## Problems

2.1 10 kg/s of steam (7 = 1.29) at 2 MPa are delivered to an adiabatic turbine (100% efficient). The exhaust steam is at 0.2 MPa and 400 K.

1. What is the intake temperature?

2. What power does the turbine deliver?

2.2 Show that the cylinder and piston experiment of Section 2.15 (with the solid object inside) is reversible, provided the compression is carried out infinitely slowly. Do this numerically. Write a computer program in which compression and expansion take place in suitably small steps and are, in each step, followed by an equalization of temperature between the gas and the solid object within the cylinder.

2.3 Refer to the experiment described in Section 2.10. Show that the work done in lifting the 1-kg mass in two steps (first 2 kg, then 1 kg) is 14.2 J. Show that the 2-kg mass rises 0.444 m. Assume that the steps occur slowly enough so that the gas cooled by the expansion returns to the original temperature after each step.

2.4 Consider 10 m3 (V0) of gas (7 = 1.6) at 105 Pa (p0) and 300 K (To).

1. How many kilomoles, n, of gas are involved?

2. The gas is compressed isothermally to a pressure, pf, of 1 Mpa.

2.2 How much energy was used in the compression?

3. Now, instead of compressing the gas isothermally, start again (from V0, p0, and To) and compress the gas adiabatically to a pressure, p2. The gas will heat up to T2. Next, let it cool down isometrically (i.e., without changing the volume) to T3 = 300 K and a pressure, p3, of 1 MPa. In other words, let the state return to that after the isothermal compression.

3.1 What is the pressure, p2?

3.2 What is the temperature, T2, after the adiabatic compression?

3.3 What is the work done during the adiabatic compression?

3.4 Subtract the heat rejected during the isometric cooling from

If the expansion is adiabatic, the polytropic law is observed and the integral becomes (see Chapter 6)

the work done during the adiabatic compression to obtain the net energy change owing to the process described in Item 3.3.

2.5 When a gas expands, it does an amount of work

Show, by using the definitions of cv and of 7, that this work is equal to the energy needed to raise the temperature of the gas from To to T1 under constant volume conditions.

2.6 The domains in a nonmagnetized ferromagnetic material are randomly oriented; however, when magnetized, these domains are reasonably well aligned. This means, of course, that the magnetized state has a lower total entropy than the nonmagnetized state.

There are materials (gadolinium, Gd, for example) in which this effect is large. At 290 K, polycrystalline gadolinium (atomic mass 157.25, density 7900 kg/m3) has a total entropy of 67.6 kJ K-1 kmole-1 when unmagnetized and 65.6 kJ K-1 kmole-1 when in a 7.5 tesla field.

Assume that 10 kg of Gd are inside an adiabatic container, in a vacuum, at a temperature of 290 K. For simplicity, assume that the heat capacity of the container is negligible. The heat capacity of Gd, at 290 K, is 38.4 kJ K-1 kmole-1.

Estimate the temperature of the gadolinium after a 7.5 T field is applied.

2.7 The French engineer, Guy Negre, invented an "eco-taxi," a low-pollution vehicle. Its energy storage system consists of compressed air tanks that, on demand, operate an engine (it could be a turbine, but in the case of this car, it is a piston device).

There are several problems to be considered. Let us limit ourselves to the turnaround efficiency of the energy storage system. For comparison, consider that a lead-acid battery has a turnaround efficiency of somewhat over 70% and flywheels, more than 90%.

A very modern compressed gas canister can operate at 500 atmospheres.

1. Calculate the energy necessary to compress 1 kilomole of air (7 = 1.4) isothermally from 1 to 500 atmospheres. The temperature is 300 K.

2. One could achieve the same result by compressing the air adiabat-ically and then allowing it to cool back to 300 K. Calculate the energy necessary to accomplish this.

3. The compressed air (at 300 K) is used to drive a turbine (in the French scheme, a piston engine). Assume that the turbine is ideal— isentropic—and it delivers an amount of mechanical energy equal to the change of enthalpy the gas undergoes when expanding. How much energy does 1 kilomole of air deliver when expanding under such conditions?

To solve this problem, follow the steps suggested here. 3.1 Write an equation for the change of enthalpy across the turbine as a function of the input temperature (300 K) and the unknown output temperature.

3.2 Using the polytropic law, find the output temperature as a function of the pressure ratio across the turbine. Assume that the output pressure is 1 atmosphere.

If you do this correctly, you will find that the temperature at the exhaust of the turbine is below the liquefaction point of the gases that make up air. This would interfere with the turbine operation, but in the present problem, disregard this fact.

3.3 Once you have the exhaust temperature, calculate the mechanical energy generated by the turbine.

4. What is the turnaround efficiency of the compressed air energy storage system under the (optimistic) assumptions of this problem. That is, what is the ratio of the recovered energy to the one required to compress the air?

2.8 The cylinder in the picture initially, has, a 1-liter volume and is filled with a given gas at 300 K and 105 Pa. It is perfectly heat insulated and is in a laboratory at sea level. The frictionless piston has no mass, and the piston and cylinder, as well as the 1-ohm electric resistor, installed inside the device have negligible heat capacity.

At the beginning of the experiment, the piston is held in place, so it cannot move.

A 10-amp dc current is applied for 1 second, causing the pressure to rise to 1.5 x 105 pascals. Next, the piston is released and rises.

What is the work done by the piston?

2.9 A metallic cylinder with a 3-cm internal diameter is equipped with a perfectly gas-tight and frictionless piston massing 1 kg. It contains a gas with a y = 1.4. When in a lab at sea level and at a room temperature of 300 K, the bottom of the piston is exactly 10 cm above the closed end of the cylinder. In other words, the gas fills a cylindrical volume 10 cm high and 3 cm in diameter.

1. An additional mass is added to the piston so that the total now masses 10 kg. Clearly, the gas inside will be compressed. Assume that this compression occurs very rapidly but not rapidly enough to be considered an abrupt process. Make an educated guess as to the nature of the compression process (adiabatic, isothermal, etc.).

What is the height of the piston at the end of the compression?

2. A considerable time after the above compression, the piston has settled at a different height. What is that height?

3. Now, repeat 1.1 but arrange for the piston to descend very slowly. Again, make an educated guess as to the nature of the compression process (adiabatic, isothermal, etc.).

When the piston finally settles at an unchanging height, how high will it be above the bottom of the cylinder?

4. Which, if any, of the two processes—Process 1 & 2 and Process 3— is reversible? Demonstrate mathematically the correctness of your answer. In other words, for each case, determine, if removing the excess 9 kg from the piston causes the system to return to the initial state. Again, the expansion in the reversal of Process 1&2 is very fast, and in Process 3, it is very slow.

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