Radiation–Conduction Heat-Transfer Study for Mold Flux by Thermoviewer-Enhanced Infrared Emitter Technique

Wanlin Wang; Kaixuan Zhang; Haihui Zhang

doi:10.1007/s11663-019-01521-3

Metallurgical and Materials Transactions B

pp 1–15 | Cite as

Radiation–Conduction Heat-Transfer Study for Mold Flux by Thermoviewer-Enhanced Infrared Emitter Technique

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Wanlin Wang
Kaixuan Zhang
Haihui Zhang

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First Online: 30 January 2019

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Abstract

A thermoviewer-enhanced infrared emitter technique was developed in this article to study the heat-transfer behavior of mold flux in the continuous casting mold. Then, a radiation–conduction heat-transfer model was built to determine the radiative/conductive heat flux, thermal conductivity, and optical properties of slag. The results showed that the mold/slag interfacial thermal resistance decreases with the increasing temperature of slag surface at the mold side, and then reaches the minimum when the slag surface is the hottest, and finally increases slightly with further increase in slag crystallization. In addition, the radiative heat flux decreases with the increasing depth of radiation propagating into the slag. While the amount of heat of radiation lost is being converted to produce an additional radiative heat source to heat the slag, the increase in the conductive heat flux in the slag is realized. Furthermore, the apparent transmittivity is obtained as 0.63 for the glassy slag disk and as 0.41 for the partially crystallized slag disk. Besides, when mold heat flux decreases from the maximum to a saturation value due to the further slag crystallization, the corresponding apparent transmittivity of slag disk decreases from 0.58 to 0.41, and the ratio of the heat across the slag disk by radiation decreases from 67.3 to 50.0 pct.

Nomenclature

C: Specific heat (J/kg·K)
C₀: Stefan–Boltzmann constant, 5.670 × 10⁻⁸ W/m² K⁴
D: Thickness of slag disk (m)
E_n(x): Exponential integral function
G(τ): Incident radiation (W·m⁻²)
h_f: Convective heat-transfer coefficient (W/m² K)
I: Intensity of radiation (W/m²sr)
I_b: Blackbody intensity (Planck function) (W/m²sr)
l: Free path of phonons (m)
M_ea: Total number of measurement (–)
Mt: Point of maximum temperature in the slag (mm)
n₁: Refractive index of slag bottom surface (–)
n₂: Refractive index of slag top surface (–)
q: Heat flux (W m⁻²)
q_b: Bare copper heat flux (W m⁻²)
q_c: Conductive flux in the slag (W m⁻²)
q_lamp: Radiative flux of infrared lamp (W m⁻²)
q_m: Mold heat flux (W m⁻²)
Q_r: Radiative flux in the slag (W m⁻²)
R_int: Thermal resistance between mold and slag (m² K W⁻¹)
s: Geometric path length (m)
$ \hat{s} $: Unit vector into a given direction (–)
T: Temperature (K))
T₁: Temperature of slag bottom surface (K)
T₂: Slag top surface temperature (K)
T_f: Temperature of ambient air (K)
T_m: Temperature of mold surface (K)
V: Sonic velocity (m/s)
x, y, z: Cartesian coordinates (m)

Greek Symbols

β: Extinction coefficient (m⁻¹)
β_j: Extinction coefficient of j-th medium (m⁻¹)
ε₁: Emissivity of mold–slag interface (–)
ε₂: Emissivity of air–crystal interface (–)
ε_m: Emissivity of mold surface (–)
θ: Angle between the direction of the radiation intensity and the positive vertical direction (rad)
λ: Thermal conductivity (W/m K)
λ_r: Radiation conductivity (W/m K)
κ: Absorption coefficient (m⁻¹)
κ_j: Absorption coefficient of j-th medium (m⁻¹)
μ: Direction cosine (of polar angle), cosθ (–)
ρ: Density (kg/m³)
ρ₁: Reflectivity of mold–slag interface (–)
ρ₂: Reflectivity of air–crystal interface (–)
ρ_m: Reflectivity of mold surface (–)
σ_s: Scattering coefficient (m⁻¹)
σ_{s ,j}: Scattering coefficient of j-th medium (m⁻¹)
τ(z): Optical depth τ(z) = $\int\nolimits_{0}^{z} \beta dz$(–)
τ_s: Slag optical thickness (–)
ω: Single scattering albedo (–)
ω_j: Single scattering albedo of j-th medium (–)
ψ: Azimuthal angle (rad)
Ω: Solid angle (sr)

