Pipe, turbulent flow equations

Various theoretical and empirical models have been derived expressing either charge density or charging current in terms of flow characteristics such as pipe diameter d (m) and flow velocity v (m/s). Liquid dielectric and physical properties appear in more complex models. The application of theoretical models is often limited by the nonavailability or inaccuracy of parameters needed to solve the equations. Empirical models are adequate in most cases. For turbulent flow of nonconductive liquid through a given pipe under conditions where the residence time is long compared with the relaxation time, it is found that the volumetric charge density Qy attains a steady-state value which is directly proportional to flow velocity... 

Turbulent flow pressure loss in pipe (Equation 4-187) is... 

P0 = Np = Power number, dimensionless, Equation 5-19 Ppew = Plate coil width, one plate, ft Ap = Pressure drop, psi AP0 = Pressure drop for open pipe, psi AP, = Static mixer pressure drop in turbulent flow, psi Q = Flow rate or pumping capacity from impeller, cu l t/sec, or Ls/1... 

If at time t the liquid level is D m above the bottom of the tank, then designating point 1 as the liquid level and point 2 as the pipe outlet, and applying the energy balance equation (2.67) for turbulent flow, then ... 

HARTNETT and KOSTIC 26 have recently examined the published correlations for turbulent flow of shear-thinning power-law fluids in pipes and in non-circular ducts, and have concluded that, for smooth pipes, Dodge and Metzner S(27) modification of equation 3.11 (to which it reduces for Newtonian fluids) is the most satisfactory. 

Yooi24) has proposed a simple modification to the Blasius equation for turbulent flow in a pipe, which gives values of the friction factor accurate to within about 10 per cent. The friction factor is expressed in terms of the Metzner and Reed(I8) generalised Reynolds number ReMR and the power-law index n. 

For the heat transfer for fluids flowing in non-circular ducts, such as rectangular ventilating ducts, the equations developed for turbulent flow inside a circular pipe may be used if an equivalent diameter, such as the hydraulic mean diameter de discussed previously, is used in place of d. 

The left-hand side of equation 10.224 is referred to as the y-factor for heat transfer ( //,). Chilton and Colburn found that a plot of against Re gave approximately the same curve as the friction chart (0 versus Re) for turbulent flow of a fluid in a pipe. 

The right-hand side of equation 10.224 gives numerical values which are very close to those obtained from the Blasius equation for the friction factor (j> for the turbulent flow of a fluid through a smooth pipe at Reynolds numbers up to about 106. 

The application to pipe flow is not strictly valid because u (= fRjp) is constant only in regions close to the wall. However, equation 12.34 appears to give a reasonable approximation to velocity profiles for turbulent flow, except near the pipe axis. The errors in this region can be seen from the fact that on differentiation of equation 12.34 and putting y = r, the velocity gradient on the centre line is 2.5u /r instead of zero. 

A simple approximate form of the relation between u+ and y+ for the turbulent flow of a fluid in a pipe of circular cross-section may be obtained using the Prandtl one-seventh power law and the Blasius equation. These two equations have been shown (Section 11.4) to be mutually consistent. 

Equation 12.37 can be used in order to calculate the friction factor

turbulent flow of fluid in a pipe, It is first necessary to obtain an expression for the mean velocity u of the fluid from the relation ... 

For flow in a smooth pipe, the friction factor for turbulent flow is given approximately by the Blasius equation and is proportional to the Reynolds number (and hence the velocity) raised to a power of -2. From equations 12.102 and 12.103, therefore, the heat and mass transfer coefficients are both proportional to w 75. 

For the inlet length of a pipe in which the boundary layers are forming, the equations in the previous section will give an approximate value for the heat transfer coefficient. It should be remembered, however, that the flow in the boundary layer at the entrance to the pipe may be streamline and the point of transition to turbulent flow is not easily defined. The results therefore are, at best, approximate. 

In fully developed flow, equations 12.102 and 12.117 can be used, but it is preferable to work in terms of the mean velocity of flow and the ordinary pipe Reynolds number Re. Furthermore, the heat transfer coefficient is generally expressed in terms of a driving force equal to the difference between the bulk fluid temperature and the wall temperature. If the fluid is highly turbulent, however, the bulk temperature will be quite close to the temperature 6S at the axis. 

Liquid core temperature and velocity distribution analysis. BankofT (1961) analyzed the convective heat transfer capability of a subcooled liquid core in local boiling by using the turbulent liquid flow equations. He found that boiling crisis occurs when the core is unable to remove the heat as fast as it can be transmitted by the wall. The temperature and velocity distributions were analyzed in the singlephase turbulent core of a boiling annular flow in a circular pipe of radius r. For fully developed steady flow, the momentum equation is given as... 

For steady, uniform, fully developed flow in a pipe (or any conduit), the conservation of mass, energy, and momentum equations can be arranged in specific forms that are most useful for the analysis of such problems. These general expressions are valid for both Newtonian and non-Newtonian fluids in either laminar or turbulent flow. 

Like the von Karman equation, this equation is implicit in/. Equation (6-46) can be applied to any non-Newtonian fluid if the parameter n is interpreted to be the point slope of the shear stress versus shear rate plot from (laminar) viscosity measurements, at the wall shear stress (or shear rate) corresponding to the conditions of interest in turbulent flow. However, it is not a simple matter to acquire the needed data over the appropriate range or to solve the equation for / for a given flow rate and pipe diameter, in turbulent flow. 

Derive the relation between the friction factor and Reynolds number in turbulent flow for smooth pipe [Eq. (6-34)], starting with the von Karman equation for the velocity distribution in the turbulent boundary layer [Eq. (6-26)]. 

For fully developed turbulent flow in rough pipes,/is independent of the Reynolds number, as shown by the nearly constant friction factors at high Reynolds number in Figure 4-7. For this case Equation 4-33 is simplified to... 

Thus the velocity of the liquid discharging from the pipe is 3.66 m/s. The table also shows that the friction factor/changes little with the Reynolds number. Thus we can approximate it using Equation 4-34 for fully developed turbulent flow in rough pipes. Equation 4-34 produces a friction factor value of 0.0041. Then... 

It was noted in Section 1.3 that the frictional pressure drop for turbulent flow in a pipe varies as the square of the flow rate at very high values of Re. At lower values of Re the pressure drop varies with flow rate, and therefore with Re, to a slightly lower power which gradually increases to the value 2 as Re increases. The pressure drop in turbulent flow is also proportional to the density of the fluid. This suggests writing equation 2.7 in the form... 

Considerable effort has been expended in trying to And algebraic expressions to relate/to Re and eld,. For turbulent flow in smooth pipes, the simplest expression is the Blasius equation ... 

The most widely accepted relationship for turbulent flow in smooth pipes is the von Karman equation... 

An approximate equation for the profile of the time-averaged velocity for steady turbulent flow of a Newtonian fluid through a pipe of circular... 

Equation 2.69 fits the experimental data for turbulent flow in smooth pipes of circular cross section for y+ > 30 when 1 IK and C are given the values 2.5 and 5.5 ... 

Experimental results for the Fanning friction factor for turbulent flow of shear thinning fluids in smooth pipes have been correlated by Dodge and Metzner (1959) as a generalized form of the von Karman equation ... 

Dodge and Metzner (1959) deduced the velocity profile from their measurements of flow rate and pressure gradient for turbulent flow of power law fluids in pipes. For the turbulent core, the appropriate equation is... 

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... 

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