INDIAN INSTITUTE OF SCIENCE

Department of Mechanical Engineering

ME 271: THERMODYNAMICS Due: 4th October 2016

Assignment 2

Teaching Assistant: Santanu Pramanik (santanupramanik07@gmail.com)

Ref: Fundamentals of Engineering Thermodynamics - Moran and Shapiro

Questions:

Second Law

1. Two reversible power cycles are arranged in series. The first cycle receives energy by heat transfer
from a reservoir at temperature TH and rejects energy to a reservoir at an intermediate temperature T.
The second cycle receives the energy rejected by the first cycle from the reservoir at temperature T
and rejects energy to a reservoir at temperature TC lower than T. Derive an expression for the
intermediate temperature T in terms of TH and TC when

(a) the net work of the two power cycles is equal.

(b) the thermal efficiencies of the two power cycles are equal.

2. The figure shows a system consisting of a power cycle driving a heat pump. At steady state, the
power cycle receives by heat transfer at Ts from the high-temperature source and delivers to a
dwelling at Td. The heat pump receives from the outdoors at T0, and delivers to the dwelling.
Obtain an expression for the maximum theoretical value of the performance parameter ( + ) /
in terms of the temperature ratios Ts/Td and T0/Td.

Problem: 2

3. Before introducing the temperature scale now known as the Kelvin scale, Kelvin suggested a

logarithmic scale in which the function ( ) ( ) takes the form ( )

 ( ) ( ) where H and C denote, respectively, the temperatures of the hot and cold
reservoirs on this scale.

(a) Show that the relation between the Kelvin temperature T and the temperature on the logarithmic
scale is ( ) where C is a constant.

(b) On the Kelvin scale, temperatures vary from 0 to . Determine the range of temperature values
on the logarithmic scale.

(c) Obtain an expression for the thermal efficiency of any system undergoing a reversible power cycle
while operating between reservoirs at temperatures H and C on the logarithmic scale.

4. A refrigeration cycle having a coefficient of performance of 3 maintains a computer laboratory at

18 oC on a day when the outside temperature is 30 oC. The thermal load at steady state consists of
energy entering through the walls and windows at a rate of 30,000 kJ/h and from the occupants,
computers, and lighting at a rate of 6000 kJ/h. Determine the power required by this cycle and
compare with the minimum theoretical power required for any refrigeration cycle operating under
these conditions, each in kW.

5. Two kilograms of water execute a Carnot power cycle. During the isothermal expansion, the water
is heated until it is a saturated vapor from an initial state where the pressure is 40 bar and the quality is
15%. The vapor then expands adiabatically to a pressure of 1.5 bar while doing 491.5 kJ/kg of work.

(a) Sketch the cycle on pv coordinates.

(b) Evaluate the heat and work for each process, in kJ.

(c) Evaluate the thermal efficiency.

6. One kilogram of air as an ideal gas executes a Carnot power cycle having a thermal efficiency of
60%. The heat transfer to the air during the isothermal expansion is 40 kJ. At the end of the isothermal
expansion, the pressure is 5.6 bar and the volume is 0.3 m3. Determine

(a) the maximum and minimum temperatures for the cycle, in K.

(b) the pressure and volume at the beginning of the isothermal expansion in bar and m3, respectively.

(c) the work and heat transfer for each of the four processes, in kJ.

(d) Sketch the cycle on pv coordinates.

Entropy

7. An isolated system of total mass m is formed by mixing two equal masses of the same liquid
initially at the temperatures T1 and T2. Eventually, the system attains an equilibrium state. Each mass
is incompressible with constant specific heat c.

(a) Show that the amount of entropy produced is

[ ]
( )

(b) Demonstrate that must be positive.

8. At steady state, an insulated mixing chamber receives two liquid streams of the same substance at
temperatures T1 and T2 and mass flow rates m1 and m2 respectively. A single stream exits at T3 and m3.
Using the incompressible substance model with constant specific heat c, obtain an expression for

(a) T3 in terms of T1, T2, and the ratio of mass flow rates m1/m3.

(b) the rate of entropy production per unit of mass exiting the chamber in terms of c, T1/T2, and m1/m3.

(c) For fixed values of c and T1/T2, determine the value of m1/m3 for which the rate of entropy
production is a maximum.

9. A quantity of air undergoes a thermodynamic cycle consisting of three internally reversible

processes in series.

Process 12: isothermal expansion from 6.25 to 1.0 bar

Process 23: adiabatic compression to 550 K, 6.25 bar

Process 31: constant-pressure compression

Employing the ideal gas model,

(a) sketch the cycle on pv and Ts coordinates.

(b) determine T1, in K

(c) If the cycle is a power cycle, determine its thermal efficiency. If the cycle is a refrigeration cycle,
determine its coefficient of performance.

10. An electric motor operating at steady state draws a current of 10 A with a voltage of 220 V. The
output shaft rotates at 1000 RPM with a torque of 16 N m applied to an external load. The rate of heat
transfer from the motor to its surroundings is related to the surface temperature Tb and the ambient
temperature T0 by hA (Tb - T0), where h = 100 W/m2 K, A = 0.195 m2, and T0 = 293 K. Energy
transfers are considered positive in the directions indicated by the arrows on the figure.

(a) Determine the temperature Tb, in K.

(b) For the motor as the system, determine the rate of entropy production, in kW/K.

