HVAC١University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

Heat Transmission

University of BalamandALBA

By Eng. Wael Zmerly – 2007-2008

H.V.A.C .

HVAC٢University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

THERMAL CONDUCTIVITY

λ

Wood

Homogeneous

Isotropic

λvλe

Transmission by vibrations of atoms or molecules

Transmission by the free electrons

λ Thermal conductivity λ of the material (W/m.°C)

λ

“λ = constant”

Brick Copper Air

Material

Glass FiberIron0.21 85386 0.024 0.046

(W/m.°C)

0.52

Glass

0.74

One will consider the Homogènes solids (characteristic physics and identical mechanicsin 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 beingneglected.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.°CIt is a function of the temperature, but in the intervals of temperatures of currentuses one will suppose “λ = constant”.

HVAC٣University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

λ

λ

Insulator

Conductor

λ Void = o

λ Liquids<λ Solids

λ Gas < λ Liquids

THERMAL CONDUCTIVITY

λ in (W/m.°C)

METALS AND ALLOYS (at the ambient temperature)

Copper 99,9% 386 Tin 61

Aluminum 99,9% 228 Nickel 61

Aluminum 99% 203 Mild steel (1% of C) 46

Zinc 111 Lead 35

Alloy (Al 92% - Mg 8%) 104 Titanium 21

Brass (Cu 70% - Zn 30%) 99 Stainless steel (Cr 18% - Nor 8%) 16

Iron 85

NONMETAL SOLIDS (at the ambient temperature)

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

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)

Sodium at 200°C 81,20 Hydrogen 0.174

Mercury at 20°C 8,47 Air 0.024

Water at 100°C 0.67 Nitrogen 0.024

Water at 20°C 0.59 Oxygen 0.024

Benzene at 30°C 0.162 Acetylene 0.019

Dowtherm A at 20°C 0.139 Carbon dioxide 0.014

The table so above contains thermal conductivities λ out of W/m°C of variousmaterials.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 conductivewhat 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 solidsFor the vacuum λ = O

HVAC٤University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

FOURIER EQUATION:

ϕ λ ∂ ∂ ∂= × + +

∂ ∂ ∂( ) T T T - ( i j k ) x y zx

∂ ∂ ∂+ + =

∂ ∂ ∂ T T T i j k grad T x y zoù

= idTgrad Tdx

Assumptions:

- Isothermal surfaces are consisted of parallel plans.

- The side losses of heat (according to “y” and “Z”) are neglected

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 .

Statement:

ϕ λ=( )dT - . dxx

X

dx

ϑ (X)

IsothermalSurfacein T2

Isothermal Surfacewith T1

Z

n

GENERALIZATION OF THE EQUATION OF FOURIER:If one considers a solid in space (characterized by its co-ordinates “X, y, Z), L” equationis written:

With: grad T (variation in temperature) represents the variation in the temperatureaccording to all the directions. And is the derivative partial of the temperaturecompared 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:

To convention the leaving heat flow is counted negatively.Statement:

That is to say a homogeneous material length “dx” and conductivity “λ”, whose externalsurfaces are respectively at temperatures T1 and T2The density flow thermal ϕ which runs out in the matter is proportional to thevariation in the temperature and the thermal conductivity of the medium.

ϕ λ ∂ ∂ ∂ ∂ ∂ ∂= × + + + + =

∂ ∂ ∂ ∂ ∂ ∂( ) T T T T T T - ( i j k ) où i j k grad T x y z x y zx

= idTgrad Tdx

HVAC٥University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

CONDUCTION THROUGH A HOMOGENEOUS WALL

d

T1

Tx

X

Isothermal plan

T2

ϕ λΦ = = ΔA A . Td

Heat flow through the wall:

ϕ ϕ = Density of flow [W/m ²]Φ = Heat flow [W]

λ: Heat flow (W)

A: Walls Surfaces (m²)

: Temperature Difference (°C)

Φ

TΔ

d : Walls Thickness (m)

: Thermal Conductivity (W/m°C)λ

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 analysisThe heat flow “Φ”, in a tube of flow of section “S”, will be written:

ϕ λ= =dT - cstedx

λ ϕ− = dT dx

λ ϕ− =∫ ∫2

1 0T

xT e

dT dϕ

λ

= 1 2T - T efrom where

( )1 2S S . T - Te

ϕ λΦ = =

HVAC٦University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

THE THERMAL RESISTANCE OF A WALL

R

d

T1 T2λ

λ=

dR

R

Equivalent thermal resistance

λ= =

= =∑ ∑n n

ii

1 1i

d i i

R R

Electric analogy: In series, total resistance isequal to the sum of resistances.

