Lecture Objectives:

1 / 27

# Lecture Objectives: - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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

Uniform Air Temperature Assumption!

What affects the air temperature?

- h and corresponding Q - as many as surfaces

Energy balance:

Tsupply

-maircp.airΔTair= Qconvective+ Qventilation

Qconvective= ΣAihi(TSi-Tair)

Ts1

mi

Qventilation= Σmicp,i(Tsupply-Tair)

Q2

Q1

Tair

h1

h2

Air balance – steady stateConvection on internal surfaces + Infiltration = Load

Uniform temperature Assumption

• h, and Qsurfaces as many as surfaces
• infiltration – mass transfer (mi – infiltration)
• Qair= Qconvective+ Qinfiltration

T outdoor air

Qconvective= ΣAihi(TSi-Tair)

Ts1

mi

Qinfiltration= Σmicp(Toutdoor_air-Tair)

Q2

Q1

In order to keep constant air

Temperate, HVAC system needs

Tair

h1

h2

QHVAC= Qair= m·cp(Tsupply_air-Tair)

HVAC

Top view

Glass

Twest_oi

Twest_i

Tinter_surf

Tair_in

Surface

IDIR

Idif

Tnorth_i

conduction

Tnorth_o

Tair_out

Styrofoam

Idif

IDIR

Homework assignment 1

2.5 m

10 m

10 m

North

West

Homework assignment 1 Surface energy balance

1) External wall (north) node

Qsolar+C1·A(Tsky4 - Tnorth_o4)+ C2·A(Tground4 - Tnorth_o4)+hextA(Tair_out-Tnorth_o)=Ak/(Tnorth_o-Tnorth_in)

Qsolar=asolar·(Idif+IDIR)A

C1=e·asurface_long_wave·s·Fsurf_sky

2) Internal wall (north) node

C3A(Tnorth_in4- Tinternal_surf4)+C4A(Tnorth_in4- Twest_in4)+hintA(Tnorth_in-Tair_in)=

=kA(Tnorth_out--Tnorth_in)+Qsolar_to_int_surf

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

C3=eniort_in·s· ynorth_in_to_ internal surface

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

temperature constant

QHVAC= Qconvection+ Qinfiltration

If QHVAC= 0 temperature is changing – unsteady state

maircp(DTair/Dt)= Qconvection+ Qinfiltration

mi

Q2

Q1

Tair

HVAC

To - known and changes in time

Tw - unknown

Ti - unknown

Ai=Ao=6 m2

(mcp)i=648 J/K

(mcp)w=9720 J/K

Initial conditions:

To = Tw = Ti = 20oC

Boundary conditions:

hi=ho=1.5 W/m2

Tw

Ti

To

Ao=Ai

Conservation of energy:

Time step Dt=0.1 hour = 360 s

Conservation of energy equations:Explicit – Implicit methods example

Wall:

Air:

After substitution:

For which time

step to solve:

+  or  ?

Wall:

Air:

+  Implicit method

 Explicit method

Implicit methods - example

After rearranging:

2 Equations with 2 unknowns!

 =0 To Tw Ti

 =36 system of equation Tw Ti

 =72 system of equation Tw Ti

Explicit methods - example

 =36 sec

 =0 To Tw Ti

 =360 To Tw Ti

 =720 To Tw Ti

Time

There is NO system of equations!

UNSTABILITY

### Explicit method

Problems with stability !!!

Often requires very small time steps

Explicit methods - example

 =0 To Tw Ti

 =36 To Tw Ti

 =72 To Tw Ti

Stable solution obtained

by time step reduction

10 times smaller time step

Time

 =36 sec

Tw

Ti

To

q

Nodes for numerical

calculation

Dx

Discretization of a non-homogeneous wall structure

Section considered in the

following discussion

Discretization in space

Discretization in time

Internal node Finite volume method

Boundaries of control volume

For node “I” - integration through control volume

Internal node finite volume method

Left side of equation for node “I”

- Discretization in Time

Right side of equation for node “I”

- Discretization in Space

Internal node finite volume method

For uniform grid

Explicit method

Implicit method

Internal node finite volume method

Substituting left and right sides:

Explicit method

Implicit method

Internal node finite volume method

Explicit method

Rearranging:

Implicit method

Rearranging:

Dx

Dx/2

Energy balance for element’s surface node

Implicit equation:

Or if TSi and TA are known:

Energy balance for element’s surface node

After rearranging the elements for implicit

equation for surface equations:

General form for each internal surface node:

General form for each external surface node:

b1T1 + +c1T2+=f(Tair,T1,T2)

a2T1+b2T2 + +c2T3+=f(T1 ,T2, T3)

Air

1

4

3

2

5

Air

6

a3T2+b3T3+ +c3T4+=f(T2 ,T3 , T4)

………………………………..

a6T5+b6T6+ =f(T5 ,T6 , Tair)

Matrix equation

M × T = F

for each time step

M × T = F

Stability of numerical scheme
• Explicit method
• - simple for calculation
• - unstable
• Implicit method
• - complex –system of equations (matrix)
• - Unconditionally stabile

System of equation for more than one element

Roof

air

Left wall

Right wall

Floor

• Elements are connected by:
• Convection – air node