Ind. Eng. Chem. Res. 1992,31,643-658 Reinhold New York, 1991; Chapter 5; pp 78-85. Kabel, R. L. Rates. Chem. Eng. Commun. 1980,9, 15-17. LaRoche, R. D. Personal communication, 1990. Perti, D.; Kabel, R. L. Kinetics of CO Oxidation over Co3O4/yAl2O3 AZChE J. 1985,31, 1420-1440. Pfefferle, L. D. Stability, Ignition, and Pollutant Formation in a Plug Flow Thermally Stabilized Burner. Ph.D. Thesis, University of Pennsylvania, Philadelphia, 1984. Weekman, V. W. Lumps, Models, and Kinetics in Practice. AZChE Monogr. Ser. 1979, 75 ( l l ) ,3-29. Wilhelmy, L. Ueber das Gesetz, nach welchem die Einwirkung der Siuren auf den Rohrzucker stattfindet. Ann. Phys. Chem. 1850, 81,413-428.
643
Willems, P. A.; Froment, G. F. Kinetic Modeling of the Thermal Cracking of Hydrocarbons. 1. Calculation of Frequency Factors. Znd. Eng. Chem. Res. 1988a, 27, 1959-1966. Willems, P. A.; Froment, G. F. Kinetic Modeling of the Thermal Cracking of Hydrocarbons. 2. Calculation of Activation Energies. Znd. Eng. Chem. Res. 1988b, 27, 1966-1971. Zitney, S . E.; LaRoche, R. D.; Eades, R. A. Chemical Process Engineering on Cray Research Supercomputers. CACHE News 1990, 31, 19-24. Received for review January 28, 1991 Revised manuscript received September 10, 1991 Accepted September 23, 1991
The Role of Analysis in the Rate Processes Stuart W . Churchill Department of Chemical Engineering, university of Pennsylvania, 311A Towne Building, 220 South 33rd Street, Philadelphia, Pennsylvania 19104-6393
Numerical solutions are now possible for a much wider range of conditions than analytical (continuous) ones and are gradually supplanting them in our literature. The results obtained from numerical solutions as well as from analytical solutions in the form of series, integrals, or tabulated functions are more precise, regular, and coherent than experimental data but are always subject to some uncertainty owing to idealizations made in developing the model itself. Both types of solutions fail to provide a functional structure for correlation, thus giving rise to the tabulations, graphical correlations, and purely empirical equations to be found in our handbooks. On the other hand, asymptotic solutions which do provide such a functional structure can generally be derived not only for extreme conditions but for intermediate regimes as well. Comprehensive correlating equations of great accuracy can often be constructed from two or more asymptotic solutions with almost no empiricism. Examples are given of the determination of asymptotes by reduction of more general solutions, by direct derivation, and by dimensional and speculative analysis, and also of their combination. Examples from the existing literature are given of invalid or inapplicable asymptotes and combinations thereof as well.
A glance at the archival journals of our profession, such as this one, reveals that analysis, both continuous and discrete, has become dominant relative to experimentation. The results of these analyses appear, however, to have a fairly limited role in practice. The objective of this paper is to examine the role of analysis in the rate processes, both in general and by examples, and to suggest how theoretical methods and results might be used more effectively for correlation, prediction, and understanding. A large fraction of the efforts of the famous applied mathematicians of the period from roughly 1700 to 1950 were directed toward developing models and analytical (continuous) solutions for what are now classified as the rate processes-momentum, heat and mass transfer, and chemical conversions. This work has since been continued by engineering scientists,but in the past forty years it has been accompanied and gradually supplanted by numerical (discrete) methods based on finite-difference, finite-element, and stochastic models and approximations. The transition from continuous to discrete methods has of course been motivated by a desire to solve more complex models. It has been made possible by the continuing development and increasing availability of computer hardware and software. On the other hand, the literature of engineering practice in the rate processes, as epitomized by our standard handbooks and even some of our textbooks, is still largely based on experimental data. It consists primarily of dimensionless coefficients of transfer with wide bands of
scatter and of empirical equations representing straight lines drawn through the data in these plots. Such graphical representations and correlating equations have been relatively uninfluenced by theory beyond simple dimensional analysis for the primary reason that most theoretical solutions do not provide any guidance with respect to the functional behavior. This is just as true of analytical solutions in the form of complex functions, infinite series, or integrals as it is of numerical (discrete) solutions. Asymptotic solutions, including those that are evident from general solutions, are often an exception, but their utility for correlation has yet to be fully recognized or fully exploited. Experimental data are recognized as being subject to uncertainty and imprecision arising from incomplete definition and control of the environment in which the experiments are conducted, as well as from errors of measurement. Analytical solutions, both continuous and discrete, are subject to uncertainty arising primarily from simplifications and idealizations incorporated in the model but sometimes also to the process of solution. Asymptotic solutions are particularly uncertain with respect to their range of validity, if any. In addition, even nominally valid solutions are often misinterpreted or misapplied. Misuse of theoretical results in correlations is more harmful than their neglect since the error then becomes imbedded in our handbooks, textbooks, and dogmas. Particular attention is given herein, by way of examples, to questionable solutions and to misapplications of valid ones, not only for their
o a a a - ~ a a ~ ~ ~ ~ / ~ ~ ~ i -0o1992 ~ ~ American ~ ~ o ~ . Chemical o o / o Society
644 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
exposure but also for guidance in deriving improved solutions and correlations.
Elements of Correlation The elements which are required to construct a theoretically sound correlation are described briefly in this section. Attention is confined to the general characteristics which influence their role. Specific, detailed examples are given in a subsequent section. Experimentation and Experimental Results. Although this paper is concerned with the role of analysis, a few remarks on the role of experimentation are appropriate. One role is to reveal the fundamental and elemental behavior of the physical world, i.e., to guide the development of the individual terms in our theoretical models. The proportionality of the rate of heat transfer by molecular motion to the gradient of the temperature, as expressed by Fourier’s law, is an example. Such behavior is usually not rationalized on a mechanistic (in this case molecular) basis until long after its discovery by observation. In the future, just as in the past, most new or improved terms in our models can be expected to come from experimental observationsrather than from analyses. A second role for experimentation, now challenged by numerical solutions, is to provide the raw material for the construction of correlations. An extension of this role is the support of the development of empirical but elemental correlations, such as for the eddy kinematic viscosity, which can be utilized to develop semitheoretical solutions extending far beyond the range of the experiments. A third and very important role is to define the range of validity, if any, of theoretical solutions and theoretically based correlations. Experimental data often have insufficient precision and/or range to justify evaluation of all the empirical coefficients in a model proposed for their representation. This difficulty is pervasive but not always acknowledged, as for example in experimental studies of catalytic packed-bed reactors [see White and Churchill (1959) and Churchill (1979), Chapter lo]. Imprecision may also preclude discrimination between two or more proposed models. The imprecision of experimental data as well as their incoherence is often disguised by the use of log-log plots, etc. [again, see Churchill (1979), Chapter 101. Even if they are very precise, experimental data may fail, because of the impact of unrecognized or unaccounted effects, to test a model with which they are compared. Variations in the physical properties with temperature, the presence of surfactants, the misalignment of “parallel” plates, and multiplicity are examples of such effects. Similarly, the bands of experimental data in many empirical correlations are not necessarily the result of imprecise measurements. More often the scatter arises from incoherence, i.e., from unrecognized,unidentified, and/or uncontrolled differences in the environment and conditions of the experiments. For example, plots of experimental heat-transfer coefficients in the current literature often represent sets of data for different boundary conditions such as uniform wall temperature and uniform heat flux, as well as for different lengths for thermal development, without such identification. Experimentation is today avoided increasingly because it is expensive and tedious, and, particularly in academia, it is not only more difficult, and handicapped by inadequate facilities, but also less productive in terms of publications than analysis. Simple experiments, such as a photograph of streaklines, may however provide a precise and sufficient test of theoretical results, including dis-
crimination between two contradictorytheoretical d y s e a . Unless experimental results of sufficient reliability to provide a critical test are already available, combined and coordinated experimentation should accompany analysis. Otherwise the analytical results are apt to remain untested and therefore speculative. Complete Analytical Solutions. As mentioned above, analytical (continuous) solutions with some generality usually require extensive computations to obtain numerical values. Such values are analogous to experimental data, with the advantage of precision, regularity, and coherence but the disadvantage of uncertainty arising from idealizations and simplifications, known or unknown, in the model. For example, the postulate of a steady motion precludes turbulence as well as the shedding of eddies. Numerical values obtained from continuous solutions are also analogous to those obtained by numerical methods. Analytical solutions have the possible advantage with respect to both experimental data and numerical solutions of providing functional insight. However, such functional information arises only from the possible groupings of variables (which may be evident from the model itself in advance of the process of solution, as discussed below in terms of dimensional analysis) and from the limiting behavior (which is discussed below in terms of asymptotic solutions). The classical techniques of the past, such as separation of variables, expansion in Fourier series, conformal mapping, and the Laplace transform, are generally applicable only for linear or quasi-linear differential equations with simple boundary and initial conditions, thereby eliminating many important aspects of behavior such as turbulence, radiative transfer, variable physical properties, and nonfirst-order reactions. Numerical Solutions. Numerical (discrete) methods are applicable to a more general class of models than analytical methods but sometimes pose problems of stability, stiffness, convergence, uniqueness, and cost. They have gained favor over analytical methods as the speed and capacity of computers have increased and the coat of their use has decreased. The principal source of uncertainty of a numerical solution arises from the unknown validity of the model, which generally can be evaluated only by comparing the solution with experimental data. Overall comparisons will generally not constitute a critical test of a complex model. For example, agreement of computed and measured values of the NO produced by a flame does not assure that the fluid-mechanical, thermal, and chemicalkinetic elements of the model were correct in every reapect; the influence of some of these factors may be negligible, or two or more errors in the modeling may compensate to a degree. Cost will presumably remain a restriction as the complexity of models arises in response to improvements in computers and software. Many aspects of the rate processes remain beyond the capabilities of present-day numerical methods and computers. Asymptotic Solutions. Solutions in closed form can usually be derived for asymptotically limiting behavior and as such have unique value in that two or more of them can often be combined using the model of Churchill and Usagi (1972) to construct correlating equations of great accuracy and generality. This latter model is described and its applications are illustrated subsequently. Asymptotic solutions may also serve as a test for experimental values or computed values in the form of bounds. A sometimes overlooked role of asymptotic solutions is the identification of groupings of dimensionless variables beyond those apparent from dimensional analysis.
