Problems
Tank and Contents
A well-insulated copper tank of mass 12 kg at 27 ◦C is filled with 4 litres of water at 50 ◦C. The tank is heated with a 1 kW resistance heater for 2 1 minutes, and then left alone. Determine the temperature of the system after equilibrium is established. Is the process reversible or irreversible? For copper: ρ = 8.9 kg , cp = 0.386
Cooling Process
A 0.5 m3 block of steel (ρ = 7.83 kg , cp = 0.5 kJ ) initially at 250 ◦C is heated with a constant rate of Q˙ block’s temperature is 600 ◦C? = 50 kW. How long does it take until the
Equilibration of Temperature
To warm the water in your bathtub, you decide to heat it by throwing a block of hot iron into the water. When your bathtub holds 150 litres of water initially at 20 ◦C, and you can heat the iron to 400 ◦C, what mass should the iron block have so that you can have a bath at 33 ◦C? Is the process reversible or irreversible? Assume no heat loss to anywhere, and no boiling, evaporation etc.
Specific heats: cw = 4.18 kJ , ciron = 0.450 kJ .
Irreversible Expansion
An ideal gas is confined to one side of a rigid, insulated (= no heat transfer, adiabatic) container, divided by a partition. The other side is initially evacuated. The initial state of the gas is p1 = 2 bar, T1 = 400 K, V1 = 0.02 m3. When the partition is removed, the gas expands to fill the entire container and achieves a final equilibrium pressure of 1.5 bar. Determine the volume of the container.
Stirring of a Liquid
A thermally insulated 2 litre tank is filled with mercury, which is stirred. When the stirring power is 200 W, how long does it take to raise the temperature of the mercury by 10 ◦C? Is the process reversible or irreversible?
Kneading of a Pizza Dough
A high quality kitchen mixer has a 575 W electric motor. Good pizza dough should be kneaded for about 10 minutes. When 2 kg of dough is kneaded in an adiabatically insulated container, and its initial temperature was 20 ◦C, what temperature will the dough have after kneading?
Assume specific heat of dough as c = 2.73 kJ .
Measurement of Specific Heat
To measure the specific heat of light oil (incompressible liquid, mass density 0.91 kg ) two litres of oil are stirred in a well-insulated container for 12.5 minutes. The stirrer consumes a power of 75 W, and it is observed that the temperature rises from 23 ◦C to 40 ◦C. Ignore kinetic and potential energies. and determine the specific heat of the oil.
Ice Cream Maker
An ice maker stirs 5 kg of a fruit-cream-air mixture (ρ = 570 kg , cp = 1.7 kJ ). The electric motor of the stirrer consumes 575 W of power. It is
observed that after 10 minutes the temperature of the ice cream has dropped from T1 = −2 ◦C to T2 = −18 ◦C. Determine the cooling rate of the ice cream maker.
Heating of a Room
A room of 300 sq.ft. area and 8 ft height is to be maintained at a constant temperature of 68 ◦F while the outside temperature is 32 ◦F. The heat transfer rate to the outside is given by Newton’s law of cooling, Q˙ = α (T − T0) with α = 25 W .
1. Compute the heating power required to maintain the temperature constant.
2. When the heating power is doubled, how long does it take to heat the room from 68 ◦F to 77 ◦F?
Convert all results to SI units.
Isobaric Heating of an Ideal Gas
0.5 kg hydrogen gas (H2) are enclosed in a piston-cylinder system at 22 ◦C, 3 atm. In a reversible isobaric process (constant pressure), the hydrogen doubles its volume.
Determine:
1. The initial volume of the system, and the work done in the expansion.
2. The temperature at the end of the expansion, and the heat exchange with the surroundings.
Isothermal Compression of an Ideal Gas
10 kg helium are enclosed in a piston-cylinder system at 20 ◦C, 10 bar. In a reversible isothermal process (constant temperature), the helium is compressed to half the original volume. Compute:
1. The initial volume of the system.
2. The work required for compression.
3. The heat exchange with the surroundings.
Ideal Gas with Non-constant Specific Heat
In a series of experiments you have found that for temperatures in (300 K, 900 K), the specific heat at constant volume of air is cv (T ) =
(0.695 + 0.0598 T ) kJ
Make a table with the specific heats cv (T ) and cp (T ), specific internal energy u (T ), and specific enthalpy h (T ) for temperatures in the range of validity. As reference value chose u (300 K) = 215 kJ .
2. 2 kg of air are isobarically heated from 340 K to 860 K. By means of your table, determine the heat supply Q12 and the work W12.
3. Redo the calculation under the assumption that the specific heat can be approximated by its value at 300 K (so that it is constant). Determine the relative error for heat and work.
Work and Heat
A fixed mass m of carbon monoxide (CO) gas at T0 = 30 ◦C is confined in a piston-cylinder system. The gas undergoes a reversible isothermal process (constant temperature), that is the pressure changes according to the relation p = mRT0/V . The initial and final volumes are V1 = 0.1 m3 and V2 = 0.15 m3 and the initial pressure is p1 = 500 kPa.
Consider CO as ideal gas with constant specific heat and molar mass M = kmol . Determine:
1. The mass of CO in the system.
2. The pressure p2 at the end of the process.
3. The total work required for the process. Show the process in a p-V- diagram.
4. The total heat exchange.
Work and Heat
Nitrogen (ideal gas with constant specific heats) undergoes a reversible process in a closed system, where the pressure changes according to the relation p = aV 2 + b. The initial and final volumes are V1 = 0.3 m3 and V2 = 0.1 m3, and the corresponding pressures are p1 = 100 kPa and p2 = 200 kPa; the initial temperature is T1 = 30 ◦C. Determine:
1. The mass of nitrogen in the system.
2. The temperature at the end of the process.
3. The total work required for the process. Show the process in a p-V- diagram.
4. The total heat exchange.
Work and Heat
Helium, initially at temperature T1 = 0 ◦C undergoes a reversible process in a closed system, where the pressure changes according to the relation p = aV 3 + b. The initial and final volumes are V1 = 0.1 m3 and V2 = 0.2 m3, and the corresponding pressures are p1 = 100 kPa and p2 = 40 kPa. For the relevant temperature range helium behaves as an ideal gas. As for all noble gases, its specific heat is constant, cv = 3 R. Determine:
1. The mass of helium in the system.
2. The temperature at the end of the process.
3. The total work required for the process. Show the process in a p-V- diagram.
4. The total heat exchange.
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