R: Thermal resistance (m²°C/W)

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 followingexpression of thermal resistance.

R is total thermal resistance [°C/W]4 - Law of evolution T (X):

(temperature in a point of co-ordinate “X” of an isothermal surface); - λ. (T (X) – T1) = ϕ. X

Evolution T = F (E) linear.

( )1 2T - T

. SeR

λ= =

Φ

λ ϕ− =∫ ∫( )

1 0T

xxT x

dT d

λ λ− × = 1 21(x)

( T - T ) ( T - T ) . x . e

= 1 21( )

( T - T ) T - . xxTe

HVAC٧University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

TRANSMISSION THROUGH MULTI-LAYER WALLS

Φ

d1

1λ 2λ 3λ

d2 d3

A

31 2

1 2 3

+ dd dRλ λ λ

= +

iR R= Σ

Homogeneous walls

Non-Homogeneous walls

Wall in series Wall in Parallel

1Φ

2Φ

3Φ

1λ

2λ

3λ

di

A3

A2

A1

i i

i

A AR R

Σ= Σ

3 31 1 2 2

1 2 3

+ AA A AR d d d

λλ λ= +

1) Layers perpendicular to flow crossing the wall.Example, floor with insulator, cover and floor covering, concrete wall with brought backinsulation, etc…The thermal resistance of the wall is calculated according to the following formula:2) Layers parallel with flow crossing the wall.Each section I parallel with the heat flow can be in its turn made up of severalsuperimposed 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:

iR R= Σ

i i

i

A AR R

Σ= Σ

HVAC٨University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

GLOBAL HEAT TRANSMISSION COEFFICIENT U

Internal surfacetransfer

Conduction through the

wall

External surfacetransfer

=1U R

s si seR R R= +

= + Σ +R si i seR R R

Global Heat Transmission Coefficient

[W/m²°C]

Thermal Resistance

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/RThe Heat flux through this element will be: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 describedby the coefficient of surface transfer of heat external He or surface resistanceRse=1 outside/HeIf 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

U.A . TΦ = Δ

The surface resistance of walls Rs (m2°C/W) is calculated according to the following formula:H and the coefficient of exchanges per radiation and Convection:hr is the coefficient of exchanges per radiation out of W/m2°C:Mc = corrected emissivity of surface, by defect of one takes Mc = 0,9 who is an average value for materialsused in construction.hro is the coefficient of radiation of a black body:σ is the constant of Stefan-Bolzmann: σ = 5,67051 X 10-8Tm 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 = 5,15hc is the coefficient of exchange by convection out of W/m2°CFor 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- If the heat flow is Horizontal hc = 2.5 W/m2°CFor the outsides:v is the speed of the wind in m/s near surface.One definite surface resistances interior Rsi and external Rse of a wall:

and From whereAll 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°Cand 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 thetotal surface of its permanent openings on outside, with its volume, is equal or higher than 0,005 m2/m3.

HVAC٩University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

0,210,040,17

0,140,040,10

Horizontal

0,170,040,13Vertical

RsRseRsiFlowWALL

SURFACE RESISTANCES

Rsi = 0.13 m²°C/WRse = 0.04 m²°C/W

Rse = 0 m²°C/W

Air Circulation

Rsi = 0.17 m²°C/W

Rsi = 0.10 m²°C/W

1 =Sii

Rh

1 =See

Rh

1 1 = + = +S Si Sei e

R R Rh h

. r c roh M h= = +r ch h h

3m 4. . Troh σ=

4 + 4 . =ch v

1 =SRh

HVAC١٠University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

THERMAL RESISTANCE Of AIR LAYER

0.0350.0350.0351

0.230.170.163000.220.170.16100

0.210.170.16500.190.170.16250.170.170.1615

0.150.150.15100.130.130.1370.110.110.115

Thermal resistance Rg m²°C/Wthickness of the non-ventilated

air layerin mm

Rg: thermal resistance of air layers

Blade of air: Is regarded as blade of air, a layer of air of which the thickness in the direction of the heat flow does not exceed 0,30 Mr.Blade of air non-ventilated: if there is no specific provision for a flow of air crossing it(example, a double glazing).A blade of air can be regarded as non-ventilated if the openings do not allow a flow of crossing air and if they do not exceed:500 mm ² per m length counted horizontally for the vertical blades of air.500 mm ² per m ² of surface for the horizontal blades of air.Default values are given in the table above for non-ventilated blades of air. The values for a horizontal flow also apply to tilted heat fluxes until more or less 30% compared to the horizontal plane.Blade of air slightly ventilated: when the external air flow is limited because ofdimension of the openings, dimensions included/understood in the following ranges:>500 mm ² but <1500 mm ² per m length counted horizontally for the vertical blades of air.>500 mm ² but <1500 mm ² per m ² of surface for the horizontal blades of air.Resistance of a blade of air slightly ventilatedThe thermal resistance of a blade of air slightly ventilated is equal to half of thatcorresponding to a non-ventilated blade of air.Nevertheless, if the thermal resistance of the layers located between the blade of air and outside is higher than 0,15 m2°C/W, this resistance must be replaced by the value of 0,15 m2°C/W.

