Intro to PLUS by Leta Moser and Kristen Cetin • PLUS accreditation• Peer-Led Undergraduate Studying (PLUS)

– assists students enrolled by offering class-specific, weekly study groups.

– Students can attend any study group at any point in the semester to review for an exam, discuss confusing concepts, or work through practice problems.

– http://www.utexas.edu/ugs/slc/support/plus

Lecture Objectives:

• Review - Heat transfer– Convection – Conduction – Radiation

Analysis of a practical problem

Example Problem –radiant barrier in attic

Example Problem –heat transfer in window construction

Convection

Convection coefficient – h [W/m2K]

khLNu

[W] )()( TThATThAQ wairwall

Conduction

Convection

Natural convection Forced convection

wT

T

L – characteristic length

wT

T

[W/m2] )( TThq w

h – natural convectionk – air conductionL- characteristic length

or

Nusselt number:

area Specific heat fluxHeat flux

Which surface in this classroom has the largest forced convection

A. Window B. CeilingC. WallsD. Floor

Which surface has the largest natural convection

How to calculate h ?

What are the parametrs that affect h ?

What is the boundary layer ?

Laminar and Turbulent Flowforced convection

Forced convection governing equations

v 2

2

yuv

yu

xuu

0 v

yxu

1) Continuity

2) Momentumu, v – velocities – air viscosity

oooo UUuuLyyLxx v v;; ; ****

2*

*2

*

**

*

** 1 v

yu

LUyu

xuu

oo

Non-dimensionless momentum equation

Using

L = characteristic length and U0 = arbitrary reference velocity

ReL Reynolds number

Forced convection governing equations

TTTTT

w

*

2

2 v

yT

yT

xTu

Energy equation for boundary layer

Non-dimensionless energy equations

2*

*2

*

**

*

**

.Pr.Re1 v

yT

yT

xTu

L

T –temperature, – thermal diffusivity =k/cp,

k-conductivity, - density, cp –specific cap.

Wall temperature

Air temperature outside of boundary layer

LU

LRe Inertial force

Viscous force a

Pr Momentum diffusivity

Thermal diffusivity

Reynolds number Prandtl number

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

khLNu

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

v 2

2

yuvTTg

yu

xuu

0 v

yxu

2

2 v

y

TyT

xT

u

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

TTTTTUU

uuLyyLxxw

***** ;v v;; ;

2*

*2*

2*

**

*

**

Re1 v

yuT

ULTTwg

yu

xuu

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

yuTGr

yu

xuu LL

Grashof number Characteristic Number for Natural Convection

2

3

LTTwgGr

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

),(),)(( nmnmforced LTfLTTwfh

Simplified equation:

mforced 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

Combined forced and natural convention

nnforced

nnatural

nnncombined hhhhh /1/1

21 )()(

0 1 2 3 40

1

2

3

4

5h

T or ACH

n=2

n=3

n=6

h2

h1

hcombined

Churchill and Usagi approach :

This equation favors a dominant term (h1 or h2), and exponent coefficient ‘n’ determines the value for hcombined when both terms have the same order of value

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

What kind of flow is the most common for indoor surfaces

A. Laminar B. TurbulentC. TransitionalD. Laminar, transitional, and turbulent

What about outdoor surfaces?

Conduction

Conductive heat transfer• Steady-state

• Unsteady-state

• Boundary conditions

– Dirichlet Tsurface = Tknown

– Neumann

)(/ 21 SS TTLkq

sourcep

qxT

ckT

2

2

)( surfaceair TThxT

L

Tair

k - conductivity of material

TS1 TS2

h

Boundary conditions

Biot number

solidkhLBi

convention

conduction

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)/(T

o-Ts

)

hour

Ts

0

T

-L / 2 L/2

h

h

h

T o

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

Important numbers

LUo

L Re Inertial force

Viscous forceReynolds number

a

Pr Momentum diffusivity

Thermal diffusivityPrandtl number

2

3

LTTsgGr

Buoyancy forcesViscous forces

khLNu

Conduction

Convection Nusselt number

solidkhLBi thermal internal resistance

surface film resistance

Grashof number

Biot number

Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt