Equivalents - A New Concept for the Prediction of Transport

 

I recently received a written communication that asked why the prediction by an expression of Churchill  and Chu in 1975 of the Nusselt number for free convection from a cylinder in the limit of Ra → 0, namely 0.60, was twice that by an expression of  Churchill and Bernstein in 1977 for forced convection from a cylinder in the limit of Re → 0, namely 0.30.  That discrepancy was exaggerated by a misinterpretation – the limiting value in the correlating equation for free convection was actually (0.60)² = 0.36, but there should not have been any difference. Both 0.30 and 0.36 were artifacts of independent ill-conceived attempts to extend by means of an additive constant the applicability of the predictive expressions to values of Gr and Re below the limits of  validity of  thin laminar boundary layer theory. A better means of extension is presented herein, but attention is first turned to a matter of broader scope that was prompted by the inquiry, namely the possible implications and applications of a quantitative  equivalence between free and forced convection.

The dimensionless grouping which determine the relative  rates of combined free and forced convection in the thin laminar boundary layer regime in the asymptotic limits of  Pr →  0 and ∞ were noted  by Acrivos in 1966, by Morgan in 1975, and by myself in 1983, but practical  applications were not identified by any of us. The current investigation has revealed that the equivalence is functionally independent of but numerically dependent on the Prandtl number, that it may be extended for other thermal boundary conditions and for other geometries, that it may contribute to understanding in the classroom, and that it has the potential for prediction in a practical sense.   

Analogies have long had a prominent role in chemical engineering design and analysis despite their flaws in both a fundamental and a practical sense. For example, that between  momentum and energy transfer allows the heat transfer coefficient to be predicted from a correlating equation for the friction factor. Unfortunately, the predictions of the most widely used analogy are in error by as much as 40% for the very conditions for which it was devised. The newly identified equivalences are found to have a sounder rationale than the analogies, and their predictions to be more accurate.   

The Grashof number that produces the same value of the local Nusselt number as the Reynolds number for  the same geometrical configuration and the same Prandtl number was chosen as a marker of the equivalence rather than the Rayleigh number because that choice simplifies the ultimate expressions.

              The first equivalence to be examined was that for an isothermal plate in the thin laminar boundary layer regime as represented by expressions in the form of the power-mean of asymptotes for limiting values of the Prandtl number. Fortuitously, the first description of that methodology for the construction of  correlative/predictive expressions by Churchill and Usagi in 1972 included the following expression for the local Nusselt number for forced convection from an isothermal plate:

                                                                                                      (1)

The coefficients 0.5027 and 0.4914 have a theoretical basis. The value of -9/4 for the combining exponent was determined empirically but its ubiquity in such expressions implies the existence of an as yet unidentified theoretical rationale.

Although this is not the place for a detailed discussion of the idealizations of thin laminar boundary layer theory, it should at least be mentioned that some limited range of flow is presumed. In the case of forced convection that implies a particular range of the Reynolds number and in the case of free convection a particular range of the Grashof number.

Churchill and Ozoe in 1973 devised a correlative/predictive expression  for the local Nusselt number for forced convection from an isothermally plate in the thin laminar boundary layer regime. They derived the asymptotes and used their own numerical solutions for several finite values of Pr to evaluate a combining exponent of -1. The result is

                                                                                                          (2)

Just as with Eq. 1, the coefficients have a theoretical basis and, again by virtue of its ubiquity,

the combining exponent probably has an as yet unidentified rationale.

Equating the right-hand sides of Eqs. 1 and 2 results in

     

                                                     =                        (3)

    

which can be re-arranged and simplified as

                                        

                                                              Gr_x = A{Pr}Re                                                                  (4)  

where                               

                                              A{Pr}                                (5)

According to Eq. 4, Gr_x is equivalent to Retimes a function of Pr. This result suggests that the concept of equivalence has potential for  prediction and for understanding. As an example of the latter, it may be inferred that free convection generates an equivalent velocity equal to (xgβΔT/A{Pr})^1/2.          

The combination of Eqs. 4 and 5 is apparently the first expression to be identified for an equivalence between free and forced convection for all values of Pr, and its generality exceeds expectations in several respects.  First, for any finite value of Pr, the function A{Pr}becomes simply a numerical value. Second, an  equivalence for the integrated-mean Nusselt number, ,  can be derived simply by multiplying the leading coefficient for free convection by 4/3 and that for forced convection by 2. The net result is that the leading coefficient of Eq. 5  is replaced by 0.2061(3/2)⁴ = 1.043. Third, insofar as the Grashof number is defined as it is for a uniform wall temperature, that is in terms of the difference between the local wall temperature (now a dependent variable) and the ambient temperature, the expressions for uniform heating are identical to those for a uniform wall temperature except for the numerical values of the three coefficients. Fourth, the corresponding expressions for a horizontal cylinder and for a sphere are also identical to those for a flat plate except for the numerical values of the three coefficients.

          If Eq. 4 is to be predictive in a quantitative sense for other geometries and thermal boundary conditions, the function A{Pr} must be generalized and/or predictable. Considerable progress has been made in those respects and is to be reported in the CDROM version and the presentation.

The complete differential models for steady flow over a long horizontal cylinder and for heat transfer by thermal conduction from a long horizontal cylinder are ill-posed in a strict mathematical sense but the postulate of a thin laminar boundary layer and the neglect of the effect of the wake and plume on heat transfer have permitted the derivation of the afore-mentioned approximate solutions.   

The various attempts to encompass the regime of a thickening boundary layer by the addition of a limiting value to the solutions for a flat plate and a cylinder failed because the experimental values of the Nusselt number appear to approach zero as does Gr for free convection and Re for forced convection. Langmuir in 1912, in the process of modeling heat transfer within a partially evacuated light-bulb, devised an approximate solution for the effect of the cylindrical curvature of the filament that can be adapted  for the regime of thickening. He began by postulating that the heat loss from the filament by free convection could be represented by thermal conduction across a hypothetical "sheath" (annular ring) of stagnant gas of thickness d = (D_o - D_i.)/2. He correctly expressed the ensuing heat flux density at the surface of wire in terms of the logarithmic-mean area and thereby obtained

                                                                                (6)

Equation 6 can be expressed more simply and generically as

                                                                                                         (7)

Here  represents the heat transfer by conduction through a flat "sheath" of gas and  

 

through a cylindrically curved one. Equation 7  is promising in that the hypothetical thickness

of the sheath does not appear explicitly, but the prediction of the Nusselt number for convection

from a cylinder from that for a flat plate is highly inaccurate. If, however, Eq. 7 is re-

expressed as

                                                                                                                  (8)

it predicts values in good agreement with experimental data for the entire laminar boundary Nu_l,  

from those for thin laminar boundary layer theory as symbolized Nu_tlbl . Equation 8 proves to be

applicable for both flat plates and cylinders and for both free and forced convection. The latter

duality resolves the discrepancy that prompted this investigation.

Theoretically based correlating equations for the integrated-mean Nusselt number for an isolated sphere in laminar flow generally consist of the sum of a closed-form solution for convection in the thin laminar boundary layer regime and the exact solution for pure conduction to unbounded surroundings. That is

                                                       + 2                                                                 (9)                                                                                                             

Curiously. the application of Langmuir's concept of approximating convection by conduction across a stagnant film in a  spherical annulus, abetted by  the geometric-mean area, results in

   + 2                                                                (10)  The forgoing is only a sample of the potential applicability of the concept of equivalence

between free and forced convection, which is also found to be applicable to turbulent and confined flows.