Considering the ammonia as a control mass, find the amount of work done in the process. Solution: Process : P ∝ D2, with V ∝ D3 this implies P ∝ D2 ∝ V2/3 so PV -2/3 = constant, which is a polytropic process, n = −2/3 From table B.2.1: V1 = mv1 = 2(0.001566 + 0.6 × 0.28783) = 0.3485 m3 P23/2 600 3/2 V2 = V1 P = 0.3485 429.3 = 0.5758 m3 1 P2V2 - P1V1 (Equation 4.4) 1W2 = ∫ P dV = 1-n = 600 × 0.5758 - 429.3 × 0.3485 = 117.5 kJ 1 - (-2/3) P 1 W V

2

Sonntag, Borgnakke and van Wylen

4.55 Consider a piston cylinder with 0.5 kg of R-134a as saturated vapor at -10°C. It is now compressed to a pressure of 500 kPa in a polytropic process with n = 1.5. Find the final volume and temperature, and determine the work done during the process. Solution: Take CV as the R-134a which is a control mass. Process: Pv

1.5

m2 = m1 = m

= constant

until P = 500 kPa

1: (T, x) v1 = 0.09921 m3/kg,

P = Psat = 201.7 kPa from Table B.5.1 (1/1.5) 2/3 2: (P, process) v2 = v1 (P1/P2) = 0.09921× (201.7/500) = 0.05416 Given (P, v) at state 2 from B.5.2 it is superheated vapor at T2 = 79°C Process gives P = C v

1W2 = -1.5

, which is integrated for the work term, Eq.(4.4)

∫ P dV = 1 -m (P2v2 - P1v1) 1.5

2 = - 0.5 × (500 × 0.05416 - 201.7 × 0.09921) = -7.07 kJ

Sonntag, Borgnakke and van Wylen

4.56 Consider the process described in Problem 3.98. With 1 kg water as a control mass, determine the boundary work during the process. A spring-loaded piston/cylinder contains water at 500°C, 3 MPa. The setup is such that pressure is proportional to volume, P = CV. It is now cooled until the water becomes saturated vapor. Sketch the P-v diagram and find the final pressure. Solution : State 1: Table B.1.3: v1 = 0.11619 m3/kg P = C0V = C0m v = C v Process: m is constant and

P = Cv ⇒ C = P1/v1 = 3000/0.11619 = 25820 kPa kg/m3 State 2: x2 = 1 & P2 = Cv2 (on process line) P 1 2 Trial & error on T2sat or P2sat: Here from B.1.2: at 2 MPa vg = 0.09963 ⇒ C = P/vg = 20074 (low) 2.5 MPa vg = 0.07998 ⇒ C = P/vg = 31258 (high) C v 2.25 MPa vg = 0.08875 ⇒ C = P/vg = 25352 (low)

Now interpolate to match the right slope C: P2 = 2270 kPa, v2 = P2/C = 2270/25820 = 0.0879 m3/kg P is linear in V so the work becomes (area in P-v diagram) 1 1w2 = ∫ P dv = 2(P1 + P2)(v2 - v1) 1 = 2 (3000 + 2270)(0.0879 - 0.11619) = - 74.5 kJ/kg

Sonntag, Borgnakke and van Wylen

4.57 Find the work for Problem 3.106. Refrigerant-12 in a piston/cylinder arrangement is initially at 50°C, x = 1. It is then expanded in a process so that P = Cv−1 to a pressure of 100 kPa. Find the final temperature and specific volume. Solution: Knowing the process (P versus V) and states 1 and 2 allows calculation of W. State 1: 50°C, x=1 Table B.3.1: P1 = 1219.3 kPa, v1 = 0.01417 m3/kg Process: v2 P = Cv-1 ⇒ 1w2 = ∫ P dv = C ln v 1 same as Eq.4.5

State 2: 100 kPa and on process curve:

v2 = v1P1/P2 = 0.1728 m3/kg

From table B.3.2 T = - 13.2°C The constant C for the work term is P1v1 so per unit mass we get v2 0.1728 w2 = P1v1 ln v = 1219.3 × 0.01417 × ln 0.01417 = 43.2 kJ/kg 1

1

P 1 2 v

T 1 2 v

Notice T is not constant. It is not an ideal gas in this range.

Sonntag, Borgnakke and van Wylen

4.58 A piston/cylinder contains water at 500°C, 3 MPa. It is cooled in a polytropic process to 200°C, 1 MPa. Find the polytropic exponent and the specific work in the process. Solution: Polytropic process: Pvn = C Both states must be on the process line: P2v2 = C = P1v1 Take the ratio to get: P1 v2 P2 = v1 n n n

and then take ln of the ratio:

P1 v2 v2 ln P = ln v = n ln v 2 1 1

n

now solve for the exponent n P1 v2 1.0986 n = ln P ln v = 0.57246 = 1.919 2 1

/

1w2 = ∫ P dv =

P2v2 - P1v1 1-n

(Equation 4.4)

=

1000 × 0.20596 - 3000 × 0.11619 = 155.2 kJ 1 - 1.919

Sonntag, Borgnakke and van Wylen

4.59 Consider a two-part process with an expansion from 0.1 to 0.2 m3 at a constant pressure of 150 kPa followed by an expansion from 0.2 to 0.4 m3 with a linearly rising pressure from 150 kPa ending at 300 kPa. Show the process in a P-V diagram and find the boundary work. Solution: By knowing the pressure versus volume variation the work is found. If we plot the pressure versus the volume we see the work as the area below the process curve. P 300 150 1 2 V 0.1 0.2 0.4 3

2 3 W3 = 1W2 + 2W3 = ⌠ PdV + ⌠ PdV 1 ⌡ ⌡ 1 2 1 = P1 (V2 – V1) + 2 (P2 + P3)(V3-V2) 1 = 150 (0.2-1.0) + 2 (150 + 300) (0.4 - 0.2) = 15 + 45 = 60 kJ

Sonntag, Borgnakke and van Wylen

4.60 A cylinder containing 1 kg of ammonia has an externally loaded piston. Initially the ammonia is at 2 MPa, 180°C and is now cooled to saturated vapor at 40°C, and then further cooled to 20°C, at which point the quality is 50%. Find the total work for the process, assuming a piecewise linear variation of P versus V. Solution:

P 2000 1555 857 3 cb 1 2 180 o C 40 oC 20 C v o State 1: (T, P)

