{"id":181843,"date":"2025-01-11T16:23:10","date_gmt":"2025-01-11T16:23:10","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=181843"},"modified":"2025-01-11T16:23:12","modified_gmt":"2025-01-11T16:23:12","slug":"the-air-fuel-ratio-of-a-diesel-engine-is-291","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/11\/the-air-fuel-ratio-of-a-diesel-engine-is-291\/","title":{"rendered":"The air-fuel ratio of a Diesel engine is 29:1"},"content":{"rendered":"\n

The air-fuel ratio of a Diesel engine is 29:1. If the compression ratio is 16:1 and the temperature at the end of compression is 900 K, find at what cylinder volume the combustion is complete? Express this volume as a percentage of stroke. Assume that the combustion begins at the top dead centre and takes place at constant pressure. Take calorific value of the fuel as 42000 kJ\/kg, R-0.287 kJ\/kg K and Cv0.709+0.000028 TkJ\/kg K.<\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To solve this problem, we need to find the cylinder volume at which the combustion is complete, expressed as a percentage of the stroke. Here’s a step-by-step explanation and calculation:<\/p>\n\n\n\n

\n\n\n\n

Given Data:<\/strong><\/h3>\n\n\n\n
    \n
  • Air-fuel ratio, AFR=29:1AFR = 29:1<\/li>\n\n\n\n
  • Compression ratio, CR=16:1CR = 16:1<\/li>\n\n\n\n
  • Temperature at the end of compression, Tc=900\u2009KT_c = 900 \\, \\text{K}<\/li>\n\n\n\n
  • Calorific value of the fuel, CV=42000\u2009kJ\/kg\\text{CV} = 42000 \\, \\text{kJ\/kg}<\/li>\n\n\n\n
  • Specific gas constant, R=0.287\u2009kJ\/kg\\cdotpKR = 0.287 \\, \\text{kJ\/kg\u00b7K}<\/li>\n\n\n\n
  • Specific heat at constant volume, Cv=0.709+0.000028T\u2009kJ\/kg\\cdotpKC_v = 0.709 + 0.000028T \\, \\text{kJ\/kg\u00b7K}<\/li>\n<\/ul>\n\n\n\n

    Assumptions:<\/strong><\/h3>\n\n\n\n
      \n
    1. Combustion occurs at constant pressure.<\/li>\n\n\n\n
    2. Combustion begins at Top Dead Center (TDC).<\/li>\n<\/ol>\n\n\n\n
      \n\n\n\n

      Step-by-Step Solution:<\/strong><\/h3>\n\n\n\n
        \n
      1. Volume at Top Dead Center (TDC):<\/strong> At TDC, the cylinder is at its minimum volume, VTDCV_{TDC}, which corresponds to the clearance volume, VcV_c. CR=VBDCVTDC=16\u21d2VTDC=VBDC16CR = \\frac{V_{BDC}}{V_{TDC}} = 16 \\quad \\Rightarrow \\quad V_{TDC} = \\frac{V_{BDC}}{16}<\/li>\n\n\n\n
      2. Heat Released by Combustion:<\/strong> The heat released by the combustion, QQ, is: Q=CVAFR=4200029=1448.28\u2009kJ\/kgQ = \\frac{\\text{CV}}{AFR} = \\frac{42000}{29} = 1448.28 \\, \\text{kJ\/kg}<\/li>\n\n\n\n
      3. Temperature Rise Due to Combustion:<\/strong> Using Q=m\u22c5Cv\u22c5\u0394TQ = m \\cdot C_v \\cdot \\Delta T and assuming CvC_v varies with temperature, the mean value of CvC_v over the temperature range can be used. For simplicity, we approximate CvC_v at the end of compression: Cv=0.709+0.000028\u22c5900=0.7342\u2009kJ\/kg\\cdotpKC_v = 0.709 + 0.000028 \\cdot 900 = 0.7342 \\, \\text{kJ\/kg\u00b7K} Assuming unit mass of air, the temperature rise is: \u0394T=QCv=1448.280.7342\u22481973.23\u2009K\\Delta T = \\frac{Q}{C_v} = \\frac{1448.28}{0.7342} \\approx 1973.23 \\, \\text{K} The final temperature after combustion: Tf=Tc+\u0394T=900+1973.23\u22482873.23\u2009KT_f = T_c + \\Delta T = 900 + 1973.23 \\approx 2873.23 \\, \\text{K}<\/li>\n\n\n\n
      4. Volume at End of Combustion:<\/strong> Since the combustion occurs at constant pressure, the relation between temperature and volume is: TcVTDC=TfVf\u21d2Vf=VTDC\u22c5TfTc\\frac{T_c}{V_{TDC}} = \\frac{T_f}{V_f} \\quad \\Rightarrow \\quad V_f = V_{TDC} \\cdot \\frac{T_f}{T_c} Substituting values: Vf=VBDC16\u22c52873.23900=VBDC16\u22c53.192V_f = \\frac{V_{BDC}}{16} \\cdot \\frac{2873.23}{900} = \\frac{V_{BDC}}{16} \\cdot 3.192 Vf=3.19216\u22c5VBDC=0.1995\u22c5VBDCV_f = \\frac{3.192}{16} \\cdot V_{BDC} = 0.1995 \\cdot V_{BDC} The volume as a percentage of the stroke is: %Vf=19.95%\\%V_f = 19.95\\%<\/li>\n<\/ol>\n\n\n\n
        \n\n\n\n

        Final Answer:<\/strong><\/h3>\n\n\n\n

        The combustion is complete at 19.95% of the stroke<\/strong>.<\/p>\n\n\n\n

        \n\n\n\n

        Explanation:<\/strong><\/h3>\n\n\n\n
          \n
        1. At TDC, the engine cylinder is at minimum volume, corresponding to the clearance volume.<\/li>\n\n\n\n
        2. During combustion at constant pressure, the temperature rises significantly, causing the gas to expand.<\/li>\n\n\n\n
        3. Using thermodynamic relationships and given data, the final volume is determined based on the temperature rise and compression ratio.<\/li>\n\n\n\n
        4. Expressing the final volume as a percentage of the total stroke allows us to understand how much expansion has occurred relative to the engine\u2019s full range of motion.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

          The air-fuel ratio of a Diesel engine is 29:1. If the compression ratio is 16:1 and the temperature at the end of compression is 900 K, find at what cylinder volume the combustion is complete? Express this volume as a percentage of stroke. Assume that the combustion begins at the top dead centre and takes […]<\/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-181843","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/181843","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=181843"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/181843\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=181843"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=181843"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=181843"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}