Role of Universalities in Chemical Engineering - Industrial

Ind. Eng. Chem. Res. 2007, 46, 7851-7869

7851

Role of Universalities in Chemical Engineering Stuart W. Churchill* Department of Chemical Engineering, UniVersity of PennsylVania, 311A Towne Building, 220 South 33rd Street, Philadelphia, PennsylVania 19104

Chemical engineering has survived and thrived for over 100 years, while many other branches of engineering have vanished or faded. This survival may be attributed to several factors, one of which is the more pervasive and effective utilization of universalities in the form of generalizations, simplifying concepts, and analogies. At the same time, many of the generalities, simplifying concepts, and analogies in current textbooks, handbooks, and even computer design packages are wrong or obsolete and should be deleted or noted to be only of historical interest. The radical changes that are currently afoot in chemical engineering and in the world in which it is now practiced can be expected to make many more obsolete or irrelevant, thereby creating a need and/or opportunity for the formulation of new ones. The origin and the past, present, and future role, if any, of the generalizations are examined in Parts I and II; that of the simplifying concepts are examined in Part III, and that of the analogies are examined in Part IV. Introduction This survival of chemical engineering as a distinct academic discipline and as a universally recognized profession over the past century has obviously required evolution and adaptation in response to scientific, technical, political, and economic changes in the world around us. A major factor in the practice of chemical engineering, in comparison to other fields of engineering, has been the greater use of generalizations, simplifying concepts, and analogies. The role of such sweeping generalizations as the unit operations and the transport phenomena is well-known; however, that of generalizations of lesser scope is not as well-recognized. These universalities have not only provided a structure for updating education and practice within the bounds of the profession but also, perhaps even more importantly, have provided a common language for communication and have reduced the impedance mismatch between academics and practitioners, fresh graduates and their first supervisors, and from one generation to the next. The advantage of that continuity has recently been tested as the employment of chemical engineers has switched increasingly from the chemical and petroleum industries (CPI) to other industries. As an example, when chemical engineering students were first hired in large numbers by the pharmaceutical industry, they complained that their unique skills were not understood, appreciated, or properly utilized by their supervisors. That complaint vanished as soon as chemical engineers rose to the rank of supervisor. The objective of this analysis is to identify and describe some of the universalities of the past and present, to evaluate the possibilities for their continued viability, and to describe techniques and provide, by means of examples, guidance, and encouragement for the recognition and formulation of new ones. The lines drawn here between generalizations, simplifying concepts, and analogies are arbitrary and the allocation herein * Tel.: 215-898-5579. Fax: 215-573-2093. E-mail address: churchil@ seas.upenn.edu.

of specific examples to one of these categories is based primarily on editorial convenience, rather than on technical grounds. Although the practice of chemical engineering rather than education is the ultimate concern of this analysis, much of the focus is, necessarily, on academia, because that is where most of the universalities have arisen, and within the classroom, because that is where most of the universalities are introduced to future practitioners. The most direct academic response to change is the addition of new courses or the addition of new material to old courses. Such additions require deletions to a corresponding extent. Additions are relatively simple to formulate, but deletions are often painful, because of the prior investment by faculty members and book writers in perfecting the presentation of the material to be deleted, and because doubts arise in industry as to whether the students ultimately are being shortchanged by the substitution of the trendy for the proven. One of my mentors, Donald L. Katz, offered two suggestions at the beginning of my career as an academic. First, he suggested that 20% of the material in a graduate course be updated each time that it is taught. Second, he suggested that I concentrate my research in a narrow field, with the intention of becoming the recognized authority in that subject. I have faithfully followed his first suggestion throughout a 58-year-long career of teaching and even extended it to undergraduate courses. Whatever the impact on the students, I have greatly benefited by the associated self-renewal. Such continuous revision precludes the direct use of a textbook, but electronic communication now expedites the distribution of up-to-the-minute notes in semi-permanent form. Such a continuous revision in content may not be as essential in an undergraduate course, and electronic communication may then be more appropriate as a supplement, rather than as a substitute for a textbook, because the latter provides some perhaps essential sense of security for the students. An alternative to replacement of old material by new material is the utilization and development of generalizations that allow incorporation of

10.1021/ie070522o CCC: $37.00 © 2007 American Chemical Society Published on Web 11/02/2007

7852

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007

the new material simply as examples. That has been the glory of both the unit operations and the transport phenomena concepts. This form of updating is relatively seamless and painless but requires initiative and ingenuity. However, the adequacy and sufficiency of that process is currently being questioned by some. I have not followed my mentor’s second suggestion but, instead, have regularly abandoned initial objectives to explore the explanation and consequences of new findings. This, in turn, promoted an interest in structures that would accommodate them and that has prompted and provided a basis for the present analysis. The dates of origin are included with many contributions to identify their longevity or the time gap, with respect to their application, replacement, or supplementation. Part I. Generalizations of Broad Scope A focus on fundamentals is a common characteristic of modern education in all of the sciences and in all branches of engineering, but the degree of reliance on generalizations in chemical engineering is unique, as are most of the generalizations themselves. These generalizations have allowed the core of critical knowledge to remain relatively stable, despite continual changes in detail. Accordingly, it is appropriate to explore the impact of generalizations on chemical engineering in the past and present, as well as the prospect for new generalizations in the future. Generalizations are so integral to education and practice in chemical engineering that their presence is taken for granted, and it may not be realized by chemical engineers that such technological structures, at least of such a sweeping nature, do not seem to have evolved or had an equivalent impact in other branches of engineering. The explanation is best left to those more knowledgeable concerning those disciplines. What is a generalization? Here, it is taken to be a concept, structure, and/or set of rules that apply to all processes, or at least a broad class of them. By these means, individual processes can be considered to be special cases. Generalizations are very useful in teaching, learning, understanding, and remembering, as well as in codifying information. Generalizations are commonplace and universal in mathematics; a simple example of which is the general solution of quadratic algebraic equations. The topics examined immediately below are well-known, at least in an overall sense, by all chemical engineers, but some of them may not be recognized to be generalizations. Therefore, it is appropriate to re-examine them briefly in that framework. For example, the roles of the unit operations and the transport phenomena concepts in the evolution and success of our profession are well-known, but the importance of some lesssweeping concepts, such as those of chemical reactor engineering, separations, process design, and process simulation, have perhaps not been recognized as generalizations, or their role in that respect has been underestimated. 1. Chemical Engineering Itself. The rapid development of the chemical industry late in the 19th century and into the 20th century sparked the onset of chemical engineering as an academic subject, and, somewhat separately, the self-identification of mechanical engineers and industrial chemists with experience and skill in the design of chemical plants as chemical engineers. This generalization was sanctioned by the founding of both this journal and the American Institute of Chemical Engineers (AIChE). 2. Unit Operations. As mentioned previously, one role of generalizations is to provide a structure for the description of

diverse elements. The unit operations concept (namely, the commonality of heat transfer, fluid flow, evaporation, distillation, etc.), as elements of design and operation for all chemical plants irrespective of the product, provides such a structure and thereby has been a major factor in the development of the curriculum and in the practice of chemical engineering. This concept has also been a factor in the unity of our profession, not only in the United States but also throughout the world. In England, in 1901, George E. Davies1 published A Handbook of Chemical Engineering, in which the principles of the new field were organized around what we now call the unit operations. However, the effective articulation of the concept of the unit operationssnamely, the commonality of heat transfer, fluid flow, evaporation, distillation, etc.sas elements of design for all chemical plants, regardless of the product, is generally attributed to The Principles of Chemical Engineering by Walker, Lewis, and McAdams in 1923.2 That certification and fleshing-out of the concept of unit operations were major factors in the development of an identity for the profession of chemical engineering in the United States. In 1928, Alfred H. White,3 then President-Elect of the AIChE, was able to assert that “Almost all schools which (sic) teach chemical engineering now recognize these unit processes (by which he meant what we now call unit operations) as providing the framework for the engineering side of chemical engineering.” To this day, no other branch of engineering seems to have developed or benefited so greatly from such a single unifying concept. Ultimately, the very success of this concept has proved to be self-defeating to a degree; the continued identification of new unit operations eventually made this generalization unwieldy. The unit operations concept was an outcome of industrial practice but was recognized and formalized in academia. That crossover may have fostered a close relationship between industry and academia in chemical engineering that persisted for many years but is now beginning to fade. The transport phenomena concept supplemented rather than replaced the unit operations concept, and, after almost a century, the latter concept is still alive in the curriculum, either explicitly or implicitly. The concept of unit operations can be expected to be of continued utility as long as chemical engineers are involved in process and product design. 3. Transport Phenomena. The transport phenomena concept gained almost immediate acceptance, because of the confluence of the call for courses in engineering science as a requirement for accreditation and the appearance of a remarkable textbook with this approach and name by Bird et al. in 1960.4 The transport phenomena concept may be considered to be a generalization based on the partial differential equations of conservation and their solutions. This concept (and the book with its name) were both criticized initially by some, because of the focus on problems that were mathematically tractable, rather than on those of the greatest practical importance; however, these criticisms have been blunted by adaptations in both academia and industry, and after some 40 years, the transport phenomena concept and book are rivaled only by the unit operations as the cornerstone of chemical engineering past and present. One curiosity is the failure of the book Transport Phenomena in the recent second edition, as well as the first edition, to encompass the numerical solution of partial differential equationssa methodology made possible by the computer revolution.

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 7853

One casualty of the replacement of the unit operations concept by the transport phenomena concept has been the loss of leadership by chemical engineering to mechanical engineering in applied fluid flow and heat transfer, except for a few specialized topics such as packed beds, fluidized beds, and nonNewtonian fluids. 4. Unit Processes. The first generalization to incorporate chemical reactions as such was the unit process concept of Shreve in 1945.5 It is based in the commonality of some chemical processes, such as nitration and sulfonation, in the manufacture of different chemicals, and thereby was proposed as the counterpart of the unit operations for chemical processes as contrasted with physical ones. However, this concept, which has obvious limitations, was not widely or permanently adopted in the curriculum or in practice. 5. The Rate Process Concept. Remarkably, because chemical reactions are the raison d’eˆtre for chemical engineering, they were excluded from the unit operations and, by definition, from the transport phenomena concept. In 1974, in The Interpretation and Use of Rate Data. The Rate Process Concept, Churchill6 proposed a new generalization based on the commonality of the definition of rates and of the experimental determination, correlation, and use of rate data for process design. Chemical reaction rates were not only included but actually utilized as the primary illustrations. This concept has been more of an artistic success than a commercial success, perhaps, in part, because it did not mesh well with other courses in the curriculum. 6. Thermodynamics. Thermodynamics itself is not ordinarily thought of as a generalization, but chemical engineering thermodynamics, because of its generality, is perhaps the most important and unique resource of our graduating students. Its generality with respect to all materials, not just air and water, and, in particular, the inclusion of chemically reacting ones, provides them with an advantage over graduates from mechanical and aeronautical engineering and even over those from materials science, to say nothing of those from electrical and civil engineering, which do not evoke thermodynamics at all. The inclusion of open systems provides a similar advantage over chemists and physicists, because of their almost-exclusive focus on Lagrangian formulations. New topics and approaches such as irreversible thermodynamics and statistical mechanics have simply been integrated into the chemical engineering courses in thermodynamics, which is one of the most important characteristics of success for a generalization. The repetitive numerical implementation of material and energy balances by students in chemical engineering and their similar repetitive experience in applying the second law provide a skill and confidence that is superior to that of any other professionals in even everyday tasks. Chemical engineering thermodynamics has been closely coupled to the unit operations since its beginnings. For example, the chapter headings of the 1944 book of that name by Dodge7 include the following topics: expansion, compression, fluid flow, heat transfer, refrigeration, chemical reaction, vaporization, condensation, and distillation. 7. Materials in General. The consideration of materials in a broad sense can also be interpreted in its own right as a generalization. The success of chemical engineering as a curriculum and as a profession has been due, to a significant degree, to the inclusion of all materials and phases, thereby extending its grasp beyond that of mechanical engineering, aerospace engineering, and materials science. No conceptual changes in the curriculum of chemical engineering have been required to embrace environmental considerations, biomaterials,

and nanomaterialssonly the inclusion of examples in the existing courses in thermodynamics, transport, reactor engineering, etc. 8. Open Systems. The consideration of open (Eulerian) systems as well as closed (Lagrangian) systems may also be interpreted as a generalization. This inclusion extends the grasp of chemical engineering beyond that of physics and chemistry, not only in thermodynamics but generally. 9. Chemical Reactor Engineering. Because of its omission from the unit operations and subsequently from the transport phenomena, chemical reactor design and analysis developed, beginning in the 1940s, as a separate academic subject and field of practice within the discipline of chemical engineering. It is now one of the glories of the profession and provides insights and methodologies beyond the normal reach and grasp of chemists, physicists, mechanical engineers, aerospace engineers, and material scientists. Chemical reactor engineering, although a most useful generalization in itself, serves as the archetypical example of the dangers of misguided subgeneralizations. Perhaps because it is separate development and isolation, several oversimplifications made in the interests of generalization have received widespread acceptance. First, as noted by Kabel8 in 1981 and reiterated in 1990, the distinction between rates of change and rates of reaction, as proposed by Churchill,6 has not yet been wholly excised from the literature. Second, the idealizations of plug flow and constant density in tubular reactors, packed beds, and fluidized beds, as well as the idealization of perfect mixing in continuous, stirred reactors, are still presented without adequate disclaimers in most textbooks, even though they are neither necessary nor justifiable simplifications when design calculations are performed with a computer. Turbulent flow is often cited as a justification for the idealization of plug flow, but it is rarely achieved in tubular reactors, and even then, the velocity profile deviates significantly from uniformity. Radial diffusion may be significant in both the laminar and turbulent regimes. If either the radial or the longitudinal variations in the velocity are taken into account, the very concepts of space time and space velocity become meaningless. The general acceptance of these idealizations in chemical reactor engineering is hard to understand in that no practitioner in the field of heat transfer would attribute any validity to solutions for convection based on the idealization of plug flow. Third, the representation of reaction rates, in terms of concentrations rather than chemical potentials, although generally justifiable on practical grounds, constitutes a discontinuity with what the students have just been taught in thermodynamics as well as an approximation of unknown validity. Fourth, the use of global models for either homogeneous or heterogeneous reactions has a very limited and generally unknown range of validity. Even if free-radical reactions are taken into account, the usually associated postulate of a pseudo-steady state is rarely valid (see, for example, Pfefferle and Churchill9). Fifth, as a final, all-too-characteristic example, the analytical expressions derived by Denbigh in 194410 for the optimal size distribution of a series of continuous, perfectly mixed reactors in terms of the total conversion per unit volume are often presented in textbooks without emphasizing the extreme postulate on which they are based, namely, an asymptotic approach to complete reaction in each stage. Denbigh noted that the economic optimum was always for reactors of equal size; however, this reservation is also omitted from most current textbooks. The oversimplifications inherent in these five categories may be useful in providing insight and an overview, but the students are done a disservice if they finish their

