Cooling by Radiation

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss thermal radiation. If you’re a black body, the net power you radiate, w_tot, is given by Eq. 14.41

w_tot = S σ_SB (T⁴ – T_s⁴) ,                (14.41)

where S is the surface area, σ_SB is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴), T is the absolute temperature of your body (about 310 K), and T_s is the temperature of your surroundings. The T⁴ term is the radiation you emit, and the T_s⁴ term is the radiation you absorb.

The fourth power that appears in this expression is annoying. It means we must use absolute temperature in kelvins (K); you get the wrong answer if you use temperature in degrees Celsius (°C). It also means the expression is nonlinear; w_tot is not proportional to the temperature difference T – T_s.

On the absolute temperature scale, the difference between the temperature of your body (310 K) and the temperature of your surroundings (say, 293 K at 20 °C) is only about 5%. In this case, we simplify the expression for w_tot by linearizing it. To see what I mean, try Homework Problem 14.32 in IPMB.

Section 14.9 

Problem 32. Show that an approximation to Eq. 14.41 for small temperature differences is w_tot = S K_rad (T − T_s). Deduce the value of K_rad at body temperature. Hint: Factor T⁴ – T_s⁴ =  (T − T_s)(…). You should get K_rad = 6.76 W m⁻² K⁻¹.

The constant K_rad has the same units as a convection coefficient (see Homework Problem 51 in Chapter 3 of IPMB). Think of it as an effective convection coefficient for radiative heat loss. Once you determine K_rad, you can use either the kelvin or Celsius temperature scales for T − T_s, so you can write its units as W m⁻² °C⁻¹.

 

Air and Water,
by Mark Denny.

In Air and Water, Mark Denny analyzes the convection coefficient. In a stagnant fluid, the convection coefficient depends only on the fluid’s thermal conductivity and the body’s size. For a sphere, it is inversely proportional to the diameter, meaning that small bodies are more effective at convective cooling per unit surface area than large bodies. If the body undergoes free convection or forced convection (for both cases the surrounding fluid is moving), the expression for the convection coefficient is more complicated, and depends on factors such as the Reynolds number and Prandtl number of the fluid flow. Denny gives values for the convection coefficient as a function of body size for both air and water. Usually, these values are greater than the 6.76 W m⁻² °C⁻¹ for radiation. However, for large bodies in air, radiation can compete with convection as the dominate mechanism. For people, radiation is an important mechanism for cooling. For a dolphin or mouse, it isn’t. Elephants probably make good use of radiative cooling.
 

Finally, our analysis implies that when the difference between the temperatures of the body and the surroundings is small, a body whose primary mechanism for getting rid of heat is radiation will cool exponentially following Newton’s law of cooling.

The Carr-Purcell-Meiboom-Gill Pulse Sequence

The most exciting phrase to hear in science, the one that heralds new discoveries, is not “Eureka!” but “That’s funny...” 

—Isaac Asimov

In Section 18.8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe the Carr-Purcell pulse sequence, used in magnetic resonance imaging.

When a sequence of π [180° radio-frequency] pulses that nutate M [the magnetization vector] about the x' axis are applied at T_E/2, 3T_E/2, 5T_E/2, etc., a sequence of echoes are formed [in the M_x signal], the amplitudes of which decay with relaxation time T₂. This is shown in Fig. 18.19.

Fig. 18.19  The Carr-Purcell pulse sequence.
All π pulses nutate about the x' axis.
The envelope of echoes decays as e^−t/T₂.

Russ and I then discuss the Carr-Purcell-Meiboom-Gill pulse sequence.

One disadvantage of the CP [Carr-Purcell] sequence is that the π pulse must be very accurate or a cumulative error builds up in the successive pulses. The Carr-Purcell-Meiboom-Gill sequence overcomes this problem. The initial π/2 [90° radio-frequency] pulse nutates M about the x' axis as before, but the subsequent [π] pulses are shifted a quarter cycle in time, which causes them to rotate about the y' axis.

 

Fig. 18.21  The Carr-Purcell-Meiboom-Gill pulse sequence.

Meiboom, S. and Gill, D. (1958)
“Modified Spin-Echo Method for
Measuring Nuclear Relaxation Times.”
Rev. Sci. Instr. 29:688–691.

