Heat Transfer to a Flat Plate

The mean Sherwood numbers for a disk of finite radius a at high Schmidt numbers can be estimated using the formula 

Relation (3.2.13) is valid in the region of laminar flow past the disk the laminar regime occurs until Re 104 to 105, depending on the roughness of the surface. For low Reynolds numbers (Re 10), this relation is invalid, because the thickness of the hydrodynamic boundary layer becomes comparable with the disk radius and the boundary effects on the hydrodynamic flow and mass transfer become stronger. 

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Statement of the problem. Thermal boundary layer. Let us consider heat transfer to a flat plate in a longitudinal translational flow of a viscous incompressible fluid with velocity U at high Reynolds numbers. We assume that the temperature on the plate surface and remote from it is equal to the constants Ts and 7], respectively. The origin of the rectangular coordinates X, Y is at the front edge of the plate, the X-axis is tangent, and the Y-axis is normal to the plate. 

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

Fig. 13.2 Heat flux transferred to a flat plate in a boundary layer, plotted as In q, as a function of Ug.

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. 

Assume that one-half the heat transfer from a cylinder in cross flow occurs on the front half of the cylinder. On this assumption, compare the heat transfer from a cylinder in cross flow with the heat transfer from a flat plate having a length equal to the distance from the stagnation point on the cylinder. Discuss this comparison. 

Ostrach, S. An analysis of laminar free convection flow and heat transfer about a flat plate parallel to the direction of the generating body force. NACA, Techn. Report 1111 (1953)... 

When the heat removal is entirely due to the flowing gas and there is no eonveetive particle movement, Gabor (1970) proposed the following simple model. For heat transfer from a flat plate of length immersed in the paeked bed. 

The log-mean mole fraction was chosen as the best way to describe the driving force for mass transfer. Since the membrane cell Is analogous to a flat plate, counterflow, heat exchanger, a log- mean mole fraction difference, similar to a log-mean temperature difference was defined as ... 

When the fluid is flowing parallel to a flat plate and heat transfer is occurring between the whole plate of length L m and the fliiid, the is as follows for below... 

HEAT AND MASS TRANSFER TABLE 5-21 Mass Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid Concluded)... 

In all cooled appliances, the heat from the device s heat sources must first arrive via thermal conduction at the surfaces exposed to the cooling fluid before it can be transferred to the coolant. For example, as shown in Fig. 2.2, it must be conducted from the chip through the lid to the heat sink before it can be discharged to the ambient air. As can be seen, thermal interface materials (TIMs) may be used to facilitate this process. In many cases a heat spreader in the form of a flat plate with high thermal conductivity may be placed between the chip and the lid. 

Heat Conduction across a Flat Solid Slab Solve the problem of heat transfer across an infinitely large flat plate of thickness H, for the following three physical situations (a) the two surfaces are kept at T and T2, respectively (b) one surface is kept at T while the other is exposed to a fluid of temperature Tb, which causes a heat flux q,, h = 2( 2 — Tb),h2 being the heat-transfer coefficient (W/m2-K) (c) both surfaces are exposed to two different fluids of temperatures Ta and Tb with heat-transfer coefficients h and hi, respectively. 

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. 

Air at a pressure of S kPa and a temperature of —30°C flows at a Mach number of 2.5 over a flat plate that is aligned with die flow. The plate is kept at a uniform temperature of 5°C. Find the heat transfer from the plate surface to the air. 

A flat plate of length L is heated to a uniform surface temperature and dragged through water which is at a temperature of 10°C at a velocity, V. Plot the variation of power required to pull the plate through die water and of the total heat transfer rate from the plate with plate surface temperature for surface temperatures between 10 and 95°C. Comment on the results obtained. The boundary layer on the plate can be assumed to remain laminar. 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

In the discussion of the use of the Reynolds analogy for the prediction of the heat transfer rate from a flat plate it was assumed that when there was transition on the plate, the x-coordinate in the turbulent portion of the flow could be measured from the leading edge. Develop an alternative expression based on the assumption that the momentum thickness before and after transition is the same. This assumption allows an effective origin for the x-coordinate in die turbulent portion of the flow to be obtained. 

Air at standard atmospheric pressure and a temperature of 30°C flows over a flat plate at a velocity of 20 m/s. The plate is 60 cm square and is maintained at uniform temperature of 90°C. The flow is normal to a sule of the plate. Calculate the heat transfer from the plate assuming that the flow is two-dimensional. 

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. 

It turns out that Eq. (5-56) can also be applied to turbulent flow over a flat plate and in a modified way to turbulent flow in a tube. It does not apply to laminar tube flow. In general, a more rigorous treatment of the governing equations is necessary when embarking on new applications of the heat-trans-fer-fluid-friction analogy, and the results do not always take the simple form of Eq. (5-56). The interested reader may consult the references at the end of the chapter for more information on this important subject. At this point, the simple analogy developed above has served to amplify ouf understanding of the physical processes in convection and to reinforce the notion that heat-transfer and viscous-transport processes are related at both the microscopic and macroscopic levels. 

An experiment is to be designed to demonstrate measurement of heat loss for water flowing over a flat plate. The plate is 30 cm square and it will be maintained nearly constant in temperature at 50°C while the water temperature will be about 10°C. (a) Calculate the flow velocities necessary to study a range of Reynolds numbers from 104 to 107. (b) Estimate the heat-transfer coefficients and heat-transfer rates for several points in the specified range. 

Plot hj versus x for air at 1 atm and 300 K flowing at a velocity of 30 m/s across a flat plate. Take Reonl = 5 x 10s and use semilog plotting paper. Extend the plot to an x value equivalent to Re 10. Also plot the average heat-transfer coefficient over this same range. 

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