The equation of heat propagation in it is:

1 a s 0 ∂ T ( r → , t ) ∂ T − Δ T ( r → , t ) = 0 (1)

The Green function G ( r → , r → ′ , t ) , the solution of the associated Equation (2):

1 a s o ∂ G ( r → , r → ′ , t ) ∂ T − Δ G ( r → , r → ′ , t ) = δ ( r → − r → ′ ) (2)

G ( r → , r → ′ , t ) represents, physically, the response of the medium to a heat impulse on the point r → ′ at the instant t = 0. One supposes it is defined in the domain (D), containing the points r → and r → ′ , limited by surface (S). G ( r → , r → ′ , t ) is such that it is null for t< 0 and tends towards δ ( r → − r → ′ ) at t = 0. If we apply the Laplace transform to the Equations (1) and (2) and we combine them and then we integrate on r → in the domain (D) limited by surface (S) we obtain:

T ¯ ( r → ′ , p ) = ∭ ( D ) G ¯ ( r → , r → ′ , p ) T 0 ( r → ) a s o d 3 r     + ∬ ( S t ) [ G ¯ ( r → , r → ′ , p ) ∇ → T ( r → , p ) − T ( r → , p ) ∇ → G ¯ ( r → , r → ′ , p ) ] n → d s     + ∬ ( S e ) [ G ¯ ( r → , r → ′ , p ) ∇ → T ( r → , p ) − T ( r → , p ) ∇ → G ¯ ( r → , r → ′ , p ) ] n → d s (3)

We obtain the expression for the temperature of the heat exchanger surface (tube wall):

T u ( r → ′ , t ) = 2 F ( r → ′ , t ) + 2 ∫ 0 t ∬ ( S e ) [ G ( r → , r → ′ , t − τ ) ∇ → T ( r → , τ )     − T ( r → , τ ) ∇ → G ( r → , r → ′ , t − τ ) ] n → d s d τ (4)

where:

F ( r → ′ , t ) = ∭ ( D ) G ( r → , r → ′ , t ) T 0 ( r → ) a s o d 3 r − ∫ 0 t ∬ ( S t ) T ( r → , τ ) ∇ → G ( r → , r → ′ , t − τ ) n → d s d τ (5)

We consider on (St) the case where the surface temperature is imposed and equal to T a ( r → , t ) : This boundary condition imposes to G ( r → , r → ′ , t ) to be null on the level of the ground surface (St). We evoke the particular case where the temperature imposed on (St) is equal to the reference temperature T₀ supposed as uniform. Consequently, F ( r → ′ , t ) is null.

T u ( r → , t ) = T f ( r → , t ) P a r o i (6)

The quantity of heat yielded by the calorific fluid is completely received by the ground. The density of heatflux, at the wall of the exchanger, is then continuous.

− k f [ ∂ T f ( r → , t ) ∂ n ] p a r o i = − k s [ ∂ T ( r → , t ) ∂ n ] p a r o i (7)

If we replace the intervening terms in the Relations (6) and (7) in the expression of the temperature at the wall of the Relation (4), we obtain:

T u ( r → ′ , t ) = 2 ∫ 0 t ∬ ( S e ) [ k f k s G ( r → , r → ′ , t − τ ) ∇ → T f ( r → , τ ) ∇ → G ( r → , r → ′ , t − τ ) ] n → d s d τ (8)

The fluid density ρ_f is supposed constant. The axial symmetry of the system imposes that the flow is plane (v = 0) and two-dimensional (r, x). The equations governing the fluid flow can be written as follows:

{ ∂ u ∂ x + u r + ∂ w ∂ r = 0 ∂ P ∂ x = 0 u ∂ w ∂ x + w ∂ w ∂ r = − 1 ρ f ∂ P ∂ r + ( ϑ f + ϑ t f ) [ ∂ 2 w ∂ r 2 + 1 r ∂ w ∂ r ] ∂ T f ∂ t + u ∂ T f ∂ r + w ∂ T f ∂ x = ( a f + a t f ) [ ∂ 2 T f ∂ r 2 + 1 r ∂ T f ∂ r ] (9)

No simplifying condition is imposed on the velocity or the temperature on the base level (annular space inlet). These boundary conditions are general and close to the practical. Uniform velocity and temperature are imposed at the inlet of the interior tube. Evidently, the regime is supposed established at the exit of the annular space because of the importance of the length of buried exchangers. Thus, the boundary conditions will be for the velocity field.

