HW-03 (25 points) i) Given: for writing Given, Find, Basic equations Rigid tank containing nitrogen gas in two sections initially separated by a membrane. Find: Initial density (kg/m3) of nitrogen gas in section B. EFD: 2 Assumptions: None. Basic Equations: None : Total mass after mixing: m = ρ final V =.8 kg/m 3 5 m 3 = 27 kg 2
Initial mass in section A: m A = m m B = 27 kg 6 kg = 2 kg 2 Volume of gas in section A: V A = m A = ρ A 2 kg.6 kg/m 3 = 3.25 m 3 2 Volume of gas in section B: V B = V V A = 5 m 3 3.25 m 3 =.875 m 3 2 Density of gas in section B: ρ B = m B V B = 6 kg.875 m 3 = 3.20 kg/m 3 3 2
ii) Given: for writing Given, Find, Basic equations A body accelerated from rest on a inclined surface with known amount of work. Find: Mass (kg) of the body EFD: 3 Assumptions: 2 Constant acceleration due to gravity. Neglect friction between the body and surface. Ignore any other resisting force. Basic Equations: None : Work is required to accelerate the body (change in its KE) and to climb up the surface (change in its PE). W 2 = 2 m(v 2 2 V 2 ) + mg(z 2 z ) 2 W m = 2 (V 2 2 V 2 ) + g(z 2 z ) 200 0 3 = 2 (2002 ) + 9.8 0 sin45 o = 9.97 kg 3 3
HW-04 (25 points) Given: for writing Given, Find A gas inside a cylinder with spring-attached piston expanded by heat transfer to the gas. Find: a) Final absolute pressure (kpa) of the gas. b) Final volume (cm 3 ) of the gas. c) p-v diagram. d) Work (J) during expansion. EFD: 3 Assumptions: 2 Quasi-equilibrium process. Massless piston. Spring is linear. Ignore area difference between two sides of piston. Neglect friction between piston and cylinder. Basic Equations: W boundary = pdv : a) Consider the force balance on the piston before adding energy to the gas : 4
The gas expands due to heat transfer and the spring gets compressed: F spring = k X Consider force balance on the piston after adding energy to the gas 3 : p 2 = (p atm = p gas ) + F spring A p = 00 kp a + 20 N cm 2 cm 0.0004 m 2 000 kp a 2 5
Final absolute pressure of the gas in the cylinder: p 2 = 200 kp a (absolute) 3 b) Initial volume of the gas: V = 32 cm 3. Final volume of the gas: V 2 = V + A p X = 32 + (0.0004 0 4 )cm 2 2 cm = 40 cm 3 3 c) 3 d) Moving boundary work during expansion of gas: W 2 = 2 pdv = Area under the p V diagram = 2 (p gas + p gas2 )(V 2 V ) = 2 (00 + 200)kP a (40 32) 0 6 m 3 = 0.002 kj =.2 J 4 The positive sign indicates work done by the gas. Alternatively: 200 00 m = = 2.5 kp a/cm3 40 32 00 = 2.5 (32) + b b = 300 kp a 6
Integrating: 2 40 W 2 = pdv = 2.5 x 300 dx 32 = 2.5 40 x 2 300 x 2 = 200 kp a cm 3 =.20 J 4 32 Alternatively, the pressure inside the piston may be expressed as: p = p atm + F spring A p dv = A p dx = kx A p Using the definition of work: x2 [ W 2 = p atm + kx ] A p dx = p atm A p x A p x =0 x 2 0 + k x 2 A p A p 2 = p atm A p x 2 + k x2 2 2 = 00 kn m 0.0004 2 m2 2 cm m 00 cm + 20 N cm 22 m 2 cm2 00 cm = 0.0008 kj + 0.0004 kj =.2 J x 2 0 kn 000 N 7
HW-05 (25 points) i) Given: Carbon monoxide gas within a piston-cylinder assembly undergoes three processes in series. for writing Given, Find Process -2: Constant pressure expansion at 5 bar from V = 0.2 m 3 to V 2 = m 3. Process 2-3: Constant volume cooling from state 2 to state 3 where p 3 = bar. Process 3-: Compression from state 3 to the initial state during which the pressurevolume relationship is pv = constant. Find: Sketch the processes in serioes on a p-v diagram and evaluate the work for each process. EFD: 2 Assumptions: 2 Frictionless piston. Quasi-equilibrium. Basic Equations: W = pdv : 8
Process -2 For a constant pressure process: W 2 = 2 pdv W 2 = p(v 2 V ) = 5 0 2 kp a( 0.2) m 3 = 400 kj 2 The positive sign indicates work done by the system. Process 2-3 The process is at constant volume, hence, W 23 = 0 2 Process 3- Noting that V 3 = V 2 and that pv = constant = p V = C W 3 = W 3 = 3 3 pdv C V dv = Cln V V 3 = p V ln V V 2 = 5 0 2 kp a 0.2 m 3 ln 0.2 = 6 kj 2 The negative sign indicates work done on the system. 9
2 ii) Given: for writing Given, Find Operating data are given for a 0-V battery providing current to a resistance. Find: Determine the resistance, in ohms, and the amount of energy transfer by work, in kj. EFD: a) 0
b) Assumptions: The battery power (Ẇ ) remains constant for the 30 min interval. Basic Equations: None. : a) From the definition of electrical resistance: Resistance = V oltage Current = 0 V 0.5 A = 20 Ω 3 b) Ẇ = (V oltage)(current) = (0 V olt)(0.5 A) = 5 W 3 W = 30 0 Ẇ dt = 5 W 30 min = 5 W 800 s = 9000 J = 9kJ 3