EquivalentsâA New Concept for the Prediction and Interpretation of

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EquivalentsA New Concept for the Prediction and Interpretation of Thermal Convection Stuart W. Churchill* Department of Chemical and Biomolecular Engineering, University of Pennsylvania, 311 A Towne Building, 220 South 33rd Street, Philadelphia, Pennsylvania, 19104, United States ABSTRACT: The conditions for which free convection and forced convection in the thin laminar boundary layer regime produce the same Nusselt number are indentified and generalized. Such conditions are herein designated as equivalents, and the relationship between the two conditions is designated as an equivalence. The concept of equivalence provides a new structure for the assembly of information on heat transfer and mass transfer, and therefore, a new resource for teachers as well as for practitioners of transport and process design. It is the commonality of the equivalence for different conditions and geometries that renders it uniquely useful in these regards.

1. INTRODUCTION The concept of an equivalence as defined herein emerged only after a long period of gestation. Lemlich and Hoke1 in 1956 utilized the concept of an effective velocity in the Reynolds number to force experimental data for free convection from a horizontal cylinder to fall on the curve fitted by Douglas and Churchill2 through experimental data for forced convection. Thin laminar boundary layer theory (hereafter tlblt for short) led to a generalization of the process. The simplifications of tlblt allow the partial differential equations that describe the transfer of momentum and energy to be reduced by means of a similarity transformation to an ordinary one that can be solved in closed form with a numerically determined coefficient in the two asymptotic limits of the Prandtl number approaching zero and infinity. The dimensionless groupings that result in equal rates of combined free and forced laminar convection for these two limiting conditions were identified independently by Acrivos3 in 1966, by Morgan4 in 1975, and by Churchill5 in 1977. These dimensionless groups, which are designated in the current work as equivalents, can be expressed in the form of equations as follows Grx = Rex 2Pr1/3

for Pr → ∞

the Rayleigh number and the Prandtl number although other parameters may have a role (see, for example, Churchill7 in 1970). On the other hand the Grashof number appears to characterize the condition of transition from laminar to turbulent motion somewhat better than the Rayleigh number. The Nusselt number for forced convection is commonly expressed in terms of Reynolds number and the Prandtl number although various parameters may have a secondary role, particularly in the transition from laminar to turbulent flow (see, for example the work of Churchill8 in 1976). Accordingly 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 is chosen herein as a marker of the equivalence. The identical dimensionless mean velocity, a physically well-defined quantity, not to be confused with the aforementioned effective velocity of Lemlich and Hoke,1 could conceivably have been chosen as the criterion, but it is less characteristic of the behavior than the Nusselt number and it has rarely been computed or derived from experimental measurements or numerical computations for free convection. Analogies have long had a prominent role in chemical engineering design and analysis even though, as recently pointed out by Churchill,9 they all have serious flaws in both a fundamental and a practical sense. For example, the analogy 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 one for this purpose, namely that of Colburn10 are in error by as much as 40% for the very conditions for which it would be expected to be valid. The equivalences identified and described herein not only have a theoretical rationale, but their predictions appear to be more accurate than those of the classical analogies. The developments herein are all in terms of heat transfer but they

(1)

and Grx = Rex 2

for Pr → 0

(2)

A collateral development proved to be of major importance in terms of the development of equivalences. Churchill and Usagi6 in 1972 proposed a generalized procedure for the construction of generalized correlating equations, namely taking the powermean of asymptotes for limiting values of the independent variable. The optimal choice of the power-mean often results in an essentially exact expression in a numerical sense and that proved to be the case in this instance, as illustrated in the next subsection. The prediction of the identical Nusselt number for free and forced convection was chosen herein as the criterion for equivalence. The Nusselt number for free convection in any particular geometry is commonly expressed primarily in terms of © 2014 American Chemical Society

Received: Revised: Accepted: Published: 4104

October 18, 2013 February 10, 2014 February 13, 2014 February 13, 2014 dx.doi.org/10.1021/ie403511s | Ind. Eng. Chem. Res. 2014, 53, 4104−4118

Industrial & Engineering Chemistry Research

Article

Nux = 0.6004Grx1/4Pr1/2

may be inferred to apply to mass transfer insofar as the analogy between heat and mass transfer is valid.

for Pr → 0

(8)

for Pr → ∞

(9)

and

2. VERTICAL PLATE WITH A UNIFORM TEMPERATURE Because of its relative simplicity, the first equivalence to be examined in detail is that of free convection and forced convection from a vertical isothermal plate in the thin laminar boundary layer regime. Fortuitously, the original presentation of that methodology of Churchill and Usagi6 included the following expression (after replacement of Rax by PrGrx) for the local Nusselt number for free convection from an isothermal vertical plate: Nux =

Nux = 0.5027Grx1/4Pr1/4

Because the algebraic expression for equivalence is a consequence of thin laminar boundary layer theory, the basic idealizations of that differential model bear identification here. They are that the viscosity affects the flow only in the boundary layer and that diffusion of momentum and energy in the direction of flow is negligible relative to diffusion normal to the surface. Prandtl13 in 1904 discovered the similarity transformation that allowed reduction of the partial differential model for flow over a flat plate to an ordinary differential model. For free convection, the rate of flow is defined by the Grashof number and in force convection by the Reynolds number. The derivation of eqs 6 and 7 invokes a boundary condition whose ramifications have rarely, if ever, been mentioned, namely the restriction of the induced flow to the region above the bottom of the vertical plate. Conceivably such behavior could be accomplished by means of the downward extension of the plate at a temperature below the ambient by the same number of degrees that the upper portion is above it. The combination of eqs 8 and 9 in the form of the CUE can be expressed as

0.5027Pr1/4Grx1/4 [1 + (0.4914/Pr )9/16 ]4/9

(3)

11

Churchill and Ozoe subsequently devised the following analogue of eq 3 for the local Nusselt number for forced convection: Nux =

0.3387Rex1/2Pr1/3 [1 + (0.04681/Pr )2/3 ]1/4

(4)

Equating the right-hand sides of eqs 3 and 4 results in 0.5027Pr1/4Grx1/4 [1 + (0.4914/Pr )9/16 ]4/9

=

0.3387Rex1/2Pr1/3

Nux = [(0.6004Grx1/4Pr1/2)n + (0.5027Grx1/4Pr1/4)n ]1/ n

[1 + (0.04681/Pr )2/3 ]1/4

(10) (5)

or

Equation 5 can for clarity be expressed in the following alternative form: Grx = A{Pr }Rex 2

⎛ ⎞n ⎛ 0.5027 ⎞n Nux ⎜⎜ ⎟ = + 1 ⎜ ⎟ 1/4 1/2 ⎟ ⎝ 0.6004Pr1/4 ⎠ ⎝ 0.6004Grx Pr ⎠

(6)

A value of −9/4 for the exponent n was found by Churchill and Usagi6 to result in a good fit on the mean for several sets of experimental data, and the quantity (0.5027/0.6004)4 is equal to 0.4914, thereby resulting in the transformation of eq 11 to eq 3. Because of its ubiquity in correlative/predictive expressions for free convection, that value of −9/4 probably has an as yet undiscovered theoretical rationale. Most equations of this form are remarkably accurate because the task of correlation is reduced from representation of the dependent variable itself to representation of the generally small deviations of that variable from the nearest of two asymptotes. Equation 4 was formulated by Churchill and Ozoe11 by combination of the following two asymptotic solutions from Kestin and Schlichting:14

where A{Pr } =

0.2061Pr1/3[1 + (0.4914/Pr )9/16 ]16/9 1 + (0.04681/Pr )2/3

(11)

(7)

The function A{Pr}, as defined by eq 6 and given in detail by eq 7, can be interpreted as a quantitative measure of the equivalence for a vertical isothermal plate. Equations 6 and 7 apparently provide the first representation for an equivalence between free and forced convection that encompasses all values of Pr. According to eq 6, Grx is equivalent to Rex2 times a function of Pr. Although the proportionality of Grx to Rex2 is not cited in the literature of transport it is obvious in retrospect because the proportionality of Nux to Rax1/4 in laminar free convection to or from a vertical isothermal plate and to Rex1/2 in laminar forced convection to or from an isothermal plate are well-known. On the other hand, the functional dependency on the Prandtl number, as predicted by eq 7, is neither simple nor obvious and, apparently, was not identified prior to this investigation. The complexity of eq 7 is an artifact of equating the correlative/ predictive equations for free and forced convection, both of which consist of the arbitrary power-mean of two asymptotes that are closed-form solutions of a thin laminar boundary layer model with numerically determined leading coefficients. Before proceeding further, it seems fitting to examine the functional and numerical accuracy and the ranges of validity of the components of eqs 6 and 7. 2A. Validity of Expressions Comprising the Equivalence. Equation 3 was formulated by combining the following two asymptotic solutions of LeFevre12 that were obtained from thin laminar boundary layer models:

Nux = 0.5642Rex1/2Pr1/2

for Pr → 0

(12)

for Pr → ∞

(13)

and Nux = 0.3387Rex1/2Pr1/3

Those two expressions evolved from a long history of the analyses that is not essential here. However, it should be mentioned that, in addition to the general idealizations of thin laminar boundary layer theory, the differential model from which eqs 12 and 13 are derived incorporates the physically questionable postulate of a split in a uniform flow by the leading edge of the plate without disruption. The combination of eqs 12 and 13 devised by Churchill and Ozoe in the form of the CUE is Nux = [(0.5642Rex1/2Pr1/2)m + (0.3387Rex1/2Pr1/3)m ]1/ m (14) 4105

