announcement

16
Announcement • Next class is in ECJ 302 (computer lab) • No class on October 17 – Conference – Make up will be an out of class time final project presentation (during the final week)

Upload: kamea

Post on 05-Feb-2016

27 views

Category:

Documents


0 download

DESCRIPTION

Announcement. Next class is in ECJ 302 (computer lab) No class on October 17 Conference Make up will be an out of class time final project presentation (during the final week). Lecture Objectives:. Analyze the unsteady-state heat transfer numerical calculation methods. Example:. - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: Announcement

Announcement

• Next class is in ECJ 302 (computer lab)

• No class on October 17 – Conference– Make up will be an out of class time final

project presentation (during the final week)

Page 2: Announcement

Lecture Objectives:

• Analyze the unsteady-state heat transfer numerical calculation methods

Page 3: Announcement

Unsteady-state heat transfer(Explicit – Implicit methods)

QT

mcp

• Example:

Ti To

Tw

Ao=Ai

To - known and changes in timeTw - unknownTi - unknownAi=Ao=6 m2

(mcp)i=648 J/K(mcp)w=9720 J/K

Initial conditions: To = Tw = Ti = 20oCBoundary conditions:

hi=ho=1.5 W/m2

Time [h] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

To 20 30 35 32 20 10 15 10

Time step =0.1 hour = 360 s

boundariesatp QT

mc _

Conservation of energy:

Page 4: Announcement

Explicit – Implicit methods example

wiwoww

wp TThATThATT

mc

Conservation of energy equations:

Wall:

iwii

ip TThATT

mc

Air:

wioww TTTTT 2)(3 Wall:

iwii TTTT )(3.0 Air:

After substitution: For which time step to solve:+ or ?

+ Implicit method Explicit method

Page 5: Announcement

Implicit methods - example

wioww TTTTT 2)(3

iwii TTTT )(2.0

woiw TTTT 3)23(

iiw TTT )12.0(

400 800 1200 1600 2000 24000

10

20

30

40

50

60

70

80

T[C

]

time

To Tw Ti

=0 To Tw Ti

=36 system of equation Tw Ti

=72 system of equation Tw Ti

After rearranging:

2 Equations with 2 unknowns!

Page 6: Announcement

Explicit methods - example

wioww TTTTT 2)(3

iwii TTTT )(2.0

3

)23( owi

w

TTTT

2.0

)12.0(

iw

i

TTT

=0 To Tw Ti

=360 To Tw Ti

=720 To Tw Ti

=360 sec

2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

T [C

]

time

To Tw Ti

UNSTABILE

There is NO system of equations!

Tim

e

Page 7: Announcement

Explicit method

Problems with stability !!!

Often requires very small time steps

Page 8: Announcement

Explicit methods - example

30

)230( owi

w

TTTT

2

)12(

iw

i

TTT

=0 To Tw Ti

=36 To Tw Ti

=72 To Tw Ti =36 sec

400 800 1200 1600 2000 24000

10

20

30

40

50

60

70

80

T[C

]

time

To Tw Ti

Stable solution obtainedby time step reduction

10 times smaller time step

Tim

e

Page 9: Announcement

Explicit methods information progressing during the calculation

QT

mcp Ti To

Tw

Page 10: Announcement

Unsteady-state conduction - Wall

sourcep

qx

T

c

T

2

2

q

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

Nodes for numerical calculation

x

Page 11: Announcement

Discretization of a non-homogeneous wall structure

Fa

cad

e s

lab

Insu

latio

n

Gyp

sum

Section considered in the following discussion

Discretization in space

2

2

x

T

c

T

p

Discretization in time

Page 12: Announcement

Internal node Finite volume method

2/

2/

2/

2/2

2I

I

I

I

XI

XI

XI

XI

pII dxdx

Tkdxd

Tc

2

2

x

Tk

Tcp

For node “I” - integration through the control volume

( x) I- 1 ( x)I

x I

I-1 I I+1q I -1 to I q I to I+1

Boundaries of control volume

Surface node

Page 13: Announcement

2/

2/

I

I

XI

XI

III TTxdxdT

1

111

2/2/

2/

2/

2/

2/2

2

I

III

I

iII

XIXI

XI

XI

XI

XI x

TTk

x

TTk

dx

dTk

dx

dTk

x

Tk

xx

Tk

II

I

I

I

I

Left side of equation for node “I”

Right side of equation for node “I”

dx

TTk

x

TTkdxd

x

Tk

I

III

I

IIIXI

XI

I

I

1

1112/

2/2

2

Mathematical approach(finite volume method)

- Discretization in Time

- Discretization in Space

Page 14: Announcement

Mathematical approach(finite volume method)

xx

dx

TTk

x

TTkdxd

x

Tk

I

III

I

III

XI

XI

I

I

1

111

2/

2/2

2

I

III

I

IIIIII x

TTk

x

TTkTTx 111

pc

Explicit method

and for uniform grid

Implicit method

I

III

I

IIIIII x

TTk

x

TTkTTx 111

pc

By Substituting left and right side terms of equation

2/

2/

2/

2/2

2I

I

I

I

XI

XI

XI

XI

pII dxdx

Tkdxd

Tc

we get the following equation for

Using

xdx

][1

111

I

III

I

III

x

TTk

x

TTk

Page 15: Announcement

Physical approach(finite volume method)( x) I- 1 ( x)I

x I

I-1 I I+1q I -1 to I q I to I+1

Boundaries of control volume

2

2

x

Tk

Tcp

Change of energy in x

Change of heat flux along x

xxT

kTcp

)(

qx

=

Change of energy in x

Sum of energy that goes in and out of control volume x

=

or

) (1

1 I toI I to1-I

qqx

Tcp

For finite volume x:

1 I toI I to1-I

qqT

xcp

Page 16: Announcement

( x) I- 1 ( x)I

x I

I-1 I I+1q I -1 to I q I to I+1

1 I toI I to1-I

qqT

xcp

1I toI I to1-I

qqT

xcp

)(/)(/ 111

IIII

II

p TTxkTTxkTT

xc II

)(/)(/ 11

IIII

II

p TTxkTTxkTT

xc

xx For uniform grid

Physical approach(finite volume method)