CHAPTER 2 PROBLEMS

B. CHAPTER 2 PROBLEMS

Problem B1 Is the relation h = u + Pv valid only for a constant pressure process?

Problem B2 Is the earth a closed or an open system? Problem B3 If you type this entire text on a computer, will the mass of the computer increase? Problem B4 Is ∫Pdv work boundary work or flow work?

Problem B5 What is physical interpretation of c v and c p ?

Problem B6 What is the Poincare Scheme?

Problem B7 Is it true that in a closed or an open system, work and heat transfer can occur across the system?

Problem B8 Is there a difference between a quasiequilibrium and an internally reversible process?

Problem B9 An incompressible liquid (v = constant) undergoes adiabatic internally reversible compression in a open system. If you follow a unit mass, then is the change in internal energy a) zero, or b) non-zero?

Problem B10 An incompressible liquid (v = constant) undergoes internally reversible compression in a closed system. Then is the work input per unit mass zero?

Problem B11 What does the term quasiequilibrium mean?

Problem B12 When can a nonquasiequilibrium process not be represented on a P–v diagram or a T–s diagram?

Problem B13 Is it generally true that we can use the equality δW = –PdV in the relation δQ–δW =

dU? Problem B14

When is the relation δQ–δW = dU equivalent to the expression δQ–δW = dH? Problem B15

What is the physical meaning of a characteristic time for a process involving the heating of a house?

Problem B16 For a steady state process involving an open system dm cv /dt = 0, i.e., m cv is constant. Is this always true for a closed system?

Problem B17 For a steady state open system dU cv /dt = 0 . Is it true that the reverse statement, dU cv /dt = 0, implies that the open system must be steady. The latter statement is a) true b) false. (Hint: consider the heating of ideal gas in an oven where P is constant.)

Problem B18 Gas cylinders normally use pressure regulators to control the downstream pressure at P R . At any set regulator pressure P Rrg , the mass flow leaving the regulator is given by

Reg , kg/s, P in bar, T reg in K and A in m (choked flow). Thus, the pressure downstream of the regulator is fixed at P reg and the flow from the cylin-

m ˙ = 4040 A

reg P Reg /T

der leaves at conditions P and T. As the flow leaves, the cylinder pressure (P) de- creases with time (t). Assume T reg = T, i.e., the cylinder temperature. Assume that the cylinder is adiabatic. If the initial cylinder pressure and temperature are 100 bar and 300 K, and regulator pressure is 3 bar, determine the time for pressure in the cylinder to decrease to 50 bar.

Problem B19 The space shuttle is powered by booster rockets and the main shuttle engines are fired with H 2 and O 2 . The empty weight of the system is 76000 lb and after charging the booster and shuttle tank, the weight increases to 1500,000 lbs. Given the following conditions, calculate the altitude reached by the shuttle 2 minutes after firing: Thrust (F) 3000,000 lb

f , where F = mv gas /g c ,v gas = 0.4 (kg c RT) , T = 6000 R, Firing Rate = constant. Note that the mass of the system varies during lift off due to discharge of

propellants. Problem B20

A closed adiabatic system with a weightless piston at the top contains air at 100 kPa, 227ºC. The ambient pressure is also 100 kPa (Area of piston: 100 cm 2 ; volume at state 1: 0.1 m 3 ). Suddenly a weight of 1 kN is placed and system reaches state 2. a)

Sketch the process on a PV diagram, and b) determine P 2 ,T 2 and V 2 . Problem B21

Air at 100 psia, and 40 F, is held in a tank of 20 ft 3 volume. Heat is added until the remaining air in the tank is at 240ºF, while some air is bled from the tank to hold the pressure constant at 100 psia. Determine the heat transfer, assuming the air to be an ideal gas with constant specific heats.

Problem B22

A cooker “A” of 30 cm diameter and volume 30 L is initially filled with liquid water of 4 kg. It is then heated until the pressure in the cooker rises to 5 bar at which pres- sure it contains a mixture of pure water vapor and liquid water. Then, assume that we insulate the cooker and attach a metal tube of cross sectional area A to it that is placed slightly away from the bottom surface of the cooker. Assume also that we provide a valve at the top of this metal tube. When the valve is opened, the water left in the cooker can be injected into another open adiabatic cooker B of equal dimension in or- der to conserve energy. We will neglect evaporation from cooker B. However, we would like to monitor the pressure in the first cooker. As an expert in thermodynam- ics you are asked to predict the pressure vs. time until no liquid water remains. As- sume that vapor behaves like an ideal gas with a specific heat of 1.59 kJ kg –1 K –1 .

