A horizontal pipe carries a smoothly flowing liquid with density 1.19 A1 103 kg\/m3. At locations 1 and 2 along the pipe the diameters are 6.41 cm and 2.61 cm, respectively. Is the flow speed at location 2 higher or lower than at location 1?<\/p>\n\n\n\n
The flow speed at location 1 is 2.03 m\/s. What is the pressure difference between location 2 and location 1 (including its sign)? Ignore viscosity.<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The question asks for the comparison of flow speed at two locations in a pipe and the pressure difference between those two locations. Let’s break this down step by step using principles from fluid dynamics, specifically the continuity equation and Bernoulli’s equation.<\/p>\n\n\n\n The continuity equation<\/strong> ensures mass conservation in a flowing fluid. It is given by:<\/p>\n\n\n\n [ Where:<\/p>\n\n\n\n The area of the pipe is related to its diameter by the equation:<\/p>\n\n\n\n [ Since the pipe narrows from location 1 to location 2 (the diameter decreases), the area at location 2 will be smaller than at location 1. According to the continuity equation, if the area decreases, the flow speed must increase to maintain the same volumetric flow rate.<\/p>\n\n\n\n Thus, the flow speed at location 2 will be higher than at location 1<\/strong>.<\/p>\n\n\n\n Bernoulli\u2019s equation relates the pressure, velocity, and elevation of a fluid in steady flow. It is expressed as:<\/p>\n\n\n\n [ Where:<\/p>\n\n\n\n Rearranging the equation to find the pressure difference:<\/p>\n\n\n\n [ Given that (v_2 > v_1) (as established earlier), it follows that (v_2^2 > v_1^2), so:<\/p>\n\n\n\n [ Thus, the pressure at location 2 will be lower than at location 1<\/strong>.<\/p>\n\n\n\n We are given:<\/p>\n\n\n\n First, find the areas (A_1) and (A_2):<\/p>\n\n\n\n [ [ Using the continuity equation, we can solve for (v_2):<\/p>\n\n\n\n [ Now, substitute the values into Bernoulli\u2019s equation to find the pressure difference:<\/p>\n\n\n\n [ [ [ [ So, the pressure difference is approximately -86.1 kPa<\/strong>, indicating that the pressure at location 2 is lower than at location 1.<\/p>\n\n\n\n The flow speed at location 2 is higher than at location 1, and the pressure at location 2 is lower than at location 1, with a pressure difference of approximately -86.1 kPa<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":" A horizontal pipe carries a smoothly flowing liquid with density 1.19 A1 103 kg\/m3. At locations 1 and 2 along the pipe the diameters are 6.41 cm and 2.61 cm, respectively. Is the flow speed at location 2 higher or lower than at location 1? The flow speed at location 1 is 2.03 m\/s. What […]<\/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-190507","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190507","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=190507"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190507\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190507"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190507"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190507"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Continuity Equation<\/h3>\n\n\n\n
A_1 v_1 = A_2 v_2
]<\/p>\n\n\n\n\n
A = \\pi \\left( \\frac{D}{2} \\right)^2
]<\/p>\n\n\n\nStep 2: Bernoulli\u2019s Equation<\/h3>\n\n\n\n
P_1 + \\frac{1}{2} \\rho v_1^2 = P_2 + \\frac{1}{2} \\rho v_2^2
]<\/p>\n\n\n\n\n
P_2 – P_1 = \\frac{1}{2} \\rho \\left( v_1^2 – v_2^2 \\right)
]<\/p>\n\n\n\n
P_2 – P_1 < 0
]<\/p>\n\n\n\nStep 3: Calculate the Pressure Difference<\/h3>\n\n\n\n
\n
A_1 = \\pi \\left( \\frac{D_1}{2} \\right)^2 = \\pi \\left( \\frac{0.0641}{2} \\right)^2 \\approx 3.23 \\times 10^{-3} \\, \\text{m}^2
]<\/p>\n\n\n\n
A_2 = \\pi \\left( \\frac{D_2}{2} \\right)^2 = \\pi \\left( \\frac{0.0261}{2} \\right)^2 \\approx 5.38 \\times 10^{-4} \\, \\text{m}^2
]<\/p>\n\n\n\n
A_1 v_1 = A_2 v_2 \\quad \\Rightarrow \\quad v_2 = \\frac{A_1 v_1}{A_2} = \\frac{3.23 \\times 10^{-3} \\times 2.03}{5.38 \\times 10^{-4}} \\approx 12.2 \\, \\text{m\/s}
]<\/p>\n\n\n\n
P_2 – P_1 = \\frac{1}{2} \\times 1.19 \\times 10^3 \\times \\left( 2.03^2 – 12.2^2 \\right)
]<\/p>\n\n\n\n
P_2 – P_1 = \\frac{1}{2} \\times 1.19 \\times 10^3 \\times \\left( 4.1209 – 148.84 \\right)
]<\/p>\n\n\n\n
P_2 – P_1 = \\frac{1}{2} \\times 1.19 \\times 10^3 \\times (-144.7191)
]<\/p>\n\n\n\n
P_2 – P_1 \\approx -86122.3 \\, \\text{Pa}
]<\/p>\n\n\n\nConclusion:<\/h3>\n\n\n\n