Soda-Ash Calciner with Dry Recycle: 23-Chem-A1 Process Balances and Chemical Thermodynamics, December 2016, Q1

Question 1: Soda-Ash Calciner with Dry Recycle (Part A — 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. Wet NaHCO₃ at 2000 kg/h containing 8 wt% free water is blended with a dry Na₂CO₃ recycle stream $R$ to lower the water to 5 wt% before the calciner. Decomposition: $2\,\text{NaHCO}_3 \rightarrow \text{Na}_2\text{CO}_3 + \text{CO}_2 + \text{H}_2\text{O}$. Molar masses (g/mol): NaHCO₃ 84.01, Na₂CO₃ 105.99, CO₂ 44.01, H₂O 18.02.

Quantity	Value
Wet NaHCO₃ fed	2000 kg/h
— free water in it (8%)	160 kg/h
— dry NaHCO₃ in it	1840 kg/h
Target water after mixing	5 wt%
Recycle stream	dry Na₂CO₃ (no water)

Find. (a) net soda-ash product rate; (b) off-gas rate; (c) CO₂:H₂O mole ratio in the off-gas; (d) recycle rate.

Figure 1 — Wet bicarbonate is blended with dry recycled soda ash to hit 5% moisture, then calcined. The recycle circulates soda ash internally, so the net product is only the freshly formed Na₂CO₃.

Approach. The recycle carries no water, so a single water balance on the mixing tee fixes $R$ independently of the reaction; the dry-bicarbonate reaction then gives the fresh Na₂CO₃, CO₂ and reaction water, and the free moisture leaves with the off-gas. The printed 2000 kg/h is read as the fresh wet (8% water) bicarbonate, as worded; the calciner itself then receives the 3200 kg/h blend.

Recycle rate from a water balance on the mixing tee. The only water is the 160 kg/h in the wet feed; the recycle is dry. Requiring 5% water in the blended stream $(2000+R)$: $$\frac{160}{2000+R}=0.05 \;\Rightarrow\; 2000+R=3200 \;\Rightarrow\; \boxed{R = 1200\ \text{kg/h}}.$$ This is part (d): the mixing requirement alone fixes the recycle, with no reference to the reaction.
Extent of decomposition. Only the fresh dry NaHCO₃ (1840 kg/h) decomposes; the recycled Na₂CO₃ is already calcined and passes through. Moles of bicarbonate: $$\dot n_{\text{NaHCO}_3}=\frac{1840}{84.01}=21.90\ \text{kmol/h}\;\Rightarrow\; \dot n_{\text{Na}_2\text{CO}_3}=\dot n_{\text{CO}_2}=\dot n_{\text{H}_2\text{O,rxn}}=\tfrac12(21.90)=10.95\ \text{kmol/h}.$$
Net soda ash produced (part a). The freshly formed Na₂CO₃ is the net product; the 1200 kg/h of recycled Na₂CO₃ merely circulates: $$\dot m_{\text{Na}_2\text{CO}_3}=10.95\times105.99=\boxed{1161\ \text{kg/h}}.$$
Off-gas rate (part b). The off-gas is CO₂ plus all the water vapour — reaction water plus the 160 kg/h of free moisture driven off: $$\dot m_{\text{CO}_2}=10.95\times44.01=482.0,\quad \dot m_{\text{H}_2\text{O,off}}=10.95\times18.02+160=357.3\ \text{kg/h},$$ $$\dot m_{\text{off-gas}}=482.0+357.3=\boxed{839\ \text{kg/h}}.$$
CO₂-to-water mole ratio (part c). Water vapour is reaction water (10.95 kmol/h) plus free moisture $(160/18.02=8.88$ kmol/h$)$: $$\frac{\dot n_{\text{CO}_2}}{\dot n_{\text{H}_2\text{O}}}=\frac{10.95}{10.95+8.88}=\frac{10.95}{19.83}=\boxed{0.552\ \ (\approx 1:1.81)}.$$ The ratio being below 1 : 1 is the tell that free moisture, not just stoichiometric water, is in the off-gas.

An overall check closes the process: in = 2000 kg/h; out = 1161 (net soda ash) + 839 (off-gas) = 2000 kg/h. ✓

Quantity	Result
(a) Net soda ash produced	1161 kg/h
(b) Off-gas produced	839 kg/h (CO₂ 482, H₂O 357)
(c) CO₂:H₂O mole ratio	0.552 (1 : 1.81)
(d) Soda ash recycled	1200 kg/h

Concept & Theory

Topic. Steady-state material balance on a reactor with an internal recycle and a mixing tee (drying/decomposition).

Key relations. Component balance on the tee $\;\text{water}/(2000+R)=0.05$; stoichiometry $2\,\text{NaHCO}_3\to\text{Na}_2\text{CO}_3+\text{CO}_2+\text{H}_2\text{O}$; overall balance in=out with the recycle internal to the boundary.

Why this works. Because the recycle is dry and the moisture spec is a mass fraction, the water balance on the mixing point decouples from the chemistry and solves for $R$ in one line. Drawing the boundary around the whole calciner makes the recycled soda ash cancel, so the net product is exactly the freshly generated Na₂CO₃ — the circulating stream never leaves and must not be counted as production.

Common pitfall. Adding the 1200 kg/h recycle to the 1161 kg/h fresh product (double-counting), or forgetting that free moisture leaves as vapour with the off-gas (which is why the CO₂:H₂O ratio is under 1).

Source. Felder, Rousseau & Bullard, Elementary Principles of Chemical Processes §4.5–4.7 (recycle, purge and reactive-system balances).

Question 1 of 6: Soda-Ash Calciner with Dry Recycle

Question 1: Soda-Ash Calciner with Dry Recycle (Part A — equal value)