Ind. Eng. Chem. Res. 2001, 40, 3053-3057
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Correlating Equations for Transitional Behavior Stuart W. Churchill* Department of Chemical Engineering, University of Pennsylvania, 311A Towne Building, 220 South 33rd Street, Philadelphia, Pennsylvania 19104
A transitional regime, as characterized by a point of inflection, links the asymptotic regimes for large and small values of the independent variable in many physical and chemical systems. A new methodology has been devised for representation of the overall behavior including such a transition by a single correlating equation. This development is based on the identification of one exact and one nearly exact solution in closed form for such a transition and on the recognition that these two particular solutions correspond to the same special case of the canonical correlating equation proposed by Churchill and Usagi for multiple regimes. This correspondence previously escaped recognition because it is evident only for one particular arrangement of the closed-form solutions and for one particular reduction and arrangement of the canonical correlating equation. The new generalized formulation appears to be applicable even for behavior for which the internal asymptote representing the transition is not known in advance, which is the usual situation. Introduction The need for correlation has not been eliminated or even decreased significantly by our increased capability in formulating improved and extended differential models and in solving these models numerically. First, the numerical solutions are usually in the form of tabulations of extensive sets of discrete values. Second, in most instances, the process of solution is sufficiently demanding that it discourages or precludes such calculations in the midst of some application. Third, the discrete numerical values do not necessarily provide much guidance for the construction of a closed-form functional relationship between the variables. In these three respects, the results of numerical solutions resemble experimental data except for their greater precision, better distribution, and complete independence from unidentified parametricity. (Of course, significant parameters are commonly omitted from a model due to naivete´ or a desire for simplicity.) In the past, most correlations, at least for heat, mass, and momentum transfer, were constructed by plotting experimental data in the form of dimensionless groups on logarithmic coordinates. Often, such plotted values were then represented by a straight line as determined by least squares. This procedure leads to a correlating equation in the form of a power function. Such a procedure is questionable on several grounds. First, logarithmic coordinates display percentage deviations of one point from another or from a correlating line or curve and thereby suppress the deviations perceptually compared to arithmetic plots, which display absolute deviations. Second, a fixed-power dependence never occurs physically, except in an asymptotic sense. Third, unaccounted parametric variations might be mistaken for experimental error. Fourth, the conditions, such as random absolute deviations, that are implied by the use of least squares or other statistical methodologies are seldom satisfied. Such procedures are also unjustifiable for computed values for all but the third reason. * Phone: 215-898-5579.
[email protected].
Fax:
215-573-2093.
E-mail:
Modern correlations generally involve interpolation between asymptotic expressions. The functionality of the latter can be determined by perturbation methods, speculative analyses, or analytical solutions for simplified cases or even empirically from plots of experimental data or computed values. Interpolation obviously implies a smooth, inflection-free transition from one asymptote to another. Many transitions, such as that from laminar to turbulent flow, do, however, involve a point of inflection. The representation of such behavior is the objective of the work reported herein. The examples are limited to forced convection in channels, but the new methodology appears to be applicable for many other processes. Prior Work Churchill and Usagi1 proposed, for correlation in general, the canonical expression
y{x}n ) y0{x}n + y∞{x}n
(1)
where y0{x} and y∞{x} are asymptotic expressions or limiting values for small and large values of x, respectively, and n is an arbitrary exponent. The representation by eq 1 of complex functional behavior, such as that given exactly by an infinite series, is necessarily approximate. Nevertheless, eq 1 has proven to provide representations of remarkable numerical accuracy for all values of x for economic, physiological, and a wide gamut of physical behavior. This success is primarily a consequence of the reduction of the task of correlation to that of representing the difference between y{x} and y0{x} or y∞{x}, whichever is nearer in value. A corollary is that the representation by eq 1 is insensitive to the numerical value of n, permitting the choice of an integer or the ratio of two small integers, rather than a statistically determined value. Churchill and Usagi1 proposed a sensitive graphical method for determination of n and for evaluation, at the same time, of the resulting representation for y{x} in arithmetic coordinates.