7854


schooling with only these idealized concepts, rather than a more rigorous and complete view of chemical reactor design. The purpose of this diversion is to emphasize that the gain due to a generalization may be counterbalanced by oversimplifications. 10. Separations. Although the unit operations include many methods of separation, this subject is often taught in a separate course and has always been one of the fortes of chemical engineering. It continues to develop more rapidly than many other unit operations and remains relevant in most of the current new subjects such as nanotechnology, biomedical technology, and molecular technology, as well as with new chemical and biochemical products of all types. It perhaps merits recognition as one of the generalizations that provide promise for the continuation of chemical engineering as a vital field. 11. Process Design. Process design is not ordinarily thought of as a generalization, but just as with thermodynamics, it has developed a unique character in chemical engineering education and constitutes an effective bridge between academia and industry. One reason for the employability of chemical engineers is their skill in process design, irrespective of the product, and a second reason is their preparation in the economics thereof. The common ground of the industrial chemists and mechanical engineers who founded the American Institute of Chemical Engineers (AIChE) was their self-taught skill in the design of chemical plants. Indeed, process engineering, including operation and research as well as design, could have been used as a synonym for chemical engineering for most of its first century. However, that commonality is now fading. Chemical engineers are honing and using their skills in other fields such as pharmaceuticals, nanotechnology, biomolecular engineering, and biomedical engineering that may not involve process design, but it will undoubtedly continue as a major endeavor for chemical engineers in connection with new chemicals and processes, as well as old ones. 12. Process Simulation. Process simulation may be considered an element of process design but it has developed a life of its own and may be considered to be a generalization. 13. Process Control and Dynamics. Process dynamics and process control have a unique history in chemical engineering. That role is expected to become even more important in the future in response to increasing demands on the purity of products and an increasing concern and involvement with the environment, living systems, and molecular assembly. Fortunately, that development will be supported by continual improvements in computer hardware and software. 14. Product Engineering. In its early days, chemical engineering was almost synonymous with process engineering, which proved to be to be a skill in ever-increasing demand as new products and new routes of manufacturing emerged. However, with many of the most recent products that involve chemical conversions, the scale of manufacture is so small that processing costs are less important than precision, reliability, and flexibility. Some associated changes in the curriculum are obviously necessary. Product engineering has been proposed to fulfill this need. It has great potential as a generalization, but, with all due respect to Cussler and Moggridge,11 the commonalities have not yet received widespread identification and/or acceptance. 15. Methodologies for the Analytical and Numerical Solutions of Models. Analytical and numerical methods of solution of models might be considered as generalizations in that they are, for the most part, not problem-specific. Despite their importance, they will not be discussed in detail herein

because they are currently being taught, except by default, in the framework of mathematics rather than in chemical engineering itself. 16. Sustainability. Sustainability has gained recognition in recent decades as a generalization encompassing environmental and related concerns, but only recently has it become a primary concern of chemical engineering, which is best positioned of all disciplines to take the leadership. For example, the conventional approaches are based only on the conservation of energy and chemical entities; however, as chemical engineers enter the field, the limitations imposed by the second law of thermodynamics (irreversible losses of energy) and rates (the dynamic carrying capacity of the surroundings) are also taken into account. (See, for example, Churchill and Neuman.12) 17. Scale. A major concern of chemical engineering from its very beginnings has been with the design and operation of fullscale industrial plants based on bench-scale experiments conducted in the laboratory, possibly with pilot plants and/or semiworks plants as intermediates. Mathematical modeling (including but not limited to dimensional analysis) has been of great assistance, and has been greatly abetted by the ever-improving capability for performing numerical solutions and simulations. Pharmaceuticals, nanotechnology, and environmental behavior are stretching the scales of concern both downward and upward. 18. The Future. Have all the possible broad generalizations of utility to chemical engineers already been discovered and exploited? The history of technology suggests that the answer is “no”. Can the well-known generalizations of the past, such as those previously described, be adapted for new products and processes? The history of our profession suggests that the answer is “yes”. For example, the version of thermodynamics distinct to chemical engineering is almost certain to evolve and remain an asset in radically new applications. Will broad generalizations continue to be a unique and essential characteristic of chemical engineering? It is too early to answer this question (that is, to predict success in developing generalizations for radically new applications of chemical engineering), but history suggests that new processes, products, and analyses will inspire new unifying concepts. If chemical engineering evolves completely away from process engineering, as is predicted by some writers, the role of broad generalizations may decrease, but that loss is by no means a certainty. The aforementioned broad generalizations that now, more or less, define chemical engineering evolved conceptually over a long period of time and were not universally recognized until they were certified by their appearance in textbooks. The unit operations concept was not fully articulated in a text until 30 some years after the first curricula in chemical engineering, reaction engineering after 24 more years, and the transport phenomena concept after another 13 years. Some time can be expected to elapse before equivalent generalizations and definitive textbooks emerge in the new fields of application of chemical engineering. In the interim, the best recourse is to test and determine the limits of applicability and adaptability of the well-established and broad generalizations in new fields as they arise. Although major changes in chemical engineering are occurring, they do not constitute a discontinuity. Chemical engineering students must be prepared for chemical manufacturing and for the processing of petroleum and other sources of energy on a large scale, as well as for the many new applications, which are primarily of smaller scale. The unit operations were a major factor in the early development of chemical engineering, and the other listed


generalizations have had a major role in its maturation. The generalizations have not only guided education and practice but have served from the beginnings to the present as a basis of communication between academia and industry and between different generations of chemical engineers. These generalizations are our common heritage and perhaps the main reason for the unity of our profession. If they vanish, lose their centrality, and are not replaced by others, the identity of our profession will certainly be weakened. Part II. Identification and/or Derivation of Generalizations of Limited Scope As contrasted with the generalizations of large scope, which are conceptual and nonmathematical, those of limited scope are quantitative and may be expressed in mathematical terms. Therefore, they are often derived directly by one of the following three means: observing quantitatively similar behavior in different systems; identifying dimensionless groups of variables that might be expected to characterize a broad range of behavior approximately, that is with a minimal parametric dependence; or devising comprehensive correlating equations that encompass several variables and regimes of behavior. The broad generalizations discussed in Part I are more or less unique to chemical engineering. On the other hand, the generalizations of limited scale that are discussed in Part II are shared to some extent with other branches of engineering. One other difference may be noted: the broad generalizations of Part I are somewhat limited in number, and most of the principal ones could be mentioned. Those of limited scope are countless. Even so, the representative ones that have been chosen for illustrative purposes are less in number than those chosen for broad generalizations. This is because, in part, many additional ones are presented in Part III in the context of simplifying concepts. The examples of generalizations of limited scope are limited to four, all from fluid mechanics and heat transfer. The limited number is in response to the unavoidable length of their descriptions and the choice of topic reflects the experiences of the author rather than their relative importance in chemical engineering in the past, present, or future. This disclaimer applies equally to the choice of topics, although not of number, for simplifying concepts and analogies. It is presumed that a similar set of illustrative examples could be chosen for other topics. It is proposed that this be done by someone with a greater knowledge of these. The choice of fluid mechanics and heat transfer for illustration has a shortcoming that should be mentioned. Mechanical engineering, and to a limited extent, aerospace and civil engineering share an interest in fluid mechanics. A similar set of simplifying concepts for mass transfer, reactor engineering, or thermodynamics would be almost unique to chemical engineering. 1. Hydraulic and Laminar-Equivalent Diameters. Perhaps the best-known example of an observational generalization is that of the so-called “hydraulic” diameter, which is equal to four times the cross-sectional area divided by the wetted perimeter. Because the geometrical rationale for this generalization is not very persuasive, it is classified here as observational. It fails badly for some laminar flows. For example, the predicted friction factor for a parallel-plate channel is underpredicted by 50%. The failures for two-dimensional channels such as open or rectangular ones can be explained, to some extent, by the presence of a secondary motion. For turbulent flows, the hydraulic equivalent diameter fails for many geometries but proves to be a very good approximation

for concentric circular annuli of all but very small or asymptotically large aspect ratios (see, for example, Kaneda et al.13). The failure of the hydraulic diameter concept to provide a good approximation for many turbulent flows long ago suggested an alternative, namely, the laminar equivalent diameter, that is, the diameter that predicts the exact value of the friction factor in laminar flow in a channel of the same geometrical configuration. However, this concept is unsatisfactory in two respects. First, it requires the availability of a theoretical solution for laminar flow for the geometry of interest, and second, its predictions are less accurate, on the whole, than those of the hydraulic diameter. 2. Heat-Transfer Coefficient and Its Analogues. In 1701, Newton14 observed that the rate of convective heat transfer seemed to be proportional to the temperature difference and the surface area, which, in modern terms and for unconfined flow, may be expressed as h ) q/A(Tw - T∞). The generalization consists of the replacement of four variables (namely, the heat flux, the surface area, the surface temperature, and the freestream temperature) by one variable (namely, the heat-transfer coefficient). The dependence of h on flow, etc., is implied to be the same for all values of q, A, Tw, and T∞. With minor modifications, for example, the replacement of T∞ by Tm for confined flows, this concept is applicable for almost all forms of thermal convection. The friction factor, drag coefficient, orifice coefficient, and mass-transfer coefficient serve the same role of generalization by virtue of replacing several variables by a relatively invariant one. However, these four coefficients are less successful as generalizations than the heat-transfer coefficient, because the first three are dependent on one of their components (namely, the velocity), and the fourth, in some instances, is dependent on the perturbation of the velocity by the mass transfer itself. 3. A Generic Correlating Equation. The process of correlation is an essential part of all experimental or computational investigations, except in the rare instance when the data or computed values are a sufficient and directly usable result by themselves. A correlation is inherently a generalization. Indeed, the success of a correlation may be measured by the degree of generalization that is achieved as well as by its accuracy. The pervasive use of computers has resulted in an overriding preference for correlating equations relative to graphical ones. In 1972, Churchill and Usagi15 proposed an answer to these several needs by devising a generic and very powerful method of correlation based on the algebraic expression

y{x}n ) y0{x}n + y∞{x}n

(1)

Here, y0{x} and y∞{x} are asymptotes or limiting values for small and large values of x, respectively. These asymptotes, or at least their functionality, are usually known from theory. Equation 1 has proven to produce very accurate representations for experimental data and numerically computed values of all sorts of behavior, even with multiple independent variables and multiple regimes (see Churchill16). How is this possible with such a simple expression as eq 1? First, this expression has the greatly reduced burden of representing the variance from the asymptotes rather than that of the overall expression. Second, parameters and secondary variables are often incorporated in the asymptotes. Third, eq 1 may be applied sequentially to encompass multiple independent variables and/or regimes. Figure 1, from Churchill and Usagi,15 illustrates the use of eq 1 to correlate the experimental data and numerically computed values for the dependence on the Prandt number (Pr) of the local rate of free convection in the laminar regime on a

7856


Figure 2. Correlation of the effective viscosity of a solution of polystyrene in a chlorinated diphenyl.

