Turbulent pressure drop

Full bore tfvough-corxkjit design elirrurtales turbulence. Pressure drop ii rvi greater than Ihrough an equal ler ih ol pipa. 

Transition, 2000 < Re < 5000. Flow fluctuates between laminar and turbulent. Pressure drops fluctuate and are uncertain. Avoid the transition region. 

The two terms on the right-hand side represent the contributions due to viscous and turbulent flow resistances. It was assumed that the pressure drop due to viscous resistance is negligible ky = 0). However, the parametric sensitivity of K on the critical superficial gas velocity was also studied. The turbulent pressure drop was estimated using the orifice equation... 

Equation (7) is based on the universal flow profile for turbulent flow. Values of A and G were obtained from the theoretical analysis of the laminar and turbulent pressure drop characteristics of assemblies of circular channels with different diameters. These values also proved to give accurate friction factor predictions for noncircular channels, such as symmetric and asymmetric annuli and rod bundles. 

The general scaleup relationships in Tables 3.1-3.3 are made specific for scaling in series by setting = 1 and Sl = S. The results are Srs = S for Reynolds number, Sap = for a turbulent pressure drop, and Sap = for a laminar pressure drop. 

If there is no indication about the nature of the fluid flow, both the laminar and turbulent pressure drops must be calculated in the manner as mentioned in the previous paragraphs. The larger value of friction pressure is to be used. 

Meanwhile, the inertial resistance term is dominated by turbulent flow. Pressure drop in turbulent pipe flow is proportional to a frictional coefficient times plfi. Approximating the LAD screen as a bank of closely packed pipes, the relationship between turbulent pressure drop and velocity is simply ... 

FIGURE 12.7 Plot of the Worst-Case Turbulent Pressure Drop down the Channel as a Percentage of the Maximum Pressure Drop across the Screen. 

The first term (AQ) is the pressure drop due to laminar flow, and the FQ term is the pressure drop due to turbulent flow. The A and F factors can be determined by well testing, or from the fluid and reservoir properties, if known. 

The basic concepts of a gas-fluidized bed are illustrated in Figure 1. Gas velocity in fluidized beds is normally expressed as a superficial velocity, U, the gas velocity through the vessel assuming that the vessel is empty. At a low gas velocity, the soHds do not move. This constitutes a packed bed. As the gas velocity is increased, the pressure drop increases until the drag plus the buoyancy forces on the particle overcome its weight and any interparticle forces. At this point, the bed is said to be minimally fluidized, and this gas velocity is termed the minimum fluidization velocity, The bed expands slightly at this condition, and the particles are free to move about (Fig. lb). As the velocity is increased further, bubbles can form. The soHds movement is more turbulent, and the bed expands to accommodate the volume of the bubbles. 

The overall pressure drop is expressed as the sum of a laminar term proportional to FT/DT and a turbulent term proportional to FP/DI to yield the Ergun equation (1) ... 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

Because of its small size and portabiHty, the hot-wire anemometer is ideally suited to measure gas velocities either continuously or on a troubleshooting basis in systems where excess pressure drop cannot be tolerated. Furnaces, smokestacks, electrostatic precipitators, and air ducts are typical areas of appHcation. Its fast response to velocity or temperature fluctuations in the surrounding gas makes it particularly useful in studying the turbulence characteristics and rapidity of mixing in gas streams. The constant current mode of operation has a wide frequency response and relatively lower noise level, provided a sufficiently small wire can be used. Where a more mgged wire is required, the constant temperature mode is employed because of its insensitivity to sensor heat capacity. In Hquids, hot-film sensors are employed instead of wires. The sensor consists of a thin metallic film mounted on the surface of a thermally and electrically insulated probe. 

Friction Coefficient. In the design of a heat exchanger, the pumping requirement is an important consideration. For a fully developed laminar flow, the pressure drop inside a tube is inversely proportional to the fourth power of the inside tube diameter. For a turbulent flow, the pressure drop is inversely proportional to D where n Hes between 4.8 and 5. In general, the internal tube diameter, plays the most important role in the deterrnination of the pumping requirement. It can be calculated using the Darcy friction coefficient,, defined as... 

This term is a measure of the unit s length. Sometimes it is referred to as the number of transfer units. This simply says that the optimum pressure drop increases as the heat exchanger gets longer, ie, has more transfer units. The forms of F, and F both foUow from the fact that in turbulent flow the... 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

Turbulent Flow The correlation by Grimison (Trans. ASME, 59, 583—.594 [1937]) is recommended for predicting pressure drop for turbulent flow (Re > 2,000) across staggered or in-hne tube banks for tube spacings [(a/Dt), (b/Dt)] ranging from 1.25 to 3.0. The pressure drop is given by... 

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

Proportional Element First, consider the outflow through the exit valve on the tank. If the flow through the line is turbulent, then Bernoulh s equation can be used to relate the flow rate through the valve to the pressure drop across the valve as ... 

Pressure drop in catalyst beds is governed by the same principles as in any flow system. Consequently, at very low flow, pressure drop is directly proportional to velocity, and at very high flow, to the square of velocity. These conditions correspond to the laminar and turbulent regimes of the flow. 

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