Overview
This lecture focused on the heat diffusion equation for conduction, methods to simplify it under different conditions, types of boundary conditions, and a sample problem applying energy conservation and heat transfer concepts.
Heat Diffusion Equation & Its Simplification
- The heat diffusion equation governs conduction heat transfer in solids and accounts for heat generation, spatial variation, and temporal change.
- In Cartesian coordinates: α(∂²T/∂x² + ∂²T/∂y² + ∂²T/∂z²) + q̇/k = ∂T/∂t, where α is thermal diffusivity.
- For steady-state with no heat generation, the equation reduces to ∂²T/∂x² = 0.
- The equation also applies to cylindrical and spherical coordinates with adjustments for geometry.
Boundary and Initial Conditions
- Solving the equation requires boundary conditions (two each for x, y, z) and one initial condition for time-dependent problems.
- Four main types of boundary conditions:
- Known surface temperature (Dirichlet condition): T at surface is specified.
- Known surface heat flux (Neumann condition): q'' = -k ∂T/∂x at surface.
- Adiabatic or insulated boundary: ∂T/∂x = 0, as no heat crosses the surface.
- Convective/mixed boundary: -k ∂T/∂x at surface = h(T_s - T_∞)._
Newton’s Law of Cooling and Convective Boundaries
- Convective heat transfer is described by Newton’s law of cooling: q_conv = hA(T_s - T_∞).
- The convective heat transfer coefficient (h) depends on fluid velocity, properties, and surface geometry.
- h is often a function of Reynolds and Prandtl numbers._
Problem-Solving Approach
- Identify coordinate system and governing equation based on geometry and physics.
- Simplify the equation according to the scenario (steady/unsteady, with/without generation).
- Select appropriate boundary/initial conditions (Dirichlet, Neumann, Adiabatic, Mixed).
- Solve for temperature profile and heat flux; use conservation of energy for storage calculations.
Example Problem Recap
- Wall with internal heat generation, known polynomial temperature profile, and material properties provided.
- Tasks: Find heat entering/exiting at boundaries, rate of energy stored, and time rate of temperature change at specific locations.
- Solutions used the derivative of the temperature profile, Fourier’s law, and energy balance.
Key Terms & Definitions
- Heat Diffusion Equation — Governing PDE describing conduction with possible heat generation and time dependence.
- Thermal Diffusivity (α) — k/(ρc_p), measures rate of temperature spread.
- Boundary Condition — Constraint applied at the edge of the domain (e.g., specified T, heat flux, insulation).
- Newton’s Law of Cooling — q = hA(T_s - T_∞), empirical law for convective heat transfer.
- Convective Heat Transfer Coefficient (h) — Parameter quantifying convective heat transfer efficiency._
Action Items / Next Steps
- Practice solving the given example problem on boundary conditions and energy conservation.
- Review textbook sections on boundary/initial conditions for conduction problems.
- Prepare to discuss modifications of the heat diffusion equation for various coordinate systems in next class.