Water is incompressible with a specific heat of 4.184 kJ kg –1 K –1 and v = 0.001 m 3 /kg. The cross sectional area of the metal tube is 10 mm 2 . Assume the power is off when

we open the valve and the cooker free space is occupied with vapor only. Neglect the potential energy change. a)

What is the quality when the valve in the metal tube is opened? b)

Write down the mass and energy conservation equations for the vapor phase in the system (assume no condensation of vapor or vaporization of water) and obtain a relation for P vs. v for the vapor phase.

c) What is the quality when all of the water has been expelled from cooker A? d)

Sketch the process for the cooker A on a P–v diagram.

Compressor

Tank

P-v diagram of a compressor in context of Problem B.23.

e) Write down the energy balance equation for the metal tube and obtain an ex- pression for velocity through the metal tube assuming that steady state exists for the c.v. (metal tube)

f) Obtain an expression for mass flow through the tube. g)

Derive the expression for P (t) in terms of vapor volume in the cooker? h)

What is the pressure in the cooker when all water is gone?

i) sat If ln P (bar) = 13.09 –4879/T, plot T vs. t and compare with T vs. t. Check your assumption in (b) Problem B23

sat

A tank must be charged with gases from a reciprocating compressor that is running at

a speed of N. The displacement volume is V disp and the compresion ratio is r. The outlet valve for the compressor opens only when the pressure in the cylinder exceeds

tank pressure. The compression process in the piston–cylinder follows the relation Pv n = Constant. Determine the P(t) vs. time if the tank is assumed to be adiabatic.

Problem B24 When propane is burned in a gas turbine, one produces a gaseous mixture of 6.48% CO 2 , 10.12% O 2 , 7.28% H2O and 76.12%% N 2 . Ten kmol of mixture /min. enter an adiabatic gas turbine at 1200 K, and 10 bar and leave at 700 K, 1 bar . There are no chemical reactions a)

Write down the species balance in terms of k mol of flow in and out, b)

Write down the energy balance equation, and

c) Simplify the equations in parts (a) and (b) for a steady state and steady flow process.

Calculate the work produced by the turbine in kW Problem B25

Consider an insulated rigid tank containing 2 m 3 of air. The tank is divided into two parts A and B by a partition mounted with a thick insulation (a constraint which pre- vents heat transfer) and locked in place by a pin (another constraint which prevents work transfer). The portion A consists of 1 kg of hot air at 320 K and 1.48 bar while portion B consists of cold air at 290 K and 0.42 bar. Determine the final conditions in both chambers for the following cases. a)

If the pin is removed but insulation stays with the partition i)

If QE expansion occurs for chamber A but the process in B may not

be QE, be QE,

b) If the pin is removed and the insulation is also removed from the partition Problem B26

You have been hired by PRESSCOOKER, Inc. to analyze the thermodynamic proc- esses in cookers. A pressure cooker of mass m C contains water (mass m w , volume

V w ). There is some air m a left in the cooker. The cooker is covered and tightly sealed.

A small weight is placed at its top and the cooker is heated. You must analyze the problem of water heat up on an electric range that supplies an electrical power W elec . Water starts evaporating and releases vapor (mass = m v ) to the air space above the water. After time t L , the pressure in the cooker reaches P L , the weight is lifted and steam is released. We can assume that the cooker is adiabatic. Neglect the kinetic and potential energy contributions. Assume uniform temperature and pressure inside the cooker and the temperatures of the contents are the same as that of the steel. The spe- cific heats of air(c P0,a ), vapor(c p0,v ), water(c w ) and steel cooker (c S ) are assumed to be constant. a)

Write down the mass and energy conservation equations for t < t L . Simplify. Explain the various terms (not exceeding two lines for each term).

b) Write down the mass and energy conservation equations for t > t L . Simplify. Explain the various terms (not exceeding two lines for each t the entropy generation term).

c) Mention how will you solve the problem for T vs. t, m vs. t for t < t L and t > t L Note: m denotes the total mass of cooker including the contents.

Problem B27 Assume that Mt. St. Helen erupts again and releases a rock of 2000 kg mass. The sur- face of the rock is at T = 2000 K while the interior is a liquid at 2000 K. The entire rock moves at a speed of 300 m/s and rises in altitude. As the rock moves in ambient air, there is also heat loss. Analyze the problem by applying the First Law.

Problem B28 Assume that scientists have measured the following with “inert” (nonfusion–causing) electrodes in heavy water in a nuclear reactor: Work input through a stirrer: 100 mW; Electric input through a heater: 200 mW (controllable); Power input through elec- trodes: 350 mW. What should be the rate of heat loss if the heavy water is maintained at 35ºC? Assume that scientists change the electrodes that may cause a fusion reaction to occur. Under identical conditions as before, they have found that they will have to reduce the heater input to 150 mW. The heavy water is still at 35ºC. What is the rate of “unaccountable energy” or “excess energy” or the so called “fusion energy” in mW?

Problem B29 Using Eqs (J) and (K) and ideal gas law prove Eq. (L) in Example (17).