Because author of this theory ("Constructal" theory) applying it to almost (or to any) multiscale technical issues, as either within the Thermal physics or to other sciences, technologies, we worked over and commented on the "constructal" theory (CT) many years ago in 90s. As soon as author (with co-authors) continue to promote the CT that would be useful for younger generation have some clues on Which is What in this claim. To reach that, was the one of reasons to look into this theory and assertion with desire to reach specifics and mathematical details.
The method we use here is always simple, when we do the reduction of main thesis of authors to the bricks, elements that can be discovered constituting the foundation of the ideology and its performance.
Our previous session of analysis of this claim for usefulness and applicability to the Heterogeneous Scaled Optimization for Volumetric Heat Transport Devices (VHTD) we have performed in 90s. The new declarations appeared since that.
This could be started with the paper -
Morega, A.M., Bejan, A. and Lee, S.W., (1995), "Free Stream Cooling of a Stack of Parallel Plates," Int. J. Heat Mass Transfer, Vol. 38, No. 3, pp. 519-531.
This is the work based on the numerical solution of the free convection problem over the number of (vertical) plates attached to a surface.
From the ABSTRACT:
"This paper addresses the fundamental heat transfer augmentation question of how to arrange a stack of parallel plates (e.g. fins of heat sink, printed circuit boards) in a free stream such that the thermal resistance between the stack and the stream is minimum. It is shown that the best way of positioning the plates relative to one another is by spacing them equidistantly. "
"Finally, it is shown that a stack with more plates than the optimal number can be modeled expediently as a porous block immersed in a free stream." ??
From text - page 519:
''The problem consists of minimizing the thermal resistance between a stream
of coolant
and a certain volume in space
in which heat is being generated at the rate
.
The overall thermal resistance is
,
where
is the highest temperature (the hot-spot temperature) that occurs at a certain
point in the heat generating volume.''
They optimize problem in a strange way - authors call it "structural" in which: see in Page 520:
"a) Is there an optimal way of spacing the plates relative to each other in given volume?
b) Is there an optimal number of plates that should be used (installed) in the given volume?
c) Is it possible to correlate all the individual optimization conclusions into simple (compact) formulas that have wide applicability?''
Page 525:
''we obtain an estimate for the spacing in terms of the free-stream conditions
(
,
where
is just my height of the each channel = their
is the length of the heat exchange along of the stream.
The problem stated as a non-conjugate one! Just Assigned as is that the heat is being generated at the stated rate?
No sense to seek comparison with conjugate statements - too far from what is reasonable from the Hierarchical Scaled VAT point of view.
~~~~~~~~~~~~~~~~~~~~~~~
Bejan, A. and Morega, A.M. (1993), "Optimal Arrays of Pin Fins and Plate Fins in Laminar Forced Convection," Journal of Heat Transfer, Vol. 115, pp. 75-81.
Here is a good example of the Segmented One Scale approach (almost arithmetical by using the simplest heat exchange dependencies).
In this paper given the study of two morphologies - round pin fins and staggered parallel-plate fins.
Analysis of both morphology heat transfer features was done on the basis of dimensional considerations and of the local variables.
From the abstract:
''The optimization of each array proceeds in two steps: The optimal fin thickness is selected in the first step, and the optimal thickness of the fluid channel is selected in the second.''
''The optimal design of each array is described in terms of dimensionless groups.''
The results are given by authors in a form of conventional curves for:
Fig. 2: Optimal pin fin diameter ( ) and maximum local thermal conductance
(G
)
- versus the dimensionless group b= N,M, P, T, etc....
Other figures are of the same kind.
This is the Homogeneous one scale statement and data reduction.
Not worth to analyze more scrupulously the developments - in terms of formulae, for example, etc.
The very characteristic feature:
Authors could not say which of their morphologies is still the best for heat dissipation?
And this is the regular conclusion - because authors couldn't do that. Because of the shortcomings of the approach taken.
All of this because authors do not possess the general theory, approach to this kind of problems - as great mathematician D.Gilbert said - "when we don't have a well defined general concept, an approach... we can not solve the problem..."
Research was supported by IBM.
~~~~~~~~~~~~~~~~~~~~~~
Bejan (1995) constructs the two curves - for the spacing
between staggered cylinders going to utmost large, and for the case when
cylinders are almost in touch. Author estimates the optimum based on the heat
flux
- and approximates the area of
where the biggest heat flux can occur. No consideration of the pressure drop
issue given in this text.
Then author considers the more precise technique - using the experimental data by Zhukauskas, (1987).
He got the relation like
where
where
is the bundle flow length,
- is the inlet free-stream velocity.
In this paper given also the segmented one phase, one scale Optimization approach.
~~~~~~~~~~~~~~~~~~~~~~~~~~~
Ledezma et al., (1996) in the abstract wrote: "This is an experimental,
numerical, and theoretical study of the heat transfer on a pin-finned plate
exposed to an impinging air stream. ... The base plate and the fin cross
section are square. .....The recommended correlations for optimal spacing,
and the maximum thermal conductance,
cover the range
For the numerical investigation they used the 3D one-temperature model
where dimensionless variables are
They write on the page 573: "The assumption that the fin surfaces are isothermal is clearly an approximation - a simplification without which we would have had to solve the conjugate convection and fin conduction problem. It is good approximation, for two reasons.
First, in air-cooled heat sinks tested in industry the fin efficiency is close to 100% (e.g., Nakayama et al., 1988; Matsushima et al., 1992).
Second, we just completed a numerical study (Morega et al., 1995) in which we numerically optimized the spacing between straight fins in a parallel air stream, by modeling the fins as isothermal.''
Meanwhile, as we have shown in our studies (and many other field workers) the calculations point out that this assumption makes the problem study mostly irrelevant to the physics of the process.
It is interesting to compare the SVAT Upper scale laminar statement dimensionless parameters set (developed in our studies) and the ones by Ledezma et al. (1996) those are of the Lower scale
 |
 |
 |
 |
 |
 |
|
Upper scale |
 |
 |
 |
 |
 |
 |
 Lower
scale |
 |
 |
 |
 |
 |
 |
where
 |
 |
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 |
 |
 |
 |
 |
 |
 |
 |
where
Comparing these two sets one can observe that dimensionless parameters set by Ledezma et al. (1996) has the 4 geometrical parameters from which 3 are the ratios of certain sizes of the system.
That set has no parameter which describes (reflects) the flow resistance or heat exchange conditions or relationship. It also has no parameter which describes the influence of the boundary conditions - which is for that particular statement (problem model) has no direct implications.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Dai, W., Bejan, A., Tang, X., Zhang, L., and Nassar, R., "Optimal temperature distribution in a three dimensional triple-layered skin structure with embedded vasculature," J. Appl. Phys., Vol. 99, pp. 104702-1- 104702-9, (2006).
In p. 104702-2 we can read that: "Here, the dimension and blood flow of
multilevel blood vessels are determined based on the constructal theory of
multiscale tree-shaped heat
exchangers.
"
Our comments: This is the pretty obsolete and incorrectly
stated homogeneous model for the blood distribution and heat transport in
human tissue. For example, and that is the main error:
The basic initial statement equations (4) --(7) are not
matching one to another -- meaning the first three supposed to be matching to
the last one -- (7). And the first three -- are supposed to interface in a
meaningful way between themselves -- not using the algebraic terms assignment.
Authors can not develop the models of heat transfer in the
three scale blood network imbedded into tissue, tumor's tissue!
Their model -- Bejan's based "constructal" model, is just
hand glued singled out lower scale homogeneous equations with no ability to
get the local-non-local averaged model and its mathematical statement for this task.
Among the few first coming questions after reading this paper we can ask:
2) Does the "constructal" theory, this one scale method of graph description
of heat transport and flow in porous media -- human tissue (that is much more
complicated); able to deliver the modeling and solution of any (one)
reasonably stated problem of the heat transfer in human tissue, tissue model?
Remember, this is the polyscale media and physical processes. As the HSP-VAT
does -
and
- "NIH Proposal 1994-"Hierarchical Multiphase Muscle Blood System
Simulation"
Of course, not.
Is the "constructal" theory able to deliver the solution of any (one)
classical problem in Thermal Physics of porous media? As the HSP-VAT does -
Of course, not.