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 645 Asymptotic solutions can be obtained by a variety of techniques, including reduction of a more general solution, direct derivation from a reduced model, approximate methods of solution, perturbation methods, and empirical observations. Often two techniques, such as separation of variables and the Laplace transform, yield solutions which converge rapidly for the opposing extreme values of the independent variable. Asymptotes obtained by each of these methods are included in the examples below. Approximate operational methods can be used to determine some asymptotes which might not otherwise be apparent [see Churchill (1957)l. The simplificationswhich lead to asymptotic models and solutions are usually arbitrary and often reflect considerable ingenuity. The resulting solutions are accordingly speculative and subjective to confirmation. Approximate Methods of Solution. Apart from simplifications and/or approximations in the basic model, some approximate methods of solution introduce further uncertainty. One example is the method of weighted residuals [see, for example, Finlayson (1972)l. Under some particular formulations of this method, convergence towmd the exact solution may occur if a sufficient number of terms can be evaluated. The integral-boundary-layer method, which was devised heuristically by von Kirmiin and Pohlhausen, is now recognized as a first-order form of the method of weighted residuals. This particular formulation, which requires the postulate of both an arbitrary function and arbitrary boundary conditions, no longer has any justification for processes such as thermal conduction, phase change, laminar flow, and laminar convection in view of the alternative of numerical methods, but it may still be viable for those turbulent flows and turbulent processes for which numerical solutions are not yet feasible. Dimensional, Asymptotic, and Speculative Analysis. If the variables describing a rate process are identifed or postulated, the classical techniques of dimensional analysis can be employed to determine the minimal set of dimensionless groupings that are required to describe the behavior. The number of such groups can be reduced only by the specification of additional constraints, but the groupings can be rearranged arbitrarily by combination, for example, to limit a particular variable to one grouping, while maintaining linear independence. Variables can be eliminated on asymptotic grounds, for example the viscosity for the limiting case of completely inertial flow, either before or after the dimensional analysis itself. Other variables can be eliminated speculatively one or more at a time, even without a physical rationalization. The reduced relationships obtained on asymptotic and speculative grounds must obviously be tested to determine their range of validity, if any, by comparison with experimental data. Dimensional analysis is the only theoretical resort in the absence of a model. The source of uncertainty in dimensional analysis is in the selection of variables. A redundant variable will generally be revealed in the process of correlation and probably would, in any event, be among those eliminated on asymptotic or speculative grounds. The omission of a variable from the initial listing is more serious and, in a graphical correlation,may result in scatter which is difficult to distinguish from experimental error. Dimensional analysis may also be applied to a set of algebraic, differential, and integral equations, and their boundary and initial conditions. Since the model constrains the behavior, some reduction in the number of dimensionless groups may occur relative to that obtained from a list of variables. The method of Hellums and
Churchill (1964) determines the minimal set of dimensionless variables and parameters required by the model in its chosen form, including possible combinations of independent variables that can be used to reduce the order of the model, for example from a partial to an ordinary differentialequation. Asymptotic and speculative analysis can be applied to the model by omitting terms or to the minimal set of derived dimensionless groups by omitting variables, just as with a listing. Dimensional analysis of a model is the only resort other than simplification when a model cannot be solved, as for example with most turbulent flows. It should be applied in order to simplify the process of solution even when the model can be solved. Dimensional analysis of a model often produces the complete solution for a quantity such as the Nusselt number less only the leading numerical coefficient. The uncertainty associated with dimensional analysis of a model arises wholly from the model itself. Redundant terms in the model may prevent the identification of a similarity transformation, but their impact will be apparent from asymptotic and speculative deletions. Omission of a term from the model may invalidate the results completely and may be apparent only from the failure of the groupings to correlate experimental data. Asymptotic and speculative analyses have often been neglected or applied incompletely, both with lists of variables and with models, perhaps because of the uncertainty of the results, but, as illustrated below, these two techniques often provide unique insight as well as critical assistance in correlation. Models and Their Simplification. In their most general form the models for the rate processes usually consist of a set of partial differential equations which describe the conservation of mass,momentum, energy, and components. These expressions equate the rate of accumulation of the several quantities to phenomenological terms for convection, molecular diffusion, generation, dissipation, etc. Integral terms representing, for example, radiative transfer, and algebraic equations representing, for example, an equation of state, may be included in the model. Solution of such a model in its general form is not practical even if it were possible. The task for engineers is to eliminate those terms which are negligible for a particular process and set of conditions, and, if appropriate, to approximate some of the remaining ones. Such reductions should be considered to be speculative. For example, the postulate of a steady state for flow or convection, and hence the elimination of the transient terms, may lead to false or at least nonunique solutions owing to the physical manifestation of turbulent fluctuations or the periodic shedding of vortices for the chosen conditions. Similarly, the postulate of a mean rectilinear fluid motion, and thereby the elimination of the cross-sectional components of the velocity and the cross-sectional variation in pressure, is invalid for conditions under which a cross-motion or a secondary circulation occurs physically (e.g., developing flow, flow in curved channels, and turbulent flow in two-dimensional channels). If fluid motion is involved, as it is in most industrially important rate processea, the Eulerian form of the balancea (for a fixed elemental volume in space) is easier to solve than the Lagrangian form (for a moving elemental mass). This is particularly so if diffusive processes are significant. The most effective single idealization for the equation of conservation is the postulate of invariant physical properties, which decouples the balances for mass and momentum from the others. This postulate is of course
646 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
invalid even as an approximation for processes, such as natural convection, that depend on the coupling. Even when decoupled, the momentum balance is the most difficult one to solve owing to its vectorial character and the nonlinearity of the inertial terms. Some of the idealizations and limiting cases that have been used to obtain solutions of the coupled balances for mass and momentum are (1) plug flow, which does not occur physically with fluids but sometimes serves as a valid bounding case, (2) perfectly mixed flow, which also sometimes serves as a useful limiting case, (3) rectilinear flow, which is a valid postulate for some conditions, including, for example, steady, fully developed, laminar flow in straight channels, (4) purely viscous flow, which occurs as laminar flow in straight channels at low velocities and as creeping flow over some but not all objects at low velocities, (5) slightly inertial flow, which Oseen and others have modeled by linearizing the inertial terms, (6) purely inertial (inviscid) flow, which is an asymptotic approximation for the motion over an object at some distance from the surface but is not attained even asymptotically inside channels, (7) free-streamline flow, which is a good approximation for a fluid moving at high velocity next to a nearly stagnant region of fluid, such as in flow beyond an orifice, (8) thin-boundary-layer flow, which is a good approximation near surfaces at an intermediate range of velocities, (9) free-molecular flow, which occurs at a sufficiently low pressure such that the mean free path of the molecules is of the order of the dimensions of the confinement, (10) W y developed flow (independent from length), which is approached far downstream from the inlet of a channel, (11)pseudo-steady flows, such as with the periodic shedding of vortices behind a cylinder, and (12) rippling motion, such as occurs with thin falling films and with large bubbles. Classical functions and techniques that have been used to reduce the mass and momentum balances under these conditions include (1)the stream function in two-dimensional flows, (2) the vector potential in three-dimensional flows, (3) the vorticity, (4) the velocity potential in inviscid flows, ( 5 ) the Gortler transformation, which generalizes with respect to geometry and improves the convergence of solutions for thin-laminar-boundary-layerflows, (6) the Meksyn transformation and the Merk expansion, which flow over extend solutions for thin-laminar-boundary-layer a wedge to other geometries, (7) the von Mises transformation, which converts the model for thin-laminarboundary-layer flows to the classical equation for transient thermal conduction with a variable diffusivity, and (8)time averaging for turbulent flows, which greatly reduces the momentum -balance at the expense of generating terms such as u'~u)which must be approximated empirically or solved for by using additional, arbitrary equations of conservation. A detailed discussion of these functions and their application, including references, is given by Churchill (1988). The energy and component balances are somewhat similar in form to that for the component of momentum in the direction of flow, suggesting analogies, that is, simply related solutions. Some of the special regimes and techniques for momentum transfer also have analogues for energy and components. These include (1)thin boundary layers in temperature and composition, (2) fully developed convection in channels such that the dimensionless temperature and concentration, when properly scaled, as well as the heat-transfer and mass-transfer coefficienb, do not change significantly with length, (3) equilibrium stages, which are analogous to perfectly mixed flows, and (4) the Saville-Churchill(1967)transformation for free convection
(analogous to the Gortler transformation). Lumped-parameter models are widely used to correlate experimental data and to summarize the results of solutions. An example is the heat-transfer Coefficient. Newton (1701) recognized that the heat-flux density was proportional to the difference in temperature between a surface and the bulk of fluid flowing over it and thereby formulated implicitly the concept of the heat-transfer coefficient. He also (1713) formulated implicitly the dimensionless drag coefficient by noting that the resistance of a sphere to the force of gravity was proportional to the density of the fluid, the square of the diameter, and the square of the relative velocity. Both of these concepts of Newton have since been rationalized on theoretical grounds, the heattransfer coefficient on the basis that the energy balance for the fluid is linear in temperature and the constancy of the drag coefficient as the asymptotic behavior for purely inertial flow. The refinement and generalization of these ideas, namely, the expression of a rate in terms of a coefficient times a potential difference,the expression of a rate in dimensionless form, and the utilization of asymptotic behavior as a basis of comparison, are the modern cornerstones for correlation. Incredibly, Adiutori (1989) rejects these particular concepts as counterproductive. Several approximate models which have proven useful in modeling and correlation are (1) h, = ~ u T ~ , 3 /the 4, effective heat-transfer coefficient for radiation, which provides a pseudolinearized representation, (2) the power law for the shear stress of non-Newtonian fluids, and (3) the eddy diffusivity, which represents turbulent transport as a diffusional process. Correlation in Terms of Asymptotic and Speculative Expressions. As contrasted with solutions for extended conditions, a closed form is usually possible for an asymptotic solution, at least as a limit. When an asymptotic solution cannot be derived, as with turbulent flows, an empirical asymptote can usually be devised on the basis of dimensional analysis and experimental data. Correlating equations of great accuracy and generality can usually be constructed from such asymptotic expressions using the model of Churchill and Usagi (1972, 1974): (Ylxl)" = bo{xD" + brnlxl)" (1) where yo(xJis an asymptote for small x , y,(x) is an asymptote for large x , and n is an arbitrary exponent. It is convenient to rewrite eq 1 in the canonical form r=1+xn (2) where Y = y(xl/yo(xland X = ymlx)/yo{xl,or Y = ylxl/ym{x1 and X = yolxI/ymlxl. The behavior provided by eq 1is quite insensitiveto the choice of n. Hence an integer or a ratio of integers can ordinarily be chosen to represent experimental values or computed values with sufficient accuracy. The two asymptotes must have a single intersection and both be upper or lower bounds. If not, a successful correlating equation may possibly be constructed by linking them through an intermediate asymptote. The chosen asymptotes should be free of singularities. A singularity can sometimes be removed by redefinition of the dependent variable, by simplification or modification of the asymptote, or by the use of an intermediate asymptote. In some cases a singularity may be outside the range or practical interest and therefore tolerable. The direct extension of eq 1 for three asymptotes takes either the form ym = (yon
or
+ y,")"/" + yrnm
(3)
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 647 ym = yom
+ (yi" + y,")"/n
(4)
where here the functional notation has been deleted for simplicity. Unless y,,, ye and y , are all three upper bounds or all three lower bounds,slightly anomalous behavior will occur in one of the limits of botli eqs 3 and 4. For example, if yo is a lower bound and y - an upper bound, eq 3 gives for x 0 values of y less thany,,. This anomaly is avoided with the forms
-
cy" - y0")m = yinm + (y," - yon)"
+y
y
(6)
Equations 5 and 6 are more complicated than eqs 3 and 4,but they only involve the same three functions and the same two exponents. Equations 3 and 4 may differ in their tolerance of a singularity in yo or y,; otherwise they are apt to be equally effective. This is also true of eqs 5 and 6.
