JPH08132204A - Continuous casting method - Google Patents

Abstract

(57) [Summary] [Purpose] Irrespective of changes in casting conditions and progress of solidification inside the slab, it is possible to carry out reduction under specified conditions and to effectively prevent shrinkage flow that causes center segregation. A continuous casting method capable of performing the above is provided. [Constitution] In the continuous casting method in which light reduction is applied at the final solidification part of the continuously cast slab, the reduction start point is within the range of 0.0 ≦ fs ≦ 0.2 and the reduction end point is within 0.8 ≦ fs ≦ 1.0. And the total amount of reduction at the solid-liquid interface based on the flow limit solid fraction is set to 0.6 to 1.4 mm or less, or the reduction rate at the solid-liquid interface based on the flow limit solid fraction is set to 0.15 mm. ~ 0.3mm / min or less continuous casting method.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a continuous casting method in continuous casting of steel in which center segregation of a slab is prevented by applying a light reduction to the slab before completely solidifying.

[0002]

2. Description of the Related Art When a slab is manufactured by a continuous casting method,
Often an internal defect called central segregation is a problem.
This center segregation is a phenomenon in which the molten steel components such as C, S, P, Si and Mn are positively segregated in the thickness center portion (final solidified portion) of the cast slab. The center segregation causes a decrease in toughness and hydrogen-induced cracking, and thus causes a serious problem especially in a material for thick plates.

The central segregation is caused by the fact that the molten steel left between dendrite trees at the end of solidification moves macroscopically due to bulging or solidification shrinkage, and that the molten steel enriched with the above components is locally accumulated. It is known to occur in As a measure to prevent this center segregation, it is effective to reduce the flow of the concentrated molten steel by compensating for the solidification shrinkage in the final solidification part by reducing the vicinity of the solidification tip part by some method. A method based on has been proposed.

[0004] The degree of improvement of the center segregation by the above reduction,
There is a clear correlation between the amount of reduction per unit time or length in the casting direction (hereinafter simply referred to as "reduction amount") and the reduction timing, and quantitative studies on the reduction amount and the reduction timing have also been made. There is.

For example, Japanese Examined Patent Publication (Kokoku) No. 59-39225 discloses that the degree of superheat of molten steel in a tundish is adjusted to 30 to 70 ° C., and 0.5 to 0.5 near the crater end (the tip of the unsolidified portion).
A continuous casting method with a reduction of 2.0 mm / m is shown.

Further, in Japanese Patent Publication No. 30548/1993, the amount of reduction from the time when the center of the slab reaches the liquidus temperature to the time when it reaches the fluidity limit solid fraction and the solidification time thereafter is described. A method of adjusting is proposed.

However, in these methods, the amount of reduction is constant in the vicinity of the crater end or in the center of the slab until the fluidity limit solid fraction is reached, so that insufficient reduction or overpressure reduction easily occurs. There is. The reason is as follows.

As will be described later in detail, the rate at which the set amount of reduction is transmitted to the solid-liquid interface (hereinafter referred to as "rolling efficiency (α)") decreases toward the casting downstream side. Therefore, even if the same amount of reduction is applied from the surface, the amount of reduction transmitted to the solid-liquid interface changes depending on the solidification timing.

That is, when the amount of coagulation contraction at an arbitrary position is used as a reference, the upstream side is overpressure-reduced and the downstream side is under-reduced. As in the methods disclosed in the above-mentioned publications and the like, when a reduction amount is applied at a constant reduction amount over a long range, it is easy to induce a total reduction amount or an insufficient reduction amount. If the reduction is insufficient, the effect of preventing center segregation is naturally small, while if the pressure is excessive, inverse V segregation occurs.

With respect to the above problems, Japanese Patent Laid-Open No.
90263, Japanese Patent Publication No. 5-73506, and Japanese Patent Publication No. 5
No. 73507 discloses a continuous casting method in which the rolling speed is increased toward the downstream side of casting. However, these methods still have the following problems. That is, the method disclosed in JP-A-3-90263 is
The range of conditions for increasing the rolling speed includes conditions from insufficient rolling to over rolling, and it is considered that the effect of improving center segregation cannot be stably obtained.

In the method disclosed in Japanese Examined Patent Publication No. 73506/1993, a specific condition regarding the amount of reduction for stably achieving the improvement of center segregation is not clear.

