2.4 Multidimensional and Unsteady Conduction

Multidimensional and unsteady conduction extends the one-dimensional analysis you've already seen to problems where heat flows in two or three directions, or where temperatures change over time. These situations show up constantly in real engineering: think of a corner joint in a furnace wall (heat flows in two directions) or a turbine blade cooling down after shutdown (temperature changes with both position and time).

Steady-State Conduction Problems

Multidimensional Heat Conduction Equations

When heat flows in more than one direction but temperatures aren't changing with time, you have a multidimensional steady-state problem. The governing equation comes from applying energy conservation and Fourier's law in each spatial direction simultaneously.

For two-dimensional steady-state conduction in rectangular coordinates:

$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial y}\right) + \dot{q} = 0$

where $k$ is thermal conductivity, $T$ is temperature, and $\dot{q}$ is the volumetric heat generation rate.

For three-dimensional problems, a third term is added:

$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial y}\right) + \frac{\partial}{\partial z}\left(k\frac{\partial T}{\partial z}\right) + \dot{q} = 0$

If $k$ is constant (a common simplification), these reduce to Laplace's equation (no heat generation) or Poisson's equation (with heat generation). The same forms exist in cylindrical and spherical coordinates, though the derivative terms look different due to the coordinate geometry.

Solution Methods for Multidimensional Conduction Problems

Solving these equations requires both boundary conditions on every surface and a method suited to the problem's complexity.

Analytical methods work for simple geometries with clean boundary conditions:

• Separation of variables is the most common approach for rectangular domains with homogeneous boundary conditions on at least some sides.
• Laplace transforms and Green's functions handle certain other configurations.

Numerical methods are needed for complex or irregular geometries:

• Finite difference methods replace partial derivatives with algebraic approximations on a grid.
• Finite element methods divide the domain into small elements and approximate the solution within each one.

At interfaces between different materials, two continuity conditions must hold: the temperature must be the same on both sides, and the heat flux normal to the interface must be continuous. Violating either of these in your model will give non-physical results.

Boundary Conditions for Conduction

Types of Boundary Conditions

You need boundary conditions on every surface of your domain to get a unique solution. There are four standard types:

• Specified temperature (Dirichlet): The surface temperature is fixed at a known value. Example: a wall surface held at 100°C by condensing steam.
• Specified heat flux (Neumann): The heat flux at the surface is prescribed. A constant electrical heater applying 5000 W/m² to a surface is a typical case.
• Convection (Robin): The surface exchanges heat with a fluid. This condition involves both the fluid temperature $T_\infty$ and the convective coefficient $h$, expressed as $-k\frac{\partial T}{\partial n} = h(T_s - T_\infty)$ where $n$ is the direction normal to the surface.
• Insulation (Adiabatic): No heat crosses the boundary. Mathematically, this is a special case of the Neumann condition with zero heat flux: $\frac{\partial T}{\partial n} = 0$. Symmetry planes in a problem are also treated as adiabatic boundaries.

These conditions can be uniform (constant along the surface) or non-uniform (varying with position).

Application of Boundary Conditions

A single problem will typically have different boundary conditions on different surfaces. Choosing the right ones is what connects your math to the actual physics.

Consider a metal plate being cooled:

• The bottom surface sits on a heat source at a constant 200°C (Dirichlet condition).
• The top surface is exposed to air at 25°C with $h = 50$ W/m²·K (Robin condition).
• The two edges are insulated (adiabatic condition).

Getting these wrong, even if your solution method is perfect, means your answer won't represent the real system.

Shape Factors in Conduction

Concept and Definition of Shape Factors

Shape factors provide a shortcut for calculating steady-state heat transfer in multidimensional geometries without solving the full conduction equation. They capture all the geometric complexity in a single dimensionless number $S$.

The heat transfer rate is then:

$Q = Sk(T_1 - T_2)$

where $S$ is the shape factor (units of length), $k$ is thermal conductivity, and $T_1 - T_2$ is the temperature difference between two isothermal surfaces.

You can also define an equivalent thermal resistance from the shape factor:

$R_{cond} = \frac{1}{Sk}$

This lets you plug multidimensional conduction directly into a thermal resistance network, just like you do with one-dimensional problems.

Application of Shape Factors

Shape factors have been tabulated for many common configurations. A few examples:

When using shape factors, keep these points in mind:

• They assume constant thermal conductivity and steady-state conditions. If either assumption breaks down, you can't use them directly.
• For systems with multiple heat flow paths, combine shape factors using the resistance network analogy: resistances in series add, resistances in parallel add as reciprocals.
• Shape factors are especially useful for quick estimates in design work, such as calculating heat loss from a buried pipe or through a building corner.

Transient Heat Conduction in Multidimensional Systems

Transient Heat Conduction Equations

When temperatures change with time, the governing equation picks up a storage term on the left side:

$\rho c\frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial y}\right) + \frac{\partial}{\partial z}\left(k\frac{\partial T}{\partial z}\right) + \dot{q}$

where $\rho$ is density, $c$ is specific heat capacity, and $t$ is time. The left side represents the rate of energy storage per unit volume; the right side represents net conduction in plus generation.

Transient problems require:

• Boundary conditions on every surface (same four types as steady-state).
• An initial condition specifying the temperature distribution at $t = 0$ throughout the entire domain.

The Biot number tells you whether internal temperature gradients matter:

$Bi = \frac{hL_c}{k}$

where $h$ is the surface convective coefficient, $L_c$ is a characteristic length (typically volume/surface area), and $k$ is the solid's thermal conductivity. The Biot number compares the conduction resistance inside the body to the convection resistance at its surface.

Solution Methods for Transient Multidimensional Conduction

The approach you use depends heavily on the Biot number and the geometry.

Lumped system analysis ($Bi < 0.1$):

When $Bi < 0.1$, internal conduction is so fast relative to surface convection that the entire body is essentially at one uniform temperature at any instant. The temperature then decays exponentially:

$\frac{T(t) - T_\infty}{T_i - T_\infty} = \exp\left(-\frac{hA_s}{\rho c V}t\right)$

This reduces a multidimensional problem to a simple ODE. Always check the Biot number first; if it's below 0.1, you can skip the more complex methods.

Analytical methods (simple geometries, $Bi \geq 0.1$):

Separation of variables produces infinite series solutions for standard shapes (plane walls, long cylinders, spheres). For multidimensional bodies like a short cylinder (finite length), you can use the product solution: the solution for a short cylinder equals the product of the solutions for an infinite cylinder and an infinite plane wall, each solved independently. This is a powerful technique that avoids solving the full 2D or 3D equation directly.

Numerical methods (complex geometries or boundary conditions):

Finite difference, finite element, and finite volume methods discretize both space and time. Key considerations:

• Explicit schemes are simple to implement but require small time steps for stability (governed by the Fourier number criterion).
• Implicit schemes (like Crank-Nicolson) are unconditionally stable but require solving a system of equations at each time step.
• Grid refinement studies should be performed to verify that results are independent of the mesh and time step sizes.

A classic example: analyzing how quickly the center of a large concrete block reaches a target temperature after its surfaces are suddenly exposed to hot gas. The temperature at the center lags behind the surfaces, and the Biot number, geometry, and thermal diffusivity ($\alpha = k/(\rho c)$) together determine how long the process takes.