Figure 1. Correlation of data for laminar free convection from a vertical isothermal Plate (from Churchill and Usagi15).

vertical isothermal plate. The curves correspond to the following expression with several values of n:

η ) 7600 - 7440

Nux ) (0.5027Grx Pr ) + (0.6004Grx Pr ) n

1/4

1/4

1/4 n

1/2 n

The terms in brackets correspond to the theoretical solutions for a “thin” laminar boundary at asymptotically large and small Pr values, respectively. These two expressions introduce the dependence on Grx as a bonus. A value of n ) 9/4 may be observed to result in a very good representation for both experimental data and numerically computed values. The relative insensitivity of the overall representation to the value of the exponent is also illustrated in Figure 1. The indicated dimensionless groups of variables were chosen to limit the scales of the ordinates and abscissas, and arithmetic coordinates were utilized in the interests of displaying the absolute rather than fractional deviations. Equation 2 is ordinarily written in the following more compact form:

0.5027Grx1/4Pr1/4 Nux ) 0.4914 9/16 4/9 1+ Pr

[ (

) ]

as a function of the shear stress, is shown in Figure 2. The experimentally determined limiting values of 7600 and 160 P do not provide an acceptable pair of asymptotes, because the latter do not intersect. A curve representing following empirical asymptote for τ f 0,

(2)

The denominator of eq 2 has been found to provide a fair approximation for other bodies, such as a round horizontal cylinder, for other thermal boundary conditions, such as a uniform heat flux density from the surface, and even for the turbulent regime. The denominator of eq 2 has been called the “universal dependence on Pr”, but that is overgenerous praise, because it is an approximation even for the conditions for which it was devised, and can be improved at least slightly by the choice of different values of the exponent n and the coefficient of Pr for each condition. Similar success has been attained with eq 1 for literally hundreds of diverse types of behavior, and the derivation of correlating equations in terms of simple powers or series can no longer be recommended. All that is required to derive a correlating equation in the form of eq 1 are two or more appropriate asymptotes and a single intermediate experimental datum or theoretical value for each pair of asymptotes. The principal problem in utilizing eq 1 is the identification or derivation of appropriate asymptotes. They (i) must be free of singularities, (ii) must intersect once and only once, and (iii) both be upper bounds or lower bounds. An example of the latter difficulty follows. A very successful correlating equation for the effective viscosity of a solution of polystyrene in a chlorinated diphenyl,

τ (2150 )

8/5

(3)

intersects the straight line that represents the limiting value for τ f ∞ but is incompatible in the context of eq 1, because it is an upper bound whereas η ) 160 is a lower bound. This difficulty may be eliminated by redefining the dependent variable as η -160, leading to the trial expression

(2150 τ )

(η - 160)n ) 7400n + 7400n

n8/5

(4)

A value of n ) -5/3 leads to the correlating equation in Figure 2. Additional details concerning the evaluation of the coefficients and exponents in this case may be found in the work of Churchill and Churchill.17 The “staggering” of the dependent variable, as illustrated in this application, is often useful in connection with eq 1, because of the frequent natural pairing of an upper bound with a lower bound or the converse. One conclusion that followed from the construction of eq 4 was that the power-law expression for non-Newtonian behavior (also known as the Ostwald or Ostwald-de Waele law) may have no physical basis but may only be a mathematical artifact of a sigmoidal transition, namely, the necessary existence of a tangent through the point of inflection in logarithmic coordinates. One other nuance concerning eq 1 is the determination of derivatives and integrals of the independent variable of eq 1. Differentiation of expressions such as eqs 2 and 4 invariably yields functional forms that are singular or so complicated as to be of no practical utility. A more satisfactory result may usually be derived by differentiating or integrating the asymptotes as needed and combining the resulting expressions in the form of eq 1 with a different appropriate value of n. 4. Identification of a Characteristic Dimensionless Variable. The identification of a characteristic dimensionless variable that eliminates or minimizes the parametric dependence on other dimensionless variables constitutes a generalization, at least in an approximate sense. The identification of such a variable for the correlation of the effective thermal conductivity of dispersions provides a sufficient illustration. In 1873, Maxwell18 (on page 365) used the principle of imbedding to derive an exact solution for electrical conduction through a static dispersion of uniformly sized spheres that is


sufficiently dilute so that disturbance of the electric field by one sphere does not disturb that of any of its neighbors. In 1985, Churchill18 inferred that, by virtue of the analogy between electrical and thermal conduction, this solution must be applicable to the latter process. In so doing, he discovered that Maxwell’s solution could be re-expressed in the following morecompact and more-generalized form:

ke 1 + 2φ ) kc 1-φ

(5)

Here, kc, kd, and ke are the conductivity of the continuum, the dispersed spheres, and the dispersion, respectively; φ ) X(kd - kc)/(kd + 2kc); and X is the volumetric fraction of the spheres. This observation led him to plot various numerically computed and experimental data in the form of ke/kc versus φ. The exact numerical values for various dilute orderly arrays and sets of randomly dispersed experimental data for uniformly sized spheres were represented almost exactly, as might have been expected, but experimental data for dense dispersions, fluidized beds, and even packed beds of nonuniformly sized spheres and non-spherical particles were also represented reasonably well. Furthermore, in all cases, eq 5 serves as a lower bound. This remarkable result is primarily a consequence of the identification by Maxwell of φ as a composite variable. It is improbable that this variable (or eq 5) would ever have been identified from experimental data or even precise calculated values. Such a result is inspiring, not only because of the conceptual simplification that led to the model and the derivation of the solution, but also because its full potential was not recognized for over a century, thereby offering hope and encouragement that other similar solutions and models remain to be discovered or more fully exploited. One such result was the development by Churchill19 of an analogous generalization for cylindrical dispersions with their axes in the plane normal to the flow of heat, such as that which occurs with fibrous thermal insulations. 5. The Future. The aforementioned examples demonstrate the useful role of small-scale generalizations in chemical engineering in the past, and, in particular, in transport. That exposition raises essentially the same questions as those raised for the broad generalizations of Part I. Namely, have all the possible small-scale generalizations already been discovered and exploited? Should some of the widely accepted ones be modified, replaced, or eliminated? Can these generalizations and simplifications be adapted for new fields of current interest, such as biotechnology and nanotechnology? Can new generalizations be devised or discovered for these emerging fields? An answer of “no” to the first of these questions is supported by the above-cited contributions of the last two decades—for example, the belated recognition of the potential value and extended applications of this solution of Maxwell after more than a century. The answer to the second question is “yes”, insofar as generalizations describe fundamental behavior such as fluid mechanics and chemical reactions, and insofar as this same behavior prevails, as it may or may not, at the nanodimensional level. It is difficult to answer the third and fourth questions (that is, to predict success in advance for improvements in old generalizations or for developments in the radically new applications of chemical engineering). One clue may be mentioned. Some of the generalizations of the past, and particularly those of limited scale, had their origin with analytical models. Therefore, adaptations and developments in completely

new fields may be dependent primarily on success in the formulation of models in tractable mathematical form. In any event, it would seem worthwhile to test and determine the limits of applicability of old generalizations in new fields. The broad generalizations that, now, more or less define chemical engineering evolved slowly rather than bursting on the scene and were not universally recognized until they were certified by their appearance in a textbook. Some time may be expected to elapse before equivalent generalizations of small scale and definitive textbooks emerge in the new fields of application of chemical engineering. On the other hand, generalizations of more-limited scope may readily be derived or identified, almost on demand, for both old and new processes and products. The continued vitality of chemical engineering and its role in emerging technologies may well be dependent on the adaptation and extension of the generalizations of small scale of the past and the development of new ones. The author once asked Subrahmanyan Chandrasekhar (who was belatedly awarded the Nobel Prize for his predictions regarding “black holes” as a graduate student) about how he came to write the definitive text in several seemingly remote topics such as radiative transfer and fluid-mechanical stability; he replied, overmodestly, that great unrealized potential existed in most aspects of physics at the time they were abandoned as passe´ or unfashionable, and that he enjoyed exploring, correcting, and extending the contents of these dustbins. The recent recognition of unrealized potential in some classical solutions and conjectures examined herein and to be examined in Parts III and IV suggests that the dustbins of chemical engineering and allied technologies and sciences may hold more such gems. Part III. Derivation, Identification, and Evaluation of Simplifying Concepts Some successful and some questionable simplifying concepts of the past, including ones not recognized as such, are examined in Part III, and suggestions are made for the adaptation and extension of the successful ones in new applications, as well as for the derivation of completely different ones. As discussed in Part I, a major factor in the success of chemical engineering, relative to the other engineering disciplines, has been the generation and use of broad generalizations. On the other hand, the generalizations of limited scale that are illustrated in Part II have been shared to some extent with the other engineering disciplines. The simplifying concepts that have been devised for reactor engineering, mass transfer, separations, non-Newtonian flow, fluidized beds, and packed beds are more or less unique to chemical engineering but those for fluid flow and heat transfer in general are shared with some of the other disciplines of engineering. The dividing line between generalizations and simplifications is not clear-cut, because some simplifications may lead to generalizations by providing asymptotes for comprehensive correlating equations or by identifying the relative importance of individual variables or groups of variables. Simplifying concepts might be expected to become obsolete more rapidly than do generalizations, and indeed, as noted in what follows, many classical concepts of the past are now known to be erroneous or misleading. The identification of such shortcomings is perhaps as important as identification of the successes (hence, the added word “Evaluation” in the title of Part III). At the same time, simplifying concepts that result from conjecture (for example, Einstein’s theory of relativity) may precede their experimental proof by decades.

7858


Some of the simplifications described herein have arisen from observations or conceptual thinking, rather than by means of mathematical analyses. The attribution of a conceptual simplification to an unquestioned genius such as Newton, Rayleigh, and Langmuir might imply that the discovery of simplifications is beyond ordinary mortals. However, the presence of my own name in several examples proves otherwise. Some of the classical techniques for deriving simplified models and representations (namely, dimensional, speculative, and asymptotic analysis) are first described generally and then illustrated for specific cases. Chemical engineering students are generally exposed to these several techniques but seldom practice them sufficiently to gain a working knowledge. It is essential that, in the future, they do acquire such skills, because in an era of change, they cannot rely entirely on the classical simplifications of the past, and must be able to adapt old techniques and derive new concepts on their own. Part III includes many more examples than Part II, because of (i) the need to identify the existing resources in the form of simplifying concepts, whose origin and limitations may not be well-known to chemical engineers; (ii) the need to identify obsolete simplifying concepts and the resulting obsolete expressions that still may lurk in textbooks and handbooks; and (iii) the need to provide guidance for the development of new and improved simplifying concepts. 1. Examples of Directly Conceived Simplifications. The identification of simplifications from observation is dependent somewhat on experience but primarily on a proper mindset and receptivity. The following examples illustrate the possibilities and provide implicit guidance for the recognition of others. They are also limited almost exclusively to fluid mechanics and transport, out of deference to brevity and the experience of the author. Again, it is proposed that those with better credentials identify equivalent simplifying concepts in other topics. A. Fully Developed Flow. Fully developed flowsthat is, the absence of changes in the velocity profile with length in the direction of flowsis regularly postulated or inferred in the interests of simplicity and/or convenience. However useful, fully developed flow is actually a hypothetical condition that is only approached asymptotically and that may require as many as 100 pipe diameters in axial distance, depending on the value of the Reynolds number (Re) and the configuration of the entrance, to be an acceptable approximation. B. Fully Developed Convection. The concept of fully developed convection is also regularly postulated, often without adequate justification. This asymptotic condition implies invariance of the dimensionless radial temperature profilesfor example, (T - Tm)/(Tw - Tm), (T - Tw)/(Tc - Tw), or k(T Tm)/ajwsand of the heat-transfer coefficient with axial distance, even though the temperature of the fluid at all radii continues to vary. It is commonly applied for a uniform wall temperature and for a uniform heat flux at the wall. The length of a tube for which this is an acceptable approximation is dependent on the mode of heating at the wall and on Pr as well as on Re, but is generally less than that required for the attainment of fully developed flow. Despite their limitations, which should always be kept in mind, fully developed flow and fully developed convection both qualify as simplifying concepts of great utility. C. Time Averaging. In 1895, Reynolds20 space-averaged the partial differential equations of conservation to make them more tractable for the analysis of turbulent flow. This procedure, which is now generally replaced by time-averaging, results in a great simplification without any loss of generality other than on a very short time scale, but it does introduce one or more

new independent variables, as discussed subsequently. It has been the starting point for all analyses of turbulent flow until the recent advent of direct numerical simulation (DNS). Despite questions concerning its validity by purists, it is perhaps the most powerful and useful simplifying concept of all time. D. Ohm’s Laws. In 1827, Ohm,21 based on theoretical conjectures, concluded that “the force of the current (amperage)...is as the sum of all the tensions (Voltages) and inversely as the entire reduced length (the resistance) ...of the circuit...” (words in parentheses have been added herein). Moreover, he used algebra to show that this law implies that resistances in series and the reciprocals of those in parallel are additive. The concept of the additivity of resistances has been recognized as applicable to all linear processes and has been proven invaluable in many aspects of chemical engineering, including thermal conduction, combined conduction and convection, and combined diffusion and reaction in packed beds of catalyst pellets. E. Radiative Heat-Transfer Coefficient. The introduction of the radiative heat-transfer coefficient,

hr ≡ σ

(

)

