Transport Phenomena Tutoring

Transport phenomena is often the most mathematical course in the chemical engineering core. It treats momentum, heat, and mass transfer as one subject with one method. Students who did fine in fluid mechanics and heat transfer often struggle here, because the course stops handing out formulas and asks you to derive them from a balance on a thin shell. STEM Tutor 360 drills that method until it becomes routine: choose the shell, write what flows in and out, insert the transport law, integrate, and apply the boundary conditions.

Sessions cover velocity profiles in slits, tubes, and falling films, temperature profiles with and without heat generation, and diffusion through stagnant films and with reaction. Later topics include the equations of change, dimensionless numbers, analogies among the three transports, and mass transfer coefficients. We follow your textbook, whether that is Bird, Stewart, and Lightfoot or a more applied book such as Welty or Geankoplis.

A typical transport problem, worked

Water flows in laminar flow through a horizontal tube of radius $R = 1\ \text{mm}$ and length $L = 1\ \text{m}$. The pressure drop is 500 Pa, the viscosity is $0.001\ \text{Pa s}$, and the density is $1000\ \text{kg/m}^{3}$. Find the velocity profile and the flow rate, and confirm the flow is laminar.

Take a thin cylindrical shell at radius $r$. For steady, fully developed flow, the momentum balance on the shell gives $\dfrac{d(r\tau)}{dr} = \dfrac{\Delta P}{L}\, r$, where $\tau$ is the shear stress. Integrate, using the fact that the stress stays finite at the centerline: $\tau = \dfrac{\Delta P}{2L}\, r$. Newton's law of viscosity says $\tau = -\mu\, \dfrac{dv}{dr}$. Integrate again with no slip at the wall, $v = 0$ at $r = R$, and the profile is $$v = \dfrac{\Delta P\, R^{2}}{4\mu L}\left(1 - \left(\dfrac{r}{R}\right)^{2}\right)$$

The centerline velocity is $$\dfrac{500 \times (0.001)^{2}}{4 \times 0.001 \times 1} = 0.125\ \text{m/s}$$ The average velocity is half of that, 0.0625 m/s. The flow rate is $0.0625 \times \pi \times (0.001)^{2} = 1.96 \times 10^{-7}\ \text{m}^{3}/\text{s}$, about 0.20 mL/s. The Reynolds number is $\dfrac{1000 \times 0.0625 \times 0.002}{0.001} = 125$, far below 2100, so the laminar assumption holds.

The same steps work for heat and mass: swap the shear stress for a heat flux or mass flux, and Newton's law for Fourier's or Fick's law.

Where students usually get stuck

• Choosing a shell that is thick in the wrong direction, so the balance picks up an extra term.
• In cylinders and spheres, forgetting that the area grows with r, so without generation the flux times the area stays constant, not the flux alone.
• Sign errors: the direction of the coordinate, the direction of the flux, and the minus sign in each transport law.
• Missing boundary conditions, such as no slip at a wall, symmetry at a centerline, or continuity of flux at an interface.
• Dropping the bulk flow term in Fick's law when the mixture itself is moving.