Pseudo-Science of Constructal Theory (Hierarchy) in Heat Transfer Modeling

Among numerous publications (there is the website) by A.Bejan on the subject of volumetric heat transport features, modeling and even optimization, we might be able to include here just few of many discussed and examined mostly in 1990s, following with few recent publications. The reason is simple - to explain to the professional and student communities the unsatisfactory claims on usefulness, generality and applicability of the "Constructal theory" for the heterogeneous scaled heat transfer problems.

Again as I said in - "Announcements ::Analysis of Published Studies on Heterogeneous and Hierarchical Media"

"Announcement #3: There is our intention in this website first of all to serve better to this field and to people studying and working in it. The issue of analysis of published in the field research papers, reviews, books is the inseparable activity. That is why everybody should understand and think about critical materials appearing on these pages - as of usual component of healthy movement in the area.

There is no personal critique on these pages........."

There is the old starting paper on volumetric heat transfer by Bejan, A.: "Theory of Heat Transfer From a Surface Covered with Hair," Trans. ASME, J. Heat Transfer, Vol. 112, pp. 662- 667, (1990).

In p. 662 one can read that: "The objective of this paper is to describe the fundamental heat transfer mechanism of a surface covered with hair."

The idea of the paper is good - but the mathematical models and GE (3,4), (17) are incorrect, that unfortunately precludes any further reasonable analysis.

In the p. 665 the homogeneous Linear [W/m] heat transfer rate definition is given. For the Heterogeneous Volumetric (or 2D as in this case) this kind of heat transfer characteristic is inappropriate due to loss of few transport features, see the explanation in -

  • "Thermal Physics::Heat Rejection Enhancing Surfaces" and "Thermal Physics::Semiconductor Coolers",

    where given an in-depth analysis of some two-scale thermal transport in porous medium of surface porous layers and in heat exchangers.

    The paper can be used as one of examples of - How not to treat the 2D Heterogeneous problem. Anyway, formulation of a problem, where instead of the total number of constituted parts missed one or more of them, will direct the statement and solution in an insufficient territory regarding its results.

    That is why; the conjugation of heat transfer two-phase problems has been debated in 70s-80s, rather long ago. All of that is - to improve the validity and increase the accuracy and power of solution. Well, since then we understand the power of collective polyscale phenomena better and better, that is why the HSP-VAT appeared on scene.


    Bejan, A. and Sciubba, E. (1992), "The Optimal Spacing of Parallel Plates Cooled by Forced Convection," Int. J. Heat Mass Transfer, Vol. 35, No. 12, pp. 3259-3264.

    In the study by Bejan and Sciubba (1992) authors did the problem where - the stack of one size width $D=~H$ (in our notations) MATH flat layers in which moves in x-direction in laminar regime a fluid has the average fluid velocity $u_{b}=\overline{U}$


    where $L$ is the length of the channel,

    while the mass flow rate per slit unit width is


    (when usual mass flow rate is MATH and the heat transfer rate MATH the amount of heat removed per slit unit width is


    Authors give the optimal spacing (width of the channel) for developed laminar regime in the stack of equal flat channels when the walls are heated to constant temperature $T_{w}$ as


    where $a_{f}$ is the thermal diffusivity of the fluid.

    Also, authors created few more formulae for the case of developing flow, one wall is at $T_{w}$ while other is adiabatic, uniform heat flux at the entrance part of the channel. The all analysis is obtained by using the extreme transition method - when investigated the cases of $H\rightarrow 0$ and to MATH

    The problem stated as a non-conjugate one.


    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.

    Morega et al., (1995) did the work based on the numerical solution of the free convection problem over a number of (vertical) plates attached to a surface.

    The problem stated as a non-conjugate one.


    The work by Rocha, L.A.O. and Bejan, A., "Geometric Optimization of Periodic Flow and Heat Transfer in a Volume Cooled by Parallel Tubes," J. Heat Transfer, Vol. 123, No.2, pp. 233-239, (2001).

    This is the good article for comparison and for criticism on the approach followed by Bejan,

    which here was done for the "non-conjugate" problem, which instantly deprive any sense to talk about optimization, - because authors have just thrown away the second phase !!

