Body heat conduction and Newton’s Law of cooling

Principles

Heat conduction is the transmission of heat across matter. Heat transfer is always directed from a higher to a lower temperature. The donor is refrigerating and the acceptor is warming. Denser substances are usually better conductors; metals are excellent conductors.

Fig. 1 Heat transfer in a bar, clamped between two solid objects with a homogene, constant temperature.

The law of heat conduction, also know as Fourier's law, states that the time rate of heat flow Q through a slab or a perfectly insulated bar, as shown in Fig.1, is proportional to the gradient of temperature difference:

Q = k·A·ΔT/Δx, or more formally: (1a)

dQ/dt = k·A·dT/dx (1b)

A is the transversal surface area, Δx is the distance of the body of matter through which the heat is passing, k is a conductivity constant (W/(K.m)) and dependent on the nature of the material and its temperature, and ΔT is the temperature difference through which the heat is being transferred. See for a simple example of a calculation More Info. This law forms the basis for the derivation of the heat equation. The R-value is the unit for heat resistance, the reciprocal of the conductance. Ohm’s law (voltage = current×resistance) is the electrical analogue of Fourier's law.

Conductance of the object per unit area U (=1/R) is:

U = k/Δx (2)

and so Fourier's law can also be stated as:

Q = U·A·ΔT.

Hear resistances, like electrical resistance, are additive when several conducting layers lie between the hot and cool regions, because heat flow Q is the same for all layers, supposing that A also remains the same. Consequently, in a multilayer partition, the total resistance is:

1/U = 1/U₁ + 1/U₂ +1/U₂ + …..(3)

So, when dealing with a multilayer partition, the following formula is usually used:

(4)

When heat is being conducted from one fluid to another through a barrier, it is sometimes important to consider the conductance of the thin film of fluid, which remains stationary next to the barrier. This thin film of fluid is difficult to quantify, its characteristics depending upon complex conditions of turbulence and viscosity, but when dealing with thin high-conductance barriers it can sometimes be quite significant.

Application

In problems of heat conduction, often with heat generated by electromagnetic radiation and ultrasound. Generation by radiation can be performed for instance by lasers (e.g. dermatological and eye chirurgy), by microwave radiation, IR and UV (the latter two also cosmetic). Specific applications are thermographic imaging (see Thermography), low level laser therapy (LLLT, Photobiomodulation), thermo radiotherapy and thermo chemotherapy of cancers. Than, laws of radiation also play a role (see Wien’s law and Stefan- Boltzmann law). Other fields of application are space and environmental medicine (insulating clothing), cardiochirurgy with a heart-lung machine.

Heat transfer calculations for the human body have several components: internal heat transfer by conduction and by perfusion (blood circulation) and external by conduction and convection. The transport by perfusion is complicates calculations substantially.

More info

Newton's law of cooling

Newton's law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. This form of heat loss principle, however, is not very precise; a more accurate formulation requires an analysis of heat flow based on the heat equation in an inhomogeneous medium. Nevertheless, it is easy to derive from this principle the exponential decay (or increase) of temperature of a body. If T(t) is the temperature of the body as a function of time, and T_env the temperature of the environment then its derivative is:

dT(t)/dt = – τ (T –T_env) (5)

where 1/τ is some positive constant. Solving (5) gives:

T(t) = T_env +(T_t=0 –T_env)e^–t/τ. (6)

The constant τ appears to be the time constant and the solution of (6) is the same as (de)charging a condenser in a resistance-capacitance circuit from its initial voltage V_t=0 to the final obligatory voltage V_obl.

In, fluid mechanics (liquids and gases) the Rayleigh number (see Rayleigh, Grashof and Prandtl Number) for a fluid is a dimensionless number associated with the heat transfer within the fluid. When the Rayleigh number is below the critical value for that fluid, heat transfer is primary in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection (see Body heat dissipation and related water loss).

An example of heat loss when submerged

The problem is: how many energy passes the skin of a human jumping into a swimming pool, supposing that:

· heat transfer in the body is solely determined by conduction;

· the heat gradient in the body is time invariant;

· the water temperature surrounding the body remains at the initial temperature;

· the mean heat gradient of the body is ΔT/Δx = 1.5 ^oC/cm = 150 ^oC /m

· body surface 2.0 m² (an athletic body of 75 kg and 188 cm)

· T_skin = 25 ^oC;

· T_water = 25 ^oC;

· k_water = 0.6 W/m.s

Since dQ/dt andT/dx are time and distance invariant, (1a) can directly applied:

Q = -0.6·2.0·150 = 180 W.

For a subject of 25 year basal metabolism is 43.7 W/m2, implying that Q is about twice basal metabolism. Even this very basic approach illustrates the very high energy expenditure of for instance a swimmer, even in warm water. With T_water = 13 ^oC, the skin and deeper tissues will cool down fast and soon the temperature gradient in the body is doubled, resulting in a 360 W loss. This makes clear that with normal sized people in such cool water, exhaustion, next hyperthermia and finally consciousness and drowning is a process of about an hour.

Several physical multi-compartment models have been developed to calculate heat transfer in the human body (see Literature).

Literature

ASHRAE, Fundamental Handbook, Ch. 8 Physiological Principles, Comfort and health. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1989.

http://www.ibpsa.org/%5Cproceedings%5CBS1999%5CBS99_C-11.pdf

Wikipedia