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07.mdsd_modelado_termicos_liquidos
This document discusses heat transfer concepts and equations. It defines key terms like heat flux (Q), temperature (T), thermal resistance (RTH), and thermal capacitance (CTH). It presents equations for one-dimensional heat conduction through a wall or slab using these variables. The equations relate the rate of heat transfer to the temperature difference and thermal resistance. Additional equations scale these concepts to model heat transfer in systems with multiple lumped-capacity elements connected in series.
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07.mdsd_modelado_termicos_liquidos
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- 12. • • • 𝝀 𝟎. 𝟖µ𝒎 𝟎. 𝟖 𝒎𝒎
- 14.
- 15. 𝑹𝑻𝑯 = 𝒄𝒂𝒎𝒃𝒊𝒐 𝒆𝒏𝒍𝒂 𝒅𝒊𝒇𝒓𝒆𝒏𝒄𝒊𝒂 𝒅𝒆 𝒍𝒂 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒂 𝒄𝒂𝒎𝒃𝒊𝒐 𝒆𝒏 𝒆𝒍 𝒇𝒍𝒖𝒋𝒐 𝒅𝒆 𝒄𝒂𝒍𝒐𝒓 = 𝒅𝑻 𝒅𝑸
- 16.
- 17. 𝑪𝑻𝑯 = 𝒄𝒂𝒎𝒃𝒊𝒐 𝒆𝒏𝒆𝒍 𝒄𝒂𝒍𝒐𝒓 𝒂𝒍𝒎𝒂𝒄𝒆𝒏𝒂𝒅𝒐 𝒄𝒂𝒎𝒃𝒊𝒐 𝒆𝒏 𝒍𝒂 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒂
- 18.
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- 20.
- 21. Magnitudes físicas SímboloSistema Internacional Razón de flujo calorífico ሶ 𝑸 𝒌𝒄𝒂𝒍 𝒔 Temperatura 𝑻 𝑲 Calor específico 𝒄 𝒌𝒄𝒂𝒍 𝒌𝒈 ∙ 𝑲 Resistencia Térmica 𝑹𝑻𝑯 𝑲 𝒌𝒄𝒂𝒍 Capacitancia Térmica 𝑪𝑻𝑯 𝒌𝒄𝒂𝒍 𝑲 Coeficiente de transferencia de calor 𝒉 𝒌𝒄𝒂𝒍 𝒔 ∙ 𝒎𝟐 Masa 𝒎 𝒌𝒈 Área de intercambio de calor 𝑨 𝒎𝟐 Energía calorífica 𝒆 𝒌𝒄𝒂𝒍
- 22. Energía de entrada = Razón de energía entregada + Energía desalida ሶ 𝑸 = 𝒉𝑨 𝑻𝟐 − 𝑻𝟏 = 𝑻𝟐 − 𝑻𝟏 𝑹𝑻𝑯 𝑻𝟐 𝑻𝟏 𝒉 𝑨
- 23. ∆ ሶ 𝑸 =ሶ 𝑸𝟏 − ሶ 𝑸𝟐 ሶ 𝑸𝟏 − ሶ 𝑸𝟐 = 𝒎𝒄 𝒅𝑻 𝒅𝒕 = 𝑪𝑻𝑯 𝒅𝑻 𝒅𝒕 𝝆, 𝒍 𝑨 ሶ 𝑸𝟏 − ሶ 𝑸𝟐 = 𝝆𝑨𝒍𝒄 𝒅𝑻 𝒅𝒕 𝑪𝑻𝑯 = 𝝆𝑨𝒍𝒄
- 24. ሶ 𝑸 = 𝑨𝒉 𝒍 𝑻𝟐 −𝑻𝟏 ⇒ 𝑹𝑻𝑯 = 𝒍 𝑨𝒉 ሶ 𝑸 = 𝑨𝒉 𝑻𝟐 − 𝑻𝟏 ⇒ 𝑹𝑻𝑯 = 𝟏 𝑨𝒉 ሶ 𝑸 = 𝑨𝒉 𝑻𝟐 𝟒 − 𝑻𝟏 𝟒
- 25. 𝒎𝒄 𝒅𝑻 𝒅𝒕 = ሶ 𝑸𝒂𝒑𝒐𝒓𝒕𝒂𝒅𝒐 −ሶ 𝑸𝒅𝒆𝒔𝒑𝒓𝒆𝒏𝒅𝒊𝒅𝒐
- 26.
