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•
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0 I ! > : % ! #FE$ #FA$ ?
q = −hAΔT
−kA
ΔT
q=
Δx
#FE$
#FA$
h " A ΔT k Δx ! ns & FE ij Th,11 W Th,11 Th,21 Tc,21
! ?
E A F L #2 $
)
% :
% #FF$ dTij (t)
Mc
= ṁ(t)c[Ti−nj−m (t) − Tij (t)] ± AUij (t)ΔTij (t)
dt i
ii
#FF$
iii
?
•
•
•
$ ij t
$ ij Q n = 1 m = 0 n = 0 m = 1 $ ij N ! % & As Us (t)ΔTs (t) #FF$ ! ! SELT ,
#2 $
! ?
•
•
•
•
1 1 1
4 Q I F & FE %
#FF$ M h ch
dTh,ij (t)
= ṁh (t)ch [Th,ij−1 (t) − Th,ij (t)] − Ah Uij (t)F ΔTij (t)
dt
M c cc
dTc,ij (t)
= ṁc (t)cc [Tc,i−1j (t) − Tc,ij (t)] + Ac Uij (t)F ΔTij (t)
dt
#FL$
#FK$
i = 1, ..., n j = 1, ..., s
R
ΔTij (t) ij & SGT
ΔTij (t) F F % 1 F = 1 F SGT
ΔTij (t)
#2 .
E ΔTij (t) = Th,ij (t) − Tc,ij (t)
#FG$
A ΔTij (t) = [Th,ij−1 (t) + Th,ij (t)]/2 − [Tc,i−1j (t) + Tc,ij (t)]/2
#FH$
F ΔTij (t) =
[Th,ij−1 (t) − Tc,ij (t)] − [Th,ij (t) − Tc,i−1j (t)]
ln([[Th,ij−1 (t) − Tc,ij (t)]/[Th,ij (t) − Tc,i−1j (t)]
#FC$
! SELT A F ' ?
F Ah Uij (t)
ṁh (t)ch
F Ac Uij (t)
βij (t) =
ṁc (t)cc
Mh
τh (t) =
ṁh (t)
Mc
τc (t) =
ṁc (t)
αij (t) =
#FJ$
#FEB$
#FEE$
#FEA$
#FJ$ #FEB$ #FEE$ #FEA$ #FH$ #FL$ #FK$ %
?
αij −1
dTh,ij (t)
αij −1
αij
αij
= (1−
)τh Th,ij−1 (t)−(1+
)τh Th,ij (t)+(
)Tc,i−1j (t)+(
)Tc,ij (t)
dt
2
2
2τh
2τh
#FEF$
βij −1
dTc,ij (t)
βij −1
βij
βij
= (1−
)τc Tc,i−1j (t)−(1+
)τc Tc,ij (t)+(
)Th,ij−1 (t)+(
)Th,ij (t)
dt
2
2
2τc
2τc
#FEL$
1%
#FEF$ #FEL$ d
T =
dt
(ṁ, θ)T +
+(ṁ, θ)Tin
#FEK$
6
#2 $
⎡
⎢
⎢
⎢
⎤
⎡
⎢
Th,11
⎢
⎢
⎥
⎢
⎢
⎥
⎢
⎢
⎥
⎢T
⎢
⎢ h,12 ⎥
⎢
⎥
⎢
⎢
⎥
⎢
⎢
⎢ Th,21 ⎥
⎢
⎥
⎢
⎢
⎥
⎢
⎢
⎥
d ⎢
T
⎢
⎢ h,22 ⎥
⎥ =⎢
⎢
⎢
⎥
⎢
dt ⎢ Tc,11 ⎥
⎢
⎢
⎥
⎢
⎢
⎥
⎢
⎢
⎥
⎢T
⎢
⎢ c,12 ⎥
⎢
⎥
⎢
⎢
⎥
⎢
⎢
⎢ Tc,21 ⎥
⎢
⎦
⎣
⎢
⎢
Tc,22
⎢
⎢
⎣
α
−(1+ 11 )
2
0
τh
α
α
(1− 12 ) −(1+ 12 )
2
2
τh
τh
0
0
0
0
β11
2τc
0
β12
2τc
β12
2τc
0
0
0
0
⎢
⎢
⎢
⎢
⎢
+⎢
⎢
⎢
⎢
⎢
⎣
0
α11
2τh
0
0
0
0
0
0
β11
2τc
0
0
β
(1− 11 )
2
τc
0
0
0
0
β21
2τc
0
0
0
?
