ASRAE Student Branch meeting

Speaker: Kenneth Simpson

USGBC – LEED rating systemToday at 5 pm

ECJ 5.410

Lecture Objectives:

• Review - Heat transfer

– Convection

– Conduction

– Radiation

Simplified Equation for Forced convection

Pr) (Re, fNu

LU

LRe 3/1PrRe LCNu

5/4PrRe TCNu

For laminar flow:

For turbulent flow:

For air: Pr ≈ 0.7, = viscosity is constant, k = conductivity is constant

k

hLNu

General equation

mnmforced UCLUfh ),(

Simplified equation:

mforced ACHCh

Or:

RoomVolumeACH

rate flow Volume

Natural convection

GOVERNING EQUATIONSNatural convection

Continuity

• Momentum which includes gravitational force

• Energy

v2

2

y

uvTTg

y

u

x

uu

0 v

yx

u

2

2 v

y

T

y

T

x

Tu

u, v – velocities , – air viscosity , g – gravitation, ≈1/T - volumetric thermal expansion T –temperature, – air temperature out of boundary layer, –temperature conductivity T

Characteristic Number for Natural Convection

TT

TTTUU

uuLyyLxxw

***** ;v v;; ;

2*

*2*

2*

**

*

**

Re

1 v

y

uT

U

LTTwg

y

u

x

uu

L

Non-dimensionless governing equations

Using

L = characteristic length and U0 = arbitrary reference velocity Tw- wall temperature

The momentum equation become

2

3

LTTg w

Multiplying by Re2 number Re=UL/

Gr

2*

*2*2

*

**

*

** )Re/1()Re/( v

y

uTGr

y

u

x

uu LL

Grashof number Characteristic Number for Natural Convection

2

3

LTTwg

Gr

The Grashof number has a similar significance for natural convection as the Reynolds number has for forced convection, i.e. it represents a ratio of buoyancy to viscous forces.

Buoyancy forces

Viscous forces

Pr) ,( GrfNu

General equation

Even more simple

Natural convection simplified equations

4/1Pr GrCNu L

3/1Pr GrCNu T

For laminar flow:

For turbulent flow:

For air: Pr ≈ 0.7, = constant, k= constant, = constant, g=constant

),(),)(( nmnmnatural LTfLTTwfh

Simplified equation:

mnatural TCh

Or:

T∞ - air temperature outside of boundary layer, Ts - surface temperature

Forced and/or natural convection

Gr) Pr, (Re, 1Re2 fNuGr LL

Pr) (Re, 1Re2 fNuGr LL

Pr) ,( 1Re2 GrfNuGr LL

In general, Nu = f(Re, Pr, Gr)

natural and forced convection

forced convection

natural convection

Example of general forced and natural convection

8.019.1 ACHh forced

3/138.0333.0 )19.1()12.2( ACHThcombinbed

333.0 )12.2( Thnatural

Equation for convection at cooled ceiling surfaces

n

Conduction

Conductive heat transfer

• Steady-state

• Unsteady-state

• Boundary conditions

– Dirichlet Tsurface = Tknown

– Neumann

)(/ 21 SS TTLkq

sourcep

qx

T

c

kT

2

2

)( surfaceair TThx

T

L

Tair

k - conductivity of material

TS1 TS2

h

0 1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Analytical solution Numerical -3 nodes, =60 min Numerical -7 nodes, =60 min Numerical -7 nodes, =12 min

(T-T

s)/(

To

-Ts)

hour

Ts

0

T

-L / 2 L /2

h

h

h

To

T

h omogenous wa ll

L = 0.2 mk = 0 . 5 W/ m Kc = 9 20 J/kgK

= 120 0 k g/mp

2

Importance of analytical solution

What will be the daily temperature distribution profile on internal surface

for styrofoam wall?

A.

B.

External temperature profile

T

time

What will be the daily temperature distribution profile on internal surface

for tin glass?

A.

B.

External temperature profile

T

time

Conduction equation describes accumulation

Radiation

Radiation wavelength

Short-wave & long-wave radiation

• Short-wave – solar radiation– <3m– Glass is transparent – Does not depend on surface temperature

• Long-wave – surface or temperature radiation– >3m– Glass is not transparent – Depends on surface temperature

Radiation emission The total energy emitted by a body, regardless of the wavelengths, is given by:

Temperature always in K ! - absolute temperatures

– emissivity of surface

– Stefan-Boltzmann constant

A - area

4ATQemited

Surface properties

• Emission ( is same as Absorption ( ) for gray surfaces

• Gray surface: properties do not depend on wavelength

• Black surface: Diffuse surface: emits and reflects in each direction equally

1

n

absorbed (α), transmitted (τ), and reflected (ρ) radiation

View (shape) factors

jijiji FAFA

i jA A

jiji

iij dAdA

lAF

2

coscos1

http://www.me.utexas.edu/~howell/

1j

ijF

For closed envelope – such as room

n

jijiniii FFFFF

1321 1... ni ,...,2,1

Example: View factor relations

F11=0, F12=1/2

F22=0, F12=F21

F31=1/3, F13=1/3

A1

A2A3 A1=A2=A3

Radiative heat flux between two surfaces

44,, BAABABABA TTAFQ

ψi,j - Radiative heat exchange factor

Exact equations for closed envelope

Simplified equation for non-closed envelope

44,, jiijiiji TTAQ

n

kkikjkjijji FF

1,,,, 1 nji ,...,2,1,

BB

B

ABAAA

A

BABA

AFAA

TTQ

111

44

,

Summary

• Convection– Boundary layer– Laminar transient and turbulent flow– Large number of equation for h for specific airflows

• Conduction – Unsteady-state heat transfer – Partial difference equation + boundary conditions– Numerical methods for solving

• Radiation – Short-wave and long-wave – View factors– Simplified equation for external surfaces– System of equation for internal surfaces

Building components