# Heat transfer

Everyone has always understood that something flows from hot object to cold one. It is called heat. The overall driving force for this heat flow is thermal gradient.

There are mainly three modes of heat transfer:

1) Conduction

2) Convection

## Conduction

Conduction is heat transfer by means of molecular agitation within a material without any motion of the material as a whole. In this mode of heat transfer, heat flow is due to molecular collisions in the substance. Since molecullar collisions increase with temperature (Temperature increases the kinetic energy of the molecules, which is a combined effect of rotational, translational and vibrational motion of molecules ), heat transfer due to conduction increases. Conduction heat transfer through a substance is because of a temperature gradient. The rate of heat transfer by conduction between two regions of a substance is proportional to the temperature difference between them. The constant of propotionality is called thermal conductivity of the material. The thermal conductivity of materials in general depends on Temperature. In general liquids and gases have lower thermal conductivity as compared to solids (esp metals).

Mathematically, it can be described by using the Fourier's law:

$Q_{Conduction} = -k*A*\frac{dT}{dx}$

Where

$Q = \mbox{the rate of heat conduction (W)}$
$k = \mbox{Thermal conductivity of the material (W/m K)}$
$A = \mbox{Cross-sectional area of the object perpendicular to heat conduction (m2)}$
$T = \mbox{Temperature (K)}$
$x = \mbox{Length of the object (m)}$

Conduction heat transfer depends on thermal conductivity of material. Heat carried away by a solid by condution is proportional to its thermal conductivity for a unit length and unit cross-section. Here are few materials with their conductivity:

Diamond: 2000 W/m-K
Silver: 406 W/m-K
Copper: 385 W/m-K
Aluminum: 205 W/m-K
Sn-Pb Solder: 50 W/m-K
FR-4: 0.3 W/m-K
Kapton: 0.2 W/m-K

....

## Convection

Convection is heat transfer by means of motion of the molecules in the fluid. Heat energy transfers between a solid and a fluid when there is a temperature difference between the fluid and the solid. Convection heat transfer can not be neglected when there is a significant fluid motion around the solid.

There are mainly two types of the convection heat transfer:

1) Natural or Free Convection

2) Forced Convection

• Natural or Free Convection:- The temperature of the solid due to an external field such as fluid buoyancy can induce a fluid motion. This is known as "natural convection" and it is a strong function of the temperature difference between the solid and the fluid. This type of convective heat transfer takes place due to only fluid buoyancy caused due to temperature defference between fluid layers.
• Forced Convection:- Forcing air to blow over the solid by using external devices such as fans and pumps can also generate a fluid motion. This is known as "forced convection". Some external means for fluid motion is necessary in this type of convective heat transfer.

Fluid mechanics plays a major role in determining convection heat transfer. For each kind of convection heat transfer, the fluid flow can be either laminar or turbulent. For laminar flow of fluid over solid surface, a steady boundary layer formation takes place through which conductive heat transfer occur. This reduces convective heat transfer rate. Turbulent flow forms when the boundary layer is shedding or breaking due to higher velocities or rough geometries. This enhances the heat transfer.

Newton's Law of Cooling

Heat transfer due to convection is described by Newton's Law of Cooling,

$Q_{Convection} = h*A*dT$

Where

$Q_{Convection} = \mbox{Heat convected to surrounding fluid (W)}$
$h = \mbox{Convection heat-transfer coefficient (constant of proportionality ) (W/m2 K)}$
$A = \mbox{Area of the solid in contact with fluid (m2)}$
$dT = \mbox{Temperature difference between solid and surrounding fluid (Ts-Tf) (K)}$

The rate of heat transfered to the surrounding fluid is proportional to the object's exposed area A, and the difference between the solid temperature Ts and the mean fluid temperature Tf. Velocity of the fluid over solid is also a major contributing to enhance the rate of heat transfer.

Convection heat-transfer coefficient (h) plays main role in heat transfer by convection. Heat transfer rates by convection are expressed in terms of h.

It depends on following factors (h is directly propotional to these factors):

1) Exposed area of solid

2) Temperature difference between solid and fluid

3) Fluid velocity

Convective heat transfer can also be characterised in terms of Nusselt number.

Nusselt number is dimensionless number which is useful for heat transfer calculations. Nusselt number is the dimensionless heat transfer coefficient and appears when you are dealing with convection. It, therefore, provides a measure of the convection heat transfer at the surface.

It can be defined as follows:

$Nu = \frac{h*l}{k}$

Where

$Nu = \mbox{Nusselt number}$
$h = \mbox{Convection heat-transfer coefficient (constant of proportionality ) (W/m2 K)}$
$l = \mbox{Characteristic dimension of the solid object (m)}$
$k = \mbox{Thermal conductivity of the solid (W/m-k)}$

The Nusselt number may be viewed as the ratio of heat flow by convection to conduction for a layer of fluid.

If Nu=1, we have pure conduction. Higher values of Nusselt mean that the heat transfer is enhanced by convection.

In general,$Nu = f(Re, Pr, Gr)$

Where

$\mbox{Re is Reynolds number}$
$\mbox{Pr is Prandtl number}$
$\mbox{Gr is Grashof number}$

Following conditions apply:

$Gr/Re_L ^2 = 1; Nu = f(Re, Pr, Gr)$ Mixed convection
$Gr/Re_L ^2 << 1; Nu = f(Re, Pr)$ Forced convection
$Gr/Re_L ^2 >> 1; Nu = f(Gr, Pr)$ Natural convection