Nusselt number turbulent flow

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

Heat transfer in static mixers is intensified by turbulence causing inserts. For the Kenics mixer, the heat-transfer coefficient b is two to three times greater, whereas for Sulzer mixers it is five times greater, and for polymer appHcations it is 15 times greater than the coefficient for low viscosity flow in an open pipe. The heat-transfer coefficient is expressed in the form of Nusselt number Nu = hD /k as a function of system properties and flow conditions. 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

For low values of the Reynolds number, such as 10, where sn eamline flow should certainly apply, the Nusselt number has a value of about 2, and a typical value of the average heat transfer coefficient is 10 ". For a Reynolds number of 104, where the gas is certainly in turbulent flow, the value of the Nusselt number is typically 20. Hence there is only a difference of a factor of ten in the heat transfer coefficient between tlrese two extreme cases. 

In this table the parameters are defined as follows Bo is the boiling number, d i is the hydraulic diameter, / is the friction factor, h is the local heat transfer coefficient, k is the thermal conductivity, Nu is the Nusselt number, Pr is the Prandtl number, q is the heat flux, v is the specific volume, X is the Martinelli parameter, Xvt is the Martinelli parameter for laminar liquid-turbulent vapor flow, Xw is the Martinelli parameter for laminar liquid-laminar vapor flow, Xq is thermodynamic equilibrium quality, z is the streamwise coordinate, fi is the viscosity, p is the density, <7 is the surface tension the subscripts are L for saturated fluid, LG for property difference between saturated vapor and saturated liquid, G for saturated vapor, sp for singlephase, and tp for two-phase. 

Predicted Nusselt numbers for turbulent flow with constant wall heat flux (John Wiley and Sons from Bird et al., 1964). Abbreviations Nu = Nusselt number Re = Reynolds number Pr = Prandtl number. 

Very little work has been reported on vaporization under conditions of turbulent gas flow. Ingebo (61), for example, took pains to minimize approach stream turbulence. Two exceptions are the investigations of Maisel and Sherwood (83) and Fledderman and Hanson (27). Neither went so far in analysis as insertion into the Nusselt number equations of allowance for the additional relative velocity between droplet and air stream occasioned by turbulence. In the case of Maisel and Sherwood s investigation with model droplets at fixed positions, the effect would not be expected to be extreme, because at all times there was appreciable relative velocity, discounting turbulence. However, in Fledderman and Hanson s experiments the relative velocity, discounting turbulence, fell away as the droplets accelerated up to stream velocity. Thus turbulence would eventually provide the only appreciable relative velocity. The results indicate a substantial increase in vaporization rate because of the turbulence and provide some basis for gross engineering estimates. 

Heat transfer and its counterpart diffusion mass transfer are in principle not correlated with a scale or a dimension. On a molecular level, long-range dimensional effects are not effective and will not affect the molecular carriers of heat. One could say that physical processes are dimensionless. This is essentially the background of the so-called Buckingham theorem, also known as the n-theorem. This theorem states that a product of dimensionless numbers can be used to describe a process. The dimensionless numbers can be derived from the dimensional numbers which describe the process (for example, viscosity, density, diameter, rotational speed). The amount of dimensionless numbers is equal to the number of dimensional numbers minus their basic dimensions (mass, length, time and temperature). This procedure is the background for the development of Nusselt correlations in heat transfer problems. It is important to note that in fluid dynamics especially laminar flow and turbulent flow cannot be described by the same set of dimensionless correlations because in laminar flow the density can be neglected whereas in turbulent flow the viscosity has a minor influence [144], This is the most severe problem for the scale-up of laminar micro results to turbulent macro results. 

Integrating Eq. (6.51) over the entire plate as was done w ith the Reynolds analogy equation and assuming that the flow is turbulent from the leading edge of the plate then gives the following expression for the mean Nusselt number... 

Numerically determine the local Nusselt number variation with two-dimensional turbulent boundary layer air flow over an isothermal flat plate for a maximum Reynolds number of 107. Assume that transition occurs at a Reynolds number of 5 X 105. Compare the numerical results with those given by the Reynolds analogy. 

Use the Reynolds analogy to derive an expression for the Nusselt number for fully developed turbulent flow in an annulus in which the inner wall is heated to a uniform temperature and the outer wall is adiabatic. Assume that the friction factor can be derived by introducing the hydraulic diameter concept. 

The so-called Taylor-Prandtl analogy was applied to boundary layer flow in Chapter 6. Use this analogy solution to derive an expression for the Nusselt number in fully developed turbulent pipe flow. 

Comparison of measured and predicted Nusselt number variations for turbulent natural convective flow over a vertical plate. 

