Goal-Oriented, Inverse Design Method—The Horizontal Integration of a Multistage Hot Rod Rolling System

Abstract

Steel mills are involved in the production of semi-products such as sheets or rods with certain grades of steel. Process designers are very much aware of the operating constraints and process requirements for each of the operations as they are involved in the whole process day-in and day-out.

Appendices

Appendix 1: cDSP Formulation for Pass 2 in This Chapter

In this section we describe the mathematical formulation of the compromise decision support problem (cDSP ) for Pass 2 of rod rolling. The cDSP for Pass 2 incorporates the design information passed from Pass 3 to Pass 4. The cDSP reads as follows:

Given:

• (1) Design information passed from Pass 3 to Pass 4

• (2) Requirements at Pass 2

  • Achieve target throughput (results obtained from Pass 4 design)

  • Achieve target rolling load

  • Target value for throughput , $T_{{p4,{\text{Target}}}} = 0.0006\,{\text{m}}^{ 3} / {\text{s}}$

  • Target value for rolling load , $P_{\text{Target}} = 40\,{\text{t}}$

  • Minimum value of rolling load , $P_{ \text{min} } = 35\,{\text{t}}$

  • Maximum value of rolling load , $P_{ \text{max} } = 45\,{\text{t}}$

  • Minimum value of throughput , $T_{\text{pmin}} = 0.0001\,{\text{m}}^{ 3} / {\text{s}}$

  • Maximum value of throughput , $T_{\text{pmax}} = 0.0008\,{\text{m}}^{ 3} / {\text{s}}$

• (3) Initial billet size = 42 × 42 mm

• (4) Other parameter values for passes

• (5) The regression equations and well-established empirical and theoretical correlations for the oval to round pass for Pass 2

• (6) Variability in system variables

The ranges identified for the system variables are provided in Table 5.8.

Table 5.8 System variables and ranges for Pass 2 cDSP

Find:

System Variables

$X_{1}$ :

diameter of rod after Pass 2 $\left( {h_{2} } \right)$

$X_{2}$ :

the coefficient of elongation for Pass 2 $(\lambda_{2} )$

$X_{3}$ :

the spread occurring in Pass 2 $(\beta_{2} )$

$X_{4}$ :

the exit velocity for Pass 2 $(w_{2} )$

$X_{5}$ :

the maximum radius of roll in Pass 2 $(R_{{{ \text{max} },2}} )$

$X_{6}$ :

the roll gap $(G_{2} )$

$X_{7}$ :

the coefficient of friction $(\mu_{2} )$

Deviation Variables

$$ d_{i}^{ - } , d_{i}^{ + } ,i = 1,2 $$

Satisfy:

System Constraints

• Minimum coefficient of elongation constraint: $\lambda_{2} \left( {X_{2} } \right) - 1.2 \ge 0$

• Maximum coefficient of elongation constraint: $2 - \lambda_{2} \left( {X_{2} } \right) \ge 0$

• Minimum spread constraint: $\beta_{2} \left( {X_{3} } \right) - 1.1 \ge 0$

• Maximum spread constraint: $1.7 - \beta_{2} \left( {X_{3} } \right) \ge 0$

• Exit speed constraint: $w_{2} - v_{r} \left( {X_{i} } \right) \ge 0$

• Minimum load constraint: $P\left( {X_{i} } \right) - P_{ \text{min} } \ge 0$

• Maximum load constraint: $P_{ \text{max} } - P\left( {X_{i} } \right) \ge 0$

• Maximum wear constraint: $0.0001 - \Delta R\left( {X_{i} } \right) \ge 0$

System Goals

Goal 1:

• Maximize Throughput:

  $$ \frac{{T_{p} \left( {X_{i} } \right)}}{{T_{{p,{\text{Target}}}} }} + d_{1}^{ - } - d_{1}^{ + } = 1 $$

Goal 2:

• Minimize Rolling Load:

  $$ \frac{{P_{\text{Target}} }}{{P\left( {X_{i} } \right)}} - d_{2}^{ - } + d_{2}^{ + } = 1 $$

Variable Bounds

Defined in Table 5.8

Bounds on deviation variables

$$ d_{i}^{ - } ,d_{i}^{ + } \ge 0{\text{ and }}d_{i}^{ - } *d_{i}^{ + } = 0,\;{\text{i}} = 1,2 $$

Minimize:

Minimize the deviation function

$$ Z = \mathop \sum \limits_{i = 1}^{2} W_{i} \left( {d_{i}^{ - } + d_{i}^{ + } } \right);\mathop \sum \limits_{i = 1}^{2} W_{i} = 1 $$

Appendix 2: Design Calculations (Refer to Fig. 5.2)

In this section, we describe the design calculations carried out for each pass based on the cDSP results obtained that are showcased in Tables 5.4 and 5.5. The design process is carried out following the sequential relationships that exist between passes ensuring the flow of information pattern shown in Fig. 5.3.