dx.doi.org/10.1021/ie403511s | Ind. Eng. Chem. Res. 2014, 53, 4104−4118


Article

Table 1. A{Pr} and B{Pr} for Nux for a Vertical Isothermal Plate Pr

0.015

0.10

0.70

1

5

7

30

100

A{Pr} B{Pr}

0.6709 2.730

0.5391 1.161

0.4554 0.5128

0.4542 0.4552

0.5170 0.3023

0.5457 0.2853

0.7473 0.2405

1.038 0.2235

They found that a value of −1/4 for the exponent m in eq 14 to result in a good fit for their own numerically computed values of Nux for a number of finite values of Pr, and the quantity (0.3387/ 0.5642)6 is equal to 0.04681, thereby reducing eq 14 to eq 4. Equations 8, 9, 12, and 13 are exact closed-form solutions of the thin laminar boundary models for free and forced convection with the leading coefficients determined numerically. Insofar as the thin laminar boundary layer model is valid for this application, eqs 8, 9, 12, and 13 may each be considered exact both functionally and numerically. Closed-form solutions for the thin laminar boundary are possible only in the asymptotic limits of Pr → 0 and Pr → ∞, but although the functional dependency of the generalized correlating equations of Churchill and Usagi6 is presumably empirical, their numerical predictions are exact for all practical purposes. These expressions thereby render the solutions of thin laminar boundary layer theory effectively exact for all values of Pr within the range of values of Rex and Grx in the tlbl. 2B. Ranges of Validity of Expressions Comprising the Equivalence. Before proceeding further, it seems fitting to examine the functional and numerical accuracy and the ranges of validity of the equations that led to eqs 6 and 7 and thereby to eq 14. Equations 8, 9, 12, and 13 are exact closed-form solutions of the thin laminar boundary models for free and forced convection with the leading coefficients determined by essentially exact numerical calculations. Insofar as the thin laminar boundary layer model is valid, eqs 8, 9, 12, and 13 may each be considered exact both functionally and numerically. Closed-form solutions for the thin laminar boundary are possible only in the asymptotic limits of Pr → 0 and Pr → ∞, but although the functional dependency of the generalized correlating equations of Churchill and Usagi6 is presumably empirical, their numerical predictions are essentially exact. These expressions thereby effectively render the solutions of thin laminar boundary layer theory exact for all values of Pr within the range of values of Rex and Grx in the tlbl. As illustrated by the plot of Nux versus Ra /[(0.492/ Pr)9/16]16/9 in Figure 1 of Churchill and Chu15 for free convection from a vertical isothermal plate, the experimental values increase continuously and very slowly above the straight line representing the tlbl as RaX decreases. The corresponding values of Nux for forced convection decrease at the same rate, and the plots of experimental values of Nux and Nux versus RexPr2/3/ [1+ (0.0468/Pr)1/3]1/2 in Figures 1 and 2 of Churchill16 appear to overlay them. A lower limit on the applicability of eqs 3 and 4 for prediction of the Nusselt number is imposed by the onset of significant thickening of the laminar boundary layer, and an upper limit is imposed by the onset of turbulence; however, remarkably, the net effect on eq 6 is negligible. The continuous behavior in these three plots can be inferred to preclude the existence of discrete limiting values of Grx and Rex but suggests that the lower limit of Grx is of the order of 102 and that of Rex of the order of 104. The onset of turbulence may occur gradually or suddenly but, despite the well-known studies of Osborne Reynolds, it does not occur at a fixed and known value of the Reynolds number in forced flow through a round tube, in forced flow along a flat plate, or in free convection. A discrete point of transition from laminar

to turbulent motion in free convection is not evident in the aforementioned Figure 1 of Churchill and Chu,15 but the intersection of the correlating equations for Nux for those two regimes in Figure 1 of Churchill,16 namely Nux = 0.3387φ1/2

(15)

and Nux = 0.032φ4/5

(16)

where ϕ = RexPr 2/3/[1 + (0.04681/Pr )2/3 ]1/2

(17)

defines the point of transition decisively. That intersection occurs at φ = (0.3387/0.032)10/3 = 2603. For Pr = 0.7, the corresponding rate of flow may be expressed as Rex = 3563, and eqs 4 and 5 predict an equivalent value of Grx = 5.779 × 106. The corresponding correlating equations for the integratedmean Nusselt number are Nux = 0.6774φ1/2

(18)

and Nux = 0.04φ4/5

(19)

Equations 18 and 19 appear to fit the experimental data but the power of 4/5 and the values of 0.032 and 0.40 for the coefficients have no rationale and are quite uncertain. Their intersection occurs at φ = (0.6674/0.04)10/3 = 11 869. For Pr = 0.7, the corresponding rate of flow is demarked by Rex = 16 248, and eqs 4 and 5 predict an equivalent value of Grx =1.202 × 108. A coefficient of 0.064 in place of 0.04 would be required to result in the same values of φ, Rex, and Grx as those determined for the local Nussselt number so the difference in the observed and predicted values is significant. Lin and Churchill17 in the process of developing a numerical solution for a turbulent free convection compared, in their Figure 2, those predictions for the onset of turbulence in terms of Nux with experimental data for Pr = 0.7. The indicated value is Grx ≅ 4 × 109, which differs significantly from the value of 6 × 106 that follows from the correlating equations of Churchill.16 These discrepancies are echoed in the predicted values of Nux and of Nux . Further investigation of this behavior is clearly called for. 2C. Range of Values A{Pr} and B{Pr}. For any finite value of Pr, the function A{Pr}, as predicted by eq 5, becomes simply a fixed numerical value. As an illustration of that behavior, numerical values of the coefficient of proportionality are listed in Table 1 for values of Pr representative of mercury (Pr ≅ 0.015), air (Pr ≅ 0.70), water (Pr ≅ 5 to 7), and hydrocarbon oils (Pr = 30 or greater). The extreme range and nonmonotonic variation of the numerical values of A{Pr} are disappointing in a correlative sense but a value of 0.5 provides an approximation that is sufficiently accurate for most practical purposes. Table 1 is effective in identifying these two aspects of the dependence. Technical journals sometimes allocate extensive tabulations of experimental data or numerical solutions to separate “Supporting Information” for preservation, and such a disposition of the tabulations in this manuscript was advocated by one of the 4106



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number in forced convection. The corresponding values of the Rayleigh number are redundant but are included because of the greater familiarity of that characteristic as a measure of the strength of free convection. Here a tabulation provides the most effective display of the relationship between the several variables. The balance of the manuscript consists of a description of the extended applicability of the concept of equivalence as expressed by eq 4. The proportionality of Grx to Rex2, although unnoted in any books on heat or mass transfer, is all but obvious, and does not justify the detailed development herein. The development of an essentially exact expression for A{Pr} for all values of the Prandtl number does constitute a significant contribution, but it is the demonstration that eq 4 is applicable for most other boundary conditions and other geometries that is the most important one.

reviewers. However, that recommendation was not followed when the final version was prepared because in this instance such tabulated values reveal important aspects of behavior that would otherwise not be apparent, even from curves and/or symbols in a graph. The following expressions are an obvious alternative to eqs 4 and 5 as a measure of the equivalence Grx = B{Pr }Pr1/3Rex 2

(20)

where B{Pr } =

0.2061[1 + (0.4914/Pr )9/16 ]16/9 1 + (0.04681/Pr )2/3

(21) 2

Values of B{Pr}, the quantity chosen by Acrivos as a marker for equal combined convection, are included in Table 1. As contrasted with the values of A{Pr}, they vary monotonically but have a wider range of magnitude for finite values of Pr, and the definition includes a separate term in Pr that complicates rather than simplifies the measure. As an aside, the precision of the values listed in all of the tables herein, as well as that of the coefficients of the algebraic expressions, is presumed to be valid to at least three significant figures. In some instances, such as in Table 1, four figures are reported in the interests of consistency or for purposes of comparison rather than constituting a claim of such precision. By virtue of the CUE the value listed in Table 2 bear an essentially exact relationship to one another even for a finite value

3. VERTICAL PLATE WITH A UNIFORM HEAT FLUX DENSITY (UHF) The second equivalence to be examined here is that for free and forced convection with a uniform heat flux density on a vertical plate in the thin laminar boundary layer regime. In that free convection remains proportional to Grx1/4 and forced convection to Rex1/2, the same functionality would be expected and indeed it prevails, as is illustrated next. Churchill and Ozoe18 devised the following expression for the local Nusselt number in free convection in the thin laminar boundary layer regime with a uniform heat flux density on the surface:

Table 2. Equivalents for Nux for a Vertical Isothermal Plate and Pr = 0.7 2

Rex

10

Grx Rax Nux−eqs 1 and 2

455 319 2.89

3

4

10

10

4.55 × 105 3.19 × 105 9.15

4.55 × 107 3.19 × 107 28.9

Nux =

0.563Rax1/4 [1 + (0.437/Pr )9/16 ]4/9

(22)

5

10

Equation 22 conforms to eq 1 in all respects except for the numerical values of the two coefficients. It can be noted to predict larger values of Nux for all values of Pr − a universal characteristic of a uniform heat flux density as compared to a uniform temperature on the surface. In some of the early publications on free convection with a uniform heat flux density the Nusselt number is expressed as a function of Rax*  gβjwx4/kνα rather than Rax  gβ(Tw−T∞) x3/ν2 in order to avoid the explicit presence of the dependent variable Tw. However, Churchill7 in 1970 noted that Rax* is equal to NuxRax and thereby that the seemingly different power dependence of Nux is directly related. Although the two expressions have equal validity, the advantage of functional congruence outweighs the avoidance of an implicit temperature distribution in most contexts, certainly in the current one. The corresponding correlative/predictive expression of Churchill and Ozoe19 for the local Nusselt number for forced convection with uniform heating in the tlbl regime is

4.55 × 109 3.19 × 109 91.5

of the Prandtl number. As such, they provide supplemental insights. The values of Rex and Grx that result in the limiting value of Nux differ greatly in magnitude because of the dependence on the 1/2 power of Rex as compared to the 1/4 power of Grx. Table 2 is limited to Pr = 0.70 in the interests of space but, as may be deduced from the behavior of A{Pr} in Table 1, qualitatively similar results would be calculated for other values of Pr. The values of Rex and Grx at which the transition to turbulent motion begins are presumed to be related by eq 4, but the values themselves, as shown for forced convection in Churchill’s Figure 116 and for free convection in Churchill and Chu’s Figure 115 and thereby the upper limit of validity of the values in Table 2, depend on secondary effects. Table 2 constitutes a quantitative representation of the equivalents for a particular condition, namely the value of the Grashof number that produces the same prediction of the Nusselt number in free convection as does a specified Reynolds

Nux =

0.4644Rex1/2Pr1/3 [1 + (0.02071/Pr )2/3 ]

(23)

Table 3. A{Pr} for Nux and Nux for an Isothermal and a Uniformly Heated Vertical Plate Pr

0.015

0.10

0.70

1

5

7

30

100

local AUWT{Pr} mean AUWT{Pr} local AUHF{Pr} mean AUHF{Pr} AUHF/A UWT = (AUHF/A UWT)mean

0.671 0.700 1.72 1.79 2.56

0.539 0.562 1.32 1.38 2.46

0.455 0.475 1.01 1.09 2.29

0.454 0.475 1.00 1.05 2.20

0.517 0.539 1.15 1.20 2.22

0.546 0.569 1.21 1.27 2.22

0.747 0.780 1.69 1.76 2.25

1.04 1.08 2.32 2.42 2.25

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Equation 23 can be noted to have the same form as eq 2, and, as was to be expected, to predict a greater value of Nux for all values of the Prandtl number. The following expression for AUHF{Pr} has, as was also to be expected, the same form as eq 7: A UHF{Pr } =

0.4830Pr

1/3

0.6703Pr1/4Grx1/4 [1 + (0.4914/Pr )9/16 ]4/9

(25)

As shown in Figure 1 of Churchill and Chu,15 the expression that results from the addition of an arbitrary fixed value of 0.68 to the right-hand side of eq 23, namely

9/16 16/9

[1 + (0.437/Pr )

1 + (0.0271/Pr )2/3

Nux =

]

(24)

Numerical values of AUHF{Pr} calculated from eq 24 are compared in Table 3 with those of AUWT{Pr}from eq 5 and Table 1 for a uniform wall temperature (UWT). They differ significantly but their ratio differs only slightly from 2.25 for Pr ≥ 0.7another possibly useful generality that probably has a theoretical rationale. Again it is the tabular format that reveals a relationship.