Tw4 - T∞4 = 4σTave3 Tw - T ∞

extends the applicability of the concept of the additivity of resistances to the combination of radiation and conduction and/ or convection insofar as hr may be postulated to be independent of Tw. F. Potential Flow and the Boundary Layer Concept. In 1905, Boussinesq22 derived solutions for forced convection in potential flow. These solutions, which were pioneering at the time, are now recognized (see the 1990 work of Galante and Churchill23) to have practical value, if at all, only as asymptotes for Pr f 0 (in which role they have been utilized extensively in correlations in the form of eq 1) and in the boundary layer model of Prandtl24 of 1904, in which he conceived a thin layer of slowly moving fluid inside a regime of potential flow. This “thin” boundary layer concept greatly simplifies the modeling of unconfined flow and convection and has been postulated in most theoretical solutions for the laminar regime and in some for the turbulent regime, despite the severe and often overlooked idealizations. G. Effective Roughness. In 1938, Colebrook25 ingeniously defined an effective roughness for commercial piping as the value that results in the same friction factor for asymptotically large values of Re as that given by a correlating equation of Nikuradse for uniform artificial roughness. This simplification led him to the development of a single, empirical expression for the friction factor for both smooth and naturally rough piping for all values of Re in the regime of fully turbulent flow. Although the concept avoids the introduction of a new variable, a tabulation of the effective roughness for different types of piping is required for its application. H. Models for the Local Shear Stress due to Turbulence. The price to be paid for the information lost by time-averaging is the appearance of additional dependent variables such as u′V′ in the partial differential equations of conservation. Much of the subsequent history of fluid mechanics has involved the postulating, testing, and substitution of empirical models for these terms. A few of these models are described here. Boussinesq26 in 1877 (and, therefore, prior to the identification by Reynolds20 in 1895 of the relationship, between the turbulent shear stress and the fluctuating components of the velocity) proposed the representation of the turbulent shear stress by an effective viscosity based on a postulated analogy between


molecular and eddy diffusion. This speculative concept, which is still widely used today, was shown to be invalid for annuli by Kjellstro¨m and Hedberg27 in 1966, but valid (independent of its original heuristic origin) for round tubes and for parallelplate channels with identical surfaces by Churchill28 in 1997. Prandtl29 in 1925 conceived a mixing length for turbulent eddies as an analogue of the mean-free-path of molecules and as an alternative to the eddy viscosity. The mixing length was eventually shown by Churchill28 to be singular at the centerline and invalid near the wall. Prandtl was mislead in this respect by the inaccuracy of the experimental data of Nikuradse30 and by the fortuitous derivation of a semilogarithmic expression for the radial distribution of the time-averaged velocity (see Part 3, Section 2 for a valid derivation), demonstrating that theoretically unsound models may result in reasonably accurate predictions if they are forced to conform to experimentally observed behavior by the choice of empirical constants. In 1995, Churchill and Chan31 proposed modeling the turbulent shear stress itself in terms of -F(u′V′)/τw, which they symbolized as (u′V′)+, thereby avoiding the singularity of the mixing length in all flows and that of the eddy viscosity in annuli. The closely related dimensionless variable proposed by Churchill27snamely, the local fraction of the transport due to the fluctuating velocities, -F(u′V′)/τ, which he symbolized by (u′V′)++sfurther simplifies the formulations of the differential momentum balance but is singular in annuli, although possibly in a tolerable sense. Results obtained with these several models (and, in particular, the latter one) are examined subsequently. This set of models for turbulent shear stress is limited to those that are based directly on conceptual simplifications. (See the 2000 work by Churchill32 for a more complete description.) 2. Classical Techniques of Simplification. Four such techniques are examined here: dimensional analysis of a list of variables, dimensional analysis applied to a mathematical model, speculative analysis, and asymptotical analysis. A. Dimensional Analysis of a List of Variables. This technique simplifies and generalizes the relationship between variables by combining them in dimensionless groupings that preempt the individual dependences. All undergraduate students in chemical engineering are exposed to dimensional analysis, but few attain a real understanding or a working knowledge. This is unfortunate, because they will almost certainly have the opportunity and will need to apply this technique in new situations. Indeed, this skill is only slightly less important than that required to apply the three laws of thermodynamics. The 1915 method of Rayleigh33 for a listing of dimensional variables, which is simple, straightforward and infallible, should be taught to all students of chemical engineering. (See, for example, the work of Churchill.34) All prior and subsequent methodologies are wrong, difficult to apply, and/or confusing, and should be avoided in the undergraduate classroom. The results of dimensional analysis are exact, insofar as the relevant variables are chosen. Omitting a significant variable necessarily leads to an incomplete relationship, while including a redundant variable causes little harm, because the redundancy will usually be revealed when the dimensionless groups are used for correlation. The selection of an alternative variable (for example, the pressure gradient rather than the shear stress on the wall in flow through piping) results in an alternative and equally valid set of groupings; however, the results of further simplifications for asymptotic conditions may be critically dependent on the original choice, as illustrated subsequently.

At first, students lack confidence in their ability to select the relevant variables. Churchill34 suggested that this concern can be alleviated or avoided altogether by interpreting the entire process as speculative, and simply exploring the consequences of the various choices. B. Dimensional Analysis Applied to a Mathematical Model. Hellums and Churchill35 devised a method of dimensional analysis for algebraic, integral, and/or differential models that includes the identification of similarity transformations, that is, combinations of dependent and independent variables that allow reduction of the order of the model (for example, from a partial differential equation (PDE) to an ordinary partial differential equation (ODE)). Such transformations are usually possible only for somewhat simplified models, such as that for thin-laminar-boundary-layer theory, but the application of the Fourier equation for transient one-dimensional conduction is an exception that may be utilized in the classroom as a first example without the necessity of simplifications other than the postulate of invariant physical properties. This technique, which is also illustrated in Chapter 9 of the earlier work by Churchill,6 can readily be taught to undergraduates, thereby providing them with a great boost in understanding and confidence, in that they do not need to accept similarity transformations, such as the well-known procedures used by Blasius,36 Le´veˆque,37 and others on faith. C. Speculative Analysis. The arbitrary methodical elimination of each variable, one at a time, and then perhaps two or more at a time, from a listing or from a previously derived set of dimensionless variables is a useful supplement to dimensional analysis and may lead to simplifying concepts in the form of asymptotes for limiting conditions. The scope and degree of validity of the resulting set of dimensionless groupings in generally representing the behavior and for particular ranges must be determined by reference to experimental data or computed values. Prandtl was a pioneer in the use of speculation combined with dimensional analysissa process in which he displayed great ingenuity. By this means he constructed extensive structures for both laminar and turbulent flow. Although most of his results have been superseded, they have an afterlife in the history of fluid mechanics and as a guide to the application of this technique. In the event of a model, variables and the associated term or terms may also be methodically eliminated one term at a time. A separate and supplementary procedure is to replace a term or terms by approximations on the basis of physical insight. Dimensional analysis of the differential model must be repeated for each reduction or approximation. This process is tedious, but it should not be neglected, because the potential rewards are great. Dimensional analysis, coupled with speculation, should be the starting point of all analyses. Despite the remarkable success of its application to some aspects of fluid mechanics, heat transfer, mass transfer, and catalyzed chemical conversions, it seems to have been underutilized in many subjects and, hence, to have great, unexploited potential for the development of new generalizations in both traditional and emerging applications. At the same time, it should be emphasized that not all of the results obtained by speculation are correct and that testing them with experimental data and/or “exact” computed values is essential. Furthermore, this testing must be conducted very carefully. For example, if the values scatter even slightly, it is difficult to distinguish in a logarithmic plot the success of

7860


representations attained using a 1/3-power and a 1/4-power dependence. D. Asymptotical Analysis. Speculative analysis is only one form of asymptotic analysis. Although more formal methods of asymptotic analysis are an important source of simplifying concepts and relationships that should not be overlooked, their mathematical demands and details are beyond the scope of the current presentation. 3. Examples of Dimensional, Speculative, and Asymptotic Simplifications. A. Invalid and Valid Results of Dimensional Analysis of a Listing of Variables. Nusselt38 in 1909, and thereby shortly before the publication of the aforementioned article on dimensional analysis by Rayleigh in 1915, postulated the proportionality of the dependent variable to the product of powers of the independent variables and thereby obtained the following expression for fully developed forced convection in a round tube:

( )( )

DumF hD )A k µ

a

cpµ k

b

(6a)

or

Nu ) AReaPrb

(6b)

Nusselt is justly commemorated for deriving and identifying the dimensionless group that now bears his name in the process of deriving eqs 6. However, the expression itself is erroneous, because of his speculative postulate of a power dependence. The correct result, as obtained by the method of Rayleigh, is

{

}

DumF cpµ hD )φ , k µ k

(7a)

Nu ) φ{Re,Pr}

(7b)

or

We now know that relationships in the form of the product of powers of the variables occur only for asymptotic conditions. Even so, eq 6 and it counterparts have infected all of chemical and mechanical engineering like a virus and will not soon be excised. The next generation of students should be inoculated against this misapplication of dimensional analysis and warned of the fundamental shortcomings of correlating equations with such a form. B. Dimensional and Speculative Analysis of Turbulent Flow. Because turbulent flow is so intractable in a mathematical sense but so important in process engineering (and, in particular, in heat transfer), it has been a principal focus of speculative analysis and provides several related illustrations. In 1913, Blasius,39 on the basis of a logarithmic plot of experimental values of the friction factor f versus the Reynolds number Re, correlated experimental data for flow in round tubes with the expression

f)

0.3867 Re1/4

(8)

In 1929, Prandtl40 inferred from eq 8, on the basis of an ingenious application of dimensional analysis, the following expression for the velocity distribution:

y 1/7 u ) uc a

()

This expression provides only a crude representation for experimental data and obviously is invalid at both y ) 0 and y ) a. Its only lasting value is the implicit proof that a power dependence of f on Re is fundamentally invalid. Equation 9 illustrates that even an ingenious and basically sound dimensional analysis may fall short, because of an erroneous starting point. In any event, its repetition is a moderately challenging and informative exercise for students. Blasius was deceived because of the limited range of the experimental data that were available to him and Prandtl misinterpreted the inverse integral value of the exponent as a result with theoretical significance. [As an aside, eq 8 appears without a warning about its limitations in the latest version of Bird, Stewart, and Lightfoot.4] In 1926, Prandtl41 applied dimensional analysis to the list of variables that he presumed to describe the radial distribution of the time-averaged velocity in fully developed turbulent flow in a round tube and thereby obtained

(9)

u

() { F τw

1/2

)φ

}

y(Fτw)1/2 a(Fτw)1/2 , µ µ

(10)

He introduced the mathematical shorthand abbreviations u+, y+, and a+ for these three dimensionless variables; this notation has endured to the present. He next conjectured that “near the wall”, u+ might be independent of a (and thereby of a+) for all values of Re, resulting in

u+ ) φ{y+}

(11)

Comparison of experimental measurements of the velocity by Nikuradse42 for a wide range of Re values revealed that the conjecture represented by eq 11 provides an excellent approximation except very near the centerline. Had Prandtl performed the dimensional analysis with τw replaced by its exact equivalent (namely, a(-dP/dx)), the expression replacing eq 11 would have been equally correct but the result obtained by eliminating a from that formulation would not have led to the invaluable generalization expressed by eq 11. This choice of the variables leading to eq 10 was not fortuitous: Prandtl chose τw rather than dP/dx with either foresight or aftersight. Recognizing the value of asymptotes such as eq 11, Prandtl derived several others for turbulent flow. For the region very near the wall, he postulated that the variation of the shear stress with radius and the contribution of the turbulence could be neglected. (The latter is equivalent to the postulate of a viscous sublayer.) The consequent integration yields

u + ) y+

(12)

It might seem surprising that Prandtl did not take into account the known variation of the shear stress and thereby obtain

(

u + ) y+ 1 -

)

y+ 2a+

(13)

The probable explanation is that he knew, based on the experimental data, that eq 13 did not improve eq 12 significantly. For the region near the centerline, Prandtl speculated that du/ dy might be essentially independent of the viscosity, and he thereby derived the following asymptote for that region: + u+ c -u )φ

{} y+ a+

(14)


The subtlety of switching from u to du/dy as the dependent variable in the derivation of eq 14 should be noted. Most of us are not as clever as Prandtl, but we can compensate, to some extent, by trying and testing different variables and speculations. In 1938, Millikan43 speculated that an intermediate region might exist in which both eqs 11 and 12 were applicable, and he recognized that the only expression satisfying this requirement was

u+ ) A + B ln{y+}

(15)

Equation 15 has been determined, based on the experimental measurements, to provide a good approximation for a+ > 300 and 30 < y+ < 0.9r/a, which is a region Prandtl called “the turbulent core near the wall”. The combination of eqs 14 and 15 results in + u+ c - u ) B ln

{ay}

(16)

Equation 16 is applicable only for the same region as eq 14, and thus not near the centerline, although it conforms to eq 14. Equations 10-16 are a testament to the ingenuity of Prandtl and Millikan and to the power of speculation combined with dimensional analysis. Once demonstrated and practiced, this methodology is within the capability of all of us. Even so, it does not seem to have been fully exploited outside of fluid mechanics and heat transfer. In 1955, Rothfus and Monrad44 determined a set of conditions that forced the velocity distribution in fully developed laminar flow in a round tube and in a parallel-plate channel to be congruent. In 1990, Churchill45 asserted that these specifications were excessive and that expressing the solutions in terms of u+, y+, and a+ or b+ was sufficient. (This conjecture of Rothfus and Monrad, as well as the generalization of Churchill that followed, can be interpreted, in advance of Part IV, as an analogy for laminar flow in round and parallel-plate channels.) In 1951, Rothfus and Monrad attributed to MacLeod46 the related conjecture that the same conditions that resulted in congruence of the velocity distributions in laminar flow might do so in turbulent flow. This remarkable conjecture was tested critically by Whan and Rothfus,47 who, in 1959, compared + experimental values of u+ c as a function of a graphically. The agreement is very good in the regimes of fully laminar and fully turbulent flow but fails completely in the regime of transition from laminar to turbulent flow. (This limitation is of little concern, because the regime of transition is avoided in practice.) Because the extension of the generalization of MacLeod for laminar flow by Churchill45 for turbulent flow has not been been confirmed by theoretical analysis, it must be considered to be both tentative and approximate, despite its apparent success. Because the fraction of the local shear stress due to turbulence, namely, F(u′V′)/τ [not to be confused with F(u′V′)/τw, which is the local shear stress as a fraction of that at the wall], is equal to du+/dy+ in both round tubes and parallelplate channels, in 1995, Churchill and Chan31 recognized that a commonality in u+ such as those of Prandtl and of Rothfus and Monrad must apply to F(u′V′)/τ. This expanded generalization, although tentative and approximate in a theoretical sense, and limited to these two geometries, is invaluable, because the experimental data, numerical solutions, and correlating equations are implied to be interchangeable. An important consequence of the interchangeability is that numerical solutions by direct numerical simulation (DNS), which are effectively limited to parallel-plate channels, may be adapted for round tubes.

