Question

Starting with an energy balance on a rectangular volume element, derive the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation.

Short answer

Based on the given step-by-step solution, provide a short answer explaining the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation. The one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation is derived from an energy balance considering the net heat rate into a volume element and the rate of change of energy storage within the element. It is given by: ρ * c_p * ∂T / ∂t = k * ∂²T / ∂x² where ρ is the density, c_p is the specific heat at constant pressure, k is the thermal conductivity, T is the temperature, x is the position, and t is the time.

Step-by-step solution

Define a volume element

Consider a rectangular volume element in the plane wall of thickness Δx, height h, and width w in the x-direction. Define T as the temperature of the material at position x and time t.

Perform an energy balance

Analyze the energy change within the volume element between its two faces at x and x + Δx due to heat conduction. Incoming heat flux at x is q_x and outgoing heat flux at x + Δx is q_(x+Δx). The net heat rate into the volume element, Δq_net, is the difference between the rates of incoming and outgoing heat fluxes: Δq_net = q_x - q_(x+Δx)

Apply Fourier's law of heat conduction

Apply Fourier's law of heat conduction to express the heat flux in terms of temperature gradient and thermal conductivity (k). q = -k * ∂T / ∂x Calculate the incoming and outgoing heat fluxes at x and x + Δx: q_x = -k * ∂T / ∂x |_(x) q_(x+Δx) = -k * ∂T / ∂x |_(x+Δx) Plug the heat fluxes into the net heat rate equation: Δq_net = -k * ∂T / ∂x |_(x) - (-k * ∂T / ∂x |_(x+Δx)) = -k * ( ∂T / ∂x |_(x+Δx) - ∂T / ∂x |_(x))

Relate net heat rate to energy storage

The net heat rate into the volume element equals the rate of change of energy storage within the volume element. The energy storage per unit volume in the element is given by ρ*c_p * ∂T / ∂t, where ρ is the density and c_p is the specific heat at constant pressure of the wall material. The rate of change of energy storage in the volume element = (ρ * c_p * ∂T / ∂t) * h * w * Δx

Combine energy balance with energy storage

Equate the rate of change of energy storage to the net heat rate into the volume element: (ρ * c_p * ∂T / ∂t) * h * w * Δx = -k * ( ∂T / ∂x |_(x+Δx) - ∂T / ∂x |_(x)) * h * w Divide both sides by h * w * Δx: ρ * c_p * ∂T / ∂t = -k * ( ∂T / ∂x |_(x+Δx) - ∂T / ∂x |_(x)) / Δx

Take the limit as Δx approaches 0

Take the limit of the above equation as Δx approaches 0 to obtain the one-dimensional transient heat conduction equation: ρ * c_p * ∂T / ∂t = k * ∂²T / ∂x² This is the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation.

Key concepts

Understanding Transient Heat Conduction

Transient heat conduction refers to the process of heat transfer in a material where the temperature changes with time. Unlike steady-state conduction, in transient conduction, the temperature at any given point inside a material is not constant over time.
This phenomenon occurs because the material reacts to a sudden change in thermal conditions, such as heating or cooling. Imagine a metal rod being heated at one end. The heat will move along the length of the rod and change its temperature distribution over time.
The main goal in studying transient heat conduction is to understand how the temperature evolves with time. This evolution is often described using differential equations that take into account various material properties like density and specific heat capacity.

Fourier's Law of Heat Conduction

Fourier's law is a fundamental principle describing heat conduction. It states that the rate of heat transfer through a material is proportional to the negative of the temperature gradient across the material. In simpler terms, heat flows from hot to cold regions, and the greater the temperature difference, the faster the flow.

• Mathematically, Fourier’s law is expressed as: \[ q = -k \frac{\partial T}{\partial x} \] where:
  • \( q \) is the heat flux, or rate of heat transfer per unit area.
  • \( k \) is the thermal conductivity of the material.
  • \( T \) is the temperature.
  • \( x \) is the direction of the heat flux.

High thermal conductivity materials like metals allow heat to travel quickly, whereas materials like wood have low thermal conductivity and are good insulators.

Energy Balance in Heat Conduction

The concept of energy balance is crucial in solving heat conduction problems. It involves calculating the net heat entering and leaving a volume element to determine how much heat is stored.
To conduct an energy balance, you assess the following:

• The incoming heat flux at one face of the volume element.
• The outgoing heat flux at the opposite face.
• The rate of change in energy stored within the element.

Using the principle of conservation of energy, any difference in incoming and outgoing heat must equal the rate of energy stored.
It's like filling a bathtub: the rate at which water enters minus the rate of drainage gives the rate at which water level rises. Similarly, in heat conduction, a net positive heat influx implies an increase in stored thermal energy.

Thermal Conductivity Explained

Thermal conductivity is a property of a material that indicates its ability to conduct heat. High thermal conductivity materials, like copper, rapidly spread heat through their molecules. In contrast, insulating materials like fiberglass have low thermal conductivity and resist heat flow.

• The unit of thermal conductivity in the SI system is watts per meter-kelvin (W/m·K).
• Thermal conductivity can depend on temperature, but in many problems, it is considered constant for simplicity.
• Understanding thermal conductivity helps in choosing materials for thermal insulation or conduction based on their application needs.

For example, when designing a building, materials with varying thermal conductivity can be chosen strategically to manage heat transfer, thus improving energy efficiency.