Nusselt functions

The Nusselt functions specified in the literature deviate from each other particularly with regard to the exponent of the Reynolds number this is because the exponent depends on the radial distance. At the stagnation point the flow is always laminar, and the exponent is therefore 0.5. Turbulence fully develops only at a considerable distance from the stagnation point, and as a result the exponent must be 0.8. Figure 1.6 shows the increase of the exponent related to the radius (Adler, 2004). At a distance of r/d = 3, the exponent has reached a value of only about 0.62. Different exponents are obtained depending on the size of the area considered in averaging. 

In the literature, values ranging from 0.67 to 0.7 are mostly reported and, in order to improve comparability, in all of the follo ving discussions the Reynolds exponent is approximated by 0.67. Nusselt functions are typically determined for air, vhich is the typical medium used during application. 

In comparison with respect to the Nusselt functions for single nozzle (Eqs 1.12 and 1.13), it is evident that the heat transfer in the nozzle array is higher than that of the single nozzle under the condition of similar Reynolds numbers. Regarding the average heat transfer, this increase is approximately 30%. 

The Nusselt functions for the stagnation point and the average value of the field can be approximated by (Attalla, 2005) ... 

In perforated plates, spent holes or slots must be installed so that the injected air can flow out again. The size and number of these outlets depends on the manufacturer of the perforated plates, and consequently various cross-flows can occur between the target area and the plate. How this affects the heat transfer is not yet known. Nonetheless, Martin (1977) developed a Nusselt function with which the heat transfer for a majority of perforated plates can be approximated, by providing the following correlation for aligned and hexagonaHy arranged perforations ... 

The values of the Nusselt numbers are similar to those resulting from the Nusselt function corresponding to cross-flow of cylinders if the cross-flow length (ji D)/2 is used as a characteristic dimension in the Nusselt and Reynolds numbers. 

The Nusselt function can alternatively be formed with the slot width s as the characteristic dimension resulting in... 

The Nusselt functions for the single nozzle are summarized in Tab. 1.1, and for the nozzle fields in Tab. 1.2. The Nusselt and Remolds numbers are defined with the inner nozzle diameter d regarding round nozzles, and with the width s regarding slot nozzles. The exponent of the Reynolds number in correlations for the average... 

Nozzle shape Nusselt function Scope Reference... 

Tab. 1.2 Nusselt functions for nozzle fields impinging flat surfaces.

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. 

The maximum possible heat flux, which corresponds to the maximum allowed wall temperatures is estimated. This maximum wall heat flux is determined by the difference between the permissible wall temperature and the vapor temperature in the outlet cross-section, which is a function of the Reynolds and Nusselt numbers. 

It is possible to observe that U increases with the utility flow rate and initial process fluid temperature. This corresponds to a classical evolution of the Nusselt number function of Reynolds and Prandtl numbers. 

Figure 2.27 Streamline patterns in a channel with sinusoidal walls (left) and Nusselt number as a function of Reynolds number for the same channel (right), taken from [120]. For comparison, the triangles represent the Nusselt number obtained in parallel-plates geometry.

Figure 2.35 Cross-section through a staggered arrangement of micro fins designed for heat transfer enhancement in a micro channel (above) and ratio of Nusselt and Poiseuille numbers as a function of air flow per unit area for different total fin lengths (below), taken from [127].

The dimensionless group hD/k is called the Nusselt number, /VNu, and the group Cp i/k is the Prandtl number, NPl. The group DVp/p is the familiar Reynolds number, NEe, encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as... 

Heat exchange in stirred reactors is described in [207]. By using dimensional analysis of heat flow and energy balance equations, the Nusselt number, containing hT, can be expressed as a function of the Reynolds number and the Prandtl number ... 

The experimental results obtained for a wide range of systems(96-99) are correlated by equation 6.58, in terms of the Nusselt number (Nu = hd/k) for the particle expressed as a function of the Reynolds number (Re c = ucdp/fx) for the particle, the Prandtl number Pr for the liquid, and the voidage of the bed. This takes the form ... 

The Nusselt number with respect to the tube Nu(= hdt/k) is expressed as a function of four dimensionless groups the ratio of tube diameter to length, the ratio of tube to particle diameter, the ratio of the heat capacity per unit volume of the solid to that of the fluid, and the tube Reynolds number, Rec = (ucdtp/p,). However, equation 6.59 and other equations quoted in the literature should be used with extreme caution, as the value of the heat transfer coefficient will be highly dependent on the flow patterns of gas and solid and the precise geometry of the system. 

Since the dimensionless equations and boundary conditions governing heat transfer and dilute-solution mass transfer are identical, the solutions to these equations in dimensionless form are also identical. Profiles of dimensionless concentration and temperature are therefore the same, while the dimensionless transfer rates, the Sherwood number (Sh = kL/ ) for mass transfer, and the Nusselt number (Nu = hL/K ) for heat transfer, are identical functions of Re, Sc or Pr, and dimensionless time. Most results in this book are given in terms of Sh and Sc the equivalent results for heat transfer may be found by simply replacing Sh by Nu and Sc by Pr. 

Just as there are correlations for the Nusselt number based on the Reynolds and Prandtl numbers [Eq. (4.96)], there are empirical correlations for the Sherwood number as a function of Reynolds and Schmidt numbers, which are of the general form... 

The above self-similar velocity profiles exists only for a Re number smaller than a critical value (e.g. 4.6 for a circular pipe). The self-similar velocity profiles must be found from the solution of the Navier-Stokes equations. Then they have to be substituted in Eq. (25) which must be solved to compute the local Nusselt number Nu z). The asymptotic Nusselt number 7Vm is for a pipe flow and constant temperature boundary condition is given by Kinney (1968) as a function of Rew and Prandtl (Pr) numbers. The complete Nu(z) curve for the pipe and slit geometries and constant temperature or constant flux boundary conditions were given by Raithby (1971). This author gave /Vm is as a function of Rew and fluid thermal Peclet (PeT) number. Both authors solved Eq. (25) via an eigenfunction expansion. 

The function A stands for the modification to the usual Nusselt number for non-porous wall (the so-called Graetz problem) due to the suction-distorted... 

Define and plot a Nusselt number for the surface heat transfer as a function of nondi-mensional time. Describe surface heat-transfer behavior in physical terms, explaining the shape the Nusselt-number curve. 

By analogy with the Graetz problem for the Nusselt number, determine quantitatively the behavior of the Sherwood number as a function of z, the axial distance from the start of the catalytic section... 

Assuming an initially fully developed parabolic velocity profile. Re = 1000, and Pr = 5, calculate and plot the Nusselt number as a function of the inverse Graetz number,... 

The values Qx and Q2 in our formulas may be expressed in terms of the dimensions and temperature of the heated body if the Nusselt number is known as a function of the Grashof number in the case under consideration. If Nu = i/>(Gr) then, within a numerical factor, Qx = A0oi/>(Gr), Q2 = Xd80ip(Gr), where A is the heat conductivity of the medium, 80 is the temperature, and d is the size of the heated body. 

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