Transfer Function Method - Solar Thermal | IEA-SHC

Transcription

1
MONTREAL SUMMER SCHOOL
25-29 June 2011
Transfer Function Method
Dipartimento dell’Energia
Prof. Maurizio Cellura
mcellura@dream . unipa .it
Ph +39-091-23861931; Fax +39-091-484425;
PhD Summer School
Montreal 25-29 June 2011
GENERAL INFORMATION
Analysis of the thermal performances of
building during the project allows the
developer:
 the choice of the best materials for the local
climatic characteristic;
energy saving and a correct bio-climatic
approach.
2
University of Palermo, Italy
Maurizio Cellura
PhD Summer School
Introduction
Simulation and analysis of the thermal fluxes in a building
help the developer to choose the best materials for the local
climatic characteristics
Many software packages
for the thermal dynamic
simulation of buildings
have been developed to
better
understand
the
thermal
behavior
of
buildings
3
Maurizio Cellura
PhD Summer School
Introduction
A caravan has walls with a great thermal
insulation, but the thermal comfort conditions
are often poor in summer conditions
On the contrary, in sunny hot days, inside a
massive building, with heavy walls and
without
insulations,
thermal
comfort
conditions are excellent – Radiant mean
temperature of surfaces and comfort? How
the HWAC control temperature of surfaces?
Are the north architectural styles successfully transferrable
to the south of the world?
4
Maurizio Cellura
PhD Summer School
Introduction
Many thermal process are relevant in the assessment of
building thermal behavior including:

heat conduction through exterior walls, roofs,
ceilings, floors and interior partitions;

solar radiation through transparent surfaces;

latent or sensible heat generated in the space by
occupants, lights, and appliances;
 heat transfer through ventilation and infiltration of
outdoor air and other miscellaneous heat gains.
5
Maurizio Cellura
PhD Summer School
Introduction
The Thermal Inertia – Developer could
Use the delay
6
Maurizio Cellura
PhD Summer School
Introduction
The Thermal Inertia
The thermal inertia, linked to
conductive heat transfer, can:
• Control the inner temperature
variations
• Realize the optimal comfort
conditions
• Limit the HVAC costs
By planning the right value of
thermal inertia of walls, the
maximum requested power for HVAC
during summer season can be
reduced
7
Maurizio Cellura
PhD Summer School
Introduction
Considering an homogeneous and isotropic element..
8
Maurizio Cellura
PhD Summer School
Fourier’s Law:
Recognized that the flux is a vectory quantity, we can
write a more general statement of the conduction rate
equation as:
 ∂T
∂T
∂T 

q = −k∇T = −k  i
+j
+k
∂y
∂z 
 ∂x
Where:
∇ is the three dimensional del operator
T (x, y, z ) is the scalar temperature field.
9
Maurizio Cellura
PhD Summer School
The non–steady–state heat problems in
multilayered walls
• For a homogeneous isotropic element, the change of temperature
and of flux with distance are:
∂θ ( x, t )
∂q ( x, t )
∂θ ( x, t )
1
=
− q ( x, t ) ;
=
− ρC
∂x
λ
∂x
∂t
Combination of these equation gives the Fourier equation:
∂ 2θ ( x, t ) 1 ∂θ ( x, t )
=
2
∂x
α
∂t
10
Maurizio Cellura
PhD Summer School
Methods:
To solve the dynamic heat transmission it is possible to
use:
Laplace’s Method;
Z-Transform Method;
Finite element Method;
Finite difference Method.
11
Analytics'
Methods
Numerical
Methods Discussion on
Analitical’s Methods
Maurizio Cellura
12
25-29 June 2011
LAPLACE’S METHOD
Prof. Maurizio
Cellura
cell-. +393204328228
PhD Summer School
Laplace’s Method: (continuous signals)
The Laplace transform is an algorithm commonly
used in the solution of differential equations
Given a function of time f(t) for 0 ≤ t < ∞, the
Laplace transform F(s) is:
=
F ( s ) L=
[ f (t )]
13
∫
∞
0
e
−st
f (t ) d t
Maurizio Cellura
PhD Summer School
Some fundamentals L-transformations
Step function:
1
=
f (t ) u=
(t )
F (s)
s
*
1
f (t ) Au
(t )
F (s) A
=
=
s
*
14
Maurizio Cellura
PhD Summer School
Some fundamentals L-transformations
Ramp function:
1
f (t ) t =
F (s)
=
s2
1
f (t ) A=
t
F ( s) A 2
=
s
15
Maurizio Cellura
PhD Summer School
Laplace’s Method: (discrete signals)
Other more complicated function are known only in fixed
instants of time. These functions (signals) are called discrete
16
Maurizio Cellura
PhD Summer School
To sample a continuous signal, the first step is to interpolate
data between two values.
This steps is very important
because the area under the
interpolating line is linked to the
energy. Without interpolating lines,
there will be no energy linked to the
thermal flux (that is continuous).
In the figure, a continuous signal
sampled with step interpolating
function
17
Maurizio Cellura
18
In place of linear or step interpolating
function, a train of linear ramps can be
used.
Combining
different
elementary
functions,
linear,
triangular,
parabolic, it is possible to
create more interpolating
functions.
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PhD Summer School
For a function such as temperature T(x, t), which
has transformed θ (x, s), the following relations:
 ∂T ( x, t ) 
L =
s θ ( x, s ) − T ( x, 0 )