Step 1: Formulation of cDSP for roll Pass 4 using the information from the end product to be realized and the sequential relationship existing between roll Pass 3 and 4

The cDSP for Pass 4 is formulated in terms of the end requirements of minimizing ovality, maximizing throughput , and minimizing rolling load within the system constraints and bounds defined. The cDSP is exercised for different scenarios and ternary plots are used to identify the best region and the results are summarized in Table 5.4.

Step 2: Design of Passes 4 and 3 using the design and operating setpoints identified and the information available from end product requirements

We calculate the area of the round rod using the height value obtained for rod from cDSP results. The cross-sectional area of material after Pass 4:

$$ F_{4} = \frac{{\pi h_{4}^{2} }}{4} = 532.26\,{\text{mm}}^{2} $$

Entry speed of material for roll Pass 4:

$$ v_{4} = \frac{{w_{4} }}{{\lambda_{4} }} = 0.8671\,{\text{m/s}} $$

Throughput achieved in Pass 4:

$$ T_{p4} = F_{4} \times w_{4} = 0.0005999\,{\text{m}}^{ 3} / {\text{s}} $$

We carry out the design calculations for Pass 3 based on the cross-sectional area of rod and elongation coefficient ( cDSP result) obtained after Pass 4. We also define some requirements for Pass 3 such as meeting the throughput same as that of Pass 4.

Cross-sectional area of material after Pass 3:

$$ F_{3} = F_{4} \times \lambda_{4} = 691.93\,{\text{mm}}^{2} $$

Theoretical width of oval pass after Pass 3:

$$ b_{3w} = \sqrt {4.35 \times F_{3} } = 54.86\,{\text{mm}} $$

Height of material after Pass 3 (assuming $b/h$ ratio = 3):

$$ h_{3} = \frac{{b_{3w} }}{{\left( {b/h} \right)}} = 18.28\,{\text{mm}} $$

Radius of curvature of oval Pass 3:

$$ R_{3}^{*} = \frac{{b_{3w}^{2} + h_{3}^{2} }}{{4h_{3} }} = 45.72\,{\text{mm}} $$

Exit speed of material for roll Pass 3:

$$ w_{3} = v_{4} = 0.8671\,{\text{m/s}} $$

Throughput to be maintained in Pass 3 (Given):

$$ T_{p3} = T_{p4} = 0.0005999\,{\text{m}}^{ 3} / {\text{s}} $$

We carry out design calculations for Pass 4 now with the new information generated for Pass 3.

Width of round profile (approximated rectangle) after Pass 4:

$$ b_{4} = \beta_{4} \times h_{3} = 21.03\,{\text{mm}} $$

Mean height after Pass 4:

$$ h_{4m} = \frac{{F_{4} }}{{b_{4} }} = 25.31\,{\text{mm}} $$

Theoretical diameter of roll for Pass 4:

$$ D_{t4} = 2\left( {R_{{{ \text{max} },4}} + \frac{{G_{4} }}{2}} \right) = 314\,{\text{mm}} $$

Effective diameter of roll for Pass 4:

$$ D_{w4} = D_{t4} - h_{4m} = 288.7\,{\text{mm}} $$

Step 3: Formulation of cDSP for roll Pass 2 using the design information generated for Passes 3 and 4; and the sequential information existing between Passes 1 and 2; along with information on input material (billet)

The designer formulates the cDSP for Pass 2 after finding the results from Passes 3 to 4. For example, the range of the height of rod for Pass 2 is identified based on the dimensions achieved in Passes 3 and 4. Another example is the rolling load target value. Since there is a chance of having a higher rolling load during Pass 2 due to larger stock that is being rolled than Pass 4, the target, minimum and maximum values for Pass 2 are fixed after looking at the rolling load value obtained in Pass 4. The designer also fixes the target throughput value for Pass 2 after analyzing the throughput achieved in Passes 3 and 4. Thus the designer makes judgments based on the information obtained from the information as it develops.

Step 4: Design of roll Passes 2 and 1 using the design and operating setpoints identified; the information available from input material and the information from Passes 3 to 4

The cDSP results for Pass 2 presented in Table 5.5 are used to design Pass 2.