Subscripts

c: Conduction
j: Glass (g), crystal (c)
m: Copper mold
f: Atmosphere
r: Radiation

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Notes

Acknowledgments

The financial supports from the National Natural Science Foundation of China (51661130154, 51704333, and U1760202), Newton Advanced Fellowship (NA150320), and the Fundamental Research Funds for the Central Universities of Central South University (2018zzts437) are gratefully acknowledged.

Appendix

The parameter vector C_n is obtained by solving the following matrix equation:

$$ \left\{ {B_{mn} } \right\}\left\{ {C_{n} } \right\} = \left\{ {d_{m} } \right\} $$

(A1)

where B_mn, d_m, and K(x, x’) are obtained as follows:

$$ B_{mn} = \frac{{\tau_{s}^{m + n + 1} }}{m + n + 1} - \sum\limits_{k = 0}^{M} {D_{k} } \left\{ \frac{{R_{m,n + k} }}{2} + b\rho_{1} R_{m} R_{n + k} + a_{1} \rho_{2} (R_{m} R_{n + k}^{*} + R_{m}^{*} R_{n + k} ) + \rho_{2} R_{m}^{*} R_{n + k}^{*} \right\} $$

(A1a)

$$ \begin{aligned} \frac{{d_{m} }}{2\pi } = 2\int_{0}^{{\tau_{s} }} {\int_{0}^{{\tau_{s} }} {K(\tau ,\tau^{\prime})S(\tau^{\prime})\tau^{m} } } d\tau d\tau^{\prime} + A_{1} b(R_{m} + a_{2} R_{m}^{*} ) + A_{2} b(R_{m}^{*} + a_{1} R_{m} ) \hfill \\ \, + (1 - \rho_{1} )m!\left\{ {\int_{0}^{1} {f_{1} (\mu )} \mu^{m + 1} d\mu - \sum\limits_{j = 0}^{m} {\frac{{\tau_{s}^{m - j} }}{(m - j)!}\int_{0}^{1} {f_{1} (\mu )\mu^{j + 1} } } e^{{ - \tau_{s} /\mu }} d\mu } \right\} + \hfill \\ \, (1 - \rho_{2} )m!\left\{ {\sum\limits_{j = 1}^{m} {( - 1)^{j} \frac{{\tau_{s}^{m - j} }}{(m - j)!}\int_{0}^{1} {f_{2} (\mu )\mu^{j + 1} } } d\mu - ( - 1)^{m} \int_{0}^{1} {f_{2} (\mu )} \mu^{m + 1} e^{{ - \tau_{s} /\mu }} d\mu } \right\} \hfill \\ + 2\rho_{2} b(a_{1} R_{m} + R_{m}^{*} )(1 - \rho_{1} )\int_{0}^{1} {f_{1} (\mu )} e^{{ - \tau_{s} /\mu }} \mu d\mu \hfill \\ + 2\rho_{ 1} b(R_{m} + a_{ 2} R_{m}^{*} )(1 - \rho_{ 2} )\int_{0}^{1} {f_{ 2} (\mu )} e^{{ - \tau_{s} /\mu }} \mu d\mu \hfill \\ \end{aligned} $$

