What can one do with the nose? If one were Cleopatra of Egypt, she could rule Rome. If one were the unfortunate Sphinx of Egypt, his form minus the nose could become the wonderment of the World. If one were Tycho Brahe, he could remove the nose, for polishing amidst a heated debate or duel, to distract his opponent. For, he lost his original nose in a duel and had a metal one fixed. If one were the Tamil Detective Sambu created by Devan, he could run his thumb and index finger over the nose to make a deduction that is often wrong. If one is a proboscis monkey inhabiting the island of pulau pulau bompa, it endures the ignominy of its nose looking similar to Rastapopulous, the villain of Tintin comics.

What can one do if one is a toco toucan with its nose and mouth wrapped into one big bill? Of course, it can use the bill for quick thermo-regulation by exchanging heat with the environment.

In fact, in a matter of minutes toucans can release upto 400 percent of their metabolic heat through their bill. In a study I liked very much, Prof. Tattersall and team have measured the temperature of toucan bills in several surrounding and body conditions to arrive at such conclusions. These results were published in July 2009 issue of the Science magazine [1]. Several webs news services [3 - 6] briefed on these findings before the paper was actually published. This note is to discuss the published results in some detail.

Picture Credit: Ref. [2]

Sometime back we discussed why elephants have large ear flaps [7]. Elephant ears serve as means for their thermo-regulation. Elephant ears are believed to be highly vascular (with dense blood vessels), which enable them to transport heat from the interior of their body to the ears, for exchange with the environment. Elephants control their body temperature by losing the excess metabolic heat to the environment through their ears. Toucans do a similar thing with their bills. Only quicker.

The bill of toco toucan, the largest of toucans (about 20 cm for the heaviest), is enlarged, uninsulated, and well vascularized with a network of superficial blood vessels. This enables blood perfusion as a dominant method of heat transport to them from their body. Like the ear flap of an elephant, the toucan bill has all the features of a thermal radiator.

Thermo-regulation in warm blooded animals is an active process that maintains constant body temperature. It can be considered an equivalent to a steady state heat transfer situation achieved through a balance between heat production (through metabolism), gain, and loss. Important thermo-regulation processes include shivering, sweating, vascular adjustment and breathing acceleration. Shivering induces muscle work and heats the body. Sweating and vascular adjustment are primary heat loss mechanisms.

Birds don’t sweat. So toucans can control their body temperature primarily by adjusting blood flow to the body surface and adjust the heat loss to the environment. The blood flow is varied in a tissue region through local vasodilation and vasoconstriction where blood vessels stretch and shrink respectively, to control the flow rate.

Prof. Tattersall and team have measured the surface temperature of the toucan bill to estimate the heat loss. The result of a typical experiment is shown in the figure below.

Figure 2: Toucan bill temperature measurement. Picture Credit: Ref. [1]

For all surrounding air temperature T_a, the eye region was nearly constant [30 ^°C to 36 ^°C] when compared to the body temperature of the toucan of ~ 38 ^°C to 39 ^°C. This indicated continuous blood flow to the naked skin around the eye regardless of the surrounding temperature. In the bill, the portion closer to the body (proximal) and tip portion (distal) showed more temperature variation with change in T_a.

From C and D part of the above figure one can observe the proximal region of the bill was used mainly to lose heat to the environment when 16 ^°C < T_a < 25 ^°C. As the environment temperature rose, the proximal region alone is unable to sustain the necessary heat loss. The blood vessels in the distal region receives increased blood flow through vasodilation and becomes warmer. The increased surface area helps the bird cope with the rise in the environment temperature to lose the necessary excess metabolic heat energy.

That the distal region begins to participate in the thermo-regulation is more evident in the temperature data for the juvenile toucans shown in E and F parts of the figure above. Observe how the proximal region in E tries to manage losing the heat by steadily increasing its local temperature with respect to increase in the environment temperature. However, until after T_a > 25 ^°C, the proximal temperature in E begins to drop, only to be compensated by an increase in the distal (tip) region shown in F.

I wonder how the toucan would feel as more blood flows suddenly through its bill towards the tip. I know how I feel with a running nose.

Toucans can do this thermo-regulation real fast. That is temporal changes in the adult bill’s surface temperatures are rapid and reversible, occurring within minutes. This is excellently captured in the figure below.

Figure 3: Blood perfusion in toucan bill. Picture Credit: (Ref. 1)

As the legend explains, in about two minutes the blood perfusion can be so high that the bill is all aglow with heat to be released to the environment when we compare the left and right images in the above figure.

Such quick control of thermo-regulation was most evident when the toucans were observed while they were sleeping as the video below shows (obtained from supplemental material available on web [2]). It is a series of shots of thermal images taken over a few minutes while the toucan goes to sleep. Watch how initially the bill is golden when the bird is awake and while preparing to sleep, it tucks the bill inside its feather, which is a possible insulator. As the toucan proceeds to sleep, the bill cools to blue suggesting a drop in the basal metabolic rate during sleep.

YouTube Link – Video Credit: Prof. Tattersall

In the next video (available in the webpage for [2]) another interesting aspect of toucans behavior can be observed. Sleeping toucans show transient changes in bill surface temperature without the bird actually waking up. This indicates the possible variations in the metabolic heat production and the sleep-state transitions associated with changes in thermo-regulation.

A few more interesting results are reported in the paper. In what is called as activity-induced vasodilation of the bill, the surface temperatures from a toco toucan in an outdoor aviary during flight was measured. Surface temperatures increased rapidly from ~ 30 ^°C to ~ 38 ^°C during a 10 minute flight at an average speed of 17 km/hr.

Next the heat loss from the bill was also estimated from the above temperature measurements using simplified models. These models with equations are explained in the supplementary material [2] provided with the paper. The toucan bill is assumed as a cylinder at constant temperature exchanging heat with the environment through radiation and convection heat transfer. The details of this modeling is in similar lines as already done for the elephant ears. For the equations and math, read the separate note I wrote on this [8].

Results from these calculations indicate as a proportion of total heat loss, the bill accounted for 30 to 60 percent of heat loss in adults and juvenile toucans.

As the figure below shows, adults on the other hand, could adjust bill heat loss to account for as little as 5 percent (T_bill – T_a ~ 0), and, for short periods, up to even 100 percent of total body heat loss. Juveniles are not this gifted and seem to be unable to control their vasodilation in the bill. Even 2-month-old toucans keep losing more heat from the bill than is necessary. Hence they shiver to produce additional heat even at as environment temperature as high as 26 ^°C.

Figure 4: Heat loss estimates from toucan bill. Picture Credit: (Ref. 1)

In short, toucans can lose as little as 5 percent or as much as 100 percent of their body heat through their bill, by opening or closing the embedded blood vessels.

However, heat loss estimates from the bill are highly variable. It depends on air speed and environment temperature T_a, due to strong forced convection possible at higher speeds. From the figure above one can observe that the heat loss estimates could account for as little as 25 percent (minimum) to as much as 400 percent (maximum) of resting heat production (RHP) in adults. This value, the authors mention, is the largest reported for an animal. But it is not clear whether toucan encounter velocities of 20 km/hr or 6 m/s in their habitat. Only for such wind speed values (the maximum in the above figure) heat loss is 400 percent of RHP. In fact, the present calculation can predict even higher heat loss for higher wind velocities, as forced convection continuously increase with speed. It is not clear from the paper [1] if the 6 m/s is experimentally measured or used only in the model for estimates.

A similar concern exists for the heat loss calculations for elephant ears reported earlier [9]. For instance, when the ear is modeled as a flat plate convecting heat into the surrounding air blowing with a certain speed, the heat loss is proportional to the wind speed. Estimates from such a model reported in [9] (and discussed in detail in my earlier note [8]) require a wind speed of 12 m/s for the elephant to lose about 100 percent of the resting heat production. Whether such a flap rate is physically possible for the ear remains, as far as I know, unverified.

References

1. Tattersall, G., Andrade, D., & Abe, A. (2009). Heat Exchange from the Toucan Bill Reveals a Controllable Vascular Thermal Radiator Science, 325 (5939), 468-470 DOI: 10.1126/science.1175553
2. Tattersall, G. J. et al. – Supporting Online Material
3. Toucan Beak Is New Kind of ‘Heating Bill’ – Wired post link
4. Toucans use their enormous bills to keep their cool – Guardian News link
5. Giant Toucan Bills Help Birds Keep Their Cool National Geographic news link
6. Hot secret behind toucan’s bill – BBC news link
7. Why do Elephants have Big Ear Flaps
8. More Heat Transfer from Elephant Ears
9. Phillips, P. K., and Heath, J. E., Heat exchange by the pinna of the african elephant (Loxodonta africana), Comparative Biochemistry and Physiology Part A: Physiology, v. 101, 693-699, 1992. Abstract

[Thanks to Lakshmi for correcting the draft version]

It can be argued that one of the most influential articles ever published in the Journal of Applied Physiology is the Analysis of tissue and arterial blood temperatures in the resting human forearm by Harry H. Pennes, which appeared in Volume 1, No. 2, published in August, 1948. Thus begins Prof. Wissler, his 1998 revisit to this classic paper by H. H. Pennes. In that 1948 paper, he proposed what can be identified today as the first analytical Bioheat transfer model with experimental validation from temperature variation data in human forearm. Many later models have refined what he proposed but his basic insight that blood is a carrier of heat, adding a distinct perfusion term to the the standard heat equation, remains a major contribution.

The schematic of the experiment is shown in the accompanying figure. After anesthetizing the right forearm of each human subject, Pennes measured the temperature variation from the surface skin into the deep muscle. During experiment, as Pennes reports in an understatement, ’phlegmatic subjects occasionally reported no unusual pain’.

For all the subjects, temperature measurements were done by placing their right forearm in a fixture that also controlled the exact location of the Y-type thermocouple locations inside the forearm. For the sake of those of us who pause at antique and wonder about proper infrastructure to perform any fruitful experiment, I have given below a low resolution version of the fixture that Pennes used.

Click on the above image for a larger picture with legend. In short, between the upright clamps is where you place your forearm with your elbow resting on this side while your fingers are on the far side. The thermocouple wire is inserted across your forearm and held taut between the clamps marked K. If the wire is not taut, it got deformed in the tissue inside the arm causing minor error in the measured temperatures, as Pennes observes.

The room temperature in all the experiments of Pennes varied between 26.1^∘C and 27.4^∘C (variation within ~ 1.5^∘C) and the thermocouple accuracy was within 0.01^∘C. The initial part of the paper discusses in detail, measurement of the forearm skin surface temperatures. He also measured the rectal and brachial arterial temperatures. Thermocouple readings were recorded after 3 to 4 hours as otherwise the alcohol used on the skin surface before piercing caused 4 to 5^∘C temperature drop.

In the later sections Pennes describes his extensive experiments to measure temperature data along the interior of the forearm. Typical steady state temperature measurement are as shown in the picture below. Data for three subjects are shown in the picture. In principle, the thermocouple recorded the tissue (muscle) temperatures. The trough in the middle of the curve for the atypical subject is attributed by Pennes to the possible presence of an artery near the thermocouple location.

In order to explain his forearm tissue temperature data, Pennes was able to suggest a suitable modification to the standard Heat Equation through the introduction of a blood perfusion term. Further, the Pennes bioheat equation was developed specifically for the arm treated as an axi-symmetric cylindrical medium with uniform thermo-physical properties of blood and tissue assumed along with uniform metabolic heating in arm tissue. In revised form, what is now called the Pennes Bioheat Equation is written as

The terms in the above equation are explained in the legend below.

