Working content >MacroModels>Heat and temperature >Heat transfer

If an object is warmer than its surroundings, it will give up heat to the surroundings until the object and surroundings are in equilibrium. There are three primary mechanisms by which heat can be transferred.

1)  Radiation.  A hot object will give off energy by emitting light (both visible and infrared) and this light can travel through space, air or water and be absorbed by another object, giving it warmth. This is the way in which heat is transferred from the sun, through space, to warm the earth.  While radiation from the sun is key to warming up objects, radiation is typically not the only means by which an object loses heat. 

We can calculate the rate at which heat radiates from an object.  It depends only on the object’s temperature and surface area.  For a perfect radiator, one known as a black body, the radiated heat transfer rate is given by the Stefan Boltzmann equation:

                                                                      (3)

where σ is the Stefan Boltzmann constant (5.67 x 10^-8 W/m²K⁴), T is the temperature in K, and A is the surface area of the blackbody.  Imperfect radiators will give off less heat than this and so a blackbody is a good upper limit to the amount of heat an object will radiate.

2)  Conduction.  Since biological organisms are rarely in a vacuum, they will be in constant contact with molecules of the surrounding medium (typically air or water).  Heat is readily transferred to or from this surrounding medium.  If the object is warmer, its molecules will give up some of their energy to the surroundings.  Conduction assumes that the surroundings are stable and do not move.  So the object will warm up the layer of air next to it more than the air further from the object.

The rate at which heat is transferred to the surrounding medium is dependent on the air or water molecules randomly bouncing off the object.  This is similar to the random motion that causes molecules to diffuse  and a very similar equation to Fick’s law of diffusion can be written for heat:

                                                                 (4)

Here J_cond is the heat flux density in W/m².  If we multiply J_cond by the object's surface area, A, we will get the rate of heat transfer, dQ/dt.   J_cond as well as dQ/dt, depends on the temperature gradient across the boundary layer of medium surrounding the object and the thermal conductivity, κ, of the medium.  Since the heat flux is given in W/m², and the temperature gradient is K/m, the units of thermal conductivity must be W / m K.  Thermal conductivity is going to be higher for media where the molecules are closer together and so more readily bounce into each other to transfer the kinetic energy away from the warm object.  The value of k for water is 0.60 W/m K at 20°C while that for air is only 0.026 W /m K.  So thermal conductivity is 23 times greater in water than in air.

Based on the equation 4, the rate of heat transferred, dQ/dt (in W), will therefore be equal to

                                                            (5)

This rate therefore depends on the properties of the surrounding medium (κ), the area of the object in contact with the medium (A), and the temperature gradient (dT/dx).  The hotter the object is relative to the surroundings and the more surface area it has, the faster heat is transferred.

3)  Convection.  If the surrounding medium is in motion, then the layer of medium next to an object is continually being replaced.  Therefore, the air next to an object never heats up as that layer is continually mixing in with the rest of the surroundings.  As a result, more heat is transferred more quickly from an object, as the temperature gradient remains larger than for stagnant air.

To model convection, we use an equation which is very similar to equation 5:

                                                                 (6)

Here ΔT is the difference in temperature between the object and its surroundings, A is the object’s surface area, and h_c is the heat transfer coefficent.  The units for h_conv are W/m² K.  This equation is known as Newton’s law of cooling.  This will allow us to calculate heat loss if we know the value of h_c or if we can estimate it.  Often, h_c is determined experimentally for a particular geometry.  These determinations take into account the various fluid conditions, such as fluid density, flow velocity, and the object size.

            As an aside, we should note that heat transfer is similar to charge flow in an electrical circuit.  You can think of the equation above as if they were ohm’s law.  So the flow of heat, dQ/dt is analogous to current, I, h_cA is equivalent to conductance (or 1/resistance) and temperature is the same as voltage.  If we rewrite equation (6) as

                                                                 (7)

this is analogous to the Hagen-Poiseuille equation for fluid flow, and to Ohm's law for electric currents (ΔV = IR).  In each case, a gradient (a change of something in space) drives something to flow.

So in response to a differential temperature, heat will flow (dQ/dt).  The speed of heat flow will depend on the resistance to that flow 1/h_cA.