Solving Steam Entering Turbine Problem

[gfd43tg]

Homework Statement

Exhaust gas at 400°C and 1 bar from internal-combustion engines flows at the rate of 125 mol s-1 into a waste-heat boiler where saturated steam is generated at a pressure of 1,200 kPa. Water enters the boiler at 20°C (T_σ), and the exhaust gases are cooled to within 10°C of the steam temperature. The heat capacity of the exhaust gases is C_P⁄R=3.34+1.12×〖10〗^(-3) T/K. The steam flows into an adiabatic turbine and exhausts at a pressure of 25 kPa. If the turbine efficiency η is 72%,

(a) What is (W_s ) ̇, the power output of the turbine?
(b) What is the thermodynamic efficiency of the boiler/turbine combination?
(c) Determine (S_G ) ̇ for the boiler and for the turbine.
(d) Express (W_lost ) ̇(boiler) and (W_lost ) ̇(turbine) as fractions of |(W_ideal ) ̇|, the ideal work of the process.

Homework Equations

The Attempt at a Solution

Right now I am only working on part (a)

For this problem, I have calculated the enthalpy change of the steam both entering and leaving the turbine using the steam table, as well as the enthalpy of the water entering the boiler. I calculated the change in enthalpy from entering the boiler to leaving the boiler, as well as entering the turbine to leaving the turbine.

I was able to calculate a work per unit mass of the turbine, but the question wants a rate of work term that removes the per mass basis. My next thought is to find the mass flow rate of the water. I believe that the enthalpy change of the gas entering and leaving the boiler is equal to the enthalpy change of the water entering and leaving the boiler.

However, on the bottom I am stuck in my calculation because I don't know the temperature of the gas leaving the boiler. This is troubling to me that the problem says that the gas comes to within ''10° C of the steam''. I don't know if that is supposed to help me, I need an exact temperature to calculate the enthalpy change of the gas. So I'm stuck with a mass flow rate of water that is unknown, and a temperature of gas leaving the boiler that is unknown.

 

[Chestermiller]

I haven't looked over your calculations, but the problem says that the steam is saturated at 1200 kPa. From this, you know its temperature. You also know that the exhaust gases are cooled to a temperature 10C higher than the saturation temperature of the steam. So you know the heat load.

Chet

 

[gfd43tg]

Oh, I interpreted the question to mean ''the temperature of the exit gas is anywhere between 0-10 C of the steam''

 

[Chestermiller]

Oh, I interpreted the question to mean ''the temperature of the exit gas is anywhere between 0-10 C of the steam''

I guess it could be interpreted that way, but I'm sure that's not what they meant.

Chet

 

[gfd43tg]

I mean, you could argue that it can be 10 above or 10 below the steam temperature...the wording is a bit ambiguous. Anyways, I took the temperature to be 10 above the steam temperature, so I was able to calculate the mass flow rate of water as .310 kg/s and put into my work rate term. I'll begin working on part (b) now.

 

[Chestermiller]

I mean, you could argue that it can be 10 above or 10 below the steam temperature...the wording is a bit ambiguous. Anyways, I took the temperature to be 10 above the steam temperature, so I was able to calculate the mass flow rate of water as .310 kg/s and put into my work rate term. I'll begin working on part (b) now.

Excellent. Incidentally, the exhaust gas couldn't get below the steam temperature, because it is being cooled by the water and steam.

Chet

 

[gfd43tg]

Oh I see. I just took a peek at the solution, and I made an error apparently in assuming the steam leaving the turbine is saturated vapor. They are saying there is some quality of steam, and I am wondering how I know that is the case, based off the question, and how I can find its quality. Is it one of those things where if its not explicitly stated to be saturated, one should assume there is some quality?

The mass flow rate is correct though, just my isentropic work per unit mass is wrong.


Edit: solved part (a)

 

[gfd43tg]

I'm running into a bit of a problem in part (b).

My initial thought was to do the general energy balance, with the boiler and turbine as my control volume, and calculate Q. I did this and got about 1.1 kJ for a basis of 1 second.

Then knowing η = W/Qh, the efficiency would be more than 1, which is impossible, using my calculated W = 135.66 kJ in part (a). So then I go back to the drawing board. I am not sure why this definition is wrong other than it would give an unreasonable answer.

So then I peek at the solution manual, and they do W/W_ideal = η

Okay, so then I proceed to attempt to find W_ideal using the equation

W_ideal = ΔH - T_σ(ΔS)

So now where I am stuck is that I don't know how to find the change in entropy of the gas from the combustion engine. I am treating it as a real gas, so I am not certain how to calculate the entropy change of a real gas. The steam I should be able to do with a steam table, but right now what's holding me back is the fuel gas entropy change. Since I don't know what the gas is, I can't find a reduced pressure and temperature to find the residual entropies, then go about it the usual way.