The results obtained by speculative analysis may be dependent on the arbitrary choice of variables of equal standing. For example, if the mass rate of flow, rather than the velocity, had been chosen for the listing leading to eq 11, the elimination of the radius a would not have produced a useful expression. Therefore, all such alternatives must be explored and the more useful ones determined on the basis of experimental data. C. Derivation and Utilization of Improved Asymptotes for Turbulent Flow. Equations 12 and 14 are valid asymptotes but are unsatisfactory for combination in the form of eq 1: the first one because it is too limited in range and functionality, and the second one because the functionality is unknown. Various formal and informal asymptotic analyses have predicted both third-power and fourth-power dependences of the turbulent shear stress on distance from the surface, which is directly linked to the time-averaged velocity distribution. The asymptotic dependence of the turbulent shear stress on the distance from the wall is critical for the prediction of convection as well as for flow, leading to 1/3-power and 1/4-power dependences, respectively, of Nu on Pr for asymptotically large values of Pr. After half a century of controversy, this discrepancy has finally been resolved in favor of the 1/3-power, by virtue of DNS, which also provides a theoretical value for the coefficient, resulting in the following asymptotic expression:

(u′V′)++ ) 0.00070(y+)3

(17)

Because of limitations in the current accuracy of the DNS, the coefficient of eq 17 is only known to perhaps two significant figures. The corresponding asymptote for the forced convection for asymptotically large values of Pr is

Nu ) 0.0734Re

() ( ) f 2

1/2

Pr Prt

1/3

(18)

The long-lasting, although now resolved, controversy regarding the power dependence of (u′V′)++ on y+, and, hence, regarding that of Nu on Pr for Pr f ∞, illustrates the possible uncertainty that may arise with seemingly valid asymptotic analyses. Integration of the momentum balance using eq 17 for (u′V′)++ results in the following explicit asymptote for u+ in the region near the wall:

u+ ) y+ - 0.00175(y+)4

(19)

The corresponding asymptote for the region near the centerline, namely,

( )

+ u+ c -u )R 1-

y+ a+

2

(20)

follows from the speculation that (u′V′)++ approaches a fixed value at the centerline. Equation 20 was first derived by Hinze48 in 1959, based on the postulate of a limiting value of the eddy viscosity, rather than of a limiting value for (u′V′)++ Equations 19 and 20 can be observed to conform to eqs 11 and 14, respectively. Equations 15, 19, and 20 have far-reaching consequences. For example, they and their counterparts for (u′V′)++ have been combined in the form of eq 1 to obtain predictive expressions, with truly minimal empirical uncertainty, for all values of y+ and a+ for round tubes, and presumably in terms of y+ and b+ for parallel-plate channels as well. Most of the expressions for the friction factor in the current literature are based on integration of eq 15 over the cross section,

7862


Nux ) 1.167Gz1/3

which results in

(2f )

1/2

≡ u+ m)A-

B + B ln{a+} 2

(21a)

The term A - B/2 has ordinarily been adjusted numerically to compensate empirically for the neglect of the deviations from eq 15 at small and large values of y+, as represented here by eqs 19 and 20. Yu et al.49 instead twice integrated the asymptotes for (u′V′)++ analytically and the predicted values numerically to obtain

() 2 f

1/2

≡ u+ m ) 3.2 -

()

227 1 50 2 + + + ln{a+} + 0.436 a a

(21b)

Equations 21 evoke no empiricism beyond that of the predictive equation for (u′V′)++. The three coefficients that result from B and R are given here numerically, in the interests of simplicity. The unfamiliar terms in (a+)-1 and (a+)-2 result from eq 19. Equation 21 may be re-expressed in terms of Re by noting that a+ ) Re(f/8)1/2. This subsection, and the prior one, illustrate the remarkable results that are possible by speculative dimensional analysis coupled with asymptotic analyses, even in such an inherently difficult context as turbulent flow, which is three-dimensional, unsteady state, and chaotic. D. Approximation of Two Terms by One. The so-called Boussinesq transformation for free convection involves the replacement of -g + (-∂p/∂x)/F by -gβ(T - T∞) in the differential energy balance, the replacement of F by F∞ in the differential momentum balance, and the postulate that gβ(T T∞) is small with respect to unity. Almost all analytical solutions for free convection have been derived from the so-simplified model. [See Hellums and Churchill50 for a more detailed discussion of this simplification.] E. Dimensional Analysis with Progressive Simplifications. The result of dimensional analysis of the list of variables describing developing convection in fully developed laminar flow with invariant physical properties and negligible viscous dissipation in the inlet of a round tube with isothermal heating at the wall may be expressed as

Nux ) φ{Re, Pr, x/D}

(22)

Dimensional analysis of the corresponding differential model by the method of Hellums and Churchill35 results in the following reduced result:

Nux ) φ{xRePr/D} ) φ{4Gz/π}

(23)

This result is consistent with the much earlier complete solution in series form as obtained by Graetz51 in 1883 by separation of variables, and it demonstrates the advantage of the method even when a similarity transformation is not identified. In 1928, Le´veˆque37 simplified the model for developing convection in fully developed laminar flow in an isothermally heated round tube by approximating the velocity distribution by its asymptote at the wall (namely, 4umy/a), the conduction term (namely, (k/r)(∂/∂r)(r(∂T/∂r))), by k(∂2T/∂y2), and postulating that the centerline and mixed-mean temperatures differed negligibly from their value at the inlet. He then ingeniously recognized that a similarity transform could be utilized to reduce the resulting partial differential model to an ODE, which he solved in closed form to obtain

(24)

Physical insight and ingenuity are unavoidable for such a simplification of the model; however, with the method of Hellums and Churchill, the subsequent determination of the similarity transformation and the solution of the ODE become routine. Indeed, White and Churchill52 wrote a computer program to perform both. Note that (i) the Le´veˆque solution neglects the effects of curvature, and (ii) Nu is based on the temperature of the fluid at the inlet, rather than on its mixedmean value at x. F. Dimensional Analysis with Alternative Speculations for Turbulent Free Convection. The history of speculative analysis of turbulent free convection from a heated vertical plate illustrates the risks of speculative dimensional analysis. In 1915, Nusselt53 made the apparently correct speculation (namely, independence of Nux from an upward distance along the plate beginning with the point of onset of the heating, which requires a proportionality of Nux to Grx ) gF2β(Tw - T∞)x3/µ2 to the 1/3-power. The further speculation that the inertial terms become negligible, relative to the viscous terms, as Pr f ∞ due to increasing µ implies the proportionality of Nux to Rax1/3 ) (GrxPr)1/3 ) [gF2cpβ(Tw - T∞)x3/µk]1/3, whereas the speculation that the viscous terms become negligible, relative to the inertial terms, as Pr f 0 due to decreasing µ implies the proportionality of Nux to (RaxPr)1/3 ) (GrxPr2)1/3 ) [gβcp2(Tw - T∞)x3/k2]1/3. These speculative results seem to be validated by the somewhatlimited experimental data and computed values. On the other hand, some subsequent analysts have either ignored or disputed these results. For example, in 1937, FrankKamenetskii54 postulated independence of the heat-transfer coefficient from both the viscosity and the thermal conductivity as Grx increases, which leads to the proportionality of Nux to Grx1/2Pr, while, in 1951, Eckert and Jackson55 utilized integralboundary-layer theory with a postulated velocity distribution to obtain a 0.4-power dependence of Nux on Grx. Others have postulated that the inertial terms become negligible as Pr f ∞ due to decreasing k rather than increasing µ, thereby finding Nux proportional to Grx1/3 rather than to Rax1/3. Only experimental data or numerically computed values can identify the correct dependence and thereby the correct speculation. G. Lower Limiting Values of the Nusselt Number (Nu). In 1912, Langmuir56 utilized the concept of an effective film thickness to derive the following first-order approximation for the effect of curvature on buoyant convection from the cylindrical filament of a light bulb, namely,

Nucyl )

2 ln{1 + (2/Nuflat)}

(25)

Equation 25 is based on the recognition that thermal conduction across an annular film is proportional to the logarithmic mean of the outer and inner radii. Here, Nuflat represents a correlating equation or a value for a vertical plate and Nucyl represents the corresponding values for a horizontal cylinder. The representation of convective heat transfer by the equivalent thickness for thermal conduction has long since been supplanted by the heattransfer coefficient, but eq 25 remains useful as an asymptote for vanishingly small rates of either forced or natural convection from a long horizontal cylinder. For free or forced convection from the outer surface of a sphere, Nu can be shown from theory to approach 2 as Re or Gr approach zero. Although Langmuir did not derive this value, the application of his effective-film-thickness formulation,


together with the geometric-mean area, leads directly to it. All other finite bodies have a lesser limiting value of Nu, which itself is a useful simplifying concept. The concept of an effective film thickness is also applicable for the effect of curvature on forced convection in the entrance of a round tube. The asymptotes based on this concept of Langmuir are useful to estimate numerical values of Nu, but their greatest value is as components of correlating equations for all Re or Gr. H. Speculative and Dimensional Analysis of Packed-Bed Catalytic Reactors. In 1937, Damko¨hler57 performed an exhaustive speculative dimensional analysis for chemical reactions in flow through packed beds of catalyst pellets that remains a landmark to this day. His analysis, and its exploitation by Hougen and Watson58 in 1947 and others, have a negative side. The coefficients and even some of the exponents of these models are generally not predictable from theory. In addition, the precision and accuracy of the available experimental data are generally insufficient to evaluate all of these coefficients and exponents uniquely. This limitation is often ignored, resulting in correlating equations that reproduce the data with some degree of success but that have no generality or predictive value. I. Simplified Models for Reaction Mechanisms and Reactors. Pseudo-first-order models for complex reactions may have conceptual value, but they are no longer justified in process design. The pseudo-stationary-state concept for gas-phase free-radical reactions developed in 1919 by Christiansen59 has been invaluable in providing an explanation for experimental observations of apparent fractional-order and bilinear-order reactions, but is inapplicable for combustion, because of the combination of high temperatures and short residence times. Global models for combustion and other complex reactions no longer have any justification in the age of computers, but the lumping of closely related molecules may still be a justifiable approximation for combustion and thermal cracking, because of the great reduction of the kinetic model at the price of a tolerable error. In 1954, Calderbank60 observed that the rate of thermal cracking of ethane, propane, and butane in flow through radiantly heated tubes in a furnace was essentially equal to the rate of heating, and he recognized that this was because the effect of the heat of reaction for these endothermic reactions was counterbalanced by the temperature dependence of Arrhenius, resulting in an almost-fixed temperature, characterized by the particular reactant and conditions. This simplification is of limited value, because a complete model must be solved to determine the approximately constant temperature. The models of Langmuir,61 Hougen and Watson,58 Damko¨hler,57 and others for mass transfer, adsorption, surface diffusion, and surface reaction in catalytic, packed-bed reactors are conceptually invaluable, but correlations based on them should be recognized as potentially misleading, because rarely does the accuracy of the experimental data justify the evaluation of so many empirical constants. The plug-flow reactor and the related concepts of space velocity and space time, as well as the continuous, perfectly mixed reactor, should likewise be recognized as gross idealizations. These simplifications provide insight with respect to firstorder effects, but the failure to identify them as asymptotes seriously misleads students and impedes the use of more-exact models and modern computational tools by practicing engineers. J. Height of a Transfer Unit. The energy balance for a stream of liquid passing through the central round tube of a

double-pipe heat exchanger may be expressed as

wcp dT ) U(t - T) dx

(26)

Here, w, cp, and T are the mass rate of flow, the heat capacity, and the mixed-mean temperature of the stream; t is the temperature of the fluid supplying or removing the heat, D is the inner diameter of the tube, x is the distance through the exchanger, and U is the overall heat-transfer coefficient based on D. Equation 25 may be integrated formally to obtain

L)

w πD

∫ U(tc -dTT) ) πDw (cU) ∫ t -dTT T2

T1

p

T2

p

m

T1

(27)

Here, L is the length of the exchanger, T1 and T2 the inlet and outlet temperatures, and m designates an integrated-mean value. The right-most integral, which is a measure of the difficulty or the extent of the heat exchange, is called the number of transfer units and is symbolized by NTU, whereas w(cp/U)m/πD is called the height of a transfer unit and is symbolized by HTU. If t is constant, as is almost the case for a condensing or boiling fluid, or linearly related to T, as it is approximately the case for countercurrent flow, the integration may be performed analytically to obtain (T2 - T1)/(t - T)lm, where the subscript “lm” designates the logarithmic-mean value. This formulation has some conceptual but almost no practical value, because the correlations for HTU are less developed than those for the heattransfer coefficients h and U, and, with a computer, the evaluation of the integral or the integration of eq 27 and the corresponding one for t, is not onerous. In 1935, Chilton and Colburn62 originally conceived the HTU-NTU formulation, in terms of mass transfer in packedbed or wetted-wall columns, for which the behavior is more complex, in that the rate of flow of the two streams varies and the interfacial surface for transfer may not be known with certainty. This is a simplifying concept that has not lost its conceptual validity but has lost its usefulness for numerical evaluations of heat and mass transfer, because of the advent of computers. K. Simplifying Concepts for Separations. The most two famous simplifying concepts for separation are that of an equilibrium stage for distillation and extraction, and the graphical representation of stage-wise processes devised by McCabe and Thiele63 in 1925. Although these concepts incorporate many idealizations, they retain their place in the curriculum and in practice, because they provide great insight. The concepts of the height of a transfer unit, which was discussed previously, and of an equilibrium stage were combined by Peters64 in 1922, in terms of the height of a theoretical plate (HETP) in a packed-bed column used for distillation or extraction. Despite the lack of a firm basis for this quantity, correlating equations were developed for it and experienced engineers developed a sense of its magnitude. The HETP is mentioned here as an example of a simplifying concept that has rightfully been relegated to the dustbin of history, not because of the advent of computers, but its shortcomings outweigh its conceptual value. L. Simplifying Concepts in Thermodynamics. The ideal gas law, van der Waals equation, and the law of corresponding states are examples of simplifying concepts in thermodynamics; however, they are so well-known that no discussion is needed here. Gibbs’ phase rule is one of the ultimate simplifying concepts of all time. On the other hand, equilibrium, isentropy, and the existence of a steady state, may not be recognized as