 ∂t 
 ∂T ( x, t ) 
L

∂
x


19
 ∂ 2T ( x, t )  ∂ 2θ ( x, s )
∂θ ( x, s )
=
L

2
2
∂x
∂
∂
x
x


Maurizio Cellura
PhD Summer School
Using Laplace Transform, the Fourier’s Law is:
∂T
∂ 2T
=
−α 2 0
∂t
∂x
∂ 2θ ( x, s ) s
−=
θ ( x, s ) 0
2
∂x
α
Assumption: the temperature at the initial instant is zero.
The general integral is:
βx
θ=
( x, s ) M e + N e
where
20
β=
−β x
s
α
Maurizio Cellura
PhD Summer School
For a homogeneous isotropic element, changes with
the distance of the temperature and flux are:
∂θ ( x, t )
∂q ( x, t )
∂θ ( x, t )
1
=
− q ( x, t ) ;
=
− ρC
∂x
λ
∂x
∂t
The solutions of this system was obtained using
Laplace Transform in 1964
21
Maurizio Cellura
PhD Summer School
The non–steady–state heat problems in multilayered walls
• Solution of these equations into an imaginary space was
obtained by Carlslaw and Jaeger
1



2
s

  x  q ( 0, s )
sinh
1

 α  


2
s



θ ( x, s ) cosh   x  θ ( 0, s ) −
1



α
 

 s 2


λ 

α 


1
1
1




2
2

s
s
 s 2




−λ   sinh   x  θ ( 0, s ) + cosh   x  q ( 0, s )
 q ( x, s ) =
 α  
 α  
α 





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Maurizio Cellura
23
25-29 June 2011
Z-TRANSFORM’S METHOD
Prof. Maurizio
Cellura
cell-. +393204328228
PhD Summer School
Z-transform Method: (discrete signals)
Considering that, usually the available signals in thermophysics of buildings, such as inner temperatures, insolation,
outdoor temperature, etc…, are discrete signals, it is better
to use a more suitable mathematical transformation (Z
transform method).
Furthermore, the transformation must be suitable even to be
implemented in software codes.
To solve the Carlslaw and Jager equations by the Ztransform method
24
Maurizio Cellura
PhD Summer School
Consider a function f(t), for t ≥ 0, where are known f(nΔ) for a time data
set Δ , its Z-transform function is:
Z [ f (=
t ) ] F (=
z)
∞
∑ f (n∆) z=
−n
f (0) + f (∆ ) z −1 + f (2∆ ) z −2 + f (3∆ ) z −3 + ⋅ ⋅ ⋅
n =0
where z is a complex variable
Z-transform is used for the following functions:
1
1
(
)
=
F
s
1 − z −1
s
1
1
(
)
=
f (t ) t =
F ( z)
=
F
s
z (1 − z −1 ) 2
s2
1
1
− at
(
)
=
f (t ) e=
F ( z)
=
F
s
1 − e − a∆ z −1
s+a
* (t )
=
f (t ) u=
F ( z)
25
Maurizio Cellura
PhD Summer School
The non–steady–state heat problems in multilayered walls
• The solution for a multi-layered wall can be expressed in
matrix form:
A B Ti
x
=
Qe C D Qi
Te
26
Maurizio Cellura
PhD Summer School
THE WALL TRANSMISSION MATRIX
A, B, C and D are the coefficients of the wall transmission matrix
reached through the product of transmission matrixes, of each n layers
forming the wall:
n a
b
A B
m
m
=∏
C D m =1 cm d m
27
Maurizio Cellura
PhD Summer School
where:


s
s
s
 ; cm = λ
 ;
am = cosh L
senh L


α
α
α





s

senh L

α


bm =
s
λ
α
λ
d m = am ; α =
ρC p
28
Maurizio Cellura
29
Z-TRASFORM METHOD
After many mathematical manipulations
Transfer function in a polynomial form
Very fast response – Coefficients decrease their values soon and this fact implies a limited
employment of them; usually less than 10
Phd summer school on solar NZEBs
30
METODO DELLA Z-TRASFORMATA
With the following substitution
And the following
The heat flux Qi could be calculated like the ASHRAE manual
Know temperatures till the current nΔ and thermal flux of the inner surface till (n-1)Δ
Phd summer school on solar NZEBs
PhD Summer School
The system can be described using the ZT at the LT place. For a
sampled, time-continuous function f(t), LT is,:
f (0 ) + f (∆ )e
− s∆
+ f (2∆ )e
−2 s∆
+ ....
Z-T of function f(t), is:
f (0 )z + f (∆ )z + f (2∆ )z
0
−1
−2 z
+ ....
Using Z-T implies modifications of previous wall transmission
matrix formula
29
Maurizio Cellura
32
25-29 June 2011
An Application:
Heat Balance Method
Prof. Maurizio
Cellura
cell-. +393204328228
PhD Summer School
Heat Balance Method (ASHRAE)
Conduction heat flux
through a multilayered
wall is of paramount
importance in the
ASHRAE algorithm.
31
Maurizio Cellura
PhD Summer School
Evaluation of conduction thermal flux.
The central element of the HB method is the dynamic
evaluation of conduction thermal fluxes through
boundary walls.
The methods to calculate the conduction heat
flux through a multilayered wall are manifold
and one of the most employed is the
Transfer Function Method ( TFM).
(TRNSYS, ENERGY PLUS)
33
Maurizio Cellura
PhD Summer School
 The TFM allows the calculation of the thermal flux or of the wall
surface temperature employing Conduction Transfer Function –
CTFs that depends on thermo-physical characteristics of the layers.
 exact determination of these coefficients would require a theoretical
infinite numbers of calculus; to evaluate CTFs a numerical approach
is used,
The numerical approach can be a source of error.
The choice of time step is really very important,
34
Maurizio Cellura
PhD Summer School
The thermal simulations with TFM applied to very massive walls,
characterized by high thermal mass, strongly depends on time step!
DOE-2, TRNSYS e ENERGY PLUS
Massive wall with
80% insulation
Error > 44%
Massive walls
composed by concrete
Error > 27%
Wood
35
Error < 2 %
Maurizio Cellura
PhD Summer School
The determination of the thermal flux through multilayered
walls was solved by Carslaw and Jager in 1959.
θe
θi
ϕe
ϕi
0
36
L
x
Maurizio Cellura