Table B.2.2

v1 = 0.10571 m3/kg State 2: (T, x) Table B.2.1 sat. vap. P2 = 1555 kPa, v2 = 0.08313 m3/kg

State 3: (T, x) P3 = 857 kPa, v3 = (0.001638 + 0.14922)/2 = 0.07543 m3/kg Sum the the work as two integrals each evaluated by the area in the P-v diagram. ⌠ 1W3 = ⌡ PdV ≈ ( 1 =

3

P2 + P3 P1 + P2 ) m(v2 - v1) + ( ) m(v3 - v2) 2 2

1555 + 857 2000 + 1555 1(0.08313 - 0.10571) + 1(0.07543 - 0.08313) 2 2

= -49.4 kJ

Sonntag, Borgnakke and van Wylen

4.61 A piston/cylinder arrangement shown in Fig. P4.61 initially contains air at 150 kPa, 400°C. The setup is allowed to cool to the ambient temperature of 20°C. a. Is the piston resting on the stops in the final state? What is the final pressure in the cylinder? b. What is the specific work done by the air during this process? Solution: State 1: State 2:

P1 = 150 kPa, T1 = 400°C = 673.2 K T2 = T0 = 20°C = 293.2 K

For all states air behave as an ideal gas. a) If piston at stops at 2, V2 = V1/2 and pressure less than Plift = P1 V1 T2 293.2 ⇒ P2 = P1 × V × T = 150 × 2 × 673.2 = 130.7 kPa < P1

2 1

⇒ Piston is resting on stops at state 2. b) Work done while piston is moving at constant Pext = P1.

1W2 =

∫ Pext dV = P1 (V2 - V1)

1 1

;

1 1 V2 = 2 V1 = 2 m RT1/P1

1w2 = 1W2/m = RT1 (2 - 1 ) = -2 × 0.287 × 673.2 = -96.6 kJ/kg

P 1a P1 P2 2 1 T1 T1a V T2

T 1 1a 2

V

Sonntag, Borgnakke and van Wylen

4.62 A piston cylinder has 1.5 kg of air at 300 K and 150 kPa. It is now heated up in a two step process. First constant volume to 1000 K (state 2) then followed by a constant pressure process to 1500 K, state 3. Find the final volume and the work in the process. Solution: P The two processes are: 1 -> 2: Constant volume V2 = V1 2 -> 3: Constant pressure P3 = P2 P2 P1 1 2 3

V

Use ideal gas approximation for air. State 1: T, P => State 2: V2 = V1 V1 = mRT1/P1 = 1.5×0.287×300/150 = 0.861 m3 => P2 = P1 (T2/T1) = 150×1000/300 = 500 kPa => V3 = V2 (T3/T2) = 0.861×1500/1000 = 1.2915 m3

State 3: P3 = P2 We find the work by summing along the process path. 1W3 = 1W2 + 2W3 = 2W3 = P3(V3 - V2) = 500(1.2915 - 0.861) = 215.3 kJ

Sonntag, Borgnakke and van Wylen

4.63 A piston/cylinder assembly (Fig. P4.63) has 1 kg of R-134a at state 1 with 110°C, 600 kPa, and is then brought to saturated vapor, state 2, by cooling while the piston is locked with a pin. Now the piston is balanced with an additional constant force and the pin is removed. The cooling continues to a state 3 where the R-134a is saturated liquid. Show the processes in a P-V diagram and find the work in each of the two steps, 1 to 2 and 2 to 3. Solution : CV R-134a This is a control mass. Properties from table B.5.1 and 5.2 State 1: (T,P) B.5.2 => v = 0.04943 m3/kg State 2: given by fixed volume v2 = v1 and x2 = 1.0 v2 = v1 = vg = 0.04943 m3/kg State 3 reached at constant P (F = constant) P so from B.5.1

=> T = 10°C v3 = vf = 0.000794 m3/kg

1 3 cb 2 V

Since no volume change from 1 to 2 => 1W2 = 0 Constant pressure 2W3 = ∫P dV = P(V3 -V2) = mP(v3 -v2) = 415.8 (0.000794 - 0.04943) 1 = -20.22 kJ

Sonntag, Borgnakke and van Wylen

4.64 The refrigerant R-22 is contained in a piston/cylinder as shown in Fig. P4.64, where the volume is 11 L when the piston hits the stops. The initial state is −30°C, 150 kPa with a volume of 10 L. This system is brought indoors and warms up to 15°C. a. Is the piston at the stops in the final state? b. Find the work done by the R-22 during this process. Solution: Initially piston floats, V < Vstop so the piston moves at constant Pext = P1 until it reaches the stops or 15°C, whichever is first. a) From Table B.4.2: v1 = 0.1487 m3/kg, 0.010 m = V/v = 0.1487 = 0.06725 kg Po P 2 mp R-22 V stop Check the temperature at state 1a: P1a = 150 kPa, v = Vstop/m. 0.011 v1a = V/m = 0.06725 = 0.16357 m3/kg b) Work done at constant Pext = P1.

1W2 =

P1

1

1a

V

=>

T1a = -9°C & T2 = 15°C

Since T2 > T1a then it follows that P2 > P1 and the piston is against stop.

∫ Pext dV = Pext(V2 - V1)

= 150(0.011 - 0.010) = 0.15 kJ

Sonntag, Borgnakke and van Wylen

4.65 A piston/cylinder contains 50 kg of water at 200 kPa with a volume of 0.1 m3. Stops in the cylinder restricts the enclosed volume to 0.5 m3, similar to the setup in Problem 4.7. The water is now heated to 200°C. Find the final pressure, volume and the work done by the water. Solution: Initially the piston floats so the equilibrium lift pressure is 200 kPa 1: 200 kPa, v1= 0.1/50 = 0.002 m3/kg, 2: 200°C, on line Check state 1a: vstop = 0.5/50 = 0.01 m3/kg P1 P 2 1 1a V stop V

=> Table B.1.2: 200 kPa , vf < vstop < vg State 1a is two phase at 200 kPa and Tstop ≈ 120.2 °C so as T2 > Tstop the state is higher up in the P-V diagram with v2 = vstop < vg = 0.127 m3/kg (at 200°C) State 2 two phase => P2 = Psat(T2) = 1.554 MPa, V2 = Vstop = 0.5 m3

