Kuppan T. Heat Exchanger Design Handbook
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324
Chapter
6
Thermal Effectiveness
The overall heat-transfer performance of the regenerator is most conveniently expressed as the
heat-transfer “effectiveness”
E,
which compares the actual heat-transfer rate to the thermody-
namically limited maximum possible heat-transfer rate. With this definition, the regenerator
thermal effectiveness is simply the dimensionless time-average cold fluid outlet temperature.
This is true under the condition that the cold stream is weaker, that is, has the lower heat
capacity rate. This definition (for
C,
I
eh
as is
the case for the recuperator) results in
E=----
rc,,
-
tC.1
(16)
2h.i
-
1c.1
where
tc.o
is the bulk average temperature of
the cold air stream after passage through the
regenerator. The effectiveness is expressed as
a function
of
four nondimensional parameters
given by
E
=
$[C*,
CT,
(hA)*,
NTU,]
(17)
Definition of NTU,,
C*,
CT,
(hA)*
NTU, is the modified number of transfer units, defined as
NTU,
=
NTU,
The heat capacity rate ratio
C*
is simply the ratio of the smaller to larger heat capacity
rate of the fluid steams
so
that
C*
should be less than or equal to
1.
Cmtn
(Mcp)rnm
-
C,
c*=-=
(19)
cm,,
(Mcp)mrx
Ch
CT
is the matrix heat capacity rate
C,
normalized with respect to
Cmtn:
where
M,
is the mass of the matrix,
c,
is the specific heat of the matrix material, and
N
is the
speed of the matrix in revolution per unit time.
(hA)* is the ratio of the convective conductance on the
C,,,
side to that on the
C,,,
side.
(hA)*
=
(hA) on the
C,,,
side
--
(hA),
(hA) on the
C,,,
side
-
(hA)h
Various
terms used here are defined as follows:
A,
=
matrix transfer area on cold side
Ah
=
matrix transfer area on hot side
Cc
=
fluid stream capacity rate
(Mc,)
for cold gas flow
C,,
=
fluid stream capacity rate
(Mc,)
for hot gas flow
C,
=
capacity rate of the rotor matrix
(MrcrN)
(hA)*
=
thermal conductance
h,
=
convective conductance for cold gas
hh
=
convective conductance for hot gas
M,
=
mass flow rate of cold gas
Mh
=
mass flow rate
of
hot gas
M,
=
matrix mass
(Mc,),
=
capacity rate of cold gas
Regenerators
325
(MCp)h
=
capacity rate of hot gas
M,.cr
=
thermal capacitance of matrix
rn:
=
mass of cold gas retained in the matrix length
L
rni
=
mass of hot gas retained in the matrix length
L
L
=
matrix flow length
N
=
rpm of the matrix
t,
=
temperature of the cold gas
th
=
temperature of the hot gas
t,
=
temperature of the matrix
t,,,, t,,,
=
cold fluid terminal temperatures
fh.i,
th,,
=
hot fluid terminal temperatures
z
=
flow period
For the special case of
C,./C,==J,
the behavior becomes identical in form to that of a
counterflow direct-type exchanger and its effectiveness is given by
-
NTU,
-
for
C*
=
1
(1
+
NTU,)
The following formula for thermal effectiveness was first empirically correlated by Lambertson
[21] and later modified by Kays and London [15]:
&=&,f
[
1
--
9
Equation 23 agrees with accurate numerical results within
+1%
for the following ranges: (1)
3
I
NTU,
5
9,
0.90
I
C*
5
1, 1.25
I
CT
15; (2) 2
c
NTU,
c
14,
C*
=
1,
CT
2
1.5;
(3) NTU,
I
20,
C*
=
1,
CT
2; and
(4)
the complete range of NTU,,
C*
=
1,
C'f
2
5.
In all cases, 0.25
I
(hA)* 14. Note that the value of
C*
is limited to between
0.9
and
1.0.
The regenerator effectiveness increases with
NTU,
asymptotically and approaches unity
for specified values of
C*
and
CT.
