Nusselt number parallel plates

Figure 7 Influence of Knudsen number on the transient and steady behaviors of (a) - dimensionless average temperature and (b) and (c) - local Nusselt number (parallel plates, Pe = 10, Kn =0, 0.001, 0.01 0.1 and Br = 0,...

Parallel Plates and Rectangular Ducts The limidng Nusselt number for parallel plates and flat rectangular ducts is given in Table 5-4. Norris and Streid [Tran.s. Am. Soe. Meeh. Eng., 62, 525 (1940)] report for constant wall temperature... 

The problem of axial conduction in the wall was considered by Petukhov (1967). The parameter used to characterize the effect of axial conduction is P = (l - dyd k2/k ). The numerical calculations performed for q = const, and neglecting the wall thermal resistance in radial direction, showed that axial thermal conduction in the wall does not affect the Nusselt number Nuco. Davis and Gill (1970) considered the problem of axial conduction in the wall with reference to laminar flow between parallel plates with finite conductivity. It was found that the Peclet number, the ratio of thickness of the plates to their length are important dimensionless groups that determine the process of heat transfer. 

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.

Systematically find the expression for the temperature distribution and the Nusselt number for laminar flow between two large parallel plates in the region of fully developed velocity and temperature profiles for a uniformly applied wall heat flux. 

A constant-property fluid flows in a laminar manner in the x direction between two large parallel plates. The same constant heat flux qw is maintained from the plates to the fluid for all x> 0. The fluid temperature is Tin at x = 0. Find an expression of the local Nusselt number by the integral method. What is this expression if Pr= 1 ... 

The average Nusselt number for the thermal entrance region of flow between isothermal parallel plates of length L is expressed as (Edwards et al., 1979)... 

Kohsenow (1984) have compiled the available dala under various boundary conditions, and developed correlations for the Nusselt number and optimum spacing. The characteristic length for vertical parallel plates used as fins is usually taken to he the spacing between adjacent fins S, aliltough the n height h could also be used. The Rayleigh number is expressed as... 

The recommended relation for the average Nusselt number for vertical isothermal parallel plates is... 

The average Nusselt number for vertical isolhermal parallel plates of spacing S and height L is given as... 

Kavehpour et al. [20] solved the compressible two-dimensional fluid flow and heat transfer characteristics of a gas flowing between two parallel plates under both uniform temperature and uniform heat flux boundary conditions. They compared their results with the experimental results of Arkilic [3] for Helium in a 52.25x1.33x7500 mm channel. They observed an increase in the entrance length and a decrease in the Nusselt number... 

Thermally fully developed heat transfer do to electro-osmotic fluid transport in micro parallel plate channel and micro mbe has been recently investigated by [21]. The dimensionless temperature profile and corresponding Nusselt number have been determined for imposed constant wall heat flux and constant temperature. The complement paper [22] study the effect of viscous dissipation. These two papers gives important physical details and references. The analyses of both papers is based on the classical simplifying assumptions that are avoided in the book by Mikhailov and Ozisik [20]. 

Mathematica package is developed that computes the eigenvalues, the eigenfunctions, the eigenintegrals, the dimensionless temperature, the average dimensionless temperature, and the Nusselt number for steady state and periodic heat transfer in micro parallel plate channel and micro tube taking into account the velocity slip and the temperature jump. Some results in form of tables and plots are given bellow. 

For electro-osmotic flow only the limiting Nusselt numbers for thermally fully-developed flow in parallel plate channel and circular tube are obtained as a special case from the solution for thermally developing flow. 

The limiting Nusselt number is of great practical interest. For n=0 (parallel plate micro channel) and n=l (micro tube) the limiting Nusselt number depend on 2 parameters Kn/3v and /3. The Kn/3v control mainly the velocity slip and have influence on the temperature jump. The parameter /3 control only the temperature jump. The limiting Nusselt number is shown on Fig 6. 

Local fully developed Nusselt numbers for parallel plates were reported by [31]. Two experimental cases were done under different boundary conditions two walls heated and one wall heated, the other insulated. Recovery factors as functions of dimensionless axial length, X, for both boundary conditions were introduced. Employing the recovery factors and plotted against the dimensionless axial length, Nusselt numbers, were found to be 8.235 and 5.385 for the boundary eonditions of the two heated walls and the one heated wall the other insulated, respectively. It is noted that these values are the same as those of conventional chanels. 

These definitions are also used for flow between two parallel plates, replacing 4y in the denominator by 2y. Next, the Nusselt number for flow in a rectangular channel is as follows ... 

Figure 5 Influence of Brinkman number on local Nusselt nrnnber evolution (parallel plates, Pe = 10, Kn -0.01, andBr...

