Pipe flow energy dissipation

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

Numerous researchers have studied damage to micro-organisms during flow in pipes, (Fig. 11) [87,88] Most researchers use a Fanning friction factor, f, to calculate the energy dissipation rate for fully developed flow in tubular bioreactors and capillary flow devices. There are minor differences in the equations that are used but they are generally of the following form [89,901 ... 

Fanning (Darcy) friction factor f(f or fD) e, D 2 V2L fo = 4f TW yv2 e, = friction loss (energy/mass) rw = wall stress (Energy dissipated)/ (KE of flow x 4L/D) or (Wall stress)/ (momentum flux) Flow in pipes, channels, fittings, etc. 

Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. Macroscopically, this energy dissipation requires pressure drop, or velocity decrease. The energy dissipation rate per unit mass is usually denoted e. For steady flow in a pipe, the average energy dissipation rate per unit mass is given by... 

Figure 6. Interfacial area vs. energy dissipation rate (23) 1, dual-flow column 2, pipe flow 3, bubble column 4, stirred tank 5, bubble column with 2-phase nozzle 6, co-current packed bed 7, jet tube washer (2-phase nozzle) 8, tube reactor with...

For DR surfactant solutions with thread-like micelles, Qi proposed a possible DR mechanism in pipe flow as shown in Fig. 16. At rest, thread-like micelles are distributed randomly in the solution. As Arb increases, the thread-like micelles near the wall are extended and start to align along the flow direction because of high wall-shear stress and the solution starts to show DR. Turbulent fluctuations decrease in the radial direction because of the micelle alignment and turbulent energy dissipation is reduced. 

There have been few experimental tests of the theoretical predictions of turbulent coagulation under controlled conditions. Delichaisios and Probstein (1975) measured rates of coagulation of 0.6-mm latex particles suspended in an aqueous solution in turbulent pipe flow. The Reynolds numbers ranged from 17,000 to 51,000 for flow through a 1-in. (l.D.) smooth-walled pipe. For the core of the pipe flow, the turbulence was approximately isotropic. The energy dissipation per unit mass was calculated from the relation... 

A loss coefficient can be defined for any element in which energy is dissipated (pipe, fittings, valves, etc.), although the friction factor is defined only for pipe flow. All that is necessary to describe the pressure-flow relation for pipe flows is Bernoulli s equation and a knowledge of the friction factor, which depends upon flow conditions, pipe size, and fluid properties. 

Estimating the size of the smallest length scale is relatively simple. One could use computational fluid dynamic modeling techniques or estimate them based on the power input to the system (head loss) and the mass of the fluid being powered. For example, in pipe flow, the energy dissipation rate is a function of the total head loss in the flow h, the volumetric flow rate Q, the density of the solution p, and the mass of the solution m, which in this case is the mass of fluid contained within the pipe [Equation (4.1-3)]... 

Friction loss or drag is synonymous with the dissipationof energy. For flow in a pipe, the rate of energy dissipation per unit mass of fluid can be expressed in terms of the mean- flow velocity and either the wall stress, pressure drop, or Fanning friction factor, as follows ... 

Figure 8.6. (a) Sketch of the turbulence production, diffusion, and dissipation processes within a pipe flow, in an established regime, (b) Radial profiles of turbulent kinetic energy, based on Laufer s experimental data (1954)... 

The pipe line and equipment dissipate the flow energy. In the process industry, flows are usually turbulent in this case it holds that ... 

Equations (7-13) and (7-14) is applicable only to a mixing length of five pipe diameters for systems turbulent in the tailpipe. Equations (7-13) and (7-14) have been developed for low viscosity fluids. As viscosity increases, the flow approaches transitional. Due to the high energy dissipation in the tee, the flow can be turbulent even at Reynolds numbers down to about 1000, depending on design and flows. However, once outside the mixer, the flow rapidly becomes laminar in the tailpipe. Equations (7-13) and (7-14) should be used with great care with such systems. 

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