Heat transfer

Heat Transfer
Finite Difference Approach for Two-Dimensional Steady-State Heat Transfer
Finite Element Approach

Lecture contents

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ACCESS – Lecture 2 Heat Transfer

to a temperature difference.
What is heat transfer?
Heat transfer is energy that flows from higher to lower level of temperature without any work being performed.
In which way is the amout of transferred heat described?
flow = transport coefficient x potential gradient
flow: heat flux q [W/m²] or heat transfer rate Q [W]
coefficient: depends on transfer characteristics
gradient: difference resp. derivative

Heat Transfer in General

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ACCESS – Lecture 2 Heat Transfer

via moving atoms & molecules
Solids: lattice oscillations and movement of unbound electrons (in electroconductive materials)
Description via Fourier‘s law:

Heat Transfer in General: Transfer Types

θ1

θ2

m

e-

 

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ACCESS – Lecture 2 Heat Transfer

(liquid, gas, solid)
Description due to wave theory (Maxwell) resp. photon emission (Plank)
Even through vacuum possible, in contrast to conduction and convection which require presence of material medium
Radiant heat exchange between two surfaces:

Heat Transfer in General: Transfer Types

θ1

θ2

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ACCESS – Lecture 2 Heat Transfer

 

motion of the fluid (advection)
2nd transfer due to random molecular motion (diffusion)
The faster the fluid motion, the faster the convective heat transfer
Two types: forced and natural convection
Description for exchange between fluid and adjoining surface:

Heat Transfer in General: Transfer Types

θ1

θ2

v

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ACCESS – Lecture 2 Heat Transfer

 

in chaotic way
Fast moving molecules collide with slower moving molecules: low-energy molecules/atoms absorb energy (temperature level increases) and high-energy molecules/atoms release energy (temperature level decreases)
Macroscopic view:
More kinetic energy is transferred from higher to lower temperature level than vice versa
System tends towards thermal equilibrium (homogeneous temperature level)

Heat Conduction

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ACCESS – Lecture 2 Heat Transfer

state which assumes
time- constant boundary conditions (indoor conditions, climate conditions, heat sources or sinks,…)
and thus no relevance of thermal storage effects
One- dimensional which assumes
heat flux perpendicular to the construction surface area
and thus no relevance of thermal bridge effects
Solely heat conduction related which assumes
Convection or radiation transfer can be neglected or described via thermal conductivity (e.g. air layers)

Heat Conduction- Simplified Approach

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ACCESS – Lecture 2 Heat Transfer

Heat Transfer

T1

T2

T [K]

x [m]

T2

T1

x1

x2

dT

dx

dT

dx

Temperature gradient
Resulting heat flux
Thermal conductivity

q

the length- related temperature difference (temperature gradient)
Proportional to the surface area
Depending on solely one material property, the thermal conductivity

Heat conduction: 1D

Total resulting 1-D steady- state
heat transfer rate is:

Resulting 1-d steady state heat flux is:

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ACCESS – Lecture 2 Heat Transfer

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ACCESS – Lecture 2 Heat Transfer

to conduct heat through it.
It defines the heat transfer rate [W] per distance unit [m] between two plane surfaces and per unit temperature difference [K] between these two surfaces.
What influences the thermal conductivity of a material?
Material density
Electric conduction
Porosity (see density)
Temperature
Pressure (Fluids)
Heat flow direction (anisotropic materials)

Heat Conduction: Thermal Conductivity

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ACCESS – Lecture 2 Heat Transfer

Barlett Learning, 2011

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ACCESS – Lecture 2 Heat Transfer

Heat Transfer

Heat Transfer

High-porous brick (HPB)

Low-porous brick (LPB)

Heat Transfer

Heat Transfer

Figures source: W. M. Willems: Lehrbuch der Bauphysik – Schall, Wärme, Feuchte, Licht, Brand, Klima, Springer Vieweg Verlag, Wiesbaden, 2013

neglected
Reference values (rated values) for thermal conductivity usually given for 10˚C
High-temperature values (e.g. insulation heating systems) for 40 ˚C
Values listed in DIN 4108-4 & ISO 12524
Gases and metals
For metals increasing or decreasing effect of temperature level
For gases increasing effect of temperature level

Thermal Conductivity and Temperature

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ACCESS – Lecture 2 Heat Transfer

transfer
Free electrons provide huge fraction of entire heat transfer
Higher temperature causes higher lattice vibrations
Higher lattice vibrations obstruct free electron transfer

Thermal Conductivity and Temperature

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ACCESS – Lecture 2 Heat Transfer

Figure source: M.M.Rathore: Engineering Heat Transfer, Jones & Barlett Learning, 2011

Higher temperature causes stronger lattice vibrations, oscillations obstruct
flow of free electrons

Conductivity and Temperature

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ACCESS – Lecture 2 Heat Transfer

Figure source: M.M.Rathore: Engineering Heat Transfer, Jones & Barlett Learning, 2011

Higher temperature causes higher kinetic energy, faster movement, higher impulse

does not vary with the direction of heat flow.
Anisotropic materials show a dependency of
thermal conductivity on the heat flow direction.
Examples are: wood, sedimentary rocks, metals
that have undergone heavy cold pressing, fiber-
reinforced composite structures.

