Sudden expansion

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

The sudden expansion of the gases, as they are heated in the arc plasma, causes the formation of a high-speed arc jet so that the atomic hydrogen and the reactive carbon species are transported almost instantly to the deposition surface and the chances of hydrogen recombination and of vapor-phase reactions are minimized. 

Observed premixed edge flames (a) Bunsen flame-tip opening, (b) propagating premixed flame in tube (From Jarosinski, J., Strehlow, R.A., and Azarbarzin, A., Proc. Combust. Inst., 19, 1549, 1982. With permission.), (c) slanted counterflow flame (From Liu, J.-B. and Ronney, P.D., Combust. Sci. Tech., 144,21,1999. With permission.), and (d) spinning premixed flames in sudden expansion tube [7]. 

M. J. Kwon, B. J. Lee, and S. H. Chung, An observation of near-planar spirming premixed flames in a sudden expansion tube, Combust. Flame 105 180-199,1996. 

Table 13.4 gives some typical values of the loss coefficient for various fittings9. It should be noted that values for loss coefficient will vary for the same fitting, but from different manufacturers, as a result of differences in geometry. Table 13.5 gives head losses for sudden contractions, sudden expansions and orifice plates. Note that the relationship for orifice plates in Table 13.5 relates to the overall pressure drop and not the pressure drop between the pressure tappings used to determine the flowrate. 

Table 13.5 Head losses in sudden contractions, sudden expansions and orifice plates. ...

Sudden contraction Sudden expansion Orifice plate... 

The total pressure drop for the tube-side includes the pressure drop in the tube, sudden contractions, sudden expansions and flow reversals. There are two major sources of pressure losses on the tube-side of a shell-and-tube exchanger ... 

The oil-water dynamic interfacial tensions are measured by the pulsed drop (4) technique. The experimental equipment consists of a syringe pump to pump oil, with the demulsifier dissolved in it, through a capillary tip in a thermostated glass cell containing brine or water. The interfacial tension is calculated by measuring the pressure inside a small oil drop formed at the tip of the capillary. In this technique, the syringe pump is stopped at the maximum bubble pressure and the oil-water interface is allowed to expand rapidly till the oil comes out to form a small drop at the capillary tip. Because of the sudden expansion, the interface is initially at a nonequilibrium state. As it approaches equilibrium, the pressure, AP(t), inside the drop decays. The excess pressure is continuously measured by a sensitive pressure transducer. The dynamic tension at time t, is calculated from the Young-Laplace equation... 

The pressure recovery is reduced by the friction loss, which is relatively high for the sudden expansion. The pressure recovery is therefore relatively low. 

Example 5-6 Friction Loss in a Sudden Expansion. Figure 5-7 shows the flow in a sudden expansion from a small conduit to a larger one. We assume that the conditions upstream of the expansion (point 1) are known, as well as the areas A and A2. We desire to find the velocity and pressure downstream of the expansion (V2 and P2) and the loss coefficient, Kt. As before, V2 is determined from the mass balance (continuity equation) applied to the system (the fluid in the shaded area). Assuming constant density,... 

An application of an internal momentum balance to determine the pressure drop in a sudden expansion is given in Section 2.4. 

Pipe entrance and exit pressure losses should also be calculated and added to obtain the overall pressure drop. The loss in pressure due to sudden expansion from a diameter dtl to a larger diameter dl2 is given by the equation... 

These expressions for the losses due to sudden expansion and sudden contraction can be derived by application of Bernoulli s equation and the momentum equation. Figure 2.3 shows a sudden expansion. 

That the losses for sudden expansion and sudden contraction are markedly different may surprise the reader. The reason is that the flow pattern for a sudden contraction is different from that for an expansion as shown in Figure 2.4. 

For a Newtonian liquid, there is negligible frictional loss due to the converging flow. However, a vena contracta forms in the smaller pipe and the expansion and reattachment downstream of it are similar to the flow in the sudden expansion. The difference is that it takes place from dc to dl2. The momentum equation cannot be applied over the whole flow because... 

For laminar flow of a power law fluid through a sudden expansion in a pipe of circular cross section, it is easily shown [Skelland (1967)] that the pressure drop is given by... 

A liquid flows in a steady state in a cylindrical pipe of inside diameter d, = 0.05 m at a flow rate Q = 2 x 10 3 m3/s. Calculate the head loss and the pressure drop for a sudden expansion to a pipe of inside diameter 0.1 m, if the liquid density p — 1000 kg/m3. 

FIGURE 4.52 Stabilization methods for high-velocity streams (a) vee gutter, (b) rodoz sphere, (c) step or sudden expansion, and (d) opposed jet (after Strehlow, Combustion Fundamentals, McGraw-Hill, New York, 1985). 

Let us look now at vapor-liquid systems with more than one component. A liquid stream at high temperature and pressure is flashed into a drum, i.e., its pressure is reduced as it flows through a restriction (valve) at the inlet of the drum. This sudden expansion is irreversible and occurs at constant enthalpy. If it were a reversible expansion, entropy (not enthalpy) would be conserved. If the drum pressure is lower than the bubblepoint pressure of the feed at the feed temperature, some of the liquid feed will vaporize. 

Egorov, A. G., et al. 1989. Experimental study of ignition and stabilization of powder aluminum flame in combustion chamber with sudden expansion. Izvestiya VUZ, Aviatsionnaya Tekhnika 32(2) 85-86. 

Active control has been found to be sensitive to the location of addition of oscillated fuel in disk-stabilized flames. The injection of fuel close to the outer edge of the flame-holding disk led to the largest reduction in RMS pressure fluctuation [14], and the importance of the location of addition of oscillated fuel in a sudden expansion flow has been confirmed recently [24]. Pressure oscil-... 

The tendency of premixed flames to detach from the flame holder to stabilize further downstream has also been reported close to the flammability limit in a two-dimensional sudden expansion flow [27]. The change in flame position in the present annular flow arrangement was a consequence of flow oscillations associated with rough combustion, and the flame can be particularly susceptible to detachment and possible extinction, especially at values of equivalence ratio close to the lean flammability limit. Measurements of extinction in opposed jet flames subject to pressure oscillations [28] show that a number of cycles of local flame extinction and relight were required before the flame finally blew off. The number of cycles over which the extinction process occurred depended on the frequency and amplitude of the oscillated input and the equivalence ratios in the opposed jets. Thus the onset of large amplitudes of oscillations in the lean combustor is not likely to lead to instantaneous blow-off, and the availability of a control mechanism to respond to the naturally occurring oscillations at their onset can slow down the progress towards total extinction and restore a stable flame. 

De Zilwa, S.R. N., L. Khezzar, and J. H. Whitelaw. 1998. Plane sudden expansion flows. Thermofluids Section Report TF/98/05. Imperial College, Department of Mechanical Engineering. 

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