22-Agric-A6 Physical Properties of Biological Materials and Food Products · May 2017
Question 3 of 9: Flow Behaviour Index of Tomato Catsup from Brookfield Viscometer Data
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 04-Agric-A6 Physical Properties of Biological Materials and
Food Products, National Exams May 2017 — a three-hour closed-book
exam (approved calculator permitted; one aid sheet, both sides). Nine questions are set and
candidates answer any five, each worth 20 marks, for a 100-mark paper. All nine are worked here
so the set is a complete study resource.
Reference texts. M.A. Rao, S.S.H. Rizvi, A.K. Datta and J. Ahmed,
Engineering Properties of Foods, 4th ed. (rheology of fluid and semisolid foods,
particle size, surface/interfacial properties); N.N. Mohsenin, Physical Properties of
Plant and Animal Materials, 2nd ed. (thermal properties, calorimetry, texture and
rheological testing); R.P. Singh and D.R. Heldman, Introduction to Food Engineering,
5th ed. (thermal-property measurement, freezing-point depression, particle size); J.F. Steffe,
Rheological Methods in Food Process Engineering, 2nd ed. (viscometry, viscoelasticity,
the Kelvin-Voigt/Maxwell models, time-dependent flow behaviour); R.L. Earle, Unit
Operations in Food Processing, 2nd ed. (particle-size averages, specific surface from
sieve/count data).
Question 3: Flow Behaviour Index of Tomato Catsup from Brookfield Viscometer Data
(20 marks)
Brookfield spindle No. 4 readings on tomato catsup
Rotational speed, \(N\) (rpm)
2
4
10
20
Indicator reading, %FS
53.5
67
80.5
97
Viscometer constant (full-scale spring torque): 7187 dyn·cm.
Find. The flow behaviour (power-law) index \(n\) of the power-law model
\(\tau = K\dot\gamma^{\,n}\).
Approach. For a fixed spindle in a fixed sample, torque and %FS are directly
proportional (\(T = T_{fs}\times \%\text{FS}/100\)) and shear rate is directly proportional to
rotational speed \(N\) (through spindle- and geometry-specific constants that stay fixed across
all four readings). Both proportionality constants therefore drop out of the SLOPE of a
\(\ln(\%\text{FS})\) versus \(\ln N\) plot, even though the spindle's radii are not given —
so the flow behaviour index can be read directly from that slope.
Check: the spindle geometry (radii) needed to convert %FS into an absolute
shear stress, and hence to report the consistency coefficient \(K\) in Pa·s\(^n\), is not
given for a Brookfield-type spindle (only the instrument's full-scale spring torque is). This
does not affect \(n\), which is scale-independent, but a numeric \(K\) cannot be reported without
the manufacturer's spindle multiplier — only \(n\) is required here.
Linearize. Since \(\tau \propto \%\text{FS}\) and \(\dot\gamma\propto N\)
through fixed constants, \(\tau=K\dot\gamma^{\,n}\) implies \(\%\text{FS} = K'N^{\,n}\), so
$$\ln(\%\text{FS}) = \ln K' + n\ln N$$
is linear in \(\ln N\) with slope \(n\).
Tabulate the logarithms.
\(N\) (rpm)
%FS
\(\ln N\)
\(\ln(\%\text{FS})\)
2
53.5
0.6931
3.9784
4
67.0
1.3863
4.2047
10
80.5
2.3026
4.3888
20
97.0
2.9957
4.5747
Least-squares fit. Regressing \(\ln(\%\text{FS})\) on \(\ln N\) over the
four points gives
$$\begin{aligned} n &= \boxed{0.250}, \\ \ln K' &= 3.825\ (R^2 = 0.992). \end{aligned}$$
Interpret. Since \(n<1\), tomato catsup is confirmed
pseudoplastic (shear-thinning) — consistent with its known behaviour as a
concentrated particulate suspension in a serum phase, and with the strongly sub-linear rise of
the indicator reading (53.5→97, less than doubling) over a ten-fold speed increase
(2→20 rpm).
Log-log plot of the viscometer indicator reading against rotational speed;
the four points fall on a straight line of slope \(n=0.250\), confirming power-law
(pseudoplastic) behaviour over this speed range.