The geometric and flow conditions are not the only parameters which have a considerable influence on the relationship between the Nusselt number and the other dimensionless numbers. The thermal boundary conditions also affect heat transfer. An example of this is, with the same values of Re and Pr in parallel flow over a plate, we have different Nusselt numbers for a plate kept at constant wall temperature dw, and for a plate with a constant heat flux qw at the wall, where the surface temperature adjusts itself accordingly. 

Example 3.7 Based on the solution (3.174) calculate the Nusselt number for parallel flow on a flat plate, if the thermal and velocity boundary layers are separated by xq, see Fig. 3.19. Find the Sherwood number for the case of Xq / 0. 

TABLE 1. Results for the local Nusselt number, Nu(Z), for the parallel-plates case with Knfi = 0.1 and jS = 1 Comparison against Ref [6] for prescribed wall temperature (with asterisk), Bi = CO, and reference results for Bi= 1. 

An experimental setup was designed and built for the determination of Nusselt numbers in a parallel plates channel made of brass and copper inside a PMMA (poly-methyl methacrylate) prism, with Joule effect heating on the brass side. Experimental runs for different Reynolds numbers allowed for obtaining a significant set of experimental results for a micro-channel height of 270 pm. Experimental results are then briefly discussed and presented to verify the proposed models. 

Figure 10. Comparison of measured (dots) local Nusselt numbers for parallel-plates micro-channel against limiting fully developed values for prescribed wall temperature (solid line) and heat flux (dashed line) (rearranging from 10 to 242).

Parallel Isothermal Plates. For parallel isothermal plates of either equal or different temperatures (see Fig. 4.21), Aung [11] has shown that the Nusselt number in the fully developed regime is given by... 

Uniform Heat Flux Parallel Plates. If the heat fluxes are specified as q" and q", respectively, on the surfaces of the vertical plates (where q" > q" and both q" and q" are positive, denoting heat transfer from the plate to the fluid), the Nusselt number for the fully developed regime for air is given by [11]... 

For r = 1, the concentric annular duct is reduced to a parallel plate duct. The applicable results are given in Table 5.28, the simple Nu being used for the Nusselt number at the heated wall. 

TABLE 5.28 Nusselt Numbers and Influence Coefficients for Fully Developed Thrbulent Flow in a Smooth Concentric Annular Duct With r = 1 (Parallel Plates Duct With Uniform Heat Flux at One Wall and the Other Wall Insulated [111] ... 

Parallel plate ducts, also referred to as flat ducts or parallel plates, possess the simplest duct geometry. This is also the limiting geometry for the family of rectangular ducts and concentric annular ducts. For most cases, the friction factor and Nusselt number for parallel plate ducts are the maximum values for the friction factor and the Nusselt number for rectangular ducts and concentric annular ducts. 

Laminar flow and heat transfer in parallel plate ducts are described in this section. The friction factor and Nusselt number are given for practical calculations. 

Similar to the four fundamental thermal boundary conditions for concentric annuli, the four kinds of fundamental conditions for parallel plate ducts are shown in Fig. 5.20. The fully developed Nusselt numbers for the four boundary conditions follow [1] ... 

Uniform Temperature at One Wall and Uniform Heat Flux at the Other. When the two walls of a parallel plate duct are subject to a thermal boundary condition such as uniform temperature at one wall and uniform heat flux at the other, the Nusselt numbers for fully developed laminar flow for qZ = 0 and q"w 0 are determined to be ... 

The Exponential Wall Heat Flux Boundary Condition . When both walls of parallel plate duct are subjected to the exponential heat flux of qZ = q% exp(rajt ), the fully developed Nusselt number can be obtained as follows [2] ... 

Equal and Uniform Temperatures on Both Walls. The local and mean Nusselt numbers for parallel plate ducts with equal and uniform temperatures on both walls can be computed from Nusselt s [131] solution, which is displayed in Fig. 5.21. The tabulated values for Fig. 5.21 are available in Shah and London [1]. 

FIGURE 5.21 Local and mean Nusselt numbers in the thermal entrance region of a parallel plate duct with the and (8) boundary conditions [1]. 

Equal and Uniform Temperatures at Both Walls. For simultaneously developing flow in a parallel plate duct with fluids of 0.1 < Pr < 1000, the following equations are recommended for the computation of the local and mean Nusselt numbers [2,136,137] ... 

Uniform and Equal Heat Flux at Both Walls. The local Nusselt number for heat transfer of laminar flow in a parallel plate duct with uniform and equal heat flux at both walls is displayed in Fig. 5.22 for different Prandtl numbers, Pr = 0 [34] and Pr = 0.01, 0.7,1,10, and °° 136]. The thermal entrance lengths obtained from the results presented in this figure are 0.016,0.030,0.017, 0.014,0.012, and 0.0115, for Pr = 0,0.01,0.7,1,10, and °°, respectively. 

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