Thermal Convection

Thermal convection is the process by which heat is transferred from one part of a fluid (liquid or gas) to another by the physical movement of the fluid itself. This involves the bulk transport of fluid molecules carrying thermal energy from regions of higher temperature to regions of lower temperature.

Types of Convection

There are two main types of convection:

1. Natural (Free) Convection
2. Forced Convection

a. Natural (Free) Convection

Natural convection occurs when fluid motion is induced by buoyancy forces that arise from density variations in the fluid due to temperature gradients. When a fluid is heated from below, it expands, becomes less dense, and rises, while the cooler, denser fluid descends, creating convection currents.

Example: Heating water in a pot causes the warmer water at the bottom to rise while cooler water descends, establishing convection currents.

b. Forced Convection

Forced convection involves external agents such as fans, pumps, or blowers that actively move the fluid, enhancing the heat transfer rate. This type of convection is common in engineering applications where controlled cooling or heating is required.

Example: A radiator fan blowing air over hot engine parts to cool them.

Governing Laws of Thermal Convection

Newton's Law of Cooling

Newton's law of cooling states that the rate of heat loss of a body is proportional to the temperature difference between the body and its surroundings. Let:

• $T$ = Temperature of the body (K)
• $T_\infty$ = Temperature of the surrounding fluid (K)
• $Q$ = Rate of heat loss (W)
• $h$ = Convective heat transfer coefficient (W/m²K)
• $A$ = Surface area through which heat is transferred (m²)

The law can be mathematically expressed as:

$$Q = h A (T - T_\infty)$$

The coefficient $h$ depends on the nature of the fluid, flow conditions, and surface characteristics. The negative sign (sometimes written) indicates heat loss.

Applicability

• Valid approximately for small temperature differences ($< 20-30\,K$) in still air.
• More accurate for all temperature differences during forced convection.

Heat Loss and Temperature Change of a Body

If $m$ is the mass of the body and $C$ its specific heat capacity, then the rate of change of temperature with time is related to heat loss by:

$$m C \frac{dT}{dt} = -h A (T - T_\infty)$$

This is a first-order differential equation describing cooling behavior, often solved to give exponential temperature decay.

Dulong and Petit Law (Five-Fourth Power Law)

Dulong and Petit proposed a modification to Newton's law for natural convection when the temperature difference is large ($\theta = T - T_\infty > 50\,K$) under still air conditions:

$$Q = h A (T^{\frac{5}{4}} - T_\infty^{\frac{5}{4}})$$

This empirical law accounts for increased convection rates at higher temperature differences where simple linear relations do not suffice.

Dimensionless Numbers

Dimensionless numbers characterize convection processes and help predict heat transfer coefficients in different flow regimes:

1. Grashof Number (Gr): Ratio of buoyancy to viscous forces in free convection:

   $$\mathrm{Gr} = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$$

   where:

   • $g$ = acceleration due to gravity (m/s²)
   • $\beta$ = thermal expansion coefficient (1/K)
   • $L$ = characteristic length (m)
   • $\nu$ = kinematic viscosity (m²/s)

2. Prandtl Number (Pr): Ratio of momentum diffusivity (viscosity) to thermal diffusivity:

   $$\mathrm{Pr} = \frac{\nu}{\alpha}$$

   where $\alpha$ is thermal diffusivity (m²/s).

3. Rayleigh Number (Ra): Product of Gr and Pr, governing free convection onset:

   $$\mathrm{Ra} = \mathrm{Gr} \times \mathrm{Pr}$$

4. Nusselt Number (Nu): Ratio of convective to conductive heat transfer:

   $$\mathrm{Nu} = \frac{h L}{k}$$

   where $k$ is thermal conductivity of the fluid.

Correlation formulas relate these numbers to the convective heat transfer coefficient $h$. For example, for laminar free convection over a vertical plate:

$$\mathrm{Nu} = 0.59 \, \mathrm{Ra}^{1/4}$$

Summary of Differences

Aspect	Natural Convection	Forced Convection
Driving Force	Buoyancy due to density differences	External agents (fans, pumps)
Fluid Motion	Induced by temperature gradients	Induced by external forces
Heat Transfer Coefficient $h$	Generally lower (0.1 to 10 W/m²K)	Higher (10 to 1000 W/m²K)
Examples	Warm air rising near a radiator	Air blown over a car radiator