Ask Me Help Desk - Gas laws and heat engines 2

number of cylinders = 6
maximum volume within each cylinder = 1.10 x 10-3 m3
compression ratio = 16 to 1
mass of engine = 757kg
mass-to-power ratio = 4.85 kg kW-1
power-to-volume ratio = 225 kW m-3
power rating = 165kW at 3750 r.p.m
maximum torque = 488Nm at 2500 r.p.m

pressure, volume and temp values for this engne at A, B, C and D are shown in the table 5.6. the application of the 1st law of thermodynamics to each part of the cycle gives the details shown in table 5.7.

Table 5.6

P/Pa V/m3 T/K
A 1.00x10^5 11.0x10^-4 300
B 48.5x10^5 0.69x10^-4 910
C 48.5x10^5 1.51x10^-4 2000
D 3.01x10^5 11.0x10^-4 904

Table 5.7

work done on gas/J heat supplie to gas/J increase on internal energy/J
A - B 560 0 560
B - C -400 1400 1000
C - D -1010 0 -1010
D - A 0 -550 - 550

1. universal gas constant R is 8.31 J mol-1 K-1. use this fact to find how many moles of gas are present inonecylinder during a cycle.

2. why are the mass to power ratio and the power-to-volume ratio important for a designer who wishes to incorporate an engine such as this into a vehicle?

3. how is the value 0.69 x 10^-4 m3 obtained for the volume of gas at B?

4. the molar heat capacity at constant volume for a gas such as air is 20.8 J mol-1 K-1. show that heating the gas in a cylinder in this engine requires 0.918 J K-1. hence show that the increase in the internal energy inmoving from D to A is 550 J.

5. explain why the increase in internal energy in moving from A to B is 560 J.

6. calculate the net work done during the one cycle shown on the indicator diagram, the heat energy supplied during the cycle and hence the theoretical efficiency.

7. calculate the net work actually done per cycle when the engine is giving 1565 kW output. The engine has 240 injections of fuel per second at this speed.

Now, I think it's your turn. Give it a try and post your results. I'll let you know if you did it right.

Quote:

Originally Posted by Perito

Now, I think it's your turn. Give it a try and post your results. I'll let you know if you did it right.

I do not understand this work that is why I asked for your help, your answers and explanations helped me a lot to understand. This is part of my assignments which had to be in for last week in order for me to carry on with my 2nd year of college. Please can you help me. I do not no how to work these questions out.

Quote:

1. universal gas constant R is 8.31 J mol-1 K-1. use this fact to find how many moles of gas are present in one cylinder during a cycle.

The ideal gas law is $PV=nRT$ where P=pressure, V = volume, n = number of moles (what you're trying to solve for), R is given to you, and T is the absolute temperature (in Kelvins). You need to find P, V, and T in order to fine n.

The compression ratio is 16 to 1. You assume the 1 is atmospheric pressure. Solve for the pressure.

$CR = \frac {16}{1} = \frac {x\,atm}{1\,atm}$ Y

You should be able to find V and T at the maximum compression from table 5.6.

Quote:

2. why are the mass to power ratio and the power-to-volume ratio important for a designer who wishes to incorporate an engine such as this into a vehicle?

The ratio mass/power is important because power is required to move the mass (especially up hill and down hill. I'm sure you can think of other reasons.

Quote:

3. how is the value 0.69 x 10^-4 m3 obtained for the volume of gas at B?

Since this is a volume, I assume that this is from the position of the piston in the cylinder. You could calculate that if you had a knowledge of the diameter of the cylinder and the position of the piston in the cylinder.

Quote:

4. the molar heat capacity at constant volume for a gas such as air is 20.8 J mol-1 K-1. show that heating the gas in a cylinder in this engine requires 0.918 J K-1. hence show that the increase in the internal energy in moving from D to A is 550 J.

You calculated the number of moles in #1. $20.8\,\frac {J}{mol\,K}$

multiply that by the number of moles and you should have the answer, if I understand the question correctly.