Because, for flow over a heated surface. r>ulc>x is positive and ST/ y is negative. S will normally be a negative. Hence, in assisting flow, the buoyancy forces will tend to decrease e and e, i.e., to damp the turbulence, and thus to decrease the heat transfer rate below the purely forced convective flow value. However, the buoyancy force in the momentum equation tends to increase thle mean velocity and, therefore, to increase the heat transfer rate. In turbulent assisting flow over a flat plate, this can lead to a Nusselt number variation with Reynolds number that resembles that shown in Fig. 9.22. 

Variation of Nusselt number variation with Reynolds number in turbulent assisting mixed convective flow. 

Numerically predicted variation of Nusselt number variation with Reynolds number in turbulent assisting mixed convective flow over a vertical plate. (Based on results obtained by Patel K., Armaly B.F., and Chen T.S., Transition from Turbulent Natural to Turbulent Forced Convection Adjacent to an Isothermal Vertical Plate , ASME HTD, Vol. 324, pp. 51-56, 1996. With permission.)... 

This chapter has been concerned with flows in wb ch the buoyancy forces that arise due to the temperature difference have an influence on the flow and heat transfer values despite the presence of a forced velocity. In extemai flows it was shown that the deviation of the heat transfer rate from that which would exist in purely forced convection was dependent on the ratio of the Grashof number to the square of the Reynolds number. It was also shown that in such flows the Nusselt number can often be expressed in terms of the Nusselt numbers that would exist under the same conditions in purely forced and purely free convective flows. It was also shown that in turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can be decreased by the buoyancy forces in assisting flows whereas in laminar flows the buoyancy forces essentially always increase the heat transfer rate in assisting flow. Some consideration was also given to the effect of buoyancy forces on internal flows. 

For constant wall heat flux in turbulent flow it is shown in Ref. 12 that the local Nusselt number is only about 4 percent higher than for the isothermal surface i.e.,... 

See also -> convection, -> Grashof number, - Hagen-Poiseuille, -> hydrodynamic electrodes, -> laminar flow, - turbulent flow, -> Navier-Stokes equation, -> Nusselt number, -> Peclet number, -> Prandtl boundary layer, - Reynolds number, -> Stokes-Einstein equation, -> wall jet electrode. 

For fully developed turbulent flow of liquid metals, the Nusselt number depends on the wall boundary condition. For a constant wall temperature [Notter and Sleicher, Chem. Eng. Science, 27,2073 (1972)],... 

Flow in Noncircular Ducts The length scale in the Nusselt and Reynolds numbers for noncircular ducts is the hydraulic diameter, D), = 4AJp, where A, is the cross-sectional area for flow and p is the wetted perimeter. Nusselt numbers for fully developed laminar flow in a variety of noncircular ducts are given by Mills (Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 307). For turbulent flows, correlations for round tubes can be used with D replaced by l. ... 

Coiled Tubes For turbulent flow inside helical coils, with tube inside radius a and coil radius R, the Nusselt number for a straight tube Nu is related to that for a coiled tube Nuc by (Rohsenow, Hartnett, and... 

The average Nusselt number for turbulent flow over a smooth, isothermal flat plate of length is given by (Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 315)... 

Now the dimensionless ratio hu/K is known as the Nusselt number Nu(ro), and for systems with convection it takes values of about 5 if the flow is not turbulent. (In the absence of convection /i, the heat transfer at the walls is determined by the temperature gradient at the walls, which in turn is proportional to K/tq,) It is interesting to note that the simple model which permits laminar convection gives values of 8c of about the order of 0, which is reasonably close to the value of 3.32 calculated for pure conduction. 

Wasan and Wilke (W3) developed the following expressions for the Sherwood number (i.e., the analog in mass transfer of the Nusselt number) from a cylindrical surface placed parallel to the stream in a turbulent flow ... 

B Gain a working knowledge of the dimensionless Reynolds, Prandtl, and Nusselt numbers, B Distinguish betvreen laminar and turbulent llovrs, and gain an understanding of the mechanisms of momentum and heat transfer In turbulent flow,... 

In turbulent flow, one can estimate the Nusselt number using the analogy between heat and momentum transfer (Colburn analogy). This analogy relates the Nusselt number to llie coefficient of friction, Cj, as (a) Nu = 0.5 Cy Re Pr (h) Nu = 0.5 C>Rc Pr ... 

Consider a flat plate whose heated section is maintained at a constant temperature (T = constant for.r > ). Using integral solution methods (see Kays and Crawford, 1994), the local Nusselt numbers for both laminar and turbulent flows are determined to be... 

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