TC and Heat Flow Direction

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ACCESS – Lecture 2 Heat Transfer

Figure source: H. Neuhaus: “Lehrbuch des Ingenieurholzbaus”, Springer Fachmedien Wiesbaden, 1994

gross density.
Example mineral wool: thermal conductivity
Increases slightly if the gross density increases
For values above about 50 kg/m³.
BUT
The density increases strongly below a density
of 50 kg/m³.

TC and Density

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ACCESS – Lecture 2 Heat Transfer

Figure source: W. M. Willems: “Lehrbuch der Bauphysik – Schall Wärme Feuchte Licht Brand Klima”, 7. Auflage, Springer Vieweg Verglag, Wiesbaden, 2013

Gross density ρ

Thermal Conductivity λ

Higher density causes slightly increasing conductivity

Very low density causes strongly increasing conductivity

each layer

q

dx2-3

x3

T3

dT1-2

dT2-3

dT2-3

dx2-3

T2

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ACCESS – Lecture 2 Heat Transfer

the entire element:

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ACCESS – Lecture 2 Heat Transfer

 

 

 

 

inside (warm air)

T3

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ACCESS – Lecture 2 Heat Transfer

the adjacent air mass are not equal because:
Energy is needed to transfer heat from solid state (solid envelope materials) to gaseous state (air mass)
Convection processes of air masses influence transfer process
Radiation exchange influences surface- near material layers
These influences can be summarized and simplified as an additional conduction resistance layer, the thermal surface resistance
This resistance is significant and must be considered.
(Values are around 0.05 to 0.5 m²K/W)

Heat conduction - Surface Resistance

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ACCESS – Lecture 2 Heat Transfer

– Lecture 2 Heat Transfer

mass layer inside (warm air)

d [m]

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ACCESS – Lecture 2 Heat Transfer

adjacent material layer are not equal because:
Most construction materials are rough and porous and are not in ideal contact with the adjacent material
Interface contains several air gaps that serve as an additional insulation layer (low conductivity of air)
This effect can also be summarized and simplified as an additional conduction resistance layer, the thermal contact resistance
For building materials, contact resistance can be neglected (values are around 0.000005 to 0.0005 m²K/W)

Heat conduction – Contact Resistance

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ACCESS – Lecture 2 Heat Transfer

25 cm
λ = 1.3 W/mK

External Plaster
d = 1,2 cm
λ = 1.0 W/mK

Internal Plaster
d = 1 cm
λ = 1.0 W/mK

Insulation
d = 9 cm
λ = 0.045 W/mK

Brick
d = 12 cm
λ = 0.72 W/mK

Heat flux q

θsur,i

θIndoor

θOutdoor

θsur,i

as follows:
Temperature gradients between layers are added
Resulting heat flux for a multi-layer construction

Example: 1D Steady-State

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ACCESS – Lecture 2 Heat Transfer

a multilayered construction:
Calculation heat transmission resistances of all layers
Calculation of total transmission resistance and U-value
Calculation of heat flux
Calculation of temperatures at each boundary

Example: 1D Steady-State

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ACCESS – Lecture 2 Heat Transfer

Heat Transfer

Heat Transfer

assumes
Time- constant boundary conditions (indoor conditions, climate conditions, heat sources or sinks,…)
And thus no relevance of thermal storage effects
Two- dimensional instead of one- dimensional which assumes
Heat flux can be described in x- and y-direction
Thus no relevance of three- dimensional effects
Solely heat conduction related which assumes
Convection or radiation transfer can be neglected or described via thermal conductivity value (e.g. air layers)

Heat Conduction 2D Steady-State

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ACCESS – Lecture 2 Heat Transfer

y-direction
Resulting heat flux
Density in each direction
?

T3

T4

qy

qx

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ACCESS – Lecture 2 Heat Transfer

boundary conditions can be handled with numerical methods, which
Provide solutions for discrete points
Give only an approximation
Two established numerical methods are
Finite Difference Method
Finite Element Method

Heat Conduction- Simplified 2-D

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ACCESS – Lecture 2 Heat Transfer

is divided into equal segments of Δx an Δy
Nodal points Px/y represent surrounding area of the size Δx and Δy

Finite- Difference- Method – 2d Steady St.

