The Austin-Palfrey equation could be directly applied to a pipeline where both the Reynolds number and the pipe diameter are not varied along the entire length and where there is no initial mixing volume at the start point of the pipeline. If the Reynolds number, or the pipe diameter, or both are varied along a pipeline, the pipeline should be divided into segments based on the point along the pipeline where Reynolds number or the diameter change. The Austin-Palfrey equation should be revised to include effects of the initial mixing volumes in subsequent segments.
In this study, the Austin-Palfrey equations were revised by introducing an initial pipe length or an initial length factor to incorporate the initial mixing volumes at the start point of each pipe segment. The revised equations could predict the mixing volume along the entire pipeline length. The initial length factor has an impact on the incremental mixing length in subsequent segments. When the initial length factor is larger than one, the mixing length in subsequent pipe segment tends to decrease. When the initial length factor is less than one, the mixing length in subsequent pipe segment tends to increase.