Dimensionless Equations for Heat Transfer

Transformation of the independent variables to dimensionless form uses = r/R and jz = z/L. In most reactor design calculations, it is preferable to retain the dimensions on the dependent variable, temperature, to avoid confusion when calculating the Arrhenius temperature dependence and other temperature-dependent properties. The following set of marching-ahead equations are functionally equivalent to Equations (8.25)-(8.27) but are written in dimensionless form for a circular tube with temperature (still dimensioned) as the dependent variable. For the centerline. 

The more restrictive of the following stability criteria is used to calculate 

FIGURE 8.7 Numerical versus analytical solutions to the Graetz problem with ari/R = 0.4. 

Example 8.9 Find the temperature distribution in a laminar flow, tubular heat exchanger having a uniform inlet temperature and constant wall temperature Twall- Ignore the temperature dependence of viscosity so that the velocity profile is parabolic everywhere in the reactor. Use art/P = 0.4 and report your results in terms of the dimensionless temperature 

Solution A transformation to dimensionless temperatures can be useful to generalize results when physical properties are constant, and particularly when the reaction term is missing. The problem at hand is the classic Graetz problem and lends itself perfectly to the use of a dimensionless temperature. Equation (8.52) becomes 

---

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 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]... 

Equation 5.2.18 is not conveniently used because the heat capacity of the reacting fluid, (NjCpj), is a function of the temperature and reaction extents, and consequently, it varies during the operation. To simplify the equation and obtain dimensionless quantities for heat transfer, we define a heat capacity of the reference state and relate the heat capacity of the reacting fluid at any instant to it by... 

Subsequently, Kawase and Moo-Young [51] combined the three zone concept of Levich [52] with the isotropic turbulence ideas to develop a dimensionless relation for heat transfer with power law fluids in bubble columns [53], Their final equation is ... 

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. 

Film coefficients of mass transfer inside or outside tubes are important in membrane processes using tube-type or the so-called hollow fiber membranes. In the case where flow inside the tubes is turbulent, the dimensionless Equation 6.25a, b (analogous to Equation 5.8a, b for heat transfer) provide the film coefficients of mass transfer k [7]... 

In a more sophisticated analysis these functions can be found as the solutions of the dynamic and energy balance equations for filling a mold. 0m is the dimensionless temperature of the mold To is the initial temperature of the reactive mixture co = (H2kr,T)/a is the dimensionless factor characterizing the ratio of time scales for heat transfer and the chemical reaction. Other dimensionless variables are as follows ... 

Peclet Number, Pe dimensionless number appearing in enthalpy or species mass conservation equations (defined for heat transfer and mass transfer, respectively). It is interpreted again as the ratio of the convective transport to the molecular transport and is defined as... 

In order to find an equation with a wide application range, it is expedient to combine the decisive properties for heat transfer into dimensionless groups. For the general law of heat transfer it is advisable to choose an exponential equation in these quantities, because equations of this nature have proven themselves for the representation of heat transfer. 

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. 

In Chap. 9, we considered the solution of this equation in the limit Re 1, where the velocity distribution could be approximated by means of solutions of the creeping-flow equations. When Pe 1, we found that the fluid was heated (or cooled) significantly in only a very thin thermal boundary layer of 0(Pe l/3) in thickness, immediately adjacent to the surface of a no-slip body, or () Pe l/2) in thickness if the surface were a slip surface with finite interfacial velocities. We may recall that the governing convection di ffusion equation for mass transfer of a single solute in a solvent takes the same form as (111) except that 6 now stands for a dimensionless solute concentration, and the Peclet number is now the product of Reynolds number and Schmidt number,... 

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... 

HEAT-TRANSFER COEFFICIENTS. In an agitated vessel, as shown in Chap. 9, the dimensionless group D np/fi is a Reynolds number useful in correlating data on power consumption. This same group has been found to be satisfactory as a correlating variable for heat transfer to jackets or coils in an agitated tank. The following equations are typical of those that have been offered for this purpose. 

Solutions for the heat-transfer problem, which arises in the use of packed beds as direct-contact recuperative heat exchangers, were presented by Furnas in 1930. The parameter Njj is the number of heat-transfer units. For heat transfer, the dimensionless time t is the heat capacity of the gas times the amount of gas that has passed through the bed divided by the total bed capacity. For Nu = co, the breakthrough curve of T /Tq vs. t would be a vertical line at t = 1.0, just as for mass transfer. The defining equations are... 

Obviously, the solutions of Eqs. 1.74-1.77 depend on the coefficients that appear in these equations. Solutions of Eqs. 1.74-1.77 are equally applicable to the model and prototype (where the model and prototype are geometrically similar systems of different linear dimensions in streams of different velocities, temperatures, and concentration), if the coefficients in these equations are the same for both model and prototype. These coefficients, Pr, Re, Sc, and Ec (called dimensionless parameters or similarity parameters), are defined in Table 1.10. Focusing attention now on heat transfer, from Eq. 1.14, using the dimensionless quantities, the heat transfer coefficient is given as ... 

As mentioned above, the preceding equation is also useful to calculate fluid temperature profiles via boundary layer heat transfer if one replaces Ca by T and 50a, mix by thermal diffusivity a. The dimensionless profile for mass transfer is constructed as follows ... 

Early attempts to get a correlation for similar to the correlations for pipe flow showed wide variations in the exponents for the dimensionless groups and even differences in the equations for heating and cooling [26,27]. Such variations are understandable when we consider the different mechanisms of heat transfer in the bed itself and in the gas film at the wall. 

For reactors in which heat is transferred through a wall, a may be obtained from the following dimensionless equation for stirred vessels ... 

Lev que s problem was extracted from the rescaled mass balance in Equation 8.28. As can be seen, this equation is the basis of a perturbation problem and can be decomposed into several subproblems of order 0(5 ). The concentration profile, the flux at the wall, and consequently the mixing-cup concentration (or conversion) can all be written as perturbation series on powers of the dimensionless boundary layer thickness. This series is often called as the extended Leveque solution or Lev jue s series. Worsoe-Schmidt [71] and Newman [72] presented several terms of these series for Dirichlet and Neumann boundary conditions. Gottifredi and Flores [73] and Shih and Tsou [84] considered the same problem for heat transfer in non-Newtonian fluid flow with constant wall temperature boundary condition. Lopes et al. [40] presented approximations to the leading-order problem for all values of Da and calculated higher-order corrections for large and small values of this parameter. 

The equations for the steady-state and transient diffusive mass transfer are analogous to those used for heat transfer. Here, we use as mass transfer coefficient and the dimensionless Sherwood number Sh, which is the counterpart to the Nusselt number. 

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) ... 

Using dimensionless variables allows for the development of scale-independent design equations. Note that the species balance equation is almost identical to the heat balance, and that, by replacing the Pr number in Equation (4.25) with the Schmidt number Sc, we obtain a similar correlation for mass transfer as for heat transfer. In principle they describe similar phenomena, and, if we replace heat conduction k with heat diffusion a = k/pcp containing the same dimension as diffusivity (m s ), we obtain the same expression. 

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

---