Transport processes

7. Semester

Chemical Engineering

Civil Engineering

Course plan

1. Elementary Fluid Dynamics2. Fluid Kinematics3. Finite Control Volume Analysis4. Differential Analysis of Fluid Flow5. Viscous Flow and Turbulence6. Turbulent Boundary Layer Flow7. Principles of Heat Transfer8. Internal Forces Convection9. Unsteady Heat Transfer10. Boiling and Condensation11. Mass Transfer12. Porous Media Flow13. Non-Newtonian Flow

Today's lecture

• Principles of Heat Transfer– Heat transfer mechanisms

– Conduction Heat transfer

– Thermal resistance concept

– Contact resistance

– Thermal networks

Heat transfer

• Heat is transferred whenever there is a temperature difference

Heat conduction

• Heat is the kinetic energy of molecules

• When a molecule with high kinetic energy hit one with low kinetic energy momentum is transferred Heat transfer

Heat convection

• Besides heat conduction, heat I s also transferred due to the motion of the fluid

• Forced convection– Due to flow

• Natural convection– Difference in temperature = difference in density

Convection = Conduction + Advection(fluid motion)

Radiation

• Heat energy is transferred by photons

• Everything emits radiation, i.e. inferred radiation

Thermal conductivity

• The thermal conductivity:

• How much heat (energy) per second, per meter, per degree Celsius there can be transferred I a given material

• Metals is good heat conductors

• Gasses are poor heat conductors

[ / ]k W m C⋅

intermezzo

• Use the next two minutes to discuss with the person next to you how a double plated window works?

Thermal diffusivity

• The ratio between heat conduction and heat capacity

• One does not burn the fingers by putting them in an oven momentarily, but one does burn them if you touch the pot

How fast heat is conducted through the material How much heat there can be stored in the material P

kc

αρ

= =

Iron: α=22.8m2/sceramic: α=0.75m2/swood: α=0.13m2/sAir: α=0.00004m2/s

Fouriers law

• ”Fouriers law of heat conduction”

• Heat always flows from warm to cold!

• A is the area normal to the flow direction

[ ]1 2cond

T TdTQ kA kA Wdx x

−= − =

∆

Newtons law of cooling

• ”Newton’s law of cooling”

• h is the convective heat transfer coefficient

• As is the surface area

• Ts is the surface temperature

• T∞ is the temperature of the surroundings

• h [W/m2·°C] depends on the flow! And is not a property of the fluid.

( ) [ ]conv s sQ hA T T W∞= −

Some empirical h relations

For fully developed flows with constant surface temperature

• Flow over a flat plate

• Pipe flow

0.5 1 3

0.8 1 3

Laminar: 0.664Re Pr

Turbulent: 0.037 Re Pr

L

L

hLNukhLNuk

= =

= =

0.8 1 3

Laminar: 3.66

Turbulent: 0.023Re PrL

hDNukhLNuk

= =

= =Tin

Tout

TS

Even more nondimensional numbers…

• Nusselt number

– Dimensionless ’convective heat transfer coefficient’

– States how much more heat is transferred from convection (movement) compared to pure heat conduction (standstill)

• Prandtl number

– Non-dimensional grouping of fluid properties

– Is temperature dependent! can be found in tables in the back of the book

hL convectionNuk conduction

= =

momentumPrheat

pckµ

= =

Comparison of mechanisms

Stafan-boltzmann’s law

• Stafan-boltzmann’s law

– ε is the emissivity (0<ε<1)

– σ is Boltzmanns constant (5.670·10-8W/m2·K4)

• Depends on T4

( ) [ ]4 4rad sQ T T Wεσ ∞= −

Thermal resistance concept

• Ohms-law:

• Fouriers law:

• Newtons law:

1 2Flow of electronse

V VIR−

= =

1 2 1 2Flow of heat condWall

T T T TQ kAx R− −

= = =∆

1 2cond

conv

T TQR−

=

• Chuck Norris’ law

Resistances in series

Resistances in series

• The heat flow is the same through the entire wall:

1 2

total

T TR

∞ ∞−=

,1 ,21 2

1 1total conv wall conv

LR R R Rh A kA h A

= + + = + +

Parallel resistances

Kombinationer

Intermezzo

• Use the next 5 min to discuss how to calculate how much energy you need to heat up this architectural marvel on a winters day. What assumptions do I need to take? What information do I need to find? How should the thermal network look like?

Overall heat transfer coefficient

• Often we are only interested in one number for the heat transfer resistance. For example a heat exchanger can have a very complex geometry. Only the total resistance can be measured and is given by a “U” value.

• Overall heat transfer coefficient:

U [W/m2·°C]

the heat transfer is found from:

and it can be seen that:

[ ]Q UA T W= ∆

1

total

UAR

=

Heat transfer through a hollow cylinder

• Newtons law of cooling in cylindrical coordinates

• This can be integrated to give:

, (W)cond cyldTQ kAdr

= −

( )1 2 1 2

,2 1

2ln /cond cyl

cyl

T T T TQ Lkr r R

π − −= =

Heat transfer through a hollow cylinder

• Thermal Resistance with Convection

,1 ,2

total

T TQ

R∞ ∞−

=

( )( )

( )

,1 ,2

2 1

1 1 2 2

ln /1 1 2 2 2

total conv cyl convR R R R

r rr L h Lk r L hπ π π

= + + =

= + +

Heat transfer through a hollow cylinder

• Multi-layered Cylinders

( )( ) ( ) ( )

( )

,1 ,1 ,3 ,3 ,2

2 1 3 2 4 3

1 1 1 2 3 4 2

ln / ln / ln /1 12 2 2 2 2

total conv cyl cyl cyl convR R R R R R

r r r r r rr L h Lk Lk Lk r L hπ π π π π

= + + + + =

= + + + +

Heat transfer through a hollow cylinder

• Critical radius of insulation– Adding more insulation to a wall always decreases heat transfer.

– Adding insulation to a cylindrical pipe or a spherical shell, however, is a different matter.

– Adding insulation increases the conduction resistance of the insulation layer but decreases the convection resistance of the surface because of the increase in the outer surface area for convection.

– The heat transfer from the pipe may increase or decrease, depending on which effect dominates.

Heat transfer through a hollow cylinder

• The rate of heat transfer from the insulated pipe to the surrounding air can be expressed as:

( )( )

22

1 1

2 1

2increases with r decreases with r

ln / 12 2

ins conv

T T T TQr rR R

Lk h r Lπ π

∞ ∞− −= =

++

Heat transfer through a hollow cylinder

• The radius of critical insulation is found when:

• Performing the differentiation gives:

0dQdr

=

, (m)cr cylinderkrh

=

Example

Thermal contact resistance

• When two surfaces are pressed against each other, the peaks form good material contact but the valleys form voids filled with air

• Unfortunately, no empirical formulae exists to properly to predict contact resistances correctly! (material, pressure, surface roughness…)

• Thermal grease can significant reduce the contact resistance.

Thermal contact resistance

Excercises