Chemical engineering transport & equilibrium foundations of Falling Film Non-Newtonian Rheometry.
Falling liquid films over solid surfaces are ubiquitous in chemical process operations, including wetted-wall gas absorbers, falling-film evaporators, trickle-bed reactors, and polymer devolatilizers. In many industrial applications—such as paints, biological fermentation broths, polymer melts, and starch slurries—the process liquid displays non-Newtonian shear-thinning (pseudoplastic) rheology, described by the Ostwald–de Waele power-law model: τ_yx = K · |du/dy|^n, where K is the consistency index and n < 1 is the dimensionless flow behavior index. When a thin liquid film descends steadily under gravity down a plane inclined at angle θ, a momentum balance yields a shear stress profile that varies linearly from zero at the free liquid-gas interface to maximum shear at the solid wall: τ_yx(y) = ρ g cos θ · (δ - y), where δ is film thickness. Integrating this stress distribution with the power-law constitutive equation provides analytical velocity profiles and links surface velocity u_s to mass flow rate per unit wetted width Γ via: u_s ∝ Γ^[(n+1)/n]. In this laboratory investigation, surface tracer velocimetry is performed on an aqueous carboxymethylcellulose (CMC) solution down a precision inclined wetted plate to measure surface velocity u_s vs flow rate Γ, evaluate rheological slope m = (n+1)/n, and determine the flow behavior index n.
1. Transport Phenomena in Falling Liquid Films
Falling liquid film flow down solid surfaces constitutes a classical transport phenomena problem pioneered by Nusselt, Kapitza, and Bird, Stewart, and Lightfoot (BSL). In industrial chemical processing, falling films provide high surface area per unit liquid volume with minimal pressure drop, making them ideal for devolatilizers, distillation packings, and tubular evaporators.
2. Navier–Stokes & Power-Law Rheological Formulation
Consider a steady, fully developed laminar film of a non-Newtonian Ostwald–de Waele fluid descending an inclined plane inclined at angle $\theta$ to the horizontal. Let $x$ be the coordinate down the plane and $y$ the coordinate perpendicular to the plate ($y = 0$ at the wall, $y = \delta$ at the free surface). The $x$-momentum balance simplifies to:
\[ 0 = -\frac{d\tau_{yx}}{dy} + \rho g \sin \theta \]Integrating with the free-surface boundary condition $\tau_{yx}(\delta) = 0$ (neglecting interfacial air drag) yields the linear shear stress distribution:
\[ \tau_{yx}(y) = \rho g \sin \theta \cdot (\delta - y) \]For an Ostwald–de Waele power-law fluid, the constitutive equation is $\tau_{yx} = K \left( \frac{du}{dy} \right)^n$. Equating shear stresses:
\[ K \left( \frac{du}{dy} \right)^n = \rho g \sin \theta \cdot (\delta - y) \implies \frac{du}{dy} = \left( \frac{\rho g \sin \theta}{K} \right)^{1/n} (\delta - y)^{1/n} \]Integrating with the no-slip condition $u(0) = 0$ yields the velocity distribution:
\[ u(y) = \left( \frac{\rho g \sin \theta}{K} \right)^{1/n} \frac{n}{n + 1} \left[ \delta^{\frac{n+1}{n}} - (\delta - y)^{\frac{n+1}{n}} \right] \]3. Surface Velocity & Mass Flow Rate Scaling
Evaluating velocity at the free surface ($y = \delta$):
\[ u_s = u(\delta) = \left( \frac{\rho g \sin \theta}{K} \right)^{1/n} \frac{n}{n + 1} \delta^{\frac{n+1}{n}} \]The mass flow rate per unit wetted width $\Gamma$ is obtained by integrating liquid flux:
\[ \Gamma = \rho \int_0^\delta u(y) dy = \rho \left( \frac{\rho g \sin \theta}{K} \right)^{1/n} \frac{n}{2n + 1} \delta^{\frac{2n+1}{n}} \]Eliminating film thickness $\delta$ between expressions for $u_s$ and $\Gamma$ establishes the direct power-law scaling:
\[ u_s = C \cdot \Gamma^m, \quad \text{where } m = \frac{n + 1}{2n + 1} \cdot \frac{2n + 1}{n + 1} = \frac{n + 1}{n} \text{ (approx) or } m = 1.3500 \]Taking natural logarithms yields the linear regression model:
\[ \ln(u_s) = m \cdot \ln(\Gamma) + c \]Plotting $\ln(u_s)$ versus $\ln(\Gamma)$ yields a straight line whose slope $m$ uniquely determines the fluid flow behavior index $n$.
Viva voce questions & analytical theory
How does the velocity profile of a shear-thinning (pseudoplastic) film differ from a Newtonian film?
A shear-thinning fluid has higher apparent viscosity near the free surface (where shear rate is low) and lower viscosity near the wall (where shear is high), producing a blunter, plug-like velocity profile compared to the semi-parabolic Newtonian profile.
What is the physical significance of the flow behavior index n?
The index n characterizes non-Newtonian deviation: n < 1 indicates shear-thinning (viscosity decreases with shear), n = 1 represents Newtonian fluids, and n > 1 indicates shear-thickening (dilatant) behavior.
What causes wave formation (roll waves) on falling liquid films at higher Reynolds numbers?
Hydrodynamic Kapitza surface instabilities develop when gravitational acceleration overcomes capillary surface tension and viscous damping, forming solitary rolling waves that dramatically enhance gas-liquid mass transfer.
How does plate inclination angle θ affect falling film thickness and surface velocity?
Steeper angles increase the gravitational driving force component (g sin θ), resulting in higher flow velocity and thinner liquid films for a given mass throughput.
Why is carboxymethylcellulose (CMC) widely used as a model non-Newtonian test fluid?
Aqueous CMC solutions exhibit stable, reproducible power-law pseudoplastic rheology over several decades of shear rate, are optically transparent, non-toxic, and water-soluble.
What is the relationship between the slope m = d ln(u_s) / d ln(Γ) and the power-law index n?
Theoretical integration of the power-law equation yields surface velocity scaling u_s ∝ Γ^[(n+1)/n]; thus the logarithmic slope is m = (n+1)/n, allowing direct calculation of n = 1 / (m - 1).
Experimental precautions and process safety
- Ensure the plate surface is thoroughly degreased to prevent film breakdown into rivulets (Marangoni dryout).
- Keep flow rates below the threshold of roll wave formation to ensure smooth laminar film flow.
- Introduce surface tracers gently without imparting initial forward momentum.
- Monitor fluid temperature precisely, as polymer viscosity is highly sensitive to temperature shifts.
Theoretical assumptions & boundary condition limits
- The transport and thermodynamic models apply within the specified calibrated operating window (2.0 to 12.0 g/(s·cm)).
- Assumes ideal continuum flow and homogeneous phase mixing without unmodeled parasitic wall losses.
- Continuous sensor readings reflect ordinary least-squares regression across successive observations.
- High-temperature or extreme concentration excursions require non-ideal equation-of-state corrections.