10.1021/ie000281j CCC: $20.00 © 2001 American Chemical Society Published on Web 04/06/2001
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A numerical value for n other than (1 appears to have been derived theoretically for only one condition, namely, n ) 3 for heat or mass transfer in assisting external convection in the thin-laminar-boundary-layer regime due to forced flow and buoyancy, and for a sphere due to vibration and rotation as well. There are no restrictions on the functional forms of the asymptotes y0{x} and y∞{x} for use in eq 1, except for conformity to the following conditions: (1) they must both be free of singularities; (2) they must intersect each other once and only once; and (3) they must both be upper or lower bounds for y{x}. Preferably, they both provide the same order of approximation. If the initally chosen expressions for y0{x} and y∞{x} fail to meet these conditions, it might be possible to modify one or both appropriately. The application of eq 1 implies that the physical transition from y0{x} to y∞{x} is monotonic and free of inflection. In many instances, secondary variables and parameters can be incorporated in y0{x} and y∞{x} inherently. If not, the exponent n can be correlated as a function of the second variable or parameter. Another approach is to nest a correlating equation for a secondary variable within y0{x} or y∞{x}. Equation 1 can be applied consecutively for three regimes of the primary variable as follows
y{x} ) [(y0{x}n + yi{x}n)m/n + y∞{x}m]1/m
(2)
where, here, yi{x} is an intermediate asymptote. If y0{x}, yi{x}, and y∞{x} are all lower bounds, m and n will both be positive. If y0{x} is a lower bound and y∞{x} an upper bound, n must be positive and m negative. The formulations for the converse case in which y0{x} is an upper bound and for the reverse order of combination, beginning with yi{x} and y∞{x}, follow directly and therefore need not be examined in detail herein. The successful application and reapplication of eq 2 for laminar, transitional, and turbulent fully developed flow and forced convection in round tubes for all values of Pr and two modes of heating is illustrated by Churchill.2 However, if y0{x} and y∞{x} are opposite bounds, eq 2 has a fundamental flaw, which, although unapparent and tolerable in some applications, can be apparent and intolerable in others. For example, if y0{x} is a lower bound, yi{x} will generally become negligible relative to y0{x} as x f 0. Then, in that limit, eq 2 reduces to
y{x} ) (y0{x}m + y∞{x}m)1/m
(3)
Because m is necessarily negative, eq 3 predicts values of y{x} that are less than those of y0{x}, which is contradictory. This anomalous behavior can be circumvented as follows. The initial combination of y0{x} and yi{x} is first rearranged as
y{x}n - y0{x}n ) yi{x}n
(4)
and then the left-hand side of eq 4 is interpreted as the dependent variable, leading to
(y{x}n - y0{x}n)m ) yi{x}mn + (y∞{x}n - y0{x}n)m
(5)
Equation 5 has a more complicated structure than eq 2 but involves no additional components or exponents. The asymptotes y0{x} and y∞{x} are generally known from theoretical analyses and, if not, can usually be constructed from experimental data or computed values
for small and large values of x, respectively. Then, with eq 1, only n remains to be determined. On the other hand, yi{x} is ordinarily not known from theoretical considerations. As shown by Churchill,3 yi{x} is not the tangent through the point of inflection in a plot of y versus x in logarithmic coordinates. Hence, it must be determined from eq 4 or the equivalent simultaneously with n by an iterative method. The exponent m can then be determined from eq 5 using the same methods as suggested for n in eq 1. The order of determination of n and m can, of course, be reversed. Because the representation of yi{x} will ordinarily require the empirical determination of