In particular, the _P ^′′′ is a lumped parameter called the blood perfusion source term that determines the heat transfer between the blood and the tissue. Interestingly, apart from T_t, the unknown tissue temperature present in the bio-heat equation, the blood perfusion term introduces two more unknowns, T_a, the arterial inlet blood temperature and T_v, the venous blood temperature. Pennes devised an equilibrating parameter through which he related T_v and T_t in the form T_v = T_t + λ(T_a – T_t). Here, λ is the degree of thermal equilibrium between venous blood in tissue and tissue itself. From this relation it is evident when λ = 0, venous blood and tissue are in thermal equilibrium and T_v = T_t; when when λ = 1, blood is not exchanging heat with the tissue and T_v = T_a.

Pennes further assumed Ta uniform within domain and equal to arterial blood temperature at inlet. He proposed to relate T_a to the temperature he measured at the inlet brachial artery (T_a0) of each forearm. In principle, blood can heat or cool the tissue, depending on the difference between the incoming arterial blood and the tissue temperature.

The steady state form of the above bioheat equation can be solved analytically in cylindrical coordinates for boundary conditions:

where h is a combined convection/radiation heat transfer coefficient between the skin surface and the surroundings.

The solution of the Pennes bioheat equation in these radial coordinates is

where

and

Here, I₀ and I₁ are the modified Bessel functions.

Because blood perfusion rate ^′′′ could not be measured by Pennes, he varied parameters to match the analytical solution with his experimental data. For T_a0 = 36.25^∘C, assuming λ = 0 (tissue-blood thermal equilibrium), best agreement between experimental data and analytical prediction resulted for 1.2 < ^′′′∕ρ _t < 1.8. The comparison of the analytical curves with the data is shown below.

Importantly, it can be observed that when the blood perfusion is set as zero (bottom-most curve in the figure), the analytical results markedly under-predicts the actual tissue temperature data. This conclusively proves the blood flow in and out of tissue is a major heat transport mechanism in living tissues. This is a major contribution of H. H. Pennes and his bioheat equation.

For completion, there have been enough criticisms and modifications to the Pennes Bioheat equation, some grounded on later date experiments, others on first principles. Even Pennes enunciated some of the shortcoming of his model like the assumption of uniform metabolic heating, perfusion rate and thermal conductivity. But the introduction of the concept of blood perfusion by Pennes as a carrier of heat in living tissues until date remains undisputed and original.

References

1. Introduction to Bioheat transfer
2. E. H. Wissler, Pennes 1948 paper revisited, J. Appl. Physiol. 85, 35-41, 1998. [link]
3. PENNES HH (1948). Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of applied physiology, 1 (2), 93-122 PMID: 18887578 [link]
4. Mathematical Models of Bioheat Transfer by Caleb K . Charny, pp. 19 – 156, in Bioengineering Heat Transfer, Advances in Heat Transfer, v. 22, Eds. Cho et al., Academic Press, 1992.

Bio means Life. Bioengineering applies engineering principles, laws of physics and chemistry in a general sense, to the understanding and modeling of living systems. Biotechnology identifies methods, processes and techniques resulting from bioengineering. Such technology attempt control and duplication of bio-processes (chemical, mechanical) performed by living systems. Examples include the micro-scale (molecular level) recombination of DNA to the macro-scale functioning of an artificial lung [1].

Contributions from bioengineering has been to several conventional engineering categories including instrumentation and measurements, materials, analysis, and modeling. Biomedical engineering (BME) is the application of engineering principles and techniques to the medical field. In a way, any engineering invention, application or method applied to humans and their health related issues in particular can be considered as a contribution of bioengineering. For instance, as early as the 16th century Professor Sanctorius (1561-1636) of Padua, attempted developing a thermometer for comparing temperatures of different persons [3].

Biothermology or Bioheat transfer, the study of heat transfer in biological systems, can be seen as a subdivision of bioengineering.

The objectives of heat transfer are threefold: insulate, enhance or control temperature at a place. These objectives when applied in the context of biological systems such as mammalian and plants, seek better insight into the underlying biological processes by modeling them through mathematical statements and finding particular solutions for them. Biofluid dynamics forms an inherent part of bioheat transfer, when convection or conjugate heat transfer needs to be investigated.

Specific to humans several applications of bioheat transfer processes can be listed [2]:

1. Thermoregulation; metabolic heat generation, evaporation, convection and radiation to achieve steady state;
2. Effect of increased Metabolic Heat Generation; temperature rise during exercise
3. Bioheat transfer in muscles and tissues accompanied with effects due to blood flow (perfusion)
4. Burning; skin burning as transient heat transfer process
5. Fever and Hypothermia
6. Thermal Comfort; Convection, conduction heat transfer through clothing, optimum temperature, humidity, energy transfer in artificial fittings like contact lens

Similarly, biomass transfer processes can be identified within the human body:

1. Blood as oxygen carrier; equilibrium of oxygen in blood with inhaled air
2. Metabolism; diffusive oxygen transfer in a tissue
3. Membranes as barriers to bulk flow; diffusive and ionic flows through membrane channels; porous medium models of capillaries and tissues
4. Liquid Diffusion in tissues; drug delivery to local regions inside body; diffusion of gastric juice in the stomach

A simple illustration should highlight the importance of thermoregulation and the related bioheat transfer mechanisms inside human body. The human body is homeothermal with a core temperature of 37 °C. This is about 3 °C higher than the surface skin temperature. Within a reasonable extent the thermoregulation mechanisms of the body maintain the core temperature constant in spite of variations in the surrounding (environment) and also variations in human body activity like exercising and resting. This suggests the temperature gradients in the human body to be of the order of 0.1 °C/cm . Assuming the human tissue thermal conductivity as 0.6 W/m °C (same as that of water), the heat fluxes across the body only by diffusion (conduction) is ~ 6 W/m². Assuming man as a cylinder and estimating a body surface of 1.8 m², it is evident that heat diffusion alone is not efficient to release the basic metabolic rate of about 90 W to the surrounding. Every body (everybody) becomes a ‘hot body’ in time. Other heat transfer mechanisms (convection, radiation, evaporation) must be important for thermoregulation.

Major applications of bioheat transfer include cryosurgery (i.e. surgery at low temperatures), cryopreservation, therapeutic hyperthermia (healing at high temperatures), ablation of (usually, malignant) tissues using lasers and in general, surgical processes involving lasers (such as retinopathy).

Cryosugery, for instance, use low temperatures (or cryogenics) to destroy cells. Such cell death is caused by the physical or osmotic disruption of plasma membrane and proteins by freeze-growing ice crystals within the cytoplasm of the cell. On the other hand, in therapeutic hyperthermia, heat energy is used to recover cells and tissues. Bioheat transfer mathematical models of varying complexity can be used to predict particular solutions, thereby better understanding, of such processes. Read also the separate note discussing cryopreservation of cells by freezing that was published in the ASME heat transfer gallery in 2008.

The effects of blood flow on heat transfer in living tissue and the mathematical modeling of the complex thermal interaction between the vasculature and tissue has remained a topic of interest for more than a century. H. H. Pennes in 1948 proposed the first bioheat transfer model. The next fifty years many variants have been suggested for this model. A separate note shall discuss these bioheat transfer models. Although there are enough variants of the bioheat transfer model, the development and testing of bioheat transfer models on all scales from microscopic to the entire human body is expected to be continued.

Major challenges in biothermology arise due to several factors:

1. multi component (and sometimes multiphase) media; conjugate heat transfer mode is essential for thermal equilibrium
2. variability of blood flow rate in the internal removal of heat and related vasodilation and vasoconstriction phenomena
3. laminar blood flow with sporadic turbulent bursts and flow reversals due to pulsation
4. exhibition of both Newtonian and non-Newtonian behaviour of blood due to the presence of deformable bodies (red blood cells)
5. accurate determination of thermal properties of tissue become difficult due to heterogeneities, anisotropy and ageing
6. unusual range of system size from macroscopic to microscopic scales makes modeling and simulation of even local regions within the body a challenging task
7. measurement of temperature at small scales and in general, performing in vivo experiments within human body pose difficulties

Ongoing and future interest in bioheat transfer aim to investigate centrally (and perhaps, obviously) the inter-related diffusive phenomena of heat, mass, and momentum transfer. Also, basic anatomical studies with targeted experiments are needed of the vasculature, particularly in the 50- to 500-pm diameter range, in tissues, organs, and in tumors [3]. The development of improved and noninvasive measurement techniques and devices is always encouraged.

References

1. Heat Transfer in Biology and Medicine Course Notes, Jose Lage, 2002, (unpublished)
2. Biological and Bioenvironmental Heat and Mass Transfer, Ashim K. Dutta, Marcel Dekker Inc., NY, 2002
3. Bioengineering Heat Transfer, Advances in Heat Transfer, v. 22, Eds. Cho et al., Academic Press, 1992.
4. Heat Transfer in Energy and Biology, v.1 and 2, Eds. A.Shitzer and R.C.Eberhart, Plenum Press, 1985.
5. Cryopreservation of cells by freezing

Highly Recommended

1. Biothermal Fluid Sciences – Principles and Applications, W.-J. Yang, Hemisphere Pub., New York, 1989.
2. Visual Dictionary of the Human Body, DK Publishing, New York, 1991.
3. Microscale Heat Transfer in Biological Systems at Low Temperatures, B. Rubinsky, Experimental Heat Transfer 10, 1-29, 1997.
4. Human Physiology Series: Temperature Control, M. E. Armstrong and L. C. Parsons, S-3275A US National Library of Medicine, 1979.
5. Progress in Clinical and Biological Research, v. 107, Proc. Int. Symposium, Eds. M. Gautherie and E. Albert Alan R. Liss, Inc, New York, 1981.
6. Physical Biochemistry, Kensal E. van Holde, Prentice-Hall, New Jersey, 1992.
7. Development of Ni-4wt.% Si thermoseeds for hyperthermia cancer treatment, J.-S. Chen et al., J. Biomedical Materials Research, 22, 303-319, 1988.
8. Bioheat Transfer by Liang Zhu, Chapter 2 of Mechanics of Human Body in Standard Handbook of Biomedical and Engineering Design
9. Biofluid Dynamics, Principles and Selected Applications, Clement Kleinstreuer, Taylor and Francis, 2006
10. The role of porous media in modeling flow and heat transfer in biological tissues, International Journal of Heat and Mass Transfer 46 (2003) 4989-5003, doi:10.1016/S0017-9310(03)00301-6
11. Physics of Fluids, March 2005, Volume 17, Issue 3, SPECIAL TOPIC: BIOFLUID MECHANICS [link]
12. research papers of interest are collected from various journals under ‘Bioheat transfer’ in my monthly Read List at this blog.

PS: This was just one out of the many things that we discussed that day, (or should I say argued… lol, that day)

The Discussion and hence, Conclusion- F_dA1-A2

The view factor between a differential area dA1 and finite area A2 is given by [1] 

                                                                 (1)

Where, θ₁ and θ₂ are the angles that the dA1 and a differential element dA2 on A2 make with the line joining the differential elements.

 Figure(1)

Here, only the orientation of the normal of the differential area dA1 is considered but the orientation of the dA1 on its plane is not taken into consideration at all in the Eqn. (1).

Ie.

 Figure(2)

Our Conclusion: Changing the orientation of a planar differential area dA1on its plane does not change the differential view-factor F_dA1-A2

Corollary to the above conclusion: The shape of the planar differential element dA1 does not change the differential view-factor F_dA1-A2  as long as the other parameters such as the direction cosines of dA1 remain unaltered.

Surprisingly, this point was not found to be mentioned in even one of the papers out the exhaustive list of publications we went through. It was found that the authors have predominantly used Case (1) in figure (2) without any justification.