Thus, in this case, the periodic-flow type regenerator would
have the same performance as a counterflow direct-type unit possessing the same hot-side
and cold-side transfer areas and the same convection heat-transfer coefficients, providing only
negligible thermal resistance offered by the wall structure. To provide a sense of magnitudes
for these parameters, the following tabulations were presented by these authors to indicate
extreme ranges of values to be expected in gas turbine design work:
E
=
50-90%
CJCh
=
0.90-1
-00
CJC,=
1 to 10
(hA)*
=
(hA)J(hA)h
=
0.2-1
NTU,
=
2-20
(==
twice
NTU,)
NTU,
=
1-10
Heat Transfer
For rotary regenerators, the magnitude
of
the heat transfer rate between the gas and matrix
during a flow period is necessarily the same for both flow periods, and therefore
4
=
(MCp)c(th.i
-
tc.1)
(24)
____
326
Chapter
6
Table
4
Parameter
Definitions
and
Equivalents
for
Stationary
and
Rotary
Regenerators
Stationary
Rotary
1
[
1
+
hA*]
NTU,
in which
fh.1
and
tc,,
are constant inlet temperatures
of
the hot and cold gases. For stationary
regenerators, the heat-transfer rate is
4
=
E
(MCpMth,l
-
tC.1)
(25)
6.4
Parameter Definitions
Various parameters definitions and their equivalents for the stationary and rotary regenerators
are given in Table
4.
Note:
While defining various parameters, throughout this chapter, it is assumed that the
cold fluid is the
C,,,
fluid and accordingly, subscripts c for cold fluid and h for hot fluid are
allotted. However, parameter definition can be gneralized to subcripts
a
and
b
such that the
term
6.5
Classification of Regenerator Operation
Based on the values of
C*
and (hA)*, or
A-ll,
regenerators are classified into eight types
[29]:
(
1) balanced regenerator,
(2)
unbalanced regenerator, (3) symmetric regenerator,
(4)
unsymmet-
ric regenerator,
(5)
symmetric and balanced regenerator,
(6)
unsymmetric and balanced regen-
erator, (7) symmetric and unbalanced regenerator, and
(8)
unsymmetric and unbalanced regen-
erator. For a balanced regenerator the heat capacity of the fluids blown through the regenerator
is
equal and a regenerator is termed symmetric if the reduced length of each period is equal,
The unsymmetric and unbalanced regenerator operation is the most general one, and the others
are merely subsets [30]. These designations are shown in Table
5.
6.6
Additional Formulas for Regenerator Effectiveness
Tables of thermal effectiveness are given by many sources: Kays and London
[15],
Baclic
[17],
Lamberston
[21],
Bhanke and Howard
[22],
and Romie [31], among others. Methods of
computing the thermal effectiveness are described by, for example, Baclic
[
171, Willmott and
Duggan
[26],
Baclic et al. [27], Romie and Baclic
[28],
reference
32,
and many others. Addi-
tional formulas and closed form solutions from a few of these references are given next.
Regenerators
32
7
Table
5
Designation
of
Regenerators
[29,
301
Terminology
A-11
method E-NTU, method
Balanced regenerator
Unbalanced regenerator
c*
=
1
c*
#
1
Symmetric regenerator
(hA)*
=
1
Unsymmetric regenerator
(hA)*
f
1
Symmetric and balanced regenerator
(hA)*
=
1
c*
=
1
Unsymmetric and balanced regenerator
(hA)*
z
1
c*
=
1
Symmetric and unbalanced regenerator
(hA)*
=
1
c*
#
1
Unsymmetric and unbalanced regenerator
(hA)*
#
1
c*
f
1
Balanced and Symmetric Counterflow Regenerator
Baclic
[
171 obtained a highly accurate closed form expression for the counterflow regenerator
effectiveness for the balanced and symmetric regenerator; that
is,
for
C*
=
1 and
(hA)*
=
1
by
the Galerkin method,
where
Bj
is determined by solving the set of equations
of
the
form
ZL,
ajkBj
=
1
where
For
N
=
2,
the equation for
E
simplifies
to
the following equation:
1
+
7p2
-
24{B
-
2[R1
-
AI
-
90
(NI
+
2E)]}
E
=
CT
1
+
9p2
-
24{B
-
6[R
-
A
-
20
(N-
3E)]}
where
B
=
3p3
-
13p4
+
30
(PS
-
P6)
(32)
and
R
=
p2r3p4
-
5(3pS
-
4~~11
(33)
328
Chapter
6
=
v,(~,A)/A'-'
for
i
=
2,3,
. .
