Kinetic energy, correction factor turbulent

Evaluate the kinetic energy correction factor a in Bernoulli s equation for turbulent flow assuming that the 1/7 power law velocity profile [Eq. (6-36)] is valid. Repeat this for laminar flow of a Newtonian fluid in a tube, for which the velocity profile is parabolic. 

The a s are the kinetic energy correction factors at the upstream and downstream points (recall that a = 2 for laminar flow and a = 1 for turbulent flow for a Newtonian fluid). 

In the MEB equation, kinetic energy losses can be calculated easily provided that the kinetic energy correction factor a can be determined. In turbulent flow, often, the value of a = 2 is used in the MEB equation. When the flow is laminar and the fluid is Newtonian, the value of a = 1 is used. Osorio and Steffe (1984) showed that for fluids that follow the Herschel-Bulkley model, the value of a in laminar flow depends on both the flow behavior index ( ) and the dimensionless yield stress ( o) defined above. They developed an analytical expression and also presented their results in graphical form for a as a function of the flow behavior index ( ) and the dimensionless yield stress ( o)- When possible, the values presented by Osorio and Steffe (1984) should be used. For FCOJ samples that do not exhibit yield stress and are mildly shear-thinning, it seems reasonable to use a value of a = 1. 

The kinetic-energy correction factor may be important in applying Bernoulli s theorem between stations when one is in laminar flow and the other in turbulent flow. Also, factors a and p are of some importance in certain types of compact heat-exchange equipment, where there are many changes in size of the fluid channel and where the tubes or heat-transfer surfaces themselves are short. In most practical situations both are taken as unity in turbulent flow. 

For steady flow in a pipe or tube the kinetic energy term can be written as m2/(2 a) where u is the volumetric average velocity in the pipe or tube and a is a dimensionless correction factor which accounts for the velocity distribution across the pipe or tube. Fluids that are treated as compressible are almost always in turbulent flow and a is approximately 1 for turbulent flow. Thus for a compressible fluid flowing in a pipe or tube, equation 6.4 can be written as... 

Each term has the dimensions of energy per unit of mass - in this case, ft-lbp/lbM. The factor, a, in the kinetic energy term, Av /2agc, corrects for the velocity profile across a duct. For laminar flow in a circular duct, the velocity profile is parabolic, and a = 1/2. If the velocity profile is flat, a = 1. For very rough pipes and turbulent flow, a may reach a value of 0.77 [10]. In many engineering applications, it suffices to let a = 1 for turbulent flow. 

The a factors in the kinetic energy (or velocity head) term represent a correction factor to account for the deviation from ping flow throngh the conduit. For a Newtonian flnid in laminar flow in a circular tube, the profile is parabolic and the valne of a is 2. For a highly tnrbnlent flow, the profile is much flatter and a 1.06 (depending on the Reynolds number), although for practical purposes it is usually assumed that a = 1 for turbulent flow. 

The kinetic-energy and momentnm correction factors. Values of a and p for turbulent flow are closer to unity than for laminar flow. Equations for these correction factors are readily obtained, however, by integrating Eqs. (4.13) and... 

The term (i )j /(2i , ) can be replaced by viJli, where a is the kinetic-energy velocity correction factor and is equal to ilJ(v ), . The term a has been evaluated for various flows in pipes and is y for laminar flow and close to 1.0 for turbulent flow. (See Section 2-7D.) Hence, Eq. (2.7-9) becomes... 

Kinetic-Energy Velocity Correction Factor for Turbulent Flow. Derive the equa-... 

Again, to assess the physical deviation between the average of products and the product of averages a kinetic-energy velocity correction factor can be defined by Ck = v a/ vz) - By use of the Hagen-Poiseuille law (1.359) and the power law velocity profile (1.360) it follows that at steady state Ck takes a value of about 0.95 for turbulent flow and 0.5 for laminar fiow [55]. In practice a value of 1 is used in turbulent flow so v a is simply replaced by the averaged bulk velocity vz) -Internal Energy Equation... 

---