Assignment B4: Thermodynamics and engines

The Diesel motor operates differently from a gasoline engine. Instead of an isochoric ignition in the gasoline motor, the ignition in the Diesel motor occurs at constant pressure.

(a) (b)

Figure 2. Diesel engines of ships can be huge (a). The Diesel cycle (b) Is different from the Otto cycle.

A freight ship is equipped with a two stroke diesel engine containing 12 cylinders. Each cylinder has a diameter of 0.8 m and a total length (full stroke) of 2 m. The total power delivered by the engine is 60 MW. The engine operates at 95 revolutions per minute (95 rpm).

The maximum pressure is 120 bar.

The temperature after exhaust of the burned gas is 50 degrees Celsius.

In full operation, the cut-off ratio ?? (= ??3/??2) is 1.86. The minimum volume before combustion is 140 liters.

The heat capacity ratio of the diesel fuel is 1.3 and the calorific value of diesel fuel (= the amount of heat energy released by combustion of 1 kg of diesel) is 45 MJ/kg.

Question:

1. Derive the efficiency of the diesel motor in a similar way as we have done for the four-stroke engine as a function of the variables ?? (the cut-off ratio) and ?? (the compression ratio).

??3 ??1

?? = ?? =

??2 ??2

Show the derivation and document every step. Display the result in signal fractions (so no fractions within fractions).

2. Determine the pressure, volume and temperature at each point (1, 2, 3 and 4) of the thermodynamic cycle.

3. What is the thermal efficiency of the engine?

4. How much diesel fuel per day is needed?

The Watt’s governor

Figure 1. Watt’s governor is a device that can be used to regulate the fuel injection, thereby the power generated by a steam engine.

Consider the steam engine system (figure 1) with the dimensions as given in the figure. Determine the amount of fuel injection (flow rate) as a function of the speed of Watts’ governor. Assume that the flow is laminar, that there are no pressure losses and that the fuel level doesn’t change significantly (large fuel container).

1. [MATLAB] Plot the function of fuel flow rate (l/s) as a function of the angular speed ?? (rpm) in Matlab.

2. Assume that the relation between the inflow of fuel in the boiler as a function of the rotational speed ??(rpm) is given by:

?? = 0.2 ??/?? + 10-5 ?? · ??2

Plot this function in the same graph as in question 2 and determine at what speed the turbine is turning

(??).

3. What can you do to set another working point (this is to change the speed of the turbine)?

Note that this is an example of negative feedback. We will later see that negative feedback loops are also frequently used in electronic circuits. There exists an entire mathematical framework that is concerned with the study of the stability of feedback loops and control systems. This is the field of cybernetics and is applicable to different kinds of systems (mechanical, mechatronics, electronics and biological). Block diagrams are often used and the systems are described by transfer functions. There exists a mathematical framework that is concerned with the stability of such feedback systems.

Basic chemistry

1. What is the molar mass of the copper(II) sulphate CuSO4 ?

2. How many potassium ions are there in 100 g of potassium (K) oxide?

3. How many carbon atoms are in 2 carat diamond? Diamond is pure carbon and one carat is exactly 0.2 g.

4. How many moles of H2 can be theoretically produced from 5 mol of Fe and 6 mol of H2O? The chemical reaction is:

3???? + 4??2?? ? ????3??4 + 4??2.

How much of the initial products is left after reaction?

5. How many millilitres (ml) of 0.75 M NaOH (sodium hydroxide) is needed to completely react with 50 ml of 0.3 M H2SO4 (sulfuric acid)?

??2????4 + 2???????? ? ????2????4 + 2??2??

How much gram sodium salt (????2????4) is produced?

6. Chlorine gas (????2) can be produced by reaction of hydrochloride acid with manganese oxide. What is the volume of chlorine gas is produced by adding 1 g of MnO2 to one litre of 1 M HCl?

??????2 + 4?????? ? ????????2 + ????2 + 2??2??

How much chlorine gas is produced when instead of 1 g MnO2 this time 50 g MnO2 is added?

7. At 500K, 1 mol of ???????? (g) is introduced in a 1 litre container. At equilibrium, 9% of the ???????? is dissociated:

2 ???????? (??) ? 2???? (??) + ????2 (??) What is the equilibrium constant K?

8. If 0.01 mol ??2 and 0.01 mol of ????2 are mixed in a 1 liter container at 750 degrees Celsius, what are the concentrations of all substances (in mol/l) present at equilibrium?

The reaction is ????2 + ??2 ? ???? + ??2?? and the equilibrium constant of the reaction is ?? = 0.771.

What is the partial pressure of ???? in the container at equilibrium?

9. Consider the reaction: ????2 + ??2 ? ???? + ??2?? ??? = 41.1 ????. ??????-1 What would happen if you increase the temperature?