PhD Summer School
The application of Mitalas resolution method to the matrix
system of Carslaw and Jager permits to obtain the inner
thermal flux or the inner surface temperature by means of
some coefficients called Transfer Function Coefficients,
Transfer Function Coefficients are
numerically calculated
37
Maurizio Cellura
PhD Summer School
Quality Assessment of CTFs
Time step
Thermophysical
characteristics
Resolution
of the system
Choice of poles
number
38
Quality of system
resolution
depends on many
parameters!!!
Choice of
interpolating
function
Choice of
coefficients
number
Maurizio Cellura
PhD Summer School
TFM and Fourier Transforms
To evaluate the quality of resolution by TFM we made a comparison
with the results obtained in Periodic Steady State (Fourier
Transform).
Percentage Mean Error
where:
PME
1 24/ ∆ QZ (τ ) − QF (τ )
⋅100
∑
QF (τ )
24∆ τ =1
QZ: thermal flux emerging from the inner surface of the wall calculated
with TFM ;
QF: thermal flux emerging from the inner surface of the wall calculated
with Fourier Transforms (Periodic Steady State Solution)
39
Maurizio Cellura
PhD Summer School
Accuracy of TFM
We applied the TFM to typical Mediterranean buildings.
Some considerations:
1.
2.
3.
4.
40
Simulations with the greatest number of poles not always are the
best;
Simulations with the greatest number of coefficients not always are
the best;
A larger time step (2 or 3 hours) generally decreases the PME but
nevertheless decreases the quality of information;
Choice of a larger time step or of a different number of poles and
coefficients is function of high inertia mass of the wall.
Maurizio Cellura
PhD Summer School
Who is really an expert in the field of
Transfer Function Method?
1. How to choice the best time step?
2. How to determine the optimum number
of poles?
3. How to choice the best number of
coefficients?
 The optimum choice of these parameters is crucial to obtain a good and reliable
simulation using TFM.
 Often in the commercial Building Simulation Software is impossible the “tuning“
of these parameters.
 Even if possible, normal user would not be able to choice the optimal values.
41
Maurizio Cellura
PhD Summer School
Software CATI
Client-server
structure
42
Maurizio Cellura
PhD Summer School
Software CATI
s
S
Density Asymmetry
∑ρ Ls
DA =
i i
∑ρ L
3
s
i
i =1
S
i
i =1
2
[ m]
i
s
1
S
Specific heat
Asymmetry
HA =
∑c L s
i
i =1
S
i i
∑c L
i
i =1
Thermal resistance
Asymmetry
S
RA =
Thermal Inertia
43
−1
i
i i
Ls
∑λ
i =1
o
i
∑λ
i =1
S
[ m]
−1
i
i
[ m]
L
S
TI =
∑c ρ L
i
i =1
i
i
 1
1 
3600 ⋅  + ∑ λi−1 Li +

