The heat diffusion equation is the backbone of conduction heat transfer analysis. It's derived from the conservation of energy principle and Fourier's law, describing how temperature changes in solids over time and space.

Solving the heat diffusion equation requires initial and boundary conditions. Analytical solutions work for simple cases, while numerical methods tackle complex problems. Understanding thermal diffusivity is key to grasping how quickly materials respond to temperature changes.

Heat diffusion equation derivation

Conservation of energy principle

• Apply the conservation of energy principle to a differential control volume in a solid medium
• The conservation of energy principle equates the rate of change of energy within a control volume to the net rate of energy transfer into the control volume
• Consider heat conduction, heat generation, and the rate of change of internal energy within the control volume when performing the energy balance

Fourier's law and partial differential equation

• Express the heat conduction terms in the energy balance equation using Fourier's law
• Fourier's law relates the heat flux to the temperature gradient through the thermal conductivity of the material
• The resulting partial differential equation, known as the heat diffusion equation, describes the temperature distribution in a solid as a function of space and time
• The heat diffusion equation is a second-order partial differential equation that governs transient heat conduction in solids

Transient conduction problem solutions

Initial and boundary conditions

• Specify the temperature distribution within the solid at the beginning of the transient heat transfer process (t = 0) using initial conditions
• Describe the thermal conditions at the surfaces of the solid using boundary conditions, such as prescribed temperature, prescribed heat flux, convection, or radiation
• Common boundary conditions include Dirichlet (specified temperature), Neumann (specified heat flux), and Robin (convection or radiation) conditions
• Combine the heat diffusion equation with the initial and boundary conditions to form a well-posed mathematical problem

Analytical and numerical solution methods

• Solve the heat diffusion equation with the appropriate initial and boundary conditions using analytical or numerical methods
• Analytical solutions are available for simple geometries and boundary conditions, such as the semi-infinite solid with a constant surface temperature or heat flux
• Employ numerical methods, such as the finite difference or finite element methods, for complex geometries and boundary conditions
• Numerical methods discretize the spatial and temporal domains and approximate the partial derivatives in the heat diffusion equation

Thermal diffusivity and its impact

Definition and influence on transient heat transfer

• Thermal diffusivity (α) is a material property that characterizes the rate at which heat diffuses through a substance
• Define thermal diffusivity as the ratio of thermal conductivity (k) to the product of density (ρ) and specific heat capacity (c): α = k / (ρ × c)
• Materials with high thermal diffusivity (metals) respond quickly to changes in thermal conditions, while materials with low thermal diffusivity (insulators) respond more slowly
• The thermal diffusivity appears in the heat diffusion equation and influences the rate of temperature change and the penetration depth of thermal disturbances

Experimental determination of thermal diffusivity

• Determine thermal diffusivity experimentally using transient heat transfer methods
• The laser flash method measures the temperature response of a thin sample subjected to a short laser pulse to determine thermal diffusivity
• The hot-wire method uses a thin wire embedded in a material to measure the temperature response and calculate thermal diffusivity
• Analyze the temperature-time data from these experiments to extract the thermal diffusivity value for the material

Temperature distribution in time-dependent conditions

Time-dependent boundary conditions

• Analyze the temperature distribution in solids subjected to time-dependent boundary conditions, such as periodic or transient heat fluxes or surface temperatures
• Solve the heat diffusion equation with the appropriate time-dependent boundary conditions to determine the temperature distribution
• Examples of time-dependent boundary conditions include sinusoidal surface temperature variations or pulsed laser heating
• Time-dependent boundary conditions lead to complex temperature distributions within solids that vary with both space and time

Analysis of temperature distribution characteristics

• Analyze the temperature distribution in the solid to determine key characteristics of the transient heat transfer process
• Determine the penetration depth of thermal disturbances, which represents how far the temperature change propagates into the solid
• Evaluate the time lag between the surface and interior temperatures, which indicates the delay in the temperature response at different locations within the solid
• Calculate the amplitude of temperature oscillations, which measures the magnitude of temperature variations at different points in the solid
• These characteristics provide insights into the transient thermal behavior of the solid and can guide the design of heat transfer systems