    In p. 233 from the abstract one can read that - "This paper addresses the fundamental problem of maximizing the thermal contact between an entire heat-generating volume and a pulsating stream of coolant that bathes the volume. The coolant flows through an array of round and equidistant tubes. Two laminar flow configurations are considered: stop-and go flow, where the reservoir of coolant is on one side of the volume, and back-and forth flow, where the volume is sandwiched between two reservoirs of coolant. The total heat transfer rate between the volume and the coolant is determined numerically for many geometric configurations in the pressure drop number range MATH, and $Pr\geqslant 1.$"

    How dare authors saying about heat transport "between the volume and the coolant" - if the study does not have a solid phase in the problem!

    Also in the p. 233 authors refer mostly to the Bejan's own studies - where:

    "The most recent constructal work dedicated to compact heat exchanger and the cooling of electronic packages has focused attention on geometry - how shapes and structures can be designed so that a global performance objective is met subject to global constraints [1]." - this is reference to the Bejan's (2000) book.

    "Examples that come closer to the geometric optimization, that spacing between heat generating components are optimized, so that the given volume is used to its fullest."

    That is the stretching - No one ever and up to now can tell that the Volumetric Heterogeneous Heat Transfer Optimization achieved. Not in a Single instance.

    p. 234 - given the equation for the fluid phase - inside the tube as (eq. (4))


    "where $a$ is the thermal diffusivity of the fluid. There are the three boundary conditions:


    The fluid flow statement corresponds to


    which is the usual laminar flow equation with the solution


    They have not the heat transfer rate per unit volume

    - but rather the "unit of duct volume" which is not used and known unity for heat transfer.

    In p. 234: "The quality of interest is the average heat transfer rate per unit of duct volume use."

    Noting - "per unit of duct volume use"? Is it cooled the solid or the fluid "duct volume"?

    Doing this definition, on p. 235 "the average heat transfer rate per unit volume" introduced


    where MATH is the time-averaged "heat transfer rate" in their definition and in units $\left[ W\right] $, MATH is the fluid's thermal conductivity, MATH is the thermal diffusivity (of fluid), $t_{c}$ is the time period $\left[ s\right] $, and $Q_{1}$ is the dimensionless integral of temperature gradient integrated over the tube wall and over the time period.

    Authors later, on p. 235 (expr. (27)) also give the real volumetric flux "heat transfer rate per unit volume" of medium


    where $C=$ MATH MATH and $S$ is the spacing between adjacent tubes $\left[ m\right] $.

    So, according to dimensions, this is not a "heat transfer rate per unit volume" - this is the volumetric heat flux in general, as for the same reasons the meaning of the HSP-VAT variable


    using which we can compare the current set up with the correct conjugate volumetric flux following the VAT developments for the straight stochastic diameter pore capillary morphology medium (see the definitions and explanation in

    "Semiconductor Coolers,"

    as this volumetric flux is


    The value MATH would be expected significantly different from the $q_{volB}$ because MATH obtained following the solution of the conjugate problem - meaning the two-phase problem.

    Let's explore in more detail what is meant by Bejan and co-authors when the "volumetric cooling" terms made use of?

    In this and other papers (works) by Bejan and co-authors he considers - as for the "Modeling and Optimization" in the heat transfer solid-fluid systems ONLY the Fluid system's phenomena? Non-conjugate models and mathematical statements used.

    How it can be told that the heat transfer problem and Optimization is sought and performed, when the solid phase temperature or heat flux are fixed?

    It's out of a good reasoning; it's the pseudo-science at present time. Well, 30-40 years ago when no mathematical tooling and computers mostly were utilized that might constitute the advancement, but not nowadays.

    It's said - the "duct volume," and at the same time authors saying - " a volume cooled parallel tubes.."

    Well, then this is the pseudo-statement and pseudo-claim on the distributed heat transfer "optimization".

    There is no such thing as to "optimize" the volume process and discard at the same time the part of the said volume completely.