- 27. 𝒒𝒆 = 𝒒𝒔 ሶ 𝑸𝒆𝒏𝒕𝒓𝒆𝒈𝒂𝒅𝒐= 𝒎𝒄 𝒅𝑻𝑻 𝒅𝒕 + 𝑻𝑻 − 𝑻𝒂 𝑹𝑻𝑯 + 𝝆𝑽𝒄 𝑻𝒄 − 𝑻𝒇 𝒕
- 28. 𝒎 𝑹𝑻𝑯 𝝆 𝑽 𝑻𝒇= 𝑻𝒂 𝑻𝑻 = 𝑻𝒄 ሶ 𝑸𝒆𝒏𝒕𝒓𝒆𝒈𝒂𝒅𝒐 = 𝒎𝒄 𝒅𝑻𝒄 𝒅𝒕 + 𝑻𝒄 − 𝑻𝒇 𝑹𝑻𝑯 + 𝝆𝑽𝒄 𝑻𝒄 − 𝑻𝒇 𝒕
- 29. ሶ 𝑸𝒆𝒏𝒕𝒓𝒆𝒈𝒂𝒅𝒐 = 𝑪𝑻𝑯 𝒅𝑻𝒄 𝒅𝒕 +𝑻𝒄 − 𝑻𝒇 𝟏 𝑹𝑻𝑯 + 𝝆𝒒𝒆𝒄 𝑪𝑻𝑯
- 30. 𝚫𝑸𝒆𝒏𝒕𝒓𝒆𝒈𝒂𝒅𝒐 − 𝝆𝚫𝑻𝒄𝟎𝚫𝒒𝒆 = 𝑪𝑻𝑯 𝒅𝑻𝒄 𝒅𝒕 + 𝟏 𝑹𝑻𝑯 + 𝝆𝒒𝒆𝒄 𝚫𝑻𝒄
- 31.
- 32. 𝐶𝑇1 𝑑𝑇1 𝑑𝑡 = ሶ 𝑄 −ሶ 𝑄1 𝐶𝑇1 𝑑𝑇1 𝑑𝑡 = ሶ 𝑄 − 𝑇1 − 𝑇2 𝑅𝑇1
- 33. 𝐶𝑇2 𝑑𝑇2 𝑑𝑡 = ሶ 𝑄1 −ሶ 𝑄2 𝐶𝑇2 𝑑𝑇2 𝑑𝑡 = 𝑇1 − 𝑇2 𝑅𝑇1 − 𝑇2 − 𝑇𝑎 𝑅𝑇2
- 34. 𝑣𝑖𝑛 𝑡 =𝑣𝑅 + 𝑣𝐿 𝑣𝑖𝑛 𝑡 = 𝑖 𝑡 𝑅 + 𝐿 𝑑𝑖(𝑡) 𝑑𝑡 𝐿 𝑑𝑖(𝑡) 𝑑𝑡 + 𝑖 𝑡 𝑅 = 𝑣𝑖𝑛 𝑡
- 35. 𝑃 = 𝑖2 𝑡𝑅 ⟹ ሶ 𝑄 = 𝑖2 𝑡 𝑅 𝐶𝑇1 𝑑𝑇1 𝑑𝑡 = 𝑖2 𝑡 𝑅 − 𝑇1 − 𝑇2 𝑅𝑇1
- 36. 𝑑𝑇1 𝑑𝑡 = 1 𝐶𝑇1 𝑖2 𝑡 𝑅 − 𝑇1− 𝑇2 𝑅𝑇1 𝑑𝑇2 𝑑𝑡 = 1 𝐶𝑇2 𝑇1 − 𝑇2 𝑅𝑇1 − 𝑇2 − 𝑇𝑎 𝑅𝑇2 𝑑𝑖(𝑡) 𝑑𝑡 = 1 𝐿 𝑣𝑖𝑛 𝑡 − 𝑖 𝑡 𝑅
- 38.
