Transient Heat Conduction

Heat flows from hot to cold. In a solid it is carried by conduction, and Fourier's law says the heat flux is proportional to the temperature gradient:

An energy balance on a small piece of material (heat in − heat out + heat generated = heat stored) gives the heat equation. With constant properties:

The thermal diffusivity α sets how fast temperature changes spread: copper (1.2 × 10⁻⁴ m²/s) responds about 200 times faster than concrete (7 × 10⁻⁷ m²/s). A change at a surface reaches a depth of about √(αt) after a time t, so the time to heat through a wall grows with the square of its thickness.

• Fixed temperature (Dirichlet): T = T_s at the surface, e.g. a face in contact with boiling water or ice.
• Convection (Robin): the heat conducted to the surface leaves into a fluid at T_∞: −k ∂T/∂n = h(T − T_∞). A very large h acts like a fixed temperature.
• Insulated (Neumann): no heat crosses, ∂T/∂n = 0. It is also the condition on a plane of symmetry.
• Heat flux: a set heat flow per unit area enters, −k ∂T/∂n = q″ (solar radiation, an electric heater foil).

If no side exchanges heat with a temperature (all sides insulated or with a flux), there is no steady state: the body heats or cools without limit while its temperature profile settles into a fixed shape.

• Fourier number Fo = αt/L²: dimensionless time. The temperature at a given relative position depends only on Fo and Bi, whatever the material or size. Below Fo ≈ 0.05 a change at one face has not yet reached the other: the wall behaves as a semi-infinite solid, T = T_s + (T_i − T_s) erf(x/2√(αt)).
• Biot number Bi = hL/k: the resistance to conduction inside the body compared with the resistance to convection at its surface. With small Bi the inside stays nearly uniform; with large Bi the surface follows the fluid at once and the inside lags behind.
• Lumped capacitance: when Bi_c = hL_c/k < 0.1 with L_c = V/A, the body can be treated as one temperature: T − T_∞ = (T_i − T_∞) e^−t/τ with τ = ρcV/(hA). The error is then below about 5%.

For a wall, write the temperature as the steady state plus a transient, T = T_s(x) + θ(x, t). The steady state solves −kT_s″ = q̇ with the boundary conditions; the transient has homogeneous conditions and splits into products X(x)·Γ(t):

The boundary conditions allow only certain β_n, the roots of the eigenvalue equation, e.g. βL = nπ between two fixed temperatures, or ζ tan ζ = Bi for a wall cooled on both faces (ζ = βL/2). The coefficients c_n follow from the initial temperature by orthogonality. Each term decays with its own time constant 1/(αβ_n²); after Fo ≈ 0.2 (based on the half-thickness) only the first term matters: the one-term approximation behind the Heisler charts.

For a rectangular bar with one surrounding temperature on all sides and a uniform initial temperature, the solution is the product of two wall solutions: θ/θ_i = X(x, t)·Y(y, t). The step-by-step solution below works these out for the current setup.

In a long cylinder or a sphere heated evenly over its surface, heat flows along the radius. The shells grow with r (area 2πr per metre of a cylinder, 4πr² for a sphere), which adds a term to the heat equation:

• Steady state: without generation, T_s = C₂ + C₁ ln r in a cylinder and C₂ + C₁/r in a sphere. The wall of a pipe resists heat by ln(r_o/r_i)/(2πk) per metre, a spherical shell by (1/r_i − 1/r_o)/(4πk). With generation q̇ a solid body's centre is warmer than its surface by q̇r_o²/4k (cylinder) or q̇r_o²/6k (sphere).
• Transient: separating the variables gives Bessel's equation in a cylinder, with solutions J₀(βr) and Y₀(βr), and sin(βr)/r, cos(βr)/r in a sphere. A solid body keeps only the one that stays finite at the centre. With convection at the surface (ζ = βr_o, Bi = hr_o/k) the eigenvalues solve ζ J₁(ζ)/J₀(ζ) = Bi for a cylinder and 1 − ζ cot ζ = Bi for a sphere.
• Lumped capacitance uses L_c = V/A = r_o/2 for a cylinder and r_o/3 for a sphere.

With θ = (T − T_∞)/(T_i − T_∞), x* = x/L and Fo = αt/L², the heat equation loses every material constant, and the surface conditions keep only Bi = hL/k. So θ depends only on x*, Fo, and Bi: one chart (Heisler's) serves every size and material. L is the distance from a cooled surface to the centre: half the thickness of a wall cooled on both faces, the radius of a solid cylinder or sphere. Choose Graph axes: Dimensionless to plot θ against Fo, and Symbols in the step-by-step solution to see the general solution in these variables.

• Space: the body is divided into equal cells (finite volumes). Each cell exchanges heat with its neighbours by Fourier's law; boundary cells exchange heat with the side's fixed temperature, fluid, or flux through half a cell. In a cylinder or sphere the cells are shells of equal thickness, and neighbours exchange heat through the exact conductance of the shell between their centres. This is accurate to second order: halving the cell size cuts the error about four times.
• Time: exact. Because the material is uniform and each side has one condition, the cell equations split into independent modes along x and along y. The simulation finds them once (an eigenvalue problem) and advances every mode with its exact exponential. There is no time step, so no time-step error and no stability limit, at any playback speed.
• Playback: heat conduction takes from seconds (a thin copper sheet) to days (a thick concrete wall). The simulation therefore plays each setup so that 95% of its change happens in about 10 seconds at 1× speed; the time readout gives the real time.
• Checks: the exact series solution is drawn alongside the simulation where it exists (every wall, cylinder, and sphere; a bar when the product solution applies). The tests compare the two, and also the erf solution, the Laplace-series steady state of a plate, the lumped-capacitance solution, the Heisler coefficients, and the energy balance.

• Constant properties k, ρ, c (in reality they change with temperature).
• A plate is a long bar: no heat flows along its length (perpendicular to the screen). A wall is wide: heat flows only through its thickness. A cylinder is long and a sphere is heated evenly: heat flows only along the radius.
• Each side has one condition along its whole length; h and T∞ are constant in time.
• No radiation, phase change, or contact resistance; heat sources are uniform inside their regions.
• Temperatures are cell averages: sharp features smaller than a cell (a thin hot block, the very first moments after a sudden change) are smoothed.