.
,6
Regenerator effectiveness
E,
computed from Eq. 3 1, as a function of NTU, and
CT
is presented
in Table
6.
Note that the results are valid for
C*
=
1
and (hA)*
=
1.
It is emphasized again that
the regenerator effectiveness
E
by Eq. 23 is valid for
0.9
S
C*
5
1
and
CT
>
1.25 while that
of
Eq. 31 is valid for
C*
=
1 but for all values
of
CT.
Exact asymptotic values of
E
for
A
-+
OQ
and
WA
+
0,
when
E
=
N(2
+
A)
holds are also included in this table.
6.7
Reduced Length-Reduced Period
(A-n)
Method
Galerkin Method of Baclic et al. [27] for Asymmetric-Unbalanced
Counterflow Regenerator
The asymmetric-unbalanced counterflow regenerator problem described by the classical ideal-
izations is solved by the Galerkin method by Baclic et al. [27]. Based on these assumptions,
an energy balance provides two equaitons applicable during the hot gas flow period and two
equaitons during the cold gas flow period:
The regenerator can operate with either parallel flow (plus sign in Eq. 43) or counterflow
arrangement (minus sign in Eq. 43). In Eq. 42 and Eq. 43,
5
is the fractional distance along
the flow path in the regenerator matrix of length
L
and
q
is the fractional completion of a
respective gas flow period. Temperatures of the hotter gas, colder gas, and the solid matrix are
denoted by
th,
t,,
and
t,,
respectively. The four parameters
Ah
and
A.
(reduced lengths),
nh
and
n,
(reduced periods), and their expressions are given in Table 7, and their equivalent parame-
ters for the rotary regenerators are given in Table
8.
The differential equations 42 and 43 describe the regenerator operation when the appro-
priate boundary conditions are specified. These are constant gas inlet temperature
th.1
and
tc,l
at
the opposite ends
(c=O
and
1)
of the regenerator matrix, and the condition stating that the
matrix temperature field at the end of one gas flow period is the initial matrix temperature
distribution for the subsequent gas flow period.
The subscript
1
is assigned to the weaker gas flow period such that the respective
U,
=
(Nn),
ratio is smaller of the two ratios
uh
=
(Nn),
and
Uc
=
(Nn),.
The ratio
(Nn)
is termed
the
utilization factor.
The main advantage of the utilization factors is the fact that these pararne-
Regenerators
329
Table
6
Symmetric and Balanced Counterflow Regenerator Effectiveness
E
=
@(A,ll)
(as
calculated
from
Eq.
31)
[17]
A\WA
0.0 0.1
0.2 0.3 0.4
0.5
0.6 0.7 0.8 0.9
1.0
I
.o
113
0.3332 0.3329
0.3323 0.33
15
0.3304 0.3292 0.3277 0.3260
0.3241 0.322
1
1.5
317
0.4283 0.4276 0.4264 0.4248 0.4227 0.4202 0.4173 0.4139
0.4102 0.4061
2
.o
112
0.4996 0.4986 0.4968 0.4943
0.4912 0.4874 0.4830
0.4780
0.4725 0.4665
2.5
519
0.5551
0.5537 0.5513 0.5481 0.5440 0.5391 0.5333 0.5269
0.5197 0.5120
3.0
315
0.5994 0.5977 0.5949 0.5910 0.5861
0.5802 0.5733
0.5655 0.5570
0.5477
3.5
711
1
