Given Chvorinov’s Rule for solidification: Volume Solidification time = C Surface area A riser is to be designed for a sand casting mold. The casting is a cube with a side of 30 mm. The total solidification time of the casting itself is known to be 4 min. The riser is in the shape of a circular cylinder with its height being the same as its base diameter. (1) Determine the value of the mold constant C in Chvorinov’s rule (assuming n=2). (4 pts) Determine the height of the riser so that it will take 25% longer for the riser to solidify than the main casting. (2) (4 pts)<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n Alright, let’s go through this carefully and clearly:<\/p>\n\n\n\n Given:<\/strong><\/p>\n\n\n\n First, calculate volume ( V ) and surface area ( A ) for the casting<\/strong>:<\/p>\n\n\n\n Now apply Chvorinov\u2019s rule: \u2705 Mold Constant ( C = 0.16 \\, \\text{min\/mm}^2 )<\/strong><\/p>\n\n\n\n Now, we want the riser<\/strong> to solidify 25% longer<\/strong> than the casting: Set up Chvorinov\u2019s rule for the riser: For a cylinder with height ( h ) and diameter ( d ) where ( h = d ):<\/p>\n\n\n\n Thus: Now plug into Chvorinov\u2019s Rule: Since ( d = h ) (height = diameter):<\/p>\n\n\n\n \u2705 Riser height = 33.54 mm<\/strong><\/p>\n\n\n\n Chvorinov\u2019s Rule is fundamental in casting because it relates the solidification time<\/strong> of a casting to its volume-to-surface-area ratio<\/strong>. Larger volumes and smaller surface areas delay heat loss, increasing the solidification time. Here, the cube casting has a known side length of 30 mm and solidifies in 4 minutes. Calculating its volume (( 27,000 \\, \\text{mm}^3 )) and surface area (( 5,400 \\, \\text{mm}^2 )), we used these values to find the mold constant ( C ) from Chvorinov\u2019s Rule formula. The mold constant is crucial because it captures the thermal characteristics of the mold material and metal.<\/p>\n\n\n\n The riser must solidify slower<\/strong> than the casting to ensure it remains molten longer and can feed shrinkage defects. We want the riser to solidify 25% longer<\/strong> than the casting, meaning 5 minutes. The riser shape is a special cylinder where the height equals the diameter. Calculating the riser\u2019s volume and surface area with this condition allowed simplifying the ratio ( V\/A ) into a function of just the diameter ( d ).<\/p>\n\n\n\n Substituting into Chvorinov\u2019s formula, solving for ( d ), and taking the square root provided the diameter\u2014and thus the height\u2014needed for the riser. The final height calculated is approximately 33.54 mm, slightly larger than the casting’s side of 30 mm, as expected. This ensures the riser solidifies after the casting, fulfilling its role effectively. Precise riser design is critical in manufacturing to avoid defects, minimize waste, and optimize yield in casting processes.<\/p>\n\n\n\n Given Chvorinov’s Rule for solidification: Volume Solidification time = C Surface area A riser is to be designed for a sand casting mold. The casting is a cube with a side of 30 mm. The total solidification time of the casting itself is known to be 4 min. The riser is in the shape of […]<\/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-208025","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/208025","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=208025"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/208025\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=208025"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=208025"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=208025"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\n\n
[
t_s = C \\left( \\frac{V}{A} \\right)^2
]
where:<\/li>\n\n\n\n
\n\n\n\nStep 1: Find Mold Constant ( C )<\/h3>\n\n\n\n
\n
[
V_{\\text{casting}} = L^3 = 30^3 = 27,000 \\, \\text{mm}^3
]<\/li>\n\n\n\n
[
A_{\\text{casting}} = 6L^2 = 6 \\times 30^2 = 6 \\times 900 = 5,400 \\, \\text{mm}^2
]<\/li>\n<\/ul>\n\n\n\n
[
t_{\\text{casting}} = C \\left( \\frac{V_{\\text{casting}}}{A_{\\text{casting}}} \\right)^2
]
Substituting known values:
[
4 = C \\left( \\frac{27,000}{5,400} \\right)^2
]
Simplify:
[
\\frac{27,000}{5,400} = 5
]
Thus:
[
4 = C (5)^2
]
[
4 = C (25)
]
[
C = \\frac{4}{25}
]
[
C = 0.16 \\, \\text{min}\/\\left(\\text{mm}\\right)^2
]<\/p>\n\n\n\n
\n\n\n\nStep 2: Find Riser Height<\/h3>\n\n\n\n
[
t_{\\text{riser}} = 1.25 \\times t_{\\text{casting}} = 1.25 \\times 4 = 5 \\, \\text{min}
]<\/p>\n\n\n\n
[
t_{\\text{riser}} = C \\left( \\frac{V_{\\text{riser}}}{A_{\\text{riser}}} \\right)^2
]<\/p>\n\n\n\n\n
[
V_{\\text{riser}} = \\pi \\left( \\frac{d}{2} \\right)^2 h = \\pi \\left( \\frac{d}{2} \\right)^2 d = \\frac{\\pi}{4} d^3
]<\/li>\n\n\n\n
[
A_{\\text{riser}} = 2\\pi \\left( \\frac{d}{2} \\right)^2 + \\pi d h = \\pi \\frac{d^2}{2} + \\pi d^2 = \\frac{3\\pi}{2} d^2
]<\/li>\n<\/ul>\n\n\n\n
[
\\frac{V_{\\text{riser}}}{A_{\\text{riser}}} = \\frac{\\frac{\\pi}{4}d^3}{\\frac{3\\pi}{2}d^2} = \\frac{1}{6}d
]<\/p>\n\n\n\n
[
5 = 0.16 \\left( \\frac{d}{6} \\right)^2
]
Simplify:
[
5 = 0.16 \\times \\frac{d^2}{36}
]
[
5 = \\frac{0.16 d^2}{36}
]
Multiply both sides by 36:
[
180 = 0.16 d^2
]
Divide by 0.16:
[
d^2 = 1125
]
Take the square root:
[
d = \\sqrt{1125} \\approx 33.54 \\, \\text{mm}
]<\/p>\n\n\n\n
\n\n\n\n\ud83d\udcda 300-Word Explanation<\/h1>\n\n\n\n
\n","protected":false},"excerpt":{"rendered":"