10. Fill in the unknown values for a, ??, ??, ?? in the following reaction:

?? ????3(????4)2 + ?? ??2????4 ? ?? ??3????4 + ?? ????????4

The Diesel motor operates differently from a gasoline engine. Instead of an isochoric ignition in the gasoline motor, the ignition in the Diesel motor occurs at constant pressure.

(a) (b)

Figure 2. Diesel engines of ships can be huge (a). The Diesel cycle (b) Is different from the Otto cycle.

A freight ship is equipped with a two stroke diesel engine containing 12 cylinders. Each cylinder has a diameter of 0.8 m and a total length (full stroke) of 2 m. The total power delivered by the engine is 60 MW. The engine operates at 95 revolutions per minute (95 rpm).

The maximum pressure is 120 bar.

The temperature after exhaust of the burned gas is 50 degrees Celsius.

In full operation, the cut-off ratio ?? (= ??3/??2) is 1.86. The minimum volume before combustion is 140 liters.

The heat capacity ratio of the diesel fuel is 1.3 and the calorific value of diesel fuel (= the amount of heat energy released by combustion of 1 kg of diesel) is 45 MJ/kg.

Question:

1. Derive the efficiency of the diesel motor in a similar way as we have done for the four-stroke engine as a function of the variables ?? (the cut-off ratio) and ?? (the compression ratio).

??3 ??1

?? = ?? =

??2 ??2

Show the derivation and document every step. Display the result in signal fractions (so no fractions within fractions).

2. Determine the pressure, volume and temperature at each point (1, 2, 3 and 4) of the thermodynamic cycle.

3. What is the thermal efficiency of the engine?

4. How much diesel fuel per day is needed?

The Watt’s governor

Figure 1. Watt’s governor is a device that can be used to regulate the fuel injection, thereby the power generated by a steam engine.

Consider the steam engine system (figure 1) with the dimensions as given in the figure. Determine the amount of fuel injection (flow rate) as a function of the speed of Watts’ governor. Assume that the flow is laminar, that there are no pressure losses and that the fuel level doesn’t change significantly (large fuel container).

1. [MATLAB] Plot the function of fuel flow rate (l/s) as a function of the angular speed ?? (rpm) in Matlab.

2. Assume that the relation between the inflow of fuel in the boiler as a function of the rotational speed ??(rpm) is given by:

?? = 0.2 ??/?? + 10-5 ?? · ??2

Plot this function in the same graph as in question 2 and determine at what speed the turbine is turning

(??).

3. What can you do to set another working point (this is to change the speed of the turbine)?

Note that this is an example of negative feedback. We will later see that negative feedback loops are also frequently used in electronic circuits. There exists an entire mathematical framework that is concerned with the study of the stability of feedback loops and control systems. This is the field of cybernetics and is applicable to different kinds of systems (mechanical, mechatronics, electronics and biological). Block diagrams are often used and the systems are described by transfer functions. There exists a mathematical framework that is concerned with the stability of such feedback systems.

Basic chemistry

1. What is the molar mass of the copper(II) sulphate CuSO4 ?

2. How many potassium ions are there in 100 g of potassium (K) oxide?

3. How many carbon atoms are in 2 carat diamond? Diamond is pure carbon and one carat is exactly 0.2 g.

4. How many moles of H2 can be theoretically produced from 5 mol of Fe and 6 mol of H2O? The chemical reaction is:

3???? + 4??2?? ? ????3??4 + 4??2.

How much of the initial products is left after reaction?

5. How many millilitres (ml) of 0.75 M NaOH (sodium hydroxide) is needed to completely react with 50 ml of 0.3 M H2SO4 (sulfuric acid)?

??2????4 + 2???????? ? ????2????4 + 2??2??

How much gram sodium salt (????2????4) is produced?

6. Chlorine gas (????2) can be produced by reaction of hydrochloride acid with manganese oxide. What is the volume of chlorine gas is produced by adding 1 g of MnO2 to one litre of 1 M HCl?

??????2 + 4?????? ? ????????2 + ????2 + 2??2??

How much chlorine gas is produced when instead of 1 g MnO2 this time 50 g MnO2 is added?

7. At 500K, 1 mol of ???????? (g) is introduced in a 1 litre container. At equilibrium, 9% of the ???????? is dissociated:

2 ???????? (??) ? 2???? (??) + ????2 (??) What is the equilibrium constant K?

8. If 0.01 mol ??2 and 0.01 mol of ????2 are mixed in a 1 liter container at 750 degrees Celsius, what are the concentrations of all substances (in mol/l) present at equilibrium?

The reaction is ????2 + ??2 ? ???? + ??2?? and the equilibrium constant of the reaction is ?? = 0.771.

What is the partial pressure of ???? in the container at equilibrium?

9. Consider the reaction: ????2 + ??2 ? ???? + ??2?? ??? = 41.1 ????. ??????-1 What would happen if you increase the temperature?

10. Fill in the unknown values for a, ??, ??, ?? in the following reaction:

?? ????3(????4)2 + ?? ??2????4 ? ?? ??3????4 + ?? ????????4

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