Tout =
0
0
0
0
α12
2τh
α
(1− 21 )
2
τh
0
⎤
α11
2τh
0
0
⎥
⎥
⎥
⎥
⎥
0
0
0
0
0
⎥
⎥
α21
⎥
)
−(1+
α
α
⎥
2
21
21
⎥
0
0
0
τh
2τh
2τh
⎥
⎥
α22
α22
⎥
) −(1+
)
(1−
α22
α22
⎥
2
2
0
0
⎥
τh
τh
2τh
2τh
⎥
⎥
β
⎥
−(1+ 11 )
⎥
2
0
0
0
0
0
⎥
τc
⎥
⎥
β12
⎥
)
−(1+
⎥
2
0
0
0
0
0
⎥
τc
⎥
β21
β31
⎥
⎥
)
)
(1−
−(1+
β21
2
2
⎥
0
0
0
⎥
2τc
τc
τc
β22
β22 ⎥
⎦
)
)
(1−
−(1+
β22
β22
2
2
0
0
2τc
2τc
τc
τc
⎡ (1− α11 )
2
τh
0
0
0.5 0 0.5
0 0 0
0 0
0 0
0
⎡
⎤
T
⎢ h,11
⎢
⎢
⎢ Th,12
⎢
⎢
⎢T
⎢ h,21
⎢
⎢T
⎢ h,22
⎢
⎢
⎢ Tc,11
⎢
⎢
⎢T
⎢ c,12
⎢
⎢
⎢ Tc,21
⎣
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Tc,22
⎤
⎥
⎥⎡
⎥ T ⎤
0
⎥ h,10
⎥ ⎢ Th,20 ⎥
0
⎥⎣
⎥ Tc,01 ⎦
0
⎥
β
⎥ Tc,02
(1− 12 )
2
⎥
τc
⎦
α12
2τh
#FEG$
0
0
⎡
⎢
⎢
⎢
⎢
⎢
0
0
⎢
0.5 0.5 ⎢
⎢
⎢
⎢
⎣
Th,11
Th,12
Th,21
Th,22
Tc,11
Tc,12
Tc,21
Tc,22
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
#FEH$
! ' M
J ECEC ! SEKT " " #2 ' SEKT 8 " h SEKT
h(t) = C ṁy (t)
#FEC$
SEKT
! " U "
U −1 =
1
1
+
hh
hc
#FEJ$
' #FEC$ "
hh hc " U U (t) =
C (ṁh (t)ṁc (t))y
hh hc
=
hh (t) + hc (t)
(ṁyh (t) + ṁyc (t))
#FAB$
! SEKT M ṁref 8 U α β M Uref #FAE$
" ṁh∗ ṁc∗
Uref =
hh∗ hc∗
C (ṁh∗ ṁc∗ )y
=
hc∗ + hc∗
(ṁyh∗ + ṁyc∗ )
N M #FAE$
#2 $
ṁh∗ Uij (t)
ṁh (t) Uij∗
ṁc∗ Uij (t)
βij (t) = βij∗
ṁc (t) Uij∗
ṁh∗
τh (t) = τh∗
ṁh (t)
ṁc∗
τc (t) = τc∗
ṁc (t)
αij (t) = αij∗
#FAA$
#FAF$
#FAL$
#FAK$
αij∗ , βij∗ , τh∗ , τc∗ %
#FJ$ #FEB$ #FEE$ #FEA$ #FAA$ #FAF$ #FAL$ #FAK$ #FAB$ #FAE$ y
Q " U ! U % " >
#FAG$ SGT
U=
1
Ac h c
1
+
hi =
1
Ah h h
/W H
#FAG$
N ui ki
Dhi
1
N ui = 0.023Reyi P ri3
ṁi Dhi
Rei =
ρHdi νi
Q i = h, c 8 U α β ' #FJ$ #FEB$ U α β Q U #2 #
α β I8
8 FK
!
" #FEK$ wθ (t) wθ (t)∈N (0, Qθ (t)) M d
T =
dt
(ṁ, T , θ)T +
+(ṁ, T , θ)Tin + wf (t) = f (ṁ, T , θ, Tin) + wf (t)
8 ' T (t + Δt) = Φ(ṁ, Δt)T (t) + Γ(ṁ, Δt)T in (t) + w(t + Δt)
(ṁ)
Φ(ṁ, Δt) = e
=
∞
i=1
Γ(ṁ, Δt) = [
0
Δt
e
(ṁ)s
#FAH$
(ṁ)i
i!
#FAC$
+
#FAJ$
ds] (ṁ)
w(t + Δt) M Q(t)
N
8 FL θ %
SAAT
θ M
#FFB$ %
! #FFB$ t1 %
#2 $
M t1 tN V (θ) =
N
#FFB$
ε(t)T ε(t)
t=t1
! L & FE #FJ$ #FEB$
#FEE$ #FEA$ #FAG$ ! ! y = 0.8 SGT
#
! SEGT ' α∗ β ∗ U %
#FEK$ α∗ β ∗ d
dt
α∗ (t)
β ∗ (t)
=
#FFE$
d
[θ] = w θ (t)
dt
wθ (t)∈N (0, Qθ (t)) M N #FEK$ #FFE$ d
dt
θ
T
=
0
f (ṁ, T , θ, Tin )
+
wθ
wf
#FFA$
6 wf = N (0, Qf ) #FEK$ #2 8 1
M T̂ 8 FL % %
d
T̂ (t) = f (x̂(t))
dt
*
d
(t) = (x̂(t)) (t) +
dt
#FFF$
(t)*T (x̂(t), t) + 7(t)
#FFL$
∂
F (x̂(t)) =
0
f (ṁ, T , θ, Tin )
∂x
#FFK$
|(x(t) = x̂(t))
F %
%
#FFG$
⎡
⎡
⎢
⎢
⎢
⎢
⎢
⎢
d ⎢
⎢
dt ⎢
⎢
⎢
⎢
⎢
⎢
⎣
α∗ (t)
β ∗ (t)
Th,11
Th,12
Th,21
Th,22
Tc,11
Tc,12
Tc,21
Tc,22
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
0
⎢
⎢
⎢
0
⎢
⎢
α11
α11
⎢
(1+ 2
)
(1− 2
)
α11
11
⎢ −
T
+
T
+
Th,10 + α
T
c,11
h,11
τ
2τ
τ
2τh c,01
⎢
h
h
h
⎢
α12
α12
(1+ 2 )
⎢ (1− 2 )
α12
α12
Th,11 −
Th,12 + 2τh Tc,12 + 2τh Tc,02
⎢
τh
τh
⎢
α21
α21
⎢
⎢ − (1+ 2 ) Th,21 + α21 Tc,11 + α21 Tc,21 + (1− 2 ) Th,20
⎢
τh
2τh
2τh
τh
=⎢
α22
(1+ 2
)
⎢ (1− α222 )
22
22
⎢
Th,21 −
Th,22 + α
T
+α
T
τh
τh
2τh c,12
2τh c,22
⎢
β11
β11
⎢ β
(1+
)
(1−
)
⎢
11
2
2
Tc,11 + β2τ11c Th,10 +
Tc,01
⎢ 2τc Th,11 −
τc
τc
⎢
β12
β12
⎢ β12
(1+
)
(1−
)
2
2
⎢ 2τ Th,11 + β2τ12 Th,12 −
Tc,12 +
Tc,02
τc
τc
c
c
⎢
⎢
β21
β21
(1−
)
(1+
)
β
β
⎢
2
2
Tc,11 −
Tc,21 + 2τ21c Th,20
⎢ 2τ21c Th,21 +
τc
τc
⎣
β22
β22
(1− 2 )
(1+ 2
)
β22
T
+ β2τ22c Th,22 +
Tc,12 −
Tc,22
2τc h,21
τc
τc
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥ ⎡
⎤
⎥
⎥
w
⎥ ⎢ θ ⎥
⎥+⎣
⎦
⎥
⎥
wf
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(
#2 $
⎡
0
0
f1
f2
f3
f4
f5
f6
f7
f8
⎢
⎢
⎢
⎢
⎢
⎢
=⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥ ⎥
⎥ + wθ
⎥
wf
⎥
⎥
⎥
⎥
⎦
#FFG$
& ?