Students might enjoy reading the abstract of Saul Meiboom and David Gill’s 1958 article published in the Review of Scientific Instruments (Volume 29, Pages 688-691).

A spin echo method adapted to the measurement of long nuclear relaxation times (T₂) in liquids is described. The pulse sequence is identical to the one proposed by Carr and Purcell, but the rf [radio-frequency] of the successive pulses is coherent, and a phase shift of 90° is introduced in the first pulse. Very long T₂ values can be measured without appreciable effect of diffusion.

This short paper is so highly cited that it was featured in a 1980 Citation Classic commentary, in which Meiboom reflected on the significance of the research.

The work leading to this paper was done nearly 25 years ago at the Weizmann Institute of Science, Rehovot, Israel. David Gill, who was then a graduate student… , set out to measure NMR T₂-relaxation times in liquids, using the well-known Carr-Purcell pulse train scheme. He soon found that at high pulse repetition rates adjustments became very critical, and echo decays, which ideally should be exponential, often exhibited beats and other irregularities. But he also saw that on rare and unpredictable occasions a beautiful exponential decay was observed... Somehow the recognition emerged that the chance occurrence of a 90° phase shift of the nuclear polarization [magnetization] must underlie the observations. It became clear that in the presence of such a shift a stable, self-correcting state of the nuclear polarization is produced, while the original scheme results in an unstable state, for which deviations are cumulative. From here it was an easy step to the introduction of an intentional phase shift in the applied pulse train, and the consistent production of good decays.

The key point is that the delay between the initial π/2 pulse (to flip the spins into the xy plane) and the string of π pulses (to create the echoes) must be timed carefully (the pulses must be coherent). Even adding a delay corresponding to a quarter of a single oscillation changes everything. In a two-tesla MRI scanner, the Larmor frequency is 83 MHz, so one period is 12 nanoseconds. Therefore, if the timing is off by just a few nanoseconds, the method won’t work.

Initially Gill didn’t worry about timing the pulses precisely, so usually he was using the error-prone Carr-Purcell sequence. Occasionally he got lucky and the timing was just right; he was using what’s now called the Carr-Purcell-Meiboom-Gill sequence. Meiboom and Gill “somehow” were able to deduce what was happening and fix the problem. Meiboom believes their paper is cited so often because it was the first to recognize the importance of maintaining phase relations between the different pulses in an MRI pulse sequence.

In his commentary, Meiboom notes that

Although in hindsight the 90° phase shift seems the logical and almost obvious thing to do, its introduction was triggered by a chance observation, rather than by clever a priori reasoning. I suspect (though I have no proof) that this applies to many scientific developments, even if the actual birth process of a new idea is seldom described in the introduction to the relevant paper.

If you’re a grad student working on a difficult experiment that’s behaving oddly, don’t be discouraged if you hear yourself saying “that’s funny...” A discovery might be sitting right in front of you!  

The Rest of the Story 2

Hermann in 1848.

The Rest of the Story

Hermann was born in 1821 in Potsdam, Germany. He was often sick as a child, suffering from illnesses such as scarlet fever, and started school late. He was hampered by a poor memory for disconnected facts, making subjects like languages and history difficult, so his interest turned to science. His father loaned him some cheap glass lenses that he used to build optical instruments. He wanted to become a physicist, but his family couldn’t afford to send him to college. Instead, he studied hard to pass an exam that won him a place in a government medical school in Berlin, where his education would be free if he served in the military for five years after he graduated.

The seventeen-year-old Hermann moved to Berlin in 1838. He brought his piano with him, on which he loved to play Mozart and Beethoven. He became friends with his fellow students Ernest von Brücke and Emil de Bois-Reymond, began doing scientific research under the direction of physiologist Johannes Müller, and taught himself higher mathematics in his spare time. By 1843 he graduated and began his required service as an army surgeon.