1W2 = 1Wstop = 200 (0.5 – 0.1) = 80 kJ

Sonntag, Borgnakke and van Wylen

4.66 Find the work for Problem 3.108. Ammonia in a piston/cylinder arrangement is at 700 kPa, 80°C. It is now cooled at constant pressure to saturated vapor (state 2) at which point the piston is locked with a pin. The cooling continues to −10°C (state 3). Show the processes 1 to 2 and 2 to 3 on both a P–v and T–v diagram. Solution :

P 700 290 3 cb T 2 1

80 14 -10 3 v

1 2

v

2 W3 = 1W2 + 2W3 = ⌠ PdV = P1(V2 - V1) = mP1(v2 - v1) 1 ⌡ 1 Since constant volume from 2 to 3, see P-v diagram. From table B.2 v1 = 0.2367 m3/kg, P1 = 700 kPa, v2 = vg = 0.1815 m3/kg

1w3 = P1(v2- v1) = 700 × (0.1815 - 0.2367) = -38.64 kJ/kg

Sonntag, Borgnakke and van Wylen

4.67 A piston/cylinder contains 1 kg of liquid water at 20°C and 300 kPa. Initially the piston floats, similar to the setup in Problem 4.64, with a maximum enclosed volume of 0.002 m3 if the piston touches the stops. Now heat is added so a final pressure of 600 kPa is reached. Find the final volume and the work in the process. Solution: Take CV as the water which is a control mass: Table B.1.1: 20°C => Psat = 2.34 kPa State 1: Compressed liquid v = vf(20) = 0.001002 m3/kg State 1a: vstop = 0.002 m3/kg , 300 kPa State 2: Since P2 = 600 kPa > Plift then piston is pressed against the stops v2 = vstop = 0.002 m3/kg and V = 0.002 m3 For the given P : vf < v < vg so 2-phase T = Tsat = 158.85 °C Work is done while piston moves at Plift = constant = 300 kPa so we get

1W2 =

m2 = m1 = m ;

∫ P dV = m Plift(v2 -v1) = 1 × 300(0.002 − 0.001002) = 0.30 kJ

P P o cb H2O

1

2 V

Sonntag, Borgnakke and van Wylen

4.68 10 kg of water in a piston cylinder arrangement exists as saturated liquid/vapor at 100 kPa, with a quality of 50%. It is now heated so the volume triples. The mass of the piston is such that a cylinder pressure of 200 kPa will float it. a) Find the final temperature and volume of the water. b) Find the work given out by the water. Solution: Take CV as the water Process: State 1: State 2:

m2 = m1 = m;

v = constant until P = Plift then P is constant. v1 = vf + x vfg = 0.001043 + 0.5 × 1.69296 = 0.8475 m3/kg v2, P2 ≤ Plift => v2 = 3 × 0.8475 = 2.5425 m3/kg; T2 = 829°C ; V2 = m v2 = 25.425 m3

1W2 =

∫ P dV = Plift × (V2 - V1)

= 200 kPa × 10 kg × (2.5425 – 0.8475) m3/kg = 3390 kJ

Po P2 H2O P1

P 2

cb

1

V

Sonntag, Borgnakke and van Wylen

4.69 Find the work in Problem 3.43. Ammonia at 10oC with a mass of 10 kg is in a piston cylinder arrangement with an initial volume of 1 m3. The piston initially resting on the stops has a mass such that a pressure of 900 kPa will float it. The ammonia is now slowly heated to 50oC. Find the work in the process. C.V. Ammonia, constant mass. Process: V = constant unless P = Pfloat P V 1 State 1: T = 10oC, v1 = m = 10 = 0.1 m3/kg From Table B.2.1 = 0.4828 vf < v < vg P2 P1 1 cb 1a

2

x1 = (v - vf)/vfg = (0.1 - 0.0016)/0.20381 V

State 1a: P = 900 kPa, v = v1 = 0.1 < vg at 900 kPa This state is two-phase T1a = 21.52oC Since T2 > T1a then v2 > v1a State 2: 50oC and on line(s) means 900 kPa which is superheated vapor. From Table B.2.2 linear interpolation between 800 and 1000 kPa: v2 = 0.1648 m3/kg,

1W2 =

V2 = mv2 = 1.648 m3

∫ P dV = Pfloat (V2 - V1) = 900 (1.648 - 1.0) = 583.2 kJ

Sonntag, Borgnakke and van Wylen

4.70 A piston cylinder setup similar to Problem 4.68 contains 0.1 kg saturated liquid and vapor water at 100 kPa with quality 25%. The mass of the piston is such that a pressure of 500 kPa will float it. The water is heated to 300°C. Find the final pressure, volume and the work, 1W2. Solution: P Take CV as the water: m2 = m1 = m 1a 2 Process: v = constant until P = Plift To locate state 1: Table B.1.2 v1 = 0.001043 + 0.25×1.69296 = 0.42428 m3/kg 1a: v1a = v1 = 0.42428 m3/kg > vg at 500 kPa so state 1a is Sup.Vapor T1a = 200°C State 2 is 300°C so heating continues after state 1a to 2 at constant P => 2: T2, P2 = Plift => Tbl B.1.3 v2 =0.52256 m3/kg ; V2 = mv2 = 0.05226 m3

1W2 = Plift (V2 - V1) = 500(0.05226 - 0.04243) = 4.91 kJ

Plift P1

1 cb V

Sonntag, Borgnakke and van Wylen

Other types of work and general concepts

4.71 A 0.5-m-long steel rod with a 1-cm diameter is stretched in a tensile test. What is the required work to obtain a relative strain of 0.1%? The modulus of elasticity of steel is 2 × 108 kPa. Solution : −1W2 = AEL0 π 2 2 -6 2 2 (e) , A = 4 (0.01) = 78.54 × 10 m 78.54×10-6 × 2×108 × 0.5 (10-3)2 = 3.93 J −1W2 = 2

Sonntag, Borgnakke and van Wylen

4.72 A copper wire of diameter 2 mm is 10 m long and stretched out between two posts. The normal stress (pressure) σ = E(L – Lo)/Lo , depends on the length L versus the unstretched length Lo and Young’s modulus E = 1.1 × 106 kPa. The force is F = Aσ and measured to be 110 N. How much longer is the wire and how much work was put in? Solution: F = As = A E ∆L/ Lo and ∆L = FLo /AE π π A = 4D2 = 4 × 0.0022 = 3.142 ×10-6 m2 ∆L = = 0.318 m 3.142×10-6 ×1.1 ×106 ×103 x 110 ×10