one arbitrary exponent and one arbitrary coefficient, four constants must be determined in all, not counting any in y0{x} and y∞{x}. Experimental data are seldom of sufficient precision to determine unique values for four constants. Hence, alternative expressions with different exponents and coefficients might be found to represent the data almost equally well. This process and multiplicity is illustrated by Churchill and Churchill4 for the effective viscosity of pseudoplastic and dilatant fluids. It can be inferred from the discussion therein and herein that the power-law regime of such fluids might be simply artifact of the transition from one fixed value to another, rather than necessarily a fundamental aspect of behavior. Values determined from numerical solution of a model might have sufficient precision to determine unique values for all of the coefficients and exponents of eq 5, but few such sets yet exist, at least in transport phenomena. A New Approach Example 1. Laminar Forced Convection in Channels with External Heat Losses. Hickman5 derived, using the Laplace transform, a series solution for fully developed convection in fully developed laminar flow through a round tube with heat losses due to radial conduction through the tube wall and external insulation in series and then free convection and radiation in parallel to the surroundings. His result for the heat transfer coefficient for the fluid phase can be expressed as
48 11Bi Nu ) 59 1 + 220 Bi 1+
(
)
(6)
Here, Bi ) UeD/k is the Biot number based on the internal diameter of the tube, the thermal conductivity of the fluid, and the equivalent overall external heat transfer coefficient based on the internal tube diameter. The quantity 48/11 can be recognized as the exact value of Nu for a uniformly heated tube, and 220/59 ) 3.729 as a close approximation for the corresponding exact value of 3.657... for a uniform wall temperature. The difference between the values of 3.729 and 3.657 is presumed to be due to numerical error by Hickman in compiling the solution for NuT. Hence, eq 6 can be rewritten as
NuJ Bi Nu ) 1 1 + NuT Bi 1+
(7)
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3055
and then rearranged as
Nu - NuT ) NuJ - NuT
1 1+
Bi NuT
(8)
On the other hand, for the special case of n ) -m ) 1 and fixed values of y0{x} ) y0 and y∞{x} ) y∞, eq 5 can be reduced and rearranged as
y - y0 ) y∞ - y0
1+
1 y∞ - y0
(9)
yi{x}
Comparison of eqs 8 and 9 reveals that the solution of Hickman is equivalent to the transitional behavior represented by eq 5 for the special case of n ) -m ) 1, y0 ) NuT, y∞ ) NuJ, and yi{x} ) Nui ) (NuJ - NuT)NT/ Bi. This equivalence is quite remarkable and significant. First, in the absence of the solution by Hickman, the dependence of Nu on Bi might not be recognized as “transitional” in the sense of eq 5. Indeed, none of the several other investigators of this behavior before and after Hickman referred to it as such. Second, in the absence of a direct comparison of eqs 8 and 9, the proportionality of the internal asymptote Nui to (Bi)-1 might never have been recognized, and even more likely, the functionality of the proportionality factor, namely, (NuJ - NuT)NuT, might have been missed. Third, the discovery of the equivalence itself was somewhat fortuitous in that it is only apparent for the special case of n ) -m ) 1 and the particular rearrangements of eqs 5 and 7 in the form of eqs 8 and 9. The trial-anderror process that led to these latter forms was actually prompted and guided by the subsequently presented analysis of turbulent convection. Insofar as 220/59 can be replaced by NuT ) 3.657, eq 6 is presumably an exact solution rather than a correlating equation. However, its presence herein appears to be justifiable as this form might be applicable to other related behavior for which an analytical solution is not known. Hickman also