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### U V A L U E

1. 1. University of Balamand ALBA H.V.A.C . Heat Transmission By Eng. Wael Zmerly – 2007-2008 University of Balamand - ALBA ١ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008
2. 2. THERMAL CONDUCTIVITY Homogeneous Material “λ = constant” Isotropic λ λ Thermal conductivity λ of the material (W/m.°C) λv Transmission by vibrations of atoms or molecules λ λe Transmission by the free electrons Wood Brick Copper Glass Iron Air Glass Fiber 0.21 0.52 386 0.74 85 0.024 0.046 (W/m.°C) University of Balamand - ALBA ٢ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008 One will consider the Homogènes solids (characteristic physics and identical mechanics in any point) and Isotropic (even characteristic in all the directions). Thus, some of them, will depend only on the temperature, the influence of the pressure being neglected. There are two mechanisms for conduction in the solids: a heat transfer by the vibrations of the atoms or molecules that one characterizes it by a coefficient λϖ and a heat transfer by the free electrons characterized by a coefficient λε. Thermal conductivity Λ of a body will be such as: λ = λϖ + ιτ λ is the thermal coefficient of conductivity expressed out of W/m.°C It is a function of the temperature, but in the intervals of temperatures of current uses one will suppose “λ = constant”.
3. 3. THERMAL CONDUCTIVITY METALS AND ALLOYS (at the ambient temperature) λ Insulator Copper 99,9% Aluminum 99,9% 386 228 Tin Nickel 61 61 Aluminum 99% 203 Mild steel (1% of C) 46 λ Conductor Zinc Alloy (Al 92% - Mg 8%) 111 104 Lead Titanium 35 21 Brass (Cu 70% - Zn 30%) 99 Stainless steel (Cr 18% - Nor 8%) 16 Iron 85 NONMETAL SOLIDS (at the ambient temperature) λ Gas < λ Liquids Electro graphite 116 Wood 0.21 Concrete 1.75 Polyester 0.209 Glass pyrex 1.16 Polyvinyls 0.162 Porcelain 0.928 Asbestos (sheets) 0.162 λ Liquids<λ Solids Glass 0.74 Phenoplasts 0.046 Asbestos cement 0.70 Glass Fiber 0.046 Bricks 0.52 Rock Wool 0.043 LIQUIDS GAS (at 0°C and under the normal pressure) λ Void = o Sodium at 200°C Mercury at 20°C 81,20 8,47 Hydrogen Air 0.174 0.024 Water at 100°C 0.67 Nitrogen 0.024 λ in (W/m.°C) Water at 20°C Benzene at 30°C 0.59 0.162 Oxygen Acetylene 0.024 0.019 Dowtherm A at 20°C 0.139 Carbon dioxide 0.014 University of Balamand - ALBA ٣ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008 The table so above contains thermal conductivities λ out of W/m°C of various materials. The smaller the value of λ is, the more the material will be known as INSULATING. The larger the value of λ is, the more the material will be known as DRIVER. It is noted that among the solids, metals are much more conductive than the nonmetal compounds except for graphite (used in certain exchangers of heat). The stainless steel is less conductive than the majority of other metals and alloys. Among the liquids, mercury is detached clearly, the molten metals are good conductive what explains for example the use of sodium salts like coolant for the cooling of the nuclear engines. Except for the molten metals: λ of gases < λ of the liquids < Λ of the solids For the vacuum λ = O
4. 4. FOURIER EQUATION: ∂ T ∂ T ∂ T ∂ T ∂ T ∂ T ϕ( x ) = - λ × ( i + j + k) où ∂ x i + ∂ y j + ∂ z k = grad T ∂ x ∂ y ∂ z Assumptions: - Isothermal surfaces are consisted of parallel plans. dT - The side losses of heat (according to “y” and “Z”) are neglected grad T = i dx Statement: The density of the thermal flow ϕ which runs out in the material is proportional to the variation of the temperature and the thermal conductivity of the environment . Z Isothermal Isothermal Surface Surface with T1 in T2 dT ϑ (X) n ϕ( x ) = - λ . dx X dx University of Balamand - ALBA ٤ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008 GENERALIZATION OF THE EQUATION OF FOURIER: If one considers a solid in space (characterized by its co-ordinates “X, y, Z), L” equation is written: ∂ T ∂T ∂ T ∂ T ∂ T ∂ T ϕ( x ) = - λ × ( i + j + k ) où i + j + k = grad T With: grad T x ∂ (variation in temperature) represents the variation in thez ∂y ∂ z ∂ x ∂ y ∂ temperature according to all the directions. And is the derivative partial of the temperature compared to the axis “X”. STATEMENT IN THE PLAN: Simplifying assumptions: - Isothermal surfaces are consisted parallel plans between them. - The side losses of heat (according to “y” and “Z”) are neglected. The variation in temperature is reduced to: dT grad T = To convention the leaving heat flow is counted negatively. i dx Statement: That is to say a homogeneous material length “dx” and conductivity “λ”, whose external surfaces are respectively at temperatures T1 and T2 The density flow thermal ϕ which runs out in the matter is proportional to the variation in the temperature and the thermal conductivity of the medium.