1W2 = ∫ F dx = ∫ A s dx = ∫ AE L dx o

AE = L ½ x2 o =

where x = L - Lo

3.142×10 -6 ×1.1 ×106 ×103 × ½ × 0.3182 = 17.47 J 10

Sonntag, Borgnakke and van Wylen

4.73 A film of ethanol at 20°C has a surface tension of 22.3 mN/m and is maintained on a wire frame as shown in Fig. P4.73. Consider the film with two surfaces as a control mass and find the work done when the wire is moved 10 mm to make the film 20 × 40 mm. Solution : Assume a free surface on both sides of the frame, i.e., there are two surfaces 20 × 30 mm W = −⌠ S dA = −22.3×10-3 × 2(800 − 600)×10-6 ⌡ = −8.92×10-6 J = -8.92 µJ

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4.74 Assume a balloon material with a constant surface tension of S = 2 N/m. What is the work required to stretch a spherical balloon up to a radius of r = 0.5 m? Neglect any effect from atmospheric pressure. Assume the initial area is small, and that we have 2 surfaces inside and out W = -∫ S dA = -S (A2 − A1) = - S(A2) = -S( 2× π D2 ) = -2 × 2 × π × 1 = -12.57 J Win = -W = 12.57 J

2

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4.75 A soap bubble has a surface tension of S = 3 × 10-4 N/cm as it sits flat on a rigid ring of diameter 5 cm. You now blow on the film to create a half sphere surface of diameter 5 cm. How much work was done?

1W2

= ∫ F dx = ∫ S dA = S ∆A π π = 2 × S × ( 2 D2 - 4 D 2) π = 2 × 3 × 10-4 × 100 × 2 0.052 ( 1- 0.5 ) = 1.18 × 10-4 J

Notice the bubble has 2 surfaces. π A1 = 4 D 2 , A2 = ½ π D2

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4.76 Assume we fill a spherical balloon from a bottle of helium gas. The helium gas

∫ PdV that stretches the balloon material ∫ S dA and pushes back the atmosphere ∫ Po dV. Write the incremental balance for dWhelium = dWstretch + provides work dWatm to establish the connection between the helium pressure , the surface tension S and Po as a function of radius. WHe = ∫ P dV = ∫ S dA + ∫ Po dV dWHe = P dV = S dA + Po dV π π dV = d ( 6 D3 ) = 6 × 3D2 dD dA = d ( 2 × π × D2) = 2π (2D) dD π π P 2 D2 dD = S (4π)D dD + Po2 D2 dD PHe = Po + 8 S D

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4.77 A sheet of rubber is stretched out over a ring of radius 0.25 m. I pour liquid water at 20oC on it so the rubber forms a half sphere (cup). Neglect the rubber mass and find the surface tension near the ring? Solution: F ↑ = F ↓ ; F ↑ = SL The length is the perimeter, 2πr, and there is two surfaces 1 2 S × 2 × 2πr = mH2o g = ρH2o Vg = ρH2o× 12 π (2r) 3g = ρH2o× π 3 r 3 1 1 S = ρH2o 6 r2 g = 997 × 6 × 0.252 × 9.81 = 101.9 N/m

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4.78 Consider a window-mounted air conditioning unit used in the summer to cool incoming air. Examine the system boundaries for rates of work and heat transfer, including signs. Solution : Air-conditioner unit, steady operation with no change of temperature of AC unit. Cool side Inside 25°C 15°C C Hot side Outside 30°C 37°C

- electrical work (power) input operates unit, +Q rate of heat transfer from the room, a larger -Q rate of heat transfer (sum of the other two energy rates) out to the outside air.

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4.79 Consider a hot-air heating system for a home. Examine the following systems for heat transfer. a) The combustion chamber and combustion gas side of the heat transfer area. b) The furnace as a whole, including the hot- and cold-air ducts and chimney. Solution: a) Fuel and air enter, warm products of the combustion exit, large -Q to the air in the duct system, small -Q loss directly to the room. b) Fuel and air enter, warm products exit through the chimney, cool air into the cold air return duct, warm air exit hot-air duct to heat the house. Small heat transfer losses from furnace, chimney and ductwork to the house.

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4.80 Consider a household refrigerator that has just been filled up with roomtemperature food. Define a control volume (mass) and examine its boundaries for rates of work and heat transfer, including sign. a. Immediately after the food is placed in the refrigerator b. After a long period of time has elapsed and the food is cold Solution: I. C.V. Food. a) short term.: -Q from warm food to cold refrigerator air. Food cools. b) Long term: -Q goes to zero after food has reached refrigerator T. II. C.V. refrigerator space, not food, not refrigerator system a) short term: +Q from the warm food, +Q from heat leak from room into cold space. -Q (sum of both) to refrigeration system. If not equal the refrigerator space initially warms slightly and then cools down to preset T. b) long term: small -Q heat leak balanced by -Q to refrigeration system. Note: For refrigeration system CV any Q in from refrigerator space plus electrical W input to operate system, sum of which is Q rejected to the room.

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4.81 A room is heated with an electric space heater on a winter day. Examine the following control volumes, regarding heat transfer and work , including sign. a) The space heater. b) Room c) The space heater and the room together Solution: a) The space heater. Electrical work (power) input, and equal (after system warm up) Q out to the room. b) Room Q input from the heater balances Q loss to the outside, for steady (no temperature change) operation. c) The space heater and the room together Electrical work input balances Q loss to the outside, for steady operation.

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Rates of work

4.82 An escalator raises a 100 kg bucket of sand 10 m in 1 minute. Determine the rate of work done during the process. Solution: The work is a force with a displacement and force is constant: F = mg W=

∫ F dx

= F ∫ dx = F ∆x = 100 kg × 9.80665 m/s2 × 10 m = 9807 J

The rate of work is work per unit time . W 9807 J W = = 60 s = 163 W ∆t

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4.83 A car uses 25 hp to drive at a horizontal level at constant 100 km/h. What is the traction force between the tires and the road? Solution: We need to relate the rate of work to the force and velocity dW . dx dW = F dx => dt = W = F dt = FV W = 25 hp = 25 × 0.7355 kW = 18.39 kW 1000 V = 100 × 3600 = 27.78 m/s F = W / V = (18.39 / 27.78) kN = 0.66 kN Units: kW / (ms−1) = kW s m−1 = kJ s−1s m−1 = kN m m−1 = kN

.