investigated the effect of heat losses on laminar convection in parallel-plate channels with both symmetrical and unsymmetrical external resistances. These solutions can also be represented by eq 8,with, of course, the different appropriate definitions of NuT, NuJ, and Bi. Example 2. Turbulent Forced Convection in Channels. Reichardt6 derived an analogy in algebraic form between heat and momentum transfer in fully developed turbulent flow and convection in a round pipe following a discrete step in wall temperature. This result is not exact in that several simplifications based on the presumed relative order of magnitude of the various terms in the differential model, and two postulates, one that Prt, the turbulent Prandtl number, is independent of radius and the other that Prt approaches a finite value at the wall, were made to permit closed-form integration. However, it is free of any explicit empiricism. Churchill et al.7 noted that the Reichardt analogy could be reexpressed in the generalized form
( )
(
)
Prt 1 Prt 1 1 + 1) Nu Pr Nu1 Pr Nu∞
(10)
where Nu1 is the particular value for Pr ) Prt, which is unique in being independent of Pr/Prt, and Nu∞ is the asymptotic expression for Pr f ∞. The analogy of Reichardt incorporated a functionally erroneous expression for Nu∞ (the correct one is 0.07343(Pr/Prt)l/3Re(f/ 2)1/2), but this does not affect eq 10 itself. Churchill et al.7 postulated that eq 10 might be applicable as a generalized correlating equation for other geometries and modes of heating. Accordingly, they tested eq 10 with the nearly exact computed values of Heng et al.8 for a uniformly heated round tube and of Danov et al.9 for a parallel-plate channel heated uniformly and equally on both plates and also with different uniform temperatures on the opposing surfaces. The resulting representations for Pr > Prt proved to be very good. The slight deviations are presumably a consequence of the idealizations made by Reichardt in order to permit closed-form integration. The subsequent calculations of Nu by Yu et al.10 for both uniform and isothermal heating of a round tube also conform closely to eq 10. It can be noted that these comparisons are independent of the dependence of Prt on Pr or, if significant, on Re. The appropriate expressions for Nu1 and Nu∞ are discussed in detail by Churchill et al.,7 as well as by Heng et al.,8 Danov et al.,9 and Yu et al.,10 and therefore will not be described or presented herein. Equation 10 can be rearranged as
Nu - Nu1 ) Nu∞ - Nu1
1 Prt Nu∞ 1+ Pr - Prt Nu1
(
)
(11)
Comparison with eq 9 indicates that eq 11, and thereby eq 10, has the exact form of eq 9 with y0 ) Nu1, y∞ ) Nu∞, and yi ) Nui ) [(Pr - Prt)/Prt](Nu1/Nu∞)(Nu∞ Nu1). It is doubtful that this expression for Nui, the effective intermediate asymptote, would have ever been identified in the absence of eq 10. Because Prt is almost invariant for Pr > Prt, the term (Pr - Prt)/Pr has the effect of stretching the range of the independent variable downward to zero at Pr ) Prt and thereby allowing Nu1, a discrete intermediate value of the dependent variable, to function as an asymptotic value. In the entire history of analyses of this primary problem in convection, the existence of a regime of transition has apparently never been mentioned, let alone the concept of stretching the independent variable. Equations 10 and 11 are obviously valid only for Pr g Prt. In the absence of even an approximate closedform solution for Nu for Pr e Prt, Churchill et al.7 attempted to correlate the computed values of Nu for Pr e Prt using the analogue of eq 10, namely
(
)
( )
Pr 1 Pr 1 1 + ) 1Nu Prt Nu0 Prt Nu1
(12)
which can be rearranged as
Nu - Nu0 ) Nu1 - Nu0
1 Prt - Pr Nu1 1+ Pr Nu0
(
)
(13)
Here, Nu0 is the known theoretical limiting value of Nu for Pr ) 0. Equation 13 proved successful functionally but not quantitatively. Hence, Churchill et al.