Examples of the Constructive Role of Analysis The role of analysis in the rate processes, as described above, is best illustrated with specific examples of the various elements and their combination. Since many different methods of analysis are often applied to a particular process in either an alternative or a complementary sense,these examples will herein be organized by processes rather than by techniques. Fully Developed Turbulent Flow in a Round Pipe. A fairly complete structure has been developed for fully developed turbulent flow in a round pipe by dimensional, asymptotic, and speculative analysis alone. Conventional, dimensional analysis of a list of the indicated variables yields a grouping such as
u(
5)"'
;( );
= f( Y
7,
l/'
,;I Y
( : + l ) ! $ In so far as eq 8A is valid,
(3 +
= fi+)
;)1
(5)
and
cy" - y,")" = ( y o n - y,")"
The eddy kinematic viscosity defined by Boussinesq (1877)can be expressed in general as
(7)
Also, the speculation of a finite value of ut at the center line requires that in the region near the center line u,' - U+ = E ( ~ / u ) ~ (15) and therefore that Vt
a+
Y
- 2E
-E-
If eq 10 is postulated to be a reasonable approximation for the entire cross-section, integration gives u: = A - 3/2B + B In (a+) (17) which can be reexpressed as f-l/'
=A
3 - ;B
+ B In
-
(18)
The same process can be applied to rough pipes, for which the final result corresponding to eq 18 is
Equations 18 and 19 can be combined per the method of Churchill and Usagi with an empirically based exponent n = -1 to obtain the following expression for all ReD and d1 roughness: f-l/'
The speculation that the velocity distribution near the wall is independent of a reduces eq 7 to
3 = A - -B
2
+ B In
where G is a constant whose value follows from A, B, and
C. or in conventional symbolic terms as u+ = fi+)
(8A)
Speculation that the velocity distribution near the center line is independent of the viscosity similarly leads to
u,' - u+ = f( The further speculation of a regime in which both eqs 8A and 9 are applicable requires, as the only function which satisfies both, u+ = A
+ B In b+]
(10)
The asymptotic analysis of the time-averaged equations of motion for the region near the wall yields u+ = y+ - (Y(y+)4 - B ( y + ) S + ... (11) which reduces for y+
-
0 to y+
(12) Equation 12 can of course be derived directly by asymptotic reasoning. u+
All of the above expressions have proven successful for correlation, although some question remains as to whether or not (Y in eq 11 is finite. Some of the functional dependences, such as in eqs 15 and 18, which have been confirmed experimentally, might never have been identified on the basis of somewhat imprecise data without the above structure based on dimensional, asymptotic, and speculative analysis. Rothfus et al. (1950)speculated on the basis of laminar flow that the velocity distribution and friction factor in fully developed turbulent flow through a round pipe are related through u+ = u; (21) and 8u+y+ = ReD
(22) This speculation is confiied by the empirical coefficients fitted to expressions of the form of eqs 10 and 18. Their analogy may be applied to eq 20 to obtain a unique, speculative expression for the velocity distribution for all ReD and e. Rothfus and Monrad (1955)observed that in laminar flow in a round tube the velocity distribution in terms of u+ is the same function of y+ both in a round tube and
10
--
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I
I
,
,
I
,
/
, , /I ', ,
Y
,
-
I
,
Figure 1. Experimental c o n f i i t i o n of the analogy of MacLeod by Whan and Rothfus (1959). Reprinted with permission from Whan and hthfus (1959).Copyright 1959 American Institute of Chemical Engineers.
between parallel plates if a+ = b+. They attribute to MacLeod (1951) the speculation that this analogy is applicable to turbulent flow as well. A composite plot of experimental data for u,' versus Q+ for round tubes and versus b+ for parallel plates by Whan and Rothfus (1959) provides the remarkable confirmation shown in Figure 1. Application of this analogy to all of the above expressions for round tubes yields their quantitative counterpart for parallel plates [see, for example, Churchill (1991), Chapter 101. Fully Developed Turbulent Convection in a Round Tube. The postulate of fully developed convection implies that (T,- T)/(T,- T,) is independent of length. Dimensional analysis with 7, rather than u, chosen as an independent variable yields
from which it follows that
The speculation of independence from Q then yields
analysis to identify six regimes of functional behavior. The existence of each of these regimes is validated by asymptotic solutions for creeping flow, sub-laminar-boundarylayer flow over a sphere, thin-laminar-boundary-layerflow over a sphere, thin-laminar-boundary-layer flow over a slightly deformed sphere, purely inertial flow, and wavelike motion. The method of Churchill and Usagi was used to develop overall correlating equations for the velocity and the drag coefficient incorporating all of these elements. The failure of prior empirical correlating equations to conform to asymptotic solutions demonstrates beyond question that behavior of this complexity cannot be identified from experimental data without the assistance of analysis. Laminar Flow and Forced Convection in a Helical Coil. Dean (1927, 1928) developed a solution for fully developed laminar flow in a torus (a helical coil of negligible pitch) for the asymptotic case of small curvature (a/ro 0) as a perturbation on the solution for Poiseuille flow through a straight tube. His solution has little practical value since it is limited to conditions for wk ch the deviation of the friction factor from that for a str ight pipe is less than 2%. However, the grouping ( ~ Q U ~ / Y ) ( Q / ~ ~ ) ~ / ~ , now called the Dean number, which was identified by the solution, has proven to characterize the behavior far beyond the range of convergence of the solution itself. Analysis for coils of finite pitch is impeded by the lack of an orthogonal set of coordinates. In compensation, Truesdell and Adler (1970) speculated that the effect of pitch might be approximated by substituting the radius ) ~ the ] , radius of the of curvature, r, = ro[l + ( p / 2 ~ r ~for coil r,. This substitution has proven successful in correlations for laminar flow by Manlapaz and Churchill (1980) and for turbulent flow by Mishra and Gupta (1979). Manlapaz and Churchill (1981) used the grouping (2au,/v)(~/r,)~/~ in asymptotic solutions and asymptotic correlations, together with the model of Churchill and Usagi, to develop the comprehensive correlation illustrated in Figure 2 for experimental data and computed values for laminarforced convection. Figure 2 indicates that log-log plots of Nu versus Re, Pr,a/ro,and a / r , for both uniform wall temperature and uniform heating without theoretical guidance could hardly be expected to produce a successful and general correlation for coils. Effective Thermal Conductivity of Dispersions. Maxwell (1873) derived the following solution for the effective thermal conductivity of a dilute dispersion of uniformly sized spheres _k - 2(1 - t) + a(1 + 2 4 (29) 2 + t + a(1 - e) k,
-
-
or
NuD = ReDf/2@?') (26) Independence from 1.1 as well reduces eq 26 to (27) NuD = A(ReD)(Pr)f/' which is presumed to be an asymptote for Pr 0. It is possible to reason from eq 11 with finite a that Nu = B ( R e ) ( P r ) 1 / 3 f / 2 (28) is the corresponding asymptote for Pr m. Equations 27 and 28 appear to be confirmed by the best experimental data [see, for example, Churchill (1977b)l. Motion of a Single Bubble. The motion of a bubble is complicated by the interaction and relative dominance of four forces-gravity, viscosity, inertia, and surface tension. Churchill (1989) [also see Churchill (19881, Chapter 171 used dimensional, asymptotic, and speculative
-
-
The ratio of the effective conductivity, k, to that of the continuous phase appears from eq 29 to be a function of two variables, the ratio a = k,/k, of the conductivitiesand e the volumetric fraction of the dispersed phase. However, rearrangement of eq 29 as -k= - 1 + 2 4 (30) k, 1-4 where 4 = e@ and @ = (a- l)/(a + 2) results in a single independent variable. The plot of the exact limiting solutions for @ = l, @ = 1/2, and e = T/S in Figure 3 indicates that eq 30 provides a good approximation for all but very dense dispersions but, more importantly, that the use of the combined variable 4 results in a minimal parametric dependence for all conditions, including the limiting ones of packed beds and foams. Churchill (19864 thereupon derived a correlating equation for all conditions by reex-
Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 649
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' 1 ,
-0.3 1
I
4
I
I
I
I
I
I
1
-0.2
-0.1
0
0.1
0.2
0.3
0.L
0.5
0.6
I
Figure 4. Comparison of the three-term solution with c = r / 6 and experimental data for the thermal conductivity of dispersions of uniformly sized spheres [from Churchill (1986a)I. Reprinted with permission from Churchill (1986a). Copyright 1986 Wiley Eastern Ltd.