References:

[1] Sparrow, E. M., “A New and Simpler Formulation of Radiative Angle Factors,” Journal of Heat Transfer,Vol. 85, No. 2, 1963, pp. 81–88.

Melting of polar ice caps is a topic of current interest due to global warming and its impact. But not long back in human history, in times of lesser pollution and implication, the inverse problem of solidification or growth of polar ice was of interest. During mid nineteenth century Arctic expeditions to study polar ice caps intended to obtain proper first hand information about those regions for scientific investigation. British and German expeditions to the Arctic regions recorded the time of growth of ice and the air temperature on several occasions, sitting in ships, either frozen up in winter quarters, or drifting with the ice. Data from such expeditions led to the formulation and partial solution of, what is known today as the Stefan problem or the moving boundary problem.

The working of Stefan’s diathermometer to measure the thermal conductivity of gases and how that knowledge helped him in predicting the T power fouth radiation law was explained earlier in two separate notes. This note is the third part that recounts Stefan’s analytical contribution to the understanding of solidification.

We shall use content from the recent Crepeau’s article [1] that recounts Stefan’s achievements and also two other sources, a historical perspective of the Stefan problem by Vuik, available as pdf on the web [2] and the scale analysis from Bejan’s text book [3].

In a nut shell Stefan’s problem of melting has to do with finding for known initial and boundary conditions (in terms of temperature), the ice growth rate or how thick ice grows in time. Say, the water in the Arctic ocean is at the freezing point (see figure below). For certain days the air above the water would be at a temperature much below freezing and would remain that way. So the water begins to freeze. Can one predict the growth rate of ice? The answer Stefan gave in his paper was, the thickness of ice growth depends on the square root of the time involved. Or the square of the thickness is a linear function of time. Of course, Stefan went on to write four papers on this topic detailing his analysis (relaxing many of the earlier assumptions) and compared his calculations with actual polar ice growth rate data taken during the British and German expeditions around 1865.

Before briefing on these history, here is a brief math on the simplified one dimensional melting or moving boundary problem that Stefan considered. The analysis can be found in standard modern heat transfer text books. What I show in the picture is from my class notes but adapted from Bejan’s text book [3], as it discusses the problem through scale analysis.

(click on image for a larger version)

In the figure above, the movement of the solidification front is governed by the conservation of energy at the x = δ(t) plane. The Control Volume (CV) – marked by the dashed rectangle – is assumed to move downwards at a speed equal to the solidification front progress dδ∕dt so that the liquid flow rate into the CV can be imagined to equal the solid ’flow rate’ leaving the CV. Therefore the energy balance (Eq. (1) in figure) follows as a straightforward first approximation. In Eq. (1) A is the frontal area (see schematic at the left bottom), h_f is the specific enthalpy of the liquid, h_s is the specific enthalpy of solid, k is the thermal conductivity of the liquid. In Stefan’s initial analysis, he assumed the air-ice interface at the top to remain at a constant temperature. Since the liquid that is entering the CV is solidifying, the energy released in this process can be expressed as the LHS in Eq. (1), basically mass times the latent heat of fusion times the solidification front velocity. This is equated to the conduction heat transfer through the ice (the driving thermal gradient for the soidification). Stefan assumed this conduction to result in an approximate linear temperature profile (Eq. (2)) resulting in the simplification of Eq.(1) to Eq. (3), which upon an integration and rearrangement gives Eq. (4), the conclusion that the solidification thickness or ice pack growth depends on the square root of time. Observe the rest of the parameters on the RHS of Eq. (4) is already known. (although I write that the rest of the parameters are known, the thermal conductivity of ice that Stefan used was, according to [1], 20 percent smaller than what is the currently accepted value.)

Stefan did much beyond this. He relaxed the assumption of fixed T for air-ice interface and made it a function of time. The revised model gave better comparison with the recorded data of both the British and German expeditions. This comparison with actual data is given in Vuik’s article [2]. Stefan then proceeded to formulate the transient heat diffusion model with a moving boundary at one end (ice-water interface) and solved it with reasonable assumptions via a transcendental equation. The first approximation of the equation of course, would result in the simplified result we discussed above. This analysis is detailed nicely in Crepeau’s article [1] and also in Vuik’s article [2] and I skip it for brevity of this note.

Now for some history. Firstly, the Stefan problem was analysed even before Stefan investigated it. To quote from [1]:

Unbeknownst to Stefan, some work on the moving boundary problem had already been done. In 1762, Joseph Black, a professor of medicine at the University of Glasgow in Scotland, studied the icewater phase change problem and identified the phenomenon of latent heat, while Franz Neumann presented solutions to the moving boundary problem in a series of lectures given around 1860. However, his work was not published until 1901 by Weber H. Weber, Die partiellen Differential-Gleichung der Mathematischen Physik, nach Riemanns Vorlesungen II, Braunshweig, 1901, pp. 118122.

Another interesting fact I learn from [1] is

Because Stefan’s journal of choice, the Sitzungberichte der Kaiserlichen Akademie der Wissenschaften of Vienna was not widely distributed and his results were considered important, his entire paper was reprinted in the Annalen der Physik und Chemie in 1891 J. Stefan, Ueber die Theorie der Eisbildung, insbesondere ber die Eisbildung im Polarmeere, Annalen der Physik und Chemie 42 (1891), pp. 269286., which had a higher circulation. For this reason dual references to this same work exist.

According to Crepeau [1], Soviet scientists Dacev and Rubenstein were those who gave the name The problem of Stefan in late 1940s, while attempting to solve the generalized version of it. The ratio of sensible heat to latent heat of a (phase change) material written as Ste = c_PΔT∕h_sf (first by Lock in 1969 – from [1]) is nowadays identified as the Stefan number. Interest in the moving boundary problem is picking up over the last several decades – average number of publications per year increase from 0.1 in 1931-40 to 55 in 1981-82 [see 2]. Searching Scirus, the portal collecting several hundred journals in the sciences and engineering, with a key phrase as Stefan Problem for the last decade (1999 – 2009) picks up 634 peer reviewed articles.

Melting obviously is the reverse of the solidification problem discussed. In figure above, flip vertically the bottom left schematic for the Stefan melting problem; accordingly, the left top solidification curves need to be flipped horizontally (T₀ > T_m). Melting problems find engineering interest for instance in the investigation of phase change material applications such as composite heat sinks to cool electronics and several manufacturing processes. Nowadays however the phase change process is solved using numerical methods in complex geometries. The benchmarking is nevertheless done with the Stefan analytical result, as the example figure below shows.

In summary, after going through the works of Josef Stefan – the thermal conductivity measurements for gases, radiation law from experiments, and the analysis of Arctic ice growth, one is awed by what natural talent coupled with enthusiasm and perseverance can achieve as first rate science. I think Stefan is one of those who remained faithful to who a scientist is in one important quality – not wondering about the external classification of their work by others, but worked on anything and everything that pricks their curiosity and interest with whatever resources they could mobilize. And as it happens often, have nevertheless managed to contribute both to fundamental understanding and particular applications of a subject.

References

1. Crepeau, J. (2007). Josef Stefan: His life and legacy in the thermal sciences Experimental Thermal and Fluid Science, 31 (7), 795-803 DOI: 10.1016/j.expthermflusci.2006.08.005
2. C. Vuik, Some historical notes on the Stefan problem, Nieuw Archief voor Wiskunde, 4e series, 11 (1993) 157167. Available as pdf.
3. A. Bejan, Heat Transfer (1995), John Wiley, NY.

The non-dimensional representation of the convection heat transfer coefficient ‘h’ is identified as the Nusselt number, in honor of Wilhelm Nusselt. It can be written as

In Eq. (1), L is a characteristic length scale. For instance, if one needs to define the overall convection heat transfer coefficient for a (cold) flow over a (hot) flat plate, then L would  represent the total finite length of the flat plate along the flow direction. The ‘k’ in Eq. (1) is the thermal conductivity of… let us wait and proceed.

There is another non-dimensional number in heat transfer physics called the Biot number, named after Jean-Baptiste Biot (whose other contributions include the Biot-Savart law and according to some, the Fourier Law). It is written as

where h is the convection heat transfer coefficient, L is the characteristic length scale, ‘k’ is the thermal conductivity of… wait a minute, aren’t Eq. (2) and Eq. (1) same?

In Eq. (2)  the thermal conductivity ‘k’ is that of the solid medium which is dipped in a fluid. In Eq. (1) the thermal conductivity ‘k’ is that of the fluid medium.

Nusselt number, through the non-dimensionalization of the heat transfer coefficient in Eq. (1), quantifies how much the convection heat transfer could be higher when compared with the conduction heat transfer, if the fluid were stationary.

The Biot number in Eq. (2) provides a way to compare the conduction resistance within a solid body to the convection resistance external to that body (offered by the surrounding fluid) for heat transfer.

Say an hot steel rod of diameter L is quenched by dipping into stationary air. Since the convection coefficient for stationary air at the maximum is around and the thermal conductivity of hot steel ranges between (decreases with increase in T), the Biot number in Eq. (2) would be provided the L is sufficiently small. This allows one to simplify the transient conduction heat transfer process within the steel rod by treating it as a lumped medium with a single temperature (practically no temperature difference from the center to the edge of the rod in radial direction) changing in time.

For thermal insulators (k is very small) kept in a strong convection situation, irrespective of the smallness of L, could prevail, where the conduction inside the insulator would result in a spatio-temporal temperature difference, which can neither be neglected nor allow one to ‘lump’ the insulator with a single representative temperature.

The Bi provides a way to use proper method of analysis for appropriate situations.

We stop here to take diversion.

Undergrads taking first course in Heat Transfer on many instances don’t appreciate the above difference between Nu and Bi. This is a standard googly question in any oral exam that involves testing of heat transfer basics. To set the record straight, I wasn’t aware of the difference in my undergrad either.

The first time I read and understood about the difference between Nu and Bi was in my masters from the excellent book Heat Transfer, a basic approach by M. N. Özisik. I distinctly remember the elation of this ’secret knowledge’ and the impulsive rush propelling me through the hostel corridors to my friend’s room to test his ‘ignorance’ against mine that is now slightly reduced. He later suggested to keep the book ‘hidden’ with me throughout the semester (the library had only one copy of the book) to ensure others remain protected from such treasures.

We stop this story of how I ‘lost’ the book and paid the hefty library fine acceding to my conniving spirit, to take one more diversion.

My entire note above is just a preamble to mention this: Professor Emeritus M. Necati Özisik (1923–2008) passed away in Oct 2008. An obituary has appeared in the latest issue of the International Journal of Heat and Mass Transfer. An highlight from the obituary text:

[...] Özisik dedicated his life to education and research in heat transfer.

[...] He published more than three hundred research papers in international journals and conferences. He was the author of eleven books, most of them best-sellers that were re-edited several times and published in different languages. His personal characteristics were apparent in all these books, where the material was rigorously presented in a clear, organized and systematic manner. As a result, his books became standards in graduate and undergraduate courses in many countries. His main contributions included analytical, numerical and hybrid solution techniques for direct and inverse problems, for coupled and uncoupled heat transfer modes.

Thank you Prof. M. Necati Özisik for making me appreciate the nuances of heat transfer at the right age.

Reference

[1] Professor Emeritus M. Necati Özisik 1923–2008 [doi:10.1016/j.ijheatmasstransfer.2008.12.022]

The working of Stefan’s diathermometer to measure the thermal conductivity was explained earlier. Here we recount how the diathermometer helped also in the prediction of the T to the fourth power law of electromagnetic radiation purely by experimental means. It was an instance of scientific advancement, where the experimental outcome preceded the theoretical support. We use again content from the article [1] that recounts Stefan’s achievements with a modern perspective. For some direct translation we consult [2], a nice revisit of Stefan’s original paper. I claim no originality of the following content barring the exposition and opinions.

