2.2 Steady-state conduction in various geometries

Steady-state conduction is a key concept in heat transfer, focusing on how heat moves through different shapes when temperatures don't change over time. This topic dives into the math behind heat flow in walls, pipes, and spheres, helping us understand how to keep things hot or cold.

We'll explore how heat moves through single and multi-layered materials, and even look at cases where heat is generated inside an object. This knowledge is crucial for designing everything from building insulation to nuclear reactors.

Steady-State Conduction in Different Systems

Heat Conduction Equation and Coordinate Systems

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• The general heat conduction equation is derived from the conservation of energy principle and Fourier's law of heat conduction and describes the temperature distribution in a medium as a function of space and time
• The steady-state heat conduction equation is a simplified form of the general heat conduction equation where the temperature does not change with time $∂T/∂t = 0$
• The Laplacian operator $∇²$ in the steady-state heat conduction equation represents the second-order partial derivatives of temperature with respect to the spatial coordinates
• The steady-state heat conduction equation takes different forms in various coordinate systems:
  • Cartesian (rectangular) coordinates: $∂²T/∂x² + ∂²T/∂y² + ∂²T/∂z² = 0$
  • Cylindrical coordinates: $(1/r)∂/∂r(r∂T/∂r) + (1/r²)∂²T/∂φ² + ∂²T/∂z² = 0$
  • Spherical coordinates: $(1/r²)∂/∂r(r²∂T/∂r) + (1/r²sinθ)∂/∂θ(sinθ∂T/∂θ) + (1/r²sin²θ)∂²T/∂φ² = 0$

Boundary Conditions

• Boundary conditions, such as specified temperature, heat flux, convection, or radiation, are required to solve the steady-state heat conduction equation uniquely
• Examples of boundary conditions:
  • Specified temperature (Dirichlet condition): $T(x,y,z) = T_0$ on a boundary surface
  • Specified heat flux (Neumann condition): $-k(∂T/∂n) = q_0$ on a boundary surface, where $n$ is the normal direction to the surface
  • Convection: $-k(∂T/∂n) = h(T_s - T_∞)$ on a boundary surface, where $h$ is the convective heat transfer coefficient, $T_s$ is the surface temperature, and $T_∞$ is the ambient temperature
  • Radiation: $-k(∂T/∂n) = εσ(T_s^4 - T_sur^4)$ on a boundary surface, where $ε$ is the emissivity, $σ$ is the Stefan-Boltzmann constant, and $T_sur$ is the surrounding temperature

Temperature Distributions and Rates

One-Dimensional Steady-State Heat Conduction

• One-dimensional steady-state heat conduction occurs when the temperature varies only along one spatial coordinate, such as in plane walls, cylindrical shells, or spherical shells
• For a plane wall with constant thermal conductivity and no heat generation, the steady-state heat conduction equation reduces to $d²T/dx² = 0$, which yields a linear temperature distribution
• The heat transfer rate through a plane wall is given by Fourier's law: $q = -kA(dT/dx)$, where $k$ is the thermal conductivity, $A$ is the cross-sectional area, and $dT/dx$ is the temperature gradient
• For a cylindrical shell with constant thermal conductivity and no heat generation, the steady-state heat conduction equation in radial coordinates is $(1/r)d/dr(rdT/dr) = 0$, which results in a logarithmic temperature distribution
• The heat transfer rate through a cylindrical shell is given by $q = -2πkL(ΔT/ln(r₂/r₁))$, where $L$ is the length of the cylinder, $ΔT$ is the temperature difference, and $r₁$ and $r₂$ are the inner and outer radii, respectively
• For a spherical shell with constant thermal conductivity and no heat generation, the steady-state heat conduction equation in radial coordinates is $(1/r²)d/dr(r²dT/dr) = 0$, which leads to an inverse temperature distribution
• The heat transfer rate through a spherical shell is given by $q = -4πk(ΔT/(1/r₁ - 1/r₂))$, where $ΔT$ is the temperature difference, and $r₁$ and $r₂$ are the inner and outer radii, respectively

