Energy balance on metallurgical furnace

[MexChemE]

Homework Statement

Pure iron sulfide (FeS₂) is fed into an Herreshoff furnace at 25 °C. An 83% of excess air is also fed into the furnace at 25 °C. The solid product consists only of Fe₂O₃. The solid product reaches a temperature of 1832 °C, and the exhaust gases (8.7% SO₂, 9.8% O₂, 81.5% N₂) reach a temperature of 1472 °C. Determine the amount of heat lost (Q < 0) by the furnace due to radiation, per tonne of pyrite loaded.
Given solution: Q = +3,786,595 BTU per tonne of pyrite
(First off, there seems to be an inconsistency between the problem statement and the given solution; I'm not showing all of my procedure and calculations since I got very close numerically to the given solution, what I want to discuss is said inconsistency)
Process block diagram attached

Homework Equations

Q = ΔH (Energy balance on the furnace)
FeS₂ + 2.75 O₂ → 0.5 Fe₂O₃ + 2 SO₂

The Attempt at a Solution

First off, I started by stablishing a basis of M₁ = 1 mol. I performed a mass balance on the furnace in order to determine the rest of the molar flows. These were
M₂ = 23.95 mol
M₃ = 0.5 mol
M₄ = 23.2 mol
It was a pretty straightforward balance, since I'm told the 100% of iron sulfide reacts to yield ferric oxide. Next, I set up the energy balance on the furnace
$$Q = \Delta H = \Delta H_{Rx}^° + \Delta H_3 + \Delta H_4$$
Where
$\Delta H_{Rx}^°$ is the standard enthalpy of the roasting reaction (maximum amount of heat generated in the furnace)
$\Delta H_3$ is the amount of heat needed to raise the solid product's temperature to 1832 °C
$\Delta H_4$ is the amount of heat needed to raise the exhaust gases' temperature to 1472 °C
I got the enthalpies of formation of reactants and products in charts provided by the professor. Molar heat capacities of the products were also found in another set of charts provided by the professor. For the heat capacity of the exhaust gases, I used the heat capacities of each gas and their molar fractions to calculate the average molar heat capacity of the mix. My results were
$$\Delta H_{Rx}^° = -197650 \ cal$$
$$\Delta H_3 = (0.5 \ mol) \left(43.7 \ \frac{cal}{mol \ K} \right)(1832 \ °C - 25 \ °C) = 39482.95 \ cal$$
$$\Delta H_4 = (23.2 \ mol) \left(8.3 \ \frac{cal}{mol \ K} \right)(1472 \ °C - 25 \ °C) = 278634.32 \ cal$$
Next, I calculated Q, but this first result is for 120 grams (1 mol) of iron sulfide loaded. I did some calculations in order to adjust the result for a tonne of sulfide roasted.
$$Q = -197650 \ cal + 39482.95 \ cal + 278634.32 \ cal = \frac{120467.27 \ cal}{120 \ g}$$
$$Q = \frac{478.04 \ BTU}{120 \ g} = 3.98 \ \frac{BTU}{g}$$
$$Q = 3.98 \ \frac{BTU}{g} (1,000,000 \ g) = 3,980,000 \ BTU$$
Now, if the problem statement tells me heat is being lost by the furnace due to radiation, I would expect a negative result of Q, not positive. However, this is the correct result for the problem, even though it doesn't agree with the physical situation. What I'm guessing is that the problem (and the process modeled within) was badly designed by the professor. 83% excess of air seems a lot to me; the exhaust gases alone need a greater amount of the maximum generable heat by the process in order to reach 1472 °C. My understanding is that a real furnace operating under these conditions would need a heat input of 3.9 mega BTU in order to carry out the process, but real furnaces are supposed to release a large amount of heat, not consume it. What I'd do to fix this inconsistency is to lower the amount of air fed into the furnace, so the exhaust gases don't consume more heat than the amount generated by the process. I guess it's an obvious choice. I hope my analysis of the situation is correct.

Thanks in advance for any input!

 

[Chestermiller]

The value I get for M4 is 20.9 moles. That might account for the difference in the magnitude of Q.

They can't mean heat lost in the problem statement, because the furnace certainly adds heat. Otherwise, why is it called a furnace.

Chet

 

[MexChemE]

The value I get for M4 is 20.9 moles.

Required amount of oxygen is 2.75 mol, times 1.83 yields 5.03 mol, which is the amount of O₂ fed. 5.03 mol times the ratio of nitrogen to oxygen (0.79/0.21) yields 18.92 mol of N₂. The amount of O₂ exiting the furnace is 5.03 mol - 2.75 mol = 2.28 mol, plus 18.92 mol of N₂ and 2 mol of SO₂ generated add up to 23.2 mol.

They can't mean heat lost in the problem statement, because the furnace certainly adds heat. Otherwise, why is it called a furnace.

Maybe I'm confusing the way this kind of furnaces are analyzed (from an energy balance point of view) versus industrial process furnaces (heaters), where your desired product is a large (and negative) magnitude of Q. In a roasting furnace your desired product is to make the sulfide react, by raising its temperature. But a metallurgical furnace still loses heat by radiation, doesn't it? And if this specific furnace needs 3.9 mega BTU, how do you supply the heat? What is actually happening in a roasting furnace? Is the sulfide actually burning? If it does, how does it ignite?

 

[Chestermiller]

Required amount of oxygen is 2.75 mol, times 1.83 yields 5.03 mol, which is the amount of O₂ fed. 5.03 mol times the ratio of nitrogen to oxygen (0.79/0.21) yields 18.92 mol of N₂. The amount of O₂ exiting the furnace is 5.03 mol - 2.75 mol = 2.28 mol, plus 18.92 mol of N₂ and 2 mol of SO₂ generated add up to 23.2 mol.

Oops. I forgot to include the oxygen in the exit stream. Sorry.

Maybe I'm confusing the way this kind of furnaces are analyzed (from an energy balance point of view) versus industrial process furnaces (heaters), where your desired product is a large (and negative) magnitude of Q. In a roasting furnace your desired product is to make the sulfide react, by raising its temperature. But a metallurgical furnace still loses heat by radiation, doesn't it? And if this specific furnace needs 3.9 mega BTU, how do you supply the heat? What is actually happening in a roasting furnace? Is the sulfide actually burning? If it does, how does it ignite?

It seems to me you analyzed this problem correctly. I don't know what else to say.

Chet

 

[MexChemE]

I will also add, that even if the furnace is losing heat via radiation, it is not possible to account for it just with the information provided in the problem statement. The amount of heat lost by radiation is included in the value of Q, I guess, but impossible to calculate without the proper data. The bigger the heat loss due to radiation is, the bigger the amount of heat we need to provide the furnace with. So I guess the issue lies within the problem statement's wording.

 

[Chestermiller]

I will also add, that even if the furnace is losing heat via radiation, it is not possible to account for it just with the information provided in the problem statement. The amount of heat lost by radiation is included in the value of Q, I guess, but impossible to calculate without the proper data. The bigger the heat loss due to radiation is, the bigger the amount of heat we need to provide the furnace with. So I guess the issue lies within the problem statement's wording.

I agree.

Chet