*American Journal of Physics*paper that begins “For many years, I have competed and judged in American Kennel Club obedience trials.” The title of the paper is also delightful: “What my Dogs Forced Me to Learn About Thermal Energy Transfer” (Craig Bohren,

*American Journal of Physics*, Volume 83, Pages 443−446, 2015). Bohren’s hypothesis is that an animal perceives hotness and coldness not directly from an object’s temperature, as one might naively expect, but from the flux density of thermal energy. I could follow his analysis of this idea, but I prefer to use the 5th edition of

*Intermediate Physics for Medicine and Biology*, because Russ Hobbie and I have already worked out almost all the results we need.

Chapter 4 of

*IPMB*analyzes diffusion. We consider the concentration,

*C*, of particles as they diffuse in one dimension. Initially (

*t*= 0), there exists a concentration difference

*C*

_{0}between the left and right sides of a boundary at

*x*= 0. We solve the diffusion equation in this case, and find the concentration in terms of an error function

*C*(

*x*,

*t*) =

*C*

_{0}/2 [ 1 – erf(

*x*/√4

*Dt*) ] , Eq. 4.75

where

*D*is the diffusion constant. A plot of

*C*(

*x,t*) is shown in Fig. 4.22 (we assume

*C*= 0 on the far right, but you could add a constant to the solution without changing the physics, so all that really matters is the concentration difference).

Fig. 4.22 The spread of an initially sharp boundary due to diffusion. |

*D*given by the thermal conductivity,

*κ*, divided by the specific heat capacity,

*c*, and the density,

*ρ*

*D*=

*κ/cρ .*

So, by analogy, if you start (

*t*= 0) with a uniform temperature on the left and on the right sides of a boundary at

*x*= 0, with an initial temperature difference

*ΔT*between sides, the temperature distribution

*T*(

*x*,

*t*) is (to within a additive constant temperature) the same as the concentration distribution calculated earlier.

*T*(

*x*,

*t*) =

*ΔT*/2 [ 1 – erf(

*x*/√4

*Dt*) ] .

The temperature of the interface is always

*ΔT*/2. If a dog responded to simply the temperature at

*x*= 0 (where its thermoreceptors are presumably located), it would react in a way strictly proportional to the temperature difference

*ΔT*. But Bohren’s hypothesis is that thermoreceptors respond to the energy flux density,

*κ*

*dT*/

*dx*.

Now let us look again at Fig. 4.22. The slope of the curve at

*x*= 0 is the key quantity. So, we must differentiate our expression for

*T*(

*x*,

*t*). We get

*κ dT*/

*dx*= -

*κ ΔT*/2

*d*/

*dx*[ erf(

*x*/√4

*Dt*) ] .

By the chain rule, this becomes (with

*u*=

*x*/√4

*Dt*)

*κ dT*/

*dx*= -

*κ ΔT*/ (2 √4

*Dt*)

*d*(erf(

*u*))/d

*u .*

The derivative of the error function is given in

*IPMB*on page 179

*d*/

*du*(erf(

*u*)) = 2/√π

*e*

^{-u2}.

At the interface (

*u*= 0), this becomes 2/√π. Therefore

*κ dT*/

*dx*= -

*κ ΔT*/√4

*πDt*.

Bohren comes to the same result, but by a slightly different argument.

The energy flux density depends on time, with an infinite response initially (we assume an abrupt difference of temperature on the two sides of the boundary

*x*= 0) that falls to zero as the time becomes large. The flux density depends on the material parameters by the quantity

*κ*/√

*D*, which is equivalent to √

*cρκ*and is often called the thermal inertia.

Bohren goes on to analyze the case when the two sides have different properties (for example, the left might be a piece of aluminum, and the right a dog’s tongue), and shows that you get a similar result except the effective thermal inertia is a combination of the thermal inertia on the left and right. He does not solve the entire bioheat equation (Sec. 14.11 in

*IPMB*), including the effect of blood flow. I would guess that blood flow would have little effect initially, but it would play a greater and greater role as time goes by.

Perhaps I will try Bohren’s experiment: I’ll give Auggie (my daughter Kathy’s lovable foxhound) a cylinder of aluminum and a cylinder of stainless steel, and see if he can distinguish between the two. My prediction is that, rather than either metal, he prefers rawhide.

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