pressing the extended solution of Zuzovsky and Brenner (1977) in terms of 4 and an arbitrary value of c = ~ / for 6 the small, residual parametric dependence. The success of that expression is demonstrated in Figure 4. In this example, the asymptotic solution of Maxwell revealed a useful grouping of variables not apparent from the more exact solutions. Laminar Film Condensation. Nusselt (1916) derived an expression for laminar condensation of a saturated vapor on an isothermal vertical plate by postulating (1) negligible heat capacity for the liquid, (2) negligible inertia
for the liquid, and (3) negligible drag due to the vapor. T h e solution implies invariant physical properties, negligible viscous dissipation, and nonrippling flow. He also derived an erroneous firsborder correction for the effect of the heat capacity, a correction for a fixed vapor velocity, and discussed the effect of noncondensibles semiquantitatively. He further asserted that his solution for a vertical plate was directly applicable for condensation inside and outaide vertical tubes and, by neglecting the effect of surface tension, modified his solution for condensation outside horizontal tubes. The predictions are approximately 15% too low for typical conditions owing primarily to rippling, but his basic solution has been proven to predict the
650 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
first-order effects of all of the variables included in the analysis. Churchill (1986b) developed solutions in the form of perturbations for the neglected effects, which are enumerated above, as well as for curvature. Emmons (1954) recognized that free convection, laminar film boiling, and laminar film melting were analogous to laminar film condensation and generalized the solution of Nusselt in that respect. Although the resulting correlation,which includes the turbulent regime as well, is only of first-order accuracy, it is noteworthy as perhaps the most far-reaching one ever constructed for rate processes. These several results are another illustration of the power of simple analysis in providing a structure for correlation. The ingenuity of Nusselt in devising a simple model and of Emmons in generalizing his solution should also be recognized as landmarks of analysis. Asymptotic Solutions for Convection to Spheres and Cylinders. Perhaps the most useful asymptote for any rate process is that obtained as the exact solution for steady thermal conduction from an isothermal sphere to infinite surroundings, namely, NuD = 2 (31) This result serves as an asymptote and a lower bound for both free and forced convection. It is of course adaptable for mass transfer as ShD = 2 (32) This limiting value of NuDand ShD is also directly applicable for heat and mass transfer, both separately and simultaneously, from small bubbles, droplets, and particles. As lower bounds, eqs 31 and 32 serve to identify erroneous measurements. An analogous limiting solution does not exist for an infiitely long cylinder. However, Langmuir (1912) derived the following asymptotic solution based on the concept of conduction across a fictive film: (33)
where NuDf is the solution for a flat plate. Also, Ohman (1970) derived the following solution for conduction from a cylinder of finite length to infinite surroundings:
These expressions serve as approximate counterparts of eq 31. Forced Convection in Plug Flow in an Isothermal Round Tube. Solutions for transient conduction in a cylinder are analogous to those for developing forced convection in plug flow in a tube if t is identified with :/urn. For an isothermal tube with negligible conduction in the direction of flow, the following asymptotic solution for small x: can be derived using the Laplace transform:
. .I
[ (y2+ x + (7 x3)1/2 +
(*X)'/2 1 - 4 -
where here X = kx/udca2. As X the limiting behavior
3
-
(35)
0, this solution has
-
NuD 2/ (*X)'I2 (36) If the effect of radial curvature is neglected, and T To as y is used as a boundary condition instead of aT,Jar
-
0)
= 0 at r = a, the following solution is obtained:
The term 4(X/7)'l2 in eqs 35 and 37 arises from defining in terms of T, - T, rather than T, - To,an effect which was lost in obtaining eq 36. The remaining higher order terms in eq 35 may be inferred to represent the effects of curvature and of the correct boundary condition. Equation 37 is more accurate than eq 36 for very small X, but eq 36 becomes more accurate as X increases, indicating that the omission of the factor 1 - ~ ( X / T ) 'compensates /~ fortuitously to some extent for the neglect of the effect of curvature. If the approximation of Langmuir, as described above, is extended for the effect of curvature inwardly, the result is NuD
where the subscript f indicates a solution such as eq 37 in which curvature is neglected. Substituting for NuDffrom eq 37 and expanding the logarithmic term give n
Nu =
I
(TX)'/2 - 4 x
- 1 - ...
(39)
which is more accurate than either eq 36 or 37. The small difference between eq 39 and eq 35 can be presumed to be primarily due to the effect of the false boundary condition. The limiting asymptotic solution for X a,as derived directly, is
-
= 5.784... (40) A correlation can be constructed by combining eqs 36 and 40 per eq 1 since both are lower bounds. Equations 37-39 are unsatisfactory as components of eq 1 because all three are singular at intermediate values of X. As an upper bound, eq 37 is also precluded. Equation 39 is an upper bound but barely so. This example illustrates the alternative procedures that can be followed to obtain different asymptotes, the error that may arise from different idealizations, and the applicability or inapplicability of different asymptotes for correlation. Plug flow was chosen for illustration because of the relative simplicity of the solutions, but similar aspects of behavior occur for the more practical case of laminar (parabolic) flow. Free Convection. As asymptotic approximation attributed to Boussinesq is the key to almost all analysis of free and natural convection. The change in pressure due to motion is neglected, and the pressure gradient is postulated to be constant and equal to its static value in the bulk of the fluid. The following approximate equation of state is then postulated for both gases and liquids: NuD
This expression is substituted in the equations of conservation for mass, momentum, and energy, and then j3( T - Tm)is taken to be negligible with respect to unity. This procedure reduces the equations of conservation to those for constant density except for the momentum balance in the vertical direction in which -g - (l/p)(dP/dx) reduces to gj3(T - Tm). The method of dimensional analysis of Hellums and Churchill (1964) when applied to the thin-boundary-layer
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 651 model for free convection from a vertical isothermal plate identifies a singularity transformation and indicates, even without solving or even deriving the resulting ordinary differential equations, that
has been called the “universal” dependence on Pr. Dimensional analysis of the thin-laminar-boundary-layer model for free convection from an isothermal or uniformly heated horizontal plate yields
h x / k = ( g f i ~ ~ A T / v ~ ) ~ / ~ f i v / a r J(42)
--
Nu, = A ( R u , P ~ ) ’ / for ~ Pr
0
(54)
or
and
Nu, = (Gr,)1/4fiPr) (43) Dropping the viscoub term in the differential model and repeating the analysis, or simply eliminating v in eq 42, gives the following asymptotic expression for Pr 0 hx/k = A(gfl~~AT/ar~)’/~ (44)
Nu, = B(RU,)’/~ for Pr (55) Experimental data for horizontal plates have often been miscorrelated simultaneously with data for vertical plates because in log-log coordinates the slight difference in slopes is obscured by scatter. This example has illustrated the extensive theoretical structure that has been constructed for free convection under a variety of conditions using primarily dimensional and asymptotic analysis of the equations of conservation. Experimentation or numerical solutions are required only to determine the leading coefficients. The odd-valued exponents in the literature such as 0.24 for Ra can be recognized as artifacts of scatter or mixtures of data for two different conditions such as those represented by eqs 47 and 55. Odd-value exponents for Pr similarly represent an approximation of the range of powers given by eq 53 as well as scatter. Thermal Conduction in a Composite Solid. The following solution for the transient heat flux density to a composite region made up of a slab of thermal insulation and a semiinfinite region of higher conductivity, both at a uniform initial temperature, after a step increase in the temperature at the surface of the insulation can be obtained using the Laplace transform:
-
or
Nu, = A(Ra,Pr)1/4 (45) Dropping the inertial terms in the momentum balance and repeating the analysis give hx/k = B(gfi~~A.T/va)’/~
(46)
or
Nu, = B(Ru,)’/~ (47) The latter result can also be obtained from eq 42 by expressing v as p / p and then eliminating p except where it is a multiplier of g. This procedure is not completely straightforward in that also expanding a as k/pc, does not permit the elimination of p. Applying dimensional analysis to the equations of conservation in their time-dependent form and then time averaging and rearranging the resulting expression give
The postulate by Nusselt (1915) that h becomes independent of x for large x reduces eq 48 to
Nu, = (Ra,)1/3fiPr} (49) The same considerations as for laminar flow indicate that
--
Nu, = A(Ra,Pr)ll3 for Pr and
0
(50)
Nu, = B(Ra,)1/3 for Pr (51) For a uniformly heated wall, exactly the same expressions except for different leading numerical coefficients are obtained in all cases. This commonality has often been obscured by expressing the solutions in terms of Re,* = gfijx4/kva = Nu$a,, thereby replacing the unknown AT by the specified j . Equation 47 is thus transformed to Nu, = A1/5(Ra,*)1/5 (52) Churchill and Ozoe (1973) correlated computed and experimental values for laminar free convection in the thin-laminar-boundary-layerregime on an isothermal plate with an expression of the form Nu, = A(
[1
Ra, )/4 + (0.492/Pr)g/16] W9
where fi = (a - l)/(u + 1) and u = (kpc/k’p’c?1/2.For large values of u the series converges very slowly for moderate and long times. Hence an approximation for that range of time would be useful. Two extreme asymptotes can be discerned directly from eq 56. For very short times (57)
For very long times the exponential terms all approach unity and eq 56 reduces to
Equations 57 and 58 do not overlap and provide no direct guidance to the intermediate behavior. Equation 57 can be recognized as the solution for a semiinfinite extent of insulation and eq 58 as the solution for no insulation. Four intermediate solutions can be derived from physical reasoning, or alternatively by speculation. For very large u, corresponding to very good insulation and/or a very conductive medium,0 1and eq 56 reduces to
-
(53)
For uniform heating a slightly different coefficient is required in the denominator, and of course also a different value of A. This same functional relationship between Pr and Ra has also been found to hold closely for other geometries, for turbulent flow, and even for enclosures. It
Equation 59 can be recognized as the solution for transient conduction across a finite slab with constant surface temperatures and could have been derived directly on that basis, again using the Laplace transform. Equation 59 can
652 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
be seen to reduce to eq 57 for very short times. It does not converge rapidly for large t , but the following alternative solution derived by separation of variables does:
For very long times eq 60 reduces to