- 40.
- 41. 𝐴 = 𝜋𝑟2 𝑉= 𝐴ℎ 𝑉 = 𝜋𝑟2 ℎ 𝒓 𝒉
- 42. 𝐴 = 𝜋𝑟2 𝑉= 1 3 𝜋𝑟2 ℎ ℎ = 𝑙 sin 𝛼 𝛼 = 45° → ℎ = 𝑟 𝑉 = 1 3 𝜋ℎ3 𝒓 𝒉 𝜶 𝒍
- 43.
- 44. 𝑽 𝒕 =𝑨𝒉(𝒕) 𝒅𝑽 𝒕 𝒅𝒕 = 𝑨 𝒅𝒉(𝒕) 𝒅𝒕 = 𝑨 ሶ 𝒉(𝒕)
- 45. 𝒒𝒊 𝒕 =𝒂𝒊𝑺 𝐬𝐠𝐧 ∆𝒉(𝒕) 𝟐𝒈 ∆𝒉(𝒕) 𝒂𝒊 𝑺 𝒔𝒈𝒏(𝒙): ∆𝒉(𝒕): 𝒈:
- 46. ∆𝒒 𝒕 = 𝒒𝒊𝒏 𝒕 − 𝒒𝒐𝒖𝒕(𝒕)
- 47. 𝒅𝑽 𝒕 𝒅𝒕 = ∆𝒒𝒕 𝑨 𝒅𝒉(𝒕) 𝒅 = 𝒒𝒊𝒏 𝒕 − 𝒒𝒐𝒖𝒕(𝒕) 𝑨 ሶ 𝒉 𝒕 = 𝒒𝒊𝒏 𝒕 − 𝒂𝒊𝑺 𝐬𝐠𝐧 ∆𝒉(𝒕) 𝟐𝒈 ∆𝒉(𝒕)
- 48. ሶ 𝒉 𝒕 = 𝟏 𝑨 𝒒𝒊𝒏𝒕 − 𝒂𝒊𝑺 𝐬𝐠𝐧 ∆𝒉(𝒕) 𝟐𝒈 ∆𝒉(𝒕)
- 49.
- 50. 𝑽𝟏 𝒕 =𝑨𝒉𝟏(𝒕) 𝑽𝟐 𝒕 = 𝑨𝒉𝟐(𝒕) 𝒅𝑽𝟏(𝒕) 𝒅𝒕 = 𝑨 ሶ 𝒉𝟏(𝒕) 𝒅𝑽𝟐(𝒕) 𝒅𝒕 = 𝑨 ሶ 𝒉𝟐(𝒕)
- 51. ∆𝒒𝑻𝟏 𝒕 =𝒒𝟏 − 𝒒𝟏𝟐 ∆𝒒𝑻𝟐 𝒕 = 𝒒𝟏𝟐 − 𝒒𝟐𝟎
- 52. 𝒒𝟏: 𝒒𝟏𝟐 = 𝒂𝟏𝟐𝑺𝐬𝐠𝐧𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) 𝟐𝒈 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) 𝒒𝟐𝟎 = 𝒂𝟐𝟎𝑺𝐬𝐠𝐧 𝒉𝟐(𝒕) 𝟐𝒈 𝒉𝟐(𝒕)
- 53. 𝑨 ሶ 𝒉𝟏 𝒕= 𝒒𝟏 − 𝒂𝟏𝟐𝑺𝐬𝐠𝐧 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) 𝟐𝒈 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) 𝑨 ሶ 𝒉𝟐 𝒕 = 𝒂𝟏𝟐𝑺𝐬𝐠𝐧 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) 𝟐𝒈 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) − 𝒂𝟐𝟎𝑺𝐬𝐠𝐧 𝒉𝟐(𝒕) 𝟐𝒈 𝒉𝟐(𝒕) ሶ 𝒉𝟏 𝒕 = 𝟏 𝑨 𝒒𝟏 − 𝒂𝟏𝟐𝑺𝐬𝐠𝐧 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) 