Pipe flow turbulence

The standard k-e model simulates the turbulence in the reactor. For flow within the porous catalyst bed, however, we suppress the turbulence. We enter the appropriate physical properties of the system, and employ standard boundary conditions at the impermeable walls and the reactor outlet. To represent the turbulence of the feed stream at the inlet, we treat it as pipe-flow turbulence. These model equations can then be solved for instance, via the well-known Simple algorithm [3]. To facilitate fast convergence, it is useful to make a reasonable initial guess of the pressure drop across the catalyst bed. 

The phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

In laminar flow,/is independent of /D. In turbulent flow, the friction factor for rough pipe follows the smooth tube curve for a range of Reynolds numbers (hydrauhcaUy smooth flow). For greater Reynolds numbers,/deviates from the smooth pipe cui ve, eventually becoming independent of Re. This region, often called complete turbulence, is frequently encountered in commercial pipe flows. The Reynolds number above which / becomes essentially independent of Re is (Davies, Turbulence Phenomena, Academic, New York, 1972, p. 37) 20[3.2-2.46ln( /D) ... 

The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = X/D. Molecular flow is characterized by Kn > 1.0 continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. 

Miller Internal Flow Systems, 2d ed.. Chap. 13, BHRA, Cranfield, 1990) gives the most complete information on losses in bends and curved pipes. For turbulent flow in circular cross-seclion bends of constant area, as shown in Fig. 6-14 7, a more accurate estimate of the loss coefficient K than that given in Table 6-4 is... 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

For turbulent pipe flow, the friction velocity u, = used earlier... 

Hanratty (pipe flow) Hinze (turbulence theory) -1.2 0.6 -0.6 ... 

Skin friction loss. Skin friction loss is the loss from the shear forces on the impeller wall caused by turbulent friction. This loss is determined by considering the flow as an equivalent circular cross section with a hydraulic diameter. The loss is then computed based on well-known pipe flow pressure loss equations. 

In problems of forced convection, it is usually the cooling mass flow that has to be found to determine the temperature difference between the cooling substance and the wall for a given heat flow. In turbulent pipe flow, the iol-low ing equation is valid ... 

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

Figure 2-43. Evaluation curves for friction losses of air and steam flowing turbulently in commercial pipe at low pressures. By permission, Standards for Steam Jet Ejectors, 4th Ed., Heat Exchange Institute, 1988.

Figure 10-50C. Tube-side (inside tubes) liquid film heat transfer coefficient for Dowtherm . A fluid inside pipes/tubes, turbulent flow only. Note h= average film coefficient, Btu/hr-ft -°F d = inside tube diameter, in. G = mass velocity, Ib/sec/ft v = fluid velocity, ft/sec k = thermal conductivity, Btu/hr (ft )(°F/ft) n, = viscosity, lb/(hr)(ft) Cp = specific heat, Btu/(lb)(°F). (Used by permission Engineering Manual for Dowtherm Heat Transfer Fluids, 1991. The Dow Chemical Co.)...

Turbulence is known to occur in pipe flow at a Reynolds number (Re) above 2300. Beyond this stability limit, any disturbance will grow exponentially in time and the flow becomes fully chaotic at Re 4000, where Re is defined by ... 

Laufer. J. United States National Advisory Committee for Aeronautics. Report No. 1174 (1955). The structure of turbulence in fully developed pipe flow,... 

Heywood, N. 1. and Cheng, D. C.-H. Trans Inst. Measurement and Control 6 (1984) 33. Comparison of methods for predicting head loss in turbulent pipe flow of non-Newtonian fluids. 

If the flow in the pipe is turbulent, the velocity at the axis increases from u to only about w/0.817, as given by equation 3.63. 

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. 

The ratio of t/t, which is characteristic of the possibility of vortices, does not depend on the micro-channel diameter and is fully determined by the Reynolds number and L/d. The lower value of Re at which f/fh > 1 can be treated as a threshold. As was shown by Darbyshire and Mullin (1995), under conditions of an artificial disturbance of pipe flow, a transition from laminar to turbulent flow is not possible for Re < 1,700, even with a very large amplitude of disturbances. 

Darbyshire AG, Mullin T (1995) Transition to turbulence in constant-mass-flux pipe flow. J Fluid Mech 289 83-114... 

Almost all flows in chemical reactors are turbulent and traditionally turbulence is seen as random fluctuations in velocity. A better view is to recognize the structure of turbulence. The large turbulent eddies are about the size of the width of the impeller blades in a stirred tank reactor and about 1/10 of the pipe diameter in pipe flows. These large turbulent eddies have a lifetime of some tens of milliseconds. Use of averaged turbulent properties is only valid for linear processes while all nonlinear phenomena are sensitive to the details in the process. Mixing coupled with fast chemical reactions, coalescence and breakup of bubbles and drops, and nucleation in crystallization is a phenomenon that is affected by the turbulent structure. Either a resolution of the turbulent fluctuations or some measure of the distribution of the turbulent properties is required in order to obtain accurate predictions. 

The production of turbulence is maximum close to walls, where both shear rate and turbulent viscosity, ut, are high. In pipe flow, the maximum is close to y+ = 12. A proper design of a chemical reactor for efficient mixing at low Re should allow the generated turbulence to be transported with the mean flow from the region where it is produced to the bulk of the fluid where it should dissipate. 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

Recommended u/ut for selected geometries for turbulent feed pipe flow conditions... 

A review on drag-reducing polymers is given in the literature [1359]. It has been suggested that drag reduction occurs by the interactions between elastic macromolecules and turbulent-flow macrostructures. In turbulent pipe flow, the region near the wall, composed of a viscous sublayer and a buffer layer, plays a major role in drag reduction. 

Experiments have been conducted to investigate the effect of a soapy industrial cleaner on reducing the skin friction of a Jordanian crude oil flowing turbulently in pilot-scale pipes of different sizes. Experiments showed that a concentration of only 2 ppm of the chemical additive injected into the crude oil line caused an appreciable amount of drag reduction [1165]. The effects of additive concentration and pipe diameter on drag reduction have been investigated. 

T. Moussa and C. Tiu. Factors affecting polymer degradation in turbulent pipe flow. Chem Eng Sci, 49(10) 1681-1692, May 1994. 

The first example pertains to forced convection in pipe flow. It is found that the rate of heat transfer between the pipe wall and a fluid flowing (turbulent flow) through the pipe depends on the following factors the average fluid velocity (u) the pipe diameter (d) the... 

In the second example, let the case of forced convective mass transfer in pipe flow be considered. Let it be assumed that the turbulent flow of the fluid, B, through the pipe is accompanied by a gradual dissolution of the material, A, of the pipe wall. Experimental... 

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