Sulfur Burner and Waste-Heat Boiler — Steam Raised: 23-Chem-A1 Process Balances and Chemical Thermodynamics, Undated paper, Q3

Question A3: Sulfur Burner and Waste-Heat Boiler — Steam Raised (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Basis one hour. 200 kg of pure sulfur burns completely to SO₂ with 10% excess air (21% O₂, 79% N₂). The burner gas enters the waste-heat boiler at 871 °C and leaves at 190 °C. The boiler raises saturated steam at 15 bar from 183 °C feed water, and 90% of the heat given up by the gas reaches the steam.

Datum	Value
Sulfur feed, $M_S=32.06$	200 kg/hr
Reaction / excess air	$\mathrm{S+O_2\rightarrow SO_2}$; 10% excess
Gas through boiler	871 °C (1144.15 K) → 190 °C (463.15 K)
Boiler efficiency	90%
Steam: $T_{sat}$ / $\lambda$ (15 bar)	198 °C / 1945 kJ/kg
Feed water / $C_{p,w}$	183 °C / 4.1868 kJ/kg·K

Find. the mass of saturated steam raised per hour.

Figure 2 — Sulfur burns to SO₂ with 10% excess air; the 871 °C burner gas gives up sensible heat in the waste-heat boiler (cooling to 190 °C, 90% recovery efficiency) to raise saturated steam at 15 bar from 183 °C feed water.

Data check: printed N₂ heat-capacity polynomial. As printed, $C_{p,N_2}=29.5909+51.141\times10^{-4}T-1.31829\times10^{-5}T^2-49.68\times10^{-10}T^3$ gives 29.8 kJ/kmol·K at 298 K but falls to 16.6 at 1000 K and 10.7 at 1144 K. Real N₂ rises to about 32.7 at 1000 K, so the printed form is impossible. Flipping the signs of the $T$ and $T^2$ terms, $C_{p,N_2}=29.5909-5.1141\times10^{-3}T+1.31829\times10^{-5}T^2-4.968\times10^{-9}T^3$, reproduces the reference values (29.1 at 298 K, 32.7 at 1000 K, 33.6 at 1144 K). This is the standard tabulated form, and it is used below. The SO₂ and O₂ polynomials are used exactly as printed; they give 54.6 and 34.9 kJ/kmol·K at 1000 K, which are correct.

Approach. Fix the gas flows from the combustion stoichiometry. Integrate each cubic heat-capacity polynomial between 190 °C and 871 °C to get the sensible heat the gas gives up. Apply the 90% boiler efficiency, then divide by the heat needed per kilogram of steam: sensible heat of the feed water up to saturation plus the latent heat.

Sulfur and air. $\dot n_S=200/32.06=6.238$ kmol/hr, and the theoretical O₂ for $\mathrm{S+O_2\rightarrow SO_2}$ is the same 6.238 kmol/hr. With 10% excess, the O₂ supplied is $1.10(6.238)=6.862$ kmol/hr and the accompanying N₂ is $6.862(79/21)=25.81$ kmol/hr.
Gas entering the boiler. Combustion is complete and no SO₃ forms: $$\mathrm{SO_2}=6.238,\quad \mathrm{O_2}=6.862-6.238=0.624,\quad \mathrm{N_2}=25.81\ \text{kmol/hr}\quad(\text{total }32.68\ \text{kmol/hr}).$$
Sensible heat released between 871 and 190 °C. For each gas $\displaystyle\int_{463.15}^{1144.15}C_p\,dT=a\Delta T+\tfrac{b}{2}\Delta(T^2)+\tfrac{c}{3}\Delta(T^3)+\tfrac{d}{4}\Delta(T^4)$, which gives SO₂ 35,345, O₂ 22,831 and N₂ 21,427 kJ/kmol. Weighting by the flows: $$Q_{gas}=6.238(35{,}345)+0.624(22{,}831)+25.81(21{,}427)=2.205\times10^{5}+1.42\times10^{4}+5.531\times10^{5}=\boxed{7.879\times10^{5}\ \text{kJ/hr}}.$$
Heat needed per kilogram of steam. The feed water is heated from 183 to 198 °C and then vaporized at 15 bar: $$q=C_{p,w}(T_{sat}-T_{fw})+\lambda=4.1868(198-183)+1945=62.8+1945=2007.8\ \text{kJ/kg}.$$
Steam raised. At 90% efficiency, $$\dot m_{steam}=\frac{0.90\,Q_{gas}}{q}=\frac{0.90(7.879\times10^{5})}{2007.8}=\boxed{353\ \text{kg/hr of saturated steam at 15 bar}}.$$

Assumptions stated (exam note 1): “90% efficiency” is read as the waste-heat boiler's heat-recovery efficiency. “Complete combustion” already fixes the sulfur conversion, and “no heat loss to the surroundings” is taken to apply to the burner. If all of the gas heat went to steam, the result would be $7.879\times10^{5}/2007.8=392$ kg/hr. If the N₂ polynomial were used exactly as printed, $Q_{gas}$ would be $6.15\times10^{5}$ kJ/hr and the steam 276 kg/hr; that figure is not physical, for the reason given in the data check above.

Quantity	Result
Gas to boiler (SO₂ / O₂ / N₂)	6.238 / 0.624 / 25.81 kmol/hr (total 32.68)
Sensible heat released by the gas, 871 → 190 °C	$7.879\times10^{5}$ kJ/hr
Heat per kg of steam	2007.8 kJ/kg
Steam produced (90% efficiency)	353 kg/hr (392 kg/hr at 100%)

Nitrogen carries about 70% of the recoverable heat even though the air is only 10% in excess. The steam output is therefore set mainly by the air-to-sulfur ratio, and much less by the SO₂ itself.

Concept & Theory

Topic. Combustion stoichiometry coupled to a sensible-heat (waste-heat-boiler) energy balance.

Key relations. Excess air: $\dot n_{O_2}=(1+\text{excess})\dot n_{O_2}^{theo}$ and $\dot n_{N_2}=\dot n_{O_2}(79/21)$. Sensible heat: $Q=\sum\dot n_i\int C_{p,i}\,dT$ with cubic $C_p(T)$. Steam side: $\dot m=\eta Q/[C_{p,w}(T_{sat}-T_{fw})+\lambda]$.

Why this works. The boiler moves the gas's sensible-enthalpy drop into the water and steam side, less the fraction lost ($1-\eta$). The steam rate is therefore one enthalpy divided by another. The combustion reaction is needed only to size the three gas streams whose heat capacities set that enthalpy drop.

Common pitfall. Using a heat-capacity polynomial without checking it at two temperatures. A sign typo in a printed coefficient can silently cut the duty by 20% or more. Other slips are integrating in °C when the correlation is written for kelvin, and forgetting the N₂ that comes with the excess O₂.

Source. Felder, Rousseau & Bullard, Elementary Principles §9.5 (combustion energy balances) and §8.3 (sensible heat, $\int C_p\,dT$).

Question 3 of 6: Sulfur Burner and Waste-Heat Boiler — Steam Raised

Part A — Process Mass and Energy Balances

Question A3: Sulfur Burner and Waste-Heat Boiler — Steam Raised (equal value)