Packing depth for CO₂ absorption into water: 23-Chem-A3 Heat and Mass Transfer, December 2015, Q7

Question 7: Packing depth for CO₂ absorption into water (Part C — 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. A liquid-film-controlled packed absorber (pure CO₂ gas, so no gas-phase resistance). The liquid-side volumetric coefficient $k_La$ comes from the Sherwood–Holloway correlation, and the packing depth follows from the plug-flow absorption balance with a constant saturation concentration $C^\ast=p_{CO_2}/H$.

Quantity	Value
Water rate	5 kg mol/min ($=1.5$ kg/s)
Tower diameter	0.25 m ($A_{cs}=0.0491$ m²)
Henry’s constant $H$	25.4 atm·m³/kg mol
CO₂ partial pressure	2 atm
$\rho$, $\mu$, $D_{AB}$	998.2, $9.93\times10^{-4}$, $1.77\times10^{-9}$
1-in ceramic saddles $\alpha,\ n$	170, 0.28

Find. (a) $k_La$; (b) packing depth $z$ for outlet $C_{out}=0.075$ kg mol/m³.

Figure 7. Co-current packed absorber (water and pure CO₂ both enter at top); the liquid picks up CO₂ toward saturation $C^\ast=p/H$ down the bed.

Approach. Compute the liquid mass velocity $L$, apply the Sherwood–Holloway correlation for $k_La$, evaluate the saturation concentration $C^\ast=p/H$, then integrate the plug-flow absorption balance for the depth.

Liquid mass velocity. $\dot m_L=(5/60)(18)=1.5$ kg/s over $A_{cs}=\tfrac{\pi}{4}(0.25)^2=0.0491$ m²: $$L=\dot m_L/A_{cs}=1.5/0.0491=30.6\ \text{kg/m}^2\text{s}.$$
Sherwood–Holloway $k_La$. With $Sc=\mu/(\rho D)=561$, $$k_La=\alpha\Big(\frac{L}{\mu}\Big)^{1-n}\Big(\frac{\mu}{\rho D_{AB}}\Big)^{0.5}D_{AB}=170\Big(\frac{30.6}{9.93\times10^{-4}}\Big)^{0.72}(561)^{0.5}(1.77\times10^{-9})=\boxed{0.0122\ \text{s}^{-1}}.$$
Saturation concentration. $$C^\ast=\frac{p_{CO_2}}{H}=\frac{2}{25.4}=0.0787\ \text{kg mol/m}^3,$$ and the 95%-of-saturation target is $C_{out}=0.075$ kg mol/m³ ($C_{in}=0$).
Packing depth. The plug-flow balance $Q_L\,dC=k_La(C^\ast-C)A_{cs}\,dz$ integrates to $$z=\frac{Q_L}{k_La\,A_{cs}}\ln\frac{C^\ast-C_{in}}{C^\ast-C_{out}}=\frac{1.503\times10^{-3}}{(0.0122)(0.0491)}\ln\frac{0.0787}{0.0787-0.075}=\boxed{7.67\ \text{m}},$$ where $Q_L=\dot m_L/\rho=1.503\times10^{-3}$ m³/s. Approaching 95% of saturation is what drives the ~8 m of packing.

Quantity	Value
Liquid mass velocity $L$	30.6 kg/m²s
(a) $k_La$	$0.0122$ s⁻¹
Saturation $C^\ast$	0.0787 kg mol/m³
(b) Packing depth $z$	7.67 m

Concept & Theory

Topic: Liquid-film-controlled packed-tower absorption (Sherwood–Holloway).

Key relations: Sherwood–Holloway $k_La/D_{AB}=\alpha(L/\mu)^{1-n}(\mu/\rho D_{AB})^{0.5}$; Henry’s law $C^\ast=p/H$; plug-flow depth $z=\dfrac{Q_L}{k_La\,A_{cs}}\ln\dfrac{C^\ast-C_{in}}{C^\ast-C_{out}}$.

Why this works: Because the gas is pure CO₂, there is no gas-phase resistance and the interfacial concentration equals the Henry’s-law saturation value, constant down the bed. The liquid then approaches $C^\ast$ exponentially, and the number of transfer units $\ln[C^\ast/(C^\ast-C_{out})]$ times the height of a transfer unit ($Q_L/k_LaA_{cs}$) gives the packing depth.

Common pitfall: Using the wrong saddle constants (the exam table lists $\alpha$ and $n$ by packing type and size) or forgetting that the driving force is $C^\ast-C$ (not $C$) for absorption — the sign flips relative to a stripping problem.

Source: Treybal, Mass-Transfer Operations (3rd ed.), Ch. 6 (Sherwood–Holloway); Geankoplis §10.6 (packed-tower design).

Question 7 of 7: Packing depth for CO₂ absorption into water

Question 7: Packing depth for CO₂ absorption into water (Part C — equal value)