For the sake of starting somewhere let us begin in a time in the past when it was agreed that radiation heat transfer was not proportional to the temperature difference, particularly at high temperatures. But it was not clear what the proportionality is.

In 1817 Dulong and Petit – known for their chemical law for specific heat (example) – published experimental data of temperature evolution in a mercury container kept inside another chamber maintained at θ₀. The gap between the mercury container and the outside chamber was evacuated to low pressure ’p’ (lowest achieved was 2 torr), while the preheated mercury was cooled from a high temperature θ (maximum reached was 300deg C). For given θ,θ₀, the time ’w’ for cooling varied with pressure p. By extrapolating their data to zero pressures, they concluded that this effect was solely due to radiation heat transfer (as convection and conduction were removed). Based on their data and associated cooling rates, they proposed an empirical formula for radiation emittance R(θ) as

where a = 1.0077 was a constant and μ depended on the density, form and surface area of the body that radiates. Since the outer chamber at θ₀ also radiates, the net radiation emittance or exchange can be written – if we substitute ’modern conventional’ nomenclature – as

Enter Stefan. He praised Dulong and Petit’s experiment for its simplicity but suspected the data. Because his previous experiment to measure thermal conductivity of air using his ingenious diathermometer has given him the insight that thermal conductivity doesn’t depend on pressure. This meant the extrapolated data of Dulong and Petit for air, contrary to what they believed, retains the effect of conduction heat transfer. Convection effects could be eliminated by lowering the air pressure in the gap, but not the conduction effect.

By adding a conduction term to the original energy balance equation (that only had radiation loss), Stefan was able to estimate the conduction contribution in Dulong and Petit’s data ranged between 10 and 50 percent. The 10 percent was with data obtained for un-silvered inner mercury chamber and 50 percent was for silvered inner mercury chamber. The silvering reduced the emissivity of the medium, thus revealing the conduction contribution more in the data. But this reasoning was not available to Stefan as emissivity, during his times, is yet to be delineated as a property and measured. Nevertheless, his prediction on conduction effect is objective and correct. Further, in a flight of insight (I don’t know what else to call it), by assuming both E and R to depend on T, Stefan was able to observe the data could be curve-fit using a relation the R = AT⁴. So Eq. (2) can be expressed as

where A depends on the material of the medium used (like silvered or un-silvered) and its size and shape. The notation T in Eq. (3) is to signify the temperature is in absolute scale (in Kelvins), whose significance Stefan was able to appreciate.

By dividing the temperature difference in Eq. (3) with a suitable constant X, Stefan was able to arrive at a cooling rate that agreed well with the experimental data and also with the cooling rate provided by Dulong and Petit (by using Eq. (2)). A sample is given in the table below, where ’w’ represents cooling rates and subscripts DP and S are self-explanatory. Further details on the associated discussion by Stefan in his original paper can be read in reference [2], an excellent expository of this topic.

Although his formula gave a slightly better agreement with the experiment data, Stefan was not happy with his proposal, as Eq. (3) is yet to be checked for data at high temperatures. Only then its generality could be established.

Stefan then located in a nondescript book by Wullner, the experiments of Tyndall (1865) on the differences in radiation from a heated platinum wire to its surrounding. The platinum wire was heated to different temperatures between about 525 deg C (red heat) and 1200 deg C (white heat). Stefan found that the intensity of radiation increased from 10.4 to 122 correspondingly, the ratio of which is around 12. The T power four law gave a ratio of 11.6, lending more credence to its generality. Stefan went on to compare the predictions of his model with more experimental data from Draper (1878) and Ericsson (1872) with success. In the Ericsson data, he again used his knowledge of conduction effects and subtracted them from the data before finding his T⁴ agreed with it better than the Dulong and Petit model.

Stefan also evaluated the temperature of the Sun using his T⁴ model along with the experimental data of Pouillet and Soret. Pouillet had earlier used Dulong and Petit’s model (Eq. (2) above and not the Dulong and Petit’s Law as the Wikipedia page claims) with his data to predict 1734 < T_sun < 2034K while Soret, measuring the radiant energy of the Sun, estimated 2446 < T_sun < 2546K. Stefan with his T⁴ model obtained 5859 < T_sun < 10420K for Pouillet’s data and 5580 < T_sun < 5838K. These are the first good estimates for the temperature of the Sun, the currently accepted value of which is T_sun = 5770K.

Stefan’s T⁴ model was not immediately accepted. Thankfully, his first Ph. D. student, Boltzmann was able to produce conclusive theoretical corroboration by deriving the T⁴ model using radiation pressure of light. This derivation can be read from the Wikipedia. Nevertheless, Stefan’s original paper was not viewed kindly by some even in the twentieth century. Here is an example cited in [2], a portion of commentary by Worthing and Halliday (1948) in a footnote in p.438 to their section on Boltzmanns fourth-power law (reproduced from [2]):

When, however, one considers the basis for Stefans deduction, it hardly seems fair to link his name with the law. The Irish physicist, John Tyndall (1820-1893) had reported the ratio of the radiation output of a platinum wire at 1200C to that at 525C as 11.7. Stefan noticed that the ratio of 1473 K to 798 K raised to the fourth power is 11.6, and stated the law empirically. There were at least two errors, however. Later work has shown that the ratio of the two radiances of platinum at these temperatures is more nearly 18.6 than 11.6, and that the radiation from platinum is far from black-body radiation and should not be assumed to follow the laws of blackbody radiation. Actually, Stefan applied his empirical law with some success to other data, but the same errors were always present. Stefan was an able physicist whose fame should rest on other accomplishments.

Emissivity (ε) as a property that could be measured separately and that it also depended on T was not known by Stefan. Only for a black body the total radiant emittance R is proportional to T⁴, as in R∕T⁴ = σ, the Stefan-Boltzmann constant which is σ = 5.67 × 10^-8W∕m²K⁴, 11 percent higher than what Stefan had estimated. For non-black bodies the ratio would be R∕T⁴ = εσ.

Firstly, much of Stefan’s paper is on the data of Dulong and Petit and others (the ’some success to other data’ in the above quote); Tyndall’s data receives a few paragraphs in a paper of 38 pages long – as pointed out in [2]. Secondly, Stefan’s contribution certainly shows he understood the relevance of his T⁴ suggestion, use of absolute temperatures (which Dulong and Petit missed) and was able to compare his model with more than one set of data at different temperature ranges before proposing its validity.

Being harsh on previous works in light of new knowledge that the previous researchers don’t have access to, is intellectual under-cutting. If there were two competing models in the past and one superseded the other, which was later found to be correct, then at least we have a case for criticism. But we cannot say Darwin’s evolution theory is wrong because he was unaware of genes or the Heliocentric theory of Copernicus in 1543 (or a Rutherford’s atom model around the start of the twentieth century) is simplistic because we now know of something better. Such myopic ’critique’ doesn’t appreciate the perspective of ideas.

[a pdf version of this note for download]

References

1. Crepeau, J. (2007). Josef Stefan: His life and legacy in the thermal sciences Experimental Thermal and Fluid Science, 31 (7), 795-803 DOI: 10.1016/j.expthermflusci.2006.08.005
2. Dougal, R. (1979). The centenary of the fourth-power law Physics Education, 14 (4), 234-238 DOI: 10.1088/0031-9120/14/4/312
3. Stefan-Boltzmann Law from Wikipedia
4. a pdf version of this note for download

Once in a while, it is useful to revisit the history of inventions and discoveries that we now take for granted. Such revisits to the masters and their works provide a perspective on how to think about the unknown from whatever was known – right or wrong. Instances where a correct discovery results from wrong beliefs like the Carnot’s ideal heat engine theory from Caloric theory of heat that modeled heat as a fluid substance. Or data from experiments providing conclusive shape to a model before it was analytically predicted, like the T power four law of radiation – as we will see later in a subsequent note.

We now revisit the ingenuity of an experiment performed by Josef Stefan, that measured something that was thought impossible. Much of what I write below is available in [1]. I also claim no originality of the equations discussed.

Thermal conductivity of a substance gives a measure of energy that is conducted as heat across a length maintained at a temperature difference. Fourier Law, which relates heat flux across a locally stationary substance to be in direct proportion with the associated temperature gradient, is the constitutive relation that defines thermal conductivity (the essay in that link explains it in detail).

It was a time whether gases would conduct heat remained in debate. Joseph Priestly in 1780  thought he had measured “the power to conduct heat” for gases using experiments, but it was not to be. He had rather measured the specific heat of gases and not the thermal conductivity. Specific heat was a concept not well understood until the experiments of Joseph Black in 1840.  In 1786 Count Rumsford attempted to measure the power to conduct heat by “artificial airs” or gases, but stumbled on to the discovery of a new mode of heat transfer, convection. So he thought gases cannot conduct heat. According to [1] (which borrows the insight, I presume, from [2]) , this thought remained unchallenged for several decades, largely due to the reputation of Rumsford. Until 1861, when Magnus showed conclusively that gases do conduct heat, by electrically heating platinum wires surrounded by different gases.

In 1860 Maxwell published his dynamical theory of gases. He calculated a theoretical value of the thermal conductivity of a gas and showed its dependence on the temperature and pressure.  Further, he made an assertion (as mentioned in [1]) that it would be impossible to measure by direct experiments the thermal conductivity of a gas, as the radiation heat transfer from the gas to the surrounding would always remain orders higher than the conduction within the gas even when circulation (convection) was prevented.

Two years later, Rudolph Clausius thought Maxwell had treated thermal conductivity incompletely.  He went on to show that the thermal conductivity was dependent on temperature but independent of pressure for ideal gases. He even measured the thermal conductivity of air (by treating it as an ideal gas) to be k = 0.0115 W/mK. Maxwell went on to revise his work and obtained for air a value of k = 0.0218 W/mK.

And this is where Josef Stefan entered. To defy Maxwell’s words [1].

Wikipedia has a brief life history of Josef Stefan. We quote an inspiring paragraph from [1] on his academic stature before proceeding to his experiment.

[...] In addition to his scientific and administrative talents, Stefan was a warm and beloved teacher. He gave very energetic, animated lectures and was said to be exhausted upon their completion. His students not only felt comfortable around him but were motivated to do high-level scientific research. One of his students later remarked on the collegial atmosphere that Stefan maintained at the cramped, underfunded Institute on Erdbergstrasse, far from the central campus buildings of the University, “Nothing diminishes the excellence of his character, the magic [Stefan] worked on the young academics. That magic could only be experienced personally…Erdberg stayed with me my whole life as a symbol of serious, inspired experimental activity”

That student who felt that way was Ludwig Boltzmann, Stefan’s first Ph. D. student.

Josef Stefan was aware of the works of Maxwell, Clausius and Magnus. HE was also determined to measure thermal conductivity of gases through direct experiments. He was aware of the detrimental effect convection of the gas can have on such a measurement (across a set temperature difference, convection can force the gas to physically move due to buoyancy carrying enthalpy along with it, increasing the heat flux enormously). In one of his earlier attempts to construct a device, he heated air from the top and cooled it from the bottom to negate the convection effect – a thermally stratified air column. But he was unable to control the heat loss to the surrounding.

After such imperfect subsequent attempts (one more is given in [1]), he constructed the diathermometer. In order to prevent the convection effects on stationary air column, Stefan first struck on the idea of using transient measurements. He conceived a method where the heat conduction across a small gap filled with a gas can be equated with the enthalpy gained by the gas under transient conditions. In modern parlance this equality is First Law of Thermodynamics applied to a fixed volume of gas, with the heat transfer equated to local temperature gradient through Fourier Law and the enthalpy gained by the gas calculated as a product of specific heat at constant volume and temperature raise. In equation form this unique energy balance can be written as

Here k is thermal conductivity of the gas (measured in W/mK), is the temperature difference across a gap of length, is the specific heat of the gas at constant volume.

