a_so

Diffusivity of the ground (m²/s)

T_e

Inlet temperature of the fluid in the inner tube (K)

a_f

Diffusivity of the fluid (m²/s)

T_m

Mean temperature of the fluid in annular

space at z and ( K)

D_H

Hydraulic diameter,

D_H = 2(R_i − r_ex). (m)

T_u

Temperature of the exchanger wall at z and t (K)

G

Green’s function

T₀

Initial temperature field in the soil t = 0 (K)

h

Local heat transfer coefficient at the wall (Wm⁻²/K)

u

Radial velocity (m/s)

k_f

Thermal conductivity of the fluid. (Wm⁻¹/K)

v

Orthoradial velocity (m/s)

k_s

Thermal conductivity of the ground (Wm⁻¹/K)

w

Axial velocity in annular space (m/s)

L

Length of the exchanger. (m)

u₀

Average velocity of fluid in the annularspace (m/s)

$\overrightarrow{r}$

Vector-position in the space

z

Axial coordinate (m)

${\overrightarrow{r}}^{\prime }$

Vector-position in the space where the temperature is evaluated

Greek

Symbols

R_e

Reynolds number, R_e = W₀D_Hρ_f/μ, dimensionless

δ

Time step (s)

r_ex

Radius of the central tube. (m)

Φ

Heat flux density (W/m²)

R_i

Radius of the exchanger (m)

ν

f cinematic viscosity of the fluid (m²/s)

t

Time (s)

ρ_f

Fluid density (kg/m³)