{ u ( r , x , 0 ) = w ( r , x , 0 ) = 0 u ( R i , x , t ) = u ( r e x , x , t ) = 0 w ( R i , x , t ) = w ( r e x , x , t ) = 0 u ( r , 0 , t ) = U 0 w ( r , 0 , t ) = 0 (10)

For the temperature field:

{ T ( r , x , 0 ) = T 0 T ( r , 0 , t ) = T e ( ∂ T ∂ r ) r = r e x = 0 T ( R i , x , t ) = T r ( R i , x , t ) = 2 ∫ 0 t ∬ ( S e ) [ k f k s G ( t − τ ) ∇ → T ( τ ) ∇ → G ( t − τ ) ] n → d s d τ (11)

If we consider a cross section of the exchanger (Figure 1), the mean temperature T m ( r , x , t ) is the mean of various local temperatures T ( r , x , t ) on this section.

T m ( x , t ) = 2 U 0 ( R i 2 − r e x 2 ) ∫ r e x R i T ( r , x , t ) U ( r , x ) r d r (12)

The expression for h(x, t) is obtained from the expression of the heat flux density:

Φ → = − k f ∇ → T f = − k f ( ∂ T f ( r → , t ) ∂ r ) P a r o i n → = h ( T f − T u ) P a r o i n → (13)

From Equations (12) and (13), the equation h(x, t) (14) is determined as follows:

h ( x , t ) = ( k f ∂ T ( r , x , t ) ∂ r ) r = R i T r − T m (14)

We benefited from the decoupling between the dynamical problem and the thermal problem: the dynamical equations were solved in the first step. However, the discretized energy equations in the fluid zone and on the exchanger surface were solved in a second step.

The equations governing the velocity field were solved using the finite volume method by applying the staggered grid technique. We use a non-uniform and close-meshed grid near the inner and outer walls to account for the significant variations in velocity and temperature in the boundary layer region.

The grid used is of (30 × 120). The coupling of pressure and velocity is based on the algorithm Semi-Implicit Method for Pressure Linked Revised (S. I. M. P. L. E. R).

The equations are integrated in the control volume and in the time interval [t, t + d]. We use an implicit diagram for the time discretization, which has the advantage of being unconditionally stable. The convective terms are discretized using an updraft diagram to ensure the stability of the numerical model.

For the discretization of the energy equation on the exchanger surface, the finite element method is used at the boundary. It consists in decomposing the exchanger surface into finite annular elements of height D. The base alone is considered as one element. Knowing the solution T(pd) at a time pd, we determine the solution T((p + 1)d) at the time (p + 1)d. Thus, stepwise from p = 0 to p = n, (t = nd) we obtain the solution. The discretization scheme used is an extension of the scheme used by Desmons [2]. The stability of this scheme has been studied by the same author. The iterative method used for the resolution is the line-by-line Gauss-Seidel method along the radial axis.

The coefficient h is independent of T_e (for a weak temperature variation where ρ_f remains constant; which is confirmed by the correlations based on experimental data presented in the literature. In the same way, we verified numerically that this transfer coefficient is independent of the heating duration. It is constant and maintains the same value during the transient regime.

We determined, numerically, the axial evolution of the transfer local coefficient h(x, Re) for R_i/r_ex = 2, L/D_H = 50 and various values of the Reynolds number corresponding to laminar regime (100 ≤ Re ≤ 2500) (Figure 2). We notice that:

For all Re, the curve h(x, Re) tends asymptotically towards the value 2.95 W/m²K (Figure 2), which is close to the value 2.86 W/m²K (corresponding to the Nusselt number 4.36 often quoted in most textbooks). This value is obtained for a conductivity k_f ≫ 0.0262 W/mK suitable with the air at the temperature near 27˚C (0.5 ≤ (T_m − T₀)/T₀ ≤ 2.5), T₀ being equal to the ambient temperature T_a = 19˚C. The value 2.86 W/m²K corresponds to an established flow at constant flow at the wall.

