## Columbia University – Chemical Engineering

## CHEN E4112 Transport in Fluids and Mixtures

## Spring 2023 Take Home Final

化学工程代考 mathematical relations or developments from your texts or literature can be used; if you do so, cite references properly

### Information/Instructions.

- answer TWO questions as succinctly as possible; work on a third problem will not count towards your score
- prepare a .pdf or .doc/.docx file; a word processed document is preferred
- do not collaborate with others; prepare your own solutions; documents will be screened for duplication/plagiarism
- unless otherwise indicated, symbols have their usual meaning in what appears below
- mathematical relations or developments from your texts or literature can be used; if you do so, cite references properly

### 1.Gas Absorbtion in a Falling Liquid Film: Gas absorbers are im-portant technologies for áue-gas cleanup, and have potential for direct air capture of CO2. 化学工程代考

Both applications can mitigate climate change. We discuss a primitive model for this operation. A gas A sparingly soluble in a non-volatile liquid B contacts a falling Ölm of the pure liquid in the system depicted in the attached figure. For x > 0 the solute A dissolves into the liquid at the gas/liquid interface, located at y = 0, and di§uses into the film towards the impermeable wall located at y = H: We want to know the local rate of absorption of A into the film as expressed by a local Sherwood number Sh^{loc}

where k^{loc} is a local mass transfer coeffcient, defined here by

where c_{A}(y = 0) means the liquid phase molar density of A at the gas/liquid interface (usually taken to be the solubility of A in B c_{A0}; for a sparingly soluble gas), while cA;ave means the average molar density of A in the Ölm at the downstream location x

Here v_{x}(y) is the steady laminar downstream velocity profile in the film,determined by áuid mechanics as

We want to determine Shloc for a falling film absorber from the FCMT model discussed in class in the case of large Pe^{m}= far enough downstream such that

(a) Show that the FCMT model for this system leads to the species A continuity equation

Identify the physical meaning of each term and show the equation is dimensionally consistent. Write down a set of auxiliary conditions. 化学工程代考

#### (b) Scale the equations from a). using

and simplify for the case that Pe^{m} = Assert a simplified set of auxiliary conditions.

(c) Show that the simpliÖed model from part b) has the solution

where the λ_{n }and H_{n}() are solutions to the eigenvalue (Sturm-Liouville) problem

(d) Far downstream where the first term in the expansion for dominates giving the very good approximation

Show that for this region downstream, the Sherwood number is givenby

Hint: Section 10.5 of Deen is helpful.

### 2.Sedimentation of Charged Colloidal Particles. 化学工程代考

Sedimentation in aqueous media typically involves charged species. A common scenario is the sedimentation of charged colloidal particles in an aqueous electrolyte solution. Here we develop a simple model for the steady-state distrib-ution in a gravitional field assuming Z (> 0) valent colloidal particles,with Z >> 1 dissolved at trace levels with their monovalent counterions(anions) in water, with no additional added elecrolyte (i.e. “salt free”conditions). Denote with subscripts 1; 2; and 3 the colloidal particles,the counterions and water, respectively.

(a) Ignore for the moment the e§ect of forced di§usion, or migration,and focus on the ordinary and pressure di§usion e§ects, which are responsible for sedimentation phenomena in uncharged systems. Re-call for a binary solution we developed an expression for the mass áux of the solute (component 1) relative to the volume average velocity including both these effects. 化学工程代考

where means the volume average velocity and s1(ρ1 ) is the sedimentation coeffcient. Show that in the limit of a trace solute in nearly pure solvent, this expression can be written

#### where ρ^{0} _{i }means the density of pure i and is the molar volume of solute. The superscript ∞ indicates the infinite dilution limit

(b) To the result for J y 1 in a). add a forced di§usion (migration) term,and then generalize the result for traces of two solutes (i = 1; 2) in a third component (solvent). Use the resulting áux laws to model the steady (rest) state of one directional sedimentation of the charged particles in a closed vertical column of height H. Employ a Carte sian coordinate system with origin on the bottom of the column and the x direction pointing upwards against gravity. Assume no vol-ume change on mixing, and local electrical neutrality Show that the following equations govern the steady density distributions of the colloidal particles c1(x) and their counter-ions, c2(x):

colloidal particles c1(x) and their counter-ions, c2(x):

Give a suffcient set of auxiliary conditions needed to solve for c1(x);c2(x) and Ø(x):

#### (c) Scale the system resulting from part b). using

where c_{i0 }are the initial uniform trace concentrations of the colloidal particles and counterions (note Zc10 = c20) to find

Solve these for the colloid particle density distribution, and show that for Z >> 1 the result is very nearly

How does this result compare with the case of uncharged colloidal particles? i.e. does the charge increase or decrease the extent of segregation of the particles?

### 3.Electro-osmotic Flow Around a Charged Sphere with a Thin Diffuse Layer: 化学工程代考

A rigid sphere of radius a with zeta (surface) potential ζ is held fixed in a trace solution of a 1 – 1 monovalent electrolytewith bulk salt molar density cs∞.A constant external electric feld is applied along the z direction in a spherical coordinatesystem with origin on the sphere’s center, putting a force on the diffuselayer around the sphere, causing a flow. We want to model the steadyelectro-osmotic velocity profile near the sphere surface in the limit that thediffuse layer thickness λ is small compared to the particle radius, << 1 and assuming no distortion of the diffuse layer from the equilibrium (rest)state. decribed adequately within the Debye-Huckel (DH) approximation.

(a) Recall the rest potential field, Ø^{eq}(r) in the DH limit. derived inclass

where λ is the Debye length. Assuming

show that Ø^{n}^{eq} obeys

subiect to

(Hints: What surface charge does the first boundary condition corre-spond to and why is that appropriate to Ø^{n}^{eq}? What electric field does the second boundary condition correspond to?) Show that the solution for Ø^{n}^{eq}(r; θ) is

#### (b) We want to find an approximate velocity profile for the áow in the (thin) diffuse layer <<1. Assuming v = (v_{r}(r, θ); v_{θ} (r; θ); 0) the continuity equation is

Within the (very thin) diffuse layer, the following characteristic scales

are valid

Show that this implies

where v^{*}_{i }is a characteristic value. 化学工程代考

(c) Neglecting inertia, and using the scales mentioned in b). show that the dominant viscous term in the θ component of momentum within the thin diffuse layer is resulting in

Hints: You will need the full θ component of momentum from a reference, and then make order of magnitude estimates of the various viscous terms to find the dominant one; also you will need to evaluate the θ component or the electric field body force term from the results of part a).

#### (d) The result from part c). can be simplified further by adopting the scaling

and taking the limit (λ⁄a)→ 0 with s’ ~ O(1): This is appropriate for the motion in the thin di§use layer, where we expect s’ ~ O(1)(Re:s’ is a boundary layer coordinate), and v^{*}is chosen to match the orders of the viscous and electric body force terms. Scale the θ momentum according to this to show

in the limit (λ⁄a) → 0 with s’~O(1).

(e) Solve the last equation with appropriate boundary conditions to find the scaled velocity profile

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