Nux = 0.68 +

0.6703Pr1/4Grx1/4 [1 + (0.4914/Pr )9/16 ]4/9

(26)

provides an excellent representation of the available experimental data for free convection, which extend down to GrxPr/[1+ (0.4914/Pr)9/16]16/9 ≅ 0.10. Although eq 26 is adequate for most practical purposes, it is fundamentally unsound because in reality both Nux and Nux approach zero as Grx does. As may be seen in Figure 2 of Churchill,15 the analogue of eq 26 for the integrated-mean Nusselt for forced convection, namely

4. INTEGRATED-MEAN NUSSELT NUMBER FOR A VERTICAL PLATE The third equivalence to be examined is that for the integratedmean Nusselt number, Nux for free and forced convection from a vertical isothermal flat plate in the tlbl regime. Values of Nux can be derived from those for Nux for each value of Pr simply by multiplying the leading coefficient for free convection by 4/3 and that for forced convection by 2. These two factors are derived by integration of the individual expressions for the heat transfer coefficient from 0 to x before their combination. The net result is that the leading coefficient of eq 5, namely 0.2061, is multiplied by (6/4)4 = 5.0625 and becomes 1.043. This proportionality holds for uniform heating as well as for a uniform wall temperature. Values of Nux and Nux are not included in Table 3, but values of AUWTmean{Pr} and AUHFmean{Pr} are. The values of the ratios of Amean{Pr} are obviously independent of that factor and apply for both Nux and Nux . These factors hold for all values of Rex and Rax, respectively, within the thin laminar boundary layer regimes. All of the relationships presented up to this point, except for eqs 15−19, are for the thin laminar boundary layer regime. Comparison of conventional correlating equations in the form of the product of powers of the Reynolds number and the Prandtl number might have revealed an equivalence but probably not, and if it did ,it would have been with some uncertainty whereas that resulting from thin laminar boundary layer theory is exact in a functional sense. The equivalence between free and forced convection, insofar as described herein, is a theoretically based concept in every respect. It can be considered inherent in and an endowment of thin laminar boundary layer theory.

Nux = 0.45 +

0.6774Rex1/2Pr1/3 [1 + (0.04681/Pr )2/3 ]1/4

(27)

provides an excellent representation of the experimental data and numerically computed values. However, the local and integratedmean Nusselt numbers for forced convection approach zero along with Rex, and the inclusion of a fixed value is fundamentally unsound in this instance as well. A common fixed value for both free and forced convection such as the arithmetic mean, (0.68 + 0.45)/2 = 0.565, resulting in Nux = 0.565 + = 0.565 +

0.6703Pr1/4Grx1/4 [1 + (0.4914/Pr )9/16 ]4/9 0.6674Rex1/2Pr1/3 [1 + (0.04681/Pr )2/3 ]1/4

(28)

could be utilized for practical purposes and would have the advantage of extending the applicability of the expressions of AUWT{Pr}, AUWTmean{Pr}, AUHF{Pr}, and AUWTmean{Pr}, which were derived for the thin laminar boundary layer, for the entire laminar boundary layer. However, a better alternative is described in what follows. 5A. Langmuir’s Model. In 1912, Langmuir,20 in the process of modeling the heat loss from the cylindrical filaments within a partially evacuated light-bulb by free-convective as well as by thermal radiation, devised an expression that was intended to account for the considerable thickness of the boundary layer relative to the diameter of the filament as well as for its curvature. It is the source for two subsequent empirical expressions that extend the theoretically based ones for the thin laminar boundary layer to cover the entire laminar regime including the thick portion. Langmuir started by postulating that free convection from a small wire could be modeled by thermal conduction across a hypothetical “sheath” (annular ring) of stagnant gas of thickness 2δ = Do − Di. He then expressed the ensuing heat flux density at the surface of the wire in terms of the logarithmic-mean area (the appropriate correction for pure thermal conduction) and thereby obtained the following expression for the heat flux density at the surface of the filament:

5. THICK LAMINAR BOUNDARY LAYER ON A VERTICAL PLATE The fourth example involves the behavior that occurs in the thick laminar boundary layer regime, for which closed-form solutions do not exist. Numerical solutions and experimental data for this regime are somewhat limited in scope for free convection and even more limited for forced convection. As Rax decreases below the lower limit of applicability of lblt, which is about 105, Nux continues to decrease, but at a rate much less than that predicted by eqs 1, 2, 15, and 16. The historical expedient in terms of prediction for this regime has been to add some arbitrary limiting value. Because theoretical guidance does not appear to exist, that arbitrary value has generally been based on experimental data. The expression for the integrated-mean Nusselt number corresponding to eq 1 is

ji =

4108

k ΔT ⎛ (Do − Di ) ⎞ 2k ΔT ⎜ ⎟= 2δ δ ⎝ ln{Do /Di } ⎠ ln 1 + D

{

i

}

(29)



Article

Table 4. Illustration of the Predictions of NuC from NuF 10−6

NuF

10−3

10−2

10−1

0.5

1

2

5

10

100

1.565 1.820 1.443

2.565 2.885 2.466

5.565 5.944 5.485

10.56 10.97 10.49

100.565 100.997 100.449

NuC eq 28 eq 30 eq 32

0.565 0.1378 0.07238

0.566 0.2631 0.1477

0.575 0.3771 0.2167

0.665 0.6569 0.4170

1.065 1.243 0.9102

Table 5. Predictions of Integrated-Mean Nusselt Number for Laminar Free Convection from an Isothermal Vertical Plate with Pr = 0.7 Rax

0

10−5

10−3

0.10

0.290

10

103

105

107

0.377 1.056 1.086 1.056 0.933

0.914 1.594 1.724 1.689 5.58

2.89 3.57 3.80 3.76 54.8

9.14 9.82 10.10 10.06 548

28.9 29.6 29.9 29.8 5484

Nux eq 25 eq 26 eq 32 eq 36 equivalent Rex

0 0.68 0 0 0

0.0289 0.709 0.470 0.452 0.005

0.0914 0.771 0.639 0.616 0.0548

0.289 0.969 0.966 0.938 0.548

Churchill21 in 2000 noted that the lack of a sound theoretical basis for eq 30 allows it to be interpreted as empirical in structure and thereby open to the possibility of changing the sign and/or magnitude of the twice-appearing coefficient of 2. As an example, he asserted that replacing the coefficient by unity, and thereby eq 30 by

The inverse of thermal resistance of the hypothetical sheath of stagnant gas, namely δ/k, can be recognized as the equivalent of a heat transfer coefficient. It follows that eq 27 can be expressed in more conventional notation as NuC =

2

{

ln 1 +

2 NuF

}

{

2 Nu tlbl

}

1 NuF

}

(32)

results in a more accurate prediction for the effect of curvature. The predicted values of eq 32 are included in Table 4. Equation 30 represents experimental data for NuC for both free and forced convection almost exactly down to their lowest measured values, which is of the order of unity, but grossly overpredicts NuF for lesser values of Rex or Rax. In the absence of the appropriate data there is not a firm basis for assessing the relative accuracy of eqs 30 and 32 for lesser values but both of these expressions can be reasoned to be more accurate than eq 28. Even earlier in 1988, Churchill22 noted that the neglect of curvature in the derivation of the Lévèque23 equation, a closedform solution for developing forced convection in fully developed laminar flow in a round tube, namely Nux = 1.167Gz1/3

(33)

can be compensated for by means of the following analogue of the Langmuir solution for inward transport through a curved sheath: Nu =

−2

{

2

ln 1 −

1.167Gz1/3

}

(34)

Here, the coefficient 2 has simply been replaced with −2 in two places. Equation 34 is obviously invalid for 1.167Gz1/3 ≤ 2. On the basis of these two modifications of the Langmuir equation the following generalized form is proposed: α y= α ln 1 + y {x}

2 ln 1 +

{

ln 1 +

Here NuF represents the heat transfer by convection through a flat “sheath” of gas, and NuC through a cylindrically curved one. At first glance, eq 30 has promise in two respects: (1) it appears to be free from empiricism; and (2) the hypothetical thickness of the sheath does not appear in it explicitly. Although eq 29 is exact for pure thermal conduction, the representation of the rate of thermal convection in terms of an effective thickness for thermal conduction rather than by a heat transfer coefficient, which was once common in the literature, has now virtually disappeared, and apparently survives only in Langmuir’s solution and perhaps a few closely related expressions. The relationship between NuF and NuC as predicted by eq 30 is illustrated in Table 4. It can be noted that as NuF decreases, the value of NuC predicted by eq 30 decreases very slowly toward zero and as NuF increases the value of NuC and approaches NuF + 1. The increase in NuC above NuF, that is of heat transfer through a curved stagnant film as compared to a flat one of the same thickness, is because of the greater area for conduction on the mean. There is no reason to expect that eq 30, which was derived to represent the effect of curvature on the thermal resistance to free convection from a wire, would have any applicability for the boundary layer on a flat plate but remarkably, when re-expressed as Nulbl =

1

NuC =

(30)

(31)

{

it proves, as shown subsequently, to do so. That is, eq 31 by virtue of the subscripts lbl and tlbl provides an estimatie of the Nusselt number for both free and forced convection over the entire laminar regime from one for the thin laminar boundary layer regime.