5. 5. CONDUCTION THROUGH A HOMOGENEOUS WALL Heat flow through the wall: A Φ = A.ϕ = λ ΔT Isothermal plan d Tx ϕ ϕ = Density of flow [W/m ²] Φ = Heat flow [W] λ : Thermal Conductivity (W/m°C) X T1 T2 d : Walls Thickness (m) Φ : Heat flow (W) λ A: Walls Surfaces (m²) d ΔT : Temperature Difference (°C) University of Balamand - ALBA ٥ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008 This case makes it possible to solve the majority of the problems encountered in the building. 1-assumptions: - Homogeneous and isotropic Solid - Neglected side Losses. - Low thickness compared to transverse dimensions - 2 it heat flow through the wall: By applicant the Fourier analysis The heat flow “Φ”, in a tube of flow of section “S”, will be written: dT ϕ = -λ = cste − λ dT = ϕ dx dx T1 - T2 T2 e ϕ = e −λ T1 ∫ dT = ϕ ∫ dx from where λ 0 S Φ = S.ϕ = λ ( T1 - T2 ) e
6. 6. THE THERMAL RESISTANCE OF A WALL Equivalent thermal resistance λ T2 T1 R d R d n di n R = R = ∑λ = ∑R i λ i =1 i i=1 R: Thermal resistance (m²°C/W) Electric analogy: In series, total resistance is equal to the sum of resistances. University of Balamand - ALBA ٦ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008 3 - the thermal resistance of a plane wall: As in electricity, resistance is the report/ratio of a potential difference thus here of temperature and of a flow of energy thus here Φ flow, from where the following expression of thermal resistance. R is total thermal resistance [°C/W] 4 - Law of evolution T (X): = R - T2 ) = (T1 e λ .S (temperature in a point of co-ordinate Φ of an isothermal surface) “X” ; - λ. (T (X) – T1) = ϕ. X Evolution T = F (E) linear. T( x ) x −λ T1 ∫ dT = ϕ ∫ dx 0 ( T1 - T2 ) − λ × ( T(x) - T1 ) = .x.λ e ( T1 - T2 ) T(x ) = T1 - .x e
7. 7. TRANSMISSION THROUGH MULTI-LAYER WALLS Wall in series Wall in Parallel Φ A1 Φ1 λ1 A2 Φ2 λ1 λ2 λ3 A λ2 λ3 A3 Φ3 d1 d2 d3 di d d d A Aλ A λ A λ R = 1 + 2+ 3 Homogeneous walls = 1 1 + 2 2+ 3 3 λ1 λ2 λ3 R d1 d2 d3 ΣAi Ai R = ΣRi Non-Homogeneous walls = Σ R Ri University of Balamand - ALBA ٧ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008 1) Layers perpendicular to flow crossing the wall. Example, floor with insulator, cover and floor covering, concrete wall with brought back insulation, etc… The thermal resistance of the wall is calculated according to the following formula: 2) Layers parallel with flow crossing ΣR wall. R = the i Each section I parallel with the heat flow can be in its turn made up of several superimposed layers J and perpendicular to flow. Example, blocks full with horizontal and vertical joints. The thermal resistance of the wall is calculated according to the following formula: ΣAi Ai = Σ R Ri
8. 8. GLOBAL HEAT TRANSMISSION COEFFICIENT U External surface Internal surface transfer transfer Thermal Resistance R = Rsi + ΣRi + Rse Rs = Rsi + Rse Conduction Global Heat Transmission through the Coefficient wall 1 U = R [W/m²°C] University of Balamand - ALBA ٨ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008 Elements such as floors, walls, flagstones, roofs, windows and doors are composed of several nonhomogeneous layer. The heat flow which crosses an element is defined by