F=W/V

.

.

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4.84 A piston/cylinder of cross sectional area 0.01 m2 maintains constant pressure. It contains 1 kg water with a quality of 5% at 150oC. If we heat so 1 g/s liquid turns into vapor what is the rate of work out? Vvapor = mvapor vg , Vliq = mliq vf mtot = constant = mvapor mliq Vtot = Vvapor + Vliq . . . = 0 = mvapor + mliq mtot ⇒ . . mliq = -mvapor

. . . . . Vtot = Vvapor + Vliq = mvaporvg + mliqvf . . = mvapor (vg- vf ) = mvapor vfg . . . W = PV = P mvapor vfg = 475.9 × 0.001 × 0.39169 = 0.1864 kW = 186 W

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4.85 A crane lifts a bucket of cement with a total mass of 450 kg vertically up with a constant velocity of 2 m/s. Find the rate of work needed to do that. Solution: Rate of work is force times rate of displacement. The force is due to gravity (a = 0) alone. . W = FV = mg × V = 450 kg × 9.807 ms−2 × 2 ms−1 = 8826 J/s . W = 8.83 kW

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4.86 Consider the car with the rolling resistance as in problem 4.27. How fast can it drive using 30 hp? F = 0.006 mg . Power = F × V = 30 hp = W . . W 30 ×0.7457 ×1000 V = W / F = 0.006 mg = = 271.5 m/s 0.006 ×1200 ×9.81 Comment : This is a very high velocity, the rolling resistance is low relative to the air resistance.

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4.87 Consider the car with the air drag force as in problem 4.29. How fast can it drive using 30 hp? 1 P 100 kg ρ = v = RT = = 1.2015 3 and A = 4 m2 0.287 ×290 m Fdrag = 0.225 A ρ V2 . Power for drag force: Wdrag = 30 hp × 0.7457 = 22.371 kW . Wdrag = Fdrag V = 0.225 × 4 × 1.2015 × V3 . V3 = Wdrag /(0.225 × 4 × 1.2015) = 20 688 Drag force: 3600 V = 27.452 m/s = 27.452 × 1000 = 98.8 km/h

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4.88 Consider a 1400 kg car having the rolling resistance as in problem 4.27 and air resistance as in problem 4.29. How fast can it drive using 30 hp? Ftot = Frolling + Fair = 0.006 mg + 0.225 AρV2 m = 1400 kg , A = 4 m2 ρ = P/RT = 1.2015 kg/m3 . W = FV = 0.006 mgV + 0.225 ρAV3 Nonlinear in V so solve by trial and error. . W = 30 hp = 30 × 0.7355 kW = 22.06 kW = 0.0006 × 1400 × 9.807 V + 0.225 × 1.2015 × 4 V3 = 82.379V + 1.08135 V3 . V = 25 m/s ⇒ W = 18 956 W . V = 26 m/s W = 21 148 W . V = 27 m/s W = 23508 W Linear interpolation V = 26.4 m/s = 95 km/h

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4.89 A battery is well insulated while being charged by 12.3 V at a current of 6 A. Take the battery as a control mass and find the instantaneous rate of work and the total work done over 4 hours. Solution : Battery thermally insulated ⇒ Q = 0 For constant voltage E and current i, Power = E i = 12.3 × 6 = 73.8 W W = ∫ power dt = power ∆t = 73.8 × 4 × 60 × 60 = 1 062 720 J = 1062.7 kJ

[Units V × A = W]

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4.90 A current of 10 amp runs through a resistor with a resistance of 15 ohms. Find the rate of work that heats the resistor up. Solution: . W = power = E i = R i2 = 15 × 10 × 10 = 1500 W

R

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4.91 A pressure of 650 kPa pushes a piston of diameter 0.25 m with V = 5 m/s. What is the volume displacement rate, the force and the transmitted power? π A = 4 D2 = 0.049087 m2 . V = AV = 0049087 m2 × 5 m/s = 0.2454 m3/s . . W = power = F V = P V = 650 kPa × 0.2454 m3/s = 159.5 kW

P V

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4.92 Assume the process in Problem 4.37 takes place with a constant rate of change in volume over 2 minutes. Show the power (rate of work) as a function of time. Solution: W = ∫ P dV since 2 min = 120 secs . W = P (∆V / ∆t) (∆V / ∆t) = 0.3 / 120 = 0.0025 m3/s P 300 150 1 2 V 0.1 0.2 0.4 0.1 0.2 0.4 3 kW 0.75 0.375 1 2 V W 3

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4.93 Air at a constant pressure in a piston cylinder is at 300 kPa, 300 K and a volume of 0.1 m3. It is heated to 600 K over 30 seconds in a process with constant piston velocity. Find the power delivered to the piston. Solution: . . Process: P = constant : dW = P dV => W = PV V2 = V1× (T2/T1) = 0.1 × (600/300) = 0.2 . W = P (∆V / ∆t) = 300 × (0.2-0.1)/30 = 1 kW

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4.94 A torque of 650 Nm rotates a shaft of diameter 0.25 m with ω = 50 rad/s. What are the shaft surface speed and the transmitted power? Solution: V = ωr = ωD/2 = 50 × 0.25 / 2 = 6.25 m/s Power = Tω = 650 × 50 Nm/s = 32 500 W = 32.5 kW

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Heat Transfer rates

4.95 The sun shines on a 150 m2 road surface so it is at 45°C. Below the 5 cm thick asphalt, average conductivity of 0.06 W/m K, is a layer of compacted rubbles at a temperature of 15°C. Find the rate of heat transfer to the rubbles. Solution : This is steady one dimensional conduction through the asphalt layer. . ∆T Q=k A ∆x 45-15 = 0.06 × 150 × 0.05 = 5400 W

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4.96 A pot of steel, conductivity 50 W/m K, with a 5 mm thick bottom is filled with 15°C liquid water. The pot has a diameter of 20 cm and is now placed on an electric stove that delivers 250 W as heat transfer. Find the temperature on the outer pot bottom surface assuming the inner surface is at 15°C. Solution : Steady conduction through the bottom of the steel pot. Assume the inside surface is at the liquid water temperature. . . ∆T Q=k A ⇒ ∆Τ = Q ∆x / kΑ ∆x π ∆T = 250 × 0.005/(50 × 4 × 0.22) = 0.796 T = 15 + 0.796 ≅ 15.8°C

cb

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4.97 A water-heater is covered up with insulation boards over a total surface area of 3 m2. The inside board surface is at 75°C and the outside surface is at 20°C and the board material has a conductivity of 0.08 W/m K. How thick a board should it be to limit the heat transfer loss to 200 W ? Solution : Steady state conduction through a single layer board. . . ∆T Q cond = k A ⇒ ∆x = k Α ∆Τ/Q ∆x ∆x = 0.08 × 3 × 75 − 20 200 = 0.066 m