0.6357 0.6338
0.6305 0.6261
0.6204 0.6137 0.6058 0.5970
0.5872 0.5766
4.0
213
0.6659 0.6638
0.6602 0.6553
0.6490 0.6416 0.6329 0.6232
0.6124 0.6006
4.5 911 3
0.6915 0.6892 0.6853 0.6800 0.6732 0.6652 0.6559
0.6454
0.6337 0.6210
5.0
517
0.7134 0.7109 0.7068
0.701
1
0.6940
0.6855 0.6756 0.6645
0.6521 0.6385
5.5
11/15
0.7324 0.7293 0.7255 0.7195 0.7121 0.7032 0.6928
0.681
1
0.6681 0.6537
6.0
314
0.7491 0.7463
0.7418 0.7356
0.7279 0.7187 0.7080
0.6958 0.6822 0.6672
6.5
13/17
0.7637 0.7609
0.7562 0.7498
0.7419 0.7324 0.7215 0.7089 0.6948 0.6792
7.0
719
0.7768 0.7738 0.7690
0.7625 0.7544 0.7447 0.7335 0.7206 0.7061 0.6900
7.5
15/19
0.7884 0.7854 0.7804 0.7738 0.7656 0.7558
0.7444 0.7313
0.7164 0.6997
8.0
415
0.7989 0.7958 0.7908 0.7840
0.7757 0.7658 0.7543 0.7409
0.7257 0.7036
8.5
1712
1
0.8084 0.8052 0.8001 0.7933
0.7849 0.7749
0.7633 0.7497
0.7342 0.7167
9.0
911
1
0.8171 0.8138 0.8086 0.8017
0.7933 0.7833 0.7715 0.7578 0.7421 0.7242
9.5
19/23
0.8250 0.8216 0.8164 0.8094
0.8010 0.7909
0.7791 0.7653
0.7493 0.731
1
10.0
516
0.8322 0.8288 0.8235 0.8165
0.8080 0.7980
0.7862 0.7722 0.7560 0.7375
10.5
2
I
125
0.8388 0.8354 0.8300 0.8230
0.8146 0.8046
0.7927 0.7787
0.7623 0.7435
11.0
11/13
0.8450 0.8415
0.8361
0.8290
0.8206 0.8106
0.7988 0.7847 0.7681 0.7491
11.5
23/27
0.8506 0.8471
0.8417 0.8346
0.8262 0.8163
0.8044 0.7903 0.7736 0.7543
12.0
617
0.8559 0.8524
0.8469
0.8398
0.8315 0.8216
0.8098 0.7956 0.7788 0.7592
12.5
25/29
0.8603 0.8573
0.8517 0.8447
0.8364 0.8266
0.8147
0.8005 0.7836 0.7638
13.0
13/15
0.8654 0.8618 0.8562 0.8492
0.8410 0.8312
0.8194 0.8052 0.7882 0.7682
13.5
2713
1
0.8697 0.8661
0.8605 0.8535
0.8453 0.8356 0.8239
0.8096 0.7925
0.7723
14.0
718
0.8737 0.8701
0.8644 0.8575 0.8494 0.8397
0.8280
0.8138 0.7966 0.7762
14.5
29/33
0.8775
0.8738 0.8682 0.8612 0.8532
0.8436 0.8320 0.8177 0.8004 0.7799
15.0
15/17
0.8811
0.8773 0.8717 0.8648 0.8568
0.8473 0.8358 0.8215 0.8041 0.7834
15.5
31/35
0.8844 0.8807 0.8750 0.8681
0.8602 0.8508
0.8393 0.8251 0.8076 0.7868
16.0
819
0.8876
0.8838 0.8781
0.8713 0.8635 0.8542 0.8427 0.8285 0.81
10
0.7900
17.0
17/19
0.8934 0.8896
0.8839 0.8771 0.8695
0.8604 0.8490 0.8348
0.8172 0.7959
18.0
9/10
0.8967
0.8948 0.8891 0.8824
0.8749 0.8660
0.8548 0.8406
0.8229 0.8014
19.0
1
912
I
0.9034
0.8995 0.8938
0.8872 0.8799
0.8711 0.8601 0.8459
0.8282 0.8065
20.0
1011
1
0.9077 0.9038
0.8981 0.8916 0.8844 0.8758
0.8649 0.8509
0.8331 0.8111
25
25/27
0.9245 0.9205
0.9148 0.9087 0.9024 0.8947
0.8844 0.8707 0.8528 0.8302
30
15/16
0.9360 0.9319
0.9263 0.9207 0.9151 0.9081 0.8984 0.8851 0.8673 0.8442
35
35/37
0.9445 0.9403
0.9347
0.9295 0.9246
0.9182 0.9090 0.8961 0.8785 0.8552
40
2012
1
0.9509 0.9467 0.9411 0.9363
0.9320 0.9261
0.9174 0.9048 0.8875 0.8640
45
45/47
0.9559 0.95 17 0.9462 0.941 7
0.9379
0.9325
0.9241 0.91 19 0.8948 0.87 13
50
25/26
