Friction losses for power-law and viscoplastic materials in an entrance of a tube and an abrupt contraction

Non-Newtonian fluids are widely used in the petroleum industry and there is a huge class of these materials that can be modeled either as a power-law fluid, when the material exhibits pseudo-plastic behavior, or as a Bingham fluid, when the material possesses a yield stress. Pressure losses in piping systems result from a variety of sources that can be roughly divided into the wall friction, change in the flow direction, and in the cross section of the duct. Here we use the commercial software Polyflow to measure the friction losses in two very common situations when dealing with pipe systems: the abrupt contraction and the entrance of a tube where the flow is not fully developed. Two kinds of non-Newtonian materials are tested, a power-law fluid and a Bingham material, and compared with the Newtonian result. The main objective is to investigate how the power-law index, n, and a dimensionless yield stress, τ′0, (similar to the Bingham number) influence these pressure losses. For the entrance region, we have found that as n decreases, friction loss also decreases and therefore, a pseudoplastic fluid has a lower friction factor than a Newtonian one. In the same geometry, an increase in the dimensionless yield stress decreases the friction factor. For the abrupt contraction geometry, there is an interesting change of relative position among the different power-law fluids after a critical Reynolds number. For the viscoplastic Bingham material, an increase of the dimensionless yield stress increases the friction losses. Using the numerical data obtained, we were able to construct master equations for the friction loss coefficient, K, as a function of the Reynolds number and the relevant dimensionless rheological parameter. All the cases presented a linearity with the inverse of Re. For the viscoplastic material K is linear with τ′0 in the two geometries considered. The power-law fluid presented a more complex behavior. For high Re, K exhibits a linear dependence on n. For low Reynolds numbers, in the entrance flow we found an exponential dependence on n, while in the abrupt contraction this dependence was found to be logarithmic.

Research Highlights
► We find friction coefficients (K) for viscoplastic and pseudoplastic fluids.
► The tested fittings are abrupt contraction and entrance flows.
► Analytical friction coefficients as functions of Re and rheological parameters.
► K decreases with yield number (viscoplastic fluids).
► K has a change on tendency for a critical exponent coefficient (pseudoplastic fluids).