7864


simplifying concepts of extraordinary utility. This is only a sampling of the countless ones that have been proposed in thermodynamics. 4. The Future. The aforementioned examples demonstrate the extensive and useful role of conceptual simplifications in chemical engineering in the past, and, in particular, in transport, separations, and chemical reactions. However, that leaves unanswered the same questions as for the classical generalizations. Have all the possible simplifications for these topics already been discovered and exploited? Can the conceptual simplifications of the past and present be adapted for emerging fields of current interest such as molecular technology, biotechnology, and nanotechnology? Will the new conceptual simplifications emerge in conjunction with new applications of chemical engineering? An emphatic answer of “no” to the first of these questions is supported by contributions in the last two decades. For example, the development and exploitation of simplifications in the form of asymptotes has recently had a rebirth in turbulent flow and convection after the lapse of a half-century. In essentially the same period of time, computer hardware and software have revolutionized design calculations for heat and mass transfer and led to the discarding of many of the simplifying concepts of the past. Computers have also made it possible to utilize reaction mechanisms and fluid mechanics of any complexity (in combustion in particular and in reactor design in general), but many obsolete simplifications have not yet been replaced or abandoned. The answer to the second question is “yes”, insofar as simplifying concepts describe fundamental behavior such as fluid mechanics and chemical reactions and insofar as this same behavior prevails, which it may not at the nanodimensional level. It is again too early to answer the third question, that is, to predict success in developing new simplifying concepts for the radically new applications of chemical engineering. One clue may be mentioned. The conceptual simplifications of the past have, in most cases, had their origin with analytical models. Adaptations and developments in completely new fields, therefore, may be dependent primarily on the formulation of models in a mathematical form. One feature of Part III is the identification of several simplifying equations that persist in the literature, even though they have become obsolete because of refutation or replacement. Part IV. Analogies Analogies are a special type of generalization and, in principle, could have been included in Parts I and II. They could also be considered as simplifying concepts and, indeed, most of them were generated in that context. However, some of the analogies have characteristics that justify their separate treatment. Analogies relate the behavior in one process to that in another and thereby allow the transfer and/or sharing of knowledge, and, in doing so, eliminate duplication. Many of the analogies noted herein arose from fields of engineering other than chemical engineering. The soundest method of deriving analogies is by comparison of differential or algebraic models. However, many classical ones have also evolved from the observation of similarities in experimental data or in the correlations representing them. Because the derivation of analogies is difficult to generalize, seven examples of individual or closely related sets of analogies are described as a guideline for the extension and adaptation of old ones, and the recognition or the development of new ones.

These seven examples include some well-known classical ones, some overlooked ones, and some recent ones. The accuracy and scope of an analogy are seldom apparent from its derivation or form. Therefore, a critical evaluation of these two characteristics is a necessary component of its application. 1. Analogy between Electrical Conduction, Thermal Conduction, and Molecular Diffusion. The analogy between electrical conduction, thermal conduction, and molecular diffusion is one of the oldest, best-known, and most widely applied analogies. Although it was probably first identified conceptually, it follows from the differential models for these three processes, which also provide some insight, relative to its limits of applicability by the presence of non-identical terms. In 1843, Wheatstone65 remarked, concerning Ohm’s laws that “there is scarcely any branch of experimental science in which so many and such various phenomena are expressed by formulas of such simplicity and generality.” Echoing Wheatstone, the applicability of Ohm’s laws to thermal conduction and molecular diffusion constitutes perhaps the best-known analogy in all of engineering. The applicability of Maxwell’s solution for electrical conduction through a dilute dispersion of spheres to thermal conduction was simply implied as obvious in Part II. Ohm’s laws for electrical conduction through resistances in series and parallel were similarly implied in Part II, not only to be applicable for thermal conduction but also for combined thermal conduction and convection, and insofar as thermal radiation can be represented by a radiative heat-transfer coefficient, for that mechanism of heat transfer as well. The simplification conceived by Ohm is applied without attribution to the transport and reaction of chemical species in a bed of porous catalyst particles. Thus, this analogy is so taken for granted that the primary concern should be with identification of the conditions for which it is not applicable. In 1881, Maxwell18 (on page 422) asserted that “Ohm, mislead by the analogy between heat and electricity, entertained an opinion that a body when raised to a high potential becomes electrified throughout its substance, as if electricity were compressed into it, and was thus by means of an erroneous opinion led to employ the equations of Fourier to express the true laws of conduction of electricity through a long wire, long before the real reason of the appropriateness of these equations had been suspected.” Maxwell’s comment is a warning that speculation that is useful in many applications may be invalid in others. One such condition of a chemical nature is a non-equimolar diffusion of species. 2. Analogy between Heat Transfer and Mass Transfer. Perhaps the most widely utilized analogy in chemical engineering is that between heat and mass transfer by molecular motion, free convection, forced convection, and regeneration. For example, expressions for the forced and free convection of energy are ordinarily presumed to be directly applicable for the forced convection of mass if the Nusselt number Nu is replaced by the Sherwood number Sh and the Prandtl number Pr is replaced by the Schmidt number Sc. The analogy extends to free convection if the Grashof number Gr is expressed in terms of densities. The analogy for heat and mass transfer is based on the similarity of the differential models or their solutions when appropriately expressed and arranged. Exceptions to this


analogy occur when the differential models are subject to nonsimilar boundary conditions or incorporate nonsimilar terms, for example, those representing thermal radiation, flow induced by mass transfer, or chemical reactions. In 1978, Shaw and Hanratty64 determined experimentally that the analogy failed for fully developed turbulent flow in a round tube, and in 1997, Papavassiliou and Hanratty66 subsequently confirmed the difference theoretically by DNS. 3. Analogies between Momentum and Energy Transfer in Turbulent Flow. The PDEs for the conservation for momentum and energy are similar to some extent, but also differ significantly in some respects, which suggests the possibility of analogous behavior under some but not all circumstances. On the other hand, some of the analogies between momentum and energy transfer that follow here were based on physical conjecture and the presumption that an analogy exists. Analogies may be formulated for laminar flow and laminar convection, but they serve no useful purpose, at least insofar as analytical solutions are possible for both processes. The most famous analogy between momentum and energy transfer is that of Reynolds,68 who, in 1874 (and, thus, before his introduction of space averaging), postulated the existence of a common but unspecified mechanism for the transport of energy and momentum by the turbulent eddies, and thereby derived the following expression relating the rates of heat transfer and momentum transfer to the wall of a round tube in fully developed flow:

Nu ) PrRe

(2f )

(28)

Although free from empiricism, eq 28 is not exact for any condition, and it is a useful approximation only for a very narrow range of conditions. Nevertheless, it has enduring stature, not only as the first significant theoretical expression for forced convection, but also as the cornerstone of all subsequent analogies in this field. In 1910, Prandtl69 improved upon the Reynolds analogy by incorporating speculatively the resistance due to the linear diffusion of momentum and energy across a boundary layer of thickness δ near the surface, and thereby obtained

Nu )

PrRe(f/2) +

1 + δ (Pr - 1)(f/2)1/2

(29)

Fortuitously, the dimensionless boundary layer thickness, δ+ ) δ(τwF)1/2/µ, proved to vary only slightly with Re. Although eq 29 has some of the same shortcomings as eq 28, it is greatly superior functionally, by virtue of the non-power dependence of Nu on Re as provided by f, and the coupling of the dependences on Pr and Re, and it retains historical stature on those grounds. In 1951, Reichardt70 derived a greatly improved analogy, relative to all prior ones, including those of Reynolds and Prandtl. He combined the time-averaged differential momentum and energy balances into one by eliminating longitudinal distance as a variable and integrated the resulting essentially exact expression in closed form by means of several idealizations that can be classified as brilliant speculations. His result, which is for a uniform wall temperature, is free of empiricism, as is that of Reynolds, but is not exact, because of the idealizations made to allow integration in closed form. Churchill et al.71 recognized, even prior to making several significant corrections in the Reichardt analogy, that it could be re-expressed in the generalized form

( )

(

)

Prt 1 Prt 1 1 + 1) Nu Pr Nu1 Pr Nu∞

(30)

The key to this re-expression of the corrected analogy of Reichardt was the recognition of groupings of dimensional terms equivalent to the exact expressions for Nu1 (namely, the integral solution for Pr ) Prt, as derived by Heng et al.72) and Nu∞ (namely, the well-known asymptotic solution for Pr f ∞ (eq 18)). In 2002, Churchill and Zajic73 quantitatively evaluated the errors resulting from the idealizations made by Reichardt and corrected them. In this process, they fortuitously discovered that re-expressing an analogy of Churchill,74 in terms of Nu1 and Nu∞, resulted in the following significantly improvement on eq 30:

( )

[ ( )]

Prt 1 Prt 1 + 1) Nu Pr Nu1 Pr

2/3

1 Nu∞

(31)

Equation 31 has three great merits. First, it seems to be applicable for all geometries, all thermal boundary conditions, all fully turbulent rates of flow, and all values of Pr > Prt. Second, it is completely free from any explicit empiricism. Third, it predicts the essentially exact values of Nu obtained by numerical integration within 1%. The recognition that eq 31 has the form of the staggered extension of eq 1 for three regimes allowed Churchill and Zajic to derive a speculative analogue for Pr < Prt. Equation 31 and its analogue for Pr < Pr1 qualify for inclusion in Part II as perhaps the greatest generalization ever devised for convection but are described here in Part IV because they originated as analogies. The analogy between heat and momentum transfer is implicit rather than explicit, in that Nu1 and Nu∞ both incorporate f. Equation 31 not only provides practicing engineers with a very useful predictive expression, but its detailed derivation, including that of the expressions for Nu1, Nu∞, and Prt, is an instructive exercise for students, in terms of insight and understanding. An analogy rivaling that of Reynolds in recognition is the j factor concept of Colburn75 in 1933. This analogy is based on his observation that a particular empirical correlating equation for f/2 in round tubes and another one for Nu/RePr1/3 had essentially the same functional and numerical dependence on Re. Designating this latter grouping as the j factor made it more memorable. A common explicit form of the Colburn analogy is

Nu 0.023 f )) ) 2 RePr1/3 Re0.2

(32)

The concept embodied in eq 32 was extended by Chilton and Colburn76 in 1935 to include mass transfer, in terms of Sh/(ReSc1/3), and later by others for many systems such as, for example, flow, heat transfer, and mass transfer in packed beds. Unfortunately, the great implied generality of this concept is counter-balanced by significant functional errors in almost every respect, and significant numerical errors for all conditions except for the particular, narrow ranges of values of Re and Pr for which the individual correlating equations for the friction factor and the Nusselt number Nu were formulated. The source of these errors is the choice of correlating equations with a fixed powerdependence for Re and Pr. For Pr ) 1, the Colburn analogy is almost correct, because, according to the Reichardt-Churchill-

7866


Zajic analogy, Nu is almost equal to Re(f/2) for Pr ) Prt = 0.867. However, for large values of Pr, Nu is actually proportional to Re(f/2)1/2 rather than to Re(f/2), leading to increasing numerical error as Pr increases. Incidentally, Colburn’s choice of 1/3 for the power dependence on Pr had no theoretical basis but was simply a convenient compromise between correlative values of 0.3 and 0.4 for cooling and heating of the fluid, respectively. The 1/3-power dependence has since been shown on theoretical grounds to be the correct asymptotic one for Pr f ∞. This dependence fortuitously reduces to that of the Reynolds analogy for Pr ) 1. The Colburn analogy suggests the possibility of formulating others by observation rather than from theory, while, at the same time, indicating the high risk of functional error in the absence of a theoretical rationale. When will expressions such as eq 31 be expected to replace in practice traditional correlating equations in erroneous forms such as eq 32? This cannot be expected to occur until these new analogies find their way into the widely used textbooks and handbooks and, then only after a wave of so-educated students reaches industry. This example illustrates the periodic refinement of the analogy between momentum and energy transfer in turbulent flow over a century and a quarter, and thereby suggests the possibility of productive refinements of other venerable ones. The turbulent Prandtl number (Prt) was included in eqs 30 and 31 without explanation. The very concept of a turbulent Prandtl number, which was originally formulated by analogy to the molecular Prandtl number (Prm)sthat is, as the ratio of the eddy diffusivities for momentum and energyshas been widely deprecated, because of uncertainty of the validity of these two components. At the same time, this quantity has, because of its more constrained behavior, been utilized almost universally in modern predictions of turbulent convection, including those based on the κ- model and the large eddy simulation (LES) methodology. The success of the predictions of convection with models incorporating Prt is now understood. The first step in this understanding was perhaps by Abbrecht and Churchill77 in 1960, who concluded from experimental measurements in thermally developing convection that the eddy conductivity ratio (kt/k) and, thereby, the turbulent Prandtl number (Prt ) cpµt/kt), was a function only of the eddy viscosity ratio (µt/µ). They further concluded from a comparison of their results for a round tube with prior ones for a parallel-plate channel that it was independent of geometry. Their measurements were only for air, and they conjectured (correctly) that Prt must be dependent on Pr. The significance of their observations and conclusions, which might have been cited in Part I as a generalization, was not fully appreciated even by them, and was generally overlooked by others until recently. The next step was by Churchill28 in 1997, who, by virtue of his expression of the differential momentum in terms of (u′V′)++ ≡ -F(u′V′)/τ, and, by analogy, the differential energy balance in terms of (T′V′)++ ≡ Fcp(T′V′)/j, was able to restate the generalization of Abbrecht and Churchill as follows: (T′V′)++, and thereby Prt, are the same functions of (u′V′)++ and Pr for all geometries and all thermal boundary conditions. He also recognized that Prt could be re-expressed in terms of (u′V′)++, (T′V′)++, and Pr as follows:

