Pulse Transfer Function & Its Application Related to Buildings H. Yamazaki Kyushu Institute of Design Fukuoke-City, Japan .Abstract For the digital calculation of a linear system, it is more convenient to use the Z-transform than the S-transform (Laplace transform). The transformation of the product of the system input and transfer function on the S-domain into the Z-domain is very complicated. This difficulty is overcome, however, if we assume that the output varies very slowly in response to a sudden input change. Discussed in this paper is the general technique of obtaining the Z-transform transfer function and the application of the technique to calculate room temperature of air-conditioned rooms under an automatic control system. Keywords:
Linear systems, transfer function, Z-transform 1.
Introduction
Need exists to calculate dynamic changes of room temperature and heating/cooling loads. The method of these calculations are either frequency response method or the pulse response method, The frequency response method requires the solution of simultaneous equations with complex coefficients, With a careful plan the computer time for the complicated system can be decreased considerably, On the other hand, the pulse transfer function method requires polynominal expression of time series data in power of z-1, Although the pulse transfer method is convenient for dealing with sampled data, there is no superiority or inferiority over the frequency response method, It is possible to obtain the pulse transfer function in three ways: 1.
transfer function
2.
frequency response
3.
Weighting function
Reference (1) describes the method to derive the pulse transfer function from the frequency response, This paper is based on a method of obtaining approximate pu~.se transfer function from transfer function. 2,
Input, Output and Components
The inputs may be the time changes of solar radiation, outdoor air temperature, internal heat generation in the room heat due to lighting and equipment or due to the occupants, The outputs may be the room temperature and heating/cooling load. These inputs and outputs are related by the thermal system components, which may be the structural characteristics of walls, 3.
Principles of the Method
First, a method of obtaining the approximate pulse transfer function from the transfer function in a continuous system will be described. It is assumed that: 1,
The system considered herein is linear
2,
The change of the output is gradual even iV"hen the input is abrupt
These assumptions are not too far off in the real thermal systems, Denoting the input by F(S), the output by H(S) and the transfer function by G(S), it can be shown that H(S) • G(S) • F(S)
687
(1)
With the assumptions given above, this equation can be modified to (2).
H*(S) • Gh(S) = G(S) • F(S)
where H*(S) is the modified output to suit the sampling width. pulse function
If Gh(S) is to represent the triangular
(3)
Equation (2) now may be represented in the Z-domain by H(Z) = Ga(Z) • F(Z)
(4)
Here Ga(Z) is the pulse transfer function and
=
Ga(Z)
1/~(Gh(S)/G(S))
(5)
Thus it can be shown that the product of the linear system input and the transfer function can be obtained by multiplying the sampled input by the pulse transfer function. 4. 4.1
Pulse Transfer Function
Heat Flow, Imaginary Function of a Slab
In a linear heat conduction system, for a slab of constant thermal resistance r, diffusivity a, and thickness t, when subjected to the unit surface temperature excitation at one side and zero at another, the Laplace transform of the heat flux at the excitation side is X=
JYa 1.
cot h
J""""'a!': t/rJ,
(6)
cosec
J§:.a J,/rt
(7)
The heat flux at the zero temperature side is Y
J-y_a
j,
when s is the complex parameter. The solution for the multi-layer wall will be treated in the next section as a function of X andY. 4.2
Surface Temperature Transfer Function For Multi-layer Wall as Temperature Input
Multi-layer wall temperature W0 (S) responding to the unit surface temperature and zero at another will be (figure (2)), (8)
where i is an index for the side of input and o is for the output, and Aw is expressed as follows, (9)
xk +
"k+l
(k
1, 2 ••. , n - 1
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and if we let input i as 01 and output o as 12, Bw is
b~ • b3 ••••• b~-1 2, 3, ... ,
(10)
n-1)
and if we let input i as nn+l and output o as 12, fuv is b~ ..... b~-1
B w
(ll)
b* k
and if i is 01 and o is n-ln, Bw is Bw
bl • b~ • b~ ..... b~-1
(12)
b* k
4,3
Transfer Function of Single Room Temperature as Heat Supply
Further if we make some assumptions as (3)
Heat supplied into the room diffuses instantly into the room air
(4)
Number of air changes in the room is constant
(5)
Heat radiation is considered as a part of the overall heat convection coefficient
The transfer function of the single room temperature for the supply heat Rh(S) is (13)
Rh(S)
where Qa is air capacity of the room, N is number of air changes, Aj is area of the wall of number j, 1/Ri is overall thermal coefficient corresponding to wall j and 01W1 2 1 j(S) is transfer function of wall surface temperature on the side of excitation (outside surface) and 8 Ls complex parameter, In this case, we let x 1 and Y1 as 1/Ri, and Xn and Yn as l/R 0 • Now from eq. (13), room temperature 8r(S) as supply heat Ht(S) will be in linear system, 8r(S) = Rh(S) • Ht(S)
(14)
Ht(S) includes heat exchange iVith outdoor air heat conduction through wall ivhile the room temperature assumed to be 0, If we want to obtain frequency response of room temperature as heat supply, i.w would replace S in eq. (13). Where i is an imaginary number index and w is an angular frequency of heat supply. In this system 8r(S) is output, therefore we can transform room temperature Sr(S) in accordance with assumption (2) and eq, (2) as follows, 8r (S)
8r*(S) • Gh(S)
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(15)
where
er~'<-(S)
Equation (14) is now modified to
is the sampled temperature of the room.
