District Heating Pipe Cross-Section Field

Network Parameters

Insulation thickness 80 mm

Pipe run length 800 m

Water flow rate 15 kg/s

Ground temperature 8 °C

Live Readouts

Delivery temperature—

Total heat loss—

Delivery efficiency—

Grid-solved R'—

Analytical R'—

cold

hot

How it works

This is a 2D-native counterpart to the 3D pipe fly-through: instead of rendering the pipe's surface colour from a pre-derived formula, it numerically solves steady-state heat conduction on a 2D grid — the actual physics that the closed-form log-resistance formula is an analytical approximation of.

∇·(k(x,y) ∇T) = 0   (steady 2D conduction, cross-section)
k = k_ins inside the insulation annulus, k_soil outside
T = T_water fixed on the pipe core, T = T_ground far away

The left panel shows the cross-section field (heatmap + isotherms) around the pipe at the point the amber scan-line has reached along the run; the right strip is the classic exponential decay of water temperature with distance, driven by the pipe's own thermal resistance:

T(x) = T_g + (T_s − T_g)·exp(−x / L_c)
L_c = ṁ·c_p·R'          (R' comes from the grid solve)

Because the solver is a genuine finite-volume Laplace solve (Gauss-Seidel with over-relaxation, harmonic-mean face conductivities, area-weighted sub-sampling at material boundaries), the heat flux leaving the pipe core and the flux crossing any ring further out agree to within numerical tolerance — a live conservation check, shown as "grid-solved R'" above matching the textbook analytical R' closely once insulation is a few grid cells thick.

The cross-section heatmap is a real field solve, not a projection of the 3D scene — thicker insulation visibly pushes the hot core further from the cooling isotherms.
The amber marker sweeps along the length chart to show where along the run the cross-section is currently sampled.

◀ plant (T_s) consumer (T_del) ▶

District Heating Pipe Cross-Section Field (2D)