Thermodynamics - Cooling of a boiler caused by a water leak

[thermoPat]

Homework Statement

I want a detailed understanding of the temperature drop of a boiler when water is lost from a leak. I have a temperature recording of a leak that lasted for an hour and caused the temperature to drop from 170^oC to 150^oC when 2.55m³ of water was lost. I have tried using steam tables to understand the temperature drop but my calculations show a slight temperature increase instead of decrease, so obviously my approach is wrong.

I would appreciate some guidance on the correct way to calculate the theoretical temperature drop. This is the approach that I tried:

Homework Equations

Steam table:
https://www.physicsforums.com/attachments/123054

The Attempt at a Solution

Assumptions:

• 1m³ of water weighs 1 tonne at the temperatures under consideration.
• No energy was added to or removed from the system other than the energy in the lost water and work done to push it out of the leak hole.

Initial conditions:

• 6.55m³ of water and 2.35m³ of saturated steam at 8 bar (116 psi).
• From steam table www.engineeringtoolbox.com/saturated-steam-properties-d_457.html steam density is 4.16kg/m³, so mass of steam is 2.35 * 4.16 = 9.78kg
• Enthalpy of water + steam is
  6.55 * 721 + 9.78 * 2.77 = 4723 + 27 = 4750MJ

Energy lost in the 2.55t of hot water that leaked out:

Water was cooling during the leak. Assume average temperature was 160. At that temp the specific enthalpy is 676, so total lost enthalpy is 1720MJ.​

Work done in pushing the water out of the hole:

Average pressure is 6.5bar = 0.65MP = 0.65 MN/m³
0.62MN moving 2.55m = 1.6MJ ------- Negligible​

End conditions:

After the water leak the enthalpy of the system is the original enthalpy less the losses:
4750 - 1720 - 2 = 3028MJ.
To find the temperature of the system we search the steam table for the temperature of a system with an enthapy of 3028MJ and having 4t of water and 4.9m³ of steam.
We find this at 175^oC, which is hotter than the initial temperature !??
Something is wrong.​

 

[Chestermiller]

If I used the density of liquid water from the steam tables, I get an initial internal energy of 4230 MJ. For the liquid water exiting the tank, I get 2314 kg exiting (assuming 160 C). The decrease in internal energy in the tank is estimated to be 1563 MJ, so my estimates of the final mass and internal energy are 3560 kg and 2667 MJ. The volume of the tank is the same 8.9 m^3, so the final average specific volume is estimated to be 0.00250 m^3/kg.

If I assume a final temperature of 170 C, this gives me 32.7 kg of vapor and 3527 kg of liquid. The estimated internal energy in this case would be 2312 MJ, which doesn't match the 2667 MJ. So, 170 C may be too low a temperature.

 

[Chestermiller]

There is another way of doing this that will probably give you a much more accurate approximation. If you focus on the 3560 kg of mass that remains in the tank after the leakage has occurred, provided the leakage was slow, this mass will have experienced an adiabatic reversible expansion from its original volume to its final volume of 8.9 m^3. Therefore, the entropy of this water/steam combination after the leakage should be equal to its initial entropy (i.e., the mass that has remained). Once the final state in the tank in the end has been established, the average enthalpy of the mass of liquid that leaked from the tank can be determined, as well as its temperature.

 

[Chestermiller]

I carried out the calculation I described in the previous post and found that the final temperature in the tank would be about 168.5 degrees. I am prepared to present the details if you are interested. If the temperature in the tank were lower, I would expect the temperature of the fluid that leaked from the tank would be higher. This is not consistent with the observation that the leaked fluid was actually lower, at about 150 C.

 

[thermoPat]

I carried out the calculation I described in the previous post and found that the final temperature in the tank would be about 168.5 degrees. I am prepared to present the details if you are interested. If the temperature in the tank were lower, I would expect the temperature of the fluid that leaked from the tank would be higher. This is not consistent with the observation that the leaked fluid was actually lower, at about 150 C.

