# Rate of heat flow

The rate of heat flow is the amount of heat that is transferred per unit of time in some material, usually measured in watt (joules per second). Heat is the flow of thermal energy driven by thermal non-equilibrium, so that 'heat flow' is a redundancy (i.e. a pleonasm, and the same for ‘work flow’). Heat must not be confused with stored thermal energy, and moving a hot object from one place to another must not be called heat transfer. But, in spite of all these remarks, it is common in normal parlance to say ‘heat flow’, to talk of ‘heat content’, etc.[1]

The equation of heat flow is given by Fourier's Law of Heat Conduction.

Rate of heat flow = - (heat transfer coefficient) * (area of the body) * (variation of the temperature) / (length of the material)

The formula for the rate of heat flow is:

${\displaystyle {\frac {Q}{\Delta t}}=-kA{\frac {\Delta T}{\Delta x}}}$

where

• ${\displaystyle Q}$ is the net heat (energy) transfer,
• ${\displaystyle \Delta t}$ is the time taken,
• ${\displaystyle \Delta T}$ is the difference in temperature between the cold and hot sides,
• ${\displaystyle \Delta x}$ is the thickness of the material conducting heat (distance between hot and cold sides),
• ${\displaystyle k}$ is the thermal conductivity, and
• ${\displaystyle A}$ is the surface area of the surface emitting heat.

If a piece of material whose cross-sectional area is ${\displaystyle A}$ and thickness is ${\displaystyle \Delta x}$ with a temperature difference ${\displaystyle \Delta T}$ between its faces is observed, heat flows between the two faces in a direction perpendicular to the faces. The time rate of heat flow, ${\displaystyle {\frac {Q}{\Delta t}}}$, for small ${\displaystyle Q}$ and small ${\displaystyle \Delta t}$, is proportional to ${\displaystyle A\times {\frac {\Delta T}{\Delta x}}}$. In the limit of infinitesimal thickness ${\displaystyle \Delta x}$, with temperature difference ${\displaystyle \Delta T}$, this becomes ${\displaystyle H=-kA({\frac {\Delta T}{\Delta x}})}$, where ${\displaystyle H=({\frac {Q}{\Delta t}})}$ is the time rate of heat flow through the area ${\displaystyle A}$, ${\displaystyle {\frac {\Delta T}{\Delta x}}}$ is the temperature gradient across the material, and ${\displaystyle k}$, the proportionality constant, is the thermal conductivity of the material.[2] People often use ${\displaystyle k}$, ${\displaystyle \lambda }$, or the Greek letter ${\displaystyle \kappa }$ to represent this constant.[3] The minus sign is there because the rate of heat flow is always negative—heat flows from the side at higher temperature to the one at lower temperature, not the other way around.[4]

## References

1. ^ http://webserver.dmt.upm.es/~isidoro/bk3/c11/Heat%20and%20mass%20transfer.pdf
2. ^ "52.09 -- Heat conduction in various metal rods". web.physics.ucsb.edu. Retrieved 2019-05-07.
3. ^ "Thermal conductivity", Wikipedia, 2019-04-19, retrieved 2019-05-07
4. ^ "Unit Operations in Food Processing - R. L. Earle". nzifst.org.nz. Retrieved 2019-05-07.