{"id":39667,"date":"2025-06-27T08:59:14","date_gmt":"2025-06-27T08:59:14","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=39667"},"modified":"2025-06-27T08:59:54","modified_gmt":"2025-06-27T08:59:54","slug":"design-a-disk-centrifuge-to-separate-yeast-cells-from-beer-with-the-following-characteristics","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/design-a-disk-centrifuge-to-separate-yeast-cells-from-beer-with-the-following-characteristics\/","title":{"rendered":"Design a disk centrifuge to separate yeast cells from beer with the following characteristics"},"content":{"rendered":"\n<p>Design a disk centrifuge to separate yeast cells from beer with the following characteristics: 1. Flow rate = 400 m\u00b3\/day 2. Density of beer = 1020 kg\/m\u00b3 3. Density of yeast cells = 1075 kg\/m\u00b3 4. Average size of yeast cell = 4 \u00b5m. It is needed to specify the outside diameter of the disk, inside diameter of the disk, number of disk, rotational speed, Sigma (?) value and angle of the disk with the vertical axis (?). Assume necessary data to design the centrifuge and also provide the justification of the assumptions.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Disk Centrifuge Design for Yeast Separation from Beer<\/strong><\/h3>\n\n\n\n<p><strong>Given Data:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Flow rate (Q): 400 m\u00b3\/day = 4.63 L\/s<\/li>\n\n\n\n<li>Beer density (\u03c1f): 1020 kg\/m\u00b3<\/li>\n\n\n\n<li>Yeast density (\u03c1p): 1075 kg\/m\u00b3<\/li>\n\n\n\n<li>Yeast diameter (dp): 4 \u00b5m = 4 \u00d7 10\u207b\u2076 m<\/li>\n\n\n\n<li>Viscosity of beer (\u03bc): assumed as 1.5 \u00d7 10\u207b\u00b3 Pa\u00b7s<\/li>\n\n\n\n<li>Disk angle (\u03b8): assume 45\u00b0<\/li>\n\n\n\n<li>Acceleration due to gravity (g): 9.81 m\/s\u00b2<\/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\"><strong>Assumptions and Justifications:<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Disk Angle (\u03b8) = 45\u00b0<\/strong>: Commonly used angle in disk-stack centrifuges for good compromise between sedimentation area and flow resistance.<\/li>\n\n\n\n<li><strong>Disk Radii<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Outer radius (R\u2081): 0.20 m<\/li>\n\n\n\n<li>Inner radius (R\u2082): 0.05 m<br>These are typical values allowing reasonable residence time and Sigma (\u03a3) values.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Number of Disks (n):<\/strong> 150 \u2013 balances sedimentation area and centrifuge size.<\/li>\n\n\n\n<li><strong>Rotational Speed (\u03c9):<\/strong> Chosen to provide sufficient centrifugal force. Assume 6000 rpm (628 rad\/s).<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Calculations:<\/strong><\/h3>\n\n\n\n<p><strong>1. Terminal settling velocity (Stokes\u2019 Law):<\/strong>vt=(\u03c1p\u2212\u03c1f)gdp218\u03bc=(1075\u22121020)(9.81)(4\u00d710\u22126)218\u00d71.5\u00d710\u22123\u22483.6\u00d710\u22128\u2009m\/sv_t = \\frac{(\u03c1_p &#8211; \u03c1_f) g d_p^2}{18\u03bc} = \\frac{(1075 &#8211; 1020)(9.81)(4 \\times 10^{-6})^2}{18 \\times 1.5 \\times 10^{-3}} \u2248 3.6 \\times 10^{-8} \\, \\text{m\/s}vt\u200b=18\u03bc(\u03c1p\u200b\u2212\u03c1f\u200b)gdp2\u200b\u200b=18\u00d71.5\u00d710\u22123(1075\u22121020)(9.81)(4\u00d710\u22126)2\u200b\u22483.6\u00d710\u22128m\/s<\/p>\n\n\n\n<p><strong>2. Equivalent settling area (\u03a3):<\/strong>\u03a3=2\u03c0n(R12\u2212R22)sin\u2061\u03b8=2\u03c0\u00d7150(0.202\u22120.052)sin\u206145\u00b0\u224835.3\u2009m2\u03a3 = \\frac{2\u03c0 n (R\u2081^2 &#8211; R\u2082^2)}{\\sin\u03b8} = \\frac{2\u03c0 \\times 150 (0.20^2 &#8211; 0.05^2)}{\\sin45\u00b0} \u2248 35.3 \\, \\text{m}^2\u03a3=sin\u03b82\u03c0n(R12\u200b\u2212R22\u200b)\u200b=sin45\u00b02\u03c0\u00d7150(0.202\u22120.052)\u200b\u224835.3m2<\/p>\n\n\n\n<p><strong>3. Required \u03a3 for separation:<\/strong>\u03a3required=Qvt=4.63\u00d710\u221233.6\u00d710\u22128\u2248128,600\u2009m2\u03a3_{required} = \\frac{Q}{v_t} = \\frac{4.63 \\times 10^{-3}}{3.6 \\times 10^{-8}} \u2248 128,600 \\, \\text{m}^2\u03a3required\u200b=vt\u200bQ\u200b=3.6\u00d710\u221284.63\u00d710\u22123\u200b\u2248128,600m2<\/p>\n\n\n\n<p><strong>4. Speed correction using centrifugal enhancement:<\/strong><br>Effective settling velocity increases by factor (\u03c92r)\/g(\u03c9^2 r) \/ g(\u03c92r)\/g<\/p>\n\n\n\n<p>Use average radius r=(R1+R2)\/2=0.125r = (R\u2081 + R\u2082)\/2 = 0.125r=(R1\u200b+R2\u200b)\/2=0.125 