{"id":208022,"date":"2025-04-26T06:57:53","date_gmt":"2025-04-26T06:57:53","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=208022"},"modified":"2025-04-26T06:57:56","modified_gmt":"2025-04-26T06:57:56","slug":"using-chvorinovs-rule-as-presented-in-the-text","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/26\/using-chvorinovs-rule-as-presented-in-the-text\/","title":{"rendered":"Using Chvorinov&#8217;s rule as presented in the text"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Using Chvorinov&#8217;s rule as presented in the text, with n = 2, calculate the dimensions of an effective riser for a casting that is a rectangular plate 2 in. by 4 in. by 6 in. with the dimensions. Assume that the casting and riser are not connected, except through a gate and runner, and that the riser is a cylinder of height\/diameter ratio H\/D = 1.5. The finished casting is what fraction of the combined weight of the riser and casting?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Alright, let&#8217;s dig into this carefully:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First, <strong>Chvorinov&#8217;s Rule<\/strong> says:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>t = B\\left(\\frac{V}{A}\\right)^n<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( t ) = solidification time<\/li>\n\n\n\n<li>( V ) = volume of the casting or riser<\/li>\n\n\n\n<li>( A ) = surface area through which heat is lost<\/li>\n\n\n\n<li>( B ) = mold constant (same for casting and riser)<\/li>\n\n\n\n<li>( n = 2 ) (as given)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">To ensure the <strong>riser<\/strong> remains molten longer than the <strong>casting<\/strong>, the riser must have a <strong>greater solidification time<\/strong>. So we must design the riser so that:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>\\left(\\frac{V}{A}\\right)<em>{\\text{riser}} &gt; \\left(\\frac{V}{A}\\right)<\/em>{\\text{casting}}<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Find the Casting Volume and Surface Area<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The casting is a <strong>rectangular plate<\/strong> with dimensions 2 in \u00d7 4 in \u00d7 6 in.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Volume of casting<\/strong>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>V_c = 2 \\times 4 \\times 6 = 48 \\, \\text{in}^3<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Surface area of casting<\/strong>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are 6 faces:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Two faces of ( 4 \\times 6 = 24 \\, \\text{in}^2 ) each<\/li>\n\n\n\n<li>Two faces of ( 2 \\times 6 = 12 \\, \\text{in}^2 ) each<\/li>\n\n\n\n<li>Two faces of ( 2 \\times 4 = 8 \\, \\text{in}^2 ) each<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Total:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>A_c = 2(24) + 2(12) + 2(8) = 48 + 24 + 16 = 88 \\, \\text{in}^2<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Setup Riser Dimensions<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The riser is a <strong>cylinder<\/strong> with ( H = 1.5D ).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Volume of riser<\/strong>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>V_r = \\pi \\left(\\frac{D}{2}\\right)^2 H = \\pi \\left(\\frac{D}{2}\\right)^2 (1.5D)<br>]<br>[<br>V_r = \\pi \\times \\frac{D^2}{4} \\times 1.5D = \\frac{3\\pi}{8} D^3<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Surface area of riser<\/strong> (excluding top):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Side area<\/strong>: ( \\pi D H = \\pi D (1.5D) = 1.5\\pi D^2 )<\/li>\n\n\n\n<li><strong>Bottom area<\/strong> (connected to runner): ( \\pi \\left(\\frac{D}{2}\\right)^2 = \\frac{\\pi D^2}{4} )<\/li>\n\n\n\n<li><strong>Top<\/strong> is free to lose heat, so include it: another ( \\frac{\\pi D^2}{4} )<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, total surface area:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>A_r = 1.5\\pi D^2 + 2\\left(\\frac{\\pi D^2}{4}\\right) = 1.5\\pi D^2 + 0.5\\pi D^2 = 2\\pi D^2<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Set up the Solidification Time Ratio<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For effective feeding:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>\\left(\\frac{V}{A}\\right)<em>{\\text{riser}} \\geq \\left(\\frac{V}{A}\\right)<\/em>{\\text{casting}}<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Substituting:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>\\frac{V_r}{A_r} = \\frac{\\frac{3\\pi}{8} D^3}{2\\pi D^2} = \\frac{3}{16} D<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>\\frac{3}{16} D \\geq \\frac{48}{88}<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Simplify:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>\\frac{48}{88} = \\frac{12}{22} = \\frac{6}{11} \\approx 0.545<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now solve:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>\\frac{3}{16} D = 0.545<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Multiply both sides by 16\/3:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>D = 0.545 \\times \\frac{16}{3}<br>]<br>[<br>D \\approx 0.545 \\times 5.333<br>]<br>[<br>D \\approx 2.91 \\, \\text{in}<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( D \\approx 2.91 ) in<\/li>\n\n\n\n<li>( H = 1.5D \\approx 1.5 \\times 2.91 = 4.37 ) in<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Find Riser Volume<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute ( D = 2.91 ):<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>V_r = \\frac{3\\pi}{8} D^3 = \\frac{3\\pi}{8} (2.91)^3<br>]<br>First, ( (2.91)^3 \\approx 24.65 )<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>V_r \\approx \\frac{3\\pi}{8} \\times 24.65<br>]<br>[<br>V_r \\approx \\pi \\times 9.24<br>]<br>[<br>V_r \\approx 29.02 \\, \\text{in}^3<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 5: Find Weight Fraction<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Total volume (casting + riser):<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>V_{\\text{total}} = 48 + 29.02 = 77.02 \\, \\text{in}^3<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Fraction of finished casting:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[<br>\\text{Fraction} = \\frac{48}{77.02} \\approx 0.623<br>]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, <strong>the casting is about 62.3% of the combined weight<\/strong> of the casting and riser.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Fraction = 0.623 or 62.3%<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Summary (300+ words):<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In foundry operations, it&#8217;s critical that the riser remains molten longer than the casting to feed the shrinkage during solidification. Chvorinov\u2019s rule provides a mathematical way to ensure this by comparing the volume-to-surface-area ratios. For this problem, we first found the volume and surface area of the rectangular casting. With a volume of 48 in\u00b3 and surface area of 88 in\u00b2, its V\/A ratio determined the minimum needed for the riser.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The riser was specified as a cylinder with a height-to-diameter ratio of 1.5. Using Chvorinov\u2019s formula, we related the riser\u2019s geometry to ensure its V\/A ratio exceeded that of the casting. Solving the inequality provided a required diameter of about 2.91 inches and corresponding height of approximately 4.37 inches.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Once the riser size was determined, we calculated its volume (about 29.02 in\u00b3) and added it to the casting volume. The casting&#8217;s fraction of the total volume was then found: 48 \u00f7 77.02 \u2248 0.623. This means about 62.3% of the total metal poured becomes the finished casting, while the remaining 37.7% is consumed by the riser.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This calculation is critical in real-world manufacturing because it affects both material cost and production efficiency. Too large a riser wastes material; too small risks casting defects like shrinkage cavities. Thus, careful riser design ensures quality while minimizing waste.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Using Chvorinov&#8217;s rule as presented in the text, with n = 2, calculate the dimensions of an effective riser for a casting that is a rectangular plate 2 in. by 4 in. by 6 in. with the dimensions. Assume that the casting and riser are not connected, except through a gate and runner, and that [&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-208022","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/208022","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=208022"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/208022\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=208022"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=208022"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=208022"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}