    When one can read further in the p. 235 that - "In other words, the geometric maximization of $Q_{1}$ teaches us how to select the dimensions of the tube bundle so that the entire volume $AL$ is reached most effectively by the fluid when the pressure cycle is specified MATH

    we would surely say - this is just conventional homogeneous heat transfer statement for quite different task that can not be covered by the one scale local physics concepts.

    Also, one would think again that the problem is about the "entire volume" heat dissipation rate optimization?

    And will be wrong, there is the trick here with substitution of one real thing - the total volume where a heat dissipates, by the one phase only volume - which is the very simple physics disconnected from the other phase.

    Also, the approach demonstrated by Rocha and Bejan (2001) won't allow to consider the problem with the irregular or distributed diameters of pores (and any other heat transfer elements in other similar publications). Because again - one needs to average over the space (and in time, if needed) the heat transfer characteristics while applying this principle. And they can not do that within this approach.

    See more features on Volumetric Heat Transfer Device (VHTD) modeling in our - "Semiconductor Coolers."


    Closer to our times - one can see more on how this conceptual description of the pseudo-science of "Constructal Theory" works for the Volumetric Flow and Heat exchange.

    Lorente, S., and Bejan, A., "Heterogeneous porous media as multiscale structures for maximum flow access," J. Appl. Phys., Vol. 100, pp. 114909-1- 114909-8, (2006).

    We can read from the abstract:

    "We develop completely multiscale configurations that guide the flow from one side of the porous structure to the other line to line and plane to plane and show analytically the advantages of tree structures over the usual stacks of parallel microchannels. The "porous medium" that has tree-shaped labyrinths is heterogeneous, with multiple scales that are distributed nonuniformly."

    Our comment: Well, we see now the word "multiscale" appeared in the texts! So what? What does this mean -- constructing the graph figures?

    Those figures are not connected to a scaled physics -- because each element is constituted, thought and modeled as a Lower scale only the simplified physics physical and mathematical model.

    No path to go to generalization of the Upper scale because there is no theory for the Upper scale mathematics, physics and modeling!

    p. 114909-5: "Although the structure of Fig. 5 is marginally less

    permeable this is another confirmation of the earlier work

    done on trees, Figs. 2 and 3, Fig. 5 is simpler than Fig. 1. It

    has fewer channels: this is an attractive feature for the design

    of microchannel structures."

    So, what? These figures we drew in 1992-93 and found that this way of doing scale construction and communication is going to the dead-end without the second Upper scale consideration -

  • "ARPA 1994-95 Project - "Two-phase Contaminant Transport Simulation in Five-Scale Saturated Soil Systems"

    Because, there is no way by childish drawings to develop, design, model and simulate the network of pores, tubes as the finite set of elements and to raise the physics and theoretical generalization to the Upper scale.

    There should be the way to figure out the Upper scale physics and mathematics tight to these drawings in a meaningful way - as the HSP-VAT does.

    2) For the round tubes the equations (7) -- (18) have no sense to model and simulate unless we have the tools, mathematics, models to make the generalization of those calculations.

    Which only the HSP-VAT has and demonstrated with problems and models solved.

    About Figs. (1)-(5) in the paper; I have created numerous figures like that for purposes to model and simulate the lower network scale flow process and fields. But I did that freely because I had the tools to obtain the data reduction, the data analysis for understanding --How flow is going on and how morphology affects the flow rate and pressure drop at the Upper (averaged fields) scale.

    This is not possible with the one scale modeling by "constructal" theory!

    Besides, and this is quite important thing -- the HSP-VAT gives the ability to see the second scale physical effects (phenomena- as we found in the direct modeling of those additional integral and integro-differential terms) -- that the "constructal" theory is not able to deliver, to detect, show up, or model!

    3) The same in many similar graphs of flow in porous media for the Lower scale description in HSP-VAT we had drawn in 1992-94 and presented in the proposal for the DOE grant at UCLA.

    The reasons for our exercises in drawing were different -- that were the methods of the Lower scale depiction and simulation with the imperative aim to seek and solve the Upper scale momentum transport in porous media.