The use of specific heat at constant volume is correct – when we look at the experiment performed – but perhaps fortuitous. The understanding that gases in principle have also and one must use it in an energy balance is a more recent understanding. Even some modern text books carry this error of using in the energy conservation statement while dealing with convection.

Rearranging and integrating Eq. (1) can be recast as

where ‘0′ in the subscript denotes initial temperature difference across the gap filled by the gas. Stefan, at this stage, perhaps through the works of Clausius, realized that for an ideal gas at fixed volume, the relative change in temperature is equal to the relative change in pressure (via the equation of state). This leads Eq. (2) to be revised as

Based on these equations, Stefan constructed his diathermometer – part of the schematic is shown here.

The gas in question is sent through the valve opening marked I into the small gap between concentric cylinders ABCD and GHJK. The small gap ensures minimal convection (gravity acting downwards in the picture). The pressure in the GHJK chamber is measured by a manometer connected to the limb marked M. The apparatus, after reaching internal thermal equilibrium, is kept in a constant temperature bath. In a transient (time dependent) process, heat is conducted through the walls AB and DC (actually one cylindrical surface), through the gas in the gap and into the GHJK chamber. The measured pressure difference for each time instant provides the unknowns in Eq. (3), resulting in the direct measurement of thermal conductivity of the gas in the gap.

Stefan measured the thermal conductivity of air to be k = 0.0234 W/mK, which is 11% off today’s accepted value of k = 0.0263 W/mK (at 300 K). It also compared well (about 7%) with Maxwell’s earlier theoretical predictions. Stefan went on to measure the ‘k’ of several gases including hydrogen, nitrous oxide, methane, carbon monoxide and carbon dioxide.

The radiation effect that was thought to mire such an experiment was completely negated in Stefan’s ingenuous diatermometer. Interestingly, he used it again to prove experimentally the T to the fourth power  radiation law. In a subsequent note.

References

[1] Crepeau, J. (2007). Josef Stefan: His life and legacy in the thermal sciences Experimental Thermal and Fluid Science, 31 (7), 795-803 DOI: 10.1016/j.expthermflusci.2006.08.005

[2] A.C. Burr, Notes on the history of the thermal conductivity of gases, Isis 21 (1) (1934), pp. 169–186. Full Text via CrossRef

[3] Image of Stefan’s Monument from http://www.carantha.net/science_and_literature.htm

[4] W.L. Reiter, The physical tourist Vienna: a random walk in science, Physics in Perspective 3 (2001), pp. 462–489. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (3) – An interesting read on the contributions of Vienna to Science

In channel or pipe flows, one of the engineering interest is the required pump power for pushing a fluid through the channel. For a fixed steady volumetric flow rate, higher the pressure drop across the channel, higher will be the pump power. For channel or pipe flows, a theoretical hydraulic model usually relates the longitudinal pressure drop to the flow rate [1]. For laminar flow, such a model can be analytically derived for pipe flows. For turbulent flows, empirical relations can be sought.

The situation remains more or less the same, even if we introduce a porous medium in such channel flows.

An analogous hydraulic equation can be used to estimate the pump power through porous medium channels. The existing hydraulic model for predicting the global pressure-drop through a porous medium [2] channel is known as the Forchheimer-extended Darcy equation (FD) or the Hazen-Dupuit-Darcy model. The global FD model can be written as

where the pressure drop is across the entire channel, U is the average fluid speed, K is the permeability and C the form coefficient of the porous medium (hydraulic properties), μ is the dynamic viscosity and ρ is the density of the fluid flowing through the channel. The first term in the RHS is identified as the viscous drag and the second as the form or inertial drag of the porous medium. For further details, read the summary in [3] for the basics of fluid flow through porous media.

Historically [3], the above FD model has been derived from experiments conducted under isothermal flow condition. That is, treating the fluid viscosity as constant and uniform. However, many engineering systems and devices operate based on fluid flow through heated (or cooled) porous media. The appropriateness of the FD equation for non-isothermal flow condition, in which the fluid viscosity changes, has not been completely established.

Figure 1: Schematic of Parallel-plates non-isothermal porous medium channel sustaining liquid flow

Temperature dependent viscosity effects on fluid flow has been explained earlier [4]. The schematic of a heated parallel-plates sandwiching a porous medium channel is shown in the accompanying figure. For liquid flow through such a channel, increasing the temperature (by heating, as done here) reduces the viscosity, which in turn reduces the overall pressure-drop to sustain a fixed flow rate across the channel. This is good because this means the pump power required will also reduce. But how to predict this pressure-drop?

It is shown in [5] the governing global FD model (Eq. (1)) for flow through porous medium channels fails to predict the pressure-drop for non-isothermal flow condition, in which the fluid viscosity changes largely. Such a conclusion is reached after investigating three alternatives to find the viscosity in the FD model (Eq. (1)) to predict the pressure drop for non-isothermal porous medium channel flows: (1) fluid viscosity determined at the algebraic average of inlet and outlet bulk temperature [read [6] for what is bulk temperature], (2) fluid viscosity determined at the log-mean of inlet and outlet bulk temperature, and (3) fluid viscosity function of the bulk temperature averaged along the entire channel. Further details of these options are available in the write up in [4]. All of the three options did not yield the correct prediction for the pressure-drop versus fluid-speed curve.

This led to suggestions in [5] for generalizing the existing global porous medium hydrodynamic model. Before that a brief look at the physics. Referring to the channel in the above figure, the heating of the fluid as it flows along the channel results in a non-uniform fluid temperature in the transverse (perpendicular to the heated plates) and longitudinal directions. This causes the viscosity of the fluid to vary in y, resulting also in a non-uniform velocity profile in y. As the velocity is non-uniform along y, the local viscous and form drags, dependent on u and u² respectively, are no longer uniform in y. Therefore, the cross-section averaged viscous and form drags are altered, leading to different global coefficients. This observation highlights the inappropriateness of the FD model (Eq. (1)), which does not account for these effects.

Two empirical coefficients modifying the viscous and form drags of the FD model (Eq. (1)) were suggested based on the results from computational fluid dynamic simulations and hydraulic experiments in non-isothermal porous medium channel flows. When incorporated, it would result in a modified FD (or HDD) equation of the form

where the ζ_μ and ζ_C are empirical coefficients lumping the effect of temperature dependent viscosity on the porous medium drags, hence the pressure-drop across the channel. Relationships were suggested between these coefficients and other parameters of the channel configuration such as heat flux, channel size and porous medium and fluid properties. Using such relationships (correlations) one could find the values for these coefficients and use Eq. (2) to predict pressure-drop for the porous medium channel configuration.

Now let us look at the physics of the situation. Figure below summarizes the temperature-dependent viscosity effects on the hydraulics of a liquid flow through a porous medium channel.

Figure 2: Temperature dependent viscosity effects in porous medium flows

The upper curve is the FD model (Eq. (1)) that predicts the pressure-drop for a flow rate, when the channel flow is isothermal. This is identified as the uniform viscosity limit. Starting from the upper curve, as the heat flux is progressively increased (following the arrow), as explained earlier, the reduction in viscosity reduces the pressure-drop for a flow rate. So we approach a viscous and form drag regime in which both linear and quadratic terms of the modified FD model (Eq. (2)) are affected by the viscosity dependency on temperature through the zeta coefficients.

As the heating increases, for a certain heat flux, the lumped viscous drag effect on the longitudinal pressure-drop eventually becomes insignificant because the local viscosity value becomes too small. This means, the effect of the first RHS term in the modified FD model (Eq. (2)) becomes negligible when compared to the second RHS term. However, the effect of viscosity does not disappear. It is still experienced by the pressure-drop through the second RHS term. The form-drag coefficient in Eq.(2) is influenced by the local velocity variation in the channel (read [3]) and the velocity variation is still governed by the local viscosity variation. So, beyond the transition region and in the form drag regime, the fluid is still influenced by the viscous effect locally along the channel.

As the heating increases further, the viscous effect will eventually disappear (become negligible but never zero) throughout the channel. Further heating will have no significant hydrodynamic effect through viscosity. This limit is termed the inviscid form-drag limit, as the global form-drag becomes independent of the viscous effect. Consequently, for this limiting case, the longitudinal pressure-drop required to support an average fluid speed across the channel will be equal to the form drag for the uniform viscosity (no heating) flow. That is, pressure-drop is predicted by Eq. (1) without the first term on the RHS (or by Eq. (2) with ζ_μ = 0 and ζ_C = 1).

Eventually, in association with existing advanced CFD software for simulation and design, such theories should allow consistent prediction of the thermal-hydraulic performance of a large number of porous media based systems and devices, including cold-plates and self-lubricating bearings.

References

1. Pipe Turbulence
2. Porous Medium Definition
3. Flow through Porous Medium
4. Variable Viscosity Effects in Channel Flows
5. Narasimhan, A., & Lage, J. (2001). Modified Hazen-Dupuit-Darcy Model for Forced Convection of a Fluid With Temperature-Dependent Viscosity Journal of Heat Transfer, 123 (1) DOI: 10.1115/1.1332778
6. Concept of Bulk Temperature

This note is meant for my graduate students and for those who have a passing interest in convection and back of the envelope methods.

Forced convection flow [1] over a flat plate with a leading edge is a basic convection configuration that has yielded to analytical scrutiny. A coupled system of partial differential equations – Navier Stokes equations [2] plus First Law of Thermodynamics [3] applied on a control volume – are solved to find the chief unknown, the convection heat transfer coefficient.

Scale analysis has been discussed earlier [4]. The following explains scale analysis performed on the boundary layer simplified equations, in the thick and thin thermal boundary layer limits of forced convection flow over an isothermal hot flat plate. Text books [5] contain the discussion in varied level of detail. This note fills minor missing details.

By habit, the flow is non-Arabic – is depicted from left to right, as shown in the accompanying schematic.

Schematic for forced convection over flat plate in the thick thermal boundary layer limit

The free stream flow entering the control volume of the flat plate at the leading edge on the left (not shown) at an uniform (spatially identical) velocity is deccelerated to zero velocity by the stationary hot flate plate. The plate is at an isothermal higher temperature than the temperature of the incoming and free stream flow. For the purpose of this discussion, it is assumed that the solution to the viscous fluid flow equations describing the steady, laminar, incompressible flow is given. Hence the hydrodynamic boundary layer (HBL) thickness () is known. If necessary, read about the concept of hydrodynamic boundary layer and thermal boundary layer in the Wikipedia [6].

The schematic is for the thick thermal boundary layer (TBL) limit (i.e. TBL thicker than HBL). By requirement, the fluid’s Prandtl number satisfies . The assumption of slenderness of both the boundary layers (the tenet of boundary layer concept) imply that and even when one of them is bigger than the other.

We begin with the thermal boundary layer (TBL) simplified energy equation in two dimension written in Cartesian coordinates:

That the axial diffusion (in x-dir) is negligible is the TBL simplification of the First Law of Thermodynamics. For the thick TBL shown in the figure, the temperature gradient prevails over a larger distance than that of the velocity gradient. Applying scale analysis we can write the appropriate scales for the differential equation above as

where and L is some finite length of the plate in x direction (it can even be just x, as in from to ). Without bounding the domain this way, scale analysis won’t make sense. Obviously, we need the scales for and the local velocities in x and y directions to proceed. These can be obtained by invoking the mass conservation, which for the situation discussed would be:

From the schematic for thick TBL above, since the HBL is much smaller than the TBL (), the scale for can safely be written as . Therefore, from the mass conservation scales for Eq. (3) within the TBL one can obtain the scale for as

In other words, for much of the TBL, free stream flow prevails and the role of HBL in local convection is restricted very close to the hot flat plate. Observe also the use of in this scale, as the HBL ends well within the TBL.

