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23-Chem-A3 Heat and Mass Transfer · May 2015

Question 4 of 7: Humidifying air bubbles — transient diffusion into a sphere

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

Notes on this paper

National Exams — May 2015 — 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 (diffusivities, vapour pressures, solubilities, humid-air heat capacities) 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, gas absorption and drying; Welty, Wicks, Wilson & Rorrer, Fundamentals of Momentum, Heat and Mass Transfer (6th ed., Wiley) — boundary-layer and transient diffusion; Incropera & DeWitt, Fundamentals of Heat and Mass Transfer — transient conduction/diffusion charts and the sphere in a stagnant medium; Treybal, Mass-Transfer Operations (3rd ed.) — packed-tower design; Crank, The Mathematics of Diffusion (2nd ed.) — series solutions for the sphere; supporting data from Perry's Chemical Engineers' Handbook (9th ed.).

Question 4: Humidifying air bubbles — transient diffusion into a sphere (Part B — 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 gas bubble absorbs water vapour from its (saturated) surface inward. This is transient diffusion into a sphere with a fixed surface concentration; we track an interior radial point rather than the volume average.

QuantityValue
Bubble radius $R$1.2 mm $=0.12$ cm
Interior point $r$0.2 mm $=0.02$ cm
Target at $r$99% of saturation
Inlet relative humidity10% ($c_0=0.10\,c_s$)
Diffusivity $D_{AB}$0.26 cm²/s

Find. The residence time for the vapour concentration at $r=0.2$ mm to reach 99% of saturation.

R = 1.2 mm r = 0.2 mm air (10% RH init.) surface saturated, c_s (100% RH) water
Figure 4 — Water vapour diffuses inward from the saturated bubble surface. We require the concentration at the interior radius $r=0.2$ mm (dashed) to reach 99% of the surface saturation value, starting from 10% RH throughout.

Approach. Use the one-term interior solution of the transient sphere (fixed surface concentration), which relates the unaccomplished-change ratio $Y$ at a radial point $r$ to the Fourier number, then solve for $\mathrm{Fo}$ and hence the time.

  1. Unaccomplished-change ratio at the point. With $c_0=0.10\,c_s$ and target $c=0.99\,c_s$, $$Y=\frac{c_s-c}{c_s-c_0}=\frac{1-0.99}{1-0.10}=0.0111.$$
  2. One-term interior sphere profile. For a step change at the surface, the leading eigenmode at radius $r$ is $$Y=\frac{2R}{\pi r}\sin\!\left(\frac{\pi r}{R}\right)e^{-\pi^2\mathrm{Fo}}.$$ The geometric prefactor is $$\frac{2R}{\pi r}\sin\!\frac{\pi r}{R}=\frac{2(0.12)}{\pi(0.02)}\sin\!\frac{\pi(0.02)}{0.12}=3.82\times\sin(30^\circ)=1.91.$$
  3. Solve for the Fourier number. $$\mathrm{Fo}=-\frac{1}{\pi^2}\ln\!\left(\frac{Y}{1.91}\right)=-\frac{1}{\pi^2}\ln\!\left(\frac{0.0111}{1.91}\right)=0.521.$$
  4. Residence time. $$t=\frac{\mathrm{Fo}\,R^2}{D_{AB}}=\frac{(0.521)(0.12)^2}{0.26}=\boxed{0.029\ \text{s}}.$$
QuantityResult
Ratio $Y$ at $r=0.2$ mm0.0111
Fourier number $\mathrm{Fo}$0.521
Required residence time$\approx0.029$ s (29 ms)
Check
The bubble saturates almost instantly (tens of milliseconds) because the diffusion length ($\sim$1 mm) is tiny and the vapour diffusivity in a gas is large (0.26 cm²/s). Any realistic residence time in a bubble column is far longer, so bubbling reliably saturates the air — consistent with the near-100% target being reached this quickly.