SOP: Determination of Consistency Constants Using a Tube Viscometer
GIFSQ — Global Institute for Food Safety & Quality
Principle
Fruit purees and juices are non-Newtonian fluids: their resistance to flow changes with the rate of shear, so a single viscosity figure does not describe them. In a tube viscometer, the product is forced through a capillary tube under measured pressure and the rate of flow is recorded at several pressures. From the relationship between the wall shear stress and the wall shear rate, the power-law constants — the flow behaviour index (n), the consistency coefficient (K), the yield stress (C) and the apparent viscosity — are obtained graphically. The flow is assumed to be laminar, the entrance and exit flow effects negligible (tube diameter below 0.64 cm and length-to-diameter ratio above 100), with no slip at the wall, and the shear rate at a point depending only on the shearing stress at that point.
Apparatus required
- Tube viscometer assembly: sample reservoir (3,000–3,500 ml capacity) fitted with an air pressure system, reducing valve and pressure gauge; straight tube (2.54 cm outer diameter, 12.7 cm long); 2.54 cm straight-bore ball valve; capillary tube of 0.26 cm internal diameter and 91 cm long (or 0.49 cm internal diameter and 92 cm long)
- Compressed air supply
- Stop-watch
- Balance, for weighing the collected sample
- Thermometer
- Log-log graph paper and coordinate graph paper
Procedure — Flow measurement
Place 1,000 ml of the sample in the reservoir, cover tightly and admit air from the compressor through the pressure reducing valve
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When the pressure reaches the desired level, close the valve and allow the assembly to equilibrate for 5 min
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Open the ball valve and allow the fluid to flow for a few seconds until a steady state is reached
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Note the weight of the sample flowing through the tube for a precise time interval (5–50 sec) using the stop-watch
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Measure the average pressure; note the height of the sample above the capillary tube and add it to the indicated pressure, and note the temperature
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Measure the flow rate similarly at different pressures
Calculation — Flow behaviour index (n) and consistency coefficient (K)
Convert the pressure difference (ΔP) in psi to dyne/cm² using the expression:
ΔP (dyne/cm²) = psi × 6.7 × 10⁴
On log-log paper, plot ΔP·D/4L on the Y axis against 32Q/πD³ on the X axis, where,
- ΔP = pressure drop in the tube, dyne/cm²
- D = tube diameter, cm
- L = tube length, cm
- Q = volumetric rate of flow, ml/sec
The resulting curve is a straight line; measure its slope, which gives n, the flow behaviour index. Extend the line to the X axis where 32Q/πD³ = 1.0; the intercept is the consistency coefficient K.
Calculation — Yield stress (C)
Most non-Newtonian fluids possess a yield stress: the stress that must be exerted to move one fluid layer past another. To find it, plot the volumetric flow rate Q against ΔP/ΔL on coordinate graph paper and extrapolate the curve to Q = 0. Read the corresponding ΔP/ΔL.
Calculate the yield stress as:
C = (Rᵧ/2) × (ΔP/ΔL)
Where,
- C = yield stress, dyne/cm²
- Rᵧ = radius of the tube, cm
- ΔP/ΔL = pressure drop per unit length at zero flow, dyne/cm³
Note: the book’s printed layout of this equation is illegible in the extraction; the form above follows the book’s worked example (tube of 1.2 cm diameter, ΔP/ΔL = 700 dyne/cm³ at zero flow: C = ½ × 700 × 0.6 = 210 dyne/cm²).
Calculation — Apparent viscosity
The apparent viscosity at various shear rates is calculated from the power-law expression:
ηₐₚₚ = K·γ̇ⁿ⁻¹
Where,
- ηₐₚₚ = apparent viscosity
- K = consistency coefficient
- n = flow behaviour index
- γ̇ = shear rate (stated in the book’s text as the function 32Q/πD³)
Note: the printed equation (11) is illegible in the extraction; the form above follows the book’s text description.
The consistency of thixotropic fluids cannot be measured accurately with the tube viscometer; the reported K and n values are valid only for the specific instrument and the experimental conditions used. Values of K measured with different tube diameters should be nearly the same — a large difference indicates thixotropy, possibly from wall slippage.
Precautions
- Achieve a true steady state before timing the flow.
- Keep the temperature constant throughout; note it with every reading.
- Prevent air entrainment in the capillary tube.
References
Saravacos, G.D., Food Technology, 22, 1585 (1968).
Metzner, A.B., in Advances in Chemical Engineering (Eds Drew, T.B. & J.W. Hoopes, Jr.), Academic Press, New York, 1, 77 (1956).
Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.