The inverse taper of the copper mould tube is an important parameter of the continuous casting machine. Recently, the calculation and expression methods of the inverse taper in China are very inconsistent, which can easily cause misunderstandings and bring undue losses and impacts to the work. For continuous casting machines imported from abroad, there is no report on the calculation and expression method of the inverse taper of the mold. To this end, this article will conduct some discussions on this.

**Inverse taper of copper mould tube**

Taper usually refers to the ratio of the difference in diameter between the large and small ends of a conical object to the axial length between the two ends. In engineering, it is often used to indicate the inclination of the conical surface of an object, and it is customary to regard the taper of a conical object with a small top and a large bottom as a positive taper.

On the continuous casting machine, it is considered that the molten steel gradually condenses and shrinks due to external cooling in thecopper mould tube. Therefore, the cross-section of the inner cavity of thecopper mould tube is usually made to gradually shrink from top to bottom along its entire height (that is, the lower section is slightly smaller than the upper section), making it possible to use the very familiar concept of taper. Because the space enclosed by the inner cavity of the crystallizer is a pyramid with a larger top and a smaller bottom, the taper is negative, so it is called an inverted taper.

The molten steel is continuously subjected to indirect high-speed forced water cooling in the copper mould tube, and gradually condenses into a billet shell with a certain thickness on all sides, and starts to shrink from the lower end. Since what is being studied is the slab mold, its width is much larger than its thickness, so the absolute shrinkage of the width is much greater than the thickness. In this way, the narrow surface of the slab breaks away from the inner wall of the mold earlier, creating an air gap. The thermal conductivity conditions are significantly deteriorated, which seriously affects the normal growth of the billet shell and increases the possibility of steel leakage. In order to avoid the generation of air gaps as much as possible, the inner walls of the two narrow inner walls of the copper mould tube are often made to have a certain inverse taper, while the inner walls of the two wide inner walls are made to have a smaller inverse taper.

**Calculation and expression method of inverted taper**

Based on the information currently available, it can be generally divided into two types: the overall calculation and representation method and the faceted calculation and representation method.

**The overall calculation and expression method of the inverse taper of the copper mould tube**

Its essence is to define the inverse taper according to the elongation rate from the cross-sectional area of the lower intraoral cavity of the mold to the cross-sectional area of the upper intraoral cavity. Its calculation method is:

In the formula: △—the inverse taper of the crystallizer;

F₁—— Cross-sectional area of the lower mouth of the crystallizer;

F₂——Cross-sectional area of the upper mouth of the crystallizer.

**Facet calculation and representation method of copper mould tube inverse taper**

Since the width and thickness of the slab mold are unequal, and the sizes vary greatly, its inverse taper is often calculated and expressed separately between the two wide faces and the two narrow faces.

(1) There are three ways to calculate and express the inverse taper between two wide surfaces (that is, thickness):

1) The inverse taper between the two wide surfaces is defined according to the shrinkage rate from the thickness of the upper mold to the thickness of the lower opening. Its calculation method is:

Where L₁——Thickness of the lower mouth of the copper mould tube;

L₂——Thickness of the upper mouth of the copper mould tube.

2) The inverse taper between the two wide surfaces is defined according to the elongation rate from the thickness of the lower opening of the mold to the thickness of the upper opening. Its calculation method is:

Where L₁——Thickness of the lower mouth of the continuous casting mold;

L₂——Thickness of the upper mouth of the continuous casting mold.

3) Under the condition that the nominal thickness dimensions of the upper and lower openings of the continuous casting mold are kept consistent, the upper opening thickness dimension has a given positive deviation, and the lower opening thickness dimension has a given negative deviation. To form an inverted taper between the two wide surfaces, its calculation method is:

continuous casting mold thickness: B

continuous casting mold top thickness: B+δ -0

Thickness of the lower opening of the crystallizer: B+0 -△

In total, δ and △ are given based on experience. For example, the continuous casting mold of Wuhan Iron and Steel slab continuous casting machine has an inverted taper between the two wide surfaces as shown in the following table:

(2) The calculation and expression methods of the inverse taper between two narrow surfaces (i.e. width) can be divided into the following three types:

1) The inverse taper between the two narrow surfaces is defined according to the shrinkage rate from the width of the upper opening of the continuous casting mold to the width of the lower opening. Its calculation and expression method can also be expressed by formula (2). It’s just L₁ and L0 in the formula. Represent the width of the lower and upper openings of the continuous casting mold respectively.