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incorporated an arbitrary coefficient R to obtain
Nu - Nu0 ) Nu1 - Nu0
1 Prt - Pr Nu1 1+R Pr Nu0
(
)
(14)
They next determined a functional expression for R without introducing any empiricism simply by equating the derivatives of Nu from eqs 11 and 14 at Pr ) Prt, a necessary condition. The result is
Nu - Nu0 ) Nu1 - Nu0
1 Prt - Pr Nu1 Nu1∞ - Nu1 1+ Pr Nu1 Nu1 - Nu0
(
)( )( ∞
)
(15)
where Nu1∞ is the asymptotic expression for Pr f ∞, evaluated at Pr ) Prt. Equation 15, which is free of any explicit empiricism, provides almost as good a representation for the computed values of Heng et al., Danov et al., and Yu et al. for Pr e Prt as does eq 11 for Pr g Prt. Again, it is doubtful that the functional form of the effective internal asymptote for 0 < Pr < Prt, namely, Nui ) Pr/(Prt - Pr)(Nu1∞/Nu1)(Nu1 - Nu0)2/(Nu1∞ Nu1), would ever have been identified in the absence of eqs 9, 10, and 12. Conclusions The two examples demonstrate the utility of complete solutions, either exact or approximate, in the development and interpretation of a correlating equation encompassing a distinct transitional regime, as characterized by a point of inflection. Comparison of each of these two solutions with the generalized correlating equation proposed by Churchill and Usagi1 for three arbitrary regimes reveals that the value of one of the two arbitrary exponents is unity and the other, negative unity, and also permits identification of the complete functionality of the intermediate asymptote. This equivalence previously escaped identification because it is directly apparent only in one particular arrangement and reduced case of the general correlating equation for three regimes and in the corresponding arrangement of the exact solution of Hickman and the approximate solution of Reichardt. The first example, based on an exact solution, implies that a point of inflection and an internal asymptote must exist whenever limiting values, rather than functional asymptotic expressions, are approached for both large and small values of the independent variable. An internal asymptote might or might not exist when one or both of the asymptotes are functions rather than fixed values. The second example, based on an approximate solution, required two separate but continuous correlating equations, one extending from a particular intermediate value of the independent variable to infinity and the other from that intermediate value to zero. This particular intermediate value is characterized by the vanishing of an explicit dependence of the dependent variable on the independent variable. The analysis herein reveals that, with such correlating equations, the effective independent variable for large x is (x - x1)/x1 and for small x is x/(x1 - x), where x1 is the particular value of the independent variable x mentioned above, thus extending the range from zero to infinity in both
instances, as well as converting the special intermediate value of the dependent variable to an asymptotic value. In Example 2, x1 is approximately constant for x g x1 but varies strongly with x for x < x1. Equations 11 and 15 combine to constitute a unique representation. They predict Nu for fully developed turbulent convection for a complete range of Pr and Re, presumably for all channels and all modes of heating. They invoke no explicit empiricism. They each predict a transition with a point of inflection, neither of which had ever before been identified. Expressions for Nu0, Nu1, Nu∞, Nu1∞, and Prt did not need to be specified in the context of this development because eqs 11 and 15 are independent of such details. Representative values and expressions for these five parameters for round tubes can be found in Heng et al.8 and Yu et al.10 and for parallel plates in Danov et al.9 The common expression for Nu∞ for all of these cases was included herein only because the expression given by Reichardt was in serious functional error. Correlating equations with the generic form of eq 9 might be applicable in many situations in which a closed-form solution is not known. The critical requirement is that n ) (1 and m ) -1 provide an adequate approximations for interpolation between the limiting asymptotic and internal values of the dependent