1. First Mechanics Analysis of Plug Flow
Konrad was the first to begin the analysis of plug flow using the basic mechanics approach. Much of the analysis is similar to soil mechanics.

3. Weber/Muschelknautz’s Macro Plug Analysis
Weber and Muschelknautz used a force balance approach on a single plug using the pressure drop across the plug to move the plug.

5. Exponential Pressure Rise Across a Plug
From the force balance analysis on finds that there will be an exponential rise in pressure across a plug which has a significant effect on the transport of long plugs in a pipeline.

6. Exponential Behavior of Pressure across a plug – Weber – Muschelknautz and Krambrock

7. Steigmeier Friction Factor Analysis
Steigmeier determined the friction factor of a plug moving against a pipe wall using an empirical analysis approach.

8. Dense Phase Transport
Stegmaier

9. Early Analysis of Dense Phase Conveying – Wen and Simon
Wen and Simon carried out a simple analysis on dense phase flow which probably is valid for wave like transport.

10. Wen-Simon Equation for Two Phase Flow

11. Mi and Wypych Equation for Dense Phase Conveying in Plugs

12. Enhancement of the Konrad Analysis for Wypych and Mi
Wypych and Mi started from the basic analysis of Konrad and enhanced the analysis using the follow parameters.

13. Nomenclature for Mi and Wypych Equation
ΔP : Overall horizontal pipeline pressure drop, (Pa)
Lth : The total length of horizontal pipeline, (m)
µw : Coefficient of wall friction (µw = Tan(φw))
φ w : Angle of wall friction
K : Stress transmission coefficient measured
ms : Mass flow-rate of solids, (kg/s)
A : Cross sectional area of pipe (m2)
Us : Velocity of slug, (m/s)
Fr : The Froude number for the material
D : Pipe Diameter, (m)
g : Gravitational constant