Quote:

5. explain why the increase in internal energy in moving from A to B is 560 J.

The table shows the work done on the gas. This work results in increased internal energy. Just calculate it.

Quote:

6. calculate the net work done during the one cycle shown on the indicator diagram, the heat energy supplied during the cycle and hence the theoretical efficiency.

I don't see the diagram, so I might not get this exactly. Basically, you have to calculate the work done on the system and the work done by the system. Subtract them to get the theoretical efficiency.

Quote:

7. calculate the net work actually done per cycle when the engine is giving 1565 kW output. The engine has 240 injections of fuel per second at this speed.

This is similar to #6, except you have information on actual output. I don't remember seeing anything in your data about the fuel, though chemically this could be used to calculate input enthalpy.

Hope this helps. I really can't do all the problems for you. That's against the rules.

Quote:

Originally Posted by Perito

The ideal gas law is $PV=nRT$ where P=pressure, V = volume, n = number of moles (what you're trying to solve for), R is given to you, and T is the absolute temperature (in Kelvins). You need to find P, V, and T in order to fine n.

The compression ratio is 16 to 1. You assume the 1 is atmospheric pressure. Solve for the pressure.

$CR = \frac {16}{1} = \frac {x\,atm}{1\,atm}$ Y

You should be able to find V and T at the maximum compression from table 5.6.

i need more help on this. how do i find V and T from table 5.6. and how do i find out how many moles of gas are present in one cylinder during a cycle. please can you explain it to me further.

The ratio mass/power is important because power is required to move the mass (especially up hill and down hill. I'm sure you can think of other reasons.

Since this is a volume, I assume that this is from the position of the piston in the cylinder. You could calculate that if you had a knowledge of the diameter of the cylinder and the position of the piston in the cylinder.

You calculated the number of moles in #1. $20.8\,\frac {J}{mol\,K}$

multiply that by the number of moles and you should have the answer, if I understand the question correctly.

what is the number of moles and what do i multiply it by, i am really confused about all this. can you please tell me.

The table shows the work done on the gas. This work results in increased internal energy. Just calculate it.

how would i do this, would you be able to show me please. i cannot find the work one on the gas which you say is provided on the table.

I don't see the diagram, so I might not get this exactly. Basically, you have to calculate the work done on the system and the work done by the system. Subtract them to get the theoretical efficiency.

This is similar to #6, except you have information on actual output. I don't remember seeing anything in your data about the fuel, though chemically this could be used to calculate input enthalpy.

Hope this helps. I really can't do all the problems for you. That's against the rules.

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[QUOTE=Nargis786;1794882]
Soory about that but I still need help on the questions I gave I do not understand it, can you help me please

Which question, and what don't you understand about it? Let's take them one at a time.

Quote:

Originally Posted by Perito

Which question, and what don't you understand about it? Let's take them one at a time.

1. universal gas constant R is 8.31 J mol-1 K-1. use this fact to find how many moles of gas are present in one cylinder during a cycle.

The ideal gas law is where P=pressure, V = volume, n = number of moles (what you're trying to solve for), R is given to you, and T is the absolute temperature (in Kelvins). You need to find P, V, and T in order to fine n.

The compression ratio is 16 to 1. You assume the 1 is atmospheric pressure. Solve for the pressure.

Y

You should be able to find V and T at the maximum compression from table 5.6.

I still do not get this question

OK.

$PV=nRT$

What is the pressure? Well, the compression ratio is 16.

$\frac {Uncompressed\,volume}{Compressed\,volume} = 16$

Since pressure is inversely proportional to volume

$\frac {Compressed\,pressure}{Uncompressed\,pressure} = 16$

In an automobile engine, in the uncompressed state, valves are open to the atmosphere. Therefore, the pressure is 1 atm.