2) The inverse taper between the two narrow surfaces is defined according to the elongation from the width of the lower opening of the mold to the width of the upper opening. Its calculation and expression method can also be expressed by formula (3). Just L₁ and L0 in the formula. Represent the width of the lower and upper openings of the crystallizer respectively.

3) Use the difference in width between the upper and lower openings of the mold cavity to define the inverse taper between the two narrow surfaces. Its calculation method is:

△=L0-L1 (4)

The above various calculation and expression methods for the inverse taper of the continuous casting mold are used both in books and periodicals and in design literature. But there are several problems:

First, although continuous casting workers are so familiar with the inverse taper of the mold, their specific definitions and calculation methods are very different. In particular, the value difference between the overall calculation expression and the facet calculation expression is even greater. Therefore, usually when it is said that the inverse taper of a certain mold is what percent, what calculation method should you use to calculate and determine other dimensions? For example, the inverse taper of the continuous casting mold of a slab continuous casting machine introduced from West Germany, According to Comcast’s patent description: the inverse taper between the two narrow surfaces of the continuous casting mold is 0.90%/meter, but there is no explanation on how to calculate it, and the result can only be determined based on personal understanding.

Second, analyze the calculation and expression method of the inverse taper of the continuous casting mold above. Its essence is nothing more than calculating the shrinkage or elongation of the cross-sectional area or side length of the upper and lower openings, without truly defining and calculating the expression according to the concept of taper. Because the plate ring continuous casting mold is actually an inverted pyramid frustum with a very small taper, the height (i.e. length) parameter of the continuous casting mold should be introduced when studying the calculation method of its inverted taper.

In short, with the rapid development of continuous steel casting today, it is very necessary to correctly and reasonably define and calculate such a very important parameter as the inverse taper of the mold.

**3. Conclusion**

1. When calculating the inverse taper between the two narrow faces of the continuous casting mold, the author believes that the length parameter of the continuous casting mold should and must be taken into account. And it should be defined according to the shrinkage rate from the upper opening width to the lower opening width of the continuous casting mold per meter length, and its calculation method is:

Where L₁—the length of the copper mould tube (meters);

l1—width of the lower opening of the copper mould tube;

l2—The width of the upper mouth of the copper mould tube.

The specific numerical value of the inverse taper between the two narrow surfaces needs to be further summarized from practice, and the necessary data can be obtained from scientific research and experiments. Theoretically, the size of the inverse taper of the mold mainly depends on the high-temperature billet under various cold sealing conditions (type of molten steel, temperature, thermal conductivity of the inner wall of the mold, water cooling intensity, billet drawing speed, etc.) of cold shrinkage. However, at present, there are few studies on the cold shrinkage coefficient of various steels under high temperature conditions, and the measurement is also very complicated. Therefore, empirical values are often taken in the design of the crystallizer. Generally, the inverse taper between the two narrow surfaces of the slab crystallizer is usually 0.9 to 1.3%/meter.

2. Regarding the inverse taper between the two wide surfaces of the copper mould tube, judging from many domestic and foreign design and installation practices, the two wide surfaces of the copper mould tube should be kept parallel. It is advisable to achieve inverse taper by using positive or negative deviations for the thickness of the upper and lower intraoral cavities. So, what is the positive and negative deviation of the thickness to ensure the necessary inverse taper of the copper mould tube? The author recommends that the air gap width value be appropriately selected according to the relevant parameters of the continuous casting machine and the air gap width value shown in the figure.

**Figure: Air gap value between the slab and the inner walls of the two wide surfaces of the mold**