Turbulent flow, temperature profile

In turbulent flow, properties such as the pressure and velocity fluctuate rapidly at each location, as do the temperature and solute concentration in flows with heat and mass transfer. By tracking patches of dye distributed across the diameter of the tube, it is possible to demonstrate that the liquid s velocity (the time-averaged value in the case of turbulent flow) varies across the diameter of the tube. In both laminar and turbulent flow the velocity is zero at the wall and has a maximum value at the centre-line. For laminar flow the velocity profile is a parabola but for turbulent flow the profile is much flatter over most of the diameter. 

Figure 12.41 shows the results of three experiments with a similar Archimedes number and different Reynolds numbers. The figure shows vertical temperature profiles in a room ventilated by displacement ventilation. The dimensionless profiles are similar within the flow rates shown in the figure, although the profile may involve areas with a low turbulence level in the middle of the room. A test of this type could indicate that further experiments can be performed independently of the Reynolds numbers. 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

The mean profiles of velocity, temperature and solute concentration are relatively flat over most of a turbulent flow field. As an example, in Figure 1.24 the velocity profile for turbulent flow in a pipe is compared with the profile for laminar flow with the same volumetric flow rate. As the turbulent fluxes are very high but the velocity, temperature and concentration gradients are relatively small, it follows that the effective diffusivities (iH-e), (a+eH) and (2+ed) must be extremely large. In the main part of the turbulent flow, ie away from the walls, the eddy diffusivities are much larger than the corresponding molecular diffusivities ... 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

In practice, there is always some degree of departure from the ideal plug flow condition of uniform velocity, temperature, and composition profiles. If the reactor is not packed and the flow is turbulent, the velocity profile is reasonably flat in the region of the turbulent core (Volume 1, Chapter 3), but in laminar flow, the velocity profile is parabolic. More serious however than departures from a uniform velocity profile are departures from a uniform temperature profile. If there are variations in temperature across the reactor, there will be local variations in reaction rate and therefore in the composition of the reaction mixture. These transverse variations in temperature may be particularly serious in the case of strongly exothermic catalytic reactions which are cooled at the wall (Chapter 3, Section 3.6.1). An excellent discussion on how deviations from plug flow arise is given by DENBIGH and TURNER 5 . 

The flow patterns, composition profiles, and temperature profiles in a real tubular reactor can often be quite complex. Temperature and composition gradients can exist in both the axial and radial dimensions. Flow can be laminar or turbulent. Axial diffusion and conduction can occur. All of these potential complexities are eliminated when the plug flow assumption is made. A plug flow tubular reactor (PFR) assumes that the process fluid moves with a uniform velocity profile over the entire cross-sectional area of the reactor and no radial gradients exist. This assumption is fairly reasonable for adiabatic reactors. But for nonadiabatic reactors, radial temperature gradients are inherent features. If tube diameters are kept small, the plug flow assumption in more correct. Nevertheless the PFR can be used for many systems, and this idealized tubular reactor will be assumed in the examples considered in this book. We also assume that there is no axial conduction or diffusion. 

If the velocity profile is known together with the distribution of e then, for any assumed relationship between the distributions of c and e, Eq. (7.49) can be used to deduce the temperature profile. Once this has been obtained the relation between the Nusselt and Reynolds numbers can be derived. Before illustrating this procedure, there is a simplifying assumption that can be introduced without any significant loss of accuracy. Because the velocity profile in turbulent pipe flow is relatively flat over a large portion of the pipe cross-section, it is usually sufficiently accurate to replace u in the integral in the expression for I by the constant value um, i.e., to write ... 

In using these equations, the forms of the velocity and temperature profiles in the boundary layer are assumed. Now, in turbulent forced convective boundary layer flows it has been found that the velocity profile is well described by ... 

The developed velocity profile for turbulent flow in a tube will appear as shown in Fig. 5-15. A laminar sublayer, or film, occupies the space near the surface, while the central core of the flow is turbulent. To determine the heat transfer analytically for this situation, we require, as usual, a knowledge of the temperature distribution in the flow. To obtain this temperature distribution, the... 

The simulations were performed assuming that the flow is laminar. Additionally, the contact angle is assumed to be known. The initial velocity is assumed to be zero everywhere in the domain. The initial fluid temperature profile is taken to be linear in the natural convection thermal boundary layer and the thermal boundary layer thickness, 5j, is evaluated using the correlation for the turbulent natural convection on a horizontal plate as, Jj. =1. 4(vfiCil ... 

A better analytically based equation which is valid over a wide range of Prandtl or Schmidt numbers is obtained if we presume a turbulent parallel flow, i.e. a steady-state turbulent flow with vanishing pressure gradient, and velocity, temperature and concentration profiles which are only dependent on the coordinate y normal to the wall. Then, as follows from (3.134) to (3.139),... 

Under smooth fluidization the motion, heat capacity, and small size of the particles result in a remarkably uniform temperature throughout the bed. Radial gradients, so important in fixed beds, are negligible. The transfer of heat to or from the reactor can be considered by assuming that a finite heat-transfer coefficient exists at the wall, and that the temperature across the bed is uniform. This situation is depicted in Fig. 13-18, where curve c applies to the fluidized bed. For comparison, curve b represents a homogeneous tubular reactor in turbulent flow, where the temperature profile is not so flat as in the fluidized bed but is still more uniform than for the packed bed, case (a). ... 

Temperature profile. Let us discuss qualitative specific features of convective heat and mass transfer in turbulent flow past a flat plate. Experimental evidence indicates that several characteristic regions with different temperature profiles can be distinguished in the thermal boundary layer on a flat plate. At moderate Prandtl numbers (0.5 < Pr < 2.0), it can be assumed for rough estimates that the characteristic sizes of these regions are of the same order of magnitude as those of the wall layer and the core of the turbulent stream, see Section 1.7. 

More detailed information about heat and mass transfer in turbulent flows past a flat plate, as well as various relations for determining the temperature profile and Nusselt (Sherwood) numbers, and a lot other useful information can be found in the references [184, 185, 212, 289, 406], which contain extensive literature surveys. 

---