(368j) 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.