•
•
N α
N β ! F EB EB ⎤
⎡
0
0
⎢
⎢
⎢ 0 0
⎢
⎢
⎢ ∂f1 ∂f1
⎢ dα dβ
⎢
⎢ ∂f2 ∂f2
⎢ dα dβ
⎢
⎢ ∂f3 ∂f3
⎢ dα dβ
F =⎢
⎢ ∂f4 ∂f4
⎢ dα dβ
⎢
⎢ ∂f ∂f
⎢ 5 5
⎢ dα dβ
⎢
⎢ ∂f6 ∂f6
⎢ dα dβ
⎢
⎢ ∂f7 ∂f7
⎢ dα dβ
⎣
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∂f1
∂f1
∂f1
∂f1
∂f1
∂f1
∂f1
dTh,11 dTh,12 dTh,21 dTh,22 dTc,11 dTc,12 dTc,21
∂f2
∂f2
∂f2
∂f2
∂f2
∂f2
∂f2
dTh,11 dTh,12 dTh,21 dTh,22 dTc,11 dTc,12 dTc,21
∂f3
∂f3
∂f3
∂f3
∂f3
∂f3
∂f3
dTh,11 dTh,12 dTh,21 dTh,22 dTc,11 dTc,12 dTc,21
∂f4
∂f4
∂f4
∂f4
∂f4
∂f4
∂f4
dTh,11 dTh,12 dTh,21 dTh,22 dTc,11 dTc,12 dTc,21
∂f5
∂f5
∂f5
∂f5
∂f5
∂f5
∂f5
dTh,11 dTh,12 dTh,21 dTh,22 dTc,11 dTc,12 dTc,21
∂f6
∂f6
∂f6
∂f6
∂f6
∂f6
∂f6
dTh,11 dTh,12 dTh,21 dTh,22 dTc,11 dTc,12 dTc,21
∂f7
∂f7
∂f7
∂f7
∂f7
∂f7
∂f7
dTh,11 dTh,12 dTh,21 dTh,22 dTc,11 dTc,12 dTc,21
0
⎥
⎥
0 ⎥
⎥
⎥
∂f1 ⎥
dTc,22 ⎥
⎥
∂f2 ⎥
dTc,22 ⎥
⎥
∂f3 ⎥
⎥
dTc,22 ⎥
⎥
∂f4 ⎥
dTc,22 ⎥
⎥
∂f5 ⎥
dTc,22 ⎥
⎥
∂f6 ⎥
dTc,22 ⎥
⎥
∂f7 ⎥
dTc,22 ⎥
⎦
#FFH$
∂f8 ∂f8
∂f8
∂f8
∂f8
∂f8
∂f8
∂f8
∂f8
∂f8
dα dβ dTh,11 dTh,12 dTh,21 dTh,22 dTc,11 dTc,12 dTc,21 dTc,22
! #2# )
% & N 6 8(
0 ?
•
8 •
I
•
& Q̇ = U AΔTLMT D = ṁk cp ΔTk
#FFC$
k & F ΔTLMT D F SGT
' #FFC$ ṁΔT ∝ U AF ΔTLMT D AU F M N ṁΔT ΔTLMT D N &
FA ' ' N
'
M ,
#2 $
The slope of the line is equal to UAF
90
80
Equation of the line:
y = 5.08*x − 366
mΔcpTh
UAF = 5.08
70
60
50
40
83
84
85
86
87
88
ΔTLMTD
ṁcp ΔT ΔTLM T D ! " #
SL EJ FAT SAET
EJGB >1 0 %
? SFAT D
0 M % ! #FFJ$ #FLF$ #FLL$ #FLC$ #2% 8 /
.
! α β !
$ %
%
?
E 8 A <0 ?
x̂−
k = Ax̂k−1 + Buk−1
#FFJ$
F <0 Pk− = APk−1 AT + Q
#FLB$
Kk = Pk− H T (HPk− H T + R)−1
#FLE$
L I K : zk
−
x̂k = x̂−
k + Kk (zk − H x̂k )
#FLA$
G : Pk = (I − Kk H)Pk−
#FLF$
H > A G FF
!
$ %& $'
! 1& 1& M
1& %
?