which can be identified as the solution for steady-state conduction acrose a finite slab with fmed temperatures and can also be considered as an intermediate-time approximation in the event of an adjacent medium of asymptotically large conductivity. For long times such that the heat capacity of the insulation can be neglected, the following solution is applicable:
in the list of variables leading to eq 7, the result would have been an expression such as
Speculative elimination of a, or for that matter u, or or p, would not lead to asymptotically valid results. Reduction of the groupings which result from using the pressure gradient instead of the shear stress on the wall similarly fails. Dimensional analysis starting with the shear stress at the wall as a dependent variable rather than the local velocity results in p
(66) Elimination of p gives the correct grouping for laminar (creeping) flow, namely,
-7 -3 - A
(67)
Pum
Correlating equations might be constructed from several of these asymptotic solutions. In the interests of simplicity, eqs 57,58, and 61 will be utilized. Combining first eqs 57 and 61 and then that result with eq 58 and evaluating the two exponents from values computed from eqs 59,60, and 62 yield
while combining first eqs 58 and 61 and then that result with eq 57 yields
or
f(%) = A
(68) but the analogous reduction for turbulent flow, namely, the elimination of p, implies independence of the friction factor from the Reynolds number, which has no range of validity for a smooth pipe. Thus an optimal choice of variables is essential for successful speculative analysis. Different sets of variables can of course be used speculatively as a starting point. The Mixing Length. According to Nikuradse (1933), Prandtl inferred from a plot of experimental determinations of the mixing length in both smooth and artificially roughened round tubes in the form of l / a versus r / a that 1 = 0 . 4 ~for r = a and 1 = 0 . 1 4 ~for r 0. The former inference, which lead to Prandtl’s derivation of the semilogarithmic velocity distribution (eq 10 with B = 1/ 0.4), is actually valid for 30 < y+ < O.la+ but not for y+ < 30, for which eq 11implies a proportionality of 1+ to the 3/2 or higher power of y+. The latter inference, which leads to uz - u+ = 4 . 7 6 ( r / ~ ) ~ / ~ (69) is seen by comparison with eq 15 to be invalid functionally. The mixing length 1 actually becomes unbounded at the center line. These misinterpretations, which have been adopted in most subsequent applications of mixing-length theory, resulted from the insensitivity of the plot of l / a versus r / a as well as from the inaccuracy of the experimental measurements in the very regions for which the functionality was inferred. Turbulent Flow in an Annulus. The false assumption that in turbulent flow in an annulus the zero in the velocity gradient (the maximum in the velocity) occurs at the same radial position as the zero in the shear stress, just as it does in laminar flow, infecta most theoretical solutions and correlations of experimental data in the current literature. Except in the few instances in which the separate locations of both the zero in the velocity gradient and the zero in the shear stress can be identifed retroactively,past experimental results cannot be reinterpreted in correct terms. This difference in locations occurs not only in annulibut in all turbulent flows in which the shear stress on the wall or walls is not the same everywhere, for example in flow between parallel plates of nonidentical roughness,in Poiseuille-Couette flows, in curved channels, and in all channels with a two-dimensional cross-section.
-
Quation 63 behaves anomalously in that it yields slightly lower values than eq 57, which is a lower bound. Quation 64 correspondingly yields slightly larger values than eq 58, which is an upper bound. These anomalies could be avoided by using eq 5 or 6. However, the error is so slight for large u that eqs 63 and 64 are to be preferred on grounds of simplicity. Equation 64 is slightly better in that the anomalous behavior for small u is less. In view of eq 62, the expression [l + ( ~ r x ~ ) ~ can / ~ ] be - ~noted / ~ to provide a good first-order expression for ex’ erfc (XI. This example illustrates the determination of asymptotes by various means, the use of combinations of these asymptotes to construct overall correlating equations, and the discrimination between alternative combinations.
Examples of Erroneous and Misapplied Models and Solutions As noted above, models for rate processes usually incorporate some idealizations, approximations, and deletions, and this is particularly true of the reductions which are necessary to obtain asymptotic solutions. The consequence of such simplifications is seldom evident from the solution itself. The failure of a solution to conform to experimental data is often attributed to some presumed inaccuracy in the measurements when instead simplifications made in constructing the model or application of the wrong model may be at fault. Some classical examples of such misapplications and misattributions are described below. Turbulent Flow in a Round Pipe. Had the mean velocity rather than the shear stress on the wall been used
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 653 The existence of a difference in locations is actually apparent from the time-averaged equations of motion themselves, and indeed was first predicted by Kjellstrom and Hedberg (1966) on that basis prior to its experimental identification. However, the determination of the two locations by analysis requires a complete solution. Unfortunately, one consequence of the difference in locations is the breakdown of the eddy kinematic viscosity model as well as the mixing-length model since the shear stress is finite, and hence these quantities are unbounded at the point where the velocity gradient is zero. Use of the k-e and direct shear-stress models is not, however, precluded. The error due to the postulate of equal locations, which greatly simplifies the modeling by permitting separate solutions for the two portions of the flow divided by the plane of zero shear stress, increases with the ratio of the shear stresses. Detailed discussions of this facet of turbulent flow are provided by Maubach and Rehme (1972) and Churchill (1991). Orifice Coefficient. The coefficient of contraction for 0, a planar, shape-edge orifice in the limit of A,/A1 which was apparently first derived by Kirchhoff (1869), is T / ( T + 2) = 0.6110. Trefftz (1916) subsequently derived a value of 0.61 as a fmt-order approximation for a round, sharp-edge orifice in the limit of A,/A1 0. Owing to the apparent agreement with the value for a planar orifice, and, in view of the known equality of the coefficients for planar and round Borda entrances, he inferred that the exact value for a round orifice was also T / ( T 2). However, the exact value has since been shown by numerical solutions [see,e.g., Fliigge (1960)] to be 0.5793... Detailed discussions of this topic are provided by Unger (1979) and Churchill (1991), Chapter 17. The value of T / ( T + 2) continues to appear for round orifices in many handbooks and textbooks, which is misleading theoretically if not practically. The Acoustic Velocity. The ancient history of the derivations for the velocity of sound provides a last instructive example in fluid mechanics. According to Westfall (1973), Newton first derived an expression on the presumption that the process was isothermal and then attributed the disagreement with measured values in air to their inaccuracy as well as to the presence of water vapor. The physically valid expression, which is based on the postulate of isentropy, was not derived until more than a century later by Laplace. Convection of Heat and Components in Inviscid Flow. Boussinesq (1905) utilized theoretical expressions for the velocity field in inviscid flow to derive solutions for forced convection. These expressions have often been implied to be applicable for heat transfer between immersed solids and liquid metals, and for mass transfer to or from bubbles. Galante and Churchill (1990) showed that these solutions are not applicable for heat transfer with liquid metals because of separation of the flow and the significant thermal resistance of the boundary layer even for small, finite Pr. They also showed that they are not valid for mass transfer to small bubbles because of the immobility of the interface or to large ones owing to deformation, and, indeed, not for bubbles in any regime because of the finite viscosity of the liquid. The solutions for heat transfer to solid objecta do remain valid and useful as asymptotes for Pr 0 in constructing correlations with the model of Churchill and Usagi. King (1914) misapplied his solution, which included conduction in the direction of flow, for convection from a uniformly heated plate in inviscid flow to heat transfer from a cylinder and obtained a completely erroneous expression which, however, agrees fairly well with the ex-
-
-
+
-
perimental data for air over a wide range of Re = Pe/0.7 [see Douglas and Churchill (195611. The agreement is purely fortuitous; the corrected solution for inviscid flow does not agree nearly as well with the data. The Colburn Analogy. Colburn (1933) derived the widely used "j-factor" correlation by combining the Reynolds analogy for Pr = 1, namely, Nu/Re = f (70) with a dependence on FW3,which he chose as a rounded-off compromise for slopes ranging from 0.3 to 0.4 on log-log plots of experimental data and an empirically based relationship between f and Re. The resulting expression
Nu
= f = 0.023Re4.2
provides fairly accurate predictions for a reasonable range of Re and for large Pr since Pr1/3is the theoreticallycorrect asymptote for Pr but ita use is no longer justified in view of more general and theoretically sound expressions such as that of Churchill (1977b), which is compared with experimental data for a wide gamut of Re, Pr,and Sc in Figure 5. The curves in Figure 5 represent a four-fold application of the model of Churchill and Usagi with theoretical and semitheoretical asymptotes for both uniform wall temperature (or concentration) and a uniform flux of heat (or concentration) for the laminar, translational, and turbulent regimes, combined with a similar three-fold model for the friction factor in all these regimes. Development of Turbulent Forced Convection on a Flat Plate. Kestin and Persen (1962) derived an analytical solution for developing turbulent, forced convection on a flat plate by postulating that for Pr m and Re, the thermal boundary layer develops wholly within the laminar sublayer. The resulting expression is
-
~ 3 ,
-
-
(72) Nu, = 0.1609(Pr)1/3(Re,)3/5 Equation 72 has no range of applicability. The source of error is the postulate of negligible thermal transport by turbulent fluctuations. The effective total kinematic viscosity vE in a turbulent regime can be expressed as (73) Near the surface the ratio of the eddy kinematic viscosity ut to the kinematic viscosity v is finite but small compared to unity and hence negligible. The effective total thermal diffusivity aE can similarly be expressed as (74) which can be expanded as
-
The "turbulent Prandtl number" Prt = vt/at is known experimentally to be the order of unity. As P r m, aE remains large with respect to a! even very near the wall where u t / v is small with respect to unity but finite. The same postulate of negligible turbulent transport of energy has been used to obtain similar erroneous expressions for both developing and fully developed turbulent convections in channels. This is an example of a seemingly reasonable simplification of a model leading to an erroneous solution. Forced Convection in Creeping and Slightly Inertial Flow. Levich (1962) derived a solution for forced
654 Ind. Eng. Chem. Res., Val. 31, No. 3, 1992 10
I
I
I
10
IO
v) c
b Y
z IC
1C
I
I
1 o3
I
I
lo5
10‘
106
7
Re
Figure 5. Generalized correlation for forced convection in round tubes [from Churchill (1977b)l.