Using the and scale from the above equation back in the scaled energy equation (2), one can see the LHS terms scale as

In other words, the second LHS term is times \textit{smaller} than the first term and hence can be considered negligible. This results in the simplification of the TBL energy equation further as a balance between longitudinal convection by the fluid and transverse conduction heat transfer from the flat plate, written as

resulting – after some substitution and rearrangement – in

The above equation is the physics for thick TBL when and , where and . is the kinematic viscosity or the momentum diffusivity and is the thermal diffusivity of the fluid.

In the thin TBL limit, the converse holds, i.e. and as shown in the accompanying schematic.

Schematic for forced convection over flat plate in the thin thermal boundary layer limit

Here, since the HBL extends well beyond the TBL, the scale for within the TBL is not but as can be verified by the geometry of the figure. When used in the mass conservation, this modifies the scale for as

Observe the use of as a transverse scale in the above mass balance as our domain of interest is only the TBL, beyond which even though the HBL extends, there is no heat transfer (compare with mass balance for thick TBL above).

The above observation suggests that one cannot readily start with the simplified Eq. (5) as energy balance, but should verify whether the scales for the two LHS terms in Eq. (2) are comparable. Using the modified scales for and for the thin TBL into Eq. (2), one can rewrite it as

The two LHS terms in the above equation is of the same order of scale. Setting one of it to balance the RHS in the convection ~ conduction balance, one obtains

Reducing the above equation after substituting for the HBL thickness as , one can find the TBL thickness as

a result that shows the distinct change in the effect, when compared to Eq. (6) for thick TBL limit.

For completion, using the TBL thickness of the TBL one can find the local (at a x-location, in the figures) convection heat transfer coefficient, expressed in non dimensional form as the Nusselt number of the configuration (in honour of Wilhelm Nusselt) as

for and and

for and .

From the above two equations it is evident that when , the heat transfer depends only on the square-root of the local .

References

1. Free and Paid Convection http://unrulednotebook.wordpress.com/2006/07/10/free-and-paid-convection/
2. Navier Stokes Equation http://en.wikipedia.org/wiki/Navier-Stokes_equation
3. First Law and Fourier Law http://unrulednotebook.wordpress.com/2008/07/12/first-law-and-fourier-law/
4. Scale Analysis http://unrulednotebook.wordpress.com/2008/07/14/scale-analysis/
5. the basic analysis is discussed in Convection Heat Transfer, by A. Bejan [ Amazon Link]
6. Boundary Layer http://en.wikipedia.org/wiki/Boundary_layer

If your research interests doesn’t involve convection, fluid flow, porous media, bio-heat transport, this may not be of interest to you.

Fluids

A Three-Equation Eddy-Viscosity Model for Reynolds-Averaged Navier–Stokes Simulations of Transitional Flow D. Keith Walters and Davor Cokljat, J. Fluids Eng. 130, 121401 (2008), DOI: http://dx.doi.org/10.1115/1.2979230

Grid Independence Via Automated Unstructured Adaptation Ronald J. Chila and Deborah A. Kaminski, J. Fluids Eng. 130, 121403 (2008), DOI: http://dx.doi.org/10.1115/1.3001099

Laminar-to-turbulent transition of pipe flows through puffs and slugs, MINA NISHI, BÜLENT ÜNSAL, FRANZ DURST and GAUTAM BISWAS,  Journal of Fluid Mechanics, (2008) 614 , pp 425-446, doi: http://dx.doi.org/10.1017/S0022112008003315 [Abstract]

MARTIN Z. BAZANT and OLGA I. VINOGRADOVA (2008). Tensorial hydrodynamic slip. Journal of Fluid Mechanics, 613 , pp 125-134 doi: http://dx.doi.org/10.1017/S002211200800356X

A. I. RUBAN and K. N. VONATSOS (2008). Discontinuous solutions of the boundary-layer equations. Journal of Fluid Mechanics, 614 , pp 407-424
doi: http://dx.doi.org/10.1017/S0022112008003303

H. HERWIG, D. GLOSS and T. WENTERODT (2008). A new approach to understanding and modelling the influence of wall roughness on friction factors for pipe and channel flows. Journal of Fluid Mechanics, 613 , pp 35-53 doi: http://dx.doi.org/10.1017/S0022112008003534

Instability of plane Poiseuille flow in a fluid-porous system Rong Liu, Qiu Sheng Liu, and Si Cheng Zhao, Phys. Fluids 20, 104105 (2008), DOI: http://dx.doi.org/10.1063/1.3000643

A challenge in Navier–Stokes-based continuum modeling: Maxwell–Burnett slip law Huei Chu Weng and Cha’o-Kuang Chen, Phys. Fluids 20, 106101 (2008), DOI: http://dx.doi.org/10.1063/1.2998451

Thermal

W. Escher, B. Michel, D. Poulikakos, Efficiency of optimized bifurcating tree-like and parallel microchannel networks in the cooling of electronics, International Journal of Heat and Mass TransferIn Press, Corrected Proof, , Available online 17 October 2008. doi:10.1016/j.ijheatmasstransfer.2008.07.048

Unsteady contact melting of a rectangular cross-section material on a flat plate T. G. Myers, S. L. Mitchell, and G. Muchatibaya, Phys. Fluids 20, 103101 (2008), DOI: http://dx.doi.org/10.1063/1.2990751

A Method for Determining the Heat Transfer Properties of Foam-Fins Robert J. Moffat, John K. Eaton, and Andrew Onstad, J. Heat Transfer 131, 011603 (2009), DOI: http://dx.doi.org/10.1115/1.2977599

Simple Explicit Equations for Transient Heat Conduction in Finite Solids A. G. Ostrogorsky, J. Heat Transfer 131, 011303 (2009), DOI: http://dx.doi.org/10.1115/1.2977540

Can convection induced by heating delay a thermal explosion? M. Al-Aseeri, W. Guo, L. E. Johns, and R. Narayanan, Phys. Fluids 20, 104107 (2008), DOI: http://dx.doi.org/10.1063/1.2992559 [http://link.aip.org/link/?PHFLE6/20/104107/1]

Porous Medium

Analytical Solution of Nonequilibrium Heat Conduction in Porous Medium Incorporating a Variable Porosity Model With Heat Generation M. Nazari and F. Kowsary, J. Heat Transfer 131, 014503 (2009), DOI: http://dx.doi.org/10.1115/1.2977544

The Effect of Pore Size on the Heat Transfer Between a Heated Finned Surface and a Saturated Porous Plate M. J. Schertzer, D. Ewing, C. Y. Ching, and J. S. Chang, J. Heat Transfer 131, 011501 (2009), DOI: http://dx.doi.org/10.1115/1.2977595

Developing Nonthermal-Equilibrium Convection in Porous Media With Negligible Fluid Conduction Nihad Dukhan, J. Heat Transfer 131, 014501 (2009), DOI: http://dx.doi.org/10.1115/1.2993540

Convection in deep vertically shaken particle beds. I. General features Sakon Klongboonjit and Charles S. Campbell, Phys. Fluids 20, 103301 (2008), DOI: http://dx.doi.org/10.1063/1.2996134

Indranil Ghosh, Heat transfer correlation for high-porosity open-cell foam, International Journal of Heat and Mass TransferIn Press, Corrected Proof, , Available online 6 October 2008. doi:10.1016/j.ijheatmasstransfer.2008.07.047

T. Grosan, C. Revnic, I. Pop, D.B. Ingham, Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium, International Journal of Heat and Mass TransferIn Press, Corrected Proof, , Available online 1 October 2008. doi:10.1016/j.ijheatmasstransfer.2008.08.011

Tomer Duman and Uri Shavit, An Apparent Interface Location as a Tool to Solve the Porous Interface Flow Problem, Transport in Porous Media 2008, http://dx.doi.org/10.1007/s11242-008-9286-9

S. Lorente and A. Bejan, Constructal Design of Vascular Porous Materials and Electrokinetic Mass Transfer, Transport in Porous Media 2008, http://dx.doi.org/10.1007/s11242-008-9283-z

Yasin Varol, Hakan F. Oztop, Ioan Pop, Natural convection in right-angle porous trapezoidal enclosure partially cooled from inclined wall, International Communications in Heat and Mass Transfer In Press, Uncorrected Proof, , Available online 14 October 2008. doi:10.1016/j.icheatmasstransfer.2008.09.010

R. Thiedmann, F. Fleischer, C. Hartnig, W. Lehnert and V. Schmidt, Stochastic 3D modelling of the GDL structure in PEM fuel cells, based on thin section detection. Journal of the Electrochemical Society 155 (2008), B391-B399, Available online on Prof. Schmidt’s website: http://www.mathematik.uni-ulm.de/stochastik/personal/schmidt/schmidtFrame.html [Direct PDF link: http://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/mitarbeiter/thiedmann/Segmentierung_GDL.pdf ]

Bio-Heat

Shadi Mahjoob, Kambiz Vafai, Analytical characterization of heat transport through biological media incorporating hyperthermia treatment, International Journal of Heat and Mass TransferIn Press, Corrected Proof, , Available online 1 October 2008. doi:10.1016/j.ijheatmasstransfer.2008.07.038

E. Bilgen, Thermo-bioconvection of gravitactic micro-organisms in shallow vertical cylindrical enclosures, International Communications in Heat and Mass TransferIn Press, Uncorrected Proof, , Available online 9 October 2008. doi:10.1016/j.icheatmasstransfer.2008.08.009

One may come up with efficient thermal management designs for cooling electronics in portable computers. But several things can go wrong, even after one employs such a design, resulting in poor cooling and subsequent damage to the component. As can be evidenced below:

Toshiba Laptop Heat Sink with Dust

The entire gaps in the heat sink fin assembly is blocked by accumulated dust. The laptop kept restarting, with a prompt warning about over heat every time.

The heat sink is now restored to its original cooling capacity.

Cleaned Heat Sink

If your research interests doesn’t overlap with mine, you may not find this list useful.

