Determination of the Heat Transfer Coefficient at the Metal-Mold Interface During Centrifugal Casting

Abstract

The heat transfer coefficient at the metal-mold interface (h _MM) has been determined for the first time during the centrifugal casting of a Fe-C alloy tube using the inverse solution method. To apply this method, a centrifugal casting experiment was carried out to measure cooling curves within the tube wall under a mold rotation speed of 900 rpm, imposing a centrifugal force 106 times as large as the gravity force (106 G). As part of the solution method, a comprehensive heat transfer model of the centrifugal casting was also developed and coupled to an optimization algorithm. Finally, the evolution of h _MM with time that gives the minimum squared error between measured and calculated cooling curves was obtained. The determined h _MM is approximately 870 W m⁻² K⁻¹ immediately after melt pouring, decreasing to about 50 W m⁻² K⁻¹ when the average temperature of the tube is ~973 K (700 °C), after the end of solidification. Despite the existence of a centrifugal force that could enhance the metal-mold contact, these values are lower than those generally reported for static molds with or without an insulating coating at the mold inner surface. The implemented model shows that the heat loss by radiation is dominant over that by convection at the tube inner surface, causing the formation of a solidification front that meets another front coming from the outer surface of the tube.

Appendix

The radiation component of the heat transfer coefficient, h _R, is required to calculate h _RC and the total heat flux at the inner surface of the tube, defined by Eq. [7]. To construct a radiation model, the internal surface that forms the complete tube cavity was subdivided into 5 surface patches (Figure 1b): (1) differential surface element on the inner surface of the tube in the middle of its length; (2) complete inner surface; (3) front lid; (4) backside of the mold; and (5) opening in the front lid. The boundary at which Eq. [7] is applied coincides with surface patch (1) and the net radiation heat flux leaving this patch is exactly the radiation part of the total flux calculated by Eq. [7].

Each of the 5 patches was assumed to have uniform radiosity and uniform temperature, which vary from patch to patch. Surface patch (5), the lid opening, was assumed black and patches (1) to (4), opaque, diffuse, and gray. Patches (3) and (4) are actually the surfaces of the coatings on the front lid and back of the mold, respectively, and they exchange heat by radiation and convection due to the air flow within the cavity. The heat conducted through the coatings is also conducted through the walls of the front lid and back of mold and, eventually, transferred to the environment (front lid) and to the cooling water (back of mold). These heat transfer steps are included in the model using thermal resistances. Considering these assumptions, the following system of equations can be written[23]

$$ \left( {F_{12} + \,F_{13} \, + F_{14} \, + F_{15} + \frac{{\varepsilon_{1} }}{{(1 - \varepsilon_{1} )}}} \right)\,J_{1} \, + F_{12} J_{2} \, + \,F_{13} J_{3} \, + F_{14} J_{4} \, = \,\frac{{\varepsilon_{1} \sigma }}{{(1 - \varepsilon_{1} )}}T_{\text{Ri}}^{4} \, - \,F_{15} \sigma T_{\text{a}}^{5} $$

(10)

$$ \left( {F_{23} + \,F_{24} \, + F_{25} \, + \frac{{\varepsilon_{2} }}{{(1 - \varepsilon_{2} )}}} \right)\,J_{2} \, + F_{23} J_{3} \, + \,F_{24} J_{4} \,\, = \,\frac{{\varepsilon_{2} \sigma }}{{(1 - \varepsilon_{2} )}}T_{\text{Ri}}^{4} \, - \,F_{25} \sigma T_{\text{a}}^{5} $$

(11)

$$ F_{32} \,J_{2} + \left( {\,F_{32} \, + F_{34} \, + \frac{{\varepsilon_{3} }}{{(1 - \varepsilon_{3} )}}} \right)\,J_{3} \, + \,F_{34} J_{4} \, - \,\frac{{\varepsilon_{3} \sigma T_{3}^{3} }}{{(1 - \varepsilon_{3} )}}\,T_{3} =0$$

(12)

$$ F_{42} \,J_{2} + F_{43} \,J_{3} \left( {\,F_{42} \, + F_{43} \, + F_{45}+ \frac{{\varepsilon_{4} }}{{(1 - \varepsilon_{4} )}}} \right)\,J_{4} \, - \,\frac{{\varepsilon_{4} \sigma T_{4}^{3} }}{{(1 - \varepsilon_{4} )}}\,T_{4}=\,{-}F_{45}\sigma T_{a}^{4} $$