#6
#2 $
E 8 A <0 ?
x̂−
k = f (x̂k−1 , uk−1 , 0)
#FLL$
F <0 Pk− = Ak Pk−1 ATk + Wk Qk−1 WkT
#FLK$
L I Kk = Pk− HkT (Hk Pk− HkT + Vk Rk VkT )−1
#FLG$
K : zk
−
x̂k = x̂−
k + Kk (zk − h(x̂k , 0))
#FLH$
G : Pk = (I − Kk Hk )Pk−
#FLC$
H > A G FL
$
& #2 #
$ % N SAGT I #I8$ " I8 N U 8 U ! #FAG$ U 8 U 9 " % ! I8 Q M I8 ! ! M ! I8 % ! $
& #
#2 $
I8 #FLJ$ #FKB$ SAGT
SH (i) = max[0, x̄i − (μ0 + K) + SH (i − 1)]
SL (i) = max[0, (μ0 − K) − x̄i + SL (i − 1)]
#FLJ$
#FKB$
Q SH (0) = 0 SL (0) = 0 SH (i) SL(i) K %
M I K M E A ! sH (i) sL (i) H H K !
8 " 6 <
8 = N :
5
! !
& ?
•
•
' ' ! ! FF
#%
%2 &
! x y W H dh dc %
Tc Th %
ṁc ∂
∂Tc
+
(cTc ) =
∂t
ρcHdc ∂x
ṁh ∂
∂Th
+
(cTh ) =
∂t
ρcHdh ∂y
U
(Th − Tc )
ρcdc
U
(Tc − Th )
ρcdh
#LE$
#LA$
! ρ c U Th Tc x y N
ṁc y ṁh x
" ! FB & < &
6 " Rf & M SAT SGT 0 ?
•
•
•
•
I
; = %2 * •
#
<
N SA CT SAJT " >
! SFT
& & N >
U Rf Rf (t) =
! 1
1
−
U (t)A U (0)A
#LF$
Q " U ABU KBU U !
& LE UU
! & LE 8 LA
f ouled
clean
U #LL$
Ufouling = Uclean cos(at + b)
#LL$
#(
%2 &
Changes in the overall heat transfer
coefficient, U, because of fouling
Continuously fouling heat exchanger
20% decrease in U
Ufouled/Uclean
1
0.9
0.8
0
0.2
0.4
0.6
0.8
1
Dimensionless time
Changes in the overall heat transfer
coefficient, U, because of fouling
Ufouled/Uclean
1
Continuously fouling heat exchanger
50% decrease in U
0.8
0.6
0.4
0
0.2
0.4
0.6
0.8
1
Dimensionless time
$'
' ( ) arccos(cf 1 )
m − floor(mcf2 ) + 1
b = −a(floor(mcf2 ) + 1)
a=
#LK$
cf 1 Ufouling Uclean cf 2 cf 1 cf 2 [0, 1]
%2 * #)
Evolution of the resistance
to heat transfer because of fouling
4
Continuously fouling heat exchanger
20% decrease in U
3
2
Rf=0.0001
1
0
0.644
Fouling factor, Rf [m2K/(W)]
−4
x 10
0
0.2
0.4
0.6
0.8
1
Dimensionless time
Evolution of the resistance
to heat transfer because of fouling
1.5
x 10
Continuously fouling heat exchanger
50% decrease in U
1
0.5
Rf=0.0001
0
0
0.2
0.4
0.876
Rf=0.0007
0.395
Fouling factor, Rf [m2K/(W)]
−3
0.6
0.8
1
Dimensionless time
$'
Rf ! & LA N SEHT SBBBBE BBBBHT SAHT & ABU SBGLK ET KBU SBFJK BCHGT Q #,
%2 &
! ' BBE Q E BK BBBA I F 34 4 '
! AKU !
( " ! 34 4 ABU
U U & LE "
! & LF ! ch = cc =
4200 [J/Kg◦ C] ρ = 998 [kg/m3 ] & 34 4 E R
" E R
U = 3.780 [W/m2 K] #FAG$ SGT
%2# 9 #.
Example of a typical data set
°C
100
thin
tcin
thout
tcout
50
0
0
2000
4000
6000
8000
10000
Sample no.
2
l/s
mh
mc
1
0
0
2000
4000
6000
8000
10000
Sample no.
Closer look on the data set above
°C
100
thin
tcin
thout
tcout
50
0
1000
1100
1200
1300
1400
1500
Sample no.
1.5
l/s
mh
mc
1
0.5
1000
1100
1200
1300
Sample no.
1400
1500
%6
%2 &
8 " N
8 FA L L & FE %
?
α=
Ah,ij U
(W H/4)U
=
ṁh ch
(ṁh /2)ch
β=
Ac,ij U
(W H/4)U
=
ṁc cc
(ṁc /2)cc
τh =
Mh,ij
W Hdρ/4
=
ṁh
(ṁh /2)
τc =
Mc,ij
W Hdρ/4
=
ṁc
(ṁc /2)
LE ' I θ
α
β
τh
τc
y
!
E[θ]
BEEAK
BEEAK
BALJK
BALJK
BC
( 8 34 4 U KBU U
& LE & LL %2# 9 %
Example of a typical data set
150
thin
tcin
thout
tcout
°C
100
50
0
0
2
4
6
8
Sample no.
10
4
x 10
6
mh
mc
l/s
4
2
0
0
2
4
6
8
Sample no.
10
4
x 10
Closer look on the data set above
150
thin
tcin
thout
tcout
°C
100
50
0
1000
1100
1200
1300
1400
1500
Sample no.
6
mh
mc
l/s
4
2
0
1000
1100
1200
1300
1400
Sample no.