Thermal

Liquid Single-Phase Flow in an Array of Micro-Pin-Fins—Part I: Heat Transfer Characteristics J. Heat Transfer — December 2008 — Volume 130, Issue 12, 122402 (11 pages) DOI: http://dx.doi.org/10.1115/1.2970080

Liquid Single-Phase Flow in an Array of Micro-Pin-Fins—Part II: Pressure Drop Characteristics J. Heat Transfer — December 2008 — Volume 130, Issue 12, 124501 (4 pages) DOI: http://dx.doi.org/10.1115/1.2970082

Ab Initio Molecular Dynamics Study of Nanoscale Thermal Energy Transport J. Heat Transfer — December 2008 — Volume 130, Issue 12, 122403 (7 pages) DOI: http://dx.doi.org/10.1115/1.2976562

Stable/Unstable Stratification in Thermosolutal Convection in a Square Cavity J. Heat Transfer — December 2008 — Volume 130, Issue 12, 122001 (10 pages) DOI: http://dx.doi.org/10.1115/1.2969757

Heat Exchanger Analysis Modified to Account for a Heat Source J. Heat Transfer — December 2008 — Volume 130, Issue 12, 124502 (4 pages) DOI: http://dx.doi.org/10.1115/1.2970063

Heat Transfer in Mini/Microchannels With Combustion: A Simple Analysis for Application in Nonintrusive IR Diagnostics J. Heat Transfer — December 2008 — Volume 130, Issue 12, 124504 (5 pages) DOI: http://dx.doi.org/10.1115/1.2969760

Erratum: “Use of Streamwise Periodic Boundary Conditions for Problems in Heat and Mass Transfer” [Journal of Heat Transfer, 2007, 129(4), pp. 601–605] J. Heat Transfer — December 2008 — Volume 130, Issue 12, 127001 (1 pages) DOI: http://dx.doi.org/10.1115/1.2970090

Multiobjective Optimization of a Microchannel Heat Sink Using Evolutionary Algorithm J. Heat Transfer — November 2008 — Volume 130, Issue 11, 114505 (3 pages) DOI: http://dx.doi.org/10.1115/1.2969261

Solutions for Transient Heat Conduction With Solid Body Motion and Convective Boundary Conditions J. Heat Transfer — November 2008 — Volume 130, Issue 11, 111301 (8 pages) DOI: http://dx.doi.org/10.1115/1.2944243

Natural convection in enclosures with partially thermally active side walls containing internal heat sources P. Kandaswamy, N. Nithyadevi, and C. O. Ng, Phys. Fluids 20, 097104 (2008), DOI: http://dx.doi.org/10.1063/1.2981834

Single- and Dual-Phase-Lagging Heat Conduction Models in Moving Media J. Heat Transfer — December 2008 — Volume 130, Issue 12, 121302 (6 pages) DOI: http://dx.doi.org/10.1115/1.2976787

Cornelia Giessler, Ruben Schlegel, Andre Thess, Numerical investigation of the flow of a glass melt through a long circular pipe, International Journal of Heat and Fluid Flow Volume 29, Issue 5, , October 2008, Pages 1462-1468. doi:10.1016/j.ijheatfluidflow.2008.06.001

Vinoj Kurian, Mahesh N. Varma, A. Kannan, Numerical studies on laminar natural convection inside inclined cylinders of unity aspect ratio, International Journal of Heat and Mass Transfer In Press, Corrected Proof, , Available online 13 September 2008. doi:10.1016/j.ijheatmasstransfer.2008.06.041

J. van Rij, T. Ameel, T. Harman, The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer, International Journal of Thermal Sciences In Press, Corrected Proof, , Available online 21 August 2008. doi:10.1016/j.ijthermalsci.2008.07.010

Fluids

PIV Study of Adverse and Favorable Pressure Gradient Turbulent Flows Over Transverse Ribs, M. Agelinchaab and M. F. Tachie, J. Fluids Eng. 130, 111305 (2008) (12 pages), Abstract | DOI: http://dx.doi.org/10.1115/1.2969448

Pressure Drop Through Anisotropic Porous Mediumlike Cylinder Bundles in Turbulent Flow Regime, Tongbeum Kim and Tian Jian Lu, J. Fluids Eng. 130, 104501 (2008), DOI: http://dx.doi.org/10.1115/1.2969454

A better nondimensionalization scheme for slender laminar flows: The Laplacian operator scaling method M. M. Weislogel, Y. Chen, and D. Bolleddula, Phys. Fluids 20, 093602 (2008), DOI: http://dx.doi.org/10.1063/1.2973900

Marco Lorenzini, Gian Luca Morini, Torsten Henning, Jurgen Brandner, Uncertainty assessment in friction factor measurements as a tool to design experimental set-ups, International Journal of Thermal SciencesIn Press, Corrected Proof, , Available online 14 July 2008. doi:10.1016/j.ijthermalsci.2008.06.006

Porous Media

Natural Convection Heat Transfer Enhancements From a Cylinder Using Porous Carbon Foam: Experimental Study J. Heat Transfer — December 2008 — Volume 130, Issue 12, 122502 (6 pages) DOI: http://dx.doi.org/10.1115/1.2977606

Stability and Convection in Impulsively Heated Porous Layers J. Heat Transfer — November 2008 — Volume 130, Issue 11, 112601 (9 pages) DOI: http://dx.doi.org/10.1115/1.2957484

A General Macroscopic Turbulence Model for Flows in Packed Beds, Channels, Pipes, and Rod Bundles, A. Nakayama and F. Kuwahara, J. Fluids Eng. 130, 101205 (2008), DOI: http://dx.doi.org/10.1115/1.2969461

P. A. LAKSHMI NARAYANA, P. V. S. N. MURTHY and RAMA SUBBA REDDY GORLA (2008). Soret-driven thermosolutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. Journal of Fluid Mechanics, 612 , pp 1-19 Abstract | doi:10.1017/S0022112008002619

JEROME A. NEUFELD and J. S. WETTLAUFER (2008). Shear-enhanced convection in a mushy layer. Journal of Fluid Mechanics, 612 , pp 339-361 doi:10.1017/S0022112008002991

JEROME A. NEUFELD and J. S. WETTLAUFER (2008). An experimental study of shear-enhanced convection in a mushy layer. Journal of Fluid Mechanics, 612, pp 363-385 doi:10.1017/S0022112008003339

Flow and transport in brush-coated capillaries: A molecular dynamics simulation D. I. Dimitrov, L. I. Klushin, A. Milchev, and K. Binder, Phys. Fluids 20, 092102 (2008), DOI: http://dx.doi.org/10.1063/1.2975840

Rui-Na Xu, Pei-Xue Jiang, Numerical simulation of fluid flow in microporous media, International Journal of Heat and Fluid Flow Volume 29, Issue 5, , October 2008, Pages 1447-1455. doi:10.1016/j.ijheatfluidflow.2008.05.005

Go-Long Tsai, Y.C. Lin, W.J. Ma, H.W. Wang, J.T. Yang, Transitional flow patterns behind a backstep with porous-based fluid injection, International Journal of Heat and Mass Transfer In Press, Corrected Proof, , Available online 15 September 2008. doi:10.1016/j.ijheatmasstransfer.2008.06.042

Numerical Study of Heat Transfer in a Rectangular Channel with Porous Covering Obstacles, Nuri Yucel and R. Tolga Guven, Transport in Porous Media, 2008, DOI  – http://dx.doi.org/10.1007/s11242-008-9260-6

Lattice Boltzmann method for flows in porous and homogenous fluid domains coupled at the interface by stress jump, Huixing Bai, P. Yu, S. H. Winoto, H. T. Low, International Journal for Numerical Methods in Fluids, 2008, http://dx.doi.org/10.1002/fld.1913

Boundary conditions at the interface between fluid layer and fibrous medium, H. X. Bai, P. Yu, S. H. Winoto, H. T. Low, International Journal for Numerical Methods in Fluids, 2008, http://dx.doi.org/10.1002/fld.1921

Bio Heat / Fluids

Maryam Shafahi, Kambiz Vafai, Biofilm affected characteristics of porous structures, International Journal of Heat and Mass TransferIn Press, Corrected Proof, , Available online 9 September 2008. doi:10.1016/j.ijheatmasstransfer.2008.07.013

The Limiting Radius for Freezing a Tumor During Percutaneous Cryoablation J. Heat Transfer — November 2008 — Volume 130, Issue 11, 111101 (6 pages) DOI: http://dx.doi.org/10.1115/1.2969759

Related Interests

Professor Adrian Bejan on his 60th birthday, International Journal of Heat and Mass TransferIn Press, Corrected Proof, , Available online 7 September 2008. doi:10.1016/j.ijheatmasstransfer.2008.06.027

Hubert Chanson, Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results, European Journal of Mechanics – B/FluidsIn Press, Corrected Proof, , Available online 1 July 2008. doi:10.1016/j.euromechflu.2008.06.004

History force on an asymmetrically rotating body in Poiseuille flow inducing particle migration across a slit pore Sukalyan Bhattacharya, Phys. Fluids 20, 093301 (2008), DOI: http://dx.doi.org/10.1063/1.2974827

Performance Characteristics of a Microscale Ranque–Hilsch Vortex Tube, A. F. Hamoudi, A. Fartaj, and G. W. Rankin, J. Fluids Eng. 130, 101206 (2008), DOI: http://dx.doi.org/10.1115/1.2969442

If your research interests doesn’t overlap with mine, you may not find this list useful.

Thermal

Thermal-Boundary-Layer Response to Convected Far-Field Fluid Temperature Changes Hongwei Li and M. Razi Nalim, J. Heat Transfer 130, 101001 (2008) (6 pages) Abstract | DOI http://dx.doi.org/10.1115/1.2953239

Fractal Model for Thermal Contact Conductance Mingqing Zou, Boming Yu, Jianchao Cai, and Peng Xu, J. Heat Transfer 130, 101301 (2008) (9 pages) Abstract | DOI http://dx.doi.org/10.1115/1.2953304

Small and Large Time Solutions for Surface Temperature, Surface Heat Flux, and Energy Input in Transient, One-Dimensional Conduction A. S. Lavine and T. L. Bergman J. Heat Transfer 130, 101302 (2008) (8 pages) Abstract | DOI http://dx.doi.org/10.1115/1.2945902

Multidisciplinary Design and Optimization Methodologies in Electronics Packaging: State-of-the-Art Review Hamid Hadim and Tohru Suwa, J. Electron. Packag. 130, 034001 (2008), DOI: http://dx.doi.org/10.1115/1.2957459

A.H. Ahmadi Motlagh, S.H. Hashemabadi, 3D CFD simulation and experimental validation of particle-to-fluid heat transfer in a randomly packed bed of cylindrical particles, International Communications in Heat and Mass Transfer In Press, Uncorrected Proof, , Available online 3 August 2008. doi:10.1016/j.icheatmasstransfer.2008.07.014

F. Corvaro, M. Paroncini, An experimental study of natural convection in a differentially heated cavity through a 2D-PIV system, International Journal of Heat and Mass TransferIn Press, Corrected Proof, , Available online 15 August 2008. doi:10.1016/j.ijheatmasstransfer.2008.05.039

M. Wong, I. Owen, C.J. Sutcliffe, A. Puri, Convective heat transfer and pressure losses across novel heat sinks fabricated by Selective Laser Melting, International Journal of Heat and Mass Transfer In Press, Corrected Proof, , Available online 21 July 2008. doi:10.1016/j.ijheatmasstransfer.2008.06.002

Jung-Yeul Jung, Hoo-Suk Oh, Ho-Young Kwak, Forced convective heat transfer of nanofluids in microchannels, International Journal of Heat and Mass Transfer In Press, Corrected Proof, , Available online 22 July 2008. doi:10.1016/j.ijheatmasstransfer.2008.03.033

Giulio Croce, Paola D’Agaro, Compressibility and rarefaction effect on heat transfer in rough microchannels, International Journal of Thermal Sciences In Press, Corrected Proof, , Available online 15 August 2008. doi:10.1016/j.ijthermalsci.2008.07.009

Fluids

An Experimental Procedure for Determining Both the Density and Flow Rate From Pressure Drop Measurements in a Cylindrical Pipe Ghislain Michaux, Olivier Vauquelin, and Elsa Gauger J. Fluids Eng. 130, 094501 (2008) (3 pages) Abstract | DOI http://dx.doi.org/10.1115/1.2953294

Interaction of a skewed Rankine vortex pair S. Jayavel, Pratish P. Patil, and Shaligram Tiwari, Phys. Fluids 20, 083601 (2008), DOI: http://dx.doi.org/10.1063/1.2969115 | http://link.aip.org/link/?PHFLE6/20/085101/1

Experimental and numerical studies of the flow over a circular cylinder at Reynolds number 3900 Philippe Parnaudeau, Johan Carlier, Dominique Heitz, and Eric Lamballais, Phys. Fluids 20, 085101 (2008), DOI: http://dx.doi.org/10.1063/1.2957018