HVAC١١University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

Rsi Rse = Rsi

Indoor

Clading

VentilatedAir Layer

Outoor

THERMAL RESISTANCE Of AIR LAYER

Blade of air strongly ventilated: It is about blade of air of which openings towardsoutside surplus:1500 mm ² per m length counted horizontally for the vertical blades of air.1500 mm ² per m ² of surface for the horizontal blades of air.Resistance of a blade of air strongly ventilatedIn this case, one neglects the thermal resistance of the blade of air and of all the layerslocated between the blade of air and outside and one applies not to the wall a surface thermal resistance Rse but Rsi.

HVAC١٢University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

CALCULATION OF VALUE U

λ λ λ+ + + +1 2

si se1 2

= R + ... Rn

n

dd dR

d1...dn : thickness of the layer of the corresponding material, in m

Rsi, Rse : surface resistances, in W/m²°C

… : thermal conductivity of the corresponding material, in W/m°C1λ 2λ

4

4

5

5

Rsi Rse

Indoor Outoor

= + Σ +R si i seR R R

=1U R

4

Construction of the Wall

R , d/λ(m²°C/W)

3

5

6

External Surface Resistance Rse9

8

7

2

Internal Surface Resistance Rsi1

l(W/m°C)

d(m)

Building MaterialNo

Designation of the Wall

21 = = (W/m . C) total

Valeur UR

=____totalR

HVAC١٣University of Balamand - ALBA

Eng. Wael Zmerly – 2007-2008HEAT TRANSMISSION

EXAMPLE OF CALCULATION OF U VALUE

+ + + + + + =0.015 0.15 0.16 0.12 0.02 = 0.13 0.04 5.260.7 0.44 0.36 0.44 0.87

R

Internal Plaster

Brick terra cotta

Heat insulation

Brick terra cotta

External Plaster

Rsi Rse= + Σ +R si i seR R R

=1U R

4.440.0360.16Insulation4

Construction of the Wall

R, d/λ(m²°C/W)

0.340.440.15Brick terra cotta3

0.270.440.12Brick terra cotta5

0.020.870.02External Plaster6

9

8

0.04--External Surface Resistance Rse7

0.020.70.015Interior Plaster2

0.13--Interior Surface Resistance Rsi1

λ(W/m°C)

d(m)

Building MaterialNo

Designation of the External Wall.

21 = = 0.19 (W/m . C) total

U ValueR = 5.26 totalR

MATERIALS d λ R

m W/m°C m²°C/W

Béton CONCRETE 1.200 -1.750

Béton caverneux 1.400Béton de vermex 0.240Béton cellulaire 0.160 - 0.330

Blolc de ConstructionConcrete hollow block 2 Layers 10cm 0.1 0.090

Concrete hollow block 2 Layers 12,5cm 0.125 0.100Concrete hollow block 2 Layers 15cm 0.15 0.120

Concrete hollow block 2 Layers 17,5cm 0.175 0.140Concrete hollow block 2 Layers 20cm 0.2 0.160Concrete hollow block 2 Layers 7,5cm 0.075 0.070Concrete hollow block 3 Layers 15cm 0.15 0.140

Concrete hollow block 3 Layers 17,5cm 0.175 0.160Concrete hollow block 3 Layers 20cm 0.2 0.190

Concrete hollow block 3 Layers 22,5cm 0.225 0.210Concrete hollow block 4 Layers 20cm 0.2 0.220

Concrete hollow block 4 Layers 22,5cm 0.225 0.240Concrete hollow block 4 Layers 25cm 0.25 0.260

Concrete hollow block 4 Layers 27,5cm 0.275 0.280Concrete hollow block 5 Layers 27,5cm 0.275 0.310Concrete hollow block 5 Layers 30cm 0.3 0.340

Concrete hollow block 5 Layers 32,5cm 0.325 0.360Concrete hollow block 6 Layers 32,5cm 0.325 0.400

