Lecture Objectives:

• Discuss the HW1b solution

• Learn about the connection of building physics with HVAC

• Solve part of the homework problem– Introduce Mat Cad Equation Solver

• Analyze the unsteady-state heat transfer numerical calculation methods

• Explicit – Implicit methods

Air balance - Convection on internal surfaces + Ventilation + Infiltration

h1

Q1

h2

Q2

What affects the air temperature?- h and corresponding Q - as many as surfaces

miTs1

Tair

Uniform Air Temperature Assumption!

Qconvective= ΣAihi(TSi-Tair)

Qventilation= Σmicp,i(Tsupply-Tair)

Tsupply-maircp.air ΔTair= Qconvective+ Qventilation

Energy balance:

Air balance – steady state Convection on internal surfaces + Infiltration = Load

h1

Q1

h2

Q2

- h, and Qsurfaces as many as surfaces- infiltration – mass transfer (mi – infiltration)

Qair= Qconvective+ Qinfiltration

miTs1

Tair

Uniform temperature Assumption

Qconvective= ΣAihi(TSi-Tair)

Qinfiltration= Σmicp(Toutdoor_air-Tair)

QHVAC= Qair= m·cp(Tsupply_air-Tair)

T outdoor air

HVAC

In order to keep constant air Temperate, HVAC system needsto remove cooling load

Homework assignment 1

North

10 m 10 m

2.5 m

West

conduction

Tair_in

IDIR

Idif

Glass

Tinter_surf

Tnorth_i

Tnorth_o

Twest_iTwest_oi

Tair_out

StyrofoamIDIRIdif

Surface radiation

Surfaceradiation

Top view

Homework assignment 1 Surface energy balance

1) External wall (north) node

2) Internal wall (north) node

Qsolar=solar·(Idif+IDIR) A

Qsolar+C1·A(Tsky4 - Tnorth_o

4)+ C2·A(Tground4 - Tnorth_o

4)+hextA(Tair_out-Tnorth_o)=Ak/(Tnorth_o-Tnorth_in)

C1=·surfacelong_wave··Fsurf_sky

Qsolar_to int surf =portion of transmitted solar radiation that is absorbed by internal surface

C3A(Tnorth_in4- Tinternal_surf

4)+C4A(Tnorth_in4- Twest_in

4)+ hintA(Tnorth_in-Tair_in)= =kA(Tnorth_out--Tnorth_in)+Qsolar_to_int_surf

C3=niort_in·· north_in_to_ internal surface

transmitedtotalsolarisurfisurfisurfisurfisurfisurfisurftosolar QAreaSUMAreaQ ___int__int__int__int__int__int__int___ ))((/)((

Using MathCad

Air balance steady state vs. unsteady state

Q1 Q2

QHVAC= Qconvection+ Qinfiltration

mi

Tair

HVAC

For steady state we have to bring or remove energy to keep the temperature constant

If QHVAC= 0 temperature is changing – unsteady state

maircpair= Qconvection+ Qinfiltration

Unsteady-state problemExplicit – 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/m2Time [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:

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

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!

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

=36 sec

2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

T [C

]

time

To Tw Ti

UNSTABILITY

There is NO system of equations!

Tim

e

Explicit method

Problems with stability !!!

Often requires very small time steps

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

Explicit methods information progressing during the calculation

QT

mcp Ti To

Tw

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

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

Internal node Finite volume method

2/

2/

, 2/

2/

2/

2/2

2 I

I

I

I

I

I

XI

XI

XI

XI

XI

XI

pII dxdqdxdx

Tkdxd

Tc

sourcep qx

TTc

2

2

For node “I” - integration through 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

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

Internal node finite volume method

- Discretization in Time

- Discretization in Space

Internal node finite volume method

xx

dx

TTk

x

TTkdxd

x

Tk

I

III

I

IIIXI

XI

I

I

1

1112/

2/2

2

I

III

I

III

x

TTk

x

TTk 111

Explicit method

For uniform grid

Implicit method

I

III

I

III

x

TTk

x

TTk 111

Internal node finite volume method

Explicit method

Implicit method

Substituting left and right sides:

qx

TTk

x

TTkTT

xc

I

III

I

IIIII

III 111

qx

TTk

x

TTkTT

xc

I

II

I

IIIII

III

11

Internal node finite volume method

qx

TTk

x

TTkTT

xc

I

II

I

IIIII

III

11

Explicit method

Implicit method

qx

TTk

x

TTkTT

xc

I

III

I

IIIII

III 111

FTCTBTA III

11

),,( 11

IIII TTTfTRearranging:

Rearranging:

Energy balance for element’s surface node

( x) I- 1

xI

I -1 A (Air node)I

Sur

face

q I -1 to Iq I to A

qor

A (Air node)

Si(Surface nodes)

q I to Si

(Outer Radiation)

x/2

x

IIIor

n

iSiiRCA

II

n

iiRC

IIII

cxTqThhT

x

kThh

x

kcxT

2

2 1,

11

1,

1

Implicit equation:

IIIor

n

iSiiRCA

II

n

iiRC

IIII

cxTqThhT

x

kThh

x

kcxT

2

2 1,

11

1,

1

Or if TSi and TA are known:

Energy balance for element’s surface node

General form for each internal surface node:

After rearranging the elements for implicit equation for surface equations:

FTSTRTBTAn

iSiAII

11

FTCTB II

1

General form for each external surface node:

Unsteady-state conductionImplicit method

1 2 3 4 5 6

Matrix equation

M × T = F

for each time step

Air Air

b1T1 + +c1T2

+=f(Tair,T1,T2

)

a2T1 + b2T2

+ +c2T3+=f(T1

,T2, T3

)

a3T2 + b3T3

+ +c3T4+=f(T2

,T3 , T4

)

a6T5 + b6T6

+ =f(T5 ,T6

, Tair)

………………………………..

M × T = F

Stability of numerical scheme

)2/()( 2 kxcp

Explicit method- simple for calculation- unstable

Implicit method- complex –system of equations (matrix) - Unconditionally stabile

What about accuracy ?

Unsteady-state conductionHomogeneous Wall

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

System of equation for more than one element

air

Left wall

Roof

Right wall

Floor

Elements are connected by:1) Convection – air node2) Radiation – surface nodes