Thanks very much Chestermiller.
By the way could you see "attachment 123054" that was part of my question? It should be the steam table I was using, but when I click on the link now I get an error -- strange. It would be convenient if we used the same steam table - www.engineeringtoolbox.com/saturated-steam-properties-d_457.html

I did not accept my analysis because it did not match what actually happened in practice. I have a graph that shows the temperature suddenly start dropping at the rate of about 20 degrees per hour. The boiler was unattended at the time but when I came in I confirmed the leak and the dangerously low water level and dramatically dropped temperature. My analysis above indicates that there should have been a slight temperature rise, which is obviously illogical and wrong.
Your analysis predicts a tiny temperature drop (1.5 degrees), which, while better than my attempt, is still very far from what actually happened. There must be something we are both missing.
Thanks again for your interest.

 

[Chestermiller]

Thanks very much Chestermiller.
By the way could you see "attachment 123054" that was part of my question? It should be the steam table I was using, but when I click on the link now I get an error -- strange. It would be convenient if we used the same steam table - www.engineeringtoolbox.com/saturated-steam-properties-d_457.html

I did not accept my analysis because it did not match what actually happened in practice. I have a graph that shows the temperature suddenly start dropping at the rate of about 20 degrees per hour. The boiler was unattended at the time but when I came in I confirmed the leak and the dangerously low water level and dramatically dropped temperature. My analysis above indicates that there should have been a slight temperature rise, which is obviously illogical and wrong.
Your analysis predicts a tiny temperature drop (1.5 degrees), which, while better than my attempt, is still very far from what actually happened. There must be something we are both missing.
Thanks again for your interest.

I couldn't see the diagram, but I have my own steam tables.

Were you measuring the temperature inside the tank, or were you measuring the temperature of the fluid that leaked out?

With all that liquid in there, it doesn't seem that evaporating a small amount to form vapor would be enough to cause the temperature to drop much.

 

[thermoPat]

I couldn't see the diagram, but I have my own steam tables.

Were you measuring the temperature inside the tank, or were you measuring the temperature of the fluid that leaked out?

With all that liquid in there, it doesn't seem that evaporating a small amount to form vapor would be enough to cause the temperature to drop much.

Thanks for your response.
My measurement was by thermocouple on the outside of the boiler under the insulation, at the level of the normal later level. So it wasn't the exact water temperature and has a bit of time delay, but it works well enough.

Yes I also thought about the energy required to evaporate the extra 13kg of steam to fill the enlarged steam space, but I calculated that that would drop the temperature by less than 2 degrees.
There must be something else ...

 

[Chestermiller]

Thanks for your response.
My measurement was by thermocouple on the outside of the boiler under the insulation, at the level of the normal later level. So it wasn't the exact water temperature and has a bit of time delay, but it works well enough.

Yes I also thought about the energy required to evaporate the extra 13kg of steam to fill the enlarged steam space, but I calculated that that would drop the temperature by less than 2 degrees.
There must be something else ...

Is there a control system which stops supplying heat to the boiler if the water level drops?

 

[thermoPat]

Is there a control system which stops supplying heat to the boiler if the water level drops?

It is a heritage coal-fired fire-tube boiler and was being warm up overnight when one of the tubes developed a leak. The fire was virtually out when the leak occurred -- the warming curve had flattened out with the small fire compensating for the heat loss through the boiler's insulation.
There is no circulation pump so the bottom of the boiler is a bit cooler than the top, so the average water temperature is a bit below the recorded temperature.

This question is about the first hour of the leak, when the rate of temperature decrease was high. But just for interest I will continue the story ... After an hour the the water level fell below the level of the leak, so steam leaked out instead of water. Water in its gaseous state (steam) leaks out of a hole 25 times slower than water in its liquid state, and this could be seen on the temperature graph -- the rate of temperature suddenly decreased dramatically.

 

[Chestermiller]

It is a heritage coal-fired fire-tube boiler and was being warm up overnight when one of the tubes developed a leak. The fire was virtually out when the leak occurred -- the warming curve had flattened out with the small fire compensating for the heat loss through the boiler's insulation.
There is no circulation pump so the bottom of the boiler is a bit cooler than the top, so the average water temperature is a bit below the recorded temperature.