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4.98 You drive a car on a winter day with the atmospheric air at −15°C and you keep the outside front windshield surface temperature at +2°C by blowing hot air on the inside surface. If the windshield is 0.5 m2 and the outside convection coefficient is 250 W/m2K find the rate of energy loos through the front windshield. For that heat transfer rate and a 5 mm thick glass with k = 1.25 W/m K what is then the inside windshield surface temperature? Solution : The heat transfer from the inside must match the loss on the outer surface to give a steady state (frost free) outside surface temperature. . Q conv = h A ∆Τ = 250 × 0.5 × [2 − ( −15)] = 250 × 0.5 × 17 = 2125 W This is a substantial amount of power. . . ∆T Q Q cond = k A ⇒ ∆Τ = kA ∆x ∆x ∆Τ = 2125 W 0.005 m = 17 K 1.25 W/mK × 0.5 m2

Tin = Tout + ∆T = 2 + 17 = 19°C Windshield T=?

-15 C

o

2C

o

Warm air

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4.99 A large condenser (heat exchanger) in a power plant must transfer a total of 100 MW from steam running in a pipe to sea water being pumped through the heat exchanger. Assume the wall separating the steam and seawater is 4 mm of steel, conductivity 15 W/m K and that a maximum of 5°C difference between the two fluids is allowed in the design. Find the required minimum area for the heat transfer neglecting any convective heat transfer in the flows. Solution : Steady conduction through the 4 mm steel wall. . . ∆T Q=k A ⇒ Α = Q ∆x / k∆Τ ∆x A = 100 × 10 × 0.004 / (15 × 5) = 480 m2

6

Condensing water

Sea water

cb

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4.100 The black grille on the back of a refrigerator has a surface temperature of 35°C with a total surface area of 1 m2. Heat transfer to the room air at 20°C takes place with an average convective heat transfer coefficient of 15 W/m2 K. How much energy can be removed during 15 minutes of operation? Solution : . . Q = hA ∆T; Q = Q ∆t = hA ∆T ∆t Q = 15 × 1 × (35-20) × 15 × 60 = 202500 J = 202.5 kJ

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4.101 Due to a faulty door contact the small light bulb (25 W) inside a refrigerator is kept on and limited insulation lets 50 W of energy from the outside seep into the refrigerated space. How much of a temperature difference to the ambient at 20°C must the refrigerator have in its heat exchanger with an area of 1 m2 and an average heat transfer coefficient of 15 W/m2 K to reject the leaks of energy. Solution : . Q tot = 25 + 50 = 75 W to go out . Q = hA∆T = 15 × 1 × ∆T = 75 . ∆T = Q / hA = 75/(15×1) = 5 °C OR T must be at least 25 °C

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4.102 The brake shoe and steel drum on a car continuously absorbs 25 W as the car slows down. Assume a total outside surface area of 0.1 m2 with a convective heat transfer coefficient of 10 W/m2 K to the air at 20°C. How hot does the outside brake and drum surface become when steady conditions are reached? Solution : . . Q = hA∆Τ ⇒ ∆Τ = Q / hA ∆T = ( ΤBRAKE − 20 ) = 25/(10 × 0.1) = 25 °C TBRAKE = 20 + 25 = 45°C

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4.103 A wall surface on a house is at 30°C with an emissivity of ε = 0.7. The surrounding ambient to the house is at 15°C, average emissivity of 0.9. Find the rate of radiation energy from each of those surfaces per unit area. Solution : . –8 Q /A = εσAT4, σ =5.67 × 10 . -8 a) Q/A = 0.7 × 5.67 × 10 × ( 273.15 + 30)4 = 335 W/m2 . -8 4 b) Q/A = 0.9 × 5.67 × 10 × 288.15 = 352 W/m2

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4.104 A log of burning wood in the fireplace has a surface temperature of 450°C. Assume the emissivity is 1 (perfect black body) and find the radiant emission of energy per unit surface area. Solution : . Q /A = 1 × σ T4 –8 4 = 5.67 × 10 × ( 273.15 + 450) = 15505 W/m2 = 15.5 kW/m2

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4.105 A radiant heat lamp is a rod, 0.5 m long and 0.5 cm in diameter, through which 400 W of electric energy is deposited. Assume the surface has an emissivity of 0.9 and neglect incoming radiation. What will the rod surface temperature be ? Solution : For constant surface temperature outgoing power equals electric power. . . 4 Qrad = εσAT = Qel ⇒ . 4 –8 T = Qel / εσA = 400 / (0.9 × 5.67 ×10 × 0.5 × π × 0.005) = 9.9803 ×10 K

11 4

⇒ T ≅ 1000 K OR 725 °C

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Review Problems

4.106 A vertical cylinder (Fig. P4.106) has a 61.18-kg piston locked with a pin trapping 10 L of R-22 at 10°C, 90% quality inside. Atmospheric pressure is 100 kPa, and the cylinder cross-sectional area is 0.006 m2. The pin is removed, allowing the piston to move and come to rest with a final temperature of 10°C for the R-22. Find the final pressure, final volume and the work done by the R-22. Solution: Po mp R-22 g

State 1: (T, x) from table B.4.1 v1 = 0.0008 + 0.9 × 0.03391 = 0.03132 m3/kg m = V1/v1 = 0.010/0.03132 = 0.319 kg Force balance on piston gives the equilibrium pressure 61.18 × 9.807 P2 = P0 + mPg/ AP = 100 + = 200 kPa 0.006 × 1000

State 2: (T,P) in Table B.4.2 v2 = 0.13129 m3/kg V2 = mv2 = 0.319 kg × 0.13129 m3/kg = 0.04188 m3 = 41.88 L