mvcentrifugal=vt\u22c5\u03c92rg=3.6\u00d710\u22128\u22c5(628)2\u22c50.1259.81\u22481.82\u00d710\u22123\u2009m\/sv_{centrifugal} = v_t \\cdot \\frac{\u03c9^2 r}{g} = 3.6 \\times 10^{-8} \\cdot \\frac{(628)^2 \\cdot 0.125}{9.81} \u2248 1.82 \\times 10^{-3} \\, \\text{m\/s}vcentrifugal\u200b=vt\u200b\u22c5g\u03c92r\u200b=3.6\u00d710\u22128\u22c59.81(628)2\u22c50.125\u200b\u22481.82\u00d710\u22123m\/s\u03a3new=Qvcentrifugal=4.63\u00d710\u221231.82\u00d710\u22123\u22482.54\u2009m2\u03a3_{new} = \\frac{Q}{v_{centrifugal}} = \\frac{4.63 \\times 10^{-3}}{1.82 \\times 10^{-3}} \u2248 2.54 \\, \\text{m}^2\u03a3new\u200b=vcentrifugal\u200bQ\u200b=1.82\u00d710\u221234.63\u00d710\u22123\u200b\u22482.54m2<\/p>\n\n\n\n<p>Our design \u03a3 = 35.3 m\u00b2 &gt; 2.54 m\u00b2 \u21d2 <strong>Sufficient<\/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>Final Design Parameters:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Outer Disk Diameter (D\u2081):<\/strong> 0.40 m<\/li>\n\n\n\n<li><strong>Inner Disk Diameter (D\u2082):<\/strong> 0.10 m<\/li>\n\n\n\n<li><strong>Number of Disks:<\/strong> 150<\/li>\n\n\n\n<li><strong>Rotational Speed:<\/strong> 6000 rpm<\/li>\n\n\n\n<li><strong>Sigma (\u03a3):<\/strong> 35.3 m\u00b2<\/li>\n\n\n\n<li><strong>Disk Angle with Vertical:<\/strong> 45\u00b0<\/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\"><strong> Explanation:<\/strong><\/h3>\n\n\n\n<p>This design outlines a disk centrifuge to separate yeast cells from beer at an industrial flow rate of 400 cubic meters per day. The separation relies on density differences between yeast (1075 kg\/m\u00b3) and beer (1020 kg\/m\u00b3), with yeast cells averaging 4 microns in diameter. Due to the small particle size and minimal density difference, gravitational settling is inefficient. A centrifuge increases the settling velocity using centrifugal force, greatly improving separation efficiency.<\/p>\n\n\n\n<p>The settling velocity under gravity was calculated using Stokes\u2019 Law, yielding approximately 3.6 \u00d7 10\u207b\u2078 m\/s. This would require an enormous settling area to handle the desired flow rate. Therefore, centrifugal acceleration was considered. By assuming a speed of 6000 rpm and an average disk radius of 0.125 meters, the effective settling velocity increases to approximately 1.82 \u00d7 10\u207b\u00b3 m\/s, which significantly reduces the required settling area.<\/p>\n\n\n\n<p>A disk stack with 150 disks, outer diameter 0.40 meters, inner diameter 0.10 meters, and a disk inclination angle of 45 degrees results in a Sigma value of about 35.3 m\u00b2. This exceeds the required Sigma for effective yeast removal, confirming the design\u2019s suitability. The 45\u00b0 angle balances effective sedimentation and fluid flow, while the selected diameters are practical for commercial centrifuges. The high number of disks maximizes settling area without expanding the centrifuge footprint excessively.<\/p>\n\n\n\n<p>In conclusion, the proposed centrifuge design effectively leverages centrifugal acceleration to overcome the limitations of gravitational separation. It offers a robust solution for yeast removal in beer processing while maintaining feasible dimensions and operation parameters. This ensures clarity, product quality, and efficient process throughput.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1176.jpeg\" alt=\"\" class=\"wp-image-39676\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1176.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1176-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1176-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Design a disk centrifuge to separate yeast cells from beer with the following characteristics: 1. Flow rate = 400 m\u00b3\/day 2. Density of beer = 1020 kg\/m\u00b3 3. Density of yeast cells = 1075 kg\/m\u00b3 4. Average size of yeast cell = 4 \u00b5m. It is needed to specify the outside diameter of the disk, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-39667","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39667","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=39667"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39667\/revisions"}],"predecessor-version":[{"id":39677,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39667\/revisions\/39677"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=39667"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=39667"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=39667"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}