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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.

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lecture objectives
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
Air balance - Convection on internal surfaces + Ventilation + Infiltration

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 state convection on internal surfaces infiltration load
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

to remove cooling load

Tair

h1

h2

QHVAC= Qair= m·cp(Tsupply_air-Tair)

HVAC

homework assignment 1

Top view

Glass

Twest_oi

Twest_i

Tinter_surf

Tair_in

Surface

radiation

IDIR

Idif

Tnorth_i

conduction

Tnorth_o

Tair_out

Styrofoam

Surface radiation

Idif

IDIR

Homework assignment 1

2.5 m

10 m

10 m

North

West

homework assignment 1 surface energy balance
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

air balance steady state vs unsteady state
Air balance steady state vs. unsteady state

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

unsteady state problem explicit implicit methods
Example:Unsteady-state problemExplicit – Implicit methods

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

explicit implicit methods example
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
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
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

Explicit method

Problems with stability !!!

Often requires very small time steps

explicit methods example1
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

unsteady state conduction wall
Unsteady-state conduction - Wall

q

Nodes for numerical

calculation

Dx

discretization of a non homogeneous wall structure
Discretization of a non-homogeneous wall structure

Section considered in the

following discussion

Discretization in space

Discretization in time

internal node finite volume method
Internal node Finite volume method

Boundaries of control volume

For node “I” - integration through control volume

slide18

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 method1
Internal node finite volume method

For uniform grid

Explicit method

Implicit method

internal node finite volume method2
Internal node finite volume method

Substituting left and right sides:

Explicit method

Implicit method

internal node finite volume method3
Internal node finite volume method

Explicit method

Rearranging:

Implicit method

Rearranging:

energy balance for element s surface node

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 node1
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:

unsteady state conduction implicit method
Unsteady-state conductionImplicit method

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

What about accuracy ?

system of equation for more than one element
System of equation for more than one element

Roof

air

Left wall

Right wall

Floor

  • Elements are connected by:
          • Convection – air node
          • Radiation – surface nodes