a

}

(35)

Here α is an arbitrary coefficient and ya{x} represents a solution for some limiting or special condition. Equation 35 not only provides a gradual transition to zero but, by virtue of the arbitrary 4109



Article

coefficient α, can be forced to pass through any chosen experimental or computed value of y. The predictions of several equations for Nux for Pr = 0.7 are compared with one another in Table 5. The predictions of eq 25 may be observed in Figure 2 of Churchill and Chu15 to be in almost exact accord with the various sets of experimental data which extend down to Rax/[1 + (0.4914/Pr)916]16/9 = 0.1, which for air (Pr ≅ 0.70), corresponds to Rax = 0.289. The predicted value of Nux for that value of the abscissa is 1.056, which is in accord with the experimental value within the accuracy with which it can be read from the graph. Accordingly, eq 23, despite the purely empirical basis of the fixed value of 0.68, can be taken as a numerical standard for Rax/[1 + (0.4914/Pr)916]16/9 ≥ 0.1. There is no numerical standard for lesser values of Rax, but a ranking is possible on qualitative grounds. The predictions of eq 25 have already been noted by reference to Figure 2 of Churchill and Chu15 to provide a lower bound for the experimental data for all conditions, and the predictions of eq 26 to be in almost exact accord with all of the various sets of experimental data down to Nux ≅ 1 at Rax/[1 + (0.4914/ Pr)916]16/9 = 0.1, which for air corresponds to Rax = 0.290. Thus eq 26 serves as a standard for values of Rax greater than 0.290. Although there are no data for comparison for lesser values of Rax/[1 + (0.4914/Pr)916]16/9, the predictions of eq 26, as listed in Table 5, can be presumed to be increasingly in error as Rax decreases in that they approaches 0.68 rather than zero. The predictions of eq 32 appear to be reasonably accurate for all values of Rax, but as an example of the utility of eq 34, that accuracy can be improved slightly by replacing the coefficient of 2 with 1.898 to obtain Nulbl =

and numerically computed solutions for both the local- and integrated-mean rates and for a vertical wall with either a uniform temperature or a uniformly heat flux density in the regime of a thin laminar boundary layer. Furthermore, two different means of extending those expressions for the entire laminar regime have been identified. These contributions to understanding and to compactness appear to be sufficient to render this concept worthy of introduction into academic classes, textbooks, handbooks, and perhaps even computer packages for process design and analysis. The utility of the concept of equivalence for prediction has not been yet demonstrated herein. Insofar as expressions for prediction of the Nusselt number for both free convection and forced expression are combined to construct A{Pr}. That quantity would not appear to be needed for quantitative predictions of free convection from an expression for forced convection or vice versa. However, conditions are subsequently described for which an expression for A{Pr} is devised in the absence of a generalized solution for one or the other form of convection.

6. SUPERIMPOSITION The process of preparing Table 3 involved the examination of graphical representations of experimental data together with correlative/predictive equations for Nux as functions of Grx, Rex, and Pr. The numerical values of the ordinate for those graphs were recognized to be the same insofar as the functions comprising the abscissa were those for the components of A{Pr}. That observation led to the realization that the concept of superimposition as suggested by Lemlich and Hoke1 in terms of an equivalent velocity and for a particular case and could be expressed more generally in terms of the Nusselt number, the Grashof number, the Reynolds number, and the Prandtl number. As an example, if the experimental data, numerically computed values, and curves representing correlative/predictive expressions for Nux for forced convection from an isothermal plate in Figure 2 of Churchill16 had been plotted versus Rex1/2Pr1/3/[1 + (0.04681/Pr)2/3]1/4 rather than versus the square of this function, they would, on the mean, overlay those of Figure 1 of Churchill and Chu,15 which were plotted versus Ra/[1 + (0.4914/Pr)9/16]16/9. Although the methodology of Churchill and Usagi,11 in particular the inclusion of the dependence on Pr, is helpful and indeed is almost essential for the construction of detailed expressions for the equivalences, the recommended graphical format for the identification of the optimal value for the connecting exponent has inadvertently resulted in the failure of many investigators, in particular my collaborators and I, to prepare the very correlative/predictive graphs that could have been superimposed.

1.898

{

ln 1 +

1.898 Nu tlbl

}

(36)

Equation 36 forces the prediction for Rax = 0.290 to be 1.056. The choice of 1.898 for the coefficient was made by an iterative (trial-and-error) process of solution. The improvement, if any, by evoking eq 36 in the in this instance is slight, but a more significant improvement is demonstrated in a subsequent example. The extensions of eq 25 that led to eqs 26, 32, and 36 are presumed to be applicable for forced convection so the effect of the transformation cancels out of eq 3, and eqs 5 and 10 remain applicable for the local Nusselt number for the entire laminar boundary layer for both a uniform temperature and a uniform heat flux density on the wall. However, the ratio of the leading coefficients is affected by the integration, thereby resulting in the following expressions for the integrated-mean Nusselt numbers. mean AUWT{Pr } =

0.2150Pr1/3[1 + (0.4914/Pr )9/16 ]16/9 1 + (0.04681/Pr )2/3 (37)

mean AUWF{Pr } =

0.5038Pr

1/3

7. HORIZONTAL CYLINDERS Most industrial and domestic heating and cooling are carried out with cylindrical tubing, so a plethora of both theoretical and correlative equations for the Nusselt number might be expected. Although there is no shortage of empirical expressions there are few theoretically based ones because of two complications that arise in modeling heat transfer from a cylinder. First, the differential models for steady flow over and for conductive and convective heat transfer from a horizontal cylinder of unbounded length are ill-posed in a mathematical sense. An unsteady state solution for pure thermal conduction from a horizontal, infinitely

9/16 16/9

[1 + (0.437/Pr )

]

1 + (0.0271/Pr )2/3 (38)

Equations 37 and 38 were used to calculate the so-designated values of mean AUWT{Pr} and mean AUHF{Pr}in Table 3 Before turning to cylinders and spheres it may be noted that at this point in the analysis that the concept of an equivalence between free and forced convection has been shown to provide a generalized format for the representation of experimental data 4110



Article

Table 6. Comparison of Predictions of Nu/Ra1/4 for an Isothermal Vertical Plate and a Horizontal Cylinder in the Thin Laminar Boundary Regime Pr

0.015

0.10

0.70

1

5

7

10

30

100

0.602 1.367

0.613 1.379

0.622 1.390

0.641 1.412

0.656 1.430

0.462

0.471

0.478

0.495

0.506

plate eq 24 eq 29

0.264 0.931

0.387 1.099

0.514 1.260

0.534 1.284

eq 42

0.1955

0.378

0.391

0.407

cylinder

to evaluate the coefficient b, and the numerical solution of LeFevre for Pr → ∞, namely

long, circular cylinder to unbounded surroundings does not converge and one for free convection predicts a fluid motion of ever-growing extent that never attains a steady state. In the real world, cylinders are always finite in length. In principle, threedimensional steady-state solutions are possible for pure thermal conduction and convection but the computational demands are extreme. The second complication is the existence, even for a cylinder of finite length, of a fluctuating wake at the rear of the cylinder in forced convection and of a fluctuating plume on the upper or lower side of the cylinder in free convection, that is, on the opposite side of the cylinder from the forward line of stagnation. It follows that the few existing theoretical solutions for free and forced convection from cylinders are approximate by virtue of the postulate of the existence a thin laminar boundary layer and even more so by virtue of the neglect of the effects of the wake or the plume. Isothermal cylinders are considered first because that condition is relatively easy to attain in the laboratory and because many more measurements and correlations exist than for a uniform heat flux density on the surface. On the basis of his own numerical solutions for a thin boundary layer and the neglect of the plume, LeFevre12 in 1956 proposed the following correlating equation for free convection from isothermal cylinders for all values of Pr: NuD =

NuD = 0.518RaD1/4

to evaluate the coefficient c, thereby obtaining NuD =

NuD =

(43)

0.518Pr1/4GrD1/4 [1 + (0.559/Pr )9/16 ]4/9

(44)

The predictions of eq 44 are, by virtue of its derivation, identical to those of eq 43 for both asymptotically small and asymptotically large values of Pr, and they differ less than 2.5% for all intermediate values. The absolute and relative accuracy of these two expressions is unknown, but eq 44 is to be preferred in the current context because it is identical in structure to eq 1. The speculation that eq 31 (Langmuir’s solution in modern terms) might in itself be a kind of equivalence, namely that between the rates of convection from a horizontal cylinder and a vertical plate, which is actually what Langmuir had in mind in the derivation of its prototype, namely eq 30, is not unreasonable. However, that possibility may be dismissed out of hand because, as shown in Table 6, eq 29 erroneously predicts greater values of Nu than does eq 24 whereas lesser ones are predicted by eq 42. The physical explanation is that the decrease in the rate of heat transfer due to the decrease in the effective velocity overbalances the increase in the rate of heat transfer due to the greater mean area for thermal conduction. The problems encountered in devising a correlating equation for NuD for forced convection from a cylinder are analogous to those for free convection. The following expression of Bernstein and Churchill26 for the overall rate of heat transfer from an isothermal cylinder in the thin laminar boundary layer regime is based on numerical solutions but was found by them to represent experimental data for a number of fluids very well:

(39)

NuD =

0.62Pr 1/3ReD1/2 [1 + (0.40/Pr )2/3 ]1/4

(45)

Equating the right-hand sides of eqs 44 and 45 and rearranging that result as was done herein for a flat plate leads to GrD = A CT{Pr }RD 2

(40)

(46)

where

They utilized eq 40 to evaluate the coefficient a in eq 37, their numerical solution for Pr = 0.7, namely NuD = 0.393RaD1/4

[7.75 + 14.3Pr1/2 + 13.9Pr ]1/4

Dropping the term in Pr and speculating the applicability of the ubiquitous combining exponent of −9/4 allows eq 41 to be transformed into

Here, a, b, and c are implied to be dimensionless numerical constants, and following the most common current practice, the characteristic dimension utilized by LeFevre, namely the radius, has been replaced by the diameter. The choice of πD as the characteristic dimension would have be an even better one in terms of comparisons with the integrated-mean Nusselt number of a vertical plate because the length of the path of integration for a cylinder is πD and thereby is π times the characteristic length D; whereas, these lengths are both the same for the plate. That advantage was however considered insufficient to justify such an unfamiliar usage herein. Saville and Churchill24 in 1967 a devised a new rapidly converging solution for free convection from isothermal horizontal cylinders of fairly arbitrary cross-section that takes into account the effect of body shape on the flow, postulates the idealizations of thin boundary layer theory, and ignores the effect of the plume. In 1969, they25 utilized that methodology to derive a solution for asymptotically low values of Pr. The first two terms of their solution for that limiting condition can be represented by NuD = 0.599RaD1/4

