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Effects of Boundary Conditions on Thermal Response of a Cellulose Acetate Layer Using Hottel’s Zonal Method

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Abstract

Energy can transfer internally by radiation in addition to conduction in translucent polymers. Since radiant propagation is very rapid, it can provide energy within the layer more quickly than diffusion by heat conduction. Thus, the transient thermal response of a layer for combined radiative and conduction may be extremely different from that of conduction alone. In this paper, the behavior of a heat conducting, absorbing, and emitting layer of Cellulose Acetate layer is investigated during the transient interval when both conductive and radiative heat transfer are considered. Specifically, the effects of boundary conditions on the response of the layer are considered here. These boundary conditions include both conductive boundary conditions, such as convection coefficient and convective fluid temperature, and radiation boundary conditions, like radiation surrounding temperature and specular reflectivity. To this end, the governing differential equations including the equation of radiative heat transfer within the material coupled to the transient energy equation with radiative terms are presented. The solution procedure is based on nodal analysis and Hottel’s zonal method extended by the ray tracing method. The transient energy equation including the radiative internal energy source is solved using a time marching finite difference procedure with variable space and time increments.

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Abbreviations

A k,T_i :

Fractional spectral emissive power of spectral band k at nodal temperature T i

a :

Absorption coefficient of the layer (m−1)

C :

Unit heat capacity (Jm−3K−1)

D :

Thickness of the layer (m)

H 1, H 2 :

Dimensionless convective coefficient, H 1  = h 1 / σ T 30 , H 2  = h 2 / σ T 30

h 1, h 2 :

Convection heat transfer coefficient at surface S 1 and S 2, respectively (Wm−1K−1)

k ie , k iw :

Harmonic mean thermal conductivity at the interfaces ie and iw (Wm−1K−1)

n m,k , n rf,k :

Refractive index of the layer and reference, relative to the spectral band k(Δλ k ), respectively

N :

Conduction-radiation parameter, N = k / 4σ T 30 D

NB :

Total number of spectral bands

NM :

Total number of control volumes

q cd, q cv, q r :

Conductive, convective and radiative heat fluxes, respectively (Wm−2)

\(\tilde{q}\) :

Dimensionless flux, \(\tilde {q}=q / \sigma T_0^{4}\)

S 1, S 2 :

Boundary surfaces

S −∞,S +∞ :

Black surfaces representing the surroundings

(S i S j ) k :

Radiation transfer coefficient of surface vs. surface relative to the spectral band k(Δλ k )

(S i V j ) k :

Radiation transfer coefficient of surface vs. volume relative to the spectral band k(Δλ k )

(V i V j ) k :

Radiation transfer coefficient of volume vs. volume relative to the spectral band k(Δλ k )

t :

Physical time (s)

t*:

Dimensionless time, t *  = (4σ T 30 / ρ CD)t

T i :

Absolute temperature of control volume i(K)

T f1, T f2 :

Fluid temperature for convection at x = 0, and x =  D (K)

T 0 :

Uniform initial temperature of the layer (K)

T * :

Dimensionless temperature, T *  = T / T 0

T S :

Temperature of the surrounding (K)

V i :

Volume corresponding to node i

x :

Coordinate in direction across the layer (m)

X :

Dimensionless coordinate, X =  x/D

γ:

Transmissivity of surfaces

Δ t :

Time interval

ɛ1,k , ɛ2,k :

Emissivity of surface S 1 and S 2 relative to spectral band k(Δλ k ), respectively

μ:

Direction cosine for angle dependent property, cosθ

μ c :

cos θ c

θ:

Angle of reflection

θ c :

Critical angle of reflection

κ:

Extinction coefficient (m−1)

ρ s i,k :

Specular reflectivity component

λ:

Wavwlength

σ:

Stefan-Boltzmann constant (Wm−2K−4)

τ, τ0 :

Optical depth and optical thickness, repectively

Φ r i :

Radiative source term of node i

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Authors and Affiliations

  1. Department of Mechanical Engineering, KNT University of Technology, 10, Vezvayi Street, Motahari Ave, Tehran, Iran

    Babak Safavisohi, Ehsan Sharbati, Cyrus Aghanajafi & Seyed Reza Khatami Firoozabadi

Authors
  1. Babak Safavisohi
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  2. Ehsan Sharbati
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  3. Cyrus Aghanajafi
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  4. Seyed Reza Khatami Firoozabadi
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Correspondence to Ehsan Sharbati.

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Safavisohi, B., Sharbati, E., Aghanajafi, C. et al. Effects of Boundary Conditions on Thermal Response of a Cellulose Acetate Layer Using Hottel’s Zonal Method. J Fusion Energ 25, 145–153 (2006). https://doi.org/10.1007/s10894-006-9017-6

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  • DOI: https://doi.org/10.1007/s10894-006-9017-6

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