(u′V′)++[1 - (T′V′)++ ] Prt ≡ Pr (T′V′)++ [1 - (u′V′)++]

(33)

Equation 33 reveals that Prt is a physically well-defined quantity, rather than a heuristic one. It is a more useful quantity than (T′V′)++, because of its more-constrained behavior. It is curious that Prt would probably never been identified as such a characteristic and convenient parameter if it had not been introduced as an analogue of Pr. Other quantities of questionable origin but great usefulness may yet wait to be rationalized on theoretical grounds. Also see Churchill.78 The experimental data for Prt are highly uncertain and unlikely to improve radically in the immediate future, because of intrinsic difficulties in measuring both u′V′ and T′V′, and because the current predictive expressions for this quantity have questionable antecedents. The remarkable success achieved in predicting values of Nu within 1% using these highly uncertain values was explained in part by Yu et al.79 in 2000, who demonstrated, by means of illustrative calculations with several empirical correlating equations, that the value of Nu is relatively insensitive to the expression used for Prt. Two justifications are offered for the length of this dissertation on the analogies between turbulent flow and convection. The first is the practical importance of this regime; most process heat transfer happens in turbulent flow in channels. The second is the current absence of these recent improvements from any textbook. 4. Analogies within Fluid Mechanics. In 1950, Rothfus et al.80 demonstrated that the then-accepted empirical correlating for the velocity distribution followed almost exactly from that for the mixed-mean velocity if u+ were substituted for u+ m and 8y+ for a+. They proposed a similar analogy for parallel-plate channels by substituting 12y+ for b+. In view of recent improvements in the correlating equations for u+ and u+ m, the direct utility of this analogy is limited, but just as with the Reynolds analogy, it served as a necessary precursor to the subsequent analogies devised by Rothfus and Monrad44 in 1955, MacLeod46 in 1951, and Churchill74 in 1997, and thereby the predictive equation of Yu et al.49 of 2001 for u+ m, which is based on numerical and piecewise analytical integration of the correlative equation of Churchill32 in 2000 for (u′V′)++. 5. Analogies for Laminar Buoyant Convection. In 1954, Emmons81 devised an analogy for heat transfer by several buoyant processes, specifically, free convection, film condensation, film boiling, and film melting in both the laminar and turbulent regimes. His analogy is only approximate, whereas relatively exact solutions exist for the individual laminar regime, but it is of great conceptual value, because of its simplicity and breadth. In 1967, Saville and Churchill82 derived a generalized solution for free convection in the thin-laminar-boundary-layer regime from horizontal cylinders of fairly general contour, and a similar one for vertical axisymmetric bodies. These generalized solutions may be interpreted as analogies, with respect to geometry. They also have the merit of converging more rapidly than the classical solutions for the individual geometries and thereby preclude the need for the latter. In 2002, Martin83,84 derived theoretical solutions for laminar convection in many different internal and external flows by means of re-expressing the Le´veˆque37 solution of 1928 for developing convection in fully developed laminar flow in the inlet of a channel in terms of the shear stress at the surface rather than in terms of the mixed-mean or free-stream velocity. 6. Analogies for Waves. Churchill85,86 developed in 1969 and discussed in 1980 an analogy between shock, detonation, and gravitational waves and Marquardt87 in 1989 developed one for thermal and compositional waves in regenerators, absorbers,


adsorbers, chromatographic columns, catalytic-bed reactors, sedimentation tanks, and solidifiers. These analogies are important conceptually as well as practically, because wave phenomena are relatively neglected in chemical engineering education, although they frequently occur in practice. 7. Analogy between Heat Transfer and Reaction. In 2006, Churchill88 devised an analogy between chemical reactions and external heat transfer in fully developed tubular flow. This expression differs radically in character from the classical ones for fluid flow and heat transfer in that it is for developing and interacting reaction and heat transfer, and in that it encompasses both the laminar and the turbulent regimes. The primary value of the analogy is in explaining and predicting the extreme and chaotic variations in Nu. (See the work of Yu and Churchill.89) 8. Generalities, with Respect to Analogies. Analogies may be primarily of conceptual value as is the case of that of Emmons,81 or of direct practical value, as is the case with that of Churchill and Zajic.73 Analogies may have great scope, as is again the case with that of Churchill and Zajic, or have a very limited scope but nevertheless important historical role as with that of Reynolds.68 Analogies may be derived from theoretical models, formulated speculatively, or based on the observation of similarities in experimental data or correlating equations. Whatever their origin, it is essential to evaluate the functional and numerical limitations of an analogy. It may prove to be qualitatively correct but quantitatively in significant error, as are those of Emmons,81 Reynolds,68 and Prandtl;69 qualitatively and quantitatively correct but of limited use, as is that of Rothfus et al.;80 or totally misleading, as is that of Colburn.75 Broad analogies such as that between heat and mass transfer may prove applicable or unreliable in yet-untested applications. Some analogies may be interpreted as generalizations and others as simplifying concepts. It should be possible to develop new analogies and/or adapt venerable ones in all new applications of chemical engineering. Conclusions Generalizations of both broad and narrow scope, simplifying concepts, and analogies have a significant role in the evolution and unity of chemical engineering as an academic discipline and as a field of practice. Perhaps their greatest value is that, by their very simplicity and/or breadth, they enhance understanding and reduce clutter. However, both their conceptual value and their applications come at a price. They invariably invoke approximations or simplifications that may be overlooked or underestimated. Because it is so well-known to chemical engineers of all vintages, the Colburn analogy serves as an archetypical example of the origin and eventual discard of a presumed universality. We rightfully honor Alan Colburn for conceiving this broad generalization, and recognize its almost universal application in the past. At the same time, we now recognize, or should, that it is wrong functionally in every respect, that its predictions are subject to gross numerical error, and that it is only of historical interest. In the latter sense, it may serve as an illustration of how to go wrong, namely to impute generality to empirical expressions with no theoretical basis. Generalizations, simplifying concepts, and analogies (in part, the classical ones, and in part, new ones yet to be formulated) can be expected to have a prominent role in the future. A few of the classical ones have become obsolete, because of improved theories and new experimental findings, and should be phased out of the classrooms, textbooks, handbooks, and computer

packages, but most of them remain valid and, at worst, only need to be updated. The radical changes that are currently afoot in chemical engineering and in the wider arena in which it is now practiced can be expected to make a few more of the classical generalities, simplifying concepts, and analogies obsolete or irrelevant or in need of restatement. In particular, the new areas of chemical engineering practice call for new generalizations, simplifying concepts, and analogies, as well as the updating of old ones. The primary objectives of this analysis have been to identify the obsolete ones, and, by examples from the past, to provide guidance for the generation of new ones. On the other hand, those working in new fields such as, for example, biomolecular engineering and nanotechnology are better positioned than I am to undertake that task. The history of the origin of the generalities, simplifying concepts, and analogies suggests that imagination and ingenuity, as well as a fundamental understanding of the topic of concern, will be required. As a final note, the development of user-friendly computer hardware and software has been the greatest source of change in the practice of chemical engineering over the past half century. This development does not fall neatly into the category of simplifying concepts or of analogies, but perhaps the expectation of its continual evolution and increased usefulness may be interpreted as a generalization. Nomenclature a ) arbitrary exponent a ) radius of tube (m) a+ ) a(F/τw)1/2 A ) arbitrary coefficient A ) area for heat transfer (m2) b ) arbitrary exponent B ) arbitrary coefficient cp ) specific heat capacity at constant pressure (J/(kg K)) D ) diameter (m) Df ) diffusivity (m2/s) f ) Fanning friction factor; f ) 2τw/(Fum2) g ) acceleration due to gravity (m/s2) Gz ) Graetz number; Gz ) wcp/kx h ) heat transfer coefficient; h ) jw/(Tm - Tw) (W/(m2 K)) j ) heat flux density (W/m2) k ) thermal conductivity (W/(m K)) kc ) thermal conductivity of continuous phase (W/(m K)) kd ) thermal conductivity of dispersed phase (W/(m K)) ke ) thermal conductivity of dispersion (W/(m K)) k* ) mass transfer coefficient based on concentration (m/s) L ) length of heat exchanger (m) n ) arbitrary combining exponent Nu ) Nusselt number; Nu ) hD/k p ) pressure (Pa) q ) heat flux (W/m2) Pr ) Prandtl number; Pr ) cpµ/k Re ) Reynolds number; Re ) DumF/µ Sc ) Schmidt number; Sc ) µ/FDf Sh ) Sherwood number; Sh ) k*D/Df t ) temperature of hotter fluid (K) T ) temperature (K) (T′V′)++ ) Fcp(T′V′)/j u ) velocity (m/s) U ) overall heat-transfer coefficient (W/(m2 K)) (u′V′)+ ) -Fu′V′/τw (u′V′)++ ) -Fu′V′/τ w ) mass rate of flow (kg/s)

7868


x ) independent variable, distance from inlet (m) X ) volumetric fraction of spheres y ) distance from surface (m) y{x} ) arbitrary function Greek Letters β ) coefficient of expansion with temperature (T-1) δ ) thickness of boundary layer (m) δ+ ) dimensionless thickness of boundary layer ) δ(τwF)1/2/µ ) emissivity η ) effective kinematic viscosity (m2/s) µ ) dynamic viscosity (kg/(m s)) F ) specific density (kg/m3) σ ) Stefan-Boltzmann constant (W/(m2 K4)) τ ) shear stress (Pa) φ ) X(kd - kc)/(kd - 2kc) φ ) arbitrary function of Subscripts ave ) average value c ) at centerline m ) mean value r ) radiative t ) turbulent w ) at wall x ) at x 0 ) asymptote for x f 0 ∞ ) asymptote for x f ∞ or y f ∞ 1 ) at Pr ) Prt; at inlet 2 ) at outlet Superscripts and Accents ′ ) fluctuating value ) time-averaged value

-

Acknowledgment The many constructive suggestions of J. D. Seader, Stanley Sandler, Stefan Zajic, and Phillip Westmoreland, and the anonymous reviewers are gratefully acknowledged. The identification of errors or limitations in the work of those whose names appear herein does not imply a lack of respect or appreciation, but is merely a demonstration that science and engineering progress by discarding the old, contributions, however seminal, to make room for the new. Isaacson,90 in a recent biography of Albert Einstein, concluded that one of his characteristics that resulted in the formulation of new concepts was a penchant for challenging accepted dogma. Two other characteristics of Einstein that are also relevant herein were his instinct for unification and his belief in simplicity. A disproportionate number of the examples in Parts II, III, and IV are based on the work of the author and his collaborators. This is in part a consequence of a focus on correlation and generalization throughout his career and in part of the greater certainty of the origins in these particular cases in which he has been involved. All chemical engineers should recognize and acknowledge the contribution that universalities such as those mentioned here have made to their careers, and be appreciative of those who formulated them. As Isaac Newton said, “If I have seen further it is by standing on ye sholders of Giants.” Literature Cited (1) Davis, G. E. A Handbook of Chemical Engineering; Davis Brothers: Manchester, U.K., 1904. (2) Walker, W. H.; Lewis, W. K.; McAdams, W. H. Principles of Chemical Engineering; McGraw-Hill Book Co.; New York, 1923.

