Heat convective, dimensional analysis

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)... 

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. 

By dimensional analysis, derive a relationship for the heat transfer coefficient h for natural convection between a surface and a fluid on the assumption that the coefficient is a function of the following variables ... 

Perform a dimensional analysis on the heating of a plastic plate inside a convection oven. Your target value is the heating time to reach certain surface and center temperatures. 

Fluid flow is often turbulent, and so heat transfer by convection is often complex and normally we have to resort to correlations of experimental data. Dimensional analysis will give us insight into the pertinent dimensionless groups see Chapter 6, Scale-Up in Chemical Engineering, Section 6.7.4. 

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 ... 

After a review of the customary calculation methods [97] for both essential classes of dryers (convective and contact dryers), the dimensioning methods for spray dryers [98, 99], fluidized and spouted bed dryers [100, 101, 102], cascading rotary dryers [103], pneumatic conveying dryers [104], conductive-heating agitated dryers [105] and layer dryers [106] were presented. They all confirmed the initially made conclusion that the scaling up of dryers is still made today without dimensional analysis and the model theory based thereupon. 

This chapter has been concerned with the meaning of convective heat transfer and the heat transfer coefficient . The dimensionless variables on which convective heat transfer rates depend have also been introduced using dimensional analysis. 

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. 

We begin this section by analyzing the case of free convection in an infinite medium. The example chosen is shown in Fig. 6.7. A two-dimensional surface with constant temperature tp transfers heat to the adjacent infinite media. As a result of the temperature difference between the surface and the media, a natural convection flow is induced. A dimensional analysis applied to this problem shows that ... 

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) ... 

Formulas for heat convection coefficients he can be found from available empirical correlation and/or theoretical relations and are expressed in terms of dimensional analysis with the dimensionless parameters Nusselt number Nu, Rayleigh number Ra, Grashof number Gr, Prandtl number Pr, and Reynolds number Re, which are defined as follows ... 

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. 

We must now determine the heat transfer coefficients for the heating medium and for the stirred fluid. Again, we use dimensional analysis to establish correlations for determining these heat transfer coefficients. First, consider the stirred, that is, agitated, fluid inside the mixer. The geometric variables are vessel diameter D [L] and agitator blade length Lb [L], The material variables for the heated fluid inside the vessel are heat capacity Cp [L T 0 ], heat conductivity k [LMT 0 ], convective heat transfer coefficient h [MT 0 ],... 

In the dimensional analysis of convective mass transport, three new and important dimensionless groups make their appearance, all of which have well-known heat transfer counterparts, and all of which represent ratios of transport rates or transport resistances. They are, respectively, the Sherwood number Sh, the Schmidt number Sc, and the Biot number (for mass transfer) Bi and they are defined as follows ... 

Eq.(79) is the second Pick s low. Its structure is the same as that of the differential equation of the convective h t transfer (in case of a st y state process Eq. (56)). This gives the possibility, as shown later, to calculate the heat transfer processes by means of experimental data or equations for mass transfer. The basic methods for these calculations are the similarily theory and the dimensional analysis. That is why before considering the theory of mass transfor processes, we present these important methods largely used in chemical engineering and in particular In the area of packed columns. 

Through the dimensional analysis we have obtained two of the most fundamental parameters in the theory of convective heat balance, namely ... 

---