the thermal coefficient of transmission U. The value U (W/m2°C) is the quotient of the density flux thermal which crosses, in stationary regime, the structural component considered, by the difference in temperature between two environments contiguous to this element. The thermal coefficient of transmission of an element is the reverse of its total resistance. U=1/R The Heat flux through this element will be: Φ = U.A . ΔT The following phenomena influence the value U of an element: - Heat Transfer enters the interior air and L `element. This process is described by the coefficient of transfer of surface heat interior hi, or surface resistance Rsi=1 interior/hi - Conduction of heat inside an element. The parameter determining is thermal conductivity here L (lambda) of various materials. -Heat transfer enters the element and the surrounding air. This process is described by the coefficient of surface transfer of heat external He or surface resistance Rse=1 outside/He If the element is an interior wall one applies Rsi twice. One definite surface resistance Rs total - External wall: Rs = Rse + Rsi - Interior wall: Rs = Rsi + Rsi
9. 9. SURFACE RESISTANCES WALL Flow Rsi Rse Rs Vertical 0,13 0,04 0,17 0,10 0,04 0,14 Horizontal 0,17 0,04 0,21 Rse = 0.04 m²°C/W Rsi = 0.13 m²°C/W Rse = 0 m²°C/W Rsi = 0.17 m²°C/W Air Circulation Rsi = 0.10 m²°C/W University of Balamand - ALBA ٩ HEAT TRANSMISSION HVAC Eng. Wael Zmerly – 2007-2008 The surface resistance of walls Rs (m2°C/W) is calculated according to the following formula: 1 RS = H and the coefficient of exchanges per radiation and Convection: h = hr + hc h hr is the coefficient of exchanges per radiation out of W/m2°C: hr = Mc . hro Mc = corrected emissivity of surface, by defect of one takes Mc = 0,9 who is an average value for materials used in construction. hro = 4. σ . Tm 3 hro is the coefficient of radiation of a black body: σ is the constant of Stefan-Bolzmann: σ = 5,67051 X 10-8 Tm is the average temperature of surface (Tm=273,15+température measured) Example for 10°C: hro = 4 X (5,67051 X 10-8) X (273,15 + 10) 3 = 5,15 hc is the coefficient of exchange by convection out of W/m2°C For the interior faces: - If the heat flow is Ascendant hc = 5 W/m2°C - If the heat flow is Descendant hc = 0.7 W/m2°C h = 4+4.v - If the heat flow is Horizontalc hc = 2.5 W/m2°C For the outsides: v is the speed of the wind in m/s near surface. 1 1 1 1 One Si = RSe = R definite surface resistances interior Rsi and external RseRof + RSe = RS = Si a wall: + hi he hi he and From where All times and to avoid these calculations, the values of Rsi and Rse of the table below can be used. They one obtained with emissivity a corrected of 0,9 and one temperature with dimensions interior for Rsi of 20°C and a temperature with dimensions outside for Rse of 0°C with a speed of wind of 4 m/s. If the wall gives on a room not heated, a roof, an underfloor space, Rsi applies of the 2 with dimensions ones. * Wall giving on: outside, an open passage or an open room. A room is known as open if the report/ratio of the total surface of its permanent openings on outside, with its volume, is equal or higher than 0,005 m2/m3.