{"id":204112,"date":"2025-03-21T11:39:17","date_gmt":"2025-03-21T11:39:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=204112"},"modified":"2025-03-21T11:39:19","modified_gmt":"2025-03-21T11:39:19","slug":"the-structure-of-a-house-is-such-that-it-loses-heat-at-a-rate-of-3800-kj-h-per-c-difference-between-the-indoors-and-outdoors","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/21\/the-structure-of-a-house-is-such-that-it-loses-heat-at-a-rate-of-3800-kj-h-per-c-difference-between-the-indoors-and-outdoors\/","title":{"rendered":"The structure of a house is such that it loses heat at a rate of 3800 kJ\/h per \u00b0C difference between the indoors and outdoors"},"content":{"rendered":"\n

The structure of a house is such that it loses heat at a rate of 3800 kJ\/h per \u00b0C difference between the indoors and outdoors. A heat pump that requires a power input of 4 kW is used to maintain this house at 24\u00b0C. Determine the lowest outdoor temperature for which the heat pump can meet the heating requirements of this house.<\/p>\n\n\n\n

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

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

The lowest outdoor temperature for which the heat pump can meet the heating requirements is \u221220\u00b0C<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Explanation:<\/h3>\n\n\n\n

A heat pump extracts heat from a cold environment and delivers it to a warmer space. The efficiency of the heat pump is measured using the Coefficient of Performance (COP)<\/strong> for heating, given by:<\/p>\n\n\n\n

[
COP = \\frac{Q_{in}}{W_{in}}
]<\/p>\n\n\n\n

where:<\/p>\n\n\n\n

    \n
  • ( Q_{in} ) = Heat delivered to the house (kJ\/h)<\/li>\n\n\n\n
  • ( W_{in} ) = Power input to the heat pump (kJ\/h)<\/li>\n<\/ul>\n\n\n\n

    Step 1: Heat Loss Calculation<\/h4>\n\n\n\n

    The rate of heat loss from the house is given as:<\/p>\n\n\n\n

    [
    Q_{\\text{loss}} = 3800 \\times (T_{\\text{in}} – T_{\\text{out}})
    ]<\/p>\n\n\n\n

    Given:<\/p>\n\n\n\n

      \n
    • Indoor temperature ( T_{\\text{in}} = 24^\\circ C )<\/li>\n\n\n\n
    • Outdoor temperature ( T_{\\text{out}} ) (to be determined)<\/li>\n\n\n\n
    • Heat pump power input ( W_{\\text{in}} = 4 \\text{ kW} = 14400 \\text{ kJ\/h} )<\/li>\n<\/ul>\n\n\n\n

      At steady-state, the heat delivered by the heat pump must equal the heat loss:<\/p>\n\n\n\n

      [
      Q_{\\text{in}} = Q_{\\text{loss}}
      ]<\/p>\n\n\n\n

      Step 2: COP of a Reversible Heat Pump<\/h4>\n\n\n\n

      For an ideal<\/strong> (Carnot) heat pump, the COP<\/strong> is given by:<\/p>\n\n\n\n

      [
      COP_{\\text{max}} = \\frac{T_{\\text{hot}}}{T_{\\text{hot}} – T_{\\text{cold}}}
      ]<\/p>\n\n\n\n

      where temperatures must be in Kelvin:<\/p>\n\n\n\n

      [
      T_{\\text{hot}} = 24 + 273 = 297K, \\quad T_{\\text{cold}} = T_{\\text{out}} + 273
      ]<\/p>\n\n\n\n

      Thus,<\/p>\n\n\n\n

      [
      COP_{\\text{max}} = \\frac{297}{297 – (T_{\\text{out}} + 273)}
      ]<\/p>\n\n\n\n

      [
      Q_{\\text{in}} = COP_{\\text{max}} \\times W_{\\text{in}}
      ]<\/p>\n\n\n\n

      Step 3: Solving for ( T_{\\text{out}} )<\/h4>\n\n\n\n

      Setting ( Q_{\\text{in}} = 3800 \\times (24 – T_{\\text{out}}) ):<\/p>\n\n\n\n

      [
      \\frac{297}{297 – (T_{\\text{out}} + 273)} \\times 14400 = 3800 \\times (24 – T_{\\text{out}})
      ]<\/p>\n\n\n\n

      Solving, ( T_{\\text{out}} = -20^\\circ C ).<\/p>\n","protected":false},"excerpt":{"rendered":"

      The structure of a house is such that it loses heat at a rate of 3800 kJ\/h per \u00b0C difference between the indoors and outdoors. A heat pump that requires a power input of 4 kW is used to maintain this house at 24\u00b0C. Determine the lowest outdoor temperature for which the heat pump can […]<\/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-204112","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204112","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=204112"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204112\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=204112"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=204112"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=204112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}