BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

More documents

Recommendations

Info

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
Page 1:
BULETINUL INSTITUTULUI POLITEHNIC D
Page 5:
BULETINUL INSTITUTULUI POLITEHNIC D
Page 8 and 9:
PETRU CÂRLESCU, VASILE DOBRE, IOAN
Page 10 and 11:
2 Niooleta Negoescu continue telle
Page 12 and 13:
4 Niooleta Negoescu min{ μ( Tu , T
Page 14 and 15:
6 Niooleta Negoescu Observation 8.
Page 16 and 17:
8 Ion Crăciun with large molecules
Page 18 and 19:
10 Ion Crăciun ∂ ∂ ∂ ∂ ∇
Page 20 and 21:
12 Ion Crăciun ⎧σ ji = ( μ+ α
Page 22 and 23:
14 Ion Crăciun If we introduce the
Page 24 and 25:
16 Ion Crăciun where the scalar La
Page 26 and 27:
18 Ion Crăciun and 0 0 0 2 2 ⎧
Page 28 and 29: 20 Ion Crăciun Ψ r Having assumed
Page 30 and 31: 22 Ion Crăciun Ignaczak, J., Tenso
Page 33 and 34: BULETINUL INSTITUTULUI POLITEHNIC D
Page 35 and 36: Bul. Inst. Polit. Iaşi, t. LVIII (
Page 41: Bul. Inst. Polit. Iaşi, t. LVIII (
Page 44 and 45: 36 Radu Ibănescu and Cătălin Ung
Page 52 and 53: 44 Petronela Paraschiv case of biom
Page 54 and 55: 46 Petronela Paraschiv calculation
Page 56 and 57: 48 Petronela Paraschiv interpolate
Page 88 and 89: 80 Gelu Coman and Valeriu Damian ma
Page 90 and 91: 82 Gelu Coman and Valeriu Damian h
Page 92 and 93: 84 Gelu Coman and Valeriu Damian Th
Page 94 and 95: 86 Gelu Coman and Valeriu Damian iv
Page 96 and 97: 88 Florin Popa et al. In this figur
Page 98 and 99: 90 Florin Popa et al. the notations
Page 100 and 101: 92 Florin Popa et al. Δ = 0 and C
Page 114 and 115: 106 Vlad Marţian et al there is al
Page 116 and 117: 108 Vlad Marţian et al that all th
Page 118 and 119: 110 Vlad Marţian et al The battery
Page 120 and 121: 112 Vlad Marţian et al Current Int
Page 122 and 123: 114 Vlad Marţian et al Another fac
Page 124 and 125: 116 Vlad Marţian et al În concluz
Page 126 and 127: 118 Petru Cârlescu et al indicate
Page 128 and 129:
120 Petru Cârlescu et al hs p dT d
Page 130 and 131:
122 Petru Cârlescu et al air malt
Page 132 and 133:
124 Petru Cârlescu et al a b Fig.
Page 134:
126 Petru Cârlescu et al Iguaz A.,
show all

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?