Nusselt number boundary layer heat transfer

Therefore, the average heat transfer coefficient over the boundary layer build-up region is equivalent to twice the local heat transfer coefficient in the fully developed region. The solution of the Polhausen equation clearly bounds the 13 percent increase in Nusselt number for compact heat exchanger modeling. In fact, increasing the Sellars solution (Nu=4.36) by a factor of two would also bound the proposed Nusselt number. 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

Now let us consider the mixing time, t. This will be estimated by an order of magnitude estimate for diffusion to occur across the boundary layer thickness, <5Bl- If we have turbulent natural conditions, it is common to represent the heat transfer in terms of the Nusselt number for a vertical plate of height, , as... 

As Re increases further and vortices are shed, the local rate of mass transfer aft of separation should oscillate. Although no measurements have been made for spheres, mass transfer oscillations at the shedding frequency have been observed for cylinders (B9, D6, SI2). At higher Re the forward portion of the sphere approaches boundary layer flow while aft of separation the flow is complex as discussed above. Figure 5.17 shows experimental values of the local Nusselt number Nuj c for heat transfer to air at high Re. The vertical lines on each curve indicate the values of the separation angle. It is clear that the transfer rate at the rear of the sphere increases more rapidly than that at the front and that even at very high Re the minimum Nuj. occurs aft of separation. Also shown in Fig. 5.17 is the thin concentration boundary layer... 

Fig. 5.17 Local Nusselt number for heat transfer from a sphere to air (Pr = 0.71). Experimental results of Galloway and Sage (Gl). Dashed lines are predictions of boundary layer theory by Lee and Barrow (LIO).

Once the system of equations has been solved, the nondimensional temperature gradient can be easily evaluated at the surface, providing the Nusselt number. It should be expected that the heat transfer depends on the boundary-layer thickness, which in turn depends on the flow field, which is principally governed by the Reynolds number. Figure 6.9 shows a correlation between the Nusselt number and the Reynolds number that was obtained by solving the nondimensional system for several Reynolds numbers. 

For vertical surfaces, the Nusselt and Grashof numbers are formed with L, the height of the surface as the characteristic dimension. If the boundary-layer thickness is not large compared with the diameter of the cylinder, the heat transfer may be calculated with the same relations used for vertical plates. The general criterion is that a vertical cylinder may be treated as a vertical flat plate [13] when... 

Dimensionless numbers have proved useful for analyzing relationships between heat transfer and boundary layer thickness for leaves. In particular, the Nusselt number increases as the Reynolds number increases for example, Nu experimentally equals 0.97 Re0-5 for flat leaves (Fig. 7-9). By Equations 7.18 and 7.19, d/8bl is then equal to 0.97 (vd/v)V2y so for air temperatures in the boundary layer of 20 to 25°C, we have... 

Kato and Wen (5) found, for the case of packed beds,that there was a dependency of the Sherwood and Nusselt numbers with the ratio dp/L. They proposed that the fall of the heat and mass transfer coefficients at low Reynolds numbers is due to an overlapping of the boundary layers surrounding the particles which produces a reduction of the available effective area for transfer of mass and heat. Nelson and Galloway W proposed a new model in terms of the Frossling number, to explain the fall of the heat and mass transfer coefficients in the zone of low Reynolds numbers. 

These relations are extremely valuable as they slate that for a given geometry, the friction coefficient can be expres.sed as a function of Reynolds number alone, and the Nusselt number as a function of Reynolds and Prandtl numbers alone (Fig. 6-34). Therefore, experimentalists can study a problem with a minimum number of experiments, and report their friction and heat transfer coefficient measure.ments conveniently in terms of Reynolds and Prandtl numbers. For example, a friction coefficient relation obtained with air for a given surface can also be used for water at the same Reynolds number. But it should be kepi in mind that the validity of these relations is limited by the limitations on (he boundary layer equations used in the analysis. 

The phenomena that affect drag force also affect heat transfer, and this effect appears in the Nusselt number. By nondimensionalizing the boundary layer equations, it was shown in Chapter 6 that the local and average Nusselt numbers have the functional form... 

The temperature held is dependent on this number when heat transfer takes place into a fluid. The Biot number has the same form as the Nusselt number defined by (1.36). There is however one very significant difference, A in the Biot number is the thermal conductivity of the solid whilst in the Nusselt number A is the thermal conductivity of the fluid. The Nusselt number serves as a dimensionless representation of the heat transfer coefficient a useful for its evaluation, whereas the Biot number describes the boundary condition for thermal conduction in a solid body. It is the ratio of L0 to the subtangent to the temperature curve within the solid body, cf. Fig. 2.4, whilst the Nusselt number is the ratio of a (possibly different choice of) characteristic length L0 to the subtangent to the temperature profile in the boundary layer of the fluid. 