This method of analytical calculation was developed in particular by Naili [9] who developed an analytical approach that evaluated the effect of operating parameters on the thermal power of an underground and forced convection heat exchanger, the results showed good agreement between the analytical model results and the experi-mental results. On the basis of this calculation method, Boughanmi [10] found an acceptable concordance of about 6% between the values of the measured and the analytically calculated heat exchanger output temperatures.

On the other hand, this analytical method was developed by Padet [11] regardless of the geometry of the exchanger, the mode of heat transfer, the flow and boundary conditions such as temperature and flow to the wall.

It is now necessary to make a number of assumptions in order to establish a simple analytical study that describes the physics of this problem, so the following assumptions are adopted:

· The heat flow is in steady state.

· The fluid is Newtonian and incompressible.

· The flow generated in the channel is laminar.

· The physical properties of the fluid are constant.

The local heat flux is exchanged through dS the small exchange surface and dx the length element of the cylinder. Is given by the following equation:

{ d Φ = U ( T 0 − T s ) d S d Φ = ρ C p V d T o u t = ρ C p V ( T o u t ( x + d x ) − T o u t ( x ) ) (15)

suppose that there is no phase change

d Φ = ρ C p V d T s (16)

Both Equations (15) and (16) give:

d Φ = U ( T 0 − T s ) d S = ρ C p V d T s (17)

d T s T 0 − T s = 2 U ρ C p V d S (18)

− d T s T s − T 0 = 2 U ρ C p V d S (19)

The Equation (19) is integrated from T_e to T_s for the air, we obtain:

∫ T e T s d T s T s − T 0 = − 2 U ρ C p V ∫ 0 x d S (20)

ln [ T s − T 0 ] T e T s = − 2 U ρ C p V d S (21)

ln T s − T 0 T e − T 0 = − 2 U ρ C p V d S (22)

T s − T 0 T e − T 0 = e − 2 U ρ C p V d S (23)

Hence the output exit temperature of the heat exchanger will be:

T s = ( T e − T 0 ) e − 2 h ρ C p V S + T 0 (24)

h is the convective heat transfer coefficient given by:

h = λ D H N u (25)

The main correlations in forced convection inside cylindrical ducts in laminar flow regime gave the temperature profiles theoretically for different boundary conditions. The classical hypothesis postulated at the wall is:

- Constant wall temperature;

- Constant heat flow.

For these two hypotheses, the theoretical studies, due to the origin of Graetz, show that, for very long tubes, the Nusselt criterion tends towards a constant value taking into account the following adimensional numbers:

D_H = 2(R_i − r_ex): the hydraulic diameter of the exchanger.

R_e = U₀D_Hρ/μ, or U₀ is the average speed in the annular pipe:

U 0 = ( r e x ) 2 ( R i ) 2 – ( r e x ) 2 V e

• N_u = 3.66 for a cylindrical pipe at a constant temperature at the wall;

• N_u = 4.36 for a cylindrical pipe with a constant flow at the wall.

In practice, it is assumed that these values are reached when the regime is established.

With G_z denotes the Graetz critere:

G z = R e P r D H / L

The average absolute error is 0.02 K. The results are therefore very close.

The values developed in Table 1 in Section 4-2, of the transfer coefficient as a function of Reynolds numbers are shown in Figure 3. The correlations developed by Kim [7], Shankar [3], and Sieder, Tale [4] and the FVM-BEM result develop in part 1 are represented in the same figure. There is good agreement

between the various results.

Boundary conditions:

The different applied boundary conditions for this simulation are given in the following table (Table 2).

The numerical solution consists in fixing the temperature at the inlet of the heat exchanger, varying the velocity, which is a function of the Reynolds number, and then following the evolution of the temperature at the outlet of the heat exchanger. According to Figure 4, the analytical variation follows the same trend

as the numerically solved one with an acceptable agreement of the order of 0.02%.

The inlet temperature of the heat exchanger (T_e) reaches a maximum value of 323 K. While the outlet temperature of the heat exchanger (T_s) reaches a value of 293 K, there is a gain in the form of a temperature difference of the order of 35 K, reflecting the importance of heat exchange. The validation of the analytical model is limited to the energy efficiency of the heat exchanger with the cylindrical shape and the spherical end with a length of 2 m at a temperature imposed on the wall.