$\frac {Compressed\,pressure}{1\, atm} = 16$

Compressed pressure = 16 atm. This is P. You can convert it to other pressure units, if desired.

What is the volume?

From table 5.6, the volume at maximum compression is $1.51 \times 10^{-4}\,m^3$ That's your volume.

From table 5.6, the temperature is 2000 Kelvins. That's your temperature

The universal gas constant is 8.31 J mol-1 K-1. You can convert Joules to m^3 atm:

$1\, Joule = 9.86923267 \times 10{-6} \, m^3\, atms$

Now the ideal gas equation becomes

$16\,atm \times 1.51 \times 10^{-4}\,m^3 = n \, 8.31 \frac {Joules}{mol\,K} \times \frac {9.869 \times 10^{-6}\,m^3\,atm}{Joule} 2000 K$

Solve for n. The units remaining will be "mol".

Quote:

Originally Posted by Perito

OK.

$PV=nRT$

What is the pressure? Well, the compression ratio is 16.

$\frac {Uncompressed\,volume}{Compressed\,volume} = 16$

Since pressure is inversely proportional to volume

$\frac {Compressed\,pressure}{Uncompressed\,pressure} = 16$

In an automobile engine, in the uncompressed state, valves are open to the atmosphere. Therefore, the pressure is 1 atm.

$\frac {Compressed\,pressure}{1\, atm} = 16$

Compressed pressure = 16 atm. This is P. You can convert it to other pressure units, if desired.

What is the volume?

From table 5.6, the volume at maximum compression is $1.51 \times 10^{-4}\,m^3$ That's your volume.

From table 5.6, the temperature is 2000 Kelvins. That's your temperature

The universal gas constant is 8.31 J mol-1 K-1. You can convert Joules to m^3 atm:

$1\, Joule = 9.86923267 \times 10{-6} \, m^3\, atms$

Now the ideal gas equation becomes

$16\,atm \times 1.51 \times 10^{-4}\,m^3 = n \, 8.31 \frac {Joules}{mol\,K} \times \frac {9.869 \times 10^{-6}\,m^3\,atm}{Joule} 2000 K$

Solve for n. The units remaining will be "mol".

so using that equation what would the answer be

Now, come on. You can calculate that

$\left( \frac {16\, \times\, 1.51 \times 10^{-4}}{8.31\, \times \,9.869 \times 10^{-6} \,\times\, 2000} \right) = n$

Pull out your calculator

I get 0.0147 moles.

Quote:

Originally Posted by Perito

Now, come on. You can calculate that

$\left( \frac {16\, \times\, 1.51 \times 10^{-4}}{8.31\, \times \,9.869 \times 10^{-6} \,\times\, 2000} \right) = n$

Pull out your calculator

I get 0.0147 moles.

yep thank you i got that as well, i have now understood it. My other question is:

how is the value 0.69 x 10^-4 m3 obtained for the volume of gas at B?

Quote:

maximum volume within each cylinder = 1.10 x 10-3 m3
compression ratio = 16 to 1

The maximum volume will be when there is no compression. When it's compressed to its maximum, you'll have a volume of:

$\frac {1.10 x 10^{-3} m^3}{16} = 6.875 \times 10^{-5}\,m^3 \approx 0.69 \times 10^{-4}$

so that's how they figured that out.

Quote:

Originally Posted by Perito

The maximum volume will be when there is no compression. When it's compressed to its maximum, you'll have a volume of:

$\frac {1.10 x 10^{-3} m^3}{16} = 6.875 \times 10^{-5}\,m^3 \approx 0.69 \times 10^{-4}$

so that's how they figured that out.

thank you very much that helped a lot, i would also like help on:

4. the molar heat capacity at constant volume for a gas such as air is 20.8 J mol-1 K-1. show that heating the gas in a cylinder in this engine requires 0.918 J K-1. hence show that the increase in the internal energy inmoving from D to A is 550 J.

5. explain why the increase in internal energy in moving from A to B is 560 J.