Heat transfer module - Fluid Dynamics 2019

Applied Examples of Heat Transfer

HEAT TRANSFER IN A LAVA LAMP

Description

When you flip on the light in a lava lamp, heat from the bulb works slowly to warm up the wax (or ‘lava’). This kind of heat transfer is called conduction, or movement of heat from one place to another. Heat energy flows from high heat (light bulb) to low heat (wax), and the gradient is the driver of conductive heat transfer. This heat transfer can be represented by an equation.

Heat Transfer Equation:

The heat transfer equation describes how heat changes through time.

 ∂T/∂t = κ∇²T + v∇T + A

 ∂T/∂t = conduction + advection + production

Until things start moving, conduction is the dominant heat transfer process in our lava lamp system.

• Conduction depends on:
  • thermal diffusivity of the material through which heat is moving
  • thermal gradient

The density of the wax is inversely related to temperature, meaning that at higher temperatures, the wax is less dense. When the density becomes lower than the density of the liquid in the lava lamp, the wax will tend to float upward.

Wax reaches critical density (less than liquid) and tends to float upward. That movement of wax introduces velocity into the system--advection.

• Advection depends on:
  • velocity of the fluid
  • thermal gradient

Peclét Number:

The Peclét Number is a dimensionless number which indicates whether conduction or advection dominates the system.

 Pe = uL/κ

where u = flow velocity, L = characteristic length Height of lava lamp, κ = thermal diffusivity

• Low peclet regime- dominated by heat transport by conduction
• High peclet regime- dominated by heat transport by advection

With the movement of wax in the upward direction away from the heat source at the bottom of the container and into a cooler material the wax density increases again (more than liquid). This wax density increase causes the wax to sink back down to the bottom of the container towards the heat source. This process is called convection.

Rayleigh Number:

The Rayleigh Number is a dimensionless number used to describe whether convection grows or decays.

 Ra ≡ (gρΔTαd³)/μκ

• Convection depends on driving forces and resisting forces.
  • Driving forces:
    • Acceleration due to gravity (g)
    • Density (ρ)
    • The temperature difference between the bottom and top of the convection cell (ΔT)
    • The volume coefficient of thermal expansion (α)
    • The height of the convection cell (d)
  • Resisting forces:
    • Viscosity (μ)
    • Thermal diffusivity (κ)

• When numerator terms are dominant convection grows
• When denominator terms are dominant convection decays

At some critical Rayleigh (Racr) number (dependent on the system), the convection regime shifts.

HEAT TRANSFER IN ICE STREAM SHUTDOWN AND START (based on Joughin and Alley, 2011)

Stop and start of ice streams is dependent on:

• Thinning (steepens basal temperature gradients)
• Basal freezing (puts on the brakes for a system)
• Thickening (build up traps geothermal heat)
• Ice stream activation & the cycle is repeated

Watch the video below to walk through the steps of ice stream shut-down and speed-up, and see how this relates to the heat transfer equation introduced above.