hout 
 hin i =1
S
−1
[h]
Maurizio Cellura
PhD Summer School
Il Software CATI
The software CATI records:
 the coefficients of the CTFs numerator;
 the coefficients of the CTFs
denominator;
 the Transfer Function in the Laplace
domain;
 the temperature time series (or thermal
flux) obtained as output of the simulation
with TFM;
 the temperature time series (or thermal
flux) obtained as output of the simulation
with Fourier
Other data linked to the used algorithms
44
Maurizio Cellura
PhD Summer School
Results:
The accuracy of a simulation performed with TFM strongly depends on the
number of coefficients!!
Wall
Sampled
period
Number of
poles
Number of coefficients
PME
A0:A1:B6:C11:E0
1
10
6
0,067
A0:A1:B6:C11:E0
A0:A1:B6:C11:E0
A0:A1:B6:C11:E0
A0:A1:B6:C11:E0
A0:A1:B6:C11:E0
A0:A1:B6:C11:E0
A0:A1:B6:C11:E0
A0:A1:B6:C11:E0
1
1
2
2
2
3
3
3
10
10
10
10
10
10
10
10
5
4
5
4
3
4
3
2
5,395
767,828
1,435
0,645
175,077
2,227
6,868
490,801
Reliability of performed simulations
A decrease of only two or three coefficients implies a significant
increase in the PME. A number of coefficients lower than the
threshold value makes the simulation completely unreliable.
45 University of Palermo, Italy
Maurizio Cellura
PhD Summer School
Results:
Elimination of only two or three coefficients provokes a drastic increase of PME.
Composizione parete: A0:A1:B6:C11:E0
Numero di poli: np= 10
nc= 4
767.828
800
700
600
nc= 2
490.801
500
400
175.077
nc= 5
5
nc= 3
nc= 3
6.868
200
5.395
PME (%)
300
4
3
0
1
1
1
nc= 4
2
2
2.227
1
nc= 5
0.645
nc= 6
1.435
2
0.067
nc= 4
2
3
3
3
Periodo di campionamento (h)
Wall composition: Table 1 in ASHRAE 1994 .
46
Maurizio Cellura
PhD Summer School
Results:
Furthermore, a sort of asymptotic trend in the PME can be also
observed. The increase of the number of poles or the coefficients does
not increase the quality of the simulation:
Wall
Sampled
period
Number of
poles
Number of
coefficients
PME
A0:A1:B6:C11:E0
1
16
12
0,065
A0:A1:B6:C11:E0
1
16
11
0,065
A0:A1:B6:C11:E0
1
16
10
0,065
A0:A1:B6:C11:E0
2
16
12
1,434
A0:A1:B6:C11:E0
2
16
11
1,434
A0:A1:B6:C11:E0
2
16
10
1,434
A0:A1:B6:C11:E0
3
16
9
2,229
A0:A1:B6:C11:E0
3
16
8
2,229
A0:A1:B6:C11:E0
3
16
7
2,229
47
Excerpt
Maurizio Cellura
PhD Summer School
Aim: provide an expert tool to help the
user.
1. An expert tool able to automatically select the best time
step
2. An expert tool able to suggest the optimum number of
poles
3. An expert tool able to suggest the optimum number of
coefficients
48
Maurizio Cellura
PhD Summer School
Data-driven approach
data-driven: an approach characterized by a “self-learning”
process based upon the results and the simulations performed
by CATI
Reliability of a CTF set
(Artificial Neural Network- ANN)
49
Maurizio Cellura
PhD Summer School
Neural network and Classification
“Pattern Recognition:
A neural network is able to recognize similar features of input data and
to classify them in categories
neuroni
f(.) funzione non lineare
j
Input l
wijl
wnil+1
i
n
w pesi sinaptici
Input 2
Input 3
Strato
l-1
Strato
l
Strato
l+1
A classifier is an expert tool that creates a relation between the
variables of a domain “A” and a vector of labels “B”, In other words it
gives a label “B” to every sample of domain “A”,
50 University of Palermo, Italy
Σ f(.)
Maurizio Cellura
PhD Summer School
Neural Networks Classifier
Using the software CATI, were performed more than 54000
simulations. It was possible to build a large databases to train a neural
network classifier.
 Time step
 Thermal inertia of the wall;
 Have been defined two classes (labels):
 Number of poles of CTF;
 Number
coefficients
of CTF;
1) Theofreliable
simulations
(PMEAsymmetry;
<5%);
Density
 Specific Heat Asymmetry
1) The unreliable simulations
 Thermal
Resistance Asymmetry
(PME >5%).
Input
vector
 Other data
51
Maurizio Cellura
PhD Summer School
Neural Network Classifier Output
Correlation
analysis
CLASSIFICATOR
JUDGEMENT
Confusion
Matrix
52
Reliable
simulation
TRUTH
Reliable
Simulation
Unreliable
simulation
Unreliable
simulation
98,2%
1,8%
2,8%
97,2%
Maurizio Cellura
PhD Summer School
Software development:
Evaluation of a CTF with 1 hour
time step, 11 poles and 5
coefficients. The CTF will be
UNRELIABLE!
Evaluation of a CTF with 1 hour
time step, 14 poles and 13
coefficients. The CTF will be
reliable!
53
Maurizio Cellura
55
25-29 June 2011
Thanks for your attention!
Dipartimento dell’Energia
Prof. Maurizio Cellura
Ph +39-091-23861931;
e-mail: mcellura@dream .unipa .it

Transfer Function Method - Solar Thermal | IEA-SHC

Transcription

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