Dimensional analysis forced convection

The continuity equation (8.9) and the energy equation (8.12) are identical to those for forced convective flow. The x- and y-momentum equations, i.e., Eqs. (8.10) and (8.11), differ, however, from those for forced convective flow due to the presence of the buoyancy terms. The way in which these terms are derived was discussed in Chapter 1 when considering the application of dimensional analysis to convective heat transfer. In these buoyancy terms, is the angle that the x-axis makes to the vertical as shown in Fig. 8.3. 

Now we are ready for the dimensional analysis of convection problems. We begin with forced convection because of its relative simplicity. 

While dimensional analysis is often a useful tool for dealing with a problem, it has not yet been successful for studying this phenomenon, mainly because the fluid properties of importance in forced-convection boiling have not been identified. Burn-out correlations based on dimensional analysis have appeared, e.g., Griffith (G5), Reynolds (R2), Zenkevitch (Zl), Ivashkevich (12), Tong et al. (T6), but the fluid properties used in these cases have been chosen on the basis of various assumptions without any demonstration that the properties used were the correct ones. They have, in fact, been shown in recent work by Barnett (B5), (to be considered later) to be either incorrect or incomplete. 

Obtain, by dimensional analysis, a functional relationship for the heat transfer coeflicien for forced convection at the inner wall of an annulus through which a cooling liquid is flowing. 

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection can be neglected in comparison with those of forced convection. 

Dimensional analysis shows that, in the treatment of natural convection, the dimensionless Grashof number, which represents the ratio of buoyancy to viscous forces, is often important. The definition of the Grashof number, Gr, is... 

This type of equation comes from dimensional analysis. The coefficient and exponents are found by experiment. If forced convection is used, the Reynolds number has the conventional meaning of DVp/p. If free convection is used, the Reynolds number is replaced by the Grashof number, which can be shown to have a meaning of a Reynolds number, owing to free convection (B8). 

Dimensional Analysis of Forced Convection in a Single-Phase Flow... 

For heat transfer for a fluid flowing through a circular pipe, the dimensional analysis is detailed in Section 9.4.2 and, for forced convection, the heat transfer coefficient at the wall is given by equations 9.64 and 9.58 which may be written as ... 

In forced convection, the heat transferred per unit area per unit time q, is determined by a linear dimension which characterizes the surface /, the temperature difference between the surface and the fluid, AT, the viscosity rj, the density p, and the velocity u, of the fluid, its conductivity k, and its specific heat Cp. Dimensional analysis yields Eq. (32)... 

FLUIDS FLOWING NORMALLY TO A SINGLE TUBE. The variables affecting the coefficient of heat transfer to a fluid in forced convection outside a tube are D , the outside diameter of the tube Cp, /r, and k, the specific heat at constant pressure, the viscosity, and the thermal conductivity, respectively, of the fluid and G, the mass velocity of the fluid approaching the tube, Dimensional analysis gives, then, an equation of the type of Eq, (12,27) ... 

For both forced and natural convection, relations have been obtained by dimensional analysis which suggest that a correlation of experimental data may be in terms of three variables instead of the original six. This reduction in variables has aided investigators who have developed correlations for estimating convective mass-transfer coefficients in a variety of situations. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

Kinematic similarity is concerned with the motion of phases within a system and the forces inducing that motion. For example, in the formation of boundary layers during flow past flat plates and during forced convection in regularly shaped channels, there are usually three dominant forces pressure, inertia, and viscous forces. If corresponding points in two different-sized cells show at corresponding times identical ratios of fluid velocity, the two units are said to be kinematically similar and heat and mass transfer coefficients will bear a simple relation in the two cells. It can be shown by means of dimensional analysis that for a closed system under forced convection the equation of motion for a fluid reduces to a function of Re, the Reynolds number, which we have met in Chapter 2. To preserve kinematic similarity under those circumstances, Reynolds numbers in the two cells must be identical. 

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