convection through a thin thermal boundary layer in creeping (Stokes) flow around a sphere, namely, NuD = 0.99145(PeD)’/3
(76)
and Friedlander (1957) the analogous solution for slightly inertial (Tomotika-Aoi) flow around a cylinder. A thin a. However, thermal boundary layer implies PeD creeping and slightly inertial flow both imply very small ReD. Hence, PeD = (ReD)(Pr)must be large by virtue of Pr. No ordinary fluids have a sufficiently large value of Pr to compensate for the restrictions on ReD and Pep Hence, these expressions, although sound in their own terms, have no practical range of application for heat transfer. Their analogue for mass transfer may have application since much larger values of Sc are encountered than of Pr. Thermoacoustic Convection. It has long been recognized that the simple energy balance for transient conduction implies an instantaneous response in temperature at all distances, which in turn implies an infinite rate of propagation of energy. One proposed resolution of this seeming anomaly has been to add a second-order derivative in time multiplied by the thermal conductivityand divided by the square of an arbitrary wave velocity. The resulting solution produces a finite rate of propagation of energy but a discrete step in temperature at the wave front. This formulation has often been attributed (but falsely) to Maxwell (18671, apparently to invoke authority. The minimal (sonic) velocity of a compressive wave is equal to (dP/dp)1/2. Hence, an infinite velocity of prop-
-
agation is actually consistent with the simple energy balance for thermal conduction, which implies an incompressible medium. The correct approach is then not to patch up the energy balance with an empirical term but instead to take compressibility into account. Indeed, the equations of motion for a compressible material, combined with an equation of state and an extended energy balance, including convection and the work of compression as well as conduction, generate a distributed, slightly supersonic wave in pressure and particle velocity as well as temperature with no empiricism. A comparison of the “hyperbolic” and general model and their solutions is provided by Churchill and Brown (19871, who have since confiied the predictions of the fluid-mechanical/thermal model experimentally for gases. Even so, solutions for the patched-up energy balance, which are wrong both conceptually and quantitatively still appear regularly in the literature. Effectiveness Factor of Double-Spiral Heat Exchangers. Computed values for the performance of double-spiral heat exchangers with countercurrent flow of the same fluid in both passages have been correlated by Choudhury et al. (1985) in terms of the expression ZJ=-tanh(:/ n N
(77)
where F is the total heat flux as compared to that €or a true countercurrent heat exchanger, n is the number of turns in each direction, and N = UA/wcis the number of transfer units. They note that this expression is the exact
Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 655 solution for a countercurrent cascade of identical cocunent exchangers and imply that its use has some theoretical justification for a double-spiral exchanger. Strenger et al. (1990) instead plotted their own computed values as E = FN versus N for a given n since E is a better measure than F of the performance of a double-spiral exchanger when operated as an incinerator, in that E equals the temperature rise in the incoming fluid divided by the temperature difference at the inlet (and outlet). Their plot revealed a maximum in E whose existence has since been confirmed by analytical solutions for limiting cases as well as by experimentation. A maximum is not predicted by eq 77 although one is implicit in the computed values which Choudhury et al. used to test this expression. The conclusions to be drawn from this example are that a misleading correlation can result from an insensitive choice of coordinate and that an apparent similarity with some other process may be fortuitous. Combined Free and Forced Convection. A number of investigators have correlated experimental and computed values for assisting free and forced convection in various geometries with Values of n = 2 have been rationalized on the basis of the presumed vectorial additivity of (Gr/2)1/2 and Re, and values of n = 4, on the basis of the additivity of the work of free and forced convection. On the other hand, Churchill (1977a) demonstrated that the bulk of experimental data and computed values clearly support n = 3. Martinelli and Boelter (1942) provide theoretical support for this value through an integral-boundary-layer solution for developing convection in a vertical tube, and Ruckenstein (1978) provides support through the addition of expressions for the velocity profiles. The correct choice of n = 3 by Churchill resulted from the use of sensitive (arithmetic) plots as compared to insensitive (logarithmic) plots by earlier investigators. This example, as well as that concerning double-spiral heat exchangers, demonstrates how erroneous conclusions can often be rationalized on a mechanistic basis or inferred from insensitive graphical forms. Turbulent-FreeConvection. Equations 49-51, which are based on the speculation by Nusselt (1915) of the attainment of an asymptotic value for the heat-transfer coefficient, appear to be in accord with experimental measurements. Nonetheless, several erroneous expressions have subsequently been proposed. For example, FrankKamenetskii (1937) speculated that the heat-transfer coefficient should approach independence from both k and p , leading to Nu, = A(Refl)1/2
(79)
The failure of eq 79 to agree with experimental measurements is obviously a consequence of the wrong speculation-but one which could not necessarily be rejected in advance. Ekkert and Jackson (1951) used integral-boundary-layer theory together with a l/,th power distribution for the temperature, a modified l/,th power distribution for the velocity, and the Blasius expression for the shear stress on the wall, to obtain Nu, =
0.0295(Ra,) 2/5(Pr) l / l5 [ l 0.495Pr2/3]2/5
+
(80)
Equation 80 has been given some credence because it agrees grossly with experimental data for air. However,
the indicated dependence on A. is obviously absurd in that N u decreases with increasing R.Furthermore, the dependence on Ra is not in as good agreement with the data as eq 49. The first-order numerical agreement is a consequence of the considerable empirical input via the velocity and temperature distributions and the shear stress, plus the smoothing accomplished by integration. Formation of Thermal NO,. Zel'dovich (1946) postulated a simple chemical kinetic mechanism for the formation of thermal NO, downstream from the flame front in premixed combustion in terms of the concentrations of 0 and N. He then further postulated a pseudo-steadystate concentration for these two free radicals. This latter postulate was accepted by a generation of subsequent investigators who attributed its gross underprediction of the formation of NO, to experimental errors in sampling and chemical analysis. Detailed kinetic calculations with a complete set of free-radical mechanisms eventually revealed that a pseudo steady state is not attained under these conditions [see, for example, Tang and Churchill (1981)l. This detailed modeling also predicted a negligible concentrationof NO2in the products of combustion at high temperature; subsequent improved experimental measurements confirmed that the measured concentrations of NO2were actually formed in the sampling tube. These two results represent a triumph of detailed analysis over both approximate analysis and inaccurate experimentation. Laminar Flow Reactors. As noted by Churchill and Pfefferle (1985), most tubular homogeneous reactors operate in the laminar regime, whereas most analyses of experimental data and most theoretical solutions postulate plug flow. Ark (1965) demonstrated theoretically that the error in conversion due to the assumption of plug flow cannot exceed 11% for a fist-order reaction and must be even less for higher order reactions. Collins and Churchill (1990) have recently shown that this result, while correct on its own terms, is misleading if the residual concentration of a reactant is of interest, as is the case for incineration of a pollutant. The error in the residual concentration is actually unbounded for a first-order reaction and significant for other finite orders. This is a warning that a theoretically based conclusion may be quite valid for some conditions but not for others which differ subtly.
Conclusions Numerical methods and computing machinery have greatly extended our capabilities for obtaining theoretical solutions for the rate processes. The results are, however, analogous to experimental data in that they do not prlovide any insight as to the functional form of the dependence on the independent variables and parameters. Numerical solutions ordinarily have greater precision, regularity, and coherence than experimental results but are always subject to some doubt concerning the validity of the model itself. With rare exceptions, experimental and theoretical results do not follow a simple power law over an extended range. Correspondingly, the appearance of an odd-valued exponent in a one-term correlating equation identifies the structure of the expression as unsound theoretically. Asymptotic solutions, if valid, provide a sound basis for testing experimental data and even computed values. More importantly, if expressed in elementary terms, they can usually be combined to yield a correlating equation which involves a minimum of empiricism and has the correct structure in all distinct regimes. Asymptotic solutions often have the form of a power dependence. The power dependence of an intermediate regime, as indicated clearly by an asymptotic solution, may be difficult if not impossible to determine from either experimental data or
666 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
numerical solutions, particularly if it corresponds, as is usual, to the tangent through a point of inflection. Asymptotic solutions may, however, be useless or inapplicable if the idealizations upon which they are based do not have physical validity for the conditions of intetest. An erroneous model leads to an erroneous solution no matter how elegant the technique of solution or how beautiful the structure of the solution itself. A number of erroneous or misapplied asymptotic solutions were identified above. Most of these are the work of the justifiably famous members of our profession, who, however, were not immune to error. Evocation of authority is not a substitute for experimental confirmation. While an extended range of experiments may be necessary to test a general solution, and, in particular, to determine its limits of validity, a few very precise measurements for even a very limited range may provide a more critical test of an asymptote. Bands of experimental data often represent unidentified or uncontrolled parameters rather than error. Conversely, such uncorrelated results may be valuable in identifying a new variable or aspect of behavior. Ikperimentalists rarely obtain the data required to test critically the theoretical results of others. A combination of analysis and experimentation is clearly to be preferred. Even the apparent agreement of theoretical and experimental results may be fortuitous, particularly with integral or exiting quantities. New concepts usually come from experimentation, but analysis is generally required for their explanation. Theoretical structures, and particularly those that come from asymptotic and speculative analysis, have yet been underutilized in correlation and as a guide to experimentation. A low standard deviation of data from a correlation may be less important than the correct functional behavior, particularly if the expression is to be used beyond the range of the data. New books and topical reviews make a great contribution when they identify outmoded correlations as well as new results.