    This is the real Goal of all these constructions, not to draw the figures for qualitative reasoning on their qualities.

    p. 114909-5: "The opportunity to discover the best architecture of a point-to-plane tree is illustrated by the following analysis of the construct of Fig. 7."

    Our comment -- There is no opportunity to discover anything with this approach as soon as just the mechanistic connection of the tubes (pores) (without nodes specific characteristics as pressure drop and flow turbulence) does not create the model to analyze the physical process of flow in that network as a whole for the Upper scale processes. Hydraulic estimation is used in the Hydrology for long ago -- still nobody (but Bejan) would claim that the hydraulic network calculation creates the new level of generality for better understanding and design of a pipe flow network as a porous medium.

    Also, again -- there are a number of physical effects that the Lower scale pipe network simulation can not discover and include in modeling and data reduction for the Upper physics scale.

    In Conclusions p. 114909-7 one can read:

    "The observer who looks from the outside at the side of the porous medium e.g., from the top of Figs. 1 and 6 sees a few large pores surrounded by many small pores. From this viewing position, the porous medium appears to have two scales."

    Our comment -- Really? Oh, now at last, authors can see the two scales in porous medium that was obvious from the beginning. Unfortunately, there are also the two-phase media and the 3-scale distinct means to model the process of fluid flow.

    Does the "constructal" theory, this the one scale method of graph description of flow in porous media able to deliver the solution of any (one) classical problem in fluid mechanics of porous media? As the HSP-VAT does -

  • "Classical Problems in Fluid Mechanics"

    Of course, not.

    Can this "constructal" theory to deliver the effective transport coefficients for fluid mechanics of porous media? As the HSP-VAT does -

  • "Effective Coefficients in Fluid Mechanics"

    Of course, not.

    Can this "constructal" theory deliver the transport problem solutions in fluid mechanics of porous media, as in the soil and groundwater contamination science, for example? As the HSP-VAT does -

  • "Some Problems and Advancements by HSP-VAT in the Soil and Groundwater Contamination Science"

    ARPA 1994-95 Project - "Two-phase Contaminant Transport Simulation in Five-Scale Saturated Soil Systems"

    Of course, not.


    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.$^{30-32}$"

    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 -

  • "Medicine Heterogeneous, Multiscale Applications"


    - "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 -

  • - "Classical Problems in Thermal Physics"

    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. and Lorente, S., "Constructal theory of generation of configuration in nature and engineering," J. Appl. Phys., Vol. 100, pp. 041301-1- 041301-27, (2006).

    From the p. 041301-1: "This principle was formulated in 1996 as the constructal law of the generation of flow configuration.1--5 "For a flow system to persist in time to survive it must evolve in such a way that it provides easier and easier access to the currents that flow through it."

    Also from the abstract: ". The same principle yields new designs for electronics, fuel cells, and tree networks for transport of people, goods, and information."

    Our comments: There is the law in an equally (as the "constructal" theory) great science as the philosophy and it says:

    "The sufficient amount of something (entity, events, matter, etc.) would inevitably transfer the means and values of that "something" to another qualitative state, another quality of a subject."

    The Law of Quantity-to-Quality transfer.

    That means the few facts with the mechanical simple observation as we might find in the "constructal" theory, is not qualified to be the next scale generalization ground breaking theory!

    That would be careful to say -- that the general observation is not yet a scientific conclusion, not a worthwhile ground for claiming a World actuality discovery and so on.

    In the p. 041301-16 we can read: "Optimal spacings

    have been determined for several configurations, depending

    on the shape of the heat transfer surface that is distributed

    through the volume: stacks of parallel plates e.g., Fig. 20,

    bundles of cylinders in crossflow, arrays of staggered plates,

    and arrays of pins with impinging flow."

    Our comments - These are the same arguments we dealt with in the above mentioned and analyzed studies by Bejan on the "constructal" theory applied to the heat transport!

    Read the above notes.

    This is not a science -- it is the system of desires, the theological system.

    In page 041301-25 : "In summary, the body of work reviewed for the first time in this article introduces a paradigm that is universally applicable in natural sciences physics and biology, engineering, and social sciences."

    Well, that kind is familiar from the history -- this is the self-appointment for everything without a limit!