Dynamics of drying in 3D porous media Authors: Lei Xu, Simon Davies, Andrew B. Schofield, David A. Weitz | [v1] Tue, 29 Jul 2008 23:07:46 GMT (336kb) | arXiv:0807.4757v1 [physics.flu-dyn]

CHRISTOF SODTKE, VLADIMIR S. AJAEV and PETER STEPHAN (2008). Dynamics of volatile liquid droplets on heated surfaces: theory versus experiment. Journal of Fluid Mechanics, 610, pp 343-362 doi: 10.1017/S0022112008002759

C. J. HEATON (2008). Linear instability of annular Poiseuille flow. Journal of Fluid Mechanics, 610, pp 391-406 doi:10.1017/S0022112008002577

Comparison Between Theoretical CFV Flow Models and NIST’s Primary Flow Data in the Laminar, Turbulent, and Transition Flow Regimes Aaron Johnson and John Wright, J. Fluids Eng. 130, 071202 (2008), DOI: http://dx.doi.org/10.1115/1.2903806

IAN S. SULLIVAN, JOSEPH J. NIEMELA, ROBERT E. HERSHBERGER, DIOGO BOLSTER and RUSSELL J. DONNELLY (2008). Dynamics of thin vortex rings. Journal of Fluid Mechanics, 609, pp 319-347 doi: http://dx.doi.org/10.1017/S0022112008002292

SAIKIRAN RAPAKA, SHIYI CHEN, RAJESH J. PAWAR, PHILIP H. STAUFFER and DONGXIAO ZHANG (2008). Non-modal growth of perturbations in density-driven convection in porous media. Journal of Fluid Mechanics, 609, pp 285-303 doi: http://dx.doi.org/10.1017/S0022112008002607

BABURAJ A. PUTHENVEETTIL and JAYWANT H. ARAKERI (2008). Convection due to an unstable density difference across a permeable membrane. Journal of Fluid Mechanics, 609, pp 139-170 doi: http://dx.doi.org/10.1017/S0022112008002334

Hubert Chanson, Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results, European Journal of Mechanics – B/FluidsIn Press, Corrected Proof, , Available online 1 July 2008.  | Link | doi:10.1016/j.euromechflu.2008.06.004

The friction factor of two-dimensional rough-pipe turbulent flows Authors: Nicholas Guttenberg, Nigel Goldenfeld |[v1] Mon, 11 Aug 2008 06:33:42 GMT (66kb)| arXiv:0808.1451v1 [physics.flu-dyn]

Porous Medium

Flows Between Rotating Cylinders With a Porous Lining M. Subotic and F. C. Lai J. Heat Transfer 130, 102601 (2008) (6 pages) Abstract | http://dx.doi.org/10.1115/1.2953305

Modeling the Natural Convection Heat Transfer and Dryout Heat Flux in a Porous Debris Bed R. Sinha, A. K. Nayak, and B. R. Sehgal J. Heat Transfer 130, 104503 (2008) (5 pages) Abstract | DOI http://dx.doi.org/10.1115/1.2952756

D. Jamet, M. Chandesris, On the intrinsic nature of jump coefficients at the interface between a porous medium and a free fluid region, International Journal of Heat and Mass Transfer In Press, Corrected Proof, , Available online 4 August 2008. doi:10.1016/j.ijheatmasstransfer.2008.04.072

A. Barletta, M. Celli, D.A.S. Rees, The onset of convection in a porous layer induced by viscous dissipation: A linear stability analysis, International Journal of Heat and Mass Transfer In Press, Corrected Proof, , Available online 31 July 2008. doi:10.1016/j.ijheatmasstransfer.2008.06.001

B. Zhang, T. Kim, T.J. Lu, Analytical solution for solidification of close-celled metal foams, International Journal of Heat and Mass Transfer In Press, Corrected Proof, , Available online 25 July 2008. doi:10.1016/j.ijheatmasstransfer.2008.06.006

Y.-M. Hung, and C. P. Tso, Temperature Variations of Forced Convection in Porous Media for Heating and Cooling Processes: Internal Heating Effect of Viscous Dissipation, Transport in Porous Media, Aug 2008, doi:10.1007/s11242-008-9226-8

Comment on “Radiative Effect on Natural Convection Flows in Porous Media”, A. A. Mohammadein, M. A. Mansour, Sahar M. Abd El Gaied and Rama Subba Reddy Gorla [Transport in Porous Media 32:263–283, 1998], Transport in Porous Media, Aug 2008, 10.1007/s11242-008-9269-x

Bio-Heat

E.Y.-K. Ng, A review of thermography as promising non-invasive detection modality for breast tumor, International Journal of Thermal Sciences In Press, Corrected Proof, , Available online 31 July 2008. doi:10.1016/j.ijthermalsci.2008.06.015

Here is the rambunctious boiling song from the one and only kitchen band. [Boiling Song MP3 File. Size is 1.1 MB, links to boxnet page for play and download. See Note 2 for copyright]. Listen to the song before you read this essay. The song is a mix of hard rock and grunge instrumental with high-pitched multiple wailing guitar sounds and cacophonous tempo changing beats. But I assure you, no musical instrument has been used for recording this song.

The sound is really cacophonous (what else you expect from a hard rock?). Take precautions like wearing a headphone. Now for some sleeve notes about the record.

The Kitchen Band comprises of Gas Stove, Water Vessel, Heat Energy and Mother Nature.

We all have witnessed the phenomenon of boiling. For most of us, it begins in our kitchen or bathroom and proceeds until the dining room and perhaps stops there, forever. For the purpose of this essay we shall take it a bit further. Boiling comes in several shapes and sounds, grouped under the two broad categories, pool boiling and flow boiling. Unlike an Italian Screwdriver (which is not a screwdriver made in Italy), Pool Boiling, as the name suggests is associated with a boiling pool of liquid. It is the boiling of a stationary mass of liquid. An example is the familiar water boiling in the pan on a kitchen stove. Flow boiling is means flow of the boiling fluid. An example is the flow inside the boiler tube (water wall panels) of a thermal power plant.

Pool boiling includes five regimes of reasonably distinct characterizations. Taking the familiar liquid water as the example liquid, let us start heating it in a kitchen vessel kept over our standard gas stove to explore. Along the way, we shall create the song.

As we know, water boils at 100 degree C. Not exactly true. A better way to say that is liquid water that is in contact with its vapour boils at 100 degree C. If it is not in contact with its vapour, but still in contact with a heating surface, it continues to remain in liquid form and raises its temperature to more than a few degrees above 100 degree C, for conventional machined surfaces like our kitchen vessel. Then it bubbles up. This range of heating of water from its room temperature falls within a regime that is identified with natural convection of water [1]. See how the red curve is depicted in the left bottom corner of Fig. 1 below.

An explanation of Fig. 1 is now in order. The abscissa is the excess temperature between the wall temperature (the bottom surface of the kitchen vessel kept over the stove) and the saturation temperature of the fluid (in our case, water). The ordinate is the heat flux that is released into the boiling fluid (in our case, water). The red curves are paths that characterize what happens to a fluid undergoing pool boiling in all of its five distinct stages.

This region is marked by a bubble inception point where the first bubble can be noticed in our vessel of heated water. Beyond this region, water boils and the nucleate boiling phase begins. This means, initially isolated bubbles are formed in the nucleation sites, which are nothing but gaps or imperfections in the heater surface – in our case, the bottom inner surface of the heated vessel on the stove. When the nucleation sites become aplenty, as the heating increases (see Fig. 1), the bubbles generated from these sites merge together to form vertical columns and slugs that could in principle reach the top free surface of water in the vessel.

No commercial kitchen stove is capable of supplying heat energy fast enough to the water to reach even the slugs and columns regime. So we may not see this range in our kitchen experiment, unless we conjure up a mini nuclear reactor as our kitchen stove. In which case, keep me informed of it, before you invite me over for dinner.

Proceeding to heat and boil the water beyond the slugs and columns regime in a laboratory set-up, there results a situation when a peak heat flux value is reached (marked q_max^,, in the ordinate of Fig. 1). Further increase of heating results in sudden increase of temperature difference (marked in the abscissa). This mostly leads to the melting of commonly used heater material surface. The transition boiling region connects this peak heat flux limit with the film boiling regime, wherein the heater surface is completely blanketed by a film of vapour of the liquid (water, in our case). Heat transfers across the film, into the liquid water residing unstably above the vapour layer.

Further explanations about many of these interesting phenomena are kept out of this essay. There are separate monographs of knowledge available for pool boiling alone (and more for flow boiling). Take a look at the reference [1]. We now go back to the song of our kitchen band.

We now understand the simple kitchen experiment we have performed. In this experiment, just after the bubble inception point, hot vapour bubbles form near the bottom of the vessel (close to the stove) and raise to the top surface of the vessel through the liquid water. The rest of the water column along the vertical path of the bubble is colder than the hot vapour bubble. So while raising, when the top surface of these bubbles come in contact with the colder water above, collapses suddenly by condensing. This cavitation collapse of the vapour bubble results in the high pitched noise (ping) that falls within our audible range.

The noise is a mixture of pings from several thousands of bubble collapse at the same instant. The process of bubble formation and collapse at a higher point is repeated at a high frequency resulting in us hearing the high pitched sound continuously. Further, the smaller the bubble size, the higher the pitch. Longer the vertical column of water, longer the duration of singing one can hear. This is because the vertical residence distance (hence the cooling distance) for the bubbles is increased before they reach the top free surface of the water inside the vessel.

At an instance, bubbles of several sizes are always present in the boiling process. The singing we hear is always a mixture of sounds of several frequencies influenced further by the cross dispersion before it reaches our ears. See for instance, in Fig. 2, the bubble distribution across the surface of the water inside the kitchen vessel.

[The above picture is taken in our kitchen and doctored a bit to visualize the bubbles better. Sophisticated doctoring techniques exist to better capture these bubbles.]

As the heating results in further increase of temperature, the bubble sizes also grow bigger and the sound becomes muted. The bubbles no longer condense within the water column in the vessel but reach the surface of the water and escape the vessel. Water in the vessel continues to boil with a dull gurgling sound. We don’t hear this in our recorded song as it takes a while in my experiment to reach this stage (about 4 minutes for the vessel I used). I have cut those parts out. The outro that one hears in the end part of the audio is the well isolated bubble situation, when the system cooled down after the stove is switched off.

The staccato drumming that one hears over the singing (the wailing guitars like sound) is more due to the pinging noise from the collapse of the bubbles that originally form on the inner surface of the hot side walls of the stainless steel vessel that is used in the experiment. One could reduce this drumming noise with another type of vessel. But the initial singing depends on the nucleation sites at the bottom of the vessel. The required size of these nucleation sites range between 0.005 mm to 5 micro-m [1] – a range in tune with the commercially manufactured surfaces (of cooking vessels).

All the noise explained so far and the accompanied phenomena are well within the bubble inception and isolated bubble region of Fig. 1. We don’t reach the slugs and columns range of Fig. 1 in our kitchen. There ends the sleeve notes for the boiling song from our kitchen band.

Wouldn’t you want to go back now and enjoy listening to the nucleate boiling bubble cavitation grunge again?

Notes

[1] The explanation given in this essay is not rigorous in many places. The essay should serve as only a starting point for understanding the phenomena of boiling. Consult text books (one is given in [1]) for quantitative explanations.

[2] The boiling song audio given in this essay is copy-lefted. It is available for copy-lifting under a re-creative non-commons license. For hearing a live version of it with a variation of the theme, next time stay close to the hot water that you are making.

Reference

[1] A Heat Transfer Text Book by John H. Lienhard IV and John H. Lienhard V. Website for download: http://web.mit.edu/lienhard/www/ahtt.html