" 1500
" #
! 34 ! 8 9 Thin Tcin ṁh ṁc Thout Tcout N ?
•
N α
•
N β •
% LF
%%
2 '
$
() M M ! ?
•
•
•
•
< 8
8
!
α β ! #
8
FL M % ! 6
&
X
SJT '
6
#KE$ ' JKU ˆ
2:((θ))
ˆ =
H (−1)
[(θ)]
N − (t1 + p)
#KE$
KE 2 34 %
KA & ! KE
KA JKU τ M " & $
*+, ' - 1
θ
α
β
τh
τc
y
σ(θ)
2; .)2;
BBJLL BBBBL BBJFH BBJKE
BEBFH BBBBL BEBAJ BEBLK
BBBG BAJ BKG BKH
BBE
EBC
AE
AE
BCE BBEL BHC
BCL
$
*+, ' - 1
θ
α
β
τh
τc
y
σ(θ)
2;
BEEB BBBBF BEBJL
BEEL BBBBF BEEAJ
BBA BEK
BF
BBL BCK
EG
BCE BBE
BHJ
.)2;
BEEBG
BEELA
BF
EH
BCF
& KE ! ! M &
KA KF & KF & LF M x(t + Δ) Δ x(t)
%(
2 '
100
Temperature [°C]
th,out
tc,out
th,out,pred
tc,out,pred
80
60
40
20
0
50
100
150
200
250
Observation no.
#
" )
! SB AT M α β y LE "
τh τc τh τc BBBE BBK
8
KF
! KF α β % ! #
(" ! M KL α β y 2 34 %)
Residuals of the predicted hot output
Error
1
0
−1
50
100
150
200
250
300
350
300
350
No.
Residuals of the predicted cold output
Error
0.5
0
−0.5
50
100
150
200
250
No.
. '
Autocorrelation of the residuals for the hot output
No.
1
0
−1
0
5
10
15
20
Correlation
Autocorrelation of the residuals for the cold output
No.
1
0
−1
0
5
10
15
Correlation
20
%,
2 '
' /00 - '
8
EBB θ
E[θ]
σ[θ]
[2.5%, 97.5%] α
BBCCJGAH 0.0005 SBBCCJGAA BBCCJGFJT
β
BBJKCHLF 0.0005 SBBJKCHFC BBJKCHKFT
τh BBELB
0.008
SBBBE B BAAJT
τc BBEHK
0.01
SBBBE BBACLT
y
BCAAC
0.01
SBCAAC BCAACT
"
τh τc $
- " E
A
F
L
K
G
H
C
J
EB
#!
α
BBJKB
BBJLJ
BBJKE
BBJLC
BBJLH
BBJLG
BBJKE
BBJKB
BBJKG
BBJKH
β
BEBLE
BEBLE
BEBLE
BEBLE
BEBLA
BEBLL
BEBLB
BEBLA
BEBLA
BEBLB
τh
BBBEB
BBLJL
BBFGF
BBKHA
BBAGC
BBAKE
BBALC
BBAFL
BBJJC
BFFGE
τc
BBFAK
BBGAH
BBALL
BBEJF
BBAHH
BBAJL
BBAEK
BBFFG
BAFEH
BLFFG
$
BCBHE
BCBLE
BCBGJ
BHJGA
BHJJK
BCBLF
BHJJH
BCBAJ
BCAAE
BCABG
* α β 8 α β α β α β ! #FEG$ α β ij ! 8 KE "
τ LF y BC SGT
2 34 %.
! α1 = α11 = α21 α2 = α12 = α22 β1 = β11 = β12 β2 = β21 = β22 Q ij & FE
α β N α β "
α β ##
( " Q 34 M ! M 200 = %
! 20 0
! 0 N % 0
N
8 FA α
β Q I8 8
FK K FKB FLJ I8 FB N
& & KL ! 6
2 '
AKU ! & KK α 0.77 β 0.40 & 5 ! &
KG KH &
KL KK 8 34 #+
), , ! 8 FA 20 34 ; 20 I8 1 I8 N JKU 0.35 0.84 SBBBBBF BBBBECT & LA
#•
•
•
" ! 34 & 34 SBFK BCLT SBBBBBF BBBBECT ! 8 LA N
SBBBBE BBBBHT
34 "
τ ! N N 2 34 Off−line estimation of the α values
Value
0.1
0.09
0.08
0.07
0
200
400
600
800
Estimation no.
Off−line estimation of the β values
0.11
Value
0.1
0.09
0.08
0.07
0
200
400
600
800
Estimation no.
$'
CuSum chart for the off−line method
2.5
936 CuSum values
Sampling time dt = 1
1.5
1
0.5
Shift detected in β:0.4
CuSum values
2
0
−0.5
Shift detected in α: 0.77
Fouled heat exchanger
β
0
0.2
0.4
α
0.6
0.8
1
Dimensionless time
" - 2 '
Off−line estimation of the α values
Value
0.11
0.1
0.09
0.08
0
200
400
600
800
Estimation no.
Off−line estimation of the β values
Value
0.12
0.1
0.08
0
200
400
600
800
Estimation no.
$'
CuSum chart for the off−line method
3
Fouled heat exchanger
2
Sampling time dt = 1
1.5
1
0.5
0
−0.5
Shift detected in α: 0.71
936 CuSum values
Shift detected in β:0.4
CuSum values
2.5
β
0
0.2
0.4
α
0.6
0.8
1
Dimensionless time
" - 2 4 #
M !
0
$ () ! α β 1
#1&$ α β N
8 FA α β I8 8 FK
! ?
•
•
•
•
•
8 M 8
8
8
!
α β ! # .