    This claim is for a "law" of universality and greatness.

    In the whole text we could not find some original reasonable scientific arguments -- apart of simplistic geometric observations, like the famous allometric laws in biology. Which we have noted is an analog of Darcy law in Fluid Mechanics and really should be greatly advanced when applied to field problems.

    For all the more strict comparisons and mathematically strict modeling quality we just spoke in our above remarks on the studies with the words "constructal" theory in texts, and in other parts of this website.


    Why Bejan continues to do this kind of claims well over more than 18 years?

    Our guess would be that in spite the correct 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.

    Summarizing the above said features about this approach to the heat transfer modeling - the "Constructal theory" we can enumerate the mortal highlights of it:

    1) Not included the second of at least, phase for modeling and optimization.

    2) Not included the physical phenomena of the two-phase scaled transport. It's just not possible to do that.

    It is not even possible because this formula of operation - the concept for the phenomena does not allow involving the other then the homogeneous mathematical statements for a one phase modeling.

    3) There is the wrong understanding and modeling mathematical statements as long as the GE's (governing equations) are used at the Lower (local) scale phenomena while the heat transfer design and optimization goals have taken for the Upper scale phenomena (those should be averaged, but they are not) - as the heat transfer rate, etc.

    4) But the most important is that the HSP-VAT statement gives possibilities to model and calculate the really hidden components of the heat transfer, those are not seen in the "constructal" theory works with homogeneous mathematical statements for heterogeneous scaled problems.

    5) The two-scale local-non-local VAT formulation of Heterogeneous, Hierarchical heat transfer problems have additional terms on the Upper scale. These are reflecting the additional physical phenomena which physically known, but can not be observed via the modeling through the one-scale homogeneous models.

    Otherwise, the scaled VAT formulation allows one to include and study the phenomena for the Upper scale that are not observable when conventional homogeneous modeling is used.

    Among these features in Heterogeneous Scaled Heat Transfer are:

    5.1) the heat transport due to the interface surface (could be a few percent of the overall rate for HE processes);

    5.2) an exact description of interphase transport and its connection to the morphology (could be 20% or more);

    5.3) field interaction based heat transport (as driven by fluctuations in temperature and in fluid velocity (could be as much as 10%));

    5.4) and nonlinearities resulting from physical models and their coefficients and forces interacting one with another.

    5.5) The VAT based averaged Upper scale heat sink heat transfer parameters are vector and tensor variables tied directly to the each phase performance -

    "Semiconductor Coolers"

    6) There is the slogan in the HSP-VAT:

  • "There exists no way to find out, to control or to optimize the performance of the two- or three scale volumetric heat transfer device (VHTD) while using the only Lower Scale modeling equations."


    Bejan, A., "Theory of Heat Transfer From a Surface Covered with Hair," Trans. ASME, J. Heat Transfer, Vol. 112, pp. 662- 667, (1990).

    Bejan, A. and Sciubba, E., "The Optimal Spacing of Parallel Plates Cooled by Forced Convection," Int. J. Heat Mass Transfer, Vol. 35, No. 12, pp. 3259-3264, 1992.

    Bejan, A. and Morega, A.M., "Optimal Arrays of Pin Fins and Plate Fins in Laminar Forced Convection," J. Heat Transfer, Vol. 115, pp. 75-81, 1993.

    Bejan, A., "The Optimal Spacing for Cylinders in Crossflow Forced Convection," J. 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., "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.

    Rocha, L.A.O. and Bejan, A., "Geometric Optimization of Periodic Flow and Heat Transfer in a Volume Cooled by Parallel Tubes," J. Heat Transfer, Vol. 123, No.2, pp. 233-239, (2001).

    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).

    Bejan, A. and Lorente, S., "Constructal theory of generation of configuration in nature and engineering," J. Appl. Phys., Vol. 100, pp. 041301-1- 041301-27, (2006).

    Lorente, S., and Bejan, A., "Heterogeneous porous media as multiscale structures for maximum flow access," J. Appl. Phys., Vol. 100, pp. 114909-1- 114909-8, (2006).

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