As soon as the functional relationships between the Nusselt, Reynolds and Prandtl numbers or the Sherwood, Reynolds and Schmidt numbers have been found, be it by measurement or calculation, the heat and mass transfer laws worked out from this hold for all fluids, velocities and length scales. It is also valid for all geometrically similar bodies. This is presuming that the assumptions which lead to the boundary layer equations apply, namely negligible viscous dissipation and body forces and no chemical reactions. As the differential equations (3.123) and (3.124) basically agree with each other, the solutions must also be in agreement, presuming that the boundary conditions are of the same kind. The functions (3.126) and (3.128) as well as (3.127) and (3.129) are therefore of the same type. So, it holds that... 

As the previous illustrations showed, the heat and mass transfer coefficients for simple flows over a body, such as those over flat or slightly curved plates, can be calculated exactly using the boundary layer equations. In flows where detachment occurs, for example around cylinders, spheres or other bodies, the heat and mass transfer coefficients are very difficult if not impossible to calculate and so can only be determined by experiments. In terms of practical applications the calculated or measured results have been described by empirical correlations of the type Nu = f(Re,Pr), some of which have already been discussed. These are summarised in the following along with some of the more frequently used correlations. All the correlations are also valid for mass transfer. This merely requires the Nusselt to be replaced by the Sherwood number and the Prandtl by the Schmidt number. 

This is the appropriate correlation to use when there is heat or mass (i.e., substitute Nu by Sh) transfer from a sphere immersed in a stagnant film is studied, Nu = 2. The second term in (5.294) accounts for convective mechanisms, and the relation is derived from the solution of the boundary layer equations. For higher Re3molds numbers the Nusselt number is set equal to the relation resulting from the boundary layer analysis of a flat plate ... 

In this chapter, we return to forced convection heat and mass transfer problems when the Reynolds number is large enough that the velocity field takes the boundary-layer form. For this class of problems, we find that there must be a correlation between the dimensionless transport rate (i.e., the Nusselt number for heat transfer) and the independent dimensionless parameters, Reynolds number Re and either Prandtl number Pr or Schmidt number Sc of the form... 

We have considered the thermal boundary-layer problem in this chapter for an arbitrary 2D body with no-slip boundary conditions for Re 1 and Pr (or Sc) either arbitrarily large or small. If we assume that we have a body of the exact same shape, but the surface of which is a slip surface (e.g., it is an interface, so that the surface tangential velocity is not zero), the form of the correlation between Nusselt number and Pr will change for Pr 1. Solve this problem, i.e., derive the governing boundary-layer equation, and show that it has a similarity solution. What is the resulting form of the heat transfer correlation among Nu, Re, and Pr ... 

The Nusselt number is the ratio of the tube diameter to the equivalent thickness of the laminar layer. Sometimes x is called the film thickness, and it is generally slightly greater than the thickness of the laminar boundary layer because there is some resistance to heat transfer in the buffer zone. 

On the other hand, as shown on pages 143 to 151, in the flow of fluids across a cylindrical shape boundary-layer separation occurs, and a wake develops that causes form friction. No sharp distinction is found between laminar and turbulent flow, and a common correlation can be used for both low and high Reynolds numbers. Also, the local value of the heat-transfer coefficient varies from point to point around a circumference. In Fig. 12.5 the local value of the Nusselt number is plotted radially for all points around the circumference of the tube. At low Reynolds numbers, is a maximum at the front and back of the tube and a minimum at the sides. In practice, the variations in the local coefficient hg are often of no importance, and average values based on the entire circumference are used. 

Heat Transfer on the Walls With Uniform Heat Flux. The solutions for simultaneously developing flow in circular ducts with uniform wall heat flux are reviewed by Shah and London [1], Recently, a new integral or boundary layer solution has been obtained by Al-Ali and Selim [33] for the same problem. However, the most accurate results for the local Nusselt numbers [1] are presented in Table 5.6. 

Shigechi et al. [112] conducted a boundary layer analysis of this problem and included momentum and convection effects in the condensate film. They obtained different solutions using as a boundary condition various inclination angles of the liquid-vapor interface at the plate edge. Their maximum average Nusselt number was found to agree well with Eq. 14.98. Chiou et al. [214] included surface tension in their model and showed that heat transfer decreases in relation to Eq. 14.98 as the surface tension of the condensate increases. 

The heat transfer phenomena and, in particular, the thermal boundary layer, are well understood for a cool, solid body immei l in a laminar hot gas stream in which no chemical reactions occur. The heat transfer can be predicted by the Nusselt number Nu = fi(Re, Pr). With some modifications, similar relationships hold for dissociation of gases, provided that Le, which describes the diffusion of the species, is close to unity... 

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