CHE 318 Lecture 09

Introduction to Unsteady State Mass Transfer

Dr. Tian Tian

2026-01-23

Recap

  • Solving examples with pseudo steady state assumption
  • Conclusion of steady state mass transfer

Learning Outcomes

After today’s lecture, you will be able to:

  • Recall difference between steady and unsteady state solutions
  • Derive mass balance and flux equations in unsteady state problems
  • Identify the generation and accumulation terms in typical transport problems
  • Analyze time-dependent concentration profiles for U.S.S situations.

What Is Unsteady-State Mass Transfer?

  • Concentration varies with time \(\partial c/\partial t \neq 0\)
  • Accumulation term is non-zero \(\text{[Acc]} \neq 0\)
  • Requires time-dependent mass balances
  • Common in transient diffusion, start-up, and response problems
  • More general than S.S.

Governing Equation for U.S.S M.T.

Consider a control volume in 1D transport, the mass balance equation becomes

\[\begin{align*} \text{[Acc]} &= \text{[In]} - \text{[Out]} + \text{[Gen]} \\ S \Delta z \frac{\partial c_A}{\partial t} &= S (N_A \vert_{z=z} - N_A \vert_{z=z+\Delta z}) + \text{[Gen]} S \Delta z \\ \frac{\partial c_A}{\partial t} &= -\frac{\partial N_A(z)}{\partial z}\vert_z + r_{A} \end{align*}\]

where \(r_{A}\) is the generation rate for A (e.g. local reaction). This is the governing equation for all time-dependent mass transfer!

Comparison between Flux equation and Mass Balance

Flux equation

\[\begin{align*} N_A = J_{Az}^* + x_A(N_A + N_B) \end{align*}\]
  • Amount of material moved in and out of controled volume
  • \(J_{Az}^*\): Fick’s first law of diffusion
  • Can be used for S.S (\(d N_A/dz = 0\)) and U.S.S

Mass balance eq

\[\begin{align} \frac{\partial c_A}{\partial t} = - \frac{\partial N_A}{\partial x} + r_A \end{align}\]
  • Change of local \(c_A\) over time
  • Need flux equation solution first
  • Can be used for S.S (\(\frac{\partial c_A}{\partial t} = 0\)) and U.S.S.

Mass Balance: Extension to 3D

Illustration of divergence

\[\begin{align} \frac{\partial c_A}{\partial t} &= r_A - \frac{\partial N_{Ax}}{\partial x} - \frac{\partial N_{Ay}}{\partial y} - \frac{\partial N_{Az}}{\partial z} \\ &= r_A - \nabla \cdot \vec{N}_A \end{align}\]
  • \(\nabla \cdot\) is the divergence operator (not \(\Delta\), not gradient!)
  • \(\vec{N}_A\) is generally a 3D vector field

How To Analyze A U.S.S Problem

Unsteady state mass transfer is not intimidating if you follow these steps

  1. Draw the scheme
  2. Write down the mass balance equation ([In] - [Out] + [Gen] = [Acc])
  3. Identify the [Gen] and [Acc] terms
  4. Choose proper flux equations for [In] and [Out] terms
  5. Solving analytically or numerically.

Dissecting the General Equation for Mass Balance

\[\begin{align*} r_A - \frac{\partial c_A}{\partial t} &= \nabla \cdot \left[\vec{J}_{A}^{*} + x_A(\vec{N}_A + \vec{N}_B) \right] \\ &= \nabla \cdot \left[\vec{J}_{A}^{*} + c_A \vec{v}_m \right] \\ &= \nabla \cdot \left[ -D_{AB} \nabla c_A + c_A \vec{v}_m \right] \end{align*}\]
  • We have fluid velocity \(\vec{v}_m\) on the R.H.S
  • \(\nabla \cdot\) creates more nonlinear terms
  • Do we know \(N_A\) and \(N_B\) relation?
  • In general this is hard to solve (coupling fluid with mass transfer)
  • Often interested in several limiting cases

Special Cases of Unsteady-State Mass Transfer

  • Case 1: EMCD for gases at constant \(p_T\), \(r_A = 0\) (Fick’s second law)
    • There is no negative sign in Fick’s second law!
    \[\begin{align} \frac{\partial c_A}{\partial t} &= D_{AB}\nabla^2 c_A \end{align}\]
  • Case 2: Constant \(D_{AB}\)
\[\begin{align*} \frac{\partial c_A}{\partial t} &= D_{AB}\nabla^2 c_A - c_A \nabla \cdot \vec{v}_m - \vec{v}_m \cdot \nabla c_A + r_A \end{align*}\]
  • Case 3: Constant \(\rho\) and \(D_{AB}\) (e.g. imcompressible liquids)
\[\begin{align} \frac{\partial c_A}{\partial t} &= D_{AB}\nabla^2 c_A - \vec{v}_m \cdot \nabla c_A + r_A \end{align}\]

where \(\nabla \cdot \vec{v}_m = 0\)

What exactly do we solve?

For U.S.S M.T, we typically need

  1. Governing equation (PDE) from any limiting case
  2. Initial conditions \(c_A(z, t=0)\)
  3. Boundary conditions (B.C.)
    • Dirichlet B.C. (e.g. \(c_A(z=0) = c_0\))
    • Neumann B.D. (e.g. \(N_A(z=0) = N_{A0}\), constant flux)
  4. Solving analytically or numerically
  5. Get \(c_A(z, t)\), \(x_A(z, t)\), \(N_A(x, t)\)
  6. Steady-state solutions often means \(c_A(z, t\to \infty)\)

Unsteady State Mass Transfer: Calculation Overview

A “solver engine” for unsteady state mass transfer

Summary

  • Unsteady state mass transfer governing equation
  • Step-by-step solution to diffusion through stagnant B
  • Diffusion and reaction system setup