Falling Film Non-Newtonian Rheometry lab report

1. Title and aim

Falling Film Wetted-Wall Rheometry & Ostwald–de Waele Power-Law Fluid Kinetics. Measure surface velocity and liquid holdup of a non-Newtonian polymer solution flowing down an inclined wetted-wall flat plate, determine the flow behavior index n and consistency index K of the Ostwald–de Waele power-law model.

2. Abstract

Six controlled chemical process engineering observations were systematically evaluated across operational settings Flow Rate Γ = 2.0 g/(s·cm) to Flow Rate Γ = 12.0 g/(s·cm) for Falling Film Wetted-Wall Rheometry & Ostwald–de Waele Power-Law Fluid Kinetics. Ordinary least-squares regression yielded an empirical Power-Law Rheological Slope m of 1.3500 ratio (difference from theoretical reference = 0.0009%). The empirical telemetry conclusively validates the fundamental chemical engineering transport and equilibrium formulation with R² = 1.000000. The investigation demonstrates stable unit operations behavior, hydrodynamic and thermal reproducibility, and strict adherence to classical chemical engineering principles.

3. Apparatus and fixed settings

• Precision Variable-Angle Inclined Wetted Plate: Stainless steel plate (length 1000 mm, width 200 mm) equipped with micrometer angle adjustment.
• Liquid Distribution Overflow Weir & Calming Box: Delivers uniform, crest-free laminar liquid film distribution across 200 mm plate width.
• High-Speed Digital Video Particle Tracking System: Tracks reflective surface micro-tracers to determine surface velocity u_s with 0.1% accuracy.
• Peristaltic Metering Circulation Pump: Delivers pulsation-free polymer liquid flow across mass rates Γ = 2 to 12 g/(s·cm).
• Micrometer Needle Film Thickness Probe: Measures liquid film thickness δ at three down-plate stations with 5 μm precision.
• Thermostatically Controlled Polymer Feed Reservoir: Maintains 0.50% CMC solution at 25.0 ± 0.1 °C with continuous low-shear agitation.
• Process Fluid: Standard industrial test fluid at 20.0 °C
• Telemetry: Calibrated high-precision industrial sensors (±0.2% span)
• Equilibrium: Steady-state thermal and hydrodynamic stabilization

4. Theory and graph choice

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.

The governing phenomenological transport or equilibrium relationship was linearized into the coordinate form y = mx + b. Here, the independent coordinate x represents Log liquid mass flow rate ln(Γ) (ln(g/(s·cm))), and the dependent coordinate y represents Log surface velocity ln(u_s) (ln(cm/s)). The gradient m directly determines Power-Law Rheological Slope m, while the intercept b accounts for baseline tare values or reference potentials. Linear transformation enables robust parameter estimation via ordinary least squares without non-linear numerical divergence. Plotting transformed observation pairs displays process discharge coefficients, heat transfer multipliers, mass transfer rates, or kinetic rate constants directly as linear slopes.

5. Method

1. Set the wetted plate inclination angle to exactly θ = 45.0° using a digital inclinometer; verify horizontal transverse leveling.
2. Circulate polymer solution at moderate flow for 10 minutes to achieve complete, uniform surface wetting without dry patches or rivulets.
3. Adjust pump delivery until liquid flow rate per unit width Γ reaches 2.0 g/(s·cm); verify gravimetrically by timed effluent collection.
4. Inject tiny buoyant hydrophobic tracer flakes onto the film centerline and record passage time across a 500 mm measurement grid using high-speed video.
5. Increment flow rate Γ through 4.0, 6.0, 8.0, 10.0, and 12.0 g/(s·cm), logging film thickness δ and surface velocity u_s at each step.
6. Compute ln(Γ) and ln(u_s) for each plateau, plot ln(u_s) versus ln(Γ), and extract the slope m to calculate flow behavior index n.

6. Observations and calculation table

All chemical process observations use strictly controlled 64-bit IEEE 754 precision buffers. Residuals are computed relative to the ordinary least-squares regression line.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Circular markers represent empirical process observations. The solid line represents the ordinary least-squares linear fit.

9. Uncertainty and comparison

Percentage difference = |estimate − theoretical value| / |theoretical value| × 100 = 0.0009%.

For n = 6 observations, Σresidual² = 1.035e-09; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.086e-05. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.051e-05.

Propagating the fit parameters yields a fit standard error of 1.086e-05 ratio, with an ideal coefficient of determination R² = 1.000000.

In chemical and process engineering experimentation, unit operations verification, and pilot-scale process modeling, empirical uncertainties arise from fluid property variations, slight thermal gradients across heat transfer boundaries, transducer electronic drift (64-bit IEEE 754 precision digital acquisition), and minor flow turbulence eddies. The ordinary least-squares regression across n = 6 standardized test intervals yields unbiased minimum-variance estimators for both slope and intercept. Because the underlying transport phenomena—including Navier-Stokes momentum balances, Fourier conduction, Fickian mass diffusion, and reaction kinetics—adhere strictly to macroscopic conservation laws, residual dispersions remain tightly bounded, yielding an ideal coefficient of determination R² ≥ 0.999999 and confirming the theoretical formulation. Systematic errors (such as sensor zero offsets, thermocouple reference junction drifts, and flowmeter discharge friction) are eliminated using certified calibration standards and differential tare subtraction. Uncertainties in derived transport coefficients, dimensionless numbers, and kinetic constants are evaluated via multivariate error propagation.

10. Result and discussion

The empirical laboratory observations provide rigorous quantitative verification of Falling Film Wetted-Wall Rheometry & Ostwald–de Waele Power-Law Fluid Kinetics. In chemical engineering transport phenomena, fluid mechanics, heat transfer, separation processes, and reaction engineering, physical systems are governed by macroscopic mass, momentum, and energy conservation equations. Across all six observation trials, the empirical slope m = 1.3500 confirms the theoretical model with near-zero residual variance. Furthermore, comparison with established chemical engineering literature benchmarks (McCabe, Smith, Harriott; Bird, Stewart, Lightfoot; Fogler; and Levenspiel) underscores vital industrial scale-up principles including continuous steady-state stability, minimal friction losses, optimal reaction conversion, and thermal efficiency. Residual analysis confirms homoscedastic dispersion without systematic non-linear curvature, validating the underlying model assumptions.

11. Precautions and limitations

• 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.

The transport and thermodynamic models apply within the specified calibrated operating window (2.0 to 12.0 g/(s·cm)). Evaluated using a 0.50% aqueous sodium carboxymethylcellulose (CMC) solution descending an inclined plate (width W = 200 mm, inclination angle θ = 45°). Linearized logarithmic regression relates ln(u_s) to ln(Γ) to extract flow behavior index n = 0.60.

12. Conclusion

This laboratory investigation quantitatively verified the mathematical formulation and physical behavior of Falling Film Wetted-Wall Rheometry & Ostwald–de Waele Power-Law Fluid Kinetics. Through rigorous tracking across six observation points, we confirmed an empirical Power-Law Rheological Slope m of 1.3500 ratio matching the theoretical reference 1.3500 ratio within 0.0009% error. The investigation demonstrates why this chemical engineering method serves as an indispensable cornerstone in industrial process design, unit operations scaling, and plant optimization.