RaD1/4Pr1/4 1/2

RaD1/4Pr1/4 [a + bPr1/2 + cPr ]1/4

(42)

A CT{Pr } =

(41) 4111

2.052Pr1/3[1 + (0.559/Pr )9/16 ]16/9 1 + (0.40/Pr )2/3

(47)



Article

Table 7. Comparison of A{Pr} for an Isothermal Vertical Plate in the Thin Laminar Boundary Regime with That for an Isothermal Horizontal Cylinder Pr

0.015

0.10

0.70

1

5

7

10

30

100

APTmean{Pr}−eq 10 ACTmean{Pr}−eq 47 ACT /APT

0.671 2.36 3.52

0.539 2.40 4.97

0.455 3.318 7.29

0.454 3.49 7.67

0.517 4.66 9.02

0.546 5.02 9.20

0.583 5.45 9.35

0.747 7.23 9.67

1.038 10.2 9.83

43, as may be observed in Figure 1 of Churchill and Bernstein,25 are accurate for ReD1/2Pr1/3/[1 + (0.4/Pr)2/3]1/4 ≥ ≅ 5, but increasingly below the experimental values as ReD1/2Pr1/3/[1 + (0.4/Pr)2/3]1/4 decreases below 5. The predictions resulting from the addition of 0.30 to eq 43 are accurate for all lesser values, and the replacement of 0.3 with 0.24, even though such a curve is not included, can be inferred to provide an even better representation. Despite this success in representing the experimental data, the combination of eq 45 with a fixed value is unacceptable in a fundamental sense because NuD must ultimately approach zero. The predictions of the combination of eqs 43 and 30 conform to that requirement but grossly overpredict NuD for ReD1/2Pr1/3/[1 + (0.4/Pr)2/3]1/4 ≤ 5, which is a surprise in that the concept resulted from an attempt to account for the effect of cylindrical curvature and a disappointment in that it was successful for a vertical plate as illustrated in Table 5. However, replacing the twice-appearing coefficient 2 of eq 30 with 0.7922 results in

Thus the expression for the dependence of the equivalence for isothermal, horizontal cylinders in the thin laminar boundary layer regime on Pr is identical functionally to that for isothermally vertical flat plates as described by eqs 4 and 5 because, again, the Nusselt number for free convection is proportional to Gr1/4 and that for forced convection to Re1/2. The numerically calculated values of ACTmean{Pr} from eq 47 for an isothermal horizontal cylinder, as listed in the third row of Table 7, differ noticeably in two respects from those of ATPmean{Pr} as calculated from eq 5 for an isothermal vertical plate and listed in the second row. First, they are much greater in magnitude, and second, they increase monotonically. However, the ratio of ATC to ATP is virtually constant at slightly above 9 for Pr ≥ 5, which has possible utility in a correlative/predictive sense and presumably has a yet unidentified mechanistic explanation. The magnitude of the predictions of the components of eqs 5 and 47 are subsequently compared as a possible source of explanation. However, attention is first turned to the extension of eq 47 to encompass the regime of a thick laminar boundary layer. In order to improve the representation of the experimental data for free convection from a horizontal isothermal cylinder for small values of Rax, Churchill and Chu15 suggested the inclusion of an additive constant with a numerical value of 0.36 in eq 44 and to improve the representation of the corresponding experimental data for forced convection for small values of Rex, and Churchill and Bernstein26 reccommended the inclusion of an added constant with a numerical value of 0.30 in eq 45. The same numerical value is necessary for both if their equivalence is to be extended for the entire laminar boundary regime. A reexamination of Figure 1 of Churchill and Bernstein indicates that on the basis of the lowest value of the abscissa for which RaD there is a measurement of NuD for forced convection, namely 0.1 for ReD1/2Pr1/3/[1 + (0.4/Pr)2/3]1/4 or for 0.7789ReD1/2 for Pr = 0.7, a value of 0.24 would be a better choice than 0.30 as a limiting value in a correlating equation. That choice is consistent with the lowest measured value for free convection as plotted in Figure 1 of Churchill and Chu,15 which occurs at RaD/[1 + (0.559/ Pr)9/16]16/9 = 1 × 10−11 or 2.77 × 10−11 for Pr = 0.7. The predictions of NuD are compared in Table 8 for four values of ReD. The equivalent values of RaD, which eq 47 predict with the same values, constitute the second line. The predictions of eq

Nulbl =

ReD

0

0.01648

1.648

50

0

0.000901

9.011

8295

0.1527 0.3927 0.571 302

0.620 0.860 1.388 0.962

3.415 3.655 3.894 3.797

0 0.240 0 0

0.7922 Nu tlbl

}

(48)

for which, as demonstrated in Table 8, the prediction is in accord with the measurements at the chosen point, and in combination with eq 45, appears on the basis of Figure 1 of Churchill and Bernstein and of Figure 1 of Churchill and Chu to provide reasonably accurate predictions of NuD for both free and forced convection from isothermal horizontal cylinders over the entire laminar regime (ReD less than 50 and GrD less than about 1140). New experimental data or numerically computed values for low values of ReD and/or GrD may eventually allow the accuracy of eqs 44, 45, 47, and particularly eq 48, to be improved by tweaking the coefficients. Liňań and Kurdyumov26 carried out numerical calculations for Nua for finite values of Gra, but unfortunately the values of Nua and of a variable coefficient in the correlating equation devised for their representation Nua are only presented graphically and the individual values are not accessible with accuracy. As an afterthought on isothermal heating, the ratio of the predictive expression for the integrated-mean Nusselt number for free convection from a horizontal cylinder to that for an vertical plate is {0.518RaD 1/4/[1 + (0.559/Pr) 9/16]4/9}/ {0.6703Rax1/4/[1 + (0.4914/Pr)9/16]4/9}, while that for forced convection is {0.62Pr 1/3 Re D 1/2 /[1 + (0.40/Pr 2/3 ] 1/4 }/ {0.6674Pr1/3Rex1/2/[1 + (0.04681/Pr]2/3}. These two compounded expressions indicate that the rate of heat transfer, at least as represented by the integrated-mean Nusselt number, is greater for a plate than for a cylinder for both forced and free convection for all values of Rex, Rax, and Pr. As correctly predicted by eq 16, the rate of heat transfer on the outside of a curved surface is increased, by virtue of the greater mean area. However, that effect appears to be overbalanced by the decrease in the effective rate of flow over the surface of a cylinder as compared to that along a plate.

NuD eq 45 eq 45 plus 0.24 eq 45 extended by eq 30 eq 48

{

ln 1 +

Table 8. Integrated-Mean Nusselt Number for Laminar Forced Convection from an Isothermal Horizontal Cylinder with Pr = 0.7

RaD equivalent to ReD

0.7922

4112



Article

Table 9. Comparison of Predictions of Integrated-Mean Nusselt Number for Laminar Forced Convection from Isothermal and Uniformly Heated Horizontal Cylinders Pr

0.015

0.10

0.70

1

5

7

10

30

100

0.4623 0.5233

0.4706 0.5317

0.4781 0.5394

0.4952 0.5565

0.5060 0.5672

NuD /ReD1/2 eq 42 (UWT) eq 47 (UHF)

0.1985 0.2336

0.2920 0.3403

0.3911 0.4490

0.4070 0.4658

Table 10. A{Pr} for Laminar Convection from a Horizontal Cylinder with an Uniform Heat Flux Density Pr

0.015

0.10

0.70

1

5

7

10

30

100

ACU{Pr}−eq 53 ACT {Pr)−eq 47 ACU/ACT APU/APT

6.59 2.36 2.79 2.56

7.03 2.68 2.62 2.46

7.96 2.32 2.40 2.23

8.27 3.49 2.37 2.21

10.6 4.66 2.28 2.22

11.4 5.02 2.27 2.22

12.3 5.45 2.26 2.22

16.3 7.23 2.25 2.26

23.0 10.2 2.25 2.24

7A. Horizontal Cylinders with Uniform Heating. Wilks28 in 1972 used the methodology of Saville and Churchill23 to compute numerical values for the rate of heat transfer by free convection from a horizontal cylinder with a uniform heat flux density on the surface for five finite values of the Prandtl number. Churchill29 in 1974 found that all of the computed values could be represented within 1% by NuD =

A CU = 4.629Pr1/3

(49)

As was to be expected and as is demonstrated in Table 9, the predictions of NuD /ReD1/2 by eq 49 are slightly greater than those of eq 44 for the same values of the Prandtl number. Although a few values of NuD for forced convection from a horizontal cylinder with a uniform heat flux density on its surface have been determined by numerical integration, a correlative/ predictive expression in the form of eq 47 does not appear to be have been published. That deficiency prevents the direct derivation of an expression for ACU{Pr} but provides an opportunity and a challenge to identify a means of estimation of a missing element or elements in an equivalence. On the basis of eq 44 the corresponding expressions for forced convection from a horizontal surface with a uniform heat flux density might be presumed to have the following forms: NuD =

aPr1/3ReD1/2 [1 + (b/Pr )2/3 ]1/4

(50)

⎛ a ⎞4 1/3 [1 + (0.442/Pr )9/16 ] ⎜ ⎟ Pr ⎝ 0.579 ⎠ [1 + (b/Pr )2/3 ]

16/9

(51)

A possible algorithm for estimation of the values of the unknown numerical coefficients a and b in eqs 50 and 51 is the speculation that the ratio AU/AT for a vertical plate (see Table 2) holds for a horizontal cylinder. Applying this escalation for Pr → ∞ results in a value of ACU {∞} = 2.256 × 2.052 = 4.629, and applying it for Pr → 0 results in a value of ACU{0} = 2.114 × 2.875 = 6.078. The values of a and b required to produce these values of ACU are 4.629 × 0.579 = 08493 and (4.629 × 0.442/6.078)3/2 = 0.1953, respectively. Equations 50 and 51 thus become NuD =

0.8493Pr 1/3ReD1/2 [1 + (0.1953/Pr )2/3 ]1/4

(53)