In the method disclosed in Japanese Patent Publication No. 73507/1993, the roll reaction force limits the numerical value, but the relationship between the roll reaction force and the appropriate amount of reduction is not described. The way of thinking has not been clarified. Further, the methods disclosed in the above three publications have a common problem that the influence of the temperature distribution of the slab on the rolling reduction condition is not taken into consideration.

[0013]

It is known that a light reduction at the final stage of solidification is effective for reducing the center segregation of continuously cast slabs, but it is difficult to select an appropriate reduction amount by the conventional method, and the reduction is performed. Due to insufficient pressure or overpressure, the improvement of the center segregation was not sufficient, or even the center segregation sometimes increased.

The object of the present invention is to change the casting conditions such as the casting speed and the secondary cooling conditions (the surface temperature of the slab, and hence the temperature gradient of the cross section of the slab) and the progress of solidification inside the slab (the change in the unsolidified thickness). ) Regardless of the above), it is an object of the present invention to provide a continuous casting method capable of performing reduction under a predetermined condition and more effectively preventing shrinkage flow causing center segregation more than the conventional method.

[0015]

The gist of the present invention is the following continuous casting methods (1) and (2).

(1) In the continuous casting method in which a light reduction is applied at the final solidification portion of the continuously cast slab, the reduction start point is 0.0 ≦ fs ≦
The range is 0.2, the rolling end point is 0.8 ≤ fs ≤ 1.0, continuous rolling is performed during that period, and the total rolling reduction at the solid-liquid interface based on the flow limit solid fraction is 0.6 to 1.4 m.
A continuous casting method characterized in that the length is m or less.

(2) In the continuous casting method in which a light reduction is applied at the final solidification part of the continuously cast slab, the reduction start point is 0.0 ≦ fs ≦
The range is 0.2, the rolling end point is 0.8 ≤ fs ≤ 1.0, continuous rolling is performed during that period, and the rolling-down speed at the solid-liquid interface is 0.15 to 0.3 mm / mi based on the flow limit solid fraction.
A continuous casting method characterized in that it is n or less.

In the above methods (1) and (2), fs is the thickness center solid fraction of the slab. Further, the above-mentioned "solid-liquid interface" means the position inside the slab that has reached the fluidity limit solid fraction (about 0.7 to 0.8).

FIG. 2 shows that the unsolidified portion 3 is formed inside the solidified shell 4.
FIG. 3 is a schematic sectional view of a cast piece for explaining the definition of parameters such as a rolling speed and a rolling efficiency of a solid-liquid interface 5 when a cast piece containing a steel is pressed. When a pinch roll is applied at the 10 position shown in the figure, each symbol shown in the figure is defined as follows.

Δ _S : Surface reduction amount (mm) L: Distance subjected to reduction (m) δ _I : Solid-liquid interface reduction amount (mm) V _C : Casting speed (m / min) where solid-liquid interface When the reduction speed is R _I (mm / min), R _I = (δ _I / L) × V _C. That is, the reduction rate R _I at the solid-liquid interface is obtained as the amount of reduction per unit time at the solid-liquid interface. Further, the rolling reduction efficiency α is expressed by α = (δ _I / δ _S ). Note that δ _I can be calculated, for example, from the meniscus to Fe.
It can be measured by adding a tracer, such as S or Pb, which has a large specific gravity and is deposited on the solid-liquid interface, and traces its movement.

[0021]

It has been conventionally known that there is an optimum amount of light reduction for improving the center segregation. Moreover, the amount of reduction depends on the surface temperature of the slab (in other words, the temperature distribution inside the slab), It was also empirically known that it varies depending on the degree of coagulation. Therefore, the degree of solidification of the slab when the slab with an unsolidified portion is rolled down, the secondary cooling condition (temperature gradient in the slab)
It was thought that the reduction conditions had to be changed according to

The inventor of the present invention provides an index of a light reduction condition that can correspond to various casting conditions, and in order to effectively perform a final solidification light reduction for improving the center segregation, the reduction of the solid-liquid interface is performed. The research was promoted focusing on the behavior.