Q 1& M Q ' ! ; ! & KC
c Q ! Q EBEB Q(1, 1) = Q(2, 2) = cq α β ! c c = 10−9 AFU ! c = 10−9 α
0.14
c = 10−5
0.12
0.1
0.08
0
500
1000
1500
Estimated parameters
2 '
Estimated parameters
%
0.2
β
0.18
0.16
0
α
c = 10−7
0.12
0.11
0.1
0
500
1000
1500
β
0.17
c = 10−9
0.117
0.116
0.115
0
500
1000
0.15
0
1500
500
1000
sample number
1500
Estimated parameters
Estimated parameters
c = 10−10
0.117
0
500
1000
1500
β
c = 10−9
0.163
0.162
0.161
0
500
1000
1500
sample number
0.118
0.116
c = 10−7
0.164
sample number
α
1500
sample number
Estimated parameters
Estimated parameters
0.118
1000
0.16
sample number
α
500
sample number
Estimated parameters
Estimated parameters
sample number
0.13
c = 10−5
0.164
β
c = 10−10
0.163
0.162
0.161
0
500
1000
sample number
- - ' 1
1500
2 4 Sampling time of 1 s
Standard deviation
Estimated parameters
Sampling time of 1 s
β
0.2
0.1
α
0
100
0.6
0.2
0
200
β
0.4
α
0
10
β
0.1
α
0
100
sample number
Standard deviation
Estimated parameters
0.2
20
30
sample number
Sampling time of 10 s
sample number
Sampling time of 10 s
200
0.4
β
0.2
0
α
0
10
20
30
sample number
2 α β / /0 '
'
# " ) N
34 EBB B A α β ! & KJ % EBB ! & KEB ! 1
EU 100 & 10
EU 70 & &
KJ KEB %
(
2 '
Sampling time of 1 s
Sampling time of 10 s
0
0
10
10
−1
−1
10
−2
10
Standard deviation
Standard deviation
10
β
−3
10
α
−4
10
−2
10
α
−3
10
−4
10
β
−5
−5
10
10
−6
10
−6
0
100
200
sample number
300
10
0
100
200
300
sample number
'
'
α β / /0 N ABB
% ! 100 # ( N
! 4 30 α β I8 8 FK
2 4 )
Fouled heat exchanger
0.5
3750 CuSum values
0.4
Sampling time dt = 1
0.3
0.2
0.1
Shift detected in α: 0.83
0.6
Shift detected in β:0.81
CuSum values
CuSum chart for the on line method
0
α β
−0.1
0
0.2
0.4
0.6
0.8
1
Dimensionless time
" & ! & KEE β 0.81 0.83 α ! 1s
& 5 ! &
KEA KEE ! 4 # !
" &
KEF KEL 4 ! ! ,
2 '
CuSum chart for the on line method
3750 CuSum values
0.6
CuSum values
Shift detected in β:0.81
Fouled heat exchanger
Sampling time dt = 1
0.4
0.2
Shift detected in α: 0.83
0.8
0
α β
0
0.2
0.4
0.6
0.8
1
Dimensionless time
" !
! 10
FHK I8 FHKB I8 1
I8 O
# #
), , N 4 KB JKU 0.65 0.91 SBBBBE BBBBAAT & LA
2 4 .
Shift detected in α: 0.84
CuSum values
0.3
Fouled heat exchanger
1875 CuSum values
Sampling time dt = 2
0.2
0.1
Shift detected in β:0.91
CuSum chart for the on line method
0.4
0
−0.1
α
0
0.2
0.4
0.6
β
0.8
1
Dimensionless time
Shift detected in α: 0.86
CuSum values
0.1
Fouled heat exchanger
938 CuSum values
Sampling time dt = 4
0.05
Shift detected in β:0.93
CuSum chart for the on line method
0.15
0
α
0
0.2
0.4
0.6
0.8
β
1
Dimensionless time
" dt = 2 dt = 4
# +
* α β N
34 4 α 34 β α1 = α11 = α21 α2 = α12 = α22 β1 = β11 = β12 β2 = β21 = β22 ij & FE
"
α β 9 "
α β (6
2 '
CuSum chart for the on line method
CuSum values
Shift detected in β:0.71
Fouled heat exchanger
626 CuSum values
Sampling time dt = 6
0.15
0.1
Shift detected in α: 0.89
0.2
0.05
0
β
0
0.2
0.4
0.6
α
0.8
1
0.8
1
Dimensionless time
CuSum chart for the on line method
Fouled heat exchanger
CuSum values
0.25
375 CuSum values
0.2
Sampling time dt = 10
0.15
0.1
0.05
Shift detected in α: 0.64
0.3
0
α
0
0.2
0.4
0.6
Dimensionless time
" dt = 6 dt = 10
# •
& 4 & 4
SBGK BJET SBBBBE
2# •
•
(
BBBBAET I
SBBBBE BBBBHT ! N N M & 0
"
α β $ 8 FF ! 9 34 4 N
8 LF AKU & " U KBU U & LE ! &
KEK ! & KEG AU F AU F ! I8 8 FK ΔU AF = U AF − U AF
& KEH I8 I8 0.42 ! 8 LA (
2 '
5
4
x 10
Equation of the line:
y = 1.96e+004*x − 1.4e+006
UAF = 1.96e+4
Reference slope
p
mc ΔT
h
3.5
The slope, UAF, calculated with window size=400
3
2.5
2
82
83
84
85
ΔTLMTD
86
87
88
$
Rf = 0.00016 SBBBBE BBBBHT
#
), , ! 8 FA 50 ; 50 4 JKU 0.33 0.92 SBBBBBH BBBBCT & LA
! ! ! 8 2# 5
4
x 10
(#
The slope, UAF, calculated with window size=400
Reference slope
mcpΔTh
Fouled slope
3
2
1
82
84
86
88
ΔT
90
92
94
LMTD
5
4
x 10
The slope, UAF, calculated with window size=400
Reference slope
mcpΔTh
3.5
Clean
Gradually fouling
3
2.5
2
Fouled slope
0.2
0.4
0.6
0.8
1
ΔTLMTD
3 4 AU F 3 4 ' '
' ! %
#
SBBBBBH BBBBCT "
SBBBBE BBBBHT
(%
2 '
CuSum chart for the slope method
Fouled heat exchanger
1471 CuSum values
−1000
−2000
−3000
−4000
0
0.2
Shift detected in ΔUA: 0.42
CuSum values
0
0.4
0.6
Dimensionless time
"
0.8
1
$ !