variable. In that event, the internal asymptote can be determined by plotting experimental or computed values of ln(y - y0) or ln(y∞ - y) versus ln(x), ln[x/(x1 - x)], or ln[(x - x1)/x1], as appropriate, although the complete functional dependence of the internal asymrptote on y0, y∞, y1, etc. might not be evident. Such determinations for the effective viscosity of pseudoplastic and dilatant fluids were illustrated by Churchill and Churchill.4 Acknowledgment Dedication. I was privileged as a visiting Professor to participate first-hand in the recruitment of J. Larry Duda to the faculty at Pennsylvania State University. Although our research has seldom overlapped, we have remained close professional friends, and I served for many years, during his Headship of the Chemical Engineering Department, as a member of the Industrial and Professional Advisory Committee. Accordingly, I am pleased to be able to include this paper in the special issue of Industrial and Engineering Chemistry Research in recognition of his receiving the E. V. Murphree Award. As a historical footnote, the correct expression for Nu∞ in eq 10 is based on a functional expression for the turbulent shear stress first derived by E. V. Murphree (Ind. Eng. Chem. 1932, 24, 726). Nomenclature Bi ) UeD/k, Biot number for a round tube D ) internal diameter of tube, m f ) Fanning friction factor h ) heat transfer coefficient for fluid, W/(m2 K) k ) thermal conductivity of fluid, W/(m K) m ) arbitrary exponent in eq 2 n ) arbitrary exponent in eqs 1 and 2 Nu ) hD/k, Nusselt number Nui ) internal asymptote NuJ ) Nusselt number for uniform heating NuT ) Nusselt number for uniform wall temperature Nu0 ) limiting Nusselt number for Pr ) 0 Nu1 ) Nusselt number for Pr ) Prt
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3057 Nu∞ ) 0.07343(Pr/Prt)1/3Re(f/2)1/2, asymptotic Nusselt number for Pr f ∞ Nu1∞ ) Nu∞{Pr ) Prt} ) 0.07343Re(f/2)1/2 Pr ) Prandtl number of fluid Prt ) turbulent Prandtl number Re ) Reynolds number Ue ) equivalent overall heat transfer coefficient corresponding to external resistances, W/(m2 K) x ) arbitrary independent variable x1 ) value of x for intermediate value of y{x} independent from x y{x} ) arbitrary function of x yi{x} ) internal asymptote y1 ) y{x ) x1} y0{x} ) asymptote for x f 0 y∞{x} ) asymptote for x f 0 Greek Symbols R ) arbitrary coefficient in eq 14
Literature Cited (1) Churchill, S. W.; Usagi, R. A General Expression for the Correlation of Rates of Transfer and Other Phenomena. AIChE J. 1972, 18, 1121. (2) Churchill, S. W. Comprehensive Correlating Equations for Heat, Mass, and Momentum Transfer in Fully Developed Flow in Smooth Tubes. Ind. Eng. Chem. Fundam. 1977, 16, 109. (3) Churchill, S. W. A Theoretical Structure and Correlating Equation for the Motion of Single Bubbles. Chem. Eng. Process. 1989, 26, 269.
(4) Churchill, S. W.; Churchill, R. U. A General Model for the Effective Viscosity of Pseudoplastic and Dilatant Fluids. Rheol. Acta 1975, 14, 404. (5) Hickman, H. J. An Asymptotic Study of the Nusselt-Graetz Problem. Part 1: Large x Behavior. Trans. ASME, J. Heat Transfer 1974, 96C, 354. (6) Reichardt, H. Die Grundlagen des Turbulenten Wa¨rmeu¨berganges. Archiv. Wa¨ rmetechnik. 1951, 6/7, 129; English translation: The Principles of Turbulent Heat Transfer; Report TM 1408; National Advisory Committee for Aeronautics: Washington, D. C., Sept 1957. (7) Churchill, S. W.; Shinoda, M.; Arai, N. A New Concept of Correlation for Turbulent Convection. Therm. Sci. Eng. 2000, 8 (4), 49. (8) Heng, L.; Chan, C.; Churchill, S. W. Essentially Exact Characteristics of Turbulent Convection in a Round Tube. Chem. Eng. J. 1998, 71, 163. (9) Danov, S. L.; Arai, N.; Churchill, S. W. Exact Formulations and Nearly Exact Solutions for Convection in Turbulent Flow Between Parallel Plates. Int. J. Heat Transfer 2000, 43, 2767. (10) Yu, Bo; Ozoe, H.; Churchill, S. W. The Characteristics of Fully Developed Convection in a Round Tube. Chem. Eng. Sci. 2001, 56, 1. (11) Churchill, S. W.; Usagi, R. Standardized Procedure for the Production of Correlations in the Form of a Common Empirical Equation. Ind. Eng. Chem. Fundam. 1974, 13, 39.
Received for review February 14, 2000 Revised manuscript received August 11, 2000 Accepted August 11, 2000 IE000281J