Cross-sectional area of material after Pass 2:

$$ F_{2} = \frac{{\pi h_{2}^{2} }}{4} = 755.07\,{\text{mm}}^{2} $$

Entry speed of material for roll Pass 2:

$$ v_{2} = \frac{{w_{2} }}{{\lambda_{2} }} = 0.611\,{\text{m/s}} $$

Throughput achieved in Pass 2:

$$ T_{p2} = F_{2} \times w_{2} = 0.0005997\,{\text{m}}^{ 3} / {\text{s}} $$

Next, the design calculations for Pass 1 is carried out using Pass 2 design results and initial billet information from caster.

Cross-sectional area of material after Pass 1:

$$ F_{1} = F_{2} \times \lambda_{2} = 981.59\,{\text{mm}}^{2} $$

Theoretical width of oval pass after Pass 1:

$$ b_{1w} = \sqrt {4.35 \times F_{1} } = 65.345\,{\text{mm}} $$

Height of material after Pass 1 (assuming $b/h$ = 3):

$$ h_{1} = \frac{{b_{1w} }}{{\left( {b/h} \right)}} = 21.78\,{\text{mm}} $$

Radius of curvature of oval Pass 1:

$$ R_{1}^{*} = \frac{{b_{1w}^{2} + h_{1}^{2} }}{{4h_{1} }} = 54.45\,{\text{mm}} $$

Exit speed of material for roll Pass 1:

$$ w_{1} = v_{2} = 0.611\,{\text{m/s}} $$

Given initial billet size from caster:

$$ h_{0} \times b_{0} = 42 \times 42\;\left( {\text{mm}} \right) $$

Cross-sectional area of initial billet:

$$ F_{0} = 42 \times 42 = 1764\,{\text{mm}}^{2} $$

Coefficient of elongation for Pass 1:

$$ \lambda_{1} = \frac{{F_{0} }}{{F_{1} }} = 1.797 $$

Width of oval profile (approximated rectangle) after Pass 1 (assuming $\beta_{1} = 1.4$):

$$ b_{1} = \beta_{1} \times b_{0} = 58.8\,{\text{mm}} $$

Mean height of material after Pass 1:

$$ h_{1m} = \frac{{F_{1} }}{{b_{1} }} = 16.69\,{\text{mm}} $$

Effective diameter of roll for Pass 1 (assuming a theoretical diameter for rolls in Pass 1, $D_{t1} = 350\,{\text{mm}}$):

$$ D_{w1} = D_{t1} - h_{1m} = 333.3\,{\text{mm}} $$

Entry speed of material for roll Pass 1:

$$ v_{1} = \frac{{w_{1} }}{{\lambda_{1} }} = 0.3401\,{\text{m/s}} $$

Throughput to be maintained in Pass 1:

$$ T_{p1} = T_{p2} = 0.0005997\,{\text{m}}^{ 3} / {\text{s}} $$

The design calculations for Pass 2 are carried out next using Pass 1 information generated followed by collecting all the results for Passes 1 and 2.

Width of round profile (approximated rectangle) after Pass 2:

$$ b_{2} = \beta_{2} \times h_{1} = 26.14\,{\text{mm}} $$

Mean height after Pass 2:

$$ h_{2m} = \frac{{F_{2} }}{{b_{2} }} = 28.88\,{\text{mm}} $$

Theoretical diameter of roll for Pass 2:

$$ D_{t2} = 2\left( {R_{{{ \text{max} },2}} + \frac{{G_{2} }}{2}} \right) = 314\,{\text{mm}} $$

Effective diameter of roll for Pass 4:

$$ D_{w2} = D_{t2} - h_{2m} = 285.1\,{\text{mm}} $$

With the information generated for Passes 1 and 2, the design calculations for Passes 3 and 4 are carried out completing design results for Passes 1, 2, 3, and 4.

Coefficient of elongation for Pass 3:

$$ \lambda_{3} = \frac{{F_{2} }}{{F_{3} }} = 1.091 $$

Width of oval profile (approximated rectangle) after Pass 3 (assuming $\beta_{3} = 1.5$):

$$ b_{3} = \beta_{3} \times b_{2} = 39.2\,{\text{mm}} $$

Mean height of material after Pass 3:

$$ h_{3m} = \frac{{F_{3} }}{{b_{3} }} = 17.65\,{\text{mm}} $$

Effective diameter of roll for Pass 3 (assuming a theoretical diameter for rolls in Pass 1, $D_{t3} = 314\,{\text{mm}}$):

$$ D_{w3} = D_{t3} - h_{3m} = 296.35\,{\text{mm}} $$

Entry speed of material for roll Pass 3:

$$ v_{3} = w_{2} = 0.7943\,{\text{m/s}} $$

Exit speed of material for roll Pass 3:

$$ w_{3} = v_{3} \times \lambda_{3} = 0.867\,{\text{m/s}} $$

This completes the design of the rolling passes with the determination of all the key dimensions presented in Figs. 5.4a, b.