Turbulent flow development

Dengler and Addoms 8 measured heat transfer to water boiling in a 6 m tube and found that the heat flux increased steadily up the tube as the percentage of vapour increased, as shown in Figure 14.4. Where convection was predominant, the data were correlated using the ratio of the observed two-phase heat transfer coefficient (htp) to that which would be obtained had the same total mass flow been all liquid (hi) as the ordinate. As discussed in Volume 6, Chapter 12, this ratio was plotted against the reciprocal of Xtt, the parameter for two-phase turbulent flow developed by Lockhart and Martinelli(9). The liquid coefficient hL is given by ... 

R. M. Olson, and E. M. Sparrow, Measurements of Turbulent Flow Development in Tubes and Annuli with Square or Rounded Entrances, AIChE J., (9) 766-770,1963. 

Figure 48.3(b) Turbulent flow develops the colored filament oscillates and breaks up. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

To avoid imposition of unrealistic exit boundary conditions in flow models Taylor et al. (1985) developed a method called traction boundary conditions . In this method starting from an initial guess, outflow condition is updated in an iterative procedure which ensures its consistency with the flow regime immediately upstream. This method is successfully applied to solve a number of turbulent flow problems. 

In general, V For laminar Newtonian flow the radial velocity profile is paraboHc and /5 = 3/4. For fully developed turbulent flow the radial... 

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... 

The minimum velocity requited to maintain fully developed turbulent flow, assumed to occur at Reynolds number (R ) of 8000, is inside a 16-mm inner diameter tube. The physical property contribution to the heat-transfer coefficient inside and outside the tubes are based on the following correlations (39) ... 

One of the most efficient implementations of the slurry process was developed by Phillips Petroleum Company in 1961 (Eig. 5). Nearly one-third of all HDPE produced in the 1990s is by this process. The reactor consists of a folded loop with four long (- 50 m) vertical mns of a pipe (0.5—1.0 m dia) coimected by short horizontal lengths (around 5 m) (58—60). The entire length of the loop is jacketed for cooling. A slurry of HDPE and catalyst particles in a light solvent (isobutane or isopentane) circulates by a pump at a velocity of 5—12 m/s. This rapid circulation ensures a turbulent flow, removes the heat of polymeriza tion, and prevents polymer deposition on the reactor walls. 

For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the Him. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vaiy according to its thickness and can range from 95 percent of the total resistance for some fluids to about I percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to beat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. 

Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. 

Inertial forces are developed when the velocity of a fluid changes direction or magnitude. In turbulent flow, inertia forces are larger than viscous forces. Fluid in motion tends to continue in motion until it meets a sohd surface or other fluid moving in a different direction. Forces are developed during the momentum transfer that takes place. The forces ac ting on the impeller blades fluctuate in a random manner related to the scale and intensity of turbulence at the impeller. 

This formula is another variation on the Affinity Laws. Monsieur s Darcy and VVeisbach were hydraulic civil engineers in France in the mid 1850s (some 50 years before Mr. H VV). They based their formulas on friction losses of water moving in open canals. They applied other friction coefficients from some private experimentation, and developed their formulas for friction losses in closed aqueduct tubes. Through the years, their coefficients have evolved to incorporate the concepts of laminar and turbulent flow, variations in viscosity, temperature, and even piping with non uniform (rough) internal. surface finishes. With. so many variables and coefficients, the D/W formula only became practical and popular after the invention of the electronic calculator. The D/W forntula is extensive and eomplicated, compared to the empirieal estimations of Mr. H W. 

Leva et al (1949) and Ergun (1952) developed similar useful equations. Later, these were refined and modified by Handley and Heggs (1968 ) and by MacDonald et al (1979). All these equations try to handle the transient regime between laminar and turbulent flow somewhat differently but are based on the same principles. 

Flow in empty tubes has a relatively narrow band of velocities—or Reynolds numbers from 2000 to 10000—wherein the character changes from laminar to turbulent. In packed beds, even the laminar flow does not mean that motion is linear or parallel to the surface. Due to the many turns between particles, stable eddies develop and therefore the difference between laminar and turbulent flow is not as pronounced as in empty tubes. 

Model Experiments in the Case of Fully Developed Turbulent Flow 1183... 

The availability of large and fast computers, in combination with numerical techniques to compute transient, turbulent flow, has made it possible to simulate the process of turbulent, premixed combustion in a gas explosion in more detail. Hjertager (1982) was the first to develop a code for the computation of transient, compressible, turbulent, reactive flow. Its basic concept can be described as follows A gas explosion is a reactive fluid which expands under the influence of energy addition. Energy is supplied by combustion, which is modeled as a one-step conversion process of reactants into combustion products. The conversion (combustion)... 

Fully developed turbulent flow, Nr over 10,000, in a tank containing four equally spaced baffles having a width of 10% of the tank diameter ... 

Fluidic Pulser System. A new type of pulser is being developed at Louisiana State University. It is based on a patent by A. B. Holmes [107]. The throttling of the mud is obtained by creating a turbulent flow in a chamber as shown in Figure 4-249. 

This may be compared with fully developed turbulent flow along a flat sheet or tube when ... 

Very rapid and highly localised pitting is sometimes observed on components exposed to very turbulent flow conditions leading to cavitation in the stream. In general, these conditions appear to induce corrosion rather than erosion on cast iron surfaces, in contradistinction to what usually happens with other metals, apparently because the erosive component of the liquid flow scours away corrosion-stifling films and allows the development of very active electrochemical cells on the exposed metal surfaces . 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

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. 

When a fluid flowing with a uniform velocity enters a pipe, a boundary layer forms at the walls and gradually thickens with distance from the entry point. Since the fluid in the boundary layer is retarded and the total flow remains constant, the fluid in the central stream is accelerated. At a certain distance from the inlet, the boundary layers, which have formed in contact with the walls, join at the axis of the pipe, and, from that point onwards, occupy the whole cross-section and consequently remain of a constant thickness. Fulty developed flow then exists. If the boundary layers are still streamline when fully developed flow commences, the flow in the pipe remains streamline. On the other hand, if the boundary layers are already turbulent, turbulent flow will persist, as shown in Figure 11.8. 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

---