FIG. 5 shows the slab surface reduction rate R _S (mm / min) when the slab containing an unsolidified portion is subjected to reduction in continuous casting of steel.
It is a figure which shows the result of having investigated the relationship of a center segregation. Here, when the slab surface temperature during rolling is 700 to 900 ℃ and 900 to 900 ℃,
It is divided into two cases in case of 1000 ℃. The center segregation was evaluated by the carbon segregation degree (C / _CO ) shown in the examples described later.

As shown in FIG. 5, the higher the surface temperature of the slab, the higher the surface rolling reduction rate (R _S ) for lowering the segregation degree (C / C _O ).

FIG. 6B shows the results of investigation of the occurrence of central segregation when light reduction is performed with three types of reduction patterns as shown in FIG. In the pattern 1 shown in FIG. 6A, that is, in the rolling mode in which the rolling gradient is increased with the progress of solidification, the degree of improvement of the central segregation is greatest. These results mean that it is necessary to select the amount of reduction according to the temperature distribution inside the slab and the degree of solidification.

The reason why the results shown in FIGS. 5 and 6 occur is considered as follows.

When the slab is rolled by a roll, the amount of reduction applied from the surface does not directly propagate to the solid-liquid interface, but the ratio of the amount of surface reduction and the amount of reduction transmitted to the solid-liquid interface (the above-mentioned reduction efficiency α). Is less than 1.0. This is because the amount of surface reduction is consumed by the spread of the slab and the advancement due to the deformation resistance, flow resistance, etc. of the unsolidified portion of the slab.

FIG. 7 shows an example of the relationship between the distance from the meniscus, that is, the distance in the casting progress direction and the rolling reduction efficiency α. α
Was determined by the method shown in FIG. 2 as described above.

The present inventor constructed a stress analysis model by the finite element method from this experimental data, and estimated the rolling behavior of the solid-liquid interface under various conditions.

As the solidification progresses, the influence of the resistance of the unsolidified portion on the solid-liquid interface increases, so that α decreases.
Further, the higher the slab temperature (that is, the smaller the temperature gradient inside the slab), the smaller the solidification shell rigidity, and the smaller the surface reduction amount, the wider the slab and the more easily it will be consumed in the advanced process, so α will be smaller. . Therefore, even if the slab is rolled down with a constant rolling down gradient, the rolling down that propagates to the solid-liquid interface due to the solidification timing (distance from the meniscus) and the temperature gradient inside the slab (slab surface temperature) as shown in FIG. The amount will be different.

From these results, the inventors of the present invention uniquely pursued the rolling-down behavior at the solid-liquid interface in order to pursue parameters that can be universally organized even if the degree of solidification and the temperature distribution in the slab are different. As a result of investigating the correlation between and the degree of center segregation, the following phenomenon was found and the present invention was completed.

FIG. 3 shows the total sum Σδ _{I of the} amount of reduction propagating to the solid-liquid interface and the carbon segregation degree (C / C) based on the fluidity limit solid fraction.
_It shows the relationship with the center segregation evaluated by _O ).
This figure shows data over a wide range of slab surface temperatures from 750 to 1100 ° C.

FIG. 3 shows the case where the rolling-down timing is changed to the following four types.

The thickness center solid fraction of the cast slab (fs above) is
Continuous reduction from 0.0 to 0.8 (○ in Fig. 3) Continuous reduction in the range of 0.2 <fs <0.8 (△ in Fig. 3) From 0 <fs to fs = 0.5 Continuous reduction (□ in Fig. 3) Continuous reduction from 0 <fs to fs = 1.0 (● in Fig. 3) In Fig. 3, the solid phase ratio forming the solid-liquid interface is set to the flow limit solid. It is taken as a phase ratio.

As is clear from FIG. 3, a large improvement in center segregation is obtained by controlling Σδ _I within the range of 0.6 to 1.4 mm regardless of the surface temperature of the slab.

[0036] From the above results, for the number of slab surface temperature is different over a wide range, it is clear that you can determine the range to reduce the center segregation in a factor called Sigma] [Delta] _I, the present invention using the Sigma] [Delta] _I It can be seen that the method is more universal than the conventional method for controlling the amount of reduction on the surface of the cast slab.

It is desirable that Σδ _I be distributed to the substantially same interfacial reduction rate R _I (mm / min) during the reduction section. However,
Even if it is distributed within the range of ± 20 to 30%, if the section is continuously rolled down, the central segregation can be greatly improved by proper control of Σδ _I. That is, even if it deviates from the proper range locally, it is sufficient to judge whether the total reduction range is insufficient or excessive.