1W2

= ⌠Pequil dV = P2(V2-V1) = 200 kPa (0.04188- 0.010) m3 = 6.38 kJ ⌡

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4.107 A piston/cylinder contains butane, C4H10, at 300°C, 100 kPa with a volume of 0.02 m3. The gas is now compressed slowly in an isothermal process to 300 kPa. a. Show that it is reasonable to assume that butane behaves as an ideal gas during this process. b. Determine the work done by the butane during the process. Solution: a) T 573.15 Tr1 = T = 425.2 = 1.35; c P 100 Pr1 = P = 3800 = 0.026 c

From the generalized chart in figure D.1 Z1 = 0.99 T 573.15 P 300 Tr2 = T = 425.2 = 1.35; Pr2 = P = 3800 = 0.079 c c From the generalized chart in figure D.1 Z2 = 0.98 Ideal gas model is adequate for both states. b) Ideal gas T = constant ⇒ PV = mRT = constant P1 100 W = ⌠P dV = P1V1 ln P = 100 × 0.02 × ln 300 = -2.2 kJ ⌡ 2

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4.108 A cylinder fitted with a piston contains propane gas at 100 kPa, 300 K with a volume of 0.2 m3. The gas is now slowly compressed according to the relation PV1.1 = constant to a final temperature of 340 K. Justify the use of the ideal gas model. Find the final pressure and the work done during the process. Solution: The process equation and T determines state 2. Use ideal gas law to say

1.1 T2 n 340 0.1 n-1 = 100 ( P2 = P1 ( T ) = 396 kPa 300 ) 1

P1 1/n 100 1/1.1 V2 = V1 ( P ) = 0.2 ( 396 ) = 0.0572 m3

2

For propane Table A.2: Tc = 370 K, Pc = 4260 kPa, Figure D.1 gives Z. Tr1 = 0.81, Pr1 = 0.023 => Z1 = 0.98 Tr2 = 0.92, Pr2 = 0.093 => Z2 = 0.95 Ideal gas model OK for both states, minor corrections could be used. The work is integrated to give Eq.4.4

1W2 = ∫ P dV =

P2V2-P1V1 (396 × 0.0572) - (100 × 0.2) = = -26.7 kJ 1-n 1 - 1.1

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4.109 The gas space above the water in a closed storage tank contains nitrogen at 25°C, 100 kPa. Total tank volume is 4 m3, and there is 500 kg of water at 25°C. An additional 500 kg water is now forced into the tank. Assuming constant temperature throughout, find the final pressure of the nitrogen and the work done on the nitrogen in this process. Solution: The water is compressed liquid and in the process the pressure goes up so the water stays as liquid. Incompressible so the specific volume does not change. The nitrogen is an ideal gas and thus highly compressible. State 1: VH O 1 = 500 × 0.001003 2 VN 1 = 4.0 - 0.5015 2 State 2: Process: = 0.5015 m3 = 3.4985 m3

VN 2 = 4.0 - 2 × 0.5015 = 2.997 m3 2 T = C gives P1V1 = mRT = P2V2

3.4985 PN 2 = 100 × 2.997 = 116.7 kPa 2 Constant temperature gives P = mRT/V i.e. pressure inverse in V for which the work term is integrated to give Eq.4.5 2 Wby N = ⌠ PN dVN = P1V1 ln(V2/V1) 2 ⌡ 2 2 1 2.997 = 100 × 3.4985 × ln 3.4985 = -54.1 kJ

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4.110 Two kilograms of water is contained in a piston/cylinder (Fig. P4.110) with a massless piston loaded with a linear spring and the outside atmosphere. Initially the spring force is zero and P1 = Po = 100 kPa with a volume of 0.2 m3. If the piston just hits the upper stops the volume is 0.8 m3 and T = 600°C. Heat is now added until the pressure reaches 1.2 MPa. Find the final temperature, show the P– V diagram and find the work done during the process. Solution: , P State 1: v1 = V/m = 0.2 / 2 = 0.1 m3/kg 2 , 3 Process: 1 → 2 → 3 or 1 → 3’ 3 State at stops: 2 or 2’ 2 v2 = Vstop/m = 0.4 m3/kg & T2 = 600°C P1 Table B.1.3 ⇒ Pstop = 1 MPa < P3 1

V1 V stop V

since Pstop < P3 the process is as 1 → 2 → 3 ⇒ T3 ≅ 770°C

State 3: P3 = 1.2 MPa, v3 = v2 = 0.4 m3/kg

1 1 W13 = W12 + W23 = 2(P1 + P2)(V2 - V1) + 0 = 2(100 + 1000)(0.8 - 0.2) = 330 kJ

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4.111 A cylinder having an initial volume of 3 m3 contains 0.1 kg of water at 40°C. The water is then compressed in an isothermal quasi-equilibrium process until it has a quality of 50%. Calculate the work done in the process splitting it into two steps. Assume the water vapor is an ideal gas during the first step of the process. Solution: C.V. Water State 2: (40°C, x = 1) Tbl B.1.1 => PG = 7.384 kPa, vG = 19.52 State 1: v1 = V1/m = 3 / 0.1 = 30 m3/kg ( > vG ) so H2O ~ ideal gas from 1-2 so since constant T vG 19.52 P1 = PG v = 7.384 × 30 = 4.8 kPa 1 V2 = mv2 = 0.1 × 19.52 = 1.952 m3

P C.P. 3 2 T C.P. Psat 7.38 1 T v 40 1 v P1

3

2

Process T = C:

2

and ideal gas gives work from Eq.4.5

V2 1.952 ⌠ 1W2 = ⌡ PdV = P1V1ln V = 4.8 × 3.0 × ln 3 = −6.19 kJ 1 1 v3 = 0.001008 + 0.5 × 19.519 = 9.7605 => V3 = mv3 = 0.976 m3 P = C = Pg:

3

This gives a work term as

2W3 = ⌠ PdV = Pg (V3−V2) = 7.384(0.976 - 1.952) = −7.21 kJ ⌡ 2

Total work:

1W3 = 1W2 + 2W3 = − 6.19 − 7.21 = -13.4 kJ

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4.112 Air at 200 kPa, 30°C is contained in a cylinder/piston arrangement with initial volume 0.1 m3 . The inside pressure balances ambient pressure of 100 kPa plus an externally imposed force that is proportional to V0.5. Now heat is transferred to the system to a final pressure of 225 kPa. Find the final temperature and the work done in the process. Solution: C.V. Air. This is a control mass. Use initial state and process to find T2 P1 = P0 + CV1/2; 225 = 100 + CV21/2 200 = 100 + C(0.1)1/2, ⇒ V2 = 0.156 m3 ⇒ C = 316.23 =>

P1V1 P2V2 = mRT2 = T T2 1 W12 = ∫ P dV =

T2 = (P2V2 / P1V1) T1 = 225 × 0.156 ×303.15 / (200 ×0.1) = 532 K = 258.9°C

∫ (P0 + CV1/2) dV

2 2

= P0 (V2 - V1) + C × 3 × (V23/2 - V13/2) = 100 (0.156 – 0.1) + 316.23 × 3 × (0.1563/2 – 0.13/2) = 5.6 + 6.32 = 11.9 kJ

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4.113 A spring-loaded piston/cylinder arrangement contains R-134a at 20°C, 24% quality with a volume 50 L. The setup is heated and thus expands, moving the piston. It is noted that when the last drop of liquid disappears the temperature is 40°C. The heating is stopped when T = 130°C. Verify the final pressure is about 1200 kPa by iteration and find the work done in the process. Solution: C.V. R-134a. This is a control mass.

