Dimensionless form heat transfer equation

C vs. the start at 100°C. Thus we have a significant effect of the level of activity upon the magnitude of intraparticle temperatures, but not upon the time response. None of this behavior shows up in the Prater Number. The key to understanding time response comes fi om a detailed solution of the transient heat transfer equation. The dimensionless time will take the form (Lee et al. [16]) ... 

Equation (21.46) is analogous to the Colburn form of the heat-transfer equations (12.52) and (12.53). A second dimensionless equation, analogous to the Nusselt form of the heat-transfer equation is obtained by multiplying Eq, (21.46) by DG/ti)(jifpD ). This gives... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

Since the dimensionless equations and boundary conditions governing heat transfer and dilute-solution mass transfer are identical, the solutions to these equations in dimensionless form are also identical. Profiles of dimensionless concentration and temperature are therefore the same, while the dimensionless transfer rates, the Sherwood number (Sh = kL/ ) for mass transfer, and the Nusselt number (Nu = hL/K ) for heat transfer, are identical functions of Re, Sc or Pr, and dimensionless time. Most results in this book are given in terms of Sh and Sc the equivalent results for heat transfer may be found by simply replacing Sh by Nu and Sc by Pr. 

Fig. 4.59 illustrates the flow pattern determined by Eqs (4.45) and (4.47). The distribution of the velocity vectors is shown for the case when the axial velocity component equals (ux - 1). Eqs. (4.45) and (4.47) allow us to find temperature and degree of conversion distributions in the frontal zone based on the fundamental balance equations. These equations differ from Eqs. (4.36) and (4.47) because they take into account convective heat transfer along the z-direction. The dimensionless forms of the main determining equations are as follows energy balance... 

G. Damkohler [113] used this possibility to develop Navier-Stockes differential equations of the mass and heat transfer for the case of an adiabatic reaction. Analytical solution of these differential equations is not possible. However, if they are made dimensionless, it becomes apparent that the pi-space is formed by the five dimensionless numbers listed below ... 

The momentum and energy equations are very difficult to solve except for simple cases. For many cases of practical interest, the convective heat transfer is studied experimentally and the results are presented in the form of empirical equations that relate dimensionless numbers. 

Representation of Heat-Transfer Coefficients Heat-transfer coefficients are usually expressed in two ways (1) dimensionless relations and (2) dimensional equations. Both approaches are used below. The dimensionless form of the heat-transfer coefficient is the Nusselt... 

Two methods have proved themselves in the reproduction of heat transfer measurements. One starts from empirical correlations for the pure substances. These correlations normally contain dimensionless numbers, that now have to be formed with the properties of the binary mixture. The reduction in heat transfer because of inhibited bubble growth caused by diffusion is taken into account by the introduction of an extra term. This type of equation has been presented by... 

Consider the heat conduction/mass transfer problem in a cylinder.[6] [9] [10] The governing equation in dimensionless form is... 

The methodology is illustrated using a Laplace equation for heat transfer in a rectangle[7] [8] using length L and height H. The governing equation for the temperature in dimensionless form can be written as [8]... 

When the dimensionless form of the thermal energy equation is compared with the dimensionless Navier-Stokes equation, it is clear that the Peclet number plays a role for heat transfer that is analogous to the Reynolds number for fluid motion. Thus it is natural to seek approximate solutions for asymptotically small values of the Peclet number, analogous to the low-Reynolds-number approximation of Chaps. 7 and 8. 

We saw in Chapter 10 that the boundary-layer structure, which arises naturally in flows past bodies at large Reynolds numbers, provides a basis for approximate analysis of the flow. In this chapter, we consider heat transfer (or mass transfer for a single solute in a solvent) in the same high-Reynolds-number limit for problems in which the velocity field takes the boundary-layer form. We saw previously that the thermal energy equation in the absence of significant dissipation, and at steady state, takes the dimensionless form... 

The lima averaged equations of mass, momentum, energy, and species conservation can he written in dimensionless fores for a fluid in turfaolem flow past a surface. If (1) radiant eentgy and chamical reaction are not pres res. (2) viscons dissipation is negligible, (3) physical properties ate jedepeedent of temperature and composition, (4) the effect of mass transfer on velocity profiles is neglected, and (5) the boundary conditions are compatible, then dimensionless Incal heat and mass tmasfer coefficients can he shown to he described by equations of the form ... 

Note that h must always be positive. Equation (11.19) can be put into a dimensionless form by multiplying by the ratio of an arbitrary length to the thermal conductivity. The choice of length depends on the situation. For heat transfer at the inner surface of a tube, the tube diameter D is the usual choice. Multiplying Eq. (11.19) by D/k gives... 

The generation of such dimensionless groups in heat transfer (known generally as dimensional analysis) is basically done (1) by using differential equations and their boundary conditions (this method is sometimes called a differential similarity) and (2) by applying the dimensional analysis in the form of the Buckingham pi theorem. 

The Bromley equation (though still widely used) can give significant errors under a variety of conditions. An extensive exercise on the correlation of pool film-boiling heat transfer from cylinders is reported by Sakurai and Shiotsu [180], whose expression for the mean convective heat transfer coefficient is expressed in dimensionless form as follows ... 

The dimensionless equation describing the transfer phenomena may be obtained either by direct reference to the ratios of the physical quantities or by recourse to the classical techniques of dimensional analysis, i.e., the Buckingham n Theorem or Rayleigh s method of indices. In addition, the basic differential equations governing the process may be reduced to dimensionless form and the coefficients identified. In general, the dimensionless equation for heat transfer through the combined film is... 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

In an adsorbent bed of relatively small diameter, heat exchange through the column wall becomes appreciable. In this case, the overall heat transfer model introduced in 8.2(ii) can replace Eq. (8-54). Then similar calculations ae possible by taking the overall heat transfer coefficient ho and/or the radius of the column / as a parameter to examine this effect. Calculation using the same example considering this effect is shown in Fig. 8.9. Obviously the heat transfer parameter, (hoi R )(CftPfUol L), which is derived from the dimensionless form of the heat balance equation, determines the effect of heat transfer through the wall. 

Using first principles and dimensionless forms, we will derive the basic format describing the heat transfer coefficient. Next, we will use experimental data or combination of analytical solutions of the Energy Equation and experimental data to obtain equation for h. 

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