Heat transfer forced convection approximation

The approximation of all fluid properties as constant, including the complete neglect of any natural convection, is known as the forced convection approximation. We shall adopt it for this chapter, as well as for Chap. 11. Unlike most approximations that are introduced in this book, the forced convection approximation is adopted initially on an ad hoc basis, without a rigorous asymptotic justification.1 We may be reassured that the resultant analysis is relevant to many conditions of practical interest by the fact that it has been adopted almost universally for analysis of heat transfer problems, in which the fluid motion is not due solely to natural convection. 

It is difficult to solve the system of Eqs. (39)—(41) for these boundary conditions. However, certain simplifying assumptions can be made, if the Prandtl number approaches large values. In this case, the thermal boundary layer becomes very thin and, therefore, only the fluid layer near the plate contributes significantly to the heat transfer resistance. The velocity components in Eq. (41) can then be approximated by the first term of their Taylor series expansions in terms of y. In addition, because the nonlinear inertial terms are negligible near the wall, one can further assume that the combined forced and free convection velocity is approximately equal to the sum of the velocities that would exist when these effects act independently. Therefore, for assisting flows at large Prandtl numbers (theoretically for Pr -> oo), Eq. (41) can be rewritten in the form ... 

In some forced convective flows it has been found that the Nusselt number is approximately proportional to the square root of the Reynolds number. If, in such a flow, it is found that h has a value of 15 W/m2K when the forced velocity has a magnitude of 5 m/s, find the heat transfer coefficient if the forced velocity is increased to 40 m/s. 

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. 

It will be seen from the results given in the above table that the heat transfer rate is within 1% of its forced convective value when the velocity is greater than approximately 0.2 m/s. 

Boundary-layer theory has been applied to solve the heat-transfer problem in forced convection laminar flow along a heated plate. The method is described in detail in numerous textbooks (El, G5, S3). Some exact solutions and approximate solutions are also obtained (B2, S3). 

The heat transfer coefficients am and rad are approximately equal. This infers that free convection to the air and radiative exchange transport almost the same amount of heat. This is not true for forced convection, where m, depending on the flow velocity, is one to two powers of ten larger than the value calculated here. However, ra[Pg.29]

The feeder and injector produced a thin pencil-like p.c. stream which passed down through the hot zone. The total combustion air supplied was approximately 3 liters/min for the bituminous coals, giving between 10 and 25 percent excess air for p.c. feed rates of 0.24 to 0.28 g/min. The flow and heat transfer conditions were modeled using the methods described by Pigford (16) for conditions of superimposed natural and forced convection at very low mass flow rates. Particle residence times were calculated by summing the centerline gas velocity and terminal velocity using Stokes s law (17). The error introduced using this method should never have exceeded 10 percent, even when pyrite was tested and particle Reynold s numbers approached one. The residence times thus calculated were found to be between one and two seconds. 

The limit Pe 0 yields the pure conduction heat transfer case. However, for a fluid in motion, we find that the pure conduction limit is not a uniformly valid first approximation to the heat transfer process for Pe 1, but breaks down far from a heated or cooled body in a flow. We discuss this in the context of the Whitehead paradox for heat transfer from a sphere in a uniform flow and then show how the problem of forced convection heat transfer from a body in a flow can be understood in the context of a singular-perturbation analysis. This leads to an estimate for the first correction to the Nusselt number for small but finite Pe - this is the first small effect of convection on the correlation between Nu and Pe for a heated (or cooled) sphere in a uniform flow. 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

The forced convection heat transfer problem [Eq. (9-7) plus boundary conditions] is linear in 6, but it still cannot be solved exactly (except for special cases) for Pe > 0(1) because of the complexity of the coefficient u. What may appear surprising at first is that simplifications arise in the limit Pe 1, which allow an approximate solution even though no analytic solutions (exact or approximate) are possible for intermediate values of Pe. This is surprising because the importance of the troublesome convection term, which is... 

Equations (12—168)—(12 170) are known as the Boussinesq equations of motion and will form the basis for the natural convection stability analyses in this chapter. In fact, the Boussinesq approximation has been used in much of the published theoretical work on natural convection flows. Although one should expect quantitative deviations from the Boussinesq predictions for systems in which the temperature differences are large (greater than 10°C-20°C), it is likely that the Boussinesq equations remain qualitatively useful over a considerably larger range of temperature differences. In any case, although the Boussinesq equations represent a very substantial simplification of the exact equations, the essential property of coupling between the thermal and velocity fields is preserved, and, even in the Boussinesq approximation, the solution of natural convection problems is more complicated than the forced convection heat transfer problems that we encountered earlier. 

The heat transfer immediately downstream of the location where heating begins will be dominated by forced convection and will depend on the velocity profile. For a parabolic inlet profile, the forced convection Nusselt number can be approximated by [249] ... 

Laminar Free Convection. When a stagnant vapor condenses on a vertical plate, the motion of the condensate will be governed by body forces, and it will be laminar over the upper part of the plate where the condensate film is very thin. In this region, the heat transfer coefficient can be readily derived following the classical approximate method of Nusselt [12], Consider the situation depicted in Fig. 14.4 where the vapor is at a saturation temperature Ts and the plate surface temperature is T . Neglecting momentum effects in the condensate film, a force balance in the z-direction on a differential element in the film yields... 

Rose [47, 48] has suggested an approximate solution for forced convection condensation along a flat plate in the presence of noncondensable gas using an analogy between heat and mass transfer. He points out that, in this situation, the diffusion problem for the vapor-gas... 

Effect of Noncondensable Gas. As described earlier, Rose [47, 48] has suggested an approximate method to calculate condensation heat transfer in the presence of a noncondensable gas. For forced convection over a single horizontal tube, he recommends the following relationship (similar to Eq. 14.52 for a flat plate) that relates the mean condensation rate to the free stream conditions ... 

J. W. Rose, Approximate Equations for Forced-Convection Condensation in the Presence of a Noncondensing Gas on a Flat Plate and Horizontal Tube, Int. J. Heat Mass Transfer, 23, pp. 539-546, 1980. 

Kenning and Cooper [233] report data for forced convective boiling of water in a vertical tube. At high qualities, the heat transfer coefficient becomes independent of heat flux and varies approximately linearly with quality as illustrated in Fig. 15.95. 

Deposition of nanoparticles was investigated in the free molecular regime approximation for thermophoretic force and the Brownian motion. The analytical solution was obtained by the Galerkin method for the heat transfer between gas flow and substrates and convective diffusion. Relative roles of two channels of nanoparticle deposition are discussed. 

In free convection, there is a new complexity in that fluid motion arises from buoyancy forces due to the density differences, and the momentum, heat and mass balance equations are therefore coupled. The published analytical results for heat transfer from plates, cylinders and spheres involve significant approximations. This work has been reviewed by Shenoy and Mashelkar [1982] and Irvine and Kami [1987] the simple expressions (which are also considered to be reliable) for heat transfer coefficients are given in the following sections. 

Cooling fluid Approximate Range of Operation Min/Max °C Temp for Ref. Data, C Density, kg/m Heat Capacity, J/kgK Thermal Conductivity, W/m K Viscosity, mPa-s Prandtl Number Reynolds Number at Nusselt Number 25 mm Tube Forced Convection Heat Transfer Coefficient, W/m -K Cooling Channel Surface Temperature, C for 424 kW/m heat flux Approximate Copper Hot Face Temperature, C... 

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