2D companion to the 3D nanofluid geothermal heat exchanger: the same effective-medium + convection + NTU pipeline, read off four live panels instead of a rendered pipe.
Effective conductivity — Maxwell model:
k_nf / k_bf = [k_p + 2k_bf + 2φ(k_p − k_bf)] / [k_p + 2k_bf − φ(k_p − k_bf)]
Mixture density, heat capacity & viscosity:
ρ_nf = (1−φ)ρ_bf + φρ_p
cp_nf = [(1−φ)ρ_bf·cp_bf + φρ_p·cp_p] / ρ_nf
μ_nf = μ_bf · (1 + 2.5φ + 6.2φ²) (Brinkman-type correction)
Convective transfer — Dittus-Boelter correlation:
Re = ρ_nf·v·D / μ_nf, Pr = μ_nf·cp_nf / k_nf
Nu = 0.023 · Re^0.8 · Pr^0.4, h = Nu·k_nf / D
Heat exchanger effectiveness (NTU method) along a pipe of length L, area A = πDL:
T(x) = T_rock − (T_rock − T_in)·exp(−hA·(x/L) / (ṁ·cp_nf))
Q = ṁ·cp_nf·(T_out − T_in)
Brownian diffusivity of the suspended particles (Stokes-Einstein), which sets the microscopic nanoparticle jitter shown in the cross-section:
D_B = k_B·T / (3π·μ_nf·d_p)
- Cross-section (top-left) — a side-on strip of the pipe with fluid and nanoparticle dots flowing along it, coloured from cold (inlet) to hot (outlet), with visible Brownian jitter scaled to D_B.
- Temperature profile (top-right) — fluid temperature T(x) rising along the pipe from inlet to outlet per the NTU exponential above, with the live outlet point marked.
- Conductivity enhancement vs. loading (bottom-left) — k_nf/k_bf − 1 vs. volume fraction φ for the selected particle material, with the live loading point marked; steeper curves (Cu) show why particle choice matters as much as loading.
- Heat extraction vs. flow (bottom-right) — Q vs. flow velocity at the current loading and rock temperature, with the live flow point marked; the curve flattens as convection becomes limited by residence time rather than by h.
- Particle type — Al₂O₃, CuO and Cu have very different bulk conductivities k_p, so the same loading gives very different enhancement.
- Volume fraction φ — more nanoparticles raise k_nf and h, but also raise viscosity, which is why real nanofluids rarely exceed ~5-6% before pumping cost outweighs the gain.
Real-world relevance: this is the same effective-medium + Dittus-Boelter + NTU pipeline used in nanofluid geothermal-drilling literature to justify 15-40% gains in downhole heat extraction from enhanced coaxial and U-tube exchangers.