Can this "constructal" theory do deliver the effective transport coefficients within Thermal Physics of porous media? Which we need for the intermediate and preliminary results tissue modeling. As the HSP-VAT does -
- "Effective Coefficients in Thermal Physics"
Of course, not.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Bejan, A., "Constructal Trees of Convective Fins," Journal of Heat Transfer, Vol. 121, pp. 675-682, (1999).
Author talks about an array of fins.
This is the second paper in the series that almost recognized that the hierarchical approach should be used in the volumetric two-phase heat exchange modeling, design and optimization.
Later on (in 2000-2004) Bejan started to claim that this goal is achieved already with the "constructal" theory?
Still, the constructions in "constructal" theory proceed with qualitative, inappropriate algorithms. The hierarchy is of the verbal Qualitative character.
From the ABSTRACT:
"This paper extends to the field of convective heat transfer the constructal theory of optimizing the access of a current that flows between one point and a finite-size volume, where the volume size is constrained. The volume is bathed by a uniform stream. A small amount of high-conductivity fin material is distributed optimally through the volume, and makes the connection between the volume and one point (fin root) on its boundary. The optimization proceeds in a series of volume subsystems of increasing sizes (elemental volume, first construct, second construct. The shape of the volume and the relative thicknesses of the fins are optimized at each level of assembly."
Wonderful writing - if only author does recognize what he is writing about?
Why would Bejan continue to do this well over more than 18 years?
Our guess would be that in spite of the proper words Bejan writes - "The optimization proceeds in a series of volume subsystems of increasing sizes (elemental volume, first construct, second construct. The shape of the volume and the relative thicknesses of the fins are optimized at each level of assembly"
in reality, he can not do neither the each scale correct modeling, nor the connection between each of the two (at least) neighboring scale models.
It is interesting to read while guessing what author does not know or does not wish to say?
1) Author does not include the second, at least, phase for modeling and optimization.
2) Does not include the physical phenomena of the two-phase scaled transport. Just not possible to do that within this approach.
It is not even possible because this formula of operation - the concepts for phenomena do not allow to involve the other then the homogeneous mathematical statements for a one phase.
3) There is the wrong understanding and modeling mathematical statements, as long as the GE's (governing equations) are used from the Lower scale phenomena while the optimization goals have taken for the Upper scale phenomena (those should be averaged, but they are not) - as the heat transfer rate, etc.
4) One can imagine, if physicists in the Manhattan project would be guided in their research ONLY by homogeneous (conventional) thermodynamics - and how many decades, definitely with no success, (the ITER program is one of examples) it would be taken for understanding the atomic - nuclear peculiarities in this situation, before making a bomb?
Too much of homogeneous thermodynamics courses were taught.
5) The optimization statement and criteria are formulated incorrectly - as for a one scale task.
6) The mathematics of 3D and 2D Distributed parameters (sources) 2 scale Optimization problem is incorrect.
7) The 3D volumetric heat exchange optimization problem presented by Bejan in p. 90 of paper (Mechanical Engineering, No. 10, pp. 90-92, 1997; for illustration the heat removal in the heat exchanger (HE) problem) is incorrectly described - the heat generation rate can not be fixed just in a volume. It depends on the volumetric heat dissipation elements morphology and their distribution, among other issues, etc.
8) In p. 92 - we can read that "... such as from large to small, through the repeated fracturing..." How is that? Exactly, how this can be done? Quantitatively, with the strict scale connected two-scale models?
Could be mentioned some other out of aim concepts of the pseudo-science of "Constructal Theory" Optimization.
+++++++
We can repeat here the prudent words regarding the "Constructal theory" that the authors of the own hierarchical hypothesis West and Brown (2004) wrote in their rebuttal on the Bejan's claim on universality of his "Constructal theory" hierarchical claim.
They wrote that: "It remains to be seen, however, whether constructal theory is sufficiently general, detailed, and mechanistic to describe these systems in a quantitative, predictive, and analytic way."...
We can add to this - that the "constructal theory" contentions are looking as just the wishful thoughts while using the simplified point of view (Bejan, 2000; other publications). In a real world we need to stick -- especially in biology, biophysics, highly priced nanotechs to the models that nature presents to us in an incontestable way, mode, form and shape.