N
8 AF %
SBBBBE BBBBHT
! 34 4 " 8 " ! Q ! !
! B ?
•
34 < JKU SBFK BCLT SBBBBBF
BBBBECT
GK
((
(2 &
< α β y % <
<
<
•
τh τc α β 4 < JKU SBGK BJET <
<
SBBBBE
BBBBAET
8" EKB α β <
<
• " < <
JKU SBFF BJAT SBBBBBH
BBBBCT
! 34 4
34 4 ! ! 34 4 !
8 ! SEGT Rf ≈ 0.00008 ! % % & & ?
•
•
•
•
:
!
1 α β ! GH
'
SET N = N' ?RR ? 6 SAT ' '
Y9 I & @ ABBK ?RR
RRABBKR
SFT Q '
I
M AK @ ABBK
?RR
RRABBKR
SLT > D ' < ZI 6 @ Q F EJJH
SKT I
>
> '
?
@ ABBK
?RR
RRABBKR
SGT Z
N I > 6 D 6 A ABBK
SHT Y 9 I 9 Y Z = 8 Y P = O [M ;
R \% GE ABBL
SCT & & N< Q
O 1
I @ ABBK
?RR
RRABBKR
SJT N D @ ' I 6 8 8 9 ' > I 6RI>I A ABBF
SEBT N8 62 =& N 8
' <@'& & I " ? EJJH
GJ
)6
++-0' =
SEET < @
;' 8 I AG @ EJJB
SEAT 6 @
! " <9 9 !
ABBG
SEFT D @
;< <
N # ! $ EEG
@ EJJL
SELT D @
;< <
80 # !
$ EEL 9 EJJA
SEKT D @
; < <
:
% % $% EJJE
SEGT D > @
8 = ; < <
' 9
:
" ! ABBG
SEHT 8 = ;< <
D> @
' 9
I
" 5!! ABBH
SECT 8 = 8 = ' 9
$
" ! FA ABBK
SEJT = @ =
?
O]
?RR
R RR=EJJHR=EJJH&^Q
SABT ' !
!
= & %
?RR
R&!61
SAET 6 ! 9 : A ABBG
SAAT 9
I D I > &'
' @ Q 8
! F ABBA
SAFT N D; 8Z N6 1 EJJJ
SALT N O0 I @
P _ 6 `8 ! EJJJ
SAKT < O N 9 N ABBK
++-0' =
)
SAGT ABBC O!8 R81N 1I6 6 8
?RR
RCJCRR
SAHT 1 < ?RRRR
SACT 88 > 8 Q 6 Y9 I ' '
N @ ABBK
?RR
RRABBKR
SAJT D >MM 6 `8 1 > ! ? ! @
ABBK ?RR
RRABBKR
SFBT 8 8 'M O 8 #$ $( ! & ABBH
SFET > 8 6
`8 M <
M " EL @ EJJF
SFAT D Q D '
N @ AL ABBG
SFFT Q ?RRR
SFLT < Q
:
%
% ( )% EJJG
( ! I K ! *
I K 8
LA ! I K & NE * () ! I K 8 KE LB 0
AB HF
)%
2 - Example of a typical data set
°C
100
thin
tcin
thout
tcout
50
0
0
0.5
1
1.5
Sample no.
2
4
x 10
2
l/s
mh
mc
1
0
0
0.5
1
1.5
Sample no.
2
4
x 10
Closer look on the data set above
thin
tcin
thout
tcout
50
0
1000
1100
1200
1300
1400
1500
Sample no.
2
mh
mc
l/s
°C
100
1
0
1000
1100
1200
1300
Sample no.
1400
1500
2# 4 )
$
*+, ' - 1
θ
α
β
τh
τc
$
/ BEBJK
BEEGH
BBAJF
BBAAH
BCBCL
V ar(θ)
2; .)2;
BBBBF BBHFF BELKC
8
BBBBL BBHGJ BEKGK
8
BHHJL EHFGL EHJKB O EAKFB AAEGE AAGEK O BBEFA BKHCC EBFCB
8
NE Q KA ! τ / ( " & NA ! AKU N
8 FA α β
! & NF α BLL β BFL * () ! I K )(
2 - Off line estimation of the α values
Value
0.12
0.1
0.08
0
200
400
600
800
1000
800
1000
Estimation no.
Off line estimation of the β values
Value
0.12
0.1
0.08
0.06
0
200
400
600
Estimation no.
$'
/
( ! & NL β BGG BHG α 2# 4 ))
CuSum chart for the off−line method
3.5
3
Sampling time dt = 1
2
1.5
1
0.5
0
−0.5
Shift detected in β:0.52
934 CuSum values
2.5
CuSum values
Shift detected in α: 0.45
Fouled heat exchanger
α
0
0.2
0.4
β
0.6
0.8
1
Dimensionless time
" - CuSum chart for the on line method
3
2
Sampling time dt = 1
1.5
1
0.5
Shift detected in β:0.64
CuSum values
17500 CuSum values
Shift detected in α: 0.77
Fouled heat exchanger
2.5
0
−0.5
β
0
0.2
0.4
0.6
α
0.8
Dimensionless time
" 1
' &
F CB #FFH$ ' % #FFG$ M
& f1 ?
