Heat Transfer

The image below shows three different ways of heat transfer: conduction, convection, and Radiation.

Conduction:

In general, heat transfer without the movement of particles is called conduction. And this type of heat transfer occurs more quickly in solids and liquids than in gases.

Since the distance between particles is more in gases, conduction occurs slowly.

Example:
Conduction of heat in atmospheric air.

Heat transfer within the liquid or gases happens due to molecular interaction/motion, and heat transfer in the solids occurs due to molecular vibration.

Heat transfer within solids due to molecular vibration is called conduction, and the free electrons transport the energy. Thermal conduction in metals is a more complex theory, but heat transfer occurs without the movement of particles.

Example:
When we heat one end of a metal rod, the other end will get the heat by conduction.

The heat transfer from one object to another through particle collision is called conduction.

Example:
When a wall separates hot and cold air, the heat is transferred through the wall from hot air to cold air due to the air particle collision on either side of the wall.

Fourier law states that heat transfer (Q) per unit area(A) is directly proportional to the temperature gradient(dT/dx).

$\dfrac{Q}{A} \space \alpha \space \dfrac{dT}{dx}$

$\dfrac{Q}{A} \space = \space -K \space \dfrac{dT}{dx}$

Where,
K => Thermal conductivity in W/m-°K or Btu/hr-ft-°F
The negative sign to k denotes the heat moving from higher to lower energy.
dT = (T1 - T2) in °K or °F
dx = X in m or ft

The above image describes the 1D heat transfer through a wall.

For 3D (x, y, & z)

$\dfrac{Q}{A} \space = \space -K \space \dfrac{\partial T}{\partial x} \space i + \space -K \space \dfrac{\partial T}{\partial y} \space j + \space -K \space \dfrac{\partial T}{\partial z} \space k$

$\dfrac{Q}{A} \space = \space -K \space (\dfrac{\partial}{\partial x} \space i + \space \dfrac{\partial}{\partial y} \space j + \space \dfrac{\partial}{\partial z} \space k) \space T$

$\dfrac{Q}{A} \space = \space -K \space \nabla T$

The above equation will give a vector called the “Gradient of Temperature.”

$\nabla = \dfrac{\partial}{\partial x} \space i + \space \dfrac{\partial}{\partial y} \space j + \space \dfrac{\partial}{\partial z} \space k$

∇ is called the Nabla Operator

When we do a dot product over this with the Nabla operator(Divergent of diffusion heat transfer), we get the “Diffusion Equation of the Navier Stroke Equation.”

$\nabla \space . (k \nabla T)$

Convection:

Heat transfer due to the movement of particles is called convection. And this type of heat transfer occurs in fluids (liquids/gases). In general, fluids are not good conductors.

The concept of “Buoyant force” explains the movement of the heated fluids.

Heat transfer from solid to fluid happens when the fluid flow over the solids.

Example:
When water flows over a hot plate, it takes the heat from it and caries.

When the hot fluid rises due to the difference in density, it is called natural convection.

When a fan or pump moves the fluid, it is called forced convection.

Newton’s law states that the convection heat transfer is directly proportional to the difference in temperature between the fluid and the solid.

$\dfrac{Q}{A} \space \alpha \space (T_s - T_f)$

$\dfrac{Q}{A} \space = \space h \space (T_s - T_f)$

Where,
h – Convective heat transfer co-efficient in W/m2-°C (or) Btu/hr-ft2-°F
Ts – Surface temperature of the solid
Tf – Temperature of the fluid
Q/A – Heat transfer per unit area

The convective heat transfer co-efficient depends on various factors like the contact area of the solid with fluid, the temperature difference between the solid and the fluid, the velocity of the fluid over the surface of the solid, and the thermophysical properties of the fluid.

The convective heat transfer coefficient is independent of the type of solid material. However, the surface roughness plays a significant role when the flow is turbulent.

Radiation:

Heat transfer due to electromagnetic waves is called convection. When the heat spreads out from a source is called radiation, and the transfer of heat does not involve particles or movement of particles.

Radiation can happen in the space of matter or a vacuum.

Stefan-Boltzmann law states that the radiation heat transfer is directly proportional to the fourth power of its temperature.

$\dfrac{Q}{A} \space \alpha \space (T^4)$

$\dfrac{Q}{A} \space = \space e \sigma (T_s^4 - T_{surr}^4)$

Where,
e – emissivity of the heat source
σ – Stefan-Boltzmann constant (5.67E-08 W/m2-K4)
Q/A – Heat transfer per unit area
Ts – Surface temperature of the heat source
Tsurr – Surrounding temperature