Examples of One-Dimensional Steady-State Heat Conduction

• Insulated pipe: A cylindrical pipe with insulation has a logarithmic temperature distribution across the insulation layer, and the heat transfer rate depends on the thermal conductivity and thickness of the insulation
• Spherical storage tank: A spherical storage tank containing a hot fluid has an inverse temperature distribution across the tank wall, and the heat transfer rate depends on the thermal conductivity and thickness of the wall material
• Plane wall with constant surface temperatures: A plane wall with constant surface temperatures on both sides has a linear temperature distribution, and the heat transfer rate depends on the thermal conductivity, cross-sectional area, and thickness of the wall

Heat Transfer in Composite Geometries

Composite Systems in Series

• Composite systems consist of multiple layers with different thermal conductivities in series or parallel arrangements
• For composite plane walls in series, the heat transfer rate is constant through each layer at steady state, and the temperature distribution is piecewise linear
• The overall heat transfer rate is determined by the sum of the thermal resistances of each layer: $q = ΔT/(ΣR)$, where $R = L/(kA)$ for each layer
• For composite cylindrical and spherical shells in series, the heat transfer rate is constant through each layer at steady state, and the temperature distribution is piecewise logarithmic or inverse, respectively
• The overall heat transfer rate is determined by the sum of the thermal resistances of each layer, with $R = ln(r₂/r₁)/(2πkL)$ for cylindrical shells and $R = (1/r₁ - 1/r₂)/(4πk)$ for spherical shells

Composite Systems in Parallel

• For composite systems in parallel, the heat transfer rate is the sum of the heat transfer rates through each parallel path, and the temperature difference across each path is the same
• The overall thermal resistance of parallel paths is determined by the reciprocal of the sum of the reciprocals of the individual resistances: $1/R_total = 1/R_1 + 1/R_2 + ...$
• Examples of composite systems:
  • Multilayer insulation: Composite plane walls with alternating layers of insulation and reflective materials to reduce heat transfer
  • Composite cylindrical pipes: Pipes with multiple layers of different materials (metal, insulation, protective coatings) to optimize heat transfer and durability
  • Composite spherical vessels: Pressure vessels with multiple layers of different materials (metal, composite, insulation) to withstand high pressures and temperatures

Steady-State Conduction with Generation and Variability

Heat Generation

• Heat generation within a medium can occur due to chemical reactions, nuclear reactions, or electrical resistance and is represented by a volumetric heat generation term $q̇$ in the heat conduction equation
• For steady-state conduction with uniform heat generation in a plane wall, the heat conduction equation becomes $d²T/dx² + q̇/k = 0$, which results in a quadratic temperature distribution
• The maximum temperature occurs at the center of the plane wall, and the heat transfer rate at the surfaces is given by $q = ±kA(dT/dx) ± q̇AL/2$, where the signs depend on the direction of heat flow
• For steady-state conduction with uniform heat generation in a cylindrical or spherical shell, the heat conduction equation includes the heat generation term, and the temperature distribution is obtained by solving the modified equation

Variable Thermal Conductivity

• Variable thermal conductivity can be a function of temperature, position, or both, and in such cases, the heat conduction equation becomes nonlinear, and analytical solutions may not be possible
• Approximate solutions for variable thermal conductivity problems can be obtained using methods such as the Kirchhoff transformation, which introduces a new variable to linearize the equation, or numerical methods like finite differences or finite elements
• Examples of heat generation and variable thermal conductivity:
  • Electric heating: Uniform heat generation in a plane wall due to electrical resistance heating (Joule heating)
  • Nuclear fuel rods: Uniform heat generation in cylindrical fuel rods due to nuclear fission reactions
  • Temperature-dependent thermal conductivity: Materials like gases and semiconductors have thermal conductivities that vary with temperature, leading to nonlinear heat conduction equations