Nomenclature A = empirical constant or area (Al, upstream area; A,, area of orifice),m2 a = radius of tube, m a+ = (a/u)(7,/p)1/2 B = empirical constant b = half-distance between plates, m b+ = ( b / v ) ( ~ , / p ) ' / ~ C = empirical constant c = specific heat capacity, J/(kgK) = specific heat capacity at constant pressure, J/(kgK) 8 = diameter, m E = (T2- T l ) / ( T 3- T J ,f w e of merit, or empirical constant e = roughness, m e+ = (e/v)(rw/p)112 F = correction factor for non-log-meantemperature difference f = rW/pum2, friction factor flz] = function of z G = empirical constant Gr, = g@x3AT/v2, Grashof number based on x g = acceleration due to gravity, m/s2 h = heat-transfer coefficient (hr,that for radiation), W / (cm2.K) j = heat flux density, W/m2 k = thermal conductivity (kc,that for continuous media; kd, that for dispersed media), W/(m.K) L = length, m 1 = mixing length, m m = arbitrary exponent
n = UA/wc, number of thermal transfer units n = arbitrary exponent or number of turns Nu = Nueselt number (Nu,, that based on x ; NuD, that baaed on D Nuh, that for free convection; Nuf&, that for forced convection) P = absolute pressure, Pa p = pitch of coil, m PeD = R e a , Peclet number Pr = cpCl/k = u/a, Prandtl number (Prt = vt/at, turbulent Prandtl number) Ra, = gflx3AT/ua,Rayleigh number based on x Ra,* = gfljx4/kua, modified Rayleigh number based on x Re = Reynolds number (Re,, that based on x ; ReD,that based on D) r = radial coordinate (ro,radius of coil; r,, radius of curvature), m Sc = Schmidt number ShD = Sherwood number based on D T = absolute temperature (T, that at wall, T,, that at infinity; To,that at inlet; To,initial temperature value; T,, mixedmean temperature value; T,,,,average temperature value), K AT = T , - T,, K t = time, s U = overall heat-transfer coefficient, W/(m2.K) u = velocity (uc,that at center line; urn,mixed-mean velocity value), m/s u+ = u(p/rw)'/2 w = maas rate of flow, kg/s X = y o l x ) / ~ ~ lorx )Y,lx)/yolx) or kx/umpca2 x = distance along wall or axis, m, or independent variable Y = Y ~ ~ ) / Y Oor~ Yl4/Ym.(xl Z) y = distance from wall, m y ( z ) = dependent variable (yo= ydx), asymptotic function for x 0;y, = y&), asymptotic function for x m; yi = yi(x), intermediate asymptote)
-
-
Greek Letters a = k d / k , or klpc,, thermal diffusivity (at, eddy thermal
diffusivity; (YE,total effective thermal diffusivity), mz/s j3 = volumetric coefficient of thermal expansion, K-l, or empirical coefficient, or ( u - 1)/(u 1) or (a - l)/(a 2) b = thickness of insulation, m E = fraction of dispersed media, or emissivity
+
p
+
= dynamic viscosity, Pa+s
v = kinematic viscosity (ut, eddy kinematic viscosity; uE, total,
effective viscosity), m2/s = density (p,, that at infinity), kg/m3 u = Stefan-Boltzmann constant, W/(m2-s.K4),or (kpc/ k'p'C?ll2 rW = shear stress on wall, Pa - l ) / b+ 2)) 9= $(z) = function of z p
Subscript f = flat Superscript ' = indication for insulation
Literature Cited Adiutori, E. F. The New Heat Transfer, 2nd ed.; Venturo Press: West Chester, OH, 1989. Aris, R. Introduction to the Analysis of Reactors; Prentice-Hall: Englewood Cliffs, NJ, 1965; p 96. Boussinesq, J. Essai sur la thhrie des eaux couranta. M6m. Acad. Sci. Znst. Fr. 1877, 23, 1-680. Boussinesq, J. Calcul du poivoir refroidieeant des couranta fluids. J. Math f i r e s Appl. 1905,60,28&332; Calculation of the Cooling Power of Fluid Currents (English Translation). Znt. Chen. Eng. 1989,29, 343-365. Choudhury, K.; Linkmeyer, H.; Bassiouny, K.; Martin, H. Analytical Studies on the Temperature Distribution in Spiral Heat Exchangers: Straightforward Design Formulae for Efficiency and Mean
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 657 Temperature Difference. Chem. Eng. Process. 1985,19,183-190. Churchill, S. W. Approximate Operational Calculus in Chemical Engineering. AIChE J. 1957,3, 289-293. Churchill, S. W. A Comprehensive Correlating Equation for Laminar, Assisting, Forced and Free Convection. AIChE J. 1977a,23, 10-15. Churchill, S. W. Comprehensive Correlating Equations for Heat, Mass and Momentum Transfer in Fully Developed Flow in Smooth Tubes. Ind. Eng. Chem. Fundam. 1977b,16,109-115. Churchill, S. W. The Interpretation and Use of Rate Data. The Rate Concept (revised printing); Hemisphere Publishing Corp.: Washington, DC, 1979. Churchill, S. W. The Thermal Conductivity of Dispersions and Packed Beds-An Illustration of the Unexploited Potential of Limiting Solutions for Correlation. In Advances in Transport Processes; Mujumdar, A. S., Mashelkar, R. A., Eds.; Wiley Eastern Ltd.: New Delhi, 1986a;Vol. IV, pp 394-418. Churchill, S. W. Laminar Film Condensation. Int. J. Heat Mass Transfer 1986b,29,1219-1226. Churchill, S. W. Viscous Flows. The Practical Use of Theory; Butterworthe: Stoneham, MA, 1988. Churchill, S. W. A Theoretical Structure and Correlating Equation for the Motion of Single Bubbles. Chem. Eng. Process. 1989,26, 269-279; 1990,27,66. Churchill, S. W. Turbulent Flows. The Practical Use of Theory; Butterworth-Heinemann: Stoneham, MA, 1991,in press. Churchill, S. W.; Usagi, R. A General Expression for the Correlation of Rates of Transfer and Other Phenomena. AIChE J. 1972,18, 1121-1128. Churchill, S. W.; Ozoe, H. A Correlation for Laminar Free Convection from a Vertical Plate. J. Heat Transfer, Trans. ASME 1973, 95C, 540-541. Churchill, S.W.; Usagi, R. A Standardized Procedure for the Production of Correlations in the Form of a Common Empirical Equation. Ind. Eng. Chem. 1974,13,39-44. Churchill, S. W.; Pfefferle, L. D. The Refractory Tube Burner as an Ideal Stationary Chemical Reactor. Imt. Chem. Eng. Symp. Ser. 1985,NO.87,287-285. Churchill, S. W.; Brown, M. A. Thermoacoustic Convection and the Hyperbolic Equation of Conduction. Int. Commun. Heat Mass Transfer 1987,14,647-655. Colburn, A. P. A Method for Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction. Trans. AIChE 1933,29,174-209. Collins, L. R.; Churchill, S. W. Effect of Laminarizing Flow on Postflame Reactions in a Thermally Stabilized Burner. Ind. Eng. Chem. Res. 1990,29,456-463. Dean, W. R. Note on the Motion of Fluid in a Curved Pipe. Philos. Mag. (Ser. 7) 1927,4,20&223. Dean, W. R. Stream-line Motion of Fluid in a Curved Pipe. Philos. Mag. (Ser. 7) 1928,5,673-695. Douglas, W. J. M.; Churchill, S.W. Recorrelation of Data for Convective Heat Transfer between Gases and Single Cylinders with Large Temperature Differences. Chem. Eng. Progr. Symp. Series 1956,52 (NO.l8),23-28. Eckert, E. R. G.; Jackson, T. W. Analysis of Turbulent Free-Convection Boundary Layer on a Flat Plate; National Advisory Committee on Aeronautics Report 1015;NACA Washington, DC, 1951. Emmons, H. Natural Convection Heat Transfer. Studies in Mathematics and Mechanics Presented to Richard von Mises by Friends, Colleagues and Pupils; Academic Press: New York, 1954;pp 232-241. Finlayson, B. A. The Method of Weighted Residuals and Variational Principles; Academic Press: New York, 1972. Fliigge, S., Ed. Handbuch der Physik, Bd 9,Strdmungsmechanik, III; Springer-Verlag: Berlin, 1960. Frank-Kamenetskii, D. A. On the Limiting Form of the Free Convection Law for High Value of the Grashof Criterion (translated title). Dokl. Acad. Nauk SSR 1937,17, 9. Friedlander, S. K. Mass Transfer to Single Spheres and Cylinders at Low Reynolds Numbers. AZChE J. 1957,3,43-48. Galante, S. K.; Churchill, S. W. Applicability of Solutions for Convection in Potential Flow. Adv. Heat Transfer 1990,20,353-388. Hellums, J. D.; Churchill, S. W. Simplification of the Mathematical Description of Boundary and Initial Value Problems. AIChE J. 1964,10,110-114. Kestin, J.; Persen, L. N. The Transfer of Heat Across a Turbulent Boundary Layer at Very High Prandtl Numbers. Int. J. Heat Mass Transfer 1962,5,355-371.