(A1b)

$$ \begin{aligned} K(\tau ,\tau^{\prime}) = \frac{1}{2}E_{1} (\left| {\tau - \tau^{\prime}} \right|) + \rho_{1} bE_{2} (\tau )[E_{2} (\tau^{\prime}) + a_{2} E_{2} (\tau_{s} - \tau^{\prime})] \hfill \\ \, + \rho_{2} bE_{2} (\tau_{s} - \tau )[E_{2} (\tau_{s} - \tau^{\prime}) + a_{1} E_{2} (\tau^{\prime})] \hfill \\ \end{aligned} $$

(A1c)

The parameters b, a₁, a₂, R_m,_n, R_n ,and R _n ^* are obtained as follows:

$$ b\left( { 1 - a_{ 1} a_{ 2} } \right) = 1,{\text{ with}}\;\;a_{i} = 2\rho_{i} E_{ 3} \left( {\tau_{s} } \right),i = 1, 2 $$

(A2)

$$ R_{m,n} = \int\limits_{0}^{{\tau_{s} }} {\int\limits_{0}^{{\tau_{s} }} {E_{1} (\left| {\tau - \tau^{\prime } } \right|)\tau^{m} \tau^{\prime n} d\tau d\tau^{\prime } } } $$

(A3)

$$ R_{n} = \int_{0}^{{\tau_{s} }} {E_{2} (\tau )\tau^{n} } d\tau $$

(A4)

$$ R_{n}^{*} = n!\left\{ {( - 1)^{n + 1} E_{n + 3} (\tau_{s} ) - \sum\limits_{j = 0}^{n} {\frac{{\tau_{s}^{n - j} }}{(n - j)!}\frac{{( - 1)^{j + 1} }}{j + 2}} } \right\} $$

(A5)

Knowing C_n, the parameters K₁ and K₂ could be obtained as follows:

$$ \begin{aligned} K_{1} = b\{ a_{2} (1 - \rho_{1} )\int_{0}^{1} {f_{1} (\mu )e^{{ - \tau_{s} /\mu }} \mu } d\mu + (1 - \rho_{2} )\int_{0}^{1} {f_{2} (\mu )e^{{ - \tau_{s} /\mu }} \mu d\mu + a_{2} E_{3} (\tau_{s} )A_{1} } \hfill \\ + E_{3} (\tau_{s} )A_{2} + \int_{0}^{{\tau_{s} }} {[E_{2} (\tau ) + a_{2} E_{2} (\tau_{s} - \tau )]S(\tau )d\tau } + \frac{1}{4\pi }\sum\limits_{k = 0}^{M} {D_{k} } \sum\limits_{n = 0}^{N} {C_{n} (R_{n + k} + a_{2} R_{n + k}^{*} )} \} \hfill \\ \end{aligned} $$

(A6)

$$ \begin{aligned} K_{2} = b\{ (1 - \rho_{1} )\int_{0}^{1} {f_{1} (\mu )e^{{ - \tau_{s} /\mu }} \mu } d\mu + a_{1} (1 - \rho_{2} )\int_{0}^{1} {f_{2} (\mu )e^{{ - \tau_{s} /\mu }} \mu d\mu + E_{3} (\tau_{s} )A_{1} } \hfill \\ \, + a_{1} E_{3} (\tau_{s} )A_{2} + \int_{0}^{{\tau_{s} }} {[a_{1} E_{2} (\tau ) + E_{2} (\tau_{s} - \tau )]S(\tau )d\tau } + \frac{1}{4\pi }\sum\limits_{k = 0}^{M} {D_{k} } \sum\limits_{n = o}^{N} {C_{n} (a_{1} R_{n + k} + R_{n + k}^{*} )} \} \hfill \\ \end{aligned} $$

(A7)

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Authors and Affiliations

Wanlin Wang
- 1
Kaixuan Zhang
- 1
Haihui Zhang
- 1
Email author

1.School of Metallurgy and EnvironmentCentral South UniversityChangshaChina

Radiation–Conduction Heat-Transfer Study for Mold Flux by Thermoviewer-Enhanced Infrared Emitter Technique

Abstract

Nomenclature

Greek Symbols

Subscripts

Notes

Acknowledgments

References

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