 

[Chestermiller]

I'm running into a bit of a problem in part (b).

My initial thought was to do the general energy balance, with the boiler and turbine as my control volume, and calculate Q. I did this and got about 1.1 kJ for a basis of 1 second.

Then knowing η = W/Qh, the efficiency would be more than 1, which is impossible, using my calculated W = 135.66 kJ in part (a). So then I go back to the drawing board. I am not sure why this definition is wrong other than it would give an unreasonable answer.

So then I peek at the solution manual, and they do W/W_ideal = η

Okay, so then I proceed to attempt to find W_ideal using the equation

W_ideal = ΔH - T_σ(ΔS)

So now where I am stuck is that I don't know how to find the change in entropy of the gas from the combustion engine. I am treating it as a real gas, so I am not certain how to calculate the entropy change of a real gas. The steam I should be able to do with a steam table, but right now what's holding me back is the fuel gas entropy change. Since I don't know what the gas is, I can't find a reduced pressure and temperature to find the residual entropies, then go about it the usual way.

I need to reread what you have written to better answer, but, regarding the exhaust gas, it is at 1 atm, so it's OK to treat it as an ideal gas. Get back to you later.

Chet

 

[gfd43tg]

Okay, I now can see why to use it as an ideal gas. However, can I assume that its a constant pressure process, and the pressure of the fuel gas exiting the boiler is equal to 1 bar? If so, why is the assumption valid?

 

[SteamKing]

The boiler only extracts heat from the exhaust gas of the IC engines. If the exhaust gas were at a pressure less than 1 bar, it could not exhaust naturally from the boiler; it would have to be pumped into the atmosphere.

 

[gfd43tg]

by that, then if the exit and entrance are both at 1 bar, why would the gas move at all?

 

[Chestermiller]

by that, then if the exit and entrance are both at 1 bar, why would the gas move at all?

These are good questions. The answer is that they expect you to assume that the pressure change, although not exactly zero, is very small. Since you are given no information on the pipe diameter, the length of the pipe, or the composition of the exhaust gas, you have no way of calculating the exact pressure change. So you are stuck assuming that it is small.

Chet

 

[Chestermiller]

Oh I see. I just took a peek at the solution, and I made an error apparently in assuming the steam leaving the turbine is saturated vapor. They are saying there is some quality of steam, and I am wondering how I know that is the case, based off the question, and how I can find its quality. Is it one of those things where if its not explicitly stated to be saturated, one should assume there is some quality?

The mass flow rate is correct though, just my isentropic work per unit mass is wrong.


Edit: solved part (a)

If you follow an isentrope for the steam from beginning to end of the turbine, the enthalpy of the steam must be decreasing because shaft work is being done. Just follow the isentrope down to 25 kPa (I don't know why the exit pressure is less than atmospheric, but that's another issue) at the initial entropy and see where it lands. It should lie between that of the saturated liquid and that of the saturated vapor at 25 kPa. That's how you know that there is some quality.

Chet

 

[Chestermiller]

I'm running into a bit of a problem in part (b).

My initial thought was to do the general energy balance, with the boiler and turbine as my control volume, and calculate Q. I did this and got about 1.1 kJ for a basis of 1 second.

Then knowing η = W/Qh, the efficiency would be more than 1, which is impossible, using my calculated W = 135.66 kJ in part (a). So then I go back to the drawing board. I am not sure why this definition is wrong other than it would give an unreasonable answer.

Was that 136 kJ/s the heat transfer rate from the exhaust gas to the steam? If so, that cancels out in the overall energy balance. All you really need to do to get the ideal shaft work in the turbine is to subtract the initial enthalpy of the steam entering the turbine from the final enthalpy of the steam exiting the turbine. You can get this from your steam tables.

Chet

 

[SteamKing]

If you follow an isentrope for the steam from beginning to end of the turbine, the enthalpy of the steam must be decreasing because shaft work is being done. Just follow the isentrope down to 25 kPa (I don't know why the exit pressure is less than atmospheric, but that's another issue) at the initial entropy and see where it lands. It should lie between that of the saturated liquid and that of the saturated vapor at 25 kPa. That's how you know that there is some quality.

Chet

I think the assumption here is that the steam cycle is closed and the turbine exhaust steam is condensed and returned to the boiler. The steam would be expanded in the turbine below ambient atmospheric pressure to extract as much work as possible, but not so low that the moisture content would cause erosion of the turbine blades.