Convection flat plate

Lin, Hsiao Tsung and Chen, Yao Han, The Analogy Between Fluid Friction and Heat Transfer of Laminar Mixed Convection Flat Plates , lnt. J. Heat and Mass Transfer, Vol. 37, Nd 11, pp. 1683-1686, 1994. 

Under conditions of limiting current, the system can be analyzed using the traditional convective-diffusion equations. For example, the correlation for flow between two flat plates is... 

C. Laminar, local, flat plate, natural convection vertical plate... 

I. Turbulent, local flat plate, natural convection, vertical plate Turbulent, average, flat plate, natural convection, vertical plate Nsk. = — = 0.0299Wg=Ws = D x(l + 0.494W ) )- = 0.0249Wg=W2f X (1 + 0.494WE )- [S] Low solute concentration and low transfer rates. Use arithmetic concentration difference. Ncr > 10 " Assumes laminar boundary layer is small fraction of total. D [151] p. 225... 

Equation (8.12) is a form of the convective dijfusion equation. More general forms can be found in any good textbook on transport phenomena, but Equation (8.12) is sufficient for many practical situations. It assumes constant diffusivity and constant density. It is written in cylindrical coordinates since we are primarily concerned with reactors that have circular cross sections, but Section 8.4 gives a rectangular-coordinate version applicable to flow between flat plates. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

Fig. 4. Migration contribution to the limiting current in acidified CuS04 solutions, expressed as the ratio of limiting current (iL) to limiting diffusion current (i ) r = h,so4/(( h,so, + cCuS(>4). "Sulfate refers to complete dissociation of HS04 ions. "bisulfate" to undissociated HS04 ions. Forced convection" refers to steady-state laminar boundary layers, as at a rotating disk or flat plate free convection refers to laminar free convection at a vertical electrode penetration to unsteady-state diffusion in a stagnant solution. [F rom Selman (S8).]...

The diffusivities thus obtained are necessarily effective diffusivities since (1) they reflect a migration contribution that is not always negligible and (2) they contain the effect of variable properties in the diffusion layer that are neglected in the well-known solutions to constant-property equations. It has been shown, however, that the limiting current at a rotating disk in the laminar range is still proportional to the square root of the rotation rate if the variation of physical properties in the diffusion layer is accounted for (D3e, H8). Similar invariant relationships hold for the laminar diffusion layer at a flat plate in forced convection (D4), in which case the mass-transfer rate is proportional to the square root of velocity, and in free convection at a vertical plate (Dl), where it is proportional to the three-fourths power of plate height. 

Forced convection, burning of a flat plate. This classical exact analysis following the well-known Blasius solution for incompressible flow was done by Emmons in 1956 [7], It includes both variable density and viscosity. Glassman [8] presents a functional fit to the Emmons solution as... 

A candle bums at a steady rate (see the drawing below). The melted wax along the wick has a diameter D — 0.5 mm and pyrolysis occurs over a length of /p = 1 mm. Treat the wick as a flat plate of width ttD and a height of Ip which has a convective heat transfer coefficient h — 3 W/m2K. Ignore all radiative effects. 

A major fallacy is made when observations obeying a known physical law are subjected to trend-oriented tests, but without allowing for a specific behaviour predicted by the law in certain sub-domains of the observation set. This can be seen in Table 11 where a partial set of classical cathode polarization data has been reconstructed from a current versus total polarization graph [28], If all data pairs were equally treated, rank distribution analysis would lead to an erroneous conclusion, inasmuch as the (admittedly short) limiting-current plateau for cupric ion discharge, albeit included in the data, would be ignored. Along this plateau, the independence of current from polarization potential follows directly from the theory of natural convection at a flat plate, with ample empirical support from electrochemical mass transport experiments. 

The details of natural convective flows over surfaces other than flat plates have only recently been studied experimentally (A7, Jl, P3, SI2). We consider a heated sphere in an infinite, stagnant medium. Flow is directed toward the surface over the bottom hemisphere and away from the surface over the top hemisphere with a stagnation point at each pole (P3, S12). The lower pole is considered the forward stagnation point. 

The mass transfer from a flat plate has been studied by Spalding (S12) for both forced and free convection. 

These relationships have been used by Spalding in the dimensionless presentation both of theoretical values obtained in his approximate solution of the boundary layer equations (58) and of the experimental data (51, 55, 60). Emmons (3), who has solved the problem of forced convection past a burning liquid plane surface in a more rigorous fashion, shows graphically the rather close correspondence between values obtained from his exact solution and that of Spalding, and between the calculated values for flat plates and the experimental values for spheres. 

Figure 9.2 Dimensionless velocity distribution for laminar free convection on a vertical flat plate. Ostrach, 1953 [3].