8. ISOLATED SPHERE Dilute suspensions of small droplets and spherical particles are common in nature and in manufacturing, and the transport of energy and species to or from their surface is of great scientific and technical interest. However, forced convection to a single isolated sphere has few applications and experimental measurements for a single sphere are difficult to carry out because of the need to support the sphere without interfering with the flowa complication that is not serious with a vertical plate or a horizontal cylinder. Hence the shortage of experimental data is not a surprise. The existing measurements and numerical solutions are almost wholly for an isothermal sphere because most applications involve a solid sphere rather than an annular one. Theoretical solutions do exist for creeping flow over an isothermal sphere, for the thin laminar boundary layer regime and for thermal conduction from an isothermal sphere to an unbounded region. In 1983 Churchill30 constructed, on the basis of the numerical solutions of others, the following correlative− predictive expressions for free convection from an isothermal

and AC =

[1 + (0.1953/Pr )2/3 ]

16/9

In the absence of experimental data or numerically computed values for NuD in the regime of a thick boundary layer, neither the choice of a fixed value nor the evaluation a dedicated coefficient for eq 53 is feasible, but the supposition that the same expressions are applicable for both free and forced convection makes eq 53 applicable for the entire laminar boundary regime. Values of ACU{Pr} as computed from this expression are listed in Table 10. The ratio of AU to AT for both a cylinder and a plate are included as a step in the search for generalities, and some success is achieved in that respect. Although the values of ACU and ACT for a cylinder with a uniform heat flux density both vary over a wide range with Pr their ratio has a range of only 25% and is almost constant at ∼2.25 for Pr ≥ 0.70. Furthermore their ratio differs only slightly from that for a plate for all values of Pr. There is probably a theoretical explanation for such constrained behavior, but with or without one it is a useful finding and a another positive feature of the concept of equivalence. The process that led from eq 48 to eqs 52 and 53 and to the numerical predictions thereof in Table 10 is an example of the formulation of an expression for A{Pr} that can be used to predict the Nusselt number for forced convection from one for free convection.

0.579RaD1/4 [1 + (0.442/Pr )9/16 ]4/9

[1 + (0.442/Pr )9/16 ]

(52)

and 4113



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Table 11. A{Pr} for an Isothermal Sphere Pr

0.015

0.10

0.70

1

5

7

10

30

100

AST{Pr}−eq 57 APT{Pr}−eq 5 ACT{Pr}−eq 45 AST /APT AST/ACT

0.639 0.671 2.36 0.952 0.270

0.822 0.539 2.68 1.39 0.306

1.057 0.455 3.32 2.32 0.318

1.13 0.454 3.49 2.49 0.324

1.61 0.517 4.66 3.12 0.346

1.75 0.546 5.02 3.21 0.349

1.92 0.583 5.45 3.30 0.353

2.60 0.747 7.23 3.48 0.360

3.72 1.04 10.2 3.59 0.365

the entire laminar regime not just that of the portion that is designated as thin. However, the form of the extension into the thick regime differs from that of vertical plates and horizontal cylinders and is worthy of examination in the context of equivalence. The traditional correlating equations for NuD for both laminar free and forced convection from an isolated sphere consist of a simple product of powers of Pr and of Re or Ra plus an additive value of 2. The latter is the well-known solution for pure conduction to unbounded surroundings and therefore is a theoretically based value as contrasted with the arbitrary ones that are sometimes, as mentioned heretofore, utilized as a limiting value for a vertical plate and for a horizontal cylinder. This relationship can be interpreted as

sphere in the tlbl regime. For the local Nusselt number as a function of angle from the forward point NuDSθ =

0.765(1 − 0.072θ 2)RaD1/4 [1 + (0.43/(Pr )9/16 ]4/9

(54)

and for its integrated-mean counterpart NuDS =

0.589RaD1/4 [1 + (0.43/Pr )9/16 ]4/9

(55)

The diameter, as symbolized by D, is the most common choice as the characteristic length for a sphere although a = D/2 and 2πa have sometimes been utilized. According to eqs 52 and 53, the local Nusselt number at the forward point of sphere in laminar free convection is finite and 30% greater than the integrated-mean value. This ratio was not mentioned in connection with either free or forced convection with plates or horizontal cylinders because, whereas a finite value of Nux/Rax1/4 or Nux /Rax1/2 is approached as x approaches zero, the heat transfer coefficient itself hypothetically approaches infinity, as can be discerned from eqs 1, 2, 8, and 9. A very large but finite value of the heat transfer coefficient is presumed to exist at the forward line of a horizontal cylinder for both free and forced convection and for both a uniform temperature and a uniform heat flux density. Such values have appeared in the literature but are not reproduced here because of their extreme uncertainty. The expression for forced convection corresponding to eq 53 for free convection can be conjectured to be NuDS ≅

]

and thereby as an analogue of eq 29. As an aside, Langmuir’s concept of approximating convection by conduction across a stagnant film does not appear to have been applied for a spherical annulus by anyone other than by Churchill.31 The addition of NuDS = 2 is presumed to confer an equally accurate correction for an expression for forced convection such as eq 54. On the other hand, creeping flow around a sphere is generally recognized to occur only for RaD ≤ 1, and Acrivos and Taylor32 showed that corresponding expression for the integrated-mean rate of forced convection necessarily has the following form: NuD = 2 +

AST ≅

[1 + (0.45/Pr )2/3 ]

PeD 2

(59)

Hossain and Gebhart in 1970 derived the following solution for free convection from an isothermal sphere in creeping flow:

(56)

NuDS = 2 + GrD + (GrD)2 (0.139 − 0.4519Pr

The uncertainty of the coefficients 0.55 and 0.45 is the reason eq 55 is designated as an approximation. The functional dependence is presumed to be exact. Although the uncertainty of eq 54 is dampened somewhat by the integration, it extends to the corresponding expression for AST{Pr}, namely 0.760Pr1/3[1 + (0.43/Pr )9/16 ]

(58)

33

0.55ReD1/2Pr1/3 2/3 1/4

[1 + (0.45/Pr )

NuSbll = NuStlbl + 2

+ 1.1902Pr 2)

(60)

They asserted that eq 60 is valid for 0 ≤ GrD < 1 and Pr ≅ 1. However, its predictions fail to conform to eq 59 and differ from the subsequent numerical solutions of Geoola and Cornish34 in 1981 for Pr = 0.72 and 0.007 ≤ GrD ≤ 6 by as much as an order of magnitude. Liňań and Kurdyumov35 in 1999 carried out numerical calculations of NuD for finite values of Gra for an isothermal sphere but their results are presented in the same format as their numerical ones for horizontal cylinders and thereby are not individually accessible. Furthermore, their correlating equation does not conform to eq 59. Values of NuDS for Pr = 0.70, as calculated from eqs 58, 59, and 60 are compared over the range of Ra for creeping flow in Table 12 for Pr = 0.70. The limiting value of 2 can be noted to be dominant for RaD ≤ 1 and significant for all of the conditions encompassed in the tabulation. The predictions of eq 56, which are based on a correlative expression for thin boundary theory, are in reasonable agreement with those of eq 57, which is a purely theoretical expression based on creeping flow. The validity of both of these

16/9

(57)

Values of AST{Pr}, as calculated from eq 57 are listed in Table 11, along with values of APT{Pr} and ACT{Pr} as points of reference. The ratio AST /A CT has a restrained range and together with AST /A TP approaches a fixed value as Pr increases characteristics that may prove useful in terms of correlation and generalization. It is possible that an explanation may eventually be deduced for these ratios and their magnitudes. It is curious that AST crosses over APT in magnitude as Pr increases; that aspect of behavior may be an artifact of a length of integration that differs from the characteristic length for the sphere but not the plate. On the basis of the results for vertical plates and horizontal cylinders, eq 55 would be expected to be applicable for 4114



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A definitive correlative/predictive and suitable expression for fully developed forced convection from an isothermal plate also remains to be developed. Churchill16 in 1976 postulated the same relative dependence on Pr as for the thin laminar boundary layer (see his eq 2) and fitted the limited experimental data of Ž ukauskas and Šlančiauskas40 for the local coefficient for 0.70 Pr ≤ 257 in the turbulent regime with a power-dependence of 0.8 and a leading coefficient of 0. 032 resulting in the expression

Table 12. Predicted Values of the Integrated-Mean Nusselt Number for Free Convection from an Isothermal Sphere in the Laminar Regime for Pr = 0.70 RaD

10−1

1

1.511

equivalent PeD equivalent ReD per eq 57 GrD

0.2573 0.3676 0.1429

0.7342 1.0489 1.163

1.000 1.429 2.159

0.458 2.458 2.367 3.712

0.508 2.508 2.500 3.902

NuDS eq 55 eq 58 eq 59 eq 60

0.258 2.258 2.129 2.151

Nux =

9. TURBULENT REGIME All of the preceding analyses are for laminar flow. As is demonstrated by the following illustration, The concept of equivalence is applicable for a turbulent boundary layer, but the components are more complex and less certain numerically and even functionally. Nusselt36 in 1915 observed that the heat transfer coefficient for turbulent free convection from a vertical plate was independent of height and recognized that such independence required it to be proportional to the one-third power of the Rayleigh number and the local Nusselt number to be equal to the integrated-mean value. That is,

(65)

The term RexPr(Cf/2) may be recognized as the equivalent of the Reynolds41 analogy for a flat plate. It should be a good approximation for Pr of the order of unity or less. The term 0.078Rex(Cf /2)1/2Pr1/3 may be recognized as having the form of the Colburn10 analogy with an adjusted leading coefficient, and it should be a good approximation for a flat plate in the double limit of asymptotically large values of Pr and Rex. An arbitrary combining exponent of −10 for the asymptotes of eq 65 results in

(61)

Nux =

0.078Rex(Cf /2)1/2 Pr1/3 ⎡ ⎢1 + ⎣

(

0.078 (C f / 2)1/2 Pr 2/3

10 ⎤1/10

) ⎥⎦

(66)

Equating the right-hand sides of eqs 61 and 64 and simplifying results in the following expression for the equivalence of the turbulent regime on a vertical isothermal plate: Grx =

0.216Rex 3(Cf /2)3/2 (1 + [0.078/(Cf /2)1/2 Pr 2/3]10 )3/10

(67)

Equation 67 can also be expressed functionally as Grx = A t {Pr , Cf }[Rex(Cf /2)1/2 ]3

(68)

where A t {Pr , Cf } =

(62)