(3) White, A. H. Chem. Eng. Educ. 1928, 65. (4) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena, 2nd Edition; Wiley: New York, 1960 (2nd Edition, 2001). (5) Shreve, R. N. The Chemical Process Industries; McGraw-Hill Book Co.: New York, 1945. (6) Churchill, S. W. The Interpretation and Use of Rate Data. The Rate Process Concept; McGraw-Hill Book Co.: New York, 1974. (Revised printing by Hemisphere Publishing Corp., Washington, DC, 1979.) (7) Dodge, B. F. Chemical Engineering Thermodynamics; McGrawHill Book Co.: New York, 1944. (8) (a) Kabel, R. L. Rates. Chem. Eng. Commun. 1981, 9, 15. (b) Kabel, R. L. Reflections on Rates. Ind. Eng. Chem. Res. 1992, 31, 641. (9) Pfefferle, L. D.; Churchill, S. W. NOx Production from the Combustion of Ethane Doped with Ammonia in a Thermally Stabilized Plug Flow Burner. Combust. Sci. Technol. 1986, 49, 239. (10) Denbigh, K. G. Velocity and Yield in Continuous Reaction Systems. Trans. Faraday Soc. 1994, 40, 352. (11) Cussler, E. L.; Moggridge, G. D. Chemical Product Design; Cambridge University Press: Cambridge, U.K., 2001. (12) Churchill, S. W.; Neuman, M. A Generalized Mathematical Model for Sustainability. In The 2005 AIChE Annual Meeting Conference Proceedings on CD-ROM, AIChE: New York, 2005; 14 pp. (13) Kaneda, M.; Yu, B.; Ozoe, H.; Churchill, S. W. The Computed Characteristics of Turbulent Flow and Convection in Concentric, Circular Annuli, Part I. Flow. Int. J. Heat Mass Transfer 2003, 46, 5045. (14) Newton, I. Scala graduum Caloris. Philos. Trans. R. Soc. London, 1701, 22, 824. (15) Churchill, S. W.; Usagi, R. A General Expression for the Correlation of Rates and Other Phenomena. AIChE J. 1972, 18, 1121. (16) Churchill, S. W. A Correlating Equation for Almost Everything. Manuscript in preparation. (17) Churchill, S. W.; Churchill, R. U. A General Model for the Effective Viscosity of Pseudoplastic and Dilatant Fluids. Rheol. Acta 1975, 14, 405. (18) Maxwell, J. C. A Treatise on Electricity and Magnetism, Vol. 1; Clarendon Press: Oxford, U.K., 1873. (19) Churchill, S. W. The Thermal Conductivity of Dispersions and Packed BedssAn Illustration of the Unexploited Potential of Limiting Solutions for Correlation. In AdVances in Transport Processing; Mujumdar, A. S., Mashelkar, R. A., Eds.; Wiley Eastern: New Delhi, India, 1985; Vol. 4, 395 pp. (20) Reynolds, O. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion, Philos. Trans. R. Soc. London 1895, 186, 123. (21) Ohm, G. S. The GalVanic Circuit InVestigated Mathematically; Berlin, 1827. (English translation by William Francis, D. Van Nostrand Co., New York, 1891.) (22) Boussinesq, J. Calcul du povoir refroidissant des courantes fluide. J. Math. Pures Appl. 1905, 60, 285. (English Translation: Calculation of the Cooling Power of Fluid Currents. Int. Chem. Eng. 1989, 29, 343.) (23) Galante, S. R.; Churchill, S. W. Application of Solutions for Convection in Potential Flow. AdV. Heat Transfer 1990, 20, 353. (24) Prandtl, L. U ¨ ber Flu¨ssigkeitsbewegung bei sehr kleiner Reibung. In Verhhandl. 3rd International Mathematics Kongress; Heidelberg, Germany, 1904; 484 pp. (25) Colebrook, C. F. Turbulent Flow in Pipes with Particular Reference to the Transition Region Between Smooth and Rough Pipe Laws, J. Inst. CiV. Eng. 1938-1939, 11, 133. (26) Boussinesq, J. Essai sur la the´orie des eaux courantes, Me´m. pre´sents divers savants. Acad. Sci. Inst. France 1877, 23, 1. (27) Kjellstro¨m, B. S.; Hedberg, S. On the Shear Stress Distributions for Flow in Smooth or Partially Rough Annuli, Aktiebolager Atomenergi, Report AE-243, Stockholm, 1966. (28) Churchill, S. W. New Simplified Models and Formulations for Turbulent Flow and Convection. AIChE J. 1997, 43, 1125. (29) Prandtl, L. Bericht ber U ¨ ntersuchungen zur ausgebildeten Turbulenz. Z. angew. Math. Mech. 1925, 5, 136. (30) Nikuradse, J. Gesetzma¨ssigkeiten der turbulenten Stro¨mung in glatten Rohren. VDI-Forschungsh. 1932, 356. (31) Churchill, S. W.; Chan, C. Turbulent Flow in Channels in Terms of Local Turbulent Shear and Normal Stresses. AIChE J. 1995, 41, 2513. (32) Churchill, S. W. The Prediction of Turbulent Flow and Convection in a Round Tube. AdV. Heat Transfer 2000, 34, 255. (33) Rayleigh, Lord (J. W. Strutt), The Principle of Similitude. Nature 1915, 95, 66. (34) Churchill, S. W. A New Approach to Teaching Dimensional Analysis. Chem. Eng. Educ. 1997, 30 (3), 158.

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 7869 (35) Hellums, J. D.; Churchill, S. W. Simplification of the Mathematical Description of Boundary and Initial Value Problems. AIChE J. 1964, 10, 110. (36) Blasius, H. Grenzschichten in Flu¨ssigkeiten mit kleiner Reibung. Z. Math. u. Phys. 1908, 56, 1. (37) Le´veˆque, A. Les lois de la transmission de la chaleur par convection. Ann. Mines 1928, 13, 201, 305, 381. (38) Nusselt, W. Der Wa¨rmeu¨bergang in Rohrleitungen. Mitt. Forsch.Arb. Ing.-Wes. 1909, 89, 1750. (39) Blasius, H. Das A ¨ hnlichkeitsgesetzebei Reibungsvorga¨ngen in Flu¨ssigkeiten, VDI-Forschungsh.-Arb. Ing.-Wes. 1913, No. 131. (40) Prandtl, L. U ¨ ber den Reibungswiderstand stro¨mender Luft, Ergebn. Aerodyn. Versanst. Go¨ttingen 1929, 3, 1. (First mentioned in I. Lieferung, 1921, p 136.) (41) Prandtl, L. U ¨ ber die ausgebildete Turbulenz. In Proceedings of the International Congress for Applied Mechanics, Zurich, 1926; p 62. (Also: see ref 29.) (42) Nikuradse, J. Gesetzma¨ssigkeit der turbulenten Stro¨mung in glatten Rohren. Ver. Dtsch. Ing.-Forschungsheft 1932, 356. (43) Millikan, C. B. A Critical Discussion of Turbulent Flows in Channels and Circular Tubes. In Proceedings of the 5th International Congress on Applied Mechanics, Cambridge, MA, 1938; p 386. (44) Rothfus, R. R.; Monrad, C. C. Correlation of Turbulent Velocities for Tubes and Parallel Plates. Ind. Eng. Chem. 1955, 47, 1144. (45) Churchill, S. W. New and Overlooked Relationships for Turbulent Flow in Channels. Chem. Eng. Technol. 1990, 13, 264; Addendum, 1991, 31, 78. (46) MacLeod, A. L. Liquid Turbulence in Gas-Liquid Absorption System, Ph.D. Thesis, Carnegie Institute of Technology, Pittsburgh, PA, 1951. (47) Whan, G. A.; Rothfus, R. R. Characteristics of Transition Flow between Parallel Plates. AIChE J. 1959, 5, 204. (48) Hinze, J. C. Turbulence; McGraw-Hill Book Co.: New York, 1959. (49) Yu, B.; Ozoe, H.; Churchill, S. W. The Characteristics of Fully Developed Turbulent Convection in a Round Tube. Chem. Eng. Sci. 2001, 56, 1781. (50) Hellums, J. D.; Churchill, S. W. Dimensional Analysis and Natural Convection. Chem. Eng. Symp. Ser. 1961, 57 (32), 75. (51) Graetz, L. U ¨ ber die Wa¨rmeleitungsfa¨higkeit von Flu¨ssigkeiten. Annu. Phys. 1883, 18, 79. (52) White, C. W., III.; Churchill, S. W. A Computer Program for Dimensional Analysis and the Identification of Similarity Transformations. Proc. Second World Congress Chem. Eng. 1981, 1, 331. (53) Nusselt, W.; Das Grundgesetz des Wa¨rmeu¨berganges. Gesund.Ing. 1915, 38, 477, 490. (54) Frank-Kamenetskii, D. A. On the Limiting Form of the Free Convection Law for High Value of Grashof Criterion. C. R. Acad. Sci. U.S.S.R. 1937, 17, 9. (55) Eckert, E. R. G.; Jackson, T. W. An Analysis of Turbulent FreeConvection Boundary Layer On Flat Plate. Natl. AdVis. Comm. Aeronaut. Tech. Report 1951, 1015. (56) Langmuir, I. Convection and Conduction of Heat in Gases. Phys. ReV. 1912, 13, 401. (57) Damko¨hler, G. Einfluss von Diffusion, Stro¨mung, und Wa¨rmetransport auf die Ausbeute in Chemisch-Technischen Reaktionen. Chem.Eng.-Tech. 1937, 3, 359. (English translation: The Influence of Diffusion, Fluid Flow, and Heat Transport on the Yield in Chemical Reactors. Int. Chem. Eng. 1988, 28 (1), 132.) (58) Hougen, O. A.; Watson, K. M. Chemical Process Principles. Part 3. Kinetics and Catalysis; Wiley: New York, 1947. (59) Christiansen, J. A. On the Reaction between Hydrogen and Bromine. Mat.-Fys. Medd.-K. Dan. Vidensk. Selsk. 1919, 1, 1. (60) Calderbank, P. H. Some Problems in the Design of LightHydrocarbon Pyrolysis Coils. Chem. Eng. Prog., Symp. Ser. 1954, 50 (9), 53. (61) Langmuir, I. The Adsorption of Gases on Plane Surfaces of Glass, Mica, and Platinum. J. Am. Chem. Soc. 1918, 40, 1361. (62) Chilton, T. H.; Colburn, A. P. Distillation and Absorption in Packed Columns. Ind. Eng. Chem. 1935, 27, 255; 904. (63) McCabe, W. L.; Thiele, E. W. Graphical Design of Fractionating Columns. Ind. Eng. Chem. 1925, 17, 605.

(64) Peters, W. A. The Efficiency and Capacity of Fractionating Columns. Ind. Eng. Chem. 1922, 14, 476. (65) Wheatstone, L. Bakerian Lecture, June 15, 1843. (66) Shaw, D. A.; Hanratty, T. J. Turbulent Mass Transfer to a Wall for Large Schmidt Numbers. AIChE J. 1997, 23, 28. (67) Papavassiliou, D. V.; Hanratty, T. J. Transport of a Passive Scalar in Turbulent Channel Flow. Int. J. Heat Mass Transfer 1997, 40, 1303. (68) Reynolds, O. On the Extent and Action of the Heating Surface of Steam Boilers. Proc. Manchester Lit. Philos. Soc. 1874, 14, 2. (69) Prandtl, L. Eine Beziehung zwischen Wa¨rmeaustausch und Stro¨mungswiderstand der Flu¨ssigkeiten. Phys. Z. 1910, 11, 1072. (70) Reichardt, H. Die Grundlagen des turbulenten Wa¨rmeu¨berganges, ArchiV ges. Wa¨rmetechn. 1951, 2, 129. (English Translation: The Principles of Turbulent Heat Transfer. NACA Tech. Memo 1957, 1408.) (71) Churchill, S. W.; Shinoda, M.; Arai, N. A New Concept of Correlation for Turbulent Convection. Therm. Sci. Eng. 2000, 8 (4), 49. (72) Heng, L.; Chan, C.; Churchill, S. W. Essentially Exact Characteristics of Turbulent Convection in a Round Tube. Chem. Eng. J. 1998, 71, 163. (73) Churchill, S. W.; Zajic, S. C. The Prediction of Fully Developed Turbulent Convection with Minimal Empiricism. AIChE J. 2002, 8, 927. (74) Churchill, S. W. New Wine in New Bottles; Unexpected Findings in Heat Transfer, Part III. The Prediction of Turbulent Convection with Minimal Explicit Empericism. Therm. Sci. Eng. 1997, 5 (3), 13. (75) Colburn, A. P. A Method for Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction. Trans. Am. Inst. Chem. Eng. 1933, 29, 174. (76) Chilton, T. H.; Colburn, A. P. Mass Transfer (Absorption) CoefficientssProduction from Data on Heat Transfer and Fluid Friction. Ind. Eng. Chem. 1934, 26, 1183. (77) Abbrecht, P. H.; Churchill, S. W. The Thermal Entrance Region in Fully Developed Turbulent Flow. AIChE J. 1960, 6, 268. (78) Churchill, S. W. A Reinterpretation of the Turbulent Prandtl Number. Ind. Eng. Chem. Res. 2002, 41, 6393. (79) Yu, Bo.; Ozoe, H.; Churchill, S. W. The Characteristics of Fully Developed Convection in a Round Tube. Chem. Eng. Sci. 2000, 56, 1781. (80) Rothfus, R. R.; Monrad, C. C.; Senecal, V. E. Velocity Distribution and Friction Factor in Smooth Concentric Annuli. Ind. Eng. Chem. 1950, 42, 2511. (81) Emmons, H. W. Natural Convection Heat Transfer. In Studies in Mathematics and Mechanics: Presented to Richard Von Mises by Friends, Colleagues, and Pupils; Academic Press: New York, 1954; p 232. (82) Saville, D. A.; Churchill, S. W. Laminar Free Convection in Boundary Layers Near Horizontal Cylinders and Vertical Axisymmetric Bodies. J. Fluid Mech. 1967, 29, 391. (83) Martin, H. The Generalized Le´veˆque Equation and Its Practical Use to Predict Heat and Mass Transfer from the Pressure Drop. Chem. Eng. Sci. 2002, 57, 3217. (84) Martin, H. The Generalized Le´veˆque Equation and Its Practical Use to Predict Heat and Mass Transfer from Fluid Friction. In Proceedings of the 12th International Heat Transfer Conference, Grenoble, France, 2002. (85) Churchill, S. W. Theories, Correlations and Uncertainties for Waves, Gradients and Fluxes. Chem. Eng. Educ. 1969, 3 (4), 178, 282. (86) Churchill, S. W. The Practical Use of Theory in Fluid Flow; Etaner Press: Thornton, PA, 1980. (87) Marquardt, W. Wellenausbreitung in verfahrenstechnischen Prozessen. Chem.-Ing.-Tech. 1989, 61, 363. (English Translation: Traveling Waves in Chemical Processes. Int. Chem. Eng. 1990, 30, 585.) (88) Churchill, S. W. An Analogy between Reaction and Heat Transfer. AIChE J. 2006, 52, 3645. (89) Yu, B.; Churchill, S. W. The Extraordinary Effects of an Energetic Reaction on Heat Transfer. In The AIChE Annual Meeting Conference Proceedings on CD-ROM; American Institute of Chemical Engineers (AIChE): New York, 2006, 17 pp. (90) Isaacson, W. Einstein; Simon & Schuster: New York, 2007.

ReceiVed for reView April 12, 2007 ReVised manuscript receiVed July 23, 2007 Accepted July 24, 2007 IE070522O

Role of Universalities in Chemical Engineering - Industrial

Recommend Documents