Problem . Derive an expression for the maximum velocity in the flee convection boundary layer on a vertical flat plate. At what position in the boundary layer does this maximum velocity occur ... 

If a flat plate is inclined with an angle p from the body force direction, show that the Nusselt number for free convection on this inclined plate is a function of Pr and Grcosp. 

S Ostrach. An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Generating Body Force. NACA Report 1111, 1953. 

EM Sparrow and J L Gregg. Laminar Free Convection from a Vertical Flat Plate with Uniform Surface Het Flux. Trans ASME, Vol 78 435-440, 1956. 

The boundary layer equations were derived in a previous chapter, or may be deduced from the general convection equations in the early part of this chapter. For two-dimensional, steady flow over a flat plate of an incompressible, constant-property fluid, the continuity, x-momentum and the energy equations are as follows ... 

Consider convection with incompressible, laminar flow of a constant-temperature fluid over a flat plate maintained at a constant temperature. With the velocity distributions found in either Prob. 10.1 or Prob. 10.2, compute the dimensionless temperature distribution within the thermal boundary layer for the Peclet number equal to 0.1,1.0,10.0,100.0. Use the ADI method. 

Consider laminar forced convective flow over a flat plate at whose surface the heat transfer rate per unit area, qw is constant. Assuming a Prandtl number of 1, use the integral equation method to derive an expression for the variation of surface temperature. Assume two-dimensional flow. 

In order to illustrate how similarity solutions are obtained for free convective flows, see [7] to [23], consideration will initially be given to two-dimensional flow over a vertical flat plate with a uniform surface temperature. The situation being considered is thus, as shown in Fig. 8.6. 

Available analyses of turbulent natural convection mostly rely in some way on the assumption that the turbulence structure is similar to that which exists in turbulent forced convection, see [96] to [105]. In fact, the buoyancy forces influence the turbulence and the direct use of empirical information obtained from studies of forced convection to the analysis of natural convection is not always appropriate. This will be discussed further in Chapter 9. Here, however, a discussion of one of the earliest analyses of turbulent natural convective boundary layer flow on a flat plate will be presented. This analysis involves assumptions that are typical of those used in the majority of available analyses of turbulent natural convection. 

Solution. The following integrals arise in the approximate solution for turbulent natural convective boundary layer flow over a flat plate discussed above ... 

Compare the heat-transfer coefficients for laminar forced and free convection over vertical flat plates. Develop an approximate relation between the Reynolds and Grashof numbers such that the heat-transfer coefficients for pure forced convection and pure free convection are equal. 

Szewczyk, A.A.. "Stability and Transition of the Free Convection Layer Along a Vertical Flat Plate", lnt. J. Heat Mass Transfer, Vol. 5, pp. 903-914, 1962. 

Eckert, E.R.G. and Jackson. T.W.. "Analysis of Turbulent Free Convection Boundary Layer on a Flat Plate , NACA Rept. 1015, 1951. 

To. W.M. and Humphrey. J.A.C.. Numerical Simulation of Buoyant. Turbulent Flow. I. Free Convection Along a Heated, Vertical, Flat Plate . Int. J. Heat Mass Transfer, Vol. 29, pp. 573-592, 1986. 

In combined convective flow over a horizontal flat surface, the buoyancy forces are at right angles to the flow direction and lead to pressure changes across the boundary layer, i.e., there is an induced pressure gradient in the boundary layer despite the fact that flow over a flat plate is involved. Under some circumstances, this can lead to complex three-dimensional flow in the boundary layer. This type of flow will not be considered here, more information being available in [17] to [23]. 

It will be seen that Eq. (9.70) does, in fact, describe the variation in the mixed convection region with assisting flow to an accuracy that is quite acceptable for most purposes. Eq. (9.66) does, therefore, apply to assisting mixed convective flow over a flat plate. It has been shown that it does, in fact, also describe experimental results for other more complex situations to a good degree of accuracy provided the value... 

Consider, for example, assisting turbulent mixed convection over a vertical flat plate. This situation is schematically shown in Fig. 9.20. If it is assumed that the boundary layer assumptions apply, the governing equations for the mean velocity components and temperature are ... 

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. 

Air at a temperature of 10°C flows upward at a velocity of 0.8 m/s over a wide vertical 15-cm high flat plate which is maintained at a uniform surface temperature of 50°C. Plot the variation of the local heat transfer rate with distance along the plate from the leading edge. Also show the variations that would exist in purely forced and purely free convective now. 

Consider mixed convective laminar boundary layer flow over a horizontal flat plate that is heated to a uniform surface temperature. In such a flow there will be a pressure change across the boundary induced by the buoyancy forces, i.e. ... 

Wickem. G., Mixed Convection from an Arbitrarily Inclined Semi-Infinite Flat Plate , lnt. J. of Heat and Mass Transfer, Vol. 34, pp. 1935-57, 1991. 

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