Lecture Objectives:

1
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

2
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.air ?Tair Qconvective Qventilation
Qconvective SAihi(TSi-Tair)
Ts1
mi
Qventilation Smicp,i(Tsupply-Tair)
Q2
Q1
Tair
h1
h2
3
Air balance steady state Convection 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 SAihi(TSi-Tair)
Ts1
mi
Qinfiltration Smicp(Toutdoor_air-Tair)
Q2
Q1
In order to keep constant air Temperate, HVAC
system needs to remove cooling load
Tair
h1
h2
QHVAC Qair mcp(Tsupply_air-Tair)
HVAC
4
Homework assignment 1
2.5 m
10 m
10 m
North
West
5
Homework assignment 1 Surface energy balance
1) External wall (north) node
QsolarC1A(Tsky4 - Tnorth_o4) C2A(Tground4 -
Tnorth_o4)hextA(Tair_out-Tnorth_o)Ak/?(Tnorth_o-
Tnorth_in)
Qsolarasolar(IdifIDIR) A
C1easurface_long_wavesFsurf_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
C3eniort_ins ynorth_in_to_ internal surface
6
Using MathCad
7
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
Q1
Q2
Tair
HVAC
8
Unsteady-state problemExplicit Implicit methods

• Example

To - known and changes in time Tw - unknown Ti
- unknown AiAo6 m2 (mcp)i648 J/K (mcp)w9720
J/K Initial conditions To Tw Ti
20oC Boundary conditions hiho1.5 W/m2
Tw
Ti
To
AoAi
Conservation of energy
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 Dt0.1 hour 360 s
9
Explicit Implicit methods example

• Conservation of energy equations

Wall
Air
After substitution
For which time step to solve ?? ? or ? ?
Wall
Air
?? ? Implicit method ? Explicit
method
10
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
11
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
12
Explicit method

• Problems with stability !!!
• Often requires very small time steps

13
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
14
Explicit methods information progressing during
the calculation
Tw
Ti
To
15
Unsteady-state conduction - Wall
q
Nodes for numerical calculation
Dx
16
Discretization of a non-homogeneous wall
structure
Section considered in the following discussion
Discretization in space
Discretization in time
17
Internal node Finite volume method
Boundaries of control volume
For node I - integration through control volume
18
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
19
Internal node finite volume method
For uniform grid
Explicit method
Implicit method
20
Internal node finite volume method
Substituting left and right sides
Explicit method
Implicit method
21
Internal node finite volume method
Explicit method
Rearranging
Implicit method
Rearranging
22
Energy balance for elements surface node
Implicit equation
Or if TSi and TA are known
23
Energy balance for elements 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
24
Unsteady-state conductionImplicit method
b1T1 ??? c1T2???f(Tair,T1?,T2?)
a2T1 ??? b2T2 ??? c2T3???f(T1 ?,T2?, T3?)
Air
1
3
4
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
25
Stability of numerical scheme
Explicit method - simple for calculation -
unstable
Implicit method - complex system of equations
(matrix) - Unconditionally stabile
What about accuracy ?
26
Unsteady-state conductionHomogeneous Wall
27
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