Heat transfer, $Q~(J)$, is energy transmission achieved by conduction, convection, or radiation. More commonly, we will work with the rate of heat transfer, $\dot Q~(W)$. Further, we will consider heat flux, the rate of heat transfer per unit area. Our preferred notation for heat flux will be $q~(\frac W{m^2})$.
Conduction is essentially how heat transfers within and between solids. It is driven by a temperature gradient and facilitated by thermal conductivity, $k~(\frac{W}{mK})$, and cross sectional area, $A~(m^2)$. Heat flows from hot to cold, opposite the temperature gradient, thus the negative proportionality.
$$ \dot Q=-kA\frac{dT}{dx} $$
When applied to a finite thickness, heat transfer by conduction is driven by a temperature difference and inhibited by the section thickness.
$$ \dot Q=-kA\frac{T_1-T_0}{x_1-x_0} $$
Convection is essentially how heat transfers from a solid to a fluid. It, like conduction, is driven by a temperature difference and facilitated by a heat transfer coefficient, $h~(\frac{W}{m^2K})$, and the area on which convection is occurring, $A_s~(m^2)$. We will discuss in future sections how to calculate the heat transfer coefficient.
$$ \dot Q=hA_s(T_s-T_\infin) $$
Radiation arises because all objects emit electromagnetic radiation related to their temperature, typically in the infrared and visible light frequency ranges. Physically, the net radiation varies based on how surfaces are physically spaced, each surface radiating at one another. To simplify our governing equation, we consider the radiation from a surface, fully enveloped by surroundings of much larger surface area.
Under these assumptions, radiation is driven by a non-linear temperature difference and facilitated by surface emissivity, $\varepsilon$, and area of the surface, $A_s$. Surface emissivity is a material property on the range $(0,1)$ which characterizes what portion of idealized radiation a material emits. Also relevant is the Stefan-Boltzmann constant, $5.67\small{\cdot10}^{-8} ~(\frac{W}{m^2K^4})$, which is universal.
$$ \dot Q=\varepsilon\sigma A_s(T_s^4-T_{surr}^4) $$