Next, from the viewpoint of the time when the reduction is applied (the range of the solid fraction fs at the thickness center of the slab) in FIG. 3, when the center segregation is compared with the same amount of Σδ _I , 0 <fs to fs = 0.8
When the range up to is continuously rolled down (circle mark in Fig. 3 above)
In the case of continuous rolling down in the range of 0 <fs to fs = 1.0 (marked with ● in Fig. 3), the center segregation is reduced most. Less than,
Segregation increases in the order of continuous reduction in the range of 0.2 <fs <0.8 (△ in Fig. 3 above) and continuous reduction from 0 <fs to fs = 0.5 (□ in Fig. 3 above). Has become.

This means that the solidification time for forming the central segregation is in the range from the solid fraction fs exceeds 0 to 0.8, and this range is surely reduced by the appropriate interfacial reduction. It shows that it is important to do.

However, even when the rolling-down time is set to 0.2 <fs <0.8, the effect of improving the segregation is seen as compared with, and it is considered that the minimum range of rolling-down is this range.

The above was made based on the above findings.
The invention of (1), that is, "the reduction start point is within the range of the thickness center solid fraction of the slab is 0 to 0.2, and the reduction end point is 0.8 to 1.0.
The continuous casting method is performed in that range, and the total amount of reduction at the solid-liquid interface is set to 0.6 to 1.4 mm.

Next, the invention of (2) above, that is, "the reduction start point is in the range of 0 to 0.2 in the thickness center solid fraction of the slab, and the reduction end point is in the range of 0.8 to 1.0. The invention of a continuous casting method characterized by performing continuous reduction at a reduction rate of 0.15 to 0.30 mm / min at the solid-liquid interface "will be described.

FIG. 4 shows the amount of reduction per unit time that propagates to the solid-liquid interface based on the critical flow solid fraction, that is, the interface reduction rate R _I and the center segregation (the carbon segregation degree, C /
It is the result of the investigation on the relationship of C _O ).

From the point of 0 <fs, which is the optimum rolling time for improving the center segregation determined from FIG. 3, fs =
The case up to 0.8 is shown. As is apparent from FIG. 4, there is an appropriate range for improving the center segregation in the rolling speed R _I at the solid-liquid interface. That is, R _I
A large improvement in center segregation is obtained when it is in the range of 0.15 to 0.30 mm / min.

The optimum range of R _I is from the same as the contraction amount per unit time of the slab to about twice. That is, R _I that balances the molten steel flow due to the volume contraction of the slab
It is important to select 2 compared to volume shrinkage
The reason why the center segregation is improved even under the condition of about double overpressure is that there is a phenomenon that causes molten steel flow such as bulging between rolls in addition to volume contraction, so it is necessary to perform a reduction to offset this. Because there is.

When R _I is smaller than the optimum range, the center segregation increases due to insufficient reduction, while when R _I is larger than the optimum range, the center segregation also increases due to overpressure (reverse v segregation occurs).

It is desirable that the conditions of the invention (1) and the conditions of the invention (2) are compatible with each other. That is, "The reduction rate at the solid-liquid interface is 0.15 to 0.30 mm / min over the entire range where the solid fraction at the thickness center of the slab is greater than 0 and 0.8 or less, and the total amount of reduction at the solid-liquid interface is Is 0.6 ~
It is most desirable to carry out continuous rolling down to 1.4 mm.

The method of the present invention can be applied to casting slabs of various shapes such as slabs, blooms and billets, and can be applied to any position in the width direction of a slab having a large flatness ratio.

[0049]

Example 1 A steel slab was continuously cast using a continuous casting machine whose schematic structure is shown in FIG.

This continuous casting machine is an S-type continuous casting machine having a bending radius of 12.5 m, and the length of the reduction zone is 5 m.

In FIG. 1, the molten steel 3 cast into the mold 1 from the dipping nozzle 2 is solidified through the support roll group 6, the reduction roll group 7, and the pinch roll 8 and then drawn. The reduction roll group is composed of a plurality of reduction rolls, and the amount of reduction (the reduction speed) can be adjusted by controlling the hydraulic pressure applied to the rolls. Casting conditions in Table 1 (other than reduction conditions)
Table 2 and Table 3 show the rolling reduction conditions.