This question is about the first hour of the leak, when the rate of temperature decrease was high. But just for interest I will continue the story ... After an hour the the water level fell below the level of the leak, so steam leaked out instead of water. Water in its gaseous state (steam) leaks out of a hole 25 times slower than water in its liquid state, and this could be seen on the temperature graph -- the rate of temperature suddenly decreased dramatically.

OK. So there could have been heat loss from the boiler during the first hour of the leak, because the boiler was no longer receiving heat. I'm going to calculate the apparent amount of heat loss using the open system version (control volume) of the first law of thermodynamics:
$$\Delta U=Q-m_{out}\bar{h}_{out}$$where $m_{out}$ is the amount of leakage (assumed to be ~2314 kg) and $\bar{h}_{out}$ is the average enthalpy per kg of the leakage (assuming an average temperature of 160 C). To get $\Delta U$, I will assume that the temperature of the saturated tank contents drops from 170 C to 150 C, while the mass decreases by the amount of leakage. The idea is to see if the calculated Q makes sense.

 

[Chestermiller]

I calculate a heat loss on the order of about 500 MJ in the 1 hr. That seems awfully high. What is the surface area of the boiler available for heat transfer?

 

[thermoPat]

I calculate a heat loss on the order of about 500 MJ in the 1 hr. That seems awfully high. What is the surface area of the boiler available for heat transfer?

It is a horizontal cylinder 3m long and 2.5m diam with the top 3/4 covered with insulating material. Inside the boiler there are 2 furnaces, 2 combustion chambers and 5 rows of fire tubes.
In that hour a little heat would have been coming from the remains of the fire and some would have been lost from the outside surface of the boiler.

 

[Chestermiller]

It is a horizontal cylinder 3m long and 2.5m diam with the top 3/4 covered with insulating material. Inside the boiler there are 2 furnaces, 2 combustion chambers and 5 rows of fire tubes.
In that hour a little heat would have been coming from the remains of the fire and some would have been lost from the outside surface of the boiler.

Well I assumed that the heat transfer area is about 20 m^2 and the driving force for heat transfer is 160 - 20 = 140 C. For a heat loss rate of 500 MJ/hr, this indicates a heat transfer coefficient of about 50 W/m^2 C. This is much higher than one would expect. So it still doesn't seem that the observed drop in temperature can be reconciled.

 

[thermoPat]

Very puzzling. Thanks for your interest.

 

[Nidum]

Do you have a record of the boiler pressure gauge readings during this event ?

How did you estimate the amount of escaped water ?

 

[thermoPat]

Do you have a record of the boiler pressure gauge readings during this event ?

How did you estimate the amount of escaped water ?

No recorded pressure, but the steam is saturated so the pressure can be calculated from the temperature.

I made a rough mathematical model of the boiler by measuring everything inside the boiler and recording their vertical positions. In a spreadsheet I can input a water level and determine the corresponding volume of water and steam in the boiler. I am reasonable confident about its accuracy.

I knew the water level at the start of the warm-up because it was visible in the gauge glass. The water level after an hour of leaking was at the level of the hole, which was was on the bottom of a tube in the third row from the top, the height of which I know from the mathematical model.

There are some inaccuracies, like not accounting for the rise in level in the gauge glass as the water heats up and expands more than the boiler shell expands, so I wouldn't be worried if the theory deviated a bit from what was recorded, but so far the theory is way way out.

 

[thermoPat]

I have an explanation. The leaked water entering the fire tube could have flowed in two directions - to the smoke box and onto the floor, or "backwards" into the combustion chamber and then overflowed into the furnace and then onto the floor. The water in the combustion chamber would have boiled because of the hot boiler water surrounding the combustion chamber. If 1/4t of leaked water per hour was converted to steam it would exactly account for the temperature drop observed on the graph.
Water flowing into the combustion chamber is highly feasible because the leak was very near that end of the tube.