0.9600 0.9557 0.9503 0.9461 0.9427 0.9378
0,9297 0.9177 0.9009 0.8775
60
3013
1
0.9662 0.9619 0.9565 0.9529 0.9502
0.9459 0.9383
0.9269
0.9106 0.8874
70
35/36
0.9707 0.9663 0.961
1
0.9578 0.9558
0.9519
0.9447 0.9337
0.9180 0.8951
80
4014
1
0.9740
0.9697
0.9645 0.9616 0.9600
0.9565 0.9497 0.9390 0.9239 0.9014
90
45/46
0.9767 0.9723 0.9672 0.9646 0.9634
0.9602 0.9536 0.9432 0.9236 0.9065
(continued)
330
Chapter
6
Table
6
Continued
0.0
0.1
0.2 0.3 0.4
0.5
0.6 0.7 0.8
0.9
1
.o
100
5015
1
0.9788
0.9744 0.9693 0.9670 0.9662 0.9632 0.9568
0.9467 0,9325 0.9 109
150
75/76
0.9852
0.9808
0.9759 0.9744 0.9747 0.9726 0.9667
0.9574 0.9452 0.9258
200
1001101
0.9835
0.9840
0.9792 0.9782 0.979
1
0.9774 0.97 19
0.9629 0.9520 0.9348
300
1501
15
1
0.99 17
0.9873
0.9826 0.982
1
0.9837
0.9824 0.9772
0.9686
0.9592 0.9456
400
200120
I
0.9934
0.9889
0.9843
0.984
1
0.9860
0.9850 0.9799
0.97 14
0.9629 0.9522
500
250125
1
0.9944
0.9899
0.9853
0.9853
0.9874
0.9865
0.98
15
0.973
1
0.965
1
0.9568
600
300130
1
0.9950
0.9905
0.9860
0.986
1
0.9884
0.9876
0.9826 0.9743 0.9665
0.9602
800
400/40
I
0.9959
0.9914
0.9869
0.987
1
0.98%
0.9889
0.9840 0.9758 0.9682
0.9650
loo0
500150
1
0.9964
0.9919
0.9874
0.9877
0.9903
0.9897
0.9848 0.9766 0.9692
0.9634
2000
1o00/1001
0.9974
0.9928
0.9884
0.9890
0.9917
0.99 12
0.9865 0.9784
0.97
1 1
0.9770
00
I
.oooo
1
.m
1
.oooo
1
.oooo
1
.m
I
.m
1
.oooo
1
.oooo
1
.m
1
.oooo
1
.oooo
~~~ ~
I
Note:For
the
E-NTU
method,
WA
=
and
A
=
2NTU,.
-
cjl
ters do not contain the fluid to matrix heat-transfer coefficients of the respective flow periods.
Thus,
UI
is defined
as
U,
=
min{
Uc,
U,,}
(
44)
The effectiveness is given
by
i
1
M
~ ~
-
n
X\m
for
rn
=
0,1,2,3,
.
.
.
,
M
(45
1
E=--&----
U,
(m
+
I)!
The unknown coefficients
xI,
and
x2,
are determined from the following set
of
algebraic equa-
tions:
Table
7
Expressions
for
hh,
A.,
nC,
and
nh
Regenerators
331
For any fixed
M,
one needs
2(M+
1)
equations for
M+
1
unknowns
xI,
and for
M+
1
un-
knowns
x:,,~.
For any practical purpose
M
=
2
will be adequate.
6.8
Razelos Method for Asymmetric-Unbalanced
Counterflow Regenerator
The forms produced by the Galerkin method of Baclic et al.
[27]
are not amenable for hand
calculation methods. They can be solved by either a computer or a programmable calculator.
A hand calculation method was presented by Razelos
[33].
In this approach, the unsymmetric
regenerator is approximated using an “equivalent” symmetric regenerator. The Razelos approx-
imate method can be used to calculate the counterflow regenerator effectiveness for the com-
plete range of
C*
and NTU,, and
CT
2
I,
0.25
I
(hA)*
24.
The method follows these steps.
1.
For the E-NTU,, method, compute the range of NTU,, and
CT.,
for
an
equivalent balanced
regenerator as follows from the specified values of NTU,,,
CT,
and
C*.
2
c: c*
CT.m
=
_____
1
+
c*
2.