Nevertheless, we also need to follow the shape and structure of the natural subjects -- real polyscale physical medium, the cell, etc.
Of course, by not using the verbal assurances and estimations mostly.
And we need to begin with understanding, explanation, and model what is set up in front us - then we might be able to contest the nature and try to modify (or create from scratch) the new entity, matter, material, device (as a HE), "cell", single cell organisms first of all, tissue, organ, etc.
We would allocate this theory to the kind of an "ad-hoc Optimization", "ad-hoc science" supposition.
~~~~~~~~~~~~~~~~~~~~~
For the models and differential equations describing HE's to be useful, the additional integral and integro-differential terms used to be present in the scaled statements reflecting the interscale transport in the GEs that need to be addressed for scaled physics features in a systematic way. VAT has the unique ability to enable the combination of direct general physical and mathematical problem statement analysis with the convenience of segmented analysis usually employed in HE design.
On the other side, a segmented approach is a method where overall physical process or group of phenomena are divided into selected sub-processes or phenomena that are interconnected each to others by an adopted chain or set of dependencies. A few of the obvious steps that need to be taken are the following:
1) model that increases the heat transfer rate;
2) model that decreases of flow resistance (pressure drop);
3) combining the transport (thermal / mass transfer) analysis and structural analysis (spatial) and design;
4) finding the minimum volume ( the combination of parameters yielding a minimum weight HE);
5) model that includes nonlinear conditions and nonlinear physical characteristics into analysis and design procedures.
The power and convenience of this method is clear, but its credibility is greatly undermined by variability and freedom of choice in selection of sub-portions of the whole system or process. The greatest weakness is that the whole process of phenomena described by voluntarily assigned set of rules for the description of each segment is done without serious consideration of implications followed by such segmentation.
Strict physical analysis and consideration of the consequences of segmentation
is not possible without a strict formulation of the problem which
the HSP-VAT based modeling supplies for Heterogeneous, Hierarchical problems
as the Heat Exchangers Modeling and Simulation. Structural optimization of a
plate HE, for example, using the HSP-VAT approach might consist of the
following steps:
1) Optimization of the number of plates, plate spacing and fin spacing;
2) Optimization of the fin shape;
3) Simultaneous optimization of multiple mathematical statements.
This approach allows also the consideration and description of hydraulically and thermally developing processes by representing them through the distributed partial differential systems.
Adrian Bejan, Alexandru Morega, Geoffrey B. West, and James H. Brown, "Constructing a Theory for Scaling and More," Physics Today, Vol. 58, Iss. 7, pp. 20-21, (2005).
Bejan, A. and Morega, A.M., "Optimal Arrays of Pin Fins and Plate Fins in Laminar Forced Convection," Journal of Heat Transfer, Vol. 115, pp. 75-81, (1993).
Bejan, A., "The Optimal Spacing for Cylinders in Crossflow Forced Convection," Journal of Heat Transfer, Vol. 117, pp. 767-770, (1995).
Morega, A.M., Bejan, A. and Lee, S.W., "Free Stream Cooling of a Stack of Parallel Plates," Int. J. Heat Mass Transfer, Vol. 38, No. 3, pp. 519-531, (1995).
Ledezma, G., Morega, A.M., and Bejan, A., "Optimal Spacing Between Pin Fins with Impinging Flow," Journal of Heat Transfer, Vol. 118, pp. 570-577, (1996).
Bejan, A., "How Nature Takes Shape," Mech. Enginering, No. 10, pp. 90-92, (1997).
Bejan, A., "Constructal Trees of Convective Fins," Journal of Heat Transfer, Vol. 121, pp. 675-682, (1999).
Bejan, A., Shape and Structure: From Engineering to Nature, Cambridge U. Press, Cambridge, (2000), and references therein.
Dai, W., Bejan, A., Tang, X., Zhang, L., and Nassar, R., "Optimal temperature distribution in a three dimensional triple-layered skin structure with embedded vasculature," J. Appl. Phys., Vol. 99, pp. 104702-1- 104702-9, (2006).
West, Geoffrey B. and Brown, James H., "Life's Universal Scaling Laws," Physics Today, Vol. 57, Issue 9, pp. 36-42, (2004).
