Flow over a plane surface

The simple Reynolds analogy gives a relation between the friction factor R/pu and the Stanton number for heat transfer  

This equation can be used for calculating the point value of the heat transfer coelScient by substituting for Rlpu in terms of the Reynolds group Re using equation 11.39  

Equation 12.132 gives the point value of the heat transfer coefficient. If the whole surface is effective for heat transfer, the mean value is given by  

These equations take no account of the existence of the laminar sub-layer and therefore give unduly high values for the transfer coefficient, especially with liquids. The effect of the laminar sub-layer is allowed for by using the Taylor-Prandtl modification  

This expression will give the point value of the Stanton number and hence of the heat transfer coefficient. The mean value over the whole surface is obtained by inte atfion. No general expression for the mean coefficient can be obtained and a graphical or numerical integration must be carried out after die insertion of the aj ropriate values of tibe coiKtants. 

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This expression is applicable only to the region of fully developed flow. The heat transfer coefficient for the inlet length can be calculated approximately, using the expressions given in Chapter 11 for the development of the boundary layers for the flow over a plane surface. It should be borne in mind that it has been assumed throughout that the physical properties of the fluid are not appreciably dependent on temperature and therefore the expressions will not be expected to hold accurately if the temperature differences are large and if the properties vary widely with temperature. 

It will be assumed that a fluid of density p and viscosity /r flows over a plane surface and the velocity of flow outside the boundary layer is us. A boundary layer of thickness S forms near the surface, and at a distance y from the surface the velocity of the fluid is reduced to a value ux. 

It may be noted that no assumptions have been made concerning the nature of the flow within the boundary layer and therefore this relation is applicable to both the streamline and the turbulent regions. The relation between ux and y is derived for streamline and turbulent flow over a plane surface and the integral in equation 11.9 is evaluated. 

The discrepancy between the coefficients in equations 11.45 and 11,46 is attributable to the fact that the effect of the curvature of the pipe wall has not been taken into account in applying the equation for flow over a plane surface to flow through a pipe. In addition, it takes no account of the existence of the laminar sub-layer at the walls. 

Heat transfer for streamline flow over a plane surface—constant surface temperature... 

The intensity of turbulence will vary with the geometry of the flow system. Typically, for a fluid flowing over a plane surface or through a pipe, it may have a value of between... 

In steady state flow over a plane surface, or close to the wall for flow in a pipe, u is constant and equation 12.26 can be integrated provided that the relation between XE and y is known. XE will increase with y and, if a linear relation is assumed, then ... 

The quantity a, which is the ratio of the velocity at the edge of the laminar sub-layer to the stream velocity, was evaluated in Chapter 11 in terms of the Reynolds number for flow over the surface. For flow over a plane surface, from Chapter 11 ... 

For flow over a plane surface, substitution from equation 11.32 gives ... 

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

In order to illustrate how the boundary layer equations are derived, [7],[9],[13], [ 14],[ 15], consider two-dimensional constant fluid property flow over a plane surface which is set parallel to the x axis. The following are then defined ... 

These equations were derived for flow over a plane surface. They may be applied to flow over a curved surface provided that the boundary layer thickness remains small compared to the radius of curvature of the surface. When applied to flow over a curved surface, x is measured along the surface and y is measured normal to it at all points as shown in Fig. 2.15, i.e., body-fitted coordinates are used. 

A dilute polymer solution at 293 K flows over a plane surface (250 mm wide X 500 mm long) maintained at 301K. The thermophysical properties (density, heat capacity and thermal conductivity) of the polymer solution are close to those of water at the same temperature. The rheological behaviour of this solution can be approximated by a power-law model with n=0.43 and m = 0.3 — 0.000 33 T, where m is in Pa-s" and J is in K. 

A china clay suspension (density 1200kg/m, n = 0.42, m = 2.3Pa S") flows over a plane surface at a mean velocity of 2.75m/s. The plate is 600 mm wide normal to the direction of flow. What is the mass flow rate within the boundary layer at a distance of 1 m from the leading edge of the plate ... 

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