(13)

$$ \begin{gathered} \frac{\varepsilon_{3} }{{\left( {1 - \varepsilon_{3} } \right)}}J_{3} - \left[ {\left( {\frac{{l_{{ 3 {\text{c}}}} }}{{\kappa_{{ 3 {\text{c}}}} }} + \frac{{l_{{ 3 {\text{m}}}} }}{{\kappa_{{ 3 {\text{m}}}} }} + \frac{1}{{h_{{ 3 {\text{ca}}}} }}} \right)^{ - 1} + h_{{ 3 {\text{c}}}} + \frac{{\varepsilon_{3} \sigma T_{3}^{3} }}{{\left( {1 - \varepsilon_{3} } \right)}}} \right]T_{3} = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - h_{{ 3 {\text{c}}}} T_{\text{c}} - \left( {\frac{{l_{{ 3 {\text{c}}}} }}{{\kappa_{{ 3 {\text{c}}}} }} + \frac{{l_{{ 3 {\text{m}}}} }}{{\kappa_{{ 3 {\text{m}}}} }} + \frac{1}{{h_{{ 3 {\text{ca}}}} }}} \right)^{ - 1} T_{\text{a}} \hfill \\ \end{gathered} $$

(14)

$$ \begin{gathered} \frac{{\varepsilon_{4} }}{{\left( {1 - \varepsilon_{4} } \right)}}J_{4} - \left[ {\left( {\frac{{l_{{ 4 {\text{c}}}} }}{{\kappa_{{ 4 {\text{c}}}} }} + \frac{{l_{{ 4 {\text{m}}}} }}{{\kappa_{{ 4 {\text{m}}}} }} + \frac{1}{{h_{{ 4 {\text{ca}}}} }}} \right)^{ - 1} + h_{{ 4 {\text{c}}}} + \frac{{\varepsilon_{4} \sigma T_{4}^{3} }}{{\left( {1 - \varepsilon_{4} } \right)}}} \right]T_{4} = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - h_{{4{\text{c}}}} T_{\text{c}} - \left( {\frac{{l_{{ 4 {\text{c}}}} }}{{\kappa_{{ 4 {\text{c}}}} }} + \frac{{l_{{ 4 {\text{m}}}} }}{{\kappa_{{ 4 {\text{m}}}} }} + \frac{1}{{h_{{ 4 {\text{ca}}}} }}} \right)^{ - 1} T_{\text{a}} \hfill \\ \end{gathered} $$

(15)

The view factor between surface patches j and i, denoted as $F_{ji}$, is calculated as explained by Modest.[24] Note that the temperature of surface patch (5) is the environment temperature, T _a, and the temperature of patches (1) and (2) are equal to T _Ri, obtained as part of the solution of Eq. [1], i.e., T at r = R _i.

The convective heat transfer coefficients $h_{{ 3 {\text{c}}}}$, $h_{{ 4 {\text{c}}}}$, and $h_{{ 3 {\text{ca}}}}$ were calculated using the following relation developed for the heat exchanged between a rotating disk and the surrounding fluid[37]

$$ Nu_{\text{c}} = 0.33\;Re_{\text{c}}^{0.5} \quad{\text{for}}\;Re_{\text{c}} < 2.5 \times 10^{5} \; \left( {\text{laminar flow}} \right) $$

(16)

The heat transfer between the external surface of the mold back and the cooling water of the water jacket was approximated by the heat transfer between a static surface and an impinging jet,[23] enabling the calculation of $h_{{ 4 {\text{ca}}}}$. In this calculation, the diameter of the water inlet was 5 cm, the distance between inlet exit and backside of mold was 10 cm, and the average velocity at the inlet was 3 ms⁻¹.

The system of Eqs. [10] through [15] was solved by the iterative Gauss–Seidel method for six unknowns, namely $J_{1} ,\;J_{2} ,\;J_{3} ,\;J_{4} ,\;T_{3}$, and $T_{4}$, from which $J_{1}$ is the necessary variable to finally calculated $h_{\text{R}}$ as follows

$$ h_{\text{R}} = \frac{\varepsilon_{1} }{{\left( {1 - \varepsilon_{1} } \right)}}\frac{{\left( {\sigma T_{\text{Ri}}^{4} - J_{1} } \right)}}{{\left( {T_{\text{Ri}} - T_{\text{a}} } \right)}}. $$

(17)

The temperature at surface patch (1), $T_{\text{Ri}}$, and the temperatures of the air in the environment, T _a, and in the cavity, $T_{\text{c}}$, should be given to enable a unique solution of the equation system, i.e., $h_{\text{R}} = h_{\text{R}} \left( {T_{\text{Ri}} ,T_{\text{a}} ,T_{\text{c}} } \right)$. In the present work, T _a = 25 °C and, since the temperature of the air in the internal cavity, T _c, is unknown, the approximation $T_{\text{c}} = \left( {T_{\text{Ri}} + T_{\text{a}} } \right)/2$ was considered. A few tests with the model revealed that changes in T _c in the range from T _Ri to T _a caused negligible variation in h _R. Therefore, if T _Ri is given, h _R can be obtained, i.e., $h_{\text{R}} = h_{\text{R}} \left( {T_{\text{Ri}} } \right)$. For the specific conditions of the present simulations, the complete radiation model was solved and this function was approximated by a polynomial of degree 2.