Life in the army required long hours, and Hermann was isolated from the scientific establishment in Berlin. But with the help of Brücke and de Bois-Reymond he somehow continued his research. His constitution was still delicate, and sometimes he would take time off to restore his health. Near the end of his five-year commitment to the army he fell in love with Olga von Velten, who would sing while he accompanied her on the piano. They became engaged, but he knew they could not marry until he found an academic job after his military service ended, and for that he needed to establish himself as a first-rank scientist. This illness-prone, cash-strapped, over-worked army doctor with a poor memory and a love for music needed to find a research project that would propel him to the top of German science.

Hermann rose to the challenge. He began a careful study of the balance between muscle metabolism and contraction. Using both experiments and mathematics he established the conservation of energy, and in the process showed that no vital force was needed to explain life. On July 23, 1847 he announced his discovery at a meeting of the German Physical Society.  

This research led to a faculty position in Berlin and his marriage to Olga. His career took off, and he later made contributions to the study of vision, hearing, nerve conduction, and ophthalmology. Today, the largest Association of German Research Centers bears his name. Many consider Hermann von Helmholtz to be the greatest biological physicist of all time.

And now you know THE REST OF THE STORY.

Good day! 

_____________________________________________________________

This blog post was written in the style of Paul Harvey’s “The Rest of the Story” radio program. The content is based on a biography of Helmholtz written by his friend and college roommate Leo Koenigsberger. You can read about nerve conduction and Helmholtz’s first measurement of its propagation speed in Chapter 6 of Intermediate Physics for Medicine and Biology. This August we will celebrate the 200th anniversary of Hermann von Helmholtz’s birth. 

Click here for another IPMB “The Rest of the Story” post.

 

Charles Osgood pays tribute to the master storyteller Paul Harvey.

https://www.youtube.com/watch?v=S5_OIoMBjSk

Estimating the Properties of Water

Water
from: www.middleschoolchemistry.com

 

I found a manuscript on the arXiv by Andrew Lucas about estimating macroscopic properties of materials using just a few microscopic parameters. I decided to try a version of this analysis myself. It’s based on Lucas’s work, with a few modifications. I focus exclusively on water because of its importance for biological physics, and make order-of-magnitude calculations like those Russ Hobbie and I discuss in the first section of Intermediate Physics for Medicine and Biology.

My goal is to estimate the properties of water using three numbers: the size, mass, and energy associated with water molecules. We take the size to be the center-to-center distance between molecules, which is about 3 Å, or 3 × 10⁻¹⁰ m. The mass of a water molecule is 18 (the molecular weight) times the mass of a proton, or about 3 × 10⁻²⁶ kg. The energy associated with one hydrogen bond between water molecules is about 0.2 eV, or 3 × 10⁻²⁰ J. This is roughly eight times the thermal energy kT at body temperature, where k is Boltzmann’s constant (1.4 × 10⁻²³ J K⁻¹) and T is the absolute temperature (310 K). A water molecule has about four hydrogen bonds with neighboring molecules.

Density

Estimating the density of water, ρ, is Homework Problem 4 in Chapter 1 of IPMB. Density is mass divided by volume, and volume is distance cubed

ρ = (3 × 10⁻²⁶ kg)/(3 × 10⁻¹⁰ m)³ = 1100 kg m⁻³ = 1.1 g cm⁻³.

The accepted value is ρ = 1.0 g cm⁻³, so our calculation is about 10% off; not bad for an order-of-magnitude estimate.

Compressibility

The compressibility of water, κ, is a measure of how the volume of water decreases with increasing pressure. It has dimensions of inverse pressure. The pressure is typically thought of as force per unit area, but we can multiply numerator and denominator by distance and express it as energy per unit volume. Therefore, the compressibility is approximately distance cubed over the total energy of the four hydrogen bonds

κ = (3 × 10⁻¹⁰ m)³/[4(3 × 10⁻²⁰ J)] = 0.25 × 10⁻⁹ Pa⁻¹ = 0.25 GPa⁻¹ ,

implying a bulk modulus, B (the reciprocal of the compressibility), of 4 GPa. Water has a bulk modulus of about B = 2.2 GPa, so our estimate is within a factor of two.

Speed of Sound

Once you know the density and compressibility, you can calculate the speed of sound, c, as (see Eq. 13.11 in IPMB)

c = (ρ κ)^−1/2 = 1/√[(1100 kg m⁻³) (0.25 × 10⁻⁹ Pa⁻¹)] = 1900 m s⁻¹ = 1.9 km s⁻¹.