1
∂f1
=
(−Th,10 − Th,11 + Tc,01 + Tc,11 )
∂α
2τh
∂f1
1 + α/2
=−
∂Th,11
τh
∂f1
α
=
∂Tc,11
2τh
∂f1
∂f1
∂f1
∂f1
∂f1
∂f1
∂f1
=
=
=
=
=
=
=0
∂β
∂Th,12
∂Th,21
∂Th,22
∂Tc,12
∂Tc,21
∂Tc,22
& f2 ?
1
∂f2
=
(−Th,11 − Th,12 + Tc,02 + Tc,12 )
∂α
2τh
∂f2
1 − α/2
=
∂Th,11
τh
∂f2
1 + α/2
=−
∂Th,12
τh
∂f2
α
=
∂Tc,12
2τh
∂f2
∂f2
∂f2
∂f2
∂f2
∂f2
=
=
=
=
=
=0
∂β
∂Th,21
∂Th,22
∂Tc,11
∂Tc,21
∂Tc,22
HJ
,6
+2 *
& f3 ?
1
∂f3
(−Th,20 − Th,21 + Tc,11 + Tc,21 )
=
∂α
2τh
1 + α/2
∂f3
=−
∂Th,21
τh
∂f3
α
=
∂Tc,11
2τh
∂f3
α
=
∂Tc,21
2τh
∂f3
∂f3
∂f3
∂f3
∂f3
∂f3
=
=
=
=
=
=0
∂β
∂Th,11
∂Th,12
∂Th,22
∂Tc,12
∂Tc,22
& f4 ?
1
∂f4
=
(−Th,21 − Th,22 + Tc,12 + Tc,22 )
∂α
2τh
1 − α/2
∂f4
=
∂Th,21
τh
∂f4
1 + α/2
=−
∂Th,22
τh
∂f4
α
=
∂Tc,12
2τh
∂f4
α
=
∂Tc,22
2τh
∂f4
∂f4
∂f4
∂f4
∂f4
=
=
=
=
=0
∂β
∂Th,11
∂Th,12
∂Tc,11
∂Tc,21
& f5 ?
1
∂f5
=
(Th,10 + Th,11 − Tc,01 − Tc,11 )
∂β
2τc
∂f5
β
=
∂Th,11
2τc
∂f5
1 + β/2
=−
∂Tc,11
τc
∂f5
∂f5
∂f5
∂f5
∂f5
∂f5
∂f5
=
=
=
=
=
=
=0
∂α
∂Th,12
∂Th,21
∂Th,22
∂Tc,12
∂Tc,21
∂Tc,22
,
& f6 ?
∂f6
1
=
(Th,11 + Th,12 − Tc,02 − Tc,12 )
∂β
2τc
∂f6
β
=
∂Th,11
2τc
∂f6
β
=
∂Th,12
2τc
∂f6
1 + β/2
=−
∂Tc,12
τc
∂f6
∂f6
∂f6
∂f6
∂f6
∂f6
=
=
=
=
=
=0
∂α
∂Th,21
∂Th,22
∂Tc,11
∂Tc,21
∂Tc,22
& f7 ?
1
∂f7
=
(Th,20 + Th,21 − Tc,11 − Tc,21 )
∂β
2τc
∂f7
β
=
∂Th,21
2τc
∂f7
1 − β/2
=
∂Tc,11
τc
∂f7
1 + β/2
=−
∂Tc,21
τc
∂f7
∂f7
∂f7
∂f7
∂f7
∂f7
=
=
=
=
=0
=
∂α
∂Th,11
∂Th,12
∂Th,22
∂Tc,12
∂Tc,22
& f8 ?
1
∂f8
=
(Th,21 + Th,22 − Tc,12 − Tc,22 )
∂β
2τc
β
∂f8
=
∂Th,21
2τc
∂f8
β
=
∂Th,22
2τc
∂f8
1 − β/2
=
∂Tc,12
τc
∂f8
1 + β/2
=−
∂Tc,22
τc
∂f2
∂f2
∂f2
∂f2
∂f2
=
=
=
=
=0
∂α
∂Th,11
∂Th,12
∂Tc,11
∂Tc,21
(
&
AE < H
AA
< H
AF I C
AL
I C
AK I
J
AG I
J
AH 8 Q EE
AC < 9 6 1
EA
FE I
L EH
FA
ṁcp ΔT ΔTLMT D AC
FF 1
FB
FL 1
FE
CF
,%
-" * *09'!"
LE 1 " : FG
LA 1 Rf FH
LF 8 FJ
LL 9 LE
KE < LG
KA >
LH
KF N LH
KL 1 KE
KK 9 KE
KG 1 KA
KH 9 KA
KC
P KL
KJ 5
α β E EB KK
KEB α β E EB KG
KEE 9 4 KH
KEA 9 4 KC
KEF 9 dt = 2 dt = 4 KJ
KEL 9 dt = 6 dt = 10 GB
KEK 1
GA
-" * *09'!"
,
KEG & $ AU F & $ GF
KEH 9 GL
NE 8 HL
NA 1 HG
NF 9 HH
NL 9 HH
(
LE I LB
KE 1
JKU LK
KA 1
JKU LK
KF 8
EBB LC
KL 1
LC
NE 1
JKU HK
CH