BricksBrick 5cm 0.05 0.100

Brick 7,5cm 0.075 0.160Brick 10cm 0.1 0.200

Brick 12,5cm 0.125 0.270Brick 15cm 0.15 0.300

Brick 17,5cm 0.175 0.330Brick 20cm 0.2 0.390

Brick 22,5cm 0.225 0.420Brick 25cm 0.25 0.450

Brick 27,5cm 0.275 0.520Brick 30cm 0.3 0.590

HourdiHourdi Concrete 8+4 0.12 0.110Hourdi Concrete 12+4 0.16 0.130Hourdi Concrete 16+4 0.2 0.150Hourdi Concrete 20+5 0.25 0.180Hourdi Concrete 25+5 0.3 0.210Hourdi Concrete 12+4 0.16 0.130

Hourdi Terra Cotta 5+3 0.08 0.110Hourdi Terra Cotta 8+4 0.12 0.140Hourdi Terra Cotta 12+4 0.16 0.230Hourdi Terra Cotta 16+4 0.2 0.260Hourdi Terra Cotta 20+5 0.25 0.310Hourdi Terra Cotta 25+5 0.3 0.400

Insulation MaterialsRock Wool 0.038 - 0.047Fiber Glass 0.031 - 0.055

Liège expansé 0.040 - 0.047Expanded Polystyrene (EPS) 0.032 - 0.048Extruded Polystyrene (XPS) 0.028 - 0.036

Polyurethane 0.022 - 0.038

PlasterPlaco-plâtre 0.01 0.350

Plâtre d'enduit 0.01 0.300Carreau de plâtre 0.05 0.350Cement Plaster 0.01 0.700

WoodWood (chêne, hêtre, frêne, pichpin) 0.230

Wood (pine) 0.150Plancher wood (pine) 0.027 0.150

Wood (sapin, peuplier, okoumé) 0.120Mouchette bois 0.015 0.150

StonesStone, granite, gneiss, porphyre 3.200

Stone shistes, ardoise 2.200Basaltes 1.600

Laves, trachytes, andesites 1.100Lime Stone 2.400

TilingCeramic 1.000

Vinyl 0.020Granite 3.200Linoleum 0,180 Marble 3.400Carpets 0.006 - 0.010 0.060-0.150Parquet 0.200

PVC 0.230

FloorClay or lime 1.500

Rock 3.500Sand et Gravel (tout venant) 2.000

MetalsSteel 50

Stainless Steel 17Aluminium Alloy 160

Aluminium 230Bronze 65Copper 380

Pure Iron 72Iron, Cast Iron 50

Brass 120Plomb 35Zinc 380

GasAir 0.025

Argon 0.017Krypton 0.009Xenon 0.0054

OtherGlass 1.150

Water Profing 0.230Mortar 1.200

Hollow Block Insulation Air Layer Hollow Block U (W/m².°C) U (W/m².°C)10cm - - - 3.53 2.2615cm - - - 3.19 2.5420cm - - - 2.83 2.9910cm - 3cm 10cm 0.75 3.6810cm 2cm - 10cm 0.94

10cm 2.5cm - 10cm 0.83

10cm 5cm - 10cm 0.49 U (W/m².°C)10cm 3cm 2cm 10cm 0.65 2.2510cm - 5cm 10cm 1.84 1.7215cm 5cm - 10cm 0.4815cm 3cm 2cm 10cm 0.64

15cm - 5cm 10cm 1.74

U (W/m².°C)6.40

3.30

Hollow Block Insulation Air Layer Hollow Block U (W/m².°C) 2.2010cm - - - 3.33 2.0115cm - - - 3.0320cm - - - 2.70

10cm 2.5cm - 10cm 0.82

10cm 5cm - 10cm 0.49 U (W/m².°C)10cm 3cm 2cm 10cm 0.64 2.7110cm - 5cm 10cm 1.78 2.5115cm 5cm - 10cm 0.48 2.2815cm 3cm 2cm 10cm 0.6315cm - 5cm 10cm 1.69

Hollow Block Insulation Air Layer Hollow Block U (W/m².°C)10cm - - - 2.7115cm - - - 2.5120cm - - - 2.2810cm 2.5cm - 10cm 0.7710cm 5cm - 10cm 0.4710cm 3cm 2cm 10cm 0.6110cm - 5cm 10cm 1.5915cm 5cm - 10cm 0.4615cm 3cm 2cm 10cm 0.6015cm - 5cm 10cm 1.52

External Walls with mechanical Clading

DescriptionSingle Glass

Triple Glass (6-12-6-12-6) en mm

Hollow Block

15cm20cm

10cm

CeillingExternal Walls

Internal Wall

Intermediate FloorDescription

Last Floor with attic

Description

External Walls with CladingTriple Glass (6-8-6-8-6) en mm

Ground Floor

Glass

Double Glass

Floors

RoofToilet (Atic)

Intermediate Floor