The measured value of the speed of sound in water is about c = 1.5 km s⁻¹, which is pretty close for a back-of-the-envelope estimate.

Latent Heat

A homework problem about vapor pressure in Chapter 3 of IPMB uses water’s latent heat of vaporization, L, which is the energy required to boil water per kilogram. We estimate it as

L = 4(3 × 10⁻²⁰ J)/(3 × 10⁻²⁶ kg) = 4.0 × 10⁶ J kg⁻¹ = 4 MJ kg⁻¹.

The known value is L = 2.5 MJ kg⁻¹. Not great, but not bad.

Surface Tension

The surface tension, γ, is typically expressed as force per unit length, which is equivalent to the energy per unit area. At a surface, we estimate one of the four hydrogen bonds is missing, so

γ = (3 × 10⁻²⁰ J)/(3 × 10⁻¹⁰ m)² = 0.33 J m⁻² .

The measured value is γ = 0.07 J m⁻², which is about five times less than our calculation. This is a bigger discrepancy than I’d like for an order-of-magnitude estimate, but it’s not horrible.

Viscosity

The coefficient of viscosity, η, has units of kg m⁻¹ s⁻¹. We can use the mass of the water molecule in kilograms, and the distance between molecules in meters, but we don’t have a time scale. However, energy has units of kg m² s⁻², so we can take the square root of mass times distance squared over energy and get a unit of time, τ

τ = √[(3 × 10⁻²⁶ kg) (3 × 10⁻¹⁰ m)²/4(3 × 10⁻²⁰ J)] = 0.15 × 10⁻¹² s = 0.15 ps.

We can think of this as a time characterizing the vibrations about equilibrium of the molecules. 

 

The viscosity of water should therefore be on the order of

η = (3 × 10⁻²⁶ kg)/[(3 × 10⁻¹⁰ m) (0.15 × 10⁻¹² s)] = 0.67 × 10⁻³ kg m⁻¹ s⁻¹.

Water has a viscosity coefficient of about η = 1 × 10⁻³ kg m⁻¹ s⁻¹. I admit this analysis provides little insight into the mechanism underlying viscous effects, and it doesn’t explain the enormous temperature dependence of η, but it gets the right order of magnitude.

Specific Heat

The heat capacity is the energy needed to raise the temperature of water by one degree. Thermodynamics implies that the heat capacity is typically equal to Boltzmann’s constant times the number of degrees of freedom per molecule times the number of molecules. The number of degrees of freedom is a subtle thermodynamic concept, but we can approximate it as the number of hydrogen bonds per molecule; about four. Often heat capacity is expressed as the specific heat, C, which is the heat capacity per unit mass. In that case, the specific heat is

C = 4 (1.4 × 10⁻²³ J K⁻¹)/(3 × 10⁻²⁶ kg) = 1900 J K⁻¹ kg⁻¹.

The measured value is C = 4200 J K⁻¹ kg⁻¹, which is more than a factor of two larger than our estimate. I’m not sure why our value is so low, but probably there are rotational degrees of freedom in addition to the four vibrational modes we counted.

Diffusion

The self-diffusion constant of water can be estimated using the Stokes-Einstein equation relating diffusion and viscosity, D = kT/(6πηa), where a is the size of the molecule. The thermal energy kT is about one eighth of the energy of a hydrogen bond. Therefore,

D = [(3 × 10⁻²⁰ J)/8]/[(6)(3.14)(0.67 × 10⁻³ kg m⁻¹ s⁻¹)(3 × 10⁻¹⁰ m)] = 10⁻⁹ m² s⁻¹.

Figure 4.11 in IPMB suggests the measured diffusion constant is about twice this estimate: D = 2 × 10⁻⁹ m² s⁻¹. 

 

Air and Water,
by Mark Denny.

We didn’t do too bad. Three microscopic parameters, plus the temperature, gave us estimates of density, compressibility, speed of sound, latent heat, surface tension, viscosity, specific heat, and the diffusion constant. This is almost all the properties of water discussed in Mark Denny’s wonderful book Air and Water. Fermi would be proud.