Examples 1 to 4 are examples of the present invention (1), in which continuous rolling is applied over the entire range of the thickness center solid fraction fs of the width center portion of the cast piece in the range of 0.0 to 0.8, and solidification is applied between them. This is the case when the set reduction amount is adjusted so that the total reduction amount Σδ _I received at the liquid interface is 0.6 to 1.4 mm. However, the interfacial rolling speed R _I is not necessarily within the range of the present invention (2).

Examples 5 to 8 are examples of the present invention (2), in which continuous reduction is applied in the entire region of the thickness center solid fraction fs of the width center portion of the slab to 0.0 to 0.8, and This is a case where the set reduction amount is adjusted so that the interface reduction velocity R _I in the phase ratio section is 0.15 to 0.30 mm / min.

On the other hand, Comparative Examples 1 to 4 are conventional examples, and when the reduction rate R _{S of the} surface of the slab is controlled, in Comparative Example 5, the interfacial reduction rate R _I is controlled to an almost appropriate value. When the solid center fs of the slab is in the range of 0.1 to 0.7, Comparative Example 6 controls the total interface reduction amount Σδ _I to an appropriate value. Solid fraction fs is 0.
The case is in the range of 3 to 0.7.

[0055]

[Table 1]

[0056]

[Table 2]

[0057]

[Table 3]

Table 4 shows the results of investigation of center segregation in Examples and Comparative Examples. The carbon segregation degree, which indicates the degree of center segregation, was measured by measuring the carbon concentration (C) at the center segregation part at the center of the slab width by optical emission spectrometry at 10 points. The peak value and the average concentration of steel
It was evaluated as (C _O) and a ratio of (C / C _O).

Further, in order to estimate the rolled state of the slab,
The segregation morphology at the center of the width of the slab was investigated. It is considered that V segregation corresponds to the state of insufficient reduction and reverse V segregation corresponds to the state of overpressure. Therefore, the segregation morphology is preferably neither V segregation nor inverse V segregation.

Examples 1 to 4 are various slab surface temperatures (Tsu)
In this range, the total reduction amount of the interface Σδ _I was controlled to an appropriate value. In this case as well, the carbon segregation degree (C / C _O ) was extremely low at 1.10 or less, and V segregation was observed on the longitudinal section of the slab. Neither reverse V segregation was observed.

In each of Examples 5 to 8, the interfacial reduction rate R _I was controlled to an appropriate value within a range of the slab surface temperature (Tsu) on the entry side of various reduction zones. In all cases, the degree of carbon segregation was controlled. (C / C _O ) showed a low segregation degree of 1.10 or less. Further, neither V segregation nor inverse V segregation was observed in the longitudinal section of the slab, and it was inferred that the steel was in a good reduction state with neither reduction insufficiency nor overpressure.

In Examples 1 to 8, a low degree of center segregation was obtained over a wide slab surface temperature range so that the solid-liquid interfacial reduction rate or the total amount of solid-liquid interfacial reduction was in the proper range. This is because the surface rolling speed is set.

On the other hand, in Comparative Examples 1 and 2, the reduction rate R _S from the surface of the slab was set to a constant value, but the carbon segregation degree was higher than those of the Examples. Comparative Example 1
At the position where fs was 0.4 or more, the interfacial reduction rate R _I was smaller than the proper value, and the total interfacial reduction amount Σδ _I was also smaller than the proper value. Therefore, V segregation was caused due to insufficient reduction and the center segregation was deteriorated.

In Comparative Example 2, the interfacial reduction rate R _{I in} the range of fs = 0.0 to 0.7 was large, and the total interfacial reduction amount Σδ _I was also larger than the appropriate value, so the center segregation deteriorated due to overpressure. As described above, if the rolling reduction in the range of fs = 0.0 to 0.8 is performed under the condition that R _{S is} constant, insufficient rolling reduction or overpressure reduction is likely to occur, and improvement of the center segregation cannot be obtained.

In Comparative Examples 3 to 4, the surface reduction rate R _S was increased toward the casting downstream side in consideration of the fact that the reduction efficiency from the surface of the slab becomes smaller toward the casting downstream side. The carbon segregation degree was higher than that in the examples.