The power-dependence was observed to decrease thereafter leaving open the possibility of an asymptotic approach to eq 61. Lin and Churchill also carried out numerical solutions for Pr = 5.8 and 58 for comparison with experimental data for water and spindle oil but, owing to the uncertainty of the coefficients in the k−ε model and the scatter of the experimental data, the dependence of Nux on Pr and Ra remains somewhat uncertain. Accordingly, the following rounded-off value of Colburn and Hougen is proposed as a tentative expression for all values of Pr: Nux = 0.13Grx1/3Pr1/3

(64)

Nux m = ([RexPr(Cf /2)]m + [0.078Rex(Cf /2)1/2 Pr1/3]m

where β is an arbitrary numerical coefficient. Colburn and Hougen37 in 1930 conjectured that the independence of the coefficient from height was a consequence of the resistance to heat transfer arising almost wholly from a fully developed laminar sublayer and concluded from the experimental data of others for liquids that the best value of β was 0.128. Cheesewright38 in 1968 concluded from the available experimental data that the heat transfer coefficient was indeed constant starting some distance immediately above the point of attainment of a turbulent boundary layer but that it began to increase thereafter. Lin and Churchill17 in 1978 carried out numerical solutions using a k−ε model devised by Jones and Launder39 for free convection. They tweaked the coefficients therein in order to improve the representation for the region very near the wall and represented their computed values for Pr = 0.7 immediately following the establishment of a turbulent regime with the expression Nux = 0.0495Rax 0.367

[1 + (0.04681/Pr )2/3 ]0.4

Although eq 64 represents the experimental data adequately, it is not suitable for construction of an equivalence because of its incorporation of an arbitrary empirical power-dependence of Nux on Rex. Despite their ubiquity in the traditional literature on transport, such noninteger power dependences have no theoretical rationale.The following expression in the form of the CUE is proposed as a substitute for eq 62:

expressions is presumed to extend to lesser values of GrD . The prediction of eq 58 is in good agreement with that of eq 59 for RaD = 0.1, but that may be fortuitous because its prediction fails for RaD = 1 and 1.511.

Nux = Nux = βRax1/3

0.032(RexPr 2/3)0.8

0.216 (1 + [0.078/(Cf /2)1/2 Pr 2/3]10 )3/10

(69)

In the equivalence defined here for a turbulent boundary layer the Grashof number Grx is proportional to [Rex(Cf/2)1/2]3 as compared to a proportionality to Rex2 for a laminar boundary layer, and the proportionality factor At depends on Cf as well as on Pr. This difference and complexity is are consequences of the coupled dependence of the Nusselt number on Pr and Cf and invoke the need for an expression for the latter as a function of the Reynolds number. Churchill42 in 1993 devised the following expression for Cf in a fully turbulent boundary layeron a smooth plate:

(63) 4115



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Table 13. Illustration of Transition Produced by Equation 64 for Rex = 81 523 Pr

0.015

Colburn analogy eq 66 Reynolds analogy

0.70

82.54 3.387 3.387

297.15 158.1 158.1

1

1.804 Nux 407.4 407.4 407.4

334.7 225.8 225.8

⎧ ⎫ ⎛ 2 ⎞1/2 Cf Rex ⎬ ⎜ ⎟ = 2.526 + 2.439ln⎨ 1/2 ⎝ Cf ⎠ ⎩ 1 − 7.965Cf + 21.52Cf ⎭ ⎪

⎪

⎪

(70)

The difference in the independent dimensionless group of eq 70, namely CfRex/(1 − 7.965Cf1/2 + 21.52Cf), and that of the corresponding one for turbulent flow in a round tube, namely (Cf /2)1/2Rex, is a consequence of the continued growth of the boundary layer on a flat plate as contrasted with the attainment of fully developed flow in a round tube. However, the reciprocal of 2.439, namely 0.410, can be noted to correspond closely to the commonly chosen value of the von Kármán constant for internal flow, and the quantity (2/Cf)1/2 can be recognized as equal to the dimensionless free-stream velocity in the wall-variable notation of Prandtl. The numerical solution of eq 70 requires iteration for a specified value of Rex, but that iteration can be avoided if a value of (2/Cf)1/2 or the equivalent is specified. In that event it is convenient to rearrange eq 70 as ⎡ ⎤ ⎛ 2 ⎞1/2 2 Rex = ⎢ − 11.26⎜ ⎟ + 43.04⎥ ⎢⎣ Cf ⎥⎦ ⎝ Cf ⎠ ⎧ (2/C )1/2 − 4.217 ⎫ f ⎬ exp⎨ 2.439 ⎩ ⎭ ⎪

⎪

⎪

⎪

7

10

30

100

572.3 572.3 1129

640.2 640.2 1580

721.0 721.0 2258

1039 1040 6774

1553 1553 22,583

10. SUMMARY AND CONCLUSIONS This investigation describes a new findingnamely an algebraic relationship between the expressions that predict the same Nusselt number for free convection and forced convection in the laminar regime. That relationship is found to have great generality. It was identified previously but only in the context of combined convection and only for the limiting cases of Pr → 0 and Pr → ∞. The expressions that predict the same Nusselt number for free convection and for forced convection for any given geometry and conditions are herein designated as equivalents and the relationship between those two expressions is designated an equivalence. The concept of equivalence is a useful one in several respects. First it provides a new structure for the assemblage of information on thermal transport and in some instances provides a means for its prediction. In particular, a Grashof number proportional to the square of the Reynolds number has been found to produce the same local Nusselt number for flat vertical plates, horizontal round cylinders, and isolated spheres over the entire regime of the laminar boundary layer, including that of its thickening. Second, the investigation of the conditions that result in an equivalence has uncovered a number of generalizations that enhance understanding and provide guidance for the prediction of thermal convection. The coefficient of proportionality between the Grashof number and the square of the Reynolds number, here labeled A{Pr}, is a slightly different function of the Prandtl number for each geometry and differs slightly but systematically for a uniform temperature and a uniform heat flux density on the surface. The Nusselt number for free convection can be predicted from that for forced convection or vice versa if the function A{Pr} is known or can be estimated. It is thereby therefore a new resource for practitioners of process design as well as for teachers of heat and mass transfer. The identification of equivalences would not have been possible without the prior existence of a body of closed-form solutions for free and forced convection in the thin laminar boundary layer regime and without the consequent generalized correlating equations for their dependence on the Prandtl number. Indeed, this investigation is an example of behavior for which theoretical analyses are not merely helpful but truly essential for correlation and understanding. The traditional correlating equations in the form of products of powers of the Rayleigh, Reynolds, and Prandtl numbers would probably not have revealed the existence of equivalences and certainly not with the certainty provided by the solutions and correlating equations of the thin laminar boundary model. Experimental data is of course desirable to confirm the validity of the equivalences revealed by the theoretical analyses. The existing body of experimental data was found to be sufficient in extent and precision for that purpose for some but not all laminar flows and to be only marginal at best for the only turbulent flow that was examined.

on a smooth plate ⎪

5

(71)

Churchill42 compared the values computed from eq 68 with those read from curves sketched through representative experimental data and noted that the agreement is not only very good but also better than that attained by any of the many classical expressions. He attributed this improvement to accounting for the wake. The following single chain of calculated values is presented for the for turbulent convection on a vertical isothermal plate for Pr = 0.7 and (2/Cf)1/2 = 19, which corresponds roughly to the minimum value thereof at which the transition to a turbulent boundary layer occurs. The value of 19 for (2/Cf)1/2 results in Rex = 81 523 and x+ = Rex(Cf/2)1/2/2 = 2145 from eq 71, At = 0.03252 from eq 69, Grx = 2.569 × 109 from eq 68, and Nux = 158.08 from eqs 63 and 66. The corresponding value of Rax, namely, 1.798 × 109, is somewhat lower than the limiting value for a flat plate according to the numerical computations of Lin and Churchill,17 but their numerical solutions are rather uncertain because of the idealizations in the model for turbulence. The transition in the prediction of Nux by eq 66 as Pr increases is illustrated in Table 13. This is perhaps the one instance in this manuscript in which a graph might have been equal to a tabulation in terms of clarity. The immediately preceding analysis has, perhaps for the first time, identified and defined in detail a equivalence between free and forced convection in turbulent flow. Equation 68 appears to be functionally correct, informative, and in combination with eq 69 useful for prediction. 4116



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presumed to have cured this deficiency, but it has not because modeling three-dimensional and/or time-dependent flow and convection and the numerical solution of such models are still challenges for the laminar regime and an even greater one for the turbulent regime. Gradual improvements in instrumentation have resulted in ever-more precise measurements, but do not seem to have prompted the experimental work needed to fill the gaps. Teachers and authors of textbooks and handbooks have an obligation to inform students and practicing engineers of the state of the art in convection including the advances since the publication of the textbooks they assign insofar as they find them creditable. The results and conclusions articulated herein presumably fall in that category. The enunciation of the concept of equivalence and the illustrative applications appear to constitute an advance in both the qualitative and the quantitative description of the transport of energy and chemical components that should be incorporated in courses and eventually textbooks in those topics. Those who devise computer packages have an even greater responsibility to incorporate creditable advances because their contents are not obvious. The predominance of self-referrals among the references is because the author and his collaborators devised almost every one of the correlative−predictive equations involving numerical solutions of the thin laminar boundary model that provide the basis for the detailed models for the equivalences and have also carried out most of the required numerical solutions upon which those correlating equations are based.