Δx

Δy

qy

qx

y3

y2

y1

P2/2

P2/1

P2/3

P1/2

P1/3

P3/3

P3/2

P3/1

P1/1

x1

x2

x3

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ACCESS – Lecture 2 Heat Transfer

form
Equations for nodes in x-direction are:

Finite- Difference- Method – 2d Steady St.

x

θ

x1

x2

x3

θ2/2

θ3/2

θ1/2

Δx2-1

Δx3-2

Θ2.5/2

Θ1.5/2

Δθx3-2

Δθx2-1

P2/2

P3/2

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ACCESS – Lecture 2 Heat Transfer

form in
the same way
Equation for red points on y-axis are:

Finite- Difference- Method – 2d Steady St.

y

θ

y1

y2

y3

θ2/2

θ3/2

θ1/2

Δθy3-2

Δθy2-1

Δy2-1

Δy3-2

Θ2.5/2

Θ1.5/2

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ACCESS – Lecture 2 Heat Transfer

Difference- Method – 2d Steady St.

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ACCESS – Lecture 2 Heat Transfer

both heat flux directions is zero in steady- state case):
Gives this FD solution:
Which can be reduced to an approximate algebraic equation if dx and dy are equal:
For given example: temperature at nodal
point equals arithmetic average of the
four adjacent nodes

Finite- Difference- Method – 2d Steady St.

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ACCESS – Lecture 2 Heat Transfer

state in each volume element
Construction is divided by a (triangular) mesh
Control volumes are generated by connecting the centers of adjacent elements

Finite- Element- Method – 2d Steady St.

qy

qx

Control Volume

Element

Node

Triangular Mesh

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ACCESS – Lecture 2 Heat Transfer

is integration of general 2D SS heat conduction over two- dimensional control volume:
Volume integrals are rewritten as integrals over entire bounding surface of the control volume by using Gauss divergence theorem

Finite- Difference- Method – 2d Steady St.

Direction cosine of unit vector of bounding surface in x- direction

Direction cosine of unit vector of bounding surface in y- direction

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ACCESS – Lecture 2 Heat Transfer

be calculated via shape functions Nx of linear triangular elements:
Discretized equation for temperature of all control volumes is then given as:
Resulting system of algebraic equations is incorporated with boundary conditions into numerical solver

Finite- Difference- Method – 2d Steady St.

Face area = portion of entire surface
around control volume

Number of faces (1 to NI)

Shape Functions

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ACCESS – Lecture 2 Heat Transfer

distribution within an area or volume, depending on boundary conditions (steady state or transient)
Appropriate solution can only be found if boundary conditions are known
There are three common types
of boundary conditions:
Defined surface temperatures
Defined surface heat flux
Specified surface heat balance

Boundary Conditions

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ACCESS – Lecture 2 Heat Transfer

of the body is
given as a function of time and position or simply
as a constant value
Description as:

Boundary Conditions

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ACCESS – Lecture 2 Heat Transfer

the boundary of the body is given
Description as:
Allows us to determine the partial derivative of
the temperature with respect to outward normal vector n:
Special case: adiabatic boundary conditions:

Boundary Conditions

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ACCESS – Lecture 2 Heat Transfer

value and derivative of solution
Assumption: heat flux into body surface must equal heat flux out of the body surface (d=0, C=0)
For convective transfer given as:
For radiant transfer:
Combined:

Boundary Conditions

Temperature of the surrounding fluid
(also called T_infinity)

Convective heat transfer coefficient

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ACCESS – Lecture 2 Heat Transfer

P. Freudenberg Contributors: P. Freudenberg, H. Fechner, J. Grunewald

Dresden, 18.04.2019

Transfer

cover the whole region
Well conditioned finite element mesh requires sophisticated FEM solution method knowledge, therefore:
THERM provides automatic mesh generation via Finite Quadtree- algorithm:
Object domain is divided into set of
squares (hierarchic quadrants tree)
Subdivision is performed until:
Each quadrant contains only one material
Size difference between adjacent elements is balanced
Entire domain is subdivided into triangles resp. quadrilaterals

Therm – Mesh Generation

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ACCESS – Lecture 2 Heat Transfer

and FACET
Method assumes constant boundary conditions (steady- state) and physical properties
Governing partial differential equation for two- dimensional heat conduction including internal heat generation:
Finite- element analysis is based on method of weighted residuals (Galerkin form which relies on algebraic shape functions)

Therm – Finite Element Method

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ACCESS – Lecture 2 Heat Transfer

wall example geometry in THERM
Create all materials which are needed for this construction in THERM material library
Create all boundary conditions which are needed for this construction in THERM boundary conditions library
Assign materials and create (F10) and assign boundary conditions to the modeled construction
Create U-Factor name for y- direction and assign it by double clicking the created boundary conditions (vertial inside)
Run the calculation (F9)
Check your resulting U-Factor
Get the U-Value for this construction via manual calculation
Compare both U-Values

Therm – Exercise 1D

in

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ACCESS – Lecture 2 Heat Transfer

an edge
Model wall example geometry in THERM: notice that minimum distance from end to inner edge is 1m
Assign materials and create (F10) and assign boundary conditions to the modeled construction
Create U-Factor names for x- direction and assign it to horizontal element, do the same for U-Factor name in y-direction
Run the calculation (F9)
Check your resulting U-Factor in x- and in y-direction
Compare these values with your manual calculation result

Therm – Exercise 2D

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ACCESS – Lecture 2 Heat Transfer