In Comparative Example 3, since the interfacial reduction rate R _I did not reach the proper value over the entire reduction range and the total interface reduction amount Σδ _I was smaller than the proper value, the center segregation was caused by insufficient reduction (V segregation). It got worse.

In Comparative Example 4, the interface reduction rate R _I at the position where fs is 0.4 or more is larger than the proper value, and the total interface reduction amount Σδ.
_{Since I} was also larger than the appropriate value, central segregation deteriorated due to overpressure. Thus, even if the surface reduction rate R _S is increased toward the downstream side of casting, unless the reduction amount is set so that the interface reduction rate R _I or the interface total reduction amount Σδ _I becomes an appropriate value, the center segregation There is no improvement.

In Comparative Example 5, the interfacial rolling velocity R _I in the range of fs = 0.1 to 0.7 was controlled to an appropriate value, but fs = 0.0 to 0.1,
Since the range of fs = 0.7 to 0.8 was not reduced, the degree of carbon segregation was higher than that of the examples, and the segregation morphology of the longitudinal section was V segregation showing insufficient reduction.

In Comparative Example 6, the total reduction amount of the interface Σδ _I was controlled to an appropriate value by rolling down only in the range of fs = 0.3 to 0.7, but in the range of fs = 0.0 to 0.30 and fs = 0.7 to 0.8. Since no reduction was performed, the interfacial reduction rate at fs = 0.3 to 0.7 became large, and over-compression occurred in this range, so the carbon segregation degree was high. From the results of Comparative Examples 5 and 6, fs = 0.
It can be seen that improvement of center segregation cannot be obtained unless proper continuous reduction is performed in the entire range of 0 to 0.8.

From the above results, it is clear that the continuous casting method of the present invention is more effective in reducing center segregation than the conventional method.

[0071]

[Table 4]

[0072]

According to the method of the present invention, the reduction control may be performed only in accordance with the progress of solidification of the slab regardless of the casting conditions. In addition, it is possible to perform ideal reduction without insufficient reduction or over-pressure reduction, and it is possible to obtain a great effect in reducing center segregation of the slab.

[Brief description of drawings]

FIG. 1 is a schematic structural diagram of a continuous casting machine for carrying out the method of the present invention.

FIG. 2 is a schematic view of a slab cross section for explaining the definition of various parameters.

FIG. 3 is a diagram showing an example of the relationship between the total reduction amount Σδ _{I at} the solid-liquid interface and the carbon segregation degree.

FIG. 4 is a diagram showing an example of the relationship between the solid-liquid interface reduction rate R _I and the carbon segregation degree.

FIG. 5 is a diagram showing an example of a relationship between a surface rolling speed R _S and a carbon segregation degree.

FIG. 6 is a diagram showing an example of a relationship between a reduction pattern and center segregation.

FIG. 7 is a diagram showing an example of transition of rolling efficiency in a casting direction.

FIG. 8 is a diagram showing an example of transition of the interfacial reduction rate R _{I in} the casting direction.

[Explanation of symbols]

1: Water-cooled copper mold, 2: Immersion nozzle, 3: Molten steel, 4: Solidified shell, 5: Solid-liquid interface 6: Support roll group, 7: Rolling roll group, 8: Pinch roll, 9: Non-pressing portion 10: Rolling down

Claims (2)

[Claims] 1. A continuous casting method in which a light reduction is applied at the final solidification portion of a continuously cast slab, where the reduction start point is 0.0 ≦ fs ≦ 0.2.
, The end point of the rolling reduction is 0.8 ≤ fs ≤ 1.0, continuous rolling is performed between them, and the total rolling reduction at the solid-liquid interface based on the solid limit of flow limit is 0.6 to 1.4 mm or less. A continuous casting method characterized by: However, fs is the thickness center solid fraction of the slab. 2. A continuous casting method in which a light reduction is applied at the final solidification portion of a continuously cast slab, where the reduction start point is 0.0 ≦ fs ≦ 0.2.
The end point of the reduction is 0.8 ≤ fs ≤ 1.0, continuous reduction is performed during that period, and the reduction rate at the solid-liquid interface is 0.15 to 0.30 mm / min or less based on the critical solid fraction for flow. A continuous casting method characterized by: However, fs is the thickness center solid fraction of the slab.