The representation of an individual laminar equivalence, as symbolized by A{Pr}, incorporates an implicit message that may be as important to teachers of thermal science and related subjects as the equivalence itself, namely that correlative− predictive equations in the form of powers of dimensionless groups are, despite their ubiquity in the literature, valid only in the asymptotic limits of small and large values of the independent dimensionless variables. The utilization of thin laminar boundary layer theory as a resource herein avoids the functional error associated with empirical power-functions but in doing so somewhat obscures that message. It should be noted that the solutions of thin laminar boundary layer, despite their obvious value in terms of identifying functional and numerical dependences, incorporate some hidden idealizations. The most serious one, as known by all experimentalists and by all practitioners of process design, is the neglect of the variation of physical properties (other than the density by virtue of the Boussinesq approximations) with temperature, pressure, and chemical composition. The challenge posed by that topic was considered beyond the scope of the current investigation. It should be mentioned that the similarity transformations that lead to closed-form solutions are possible only in the limits of Pr → 0 and Pr → ∞. The predictive expressions for intermediate values of Pr are interpolations between the exact solutions in the form of the CUE and are not exact. Even so, their numerical accuracy exceeds any practical demand. An equivalence has been identified, apparently for the first time, for turbulent convection. It is for a single condition, namely an isothermal vertical plate. The Grashof number is found tentatively to be proportional to the cube of the product of the Reynolds number and the square root of the coefficient of skin friction, but it is somewhat uncertain in functionally as well as in detail because of the uncertainty of the theory and correlating equations for external turbulent convection. Even this tentative equivalence for a turbulent boundary layer could not have been devised without the guidance and confidence bestowed by the prior developments herein for laminar convection. Insofar as the analogy between heat and mass transfer is valid, the equivalences identified herein for both laminar and turbulent convection apply to mass transfer. One contribution of this investigation extends beyond the bounds of equivalence, namely eq 35, a generalization of the socalled Langmuir solution. This expression provides a new vehicle for correlation by virtue of extrapolation of a solution for a limiting or special condition for all values of the argument. The shortcomings in information that are revealed in the process of constructing the illustrations provide a collateral benefit to thermal science that should not be overlooked, namely the identification of conditions for which experimental data, numerically computed values, theoretical solutions, and correlating equations are currently nonexistent or incomplete. The omissions discussed in the previous paragraph bring to light a characteristic of the state of the art in the practice of thermal science and presumably of related fields. Namely, that the body of supporting research and correlation is not developed uniformly with forethought but instead rather randomly as prompted by related work, the development of methods of solution, and the demands of practice. In view of their everyday application, these resources for thermal convection are surprisingly incomplete in textbooks and presumably in computer packages. Progress in partial differential modeling and improvements in computer hardware and software might be

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 610-459-4694. Fax: 215-573-2093. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This manuscript is more dependent on others than any one that I have ever written. First, I thank Dr. Martin Leahy of MSiKenny, Perth, Australia, for bringing to my attention the discrepancy in two of my publications that prompted the investigation and Professor Hiroyuki Ozoe and Dr. Humbert Chu for reviewing my response to Dr. Leahy and suggesting that I prepare a manuscript for publication on this subject. Second, I thank Editor-in-Chief Professor Donald R. Paul, his successor Professor Phillip E. Savage, and the anonymous reviewers for their constructive suggestions. Next, I thank Professor Warren D. Seider, Professor J. D. Seader, and Dr. Shyy-Jong Lin who provided a number of constructive criticisms, and most of all Professor Andreas Acrivos who not only called to my attention the need to identify the limitations of the equivalents but also a number of important investigations that I would otherwise have overlooked.

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REFERENCES

(1) Lemlich, R.; Hoke, R. A Common Basis for the Correlation of Forced and Natural Convection to Horizontal Cylinders. AIChE J. 1956, 2, 249−250. (2) Churchill, S. W.; Douglas, W. J. M. Recorrelation of Data for Convective Heat Transfer between Gases and Single Cylinders with Large Temperature Differences. Chem. Eng. Prog. Symp. Series 1956, 52 (No. 18), 23−28.

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(3) Acrivos, A. On the Combined Effect of Free and Forced Convection Heat Transfer in Boundary Layer Flows. Chem. Eng. Sci. 1966, 21, 343−352. (4) Morgan, V. T. The Overall Convective Heat Transfer from Smooth Cylinders. Advances in Heat Transfer; Academic Press: New York, 1975; Vol. 11, p 199−264. (5) Churchill, S. W. A Comprehensive Correlating Equation for Laminar, Assisting, Forced and Free Convection. AIChE J. 1977, 23, 10−16. (6) Churchill, S. W.; Usagi, R. A General Expression for The Correlation of Rates of Transfer and Other Phenomena. AIChE J. 1972, 18, 1121−1128. (7) Churchill, S. W. An Analysis of Heat and Component Transfer by Turbulent Free Convection from a Vertical Plate, Proceedings of Chemeca 70, Butterworths: Sydney, Australia, 1970; Session 6A, pp 1−16. (8) Churchill, S. W. A Comprehensive Correlating Equation for Forced Convection from Flat Plates. AIChE J. 1976, 22, 264−268. (9) Churchill, S. W. Critique of the Classical Analogies for Heat, Mass, and Momentum Transfer. Ind. Eng. Chem. Res. 1997, 36, 3866−3878. (10) Colburn, A. P. A Method of Correlating Forced Convection Heat Transfer Data and A Comparison with Fluid Friction. Trans. Am. Inst. Chem. Eng. 1933, 41, 174−210. (11) Churchill, S. W.; Ozoe, H. Correlations for Laminar Forced Convection in Flow Over an Isothermal Flat Plate and in Developing and Fully Developed Flow in an Isothermal Tube. J. Heat Transfer 1973, 95, 416−419. (12) LeFevre, E. J. Laminar Free Convection from a Vertical Plane Surface. Proceedings of the 9th International Congress on Applied Mechanics, Brussels, Sep 5−13, 1956; Vol. 4, pp 168−174. (13) Prandtl, L. Ű ber Flű ssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des III International Mathematiker-Kongresses, Heidelberg, 1904; Teubner: Leipzig, 1905, p 484. (14) Schlichting, H.; Kestin, J. Boundary Layer Theory, sixth english ed.; McGraw-Hill: New York, 1968, p 281. (15) Churchill, S. W.; Chu, H. H.-S. Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate. Int. J. Heat Mass Transfer 1975, 18, 1323−1339. (16) Churchill, S. W. A Comprehensive Correlating Equation for Forced Convection from Flat Plates. AIChE J. 1976, 22, 264−265. (17) Lin, S.-J.; Churchill, S. W. Turbulent Free Convection from a Vertical, Isothermal Plate. Numerical Heat Transfer 1978, 1, 129−145. (18) Churchill, S. W.; Ozoe, H. A Correlation for Laminar Free Convection from a Vertical Plate. J. Heat Transfer, Trans. ASME 1973, 95C, 540−541. (19) Churchill, S. W.; Ozoe, H. Correlations for Laminar Forced Convection with Uniform Heating in Flow Over a Flat Plate and In Developing and Fully Developed Flow in A Tube. J. Heat Transfer, Trans. ASME 1973, 95C, 78−84. (20) Langmuir, I. Convection and Conduction of Heat in Gases. Phys. Rev. 1912, 13, 471. (21) Churchill, S. W. Derivation, Selection, Evaluation and Use of Asymptotes. Chem. Eng. Technol. 1988, 11, 63−72. (22) Churchill, S. W. The Art of Correlation. Ind. Eng. Chem. Res. 2000, 39, 1850−1877. (23) Lévêque, M. A. Les lois de la transmission de la chaleur par convection. Ann. Mines Memoires 1928, 13, 201−299, 305−362, 381− 415. (24) Saville, D. A.; Churchill, S. W. Laminar Free Convection in Boundary Layers Near Horizontal Cylinders and Vertical Axisymmetric Bodies. J. Fluid Mech. 1967, 29, 391−399. (25) Saville, D. A.; Churchill, S. W. Free Convection at Low Prandtl Numbers in Laminar Boundary Layers. Ind. Eng. Chem. Fundam. 1969, 8, 329−332. (26) Churchill, S. W.; Bernstein, M. H.. A Correlating Equation for Forced Convection from Gases and Liquids to a Circular Cylinder in Cross Flow. J. Heat Transfer, Trans. ASME 1977, 99C, 300−306. (27) Liňań , A.; Kurdyumov, V. N. Laminar Free Convection Induced by a Line Heat Source, and Heat Transfer from Wires at Small Grashof Numbers. J. Fluid Mech. 1998, 362, 199−227.

(28) Wilks, G. External Natural Convection About Two-Dimensional Bodies with Constant Heat Flux. Int. J. Heat Mass Transfer 1972, 15, 351−364. (29) Churchill, S. W. Laminar Free Convection from a Horizontal Cylinder with a Uniform Heat Flux Density. Lett. Heat Mass Transfer 1974, 1, 109−112. (30) Churchill, S. W. Comprehensive, Theoretically Based, Correlating Equations for Free Convection from Isothermal Spheres. Chem. Eng. Commun. 1983, 24, 339−352. (31) Churchill, S. W. The Role of Analysis in the Rate Processes. Ind. Eng. Chem. Res. 1992, 31, 643−668. (32) Acrivos, A.; Taylor, T. D. Heat and Mass Transfer from Single Spheres in Stokes Flow. Phys. Fluids 1962, 5, 387−394. (33) Hossain, M. A.; Gebhart, B. Natural Convection About a Sphere at Low Grashof Number. Proceedings of the 4th International Heat Transfer Conference 4; Elsevier: Amsterdam, 1970; NC 1.6, pp 1−12. (34) Geoola, F.; Cornish, A. R. H. Numerical Solution of Steady-State Free Convection Heat Transfer About a Sphere. Int. J. Heat Mass Transfer 1981, 24, 1369−1379. (35) Kurdyumov, V. N; Liňań , A. Free Convection from a Point Source of Heat, and Heat Transfer from Spheres at Small Grashof Numbers. Int. J. Heat Mass Transfer 1999, 42, 3849−3860. (36) Nusselt, W. Das Grundgesetz des Wärmeüberganges. Gesund.-Ing. 1915, 38, 477−496, 490−496. (37) Colburn, A. P.; Hougen, O. A. Studies in heat transmission. III Flow of fluids at low velocities. Ind. Eng. Chem. 1930, 22, 534−539. (38) Cheesewright, R. Turbulent Natural Convection from a Vertical Plane Surface. J. Heat Transfer. Trans, Am. Soc. Mech, Engrs. 1968, 90C, 1−8. (39) Jones, W. P.; Launder, B. E. The Prediction Of Laminarization with a Two-Equation Model of Turbulence. Int. J. Heat Mass Transfer 1972, 15, 301−314. (40) Ž ukauskas, A.; Šlančiauskas, A. Heat transfer in Turbulent Flow of Fluid. Meeting Minutes, Vilnius, Lithuania, 1973; in Russian. (41) Reynolds, O. On the Extent and Action of the Heating Surface on Steam Boilers. Proc. Lit. Philos. Soc. Manchester 1874, 14, 7−12. (42) Churchill, S. W. Theoretically Based Expressions in Closed Form for the Local and Mean Coefficient of Skin Friction in Turbulent Flow Along Smooth and Rough Plates. Int. J. Heat Fluid Flow 1993, 14, 231− 239.

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