P P3 P2 P1 2 3

State 1: Table B.5.1 => v1 = 0.000817 + 0.24*0.03524 = 0.009274 P1 = 572.8 kPa, m = V/ v1 = 0.050 / 0.009274 = 5.391 kg Process: Linear Spring P = A + Bv v 1

State 2: x2 = 1, T2 ⇒ P2 = 1.017 MPa, v2 = 0.02002 m3/kg Now we have fixed two points on the process line so for final state 3: P2 - P1 P3 = P1 + v - v (v3 - v1) = RHS Relation between P3 and v3 2 1 State 3: T3 and on process line ⇒ iterate on P3 given T3 at P3 = 1.2 MPa => v3 = 0.02504 at P3 = 1.4 MPa => v3 = 0.02112 => P3 - RHS = -0.0247 => P3 - RHS = 0.3376

Linear interpolation gives : 0.0247 P3 ≅ 1200 + 0.3376 + 0.0247 (1400-1200) = 1214 kPa 0.0247 v3 = 0.02504 + 0.3376 + 0.0247 (0.02112-0.02504) = 0.02478 m3/kg W13 = ∫ P dV = 2 (P1 + P3)(V3 - V1) = 2 (P1 + P3) m (v3 - v1)

1 1

= 2 5.391(572.8 + 1214)(0.02478 - 0.009274) = 74.7 kJ

1

Sonntag, Borgnakke and van Wylen

4.114 A piston/cylinder (Fig. P4.114) contains 1 kg of water at 20°C with a volume of 0.1 m3. Initially the piston rests on some stops with the top surface open to the atmosphere, Po and a mass so a water pressure of 400 kPa will lift it. To what temperature should the water be heated to lift the piston? If it is heated to saturated vapor find the final temperature, volume and the work, 1W2. Solution: (a) State to reach lift pressure of P = 400 kPa, v = V/m = 0.1 m3/kg Table B.1.2: vf < v < vg = 0.4625 m3/kg => T = T sat = 143.63°C (b) State 2 is saturated vapor at 400 kPa since state 1a is two-phase.

P P o H2O

1a 1

2

V

v2 = vg = 0.4625 m3/kg , V2 = m v2 = 0.4625 m3, Pressure is constant as volume increase beyond initial volume.

1W2 =

∫ P dV = P (V2 - V1) = Plift (V2 – V1) = 400 (0.4625 – 0.1) = 145 kJ

Sonntag, Borgnakke and van Wylen

4.115 Two springs with same spring constant are installed in a massless piston/cylinder with the outside air at 100 kPa. If the piston is at the bottom, both springs are relaxed and the second spring comes in contact with the piston at V = 2 m3. The cylinder (Fig. P4.115) contains ammonia initially at −2°C, x = 0.13, V = 1 m3, which is then heated until the pressure finally reaches 1200 kPa. At what pressure will the piston touch the second spring? Find the final temperature and the total work done by the ammonia. Solution :

P 2 0 P0 cb 3

State 1: P = 399.7 kPa Table B.2.1 v = 0.00156 + 0.13×0.3106 = 0.0419 At bottom state 0: 0 m3, 100 kPa State 2: V = 2 m3 and on line 0-1-2 Final state 3: 1200 kPa, on line segment 2.

1

1W2 2W3

V

V3

0

1

2

Slope of line 0-1-2: ∆P/ ∆V = (P1 - P0)/∆V = (399.7-100)/1 = 299.7 kPa/ m3 P2 = P1 + (V2 - V1)∆P/∆V = 399.7 + (2-1)×299.7 = 699.4 kPa State 3: Last line segment has twice the slope. P3 = P2 + (V3 - V2)2∆P/∆V ⇒ V3 = V2+ (P3 - P2)/(2∆P/∆V) V3 = 2 + (1200-699.4)/599.4 = 2.835 m3 v3 = v1V3/V1 = 0.0419×2.835/1 = 0.1188 1 ⇒ T = 51°C 1

1W3 = 1W2 + 2W3 = 2 (P1 + P2)(V2 - V1) + 2 (P3 + P2)(V3 - V2)

= 549.6 + 793.0 = 1342.6 kJ

Sonntag, Borgnakke and van Wylen

4.116 Find the work for Problem 3.101. A piston/cylinder arrangement is loaded with a linear spring and the outside atmosphere. It contains water at 5 MPa, 400°C with the volume being 0.1 m3. If the piston is at the bottom, the spring exerts a force such that Plift = 200 kPa. The system now cools until the pressure reaches 1200 kPa. Find the mass of water, the final state (T2, v2) and plot the P–v diagram for the process. Solution : P

5000 2 a v 0 ? 0.05781 1

1: 5 MPa, 400°C ⇒ v1= 0.05781 m3/kg m = V/v1 = 0.1/0.05781 = 1.73 kg Straight line: P = Pa + Cv P2 - Pa v2 = v1 P - P = 0.01204 m3/kg 1 a v2 < vg(1200 kPa) so two-phase T2 = 188°C ⇒ x2 =

1200 200

v2 - 0.001139 = 0.0672 0.1622 The P-V coordinates for the two states are then: P1 = 5 MPa, V1 = 0.1 m3, P2 = 1200 kPa, V2 = mv2 = 0.02083 m3 P vs. V is linear so

1W2 = ⌠PdV = 2 (P1 + P2)(V2 - V1) ⌡

1

1 = 2 (5000 + 1200)(0.02083 - 0.1) = -245.4 kJ

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