9r*(S) , Gh(S)/G(S) ~ Ht(S)
(16)
Equation (16) is then transformed to Z-domain as follows, 9r(Z) • C(Gh(S)/G(S)) ~ Ht(Z)
In this case Ht(Z) corresponds to input in eq. (4). function of single room for heat supply,
(17)
And if we consider eq. (17) as pulse transfer
(18)
Rh(Z) ~ 1/C(Gh(S)/G(S))
where Rh(Z) corresponds to Ra(Z) in eq. (5), then eq. (17) becomes, 9r(Z)
~
(19)
Rh(Z) , Ht(Z)
Now, from eq, (3), eq, (13) and eq. (18), we can get Rh(Z) in the following, (20)
where 0 Jw 12 ,(Z) is calculated by assumptions (2), (3), (4), and (5) and eqs. (3), (4), (5), (6), (7), (8), (9 ' and (10), as follows
when Aw and Bw are given in eq, (9) and eq. (10), respectively. y(Z) in this case to represent x(Z) ~ (1+(2/t) , B (Z/(Z-1) y(Z)
~
k
~
k
+ Z/(Z-dk))/ck)r1
(22)
(Z/(Z-1) + Z/(Z-dk))/ck)/r 1
(1+(2/t) , B (-1)k
c
However, we denote X as x(Z) andY as
2 2 r?/(1 /a )
2
T: sampled width In these equations, slab number of multi-layer wall are omitted. as x(Z)
(1+(2/T) (B
(1-dk)/ck
+B M~1
k~1
(1-d / 1 K~1
(B
x(Z) and y(Z) may also be expressed m-1
dk
)Z/ck
-m
)r1 (23)
5.
Pulse Transfer Function of Multi-connected Rooms Temperature
Now, if we let the input as heat supply Hi(S), in which i is the room number and let output as room temperature 8i(S), and if the transfer function of this room is Rhi(S) (24)
690
and if we let the transfer function of a multiply-connected rooms as Rhji(S), where j is the room number of input and i is the room number of output, 9i(S) becomes (25)
where Rhji(S) equals Drji/Dr.
Dr and Drji are as foll01vs, Dr
==
0
0 0
Rhji' 0
0
gij
-(A/R1 ) . jWi(S), Rhi(S)
(room of number i adjacent to room of number j) 0 (non-adjacent) 1 (i
~
j)
Therefore, pulse transfer function of this system Rh .. (Z) -.;vill be, JL
(27)
We can obtain the solution of eq. (27) by using assumption (2). 6.
Temperature of an Automatically Air-conditioned Rooms
It is assumed that an office room is automatically air-conditioned during the office hours by a feed back controlled system. Temperature of the rooms during the off office hours will vary naturally in accordance with eq. (14). Now, the heat supplied into the room during the air-conditioned period is Hm(S). Heat supplied during the off-hours is Ho(S). The programmed temperature profile is Op(S), then equations will be, 9r(S) ~ Rh(S) , Ho(S)/(l+Rh(S) Hm(S)
Tf(S)) + Rh(s)-, Tf(S) Tf(S) , (9p(S) - Br(S))
and if we want to express eq. (28) by the Z-expression,
691
9p(S)/(l+Rh(S) , Tf(S))
(28) (29)
Sr(Z) = Rh(Z) , Ho(Z)/(l+Rh(Z) • Tf(Z))
+ Rh(Z)
• Tf(Z) • Sp(Z)/(l+Rh(Z) • Tf(Z))
(30)
If we let Tf(S) as p.(l+I/S), Gh(S) as
when Rh(Z) is shown in eq. (18) and Tf(Z) is 1/(Gh(S)/Tf(S)). eq. (3), then Tf(Z)
2 j-1 -j Z )/(p.I.T) (1-q- ~ (1-q) q
(31)
q = EXP(-I,T)
And the programmed temperature will be expressed in Z-expression as follows, .
Bp(Z)
= E Brjz-J
+; e
.
.
BpjZ-J + E Brjz-J
(32)
then, if 8pj and T are given, 8rj can be solved from eqs, (30), (31) and (32), 7.
References
[1]
H. Yamazaki, Method of Time-varying Load-calculation of Air Conditionied Room by Use of Frequency Response, Extra Summaries of Technical Papers of Annual Meeting of A. I, J,, 1969.
[2]
H. Yamazaki, Frequency Response of Wall and Room Temperature, E. S, T. A. I. J., 1966.
[3]
H. Yamazaki, Approximation of Double-Layer Wall Transfer Function, E. S, T. A. I. J., 1968.
[4]
H. Yamazaki, Approximate Formula on Transfer Function of Single Wall Surface Temperature on the Side of Excitation as Room Temperature Input, E. S, T, A. I. J., 1967.
[5]
H. Yamazaki, Pulse Transfer Function on Analysis of Room Temperature Variation, Technical Papers of Annual Meeting of A. I. J., Kantoh-District, 1969.
[6]
D, G. Stephenson and G, P. Mitalas, Cooling Load Calculation by Thermal Response Method, ASHRAE Trans., Vol, 73, 1967,
[7]
E, I. Jury, Sample,d-Data Control Systems, John Wiley
&
Sons, Inc.
~im;;;;ag=e.:.r._y_-j...,_ X
1 sur face temperature
y 0
imagery
F'ig(l) Heat flaw: imagery function of a slab
1
n n+l inside
n
Fig(2) System oi transfer function
of multi-layer wall
692
outside
