![]() Turbulent Flow in Pipes/Annuli – nonNewtonian Fluids Equivalent Diameter for Annular Geometry – From Momentum Equation From the momentum equation, frictional pressure drop for Newtonian fluid in the annulus is For pipe flow, d1 -> 0 then Comparing these two equations, the equivalent diameter an annulus can be obtained Turbulent Flow in Pipes/Annuli – NonNewtonian Fluids Equivalent Diameter for Annular Geometry – Hydraulic Diameter Method Hydraulic diameter is defined as: Equivalent diameter by using hydraulic diameter method: Turbulent Flow in Pipes – Newtonian Fluids Example Using Blasius approximation ,the equation becomes Pressure drop: DP = dp/dL x D = (0.0777)(10,000) = 777 psi The relative roughness e/d = 0.000013/3.826 = 0.0000034 smooth pipe Solve this equation for the Fanning friction factor: f = 0.00666 Thus the frictional pressure loss can be obtained by ![]() Turbulent Flow in Pipes – Newtonian Fluids Example From table 4.5, the absolute roughness for commercial steel pipe is e = 0.000013 inches. For smooth pipe, the relative roughness e/d 2,100, the flow is under turbulent flow conditions. Turbulent Flow in Pipes – Newtonian Fluids Determination of Friction Factor - Turbulent Flow For turbulent flow, the friction factor can be calculated by using Colebrook correlation. Turbulent Flow in Pipes – Newtonian Fluids Determination of Friction Factor - Laminar Flow Relationship between shear stress and friction factor: Pipe flow under laminar conditions: Therefore, Newtonian fluids flow in pipe under laminar flow conditions: Hence, This equation will be used to calculate the friction factor of Newtonian fluids flow in pipe under laminar flow conditions. Turbulent Flow in Pipes – Newtonian Fluids Determination of Laminar/Turbulent Flow If Re 4,000 Turbulent Note that this critical Reynolds number is correct only for Newtonian fluids Laminar occurs at low Reynolds number, where viscous forces are dominant, and is characterized by smooth, constant fluid motion turbulent flow occurs at high Reynolds number and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities. Reynolds number is used to characterize different flow regimes, such as laminar or turbulent flow. Turbulent Flow in Pipes – Newtonian Fluids Definition of Reynolds Number Reynolds number, Re, is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces. This irregular, fluctuating motion can be regarded as superimposed on the mean motion of the fluid. Turbulent Flow: In turbulent flow, there is an irregular random movement of fluid in transverse direction to the main flow. Laminar flow systems are generally represented graphically by streamlines. There is no microscopic or macroscopic intermixing of the layers. ![]() Turbulent Flow in Pipes – Newtonian Fluids Introduction Laminar Flow: In this type of flow, layers of fluid move in streamlines. Drilling Engineering – PE 311 Turbulent Flow in Pipes and Annuliįrictional Pressure Drop in Pipes and Annuli When attempting to quantify the pressure losses in side the drillstring and in the annulus it is worth considering the following matrix:
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