King, L. V. On Convection of Heat from Small Cylinders in a Stream of Fluid: Determination of the Convection Constants of Small Platinum Wires with Applications to Hot Wire Anemometry. London, Ser. A 1914,90,373-433. Philos. Trans. R. SOC. Kirchhoff, G. Zw Theorie freier Flbsigkeitsstrahlen. J. reine a. angew. Math. 1869,70,289. Kjellstrom, B.; Hedberg, S. On Shear Stress Distributions for Flow in Smooth or Partially Rough Annuli. Aktiebolaget Atomenergi, [Rapp.] AE-243: Aktiebolaget Atomenergi: Stockholm, 1966. Langmuir, I. Convection and Conduction of Heat in Gases. Phys. Rev. 1912,34,401-422. Levich, V. G. Physicochemical Hydrodynamics (English transl. by Scripta Technica, Inc.); Prentice Hak Englewood, NJ, 1962;p 85. MacLeod, A. L. Liquid Turbulence in a Gas-Liquid Absorption System. Ph.D. Thesis, Carnegie - Institute of Technoloev, - - . Pittsburgh, PA, 1951. Manlapaz, R. L.; Churchill, S. W. Fully Developed Laminar Flow in a Helically Coiled Tube on Finite Pitch. Chem. Ena. - Commun. 1980,7,5?-78. Manlapaz, R. L.; Churchill, S. W. Fully Developed Laminar Convection from a Helical Coil. Chem. Eng. Commun. 1981, 9, 185-200. Martinelli, R. C.; Boelter, L. M. K. The Analytical Prediction of Superposed Free and Forced Viscous Convection in a Vertical Pipe. Uniu. of Calif. Publ. Eng. 1942,5 (No. 2), 23-57. Maubach, K.; Rehme, K. Negative Eddy Diffusivities for Asymmetric Turbulent Velocity Profiles. Int. J. Heat Mass Transfer 1972,15,426432. Maxwell, J. C. On the Dynamical Theory of Gases. Philos. Trans. London 1867,157,49-88. R. SOC. Maxwell, J. C.A Treatise on Electicity and Magnetism, Clarendon Press: Oxford, U.K., 1873;Vol. I, p 365. Mishra, P.; Gupta, S. N. Momentum Transfer in Curved Pipes. 1. Ind. Eng. Chem. Process Des. Dev. 1979,18, 130-142. Newton, I. Scala gradium Caloris. Philos. Trans. R. SOC. London 1701,22,824-829 Newton, I. Principia, Vol. I, 2nd ed.; S. Pepys Press: London, 1713; English translation by A. Mott, 1729, as revised by F. Cajori; University of California, Press: Berkeley, CA, 1934;pp 334-336. Nikuradse, J. Strbmungsgesetze in rauhen Rohren. V.D.I-Forschungsheft 361;VDI-Verlag: Dusseldorf, Germany, 1933;Laws of Flow Zn Smooth Tubes (English translation); NACA Technical Memorandum No. TM62; NACA Washington, DC, 1950. Nusselt, W. Das Grundgesetz des Wiirmeiiberganges. Gesund.-Ing. 1915,38,477-482,490-496. Nusselt, W. Die Obeflchenkondensation des Waaserdampfes. VDZ 2 1916,60,541-546,569-575. Ohman, G. A. A Note on the Experimental Determination of Convective Heat Transfer from Wires at Extremely Small Reynolds and Grashof Numbers. Acta Acad. Abo., Ser. B 1970,30, 1-7. Rothfus, R. R.; Monrad, C. C. Correlation of Turbulent Velocities for Tubes and Parallel Plates. Ind. Eng. Chem. 1955, 47, 1144-1 148. Rothfus, R. R.; Monrad, C. C.; Senecal, V. C. Velocity Distribution and Friction Factor in Smooth Concentric Annuli. Znd. Eng. Chem. 1950,42,2511-2521. Ruckenstein, E. Interpolating Equations between Two Limiting Cases for the Heat Transfer Coefficient. AIChE J. 1978,24, 940-941. Saville, D. A.; Churchill, S. W. Laminar Free Convection in Boundary Layers Near Horizontal Bodies and AxisymmetricBodies. J. Fluid Mech. 1967,29,391-399. Strenger, M. R.; Churchill, S. W.; Retallick, W. B. Operational Characteristics of a Double-Spiral Heat Exchanger for the Catalytic Incineration of Contaminated Air. Ind. Eng. Chem. Res. 1990,29,1977-1984. Tang, S. K.; Churchill, S. W. A Theoretical Model for Combustion Reactions Inside a Refractory Tube. Chem. Eng. Commun. 1981, 9,137-180. Trefftz, E. ffber der Kontraktion kreisformiger Fliieaigkeitsetrahlen. 2.Math. Phys. 1916,64,34-61. Truesdell, L. C., Jr.; Adler, R. J. Numerical Treatment of Fully Developed Laminar Flow in Helically Coiled Tubes. AIChE J. 1970,16,1010-1015. Unger, J. Stromunginzylinderische Kilnale mit Versperrungen beihohen Reynolds Zahlen. Forsch. Ingenieurwes. 1979,45,68-80; Flow at HIgh Reynolds Numbers in Cylindrical Channels with Constrictions (Engl. Transl.) Znt. Chem. Eng. 1982,22, 1-15. Westfall, R. S.Newton and the Fudge Factor. Science 1973,179 (No. 4075,Feb 23),751-758.
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Zuzovsky, M.; Brenner, H. Effective Conductivities of Composite Materials Composed of Cubic Arrangementsof Spherical Particlea Embedded in an Isotropic Matrix. J. Appl. Math. Phys. 1977,28, 979-992. Received for reuiew January 23, 1991 Accepted May 29, 1991
Solutions in Closed Form for a Double-Spiral Heat Exchanger Matthew J. Targett, William B. Retallick,+and Stuart W. Churchill* Department of Chemical Engineering, University of Pennsylvania, 31I A Towne Building, 220 South 33rd Street, Philadelphia, Pennsylvania 19104-6393
The MACSYMA code for symbolic manipulation was used to obtain solutions in closed form for double-spiral heat exchangers of a few turns, both with and without heat losses to the surroundings. The solutions, which are for an equal rate of flow of the same fluid in both directions, reveal the existence of an optimal number of transfer units (an optimal rate of flow) for which the temperature rise for the heated stream is a maximum for a given thermal input or temperature difference. Such anomalous behavior, which is of obvious importance in the design and operation of double-spiral exchangers, has generally been overlooked or misinterpreted in prior experimental work and numerical solutions. For realistic heat losses to the surroundings, double-spiral exchangers of multiple turns are shown to be superior to true countercurrent exchangers in terms of producing a temperature rise. Minton (1970) asserted that double-spiralheat exchangers might be advantageous for a number of reasons, including enhancement of the heat-transfer coefficient, compactness,greater resistance to fouling, and relative ease of cleaning. Although not mentioned by him,doublespiral heat exchangers of many turns would appear to have a particular advantage in high-temperature and cryogenic applications in that the external area is reduced to the outer curved surface and the end plates, thereby reducing the leakage of energy to the surroundings. Strenger et al. (1990) have examined the use of a double-spiral heat exchanger for the catalytic incineration of low concentrations of contaminates, such as carbon monoxide, hydrocarbons, organic compounds, aerosols, and microorganisms, which may be present in the air of spacecraft, airliner cabins, automobiles, hospital rooms, and industrial clean rooms. In this application the heat losses from a double-pipe heat exchanger would be prohibitive despite the use of the best possible thermal insulation. Numerical solutions of a fairly exact model for doublespiral heat exchangers by Strenger et al. for different numbers of thermal transfer units, N = UA/wc,indicated the presence of a maximum in the thermal figure of merit with respect to the application of the heat exchanger as a catalytic incinerator for air, in which case the fluid flows first inward and then, after being heated incrementally, outward. This criterion is W C ( T- ~Ti) =-T2 - Ti E= (1) Q T3 - T2 where, as shown in Figure 1, T2- Ti is the increase in the temperature of the inwardly flowing stream due to heat transfer and T3- T2is the temperature increase in the fluid resulting from the imposition at the core of an external source of energy Q such as an electrical current through a coil.
* To whom correspondence should be addressed.
t Current address: William B. Retallick Associates, 1432 Johnny’s Way, West Chester, PA 19382.
In conventional heat exchangers in which the “entering” temperatures TIand T3are specified rather than Ti and the difference T3 - T2,the thermal effectiveness
is more commonly chosen as the measure of performance. , in a double-spiral catalytic incinFor ( w c ) , = - ( w c ) ~as erator, eq 2 reduces to (3)
Another related dimensionless quantity has also been used as a criterion of performance for noncountercurrent heat exchangers, namely, the correction factor F, which equals the ratio of the total amount of heat transfer to that for a true countercurrent heat exchanger. Since the latter equals U A ( A T ) , , (4)
For the operation of Figure 1and negligible losses to the surroundings,(AT), = T3- T2= T4- Ti,and eq 4 reduces to (5)
In all of the above expressions the heat capacities are implied to be mean values and in the event of eqs 3 and 5 to be equal for the two streams. For conditions such that eqs 3 and 5 are applicable, the quantities E, E, and Fobviously following from one another. As shown subsequently, E is a much more sensitive measure of the performance than e or F. Prior experimental work for double-spiral heat exchangers has almost wholly been for turbulent flow, and generally has been miscorrelated in terms of a mean overall
0 1992 American Chemical Society oaaa-5aa5/92/2~31-0~5a~03.00/0