For the
A-l7
method, calculate
A,
and
n,
as
3.
The equivalent balanced regenerator
effectiveness
E,
is
evaluated from Eq.
23
or Eq.
31
for NTU,,, and
CT,,
or from Table
6
for
A,
and
n,.
For example, the equivalent form of
Eq.
23
is
given by
4.
The acutal regenerator effectiveness
E
is
then given by
Table
8
Reduced Length and Reduced Period
Parameters: Definitions
[20]
~ ~~
Equivalent in the two
Term
thermal design models
Reduced length
&
=
NTU,
&
=
NTUh
Reduced period
n,=-
NTU,
Nmh
nh
=-
CKC
CKIl
332
Chapter
6
(55)
where
where
C*
is the original specified value, which is less than unity.
A
sample calculation
[
171
for
fib
=
18,
&
=
15.5,
nh
=
16,
and
(M),
=
1.2
yields
l&,
=
4960/347
and
A,
=
5760/347.
From Table
6,
by linear interpolation, one obtains
E,
=
0.8215,
since
(WA),,,
=
31/36
for this case, while
+
=
-0.0645833
from the equation. With these values,
E
=0.847047
from the equation. The exact value
of
the effectiveness for this example is
0.8473
1
1, which is just
0.03%
higher than the result obtained from this approximate procedure.
6.9
Influence of Longitudinal Heat Conduction in the Wall
In the foregoing analysis, the influence of longitudinal heat conduction was neglected. For
a high-thermal-effectiveness regenerator with a large axial temperature gradient in the wall,
longitudinal heat conduction reduces the regenerator effectiveness and the overall heat-transfer
rate. Bahnke and Howard
[22]
showed that
an
additional parameter
h,
the longitudinal conduc-
tion parameter, would account for the influence of longitudinal heat conduction in the wall.
Here
h
is defined as
where
k,
is
the thermal conductivity of the matrix wall, and
Ak,,
is the total solid area available
for longitudinal conduction;
Ak.l
is given by
[
141:
Ak.c
=
Ak,h
+
Ak.c
=
Afr
-
A,,
=
Afr
(1
-
0)
(59)
Introducing the longitudinal conduction parameter, a generalized expression for thermal effec-
tiveness in the functional form
is
given by
C*,
CT,
(hA)*,
h}
(60)
E
=
(NTU{,,
Lowered effectiveness because of longitudinal conduction,
AE,
can be calculated in several
ways. Parameter definitions for
h
and its equivalents for the stationary and rotary regenerators
are given in Table
9.
Table
9
Parameter Definitions
for
h
Stationary Rotary
Regenerators
333
Bahnke and Howard Method
Bahnke and Howard [22] illustrated the influence of the thermal conductivity on thermal effec-
tiveness by a factor called the
conduction
effect.
The conduction effect is defined as the ratio
of the difference between effectiveness with no conduction effect and with conduction effect
to the no conduction effectivness, as given by
Bahnke and Howard tabulated results for
E
for a wide range of NTU,,
CT,
C,,
and
h.
their
results are valid over the following range of dimensionless parameters:
1
2
Cmin/CmaX
2
0.90
1.0
5
CJC,l,
5
00
1
5
NTU,
I
100
1.0
2
(hA)*
2
0.25
1.0
1
A*
2
0.25
0.01
I
h
50.32
Bahnke and Howard’s results for
C*
=
1
can
be
accurately expressed by
[29]:
where
em
is given by
Eq.
23 or 31, and
Ck
is given by
1
+
NTU, 1
Ck
=
NTU,
[
1
-
1
+
NTU,
(1
+
hyl)/(l
+
Am,)
I
in which
LNTU,
\CI
=
[
1
+
hNTU,
r5
tan,[
[hNTUJ(
1
+
hNTU,)l0
Romie’s Solution
Romie [34] expressed the effects of longitudinal heat conduction on thermal effectiveness in
terms of a parameter GL. The factor GL
is
derived from the analysis of axial conduction in
counterflow recuperators by Hahnemann [35]. GL is given by
GL
=
1
-
[l
-
tanh(b)/b] (NTUlb)’ for
C*
=
1,
(U)*
=
1
(65)
where
b
=
NTU, L1
+
l/(hNTU,)P5
(66)
The expression for
EM
is given by