The following data were obtained when tomato catsup was...

The following data were obtained when tomato catsup was passed through a tube having an inside diameter of $1.384 \mathrm{~cm}$ and a length of $1.22 \mathrm{~m}$. $$ \begin{array}{cc} \hline \text { Flow rate }\left(\mathrm{cm}^3 / \mathrm{s}\right) & \left.P \text { (dynes } / \mathrm{cm}^2\right) \\ \hline 107.5 & 50.99 \times 10^4 \\ 67.83 & 42.03 \times 10^4 \\ 50.89 & 33.07 \times 10^4 \\ 40.31 & 29.62 \times 10^4 \\ 10.10 & 15.56 \times 10^4 \\ 8.80 & 14.49 \times 10^4 \\ 33.77 & 31.00 \times 10^4 \\ 53.36 & 35.14 \times 10^4 \\ 104.41 & 46.85 \times 10^4 \\ \hline \end{array} $$ Determine the fluid consistency index $\mathrm{K}$ and the flow behavior index $\mathrm{n}$ of this fluid.

The following data were obtained when tomato catsup was passed through a tube having an inside diameter of $1.384 \mathrm{~cm}$ and a length of $1.22 \mathrm{~m}$.
$$
\begin{array}{cc}
\hline \text { Flow rate }\left(\mathrm{cm}^3 / \mathrm{s}\right) & \left.P \text { (dynes } / \mathrm{cm}^2\right) \\
\hline 107.5 & 50.99 \times 10^4 \\
67.83 & 42.03 \times 10^4 \\
50.89 & 33.07 \times 10^4 \\
40.31 & 29.62 \times 10^4 \\
10.10 & 15.56 \times 10^4 \\
8.80 & 14.49 \times 10^4 \\
33.77 & 31.00 \times 10^4 \\
53.36 & 35.14 \times 10^4 \\
104.41 & 46.85 \times 10^4 \\
\hline
\end{array}
$$
Determine the fluid consistency index $\mathrm{K}$ and the flow behavior index $\mathrm{n}$ of this fluid.

Key Concepts

Power?Law Model for Non?Newtonian Fluids

The power?law model is used to describe non?Newtonian fluids, whose viscosity is not constant but changes with the rate of shear. According to this model, the shear stress is proportional to the shear rate raised to a power, which allows the characterization of fluids that exhibit either shear thinning (n < 1) or shear thickening (n > 1) behavior.

Fluid Consistency Index (K)

The fluid consistency index, denoted by K, is a parameter in the power?law equation that relates shear stress to shear rate. It essentially quantifies the apparent viscosity of a non?Newtonian fluid at a unit shear rate. Higher values of K indicate a higher resistance to flow under applied stress.

Flow Behavior Index (n)

The flow behavior index, represented by n, is a dimensionless factor in the power?law formulation that indicates how the viscosity of a fluid changes with the rate of shear. If n equals 1, the fluid is Newtonian with constant viscosity, whereas values of n less than 1 or greater than 1 signify shear thinning or shear thickening behavior, respectively.

Pipe Flow Dynamics in Non?Newtonian Fluids

In the context of pipe flow, experimental data such as flow rate and pressure drop can be utilized to determine the parameters of the power?law model. This involves analyzing how pressure gradients drive flow through a conduit and often employs dimensional analysis and curve fitting techniques to extract the fluid consistency and flow behavior indices from the measured data.

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