BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

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70 Bogdan Horbaniuc et al Hypotheses: 1. The soil is homogenous and isotropic in terms of thermophysical properties. 2. The soil is dry (the influence of the humidity is negligible). 3. The soil thermophysical properties are constant. 4. The tube is isothermal longitudinally. 5. The heat propagation is one-dimensional (radial). In order to get the most general form of the results, dimensionless temperatures will be used, defined by t− t∞ θ = t − t HF ∞ . (1) The equations that describe the heat transfer phenomenon are: a) the heat conduction equation in cylindrical coordinates 2 ∂θ ⎛ i ∂ θi 1 ∂θ ⎞ = ai + i ∂τ 2 ; ⎜ ∂ r r ∂r ⎟ (2) ⎝ ⎠ where a is the thermal diffusivity and subscript i means W for the wall domain, respectively S for the soil (SSM) region. b) the heat balance equation at the wall-SSM interface λ W ⎛∂θW ⎞ ⎛∂θ ⎞ ⎜ ∂r ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ S = λS r= R ∂r W r= RW , (3) where λ is the thermal conductivity of the material. The initial condition is 0 [ ] τ < 0: θ= θ ∀r∈ R , R . (4) The boundary conditions are: i) on the inner tube surface (r = R 0 ) W = : = = 0. (5) r R0 θW,0 θ R ii) at the end of the SSM region (r = R) ⎛∂θS ⎞ r = R: ⎜ = 0, τ r ⎟ ⎝ ⎠ ∂ r= R ≥ 0. (6)
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 71 2.2. Numerical Approach The heat transfer equation (2), transposed in the finite difference form, becomes (Horbaniuc et al., 2003), (Horbaniuc & Dumitraşcu, 2006) 2 ∂ θ 1∂θ 1 ⎡⎛ h ⎞ ⎛ h ⎞ + ≅ 1 θ 2θ 1 θ 2 r r 2 ⎢⎜ − ⎟ − + ⎜ + ⎟ ∂ r ∂ h ⎢⎣⎝ 2rm ⎠ ⎝ 2r ⎠ m− 1 m m+1 m ⎤ ⎥, (7) ⎥⎦ where h is the grid step of the domain (wall, or SSM) and subscript m represents the current node number. After a series of manipulations, Eq (7) becomes p p p im , − 1 im , im , m im , + 1 im , − θ + σ θ − γ θ = β , (8) where superscript p accounts for the current time step (time step number p). The coefficients σ, γ, and β are σ i,m ⎡ Ri ⎤ 2⎢Ni + m⎥ ⎡ ( ) 2 ⎤ Δ h i ⎢ i ⎥ = ⎣ ⎦ 2 R ⎢ + i ai τ ⎥ , (9) ⎡ ⎤ Δ 2⎢Ni + m⎥−1⎢ ⎣ ⎥ Δ ⎦ ⎣ i ⎦ γ m ⎡ Ri ⎤ 2⎢Ni + m⎥ + 1 Δi = ⎣ ⎦ , (10) ⎡ Ri ⎤ 2⎢Ni + m⎥ −1 ⎣ Δi ⎦ β m ⎡ Ri ⎤ 2⎢Ni + m⎥ 2 Δi hi = ⎣ ⎦ ⎡ Ri ⎤ aiΔτ 2⎢Ni + m⎥−1 ⎣ Δi ⎦ , (11) where Δ represents the depth of region i which is W or S, h is the grid step (see also Fig. 1.), Δτ is the time step, N is the total number of nodes of a domain, and m is the current node for which the equation has been written. For a time step, all of the finite difference equations of type (8), along with the heat balance and the boundary conditions equations written in terms of finite differences, form a set that has to be simultaneously solved. Since the matrix of this set is tridiagonal (Croft & Lilley, 1977), (LeVeque, 2007), the set
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BULETINUL INSTITUTULUI POLITEHNIC D
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BULETINUL INSTITUTULUI POLITEHNIC D
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