𝟐𝒈 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) ሶ 𝒉𝟐 𝒕 = 𝟏 𝑨 𝒂𝟏𝟐𝑺𝐬𝐠𝐧 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) 𝟐𝒈 𝒉𝟏(𝒕) − 𝒉𝟐(𝒕) − 𝒂𝟐𝟎𝑺𝐬𝐠𝐧 𝒉𝟐(𝒕) 𝟐𝒈 𝒉𝟐(𝒕)
- 56. 𝑉1 𝑡 =𝐴ℎ1(𝑡) 𝑉2 𝑡 = 𝐴ℎ2(𝑡) 𝑉3 𝑡 = 𝐴ℎ2(𝑡) 𝑑𝑉1(𝑡) 𝑑𝑡 = 𝐴 ሶ ℎ1(𝑡) 𝑑𝑉2(𝑡) 𝑑𝑡 = 𝐴 ሶ ℎ2(𝑡) 𝑑𝑉3(𝑡) 𝑑𝑡 = 𝐴 ሶ ℎ3(𝑡)
- 57. ∆𝑞𝑇1 𝑡 =𝑞1 − 𝑞13 ∆𝑞𝑇2 𝑡 = 𝑞2 + 𝑞32 − 𝑞20 ∆𝑞𝑇3 𝑡 = 𝑞13 − 𝑞32
- 58. 𝑞1: 𝑞2: 𝑞13 = 𝑎13𝑆sgnℎ1(𝑡) − ℎ3(𝑡) 2𝑔 ℎ1(𝑡) − ℎ3(𝑡) 𝑞32 = 𝑎32𝑆sgn ℎ3(𝑡) − ℎ2(𝑡) 2𝑔 ℎ3(𝑡) − ℎ2(𝑡) 𝑞20 = 𝑎20𝑆sgn ℎ2(𝑡) 2𝑔 ℎ2(𝑡)
- 59. 𝐴 ሶ ℎ1 𝑡= 𝑞1 − 𝑞13 𝐴 ሶ ℎ2 𝑡 = 𝑞2 + 𝑞32 − 𝑞20 𝐴 ሶ ℎ3 𝑡 = 𝑞13 − 𝑞32
- 60. 𝑻𝟏 𝐴 ሶ ℎ1 𝑡= 𝑞1 − 𝑎13𝑆sgn ℎ1(𝑡) − ℎ3(𝑡) 2𝑔 ℎ1(𝑡) − ℎ3(𝑡) 𝑻𝟐 𝐴 ሶ ℎ2 𝑡 = 𝑞2 + 𝑎32𝑆sgn ℎ3(𝑡) − ℎ2(𝑡) 2𝑔 ℎ3(𝑡) − ℎ2(𝑡) − 𝑎20𝑆sgn ℎ2(𝑡) 2𝑔 ℎ2(𝑡) 𝑻𝟑 𝐴 ሶ ℎ3 𝑡 = 𝑎13𝑆sgn ℎ1(𝑡) − ℎ3(𝑡) 2𝑔 ℎ1(𝑡) − ℎ3(𝑡) − 𝑎32𝑆sgn ℎ3(𝑡) − ℎ2(𝑡) 2𝑔 ℎ3(𝑡) − ℎ2(𝑡)
- 61. ሶ ℎ1 𝑡 = 1 𝐴 𝑞1− 𝑎13𝑆sgn ℎ1(𝑡) − ℎ3(𝑡) 2𝑔 ℎ1(𝑡) − ℎ3(𝑡) ሶ ℎ2 𝑡 = 1 𝐴 𝑞2 + 𝑎32𝑆sgn ℎ3(𝑡) − ℎ2(𝑡) 2𝑔 ℎ3(𝑡) − ℎ2(𝑡) − 𝑎20𝑆sgn ℎ2(𝑡) 2𝑔 ℎ2(𝑡) ሶ ℎ3 𝑡 = 1 𝐴 𝑎13𝑆sgn ℎ1(𝑡) − ℎ3(𝑡) 2𝑔 ℎ1(𝑡) − ℎ3(𝑡) − 𝑎32𝑆sgn ℎ3(𝑡) − ℎ2(𝑡) 2𝑔 ℎ3(𝑡) − ℎ2(𝑡)