(c) If the system boundary is located to take in enough of the nearby surroundings for heat transfer to
take place at temperature T0, determine the rate of entropy production, in kW/K, for the enlarged
system.

Problem: 10

11. An insulated cylinder is initially divided into halves by a frictionless, thermally conducting piston.
On one side of the piston is 1 m3 of a gas at 300 K, 2 bar. On the other side is 1 m3 of the same gas at
300 K, 1 bar. The piston is released and equilibrium is attained, with the piston experiencing no
change of state. Employing the ideal gas model for the gas, determine

(a) the final temperature, in K.

(b) the final pressure, in bar.

(c) the amount of entropy produced, in kJ/kg.

12. According to test data, a new type of engine takes in streams of water at 200 oC, 3 bar and 100 oC,
3 bar. The mass flow rate of the higher temperature stream is twice that of the other. A single stream
exits at 3.0 bar with a mass flow rate of 5400 kg/h. There is no significant heat transfer between the
engine and its surroundings, and kinetic and potential energy effects are negligible. For operation at
steady state, determine the maximum theoretical rate that power can be developed, in kW.

13. The figure shows a 30-ohm electrical resistor located in an insulated duct carrying a stream of air.
At steady state, an electric current of 15 amp passes through the resistor, whose temperature remains
constant at 127 oC. The air enters the duct at 15 oC, 1 atm and exits at 25 oC with a negligible change
in pressure. Kinetic and potential energy changes can be ignored.
(a) For the resistor as the system, determine the rate of entropy production, in kW/K.

(b) For a control volume enclosing the air in the duct and the resistor, determine the volumetric flow
rate of the air entering the duct, in m3/s, and rate of entropy production, in kW/K.

Why do the entropy production values of (a) and (b) differ?

Problem: 13

14. Carbon dioxide (CO2) enters a nozzle operating at steady state at 28 bar, 267 oC, and 50 m/s. At
the nozzle exit, the conditions are 1.2 bar, 67 oC, 580 m/s, respectively.

(a) For a control volume enclosing the nozzle only, determine the heat transfer, in kJ, and the change
in specific entropy, in kJ/K, each per kg of carbon dioxide flowing through the nozzle. What
additional information would be required to evaluate the rate of entropy production?

(b) Evaluate the rate of entropy production, in kJ/K per kg of carbon dioxide flowing, for an enlarged
control volume enclosing the nozzle and a portion of its immediate surroundings so that the heat
transfer occurs at the ambient temperature, 25 oC.

15. A counterflow heat exchanger operates at steady state with negligible kinetic and potential energy
effects. In one stream, liquid water enters at 15 oC and exits at 23 oC with a negligible change in
pressure. In the other stream, Refrigerant 22 enters at 12 bar, 90 oC with a mass flow rate of 150 kg/h
and exits at 12 bar, 28 oC. Heat transfer from the outer surface of the heat exchanger can be ignored.
Determine

(a) the mass flow rate of the liquid water stream, in kg/h.

(b) the rate of entropy production within the heat exchanger, in kW/K.

16. Air as an ideal gas flows through the compressor and heat exchanger shown in the figure. A
separate liquid water stream also flows through the heat exchanger. The data given are for
operation at steady state. Stray heat transfer to the surroundings can be neglected, as can all kinetic
and potential energy changes. Determine
(a) the compressor power, in kW, and the mass flow rate of the cooling water, in kg/s.

(b) the rates of entropy production, each in kW/K, for the compressor and heat exchanger.

Problem:16

17. Air enters a 3600-kW turbine operating at steady state with a mass flow rate of 18 kg/s at 800 oC,
3 bar and a velocity of 100 m/s. The air expands adiabatically through the turbine and exits at a
velocity of 150 m/s. The air then enters a diffuser where it is decelerated isentropically to a velocity of
10 m/s and a pressure of 1 bar. Employing the ideal gas model, determine

(a) the pressure and temperature of the air at the turbine exit, in bar and oC, respectively.

(b) the rate of entropy production in the turbine, in kW/K.

(c) Show the processes on a Ts diagram.

18. In a gas turbine operating at steady state, air enters the compressor with a mass flow rate of 5 kg/s
at 0.95 bar and 22 oC and exits at 5.7 bar. The air then passes through a heat exchanger before
entering the turbine at 1100 K, 5.7 bar. Air exits the turbine at 0.95 bar. The compressor and turbine
operate adiabatically and kinetic and potential energy effects can be ignored. Determine the net power
developed by the plant, in kW, if

(a) the compressor and turbine operate without internal irreversibilities.

(b) the compressor and turbine isentropic efficiencies are 82 and 85%, respectively.

19. The figure shows three devices operating at steady state: a pump, a boiler, and a turbine. The
turbine provides the power required to drive the pump and also supplies power to other devices. For
adiabatic operation of the pump and turbine, and ignoring kinetic and potential energy effects,
determine, in kJ per kg of steam flowing
(a) the work required by the pump.

(b) the net work developed by the turbine.

(c) the heat transfer to the boiler.

Problem: 19

20. As shown in the figure, water flows from an elevated reservoir through a hydraulic turbine. The
pipe diameter is constant, and operation is at steady state. Estimate the minimum mass flow rate, in
kg/s, that would be required for a turbine power output of 1 MW. The local acceleration of gravity is
9.8 m/s2.

Problem: 20