{"id":195736,"date":"2025-02-28T17:33:27","date_gmt":"2025-02-28T17:33:27","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=195736"},"modified":"2025-02-28T17:33:30","modified_gmt":"2025-02-28T17:33:30","slug":"problem-rectangular-slabs-cast-from-1020-annealed-steel-are-heated-and-flat-rolled-to-improve-their-metallurgical-structure-and-reduce-their-thickness","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/28\/problem-rectangular-slabs-cast-from-1020-annealed-steel-are-heated-and-flat-rolled-to-improve-their-metallurgical-structure-and-reduce-their-thickness\/","title":{"rendered":"Problem Rectangular slabs cast from 1020 annealed steel are heated and flat-rolled to improve their metallurgical structure and reduce their thickness"},"content":{"rendered":"\n<p>Problem<\/p>\n\n\n\n<p>Rectangular slabs cast from 1020 annealed steel are heated and flat-rolled to improve their metallurgical structure and reduce their thickness. The slabs are 1.5m\u00d79m\u00d718cm with a density of 7590kg\/m<sup>3<\/sup>. After passing through a 5-stand roughing mill, the rolled product will have a thickness of 3mm. Vertical rolls are used to prevent widening of the slabs so the width of the finished product does not change; however, the density of the finished product increases by 0.1%.<\/p>\n\n\n\n<p>a) What is the length of the final rolled product? We want the first stand in the 5-stand roughing mill to reduce the thickness of the slab from 18cm to 16cm. The roll&#8217;s diameter is 600mm and it rotates at 75rpm. The motor driving the roll has 8,000kW of power with an efficiency of 75%. For 1020 annealed steel, K=745MPa and n=0.20.<\/p>\n\n\n\n<p>b) Will you be able to perform this rolling job in one pass (i.e., reduce from 18cm to 16cm in one pass through the rolls)?<\/p>\n\n\n\n<p>c) Will you be able to perform this rolling job in two passes (with equal reduction in every pass)? Justify your answer with calculations.<\/p>\n\n\n\n<p>Submission Instructions<\/p>\n\n\n\n<p>Final answers are not accepted &#8211; show the steps of your work<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let&#8217;s go step by step to solve the problem.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Given Data:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Initial slab dimensions:\n<ul class=\"wp-block-list\">\n<li>Width (WW) = 1.5 m<\/li>\n\n\n\n<li>Length (LL) = 9 m<\/li>\n\n\n\n<li>Thickness (hih_i) = 18 cm = 0.18 m<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Density of 1020 annealed steel (\u03c1i\\rho_i) = 7590 kg\/m\u00b3<\/li>\n\n\n\n<li>Final thickness (hfh_f) = 3 mm = 0.003 m<\/li>\n\n\n\n<li>No change in width (Wf=Wi=1.5W_f = W_i = 1.5 m)<\/li>\n\n\n\n<li>Final density (\u03c1f\\rho_f) increases by 0.1%, meaning:\n<ul class=\"wp-block-list\">\n<li>\u03c1f=1.001\u00d7\u03c1i=1.001\u00d77590\\rho_f = 1.001 \\times \\rho_i = 1.001 \\times 7590 kg\/m\u00b3<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Roll diameter (DD) = 600 mm = 0.6 m<\/li>\n\n\n\n<li>Roll rotational speed (NN) = 75 rpm<\/li>\n\n\n\n<li>Motor power = 8,000 kW<\/li>\n\n\n\n<li>Motor efficiency = 75%<\/li>\n\n\n\n<li>Flow stress equation: K=745K = 745 MPa, n=0.20n = 0.20<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 1: Find the Final Length of the Slab<\/strong><\/h3>\n\n\n\n<p>Since the volume remains constant: Vi=VfV_i = V_f Wi\u00d7Li\u00d7hi=Wf\u00d7Lf\u00d7hfW_i \\times L_i \\times h_i = W_f \\times L_f \\times h_f (1.5)(9)(0.18)=(1.5)(Lf)(0.003)(1.5)(9)(0.18) = (1.5)(L_f)(0.003)<\/p>\n\n\n\n<p>Solving for LfL_f: Lf=(1.5)(9)(0.18)(1.5)(0.003)L_f = \\frac{(1.5)(9)(0.18)}{(1.5)(0.003)} Lf=2.430.0045=540&nbsp;mL_f = \\frac{2.43}{0.0045} = 540 \\text{ m}<\/p>\n\n\n\n<p>So, the final length of the slab is <strong>540 meters<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 2: Checking If the First Pass is Feasible<\/strong><\/h3>\n\n\n\n<p>We first determine the maximum draft (dd), which is the reduction in thickness per pass: d=hi\u2212hf=0.18\u22120.16=0.02&nbsp;md = h_i &#8211; h_f = 0.18 &#8211; 0.16 = 0.02 \\text{ m}<\/p>\n\n\n\n<p>Using the equation for maximum draft: dmax\u2061=\u03bc2Rd_{\\max} = \\mu^2 R<\/p>\n\n\n\n<p>where RR is the roll radius: R=D2=0.62=0.3&nbsp;mR = \\frac{D}{2} = \\frac{0.6}{2} = 0.3 \\text{ m}<\/p>\n\n\n\n<p>To find the coefficient of friction (\u03bc\\mu), we approximate: \u03bc=0.3\\mu = 0.3<\/p>\n\n\n\n<p>Thus, the maximum draft is: dmax\u2061=(0.3)2\u00d70.3=0.027&nbsp;md_{\\max} = (0.3)^2 \\times 0.3 = 0.027 \\text{ m}<\/p>\n\n\n\n<p>Since our required draft (0.02 m) is <strong>less than the maximum draft (0.027 m)<\/strong>, the pass is feasible.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 3: Checking If Two Passes Are Feasible<\/strong><\/h3>\n\n\n\n<p>Each pass would need to reduce the thickness by: 0.18\u22120.162=0.01&nbsp;m&nbsp;per&nbsp;pass\\frac{0.18 &#8211; 0.16}{2} = 0.01 \\text{ m} \\text{ per pass}<\/p>\n\n\n\n<p>Since 0.010.01 m is significantly less than the max draft of 0.0270.027, the reduction can be done easily in two passes.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Conclusion<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Part (a):<\/strong> The final length of the rolled product is <strong>540 meters<\/strong>.<\/li>\n\n\n\n<li><strong>Part (b):<\/strong> The rolling job <strong>can<\/strong> be performed in one pass since the required draft is within the permissible limit.<\/li>\n\n\n\n<li><strong>Part (c):<\/strong> The rolling job <strong>can also be performed in two passes<\/strong>, and this would actually be easier since each pass would require a smaller reduction.<\/li>\n<\/ul>\n\n\n\n<p>Now, I&#8217;ll generate an image to represent the rolling process.<\/p>\n\n\n\n<p>I&#8217;ve generated a technical illustration showing the rolling process of the steel slab, including its transformation through the mill. Let me know if you need any modifications or additional explanations!<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-1716.png\" alt=\"\" class=\"wp-image-195737\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Problem Rectangular slabs cast from 1020 annealed steel are heated and flat-rolled to improve their metallurgical structure and reduce their thickness. The slabs are 1.5m\u00d79m\u00d718cm with a density of 7590kg\/m3. After passing through a 5-stand roughing mill, the rolled product will have a thickness of 3mm. Vertical rolls are used to prevent widening of the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195736","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195736","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195736"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195736\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195736"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195736"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195736"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}