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23-Chem-A3 Heat and Mass Transfer · December 2014

Question 5 of 7: Overall gas-phase coefficient for a packed absorber

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2014 — 04-Chem-A3 Mass Transfer Operations. Three-hour, open-book exam; any non-communicating calculator permitted. Format: seven questions in three parts — answer one of Q1–Q2 (Part A), one of Q3–Q4 (Part B) and two of Q5–Q7 (Part C); four questions of equal value constitute a complete paper. All seven are solved below for completeness. Every property datum (vapour pressures, diffusivities, Henry constants) is stated in the question, so each answer is self-contained.

Reference texts: Geankoplis, Transport Processes and Separation Process Principles (4th ed., Prentice Hall) — molecular diffusion, convective mass transfer and gas absorption; Welty, Wicks, Wilson & Rorrer, Fundamentals of Momentum, Heat and Mass Transfer (6th ed., Wiley) — boundary-layer mass transfer and diffusion with reaction; Incropera & DeWitt, Fundamentals of Heat and Mass Transfer (species diffusion, sphere in a stagnant medium); Treybal, Mass-Transfer Operations (3rd ed.) — packed-tower and wetted-wall design; supporting data from Perry's Chemical Engineers' Handbook (9th ed.).

Question 5: Overall gas-phase coefficient for a packed absorber (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 dilute countercurrent packed absorber: gas up, liquid down, with a linear equilibrium line through the origin. The overall gas-phase coefficient follows from the transfer-unit relation $z=H_{OG}N_{OG}$.

QuantityValue
Packed height $z$12.0 m
Gas in / out, $y_1,y_2$0.020 / 0.0050
Gas velocity $G$0.0136 kmol/m²·s
Liquid velocity $L$0.0272 kmol/m²·s (pure, $x_2=0$)
Total pressure $P$1.2 atm $=121.6$ kPa
Equilibrium slope $m$1.5 ($y_A^*=1.5x_A$)

Find. The overall gas-phase volumetric coefficient $K_Ga$.

packing, z = 12 m gas in, y₁ = 0.020 gas out, y₂ = 0.0050 solvent in, x₂ = 0 liquid out, x₁ = 0.0075
Figure 5 — Countercurrent dilute packed absorber. Solute A leaves the rising gas into the descending pure solvent; the liquid exit composition $x_1$ closes the solute balance and sets the bottom driving force.

Approach. Close the solute balance for the liquid exit $x_1$, evaluate the log-mean gas-phase driving force, get $N_{OG}$ and $H_{OG}=z/N_{OG}$, then $K_ya=G/H_{OG}$ and finally $K_Ga=K_ya/P$.

  1. Liquid exit composition (solute balance). With $x_2=0$, $$x_1=\frac{G(y_1-y_2)}{L}=\frac{0.0136(0.020-0.0050)}{0.0272}=0.0075.$$
  2. Gas-phase driving forces at each end. Using $y^*=1.5x$: at the bottom $\Delta y_1=y_1-1.5x_1=0.020-0.01125=0.00875$; at the top $\Delta y_2=y_2-0=0.0050$.
  3. Log-mean driving force. $$\Delta y_\text{lm}=\frac{\Delta y_1-\Delta y_2}{\ln(\Delta y_1/\Delta y_2)}=\frac{0.00875-0.0050}{\ln(0.00875/0.0050)}=0.00670.$$
  4. Number and height of transfer units. $$N_{OG}=\frac{y_1-y_2}{\Delta y_\text{lm}}=\frac{0.015}{0.00670}=2.24,\qquad H_{OG}=\frac{z}{N_{OG}}=\frac{12.0}{2.24}=5.36\ \text{m}.$$
  5. Overall gas-phase coefficients. On a mole-fraction basis, $$K_ya=\frac{G}{H_{OG}}=\frac{0.0136}{5.36}=2.54\times10^{-3}\ \text{kmol/m}^3\!\cdot\!\text{s}.$$ Converting to a partial-pressure basis ($p=Py$): $$K_Ga=\frac{K_ya}{P}=\frac{2.54\times10^{-3}}{121.6}=\boxed{2.09\times10^{-5}\ \text{kmol/m}^3\!\cdot\!\text{s}\!\cdot\!\text{kPa}}.$$ Equivalently $K_Ga=2.11\times10^{-3}$ kmol/m³·s·atm.
QuantityResult
Liquid exit $x_1$0.0075
Log-mean driving force $\Delta y_\text{lm}$0.00670
$N_{OG}$ / $H_{OG}$2.24 / 5.36 m
$K_ya$ (mole-fraction basis)$2.54\times10^{-3}$ kmol/m³·s
$K_Ga$ (pressure basis)$2.09\times10^{-5}$ kmol/m³·s·kPa
Check
The system is dilute (solute $\le2$ mol%), so the constant-flow HTU–NTU form is valid and $G$, $L$ may be treated as constant. Report the coefficient in whichever unit the marker expects: $K_ya$ (mole-fraction driving force, kmol/m³·s) or $K_Ga$ (partial-pressure driving force, kmol/m³·s·kPa) — they differ only by the total pressure $P$.