1. Pressure drop and loss due to friction When the fluid is in steady-state laminar flow in a pipe, for a Newtonian fluid, the Hagen Poiseuille equation is obtained. This can be written as 1.4-20 One of the uses of HagenPoiseuille equation is in the experimental measurement of the viscosity of a fluid by measuring the pressure drop and volumetric flow rate through a tube of known length and diameter. 2. Relation between skin friction and wall shearFor horizontal pipe and constant cross section, Z2-Z1=0, and the two kinetic- energy terms can be canceled. Eq 1.3-25 becomesThe relationship between skin friction and pressure drop can be written . Then equation becomesThis is the mechanical-energy loss due to skin friction and is part of the hf term for losses in the mechanical- energy-balance equation (1.3-25). 1.4-5 This term (p1 - p2)f for skin-friction loss is different from the (p1 - p2) term, owing to velocity head or potential head changes .3. The friction factor A common parameter used in fluid flow is the Fanning friction factor, f, which is defined as shear stress w at the surface divided by the product of density and velocity head. 1.4-7 Equation (1.4-1 ) can be written for entire cross section of the tube by taking =w and r = R. Equation (1.4-1 ) then becomes: 1.4-2 Rearranging equation (1.4-2 ) givesSubstituting from equation above into equation (1.4-7) gives Rearranging the equation and let L=L, then the equation becomes1.4-9 and1.4-10thus =4fdefining as a friction coefficientThe equation above and Eq(1.4-10) is called the Fanning equation ,and the friction factor f is called the Fanning friction factor.The equation(1.4-10) is the equation usually used to calculate skin friction loss in straight pipe.For laminar flow only, combining Eqs. (1.4-20 ) and (1.4-10) .gives(1.4-22 )It is not possible to predict theoretically the Fanning friction factor f for turbulent flow as was done for laminar flow.1.4.3 Turbulent Flow in Pipes and Channels Because of the dependence of important flow parameters on the velocity distribution, theoretical and experimental study has been devoted to determining the velocity distribution in turbulent flow.Although the problem has not been completely solved, useful relationships are available. For turbulent flow the friction factor must be determined empirically, and it not only depends upon the Reynolds number but also on surface roughness of the pipe. In laminar flow the roughness has essentially no effect. A large number of experimental data on friction factors for smooth pipe and coarse pipes have been obtained and correlated. For design purposes, to predict the friction factor f and, hence, the frictional pressure drop for round pipe, the friction-factor chart can be used. It is a loglog plot of f versus Re. Completely turbulent flow zoneLaminar flow zoneFor the region with a Reynolds number below 2100, the line is the same as Eq. (1.4-22 ). For a Reynolds number above 4000 for turbulent flow, the lowest line in figure represents the friction-factor line for smooth pipes and tubes. The other lines, for higher friction factors, represent lines for different relative roughness factors, /D, where D is the inside pipe diameter and is a roughness parameter. For turbulent flow the lowest line represents the friction factor for smooth tubes.This applies over a range of Reynolds number from 50000 to 1106Another equation, applicable over a range of Reynolds numbers from 3000 to 3106, is The other curved lines in the turbulent range represent the friction coefficients for various types of pipe, each of which is characterized by a different value of k.1.4.4 Friction from Changes in Velocity or Direction Whenever the velocity of a fluid is changed, either in direction or magnitude,friction is generated in addition to the skin friction resulting from flow through a straight pipe. Friction Losses in Expansion, Contraction, and Pipe Fittings Skin-friction losses in flow through straight pipe are calculated by using the Fanning friction factor. However, if the velocity of the fluid is changed in direction or magnitude, additional friction losses occur. This results from additional turbulence which develops because of vortices and other factors. 1.Sudden enlargement losses If the cross section of a pipe changes suddenly, it results in additional losses due to eddies formed by the jet expanding in the enlarged section. The friction loss hfe from sudden expansion of cross section is proportional to the velocity head of the fluid in the small conduit .In this case the calculation of hfe can be made theoretically and satisfactory result obtained. The calculation utilizes the continuity equation , steady-flow momentum-balance equation , and Bernoulli equation .The equation was derived as followsThe momentum equation between the station 1 and 2 gives12Since Z=0, mechanical energy balance equation may be written for this situation asElimination of p1-p2, since Fr