New Insight into Combined Model and Revised Model for RTD Curves in a Multi-strand Tundish

Abstract

The analysis for the residence time distribution (RTD) curve is one of the important experimental technologies to optimize the tundish design. But there are some issues about RTD analysis model. Firstly, the combined (or mixed) model and the revised model give different analysis results for the same RTD curve. Secondly, different upper limits of integral in the numerator for the mean residence time give different results for the same RTD curve. Thirdly, the negative dead volume fraction sometimes appears at the outer strand of the multi-strand tundish. In order to solve the above problems, it is necessary to have a deep insight into the RTD curve and to propose a reasonable method to analyze the RTD curve. The results show that (1) the revised model is not appropriate to treat with the RTD curve; (2) the conception of the visual single-strand tundish and the combined model with the dimensionless time at the cut-off point are applied to estimate the flow characteristics in the multi-strand tundish; and that (3) the mean residence time at each exit is the key parameter to estimate the similarity of fluid flow among strands.

Appendix Revised Model

In the previous paper[4], the mean residence time of the C-curve is defined as follows:

$$ {\tilde{t}_{\text{C}} = \frac{{\int_{0}^{2\tau } {Ct{\text{d}}t} }}{{\int_{0}^{2\tau } {C{\text{d}}t} }}}. $$

(A1)

Then the related dimensionless mean residence time can be expressed as

$$ {\tilde{\theta }_{\text{C}} = \frac{{\tilde{t}_{\text{C}} }}{\tau } = \frac{{\int_{0}^{2\tau } {Ct{\text{d}}t} }}{{\tau \int_{0}^{2\tau } {C{\text{d}}t} }}}. $$

(A2)

So the dead volume fraction can be calculated by

$$ {\tilde{v}_{\text{d}} = 1 - \frac{{\int_{0}^{2\tau } {C{\text{d}}t} }}{{\int_{0}^{\infty } {C{\text{d}}t} }}\tilde{\theta }_{\text{C}} = 1 - \frac{{\int_{0}^{2\tau } {C{\text{d}}t} }}{{\int_{0}^{\infty } {C{\text{d}}t} }}\frac{{\int_{0}^{2\tau } {Ct{\text{d}}t} }}{{\tau \int_{0}^{2\tau } {C{\text{d}}t} }} = 1 - \frac{{{{\int_{0}^{2\tau } {Ct{\text{d}}t} } \mathord{\left/ {\vphantom {{\int_{0}^{2\tau } {Ct{\text{d}}t} } {\int_{0}^{\infty } {C{\text{d}}t} }}} \right. \kern-0pt} {\int_{0}^{\infty } {C{\text{d}}t} }}}}{\tau }}. $$

(A3)

If the dead volume fraction is expressed as the general form in the combined model

$$ {v_{\text{d}} = 1 - \frac{{\bar{t}}}{\tau }}, $$

(A4)

then the mean residence time can be rewritten as

$$ {\bar{t} = \frac{{\int_{0}^{2\tau } {Ct{\text{d}}t} }}{{\int_{0}^{